{"text": "{-# OPTIONS --without-K --rewriting --termination-depth=2 #-}\n\nopen import HoTT\nopen import cw.CW\nopen import cw.examples.Sphere\n\nmodule cw.examples.Torus where\n\n⊤-has-dec-eq : has-dec-eq ⊤\n⊤-has-dec-eq unit unit = inl idp\n\n⊤-is-set : is-set ⊤\n⊤-is-set = dec-eq-is-set ⊤-has-dec-eq\n\n\ncw-torus-skel : Skeleton {lzero} (S (S 0))\n\nCWTorus : Type₀\nCWTorus = ⟦ cw-torus-skel ⟧\n\nα₁ : Bool → S⁰ → ⊤\nα₁ _ _ = unit\n\nX₁ = attached-skeleton (⊤ , ⊤-is-set) (Bool , Bool-is-set) α₁\n\nα₂ : ⊤ → S¹ → (Attached α₁)\nα₂ unit = Pushout-rec ψ ψ (cst idp)\n where\n ψ : ⊤ → Attached α₁\n ψ unit = PushoutGeneric.from-cc (inl unit)\n\nX₂ = attached-skeleton X₁ (⊤ , ⊤-is-set) α₂\n\ncw-torus-skel = X₂\n\n\n", "meta": {"hexsha": "8e6af8922636c53e740be4023a5bf03b3bbd1069", "size": 682, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "theorems/cw/examples/Torus.agda", "max_stars_repo_name": "maxdore/hott-morse", "max_stars_repo_head_hexsha": "01bbd8841f9b9b25666b91e65b196d8f472b9978", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "theorems/cw/examples/Torus.agda", "max_issues_repo_name": "maxdore/hott-morse", "max_issues_repo_head_hexsha": "01bbd8841f9b9b25666b91e65b196d8f472b9978", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "theorems/cw/examples/Torus.agda", "max_forks_repo_name": "maxdore/hott-morse", "max_forks_repo_head_hexsha": "01bbd8841f9b9b25666b91e65b196d8f472b9978", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.4324324324, "max_line_length": 61, "alphanum_fraction": 0.6290322581, "num_tokens": 284, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.3498681907539131}} {"text": "{-# OPTIONS --safe --without-K #-}\n\nopen import Generics.Prelude hiding (lookup)\nopen import Generics.Telescope\nopen import Generics.Desc\nopen import Generics.HasDesc\n\nmodule Generics.Constructions.Case\n {P I ℓ} {A : Indexed P I ℓ}\n (H : HasDesc A) (open HasDesc H)\n {p c} (Pr : Pred′ I λ i → A′ (p , i) → Set c)\n where\n\nprivate\n variable\n V : ExTele P\n i : ⟦ I ⟧tel p\n v : ⟦ V ⟧tel p\n\nPr′ : A′ (p , i) → Set c\nPr′ {i} = unpred′ I _ Pr i\n\n\n--------------------------\n-- Types of motives\n\nlevelCase : ConDesc P V I → Level\nlevelCase (var x) = c\nlevelCase (π {ℓ′} _ _ C) = ℓ′ ⊔ levelCase C\nlevelCase (A ⊗ B) = levelIndArg A ℓ ⊔ levelCase B\n\nMotiveCon : (C : ConDesc P V I)\n → (∀ {i} → ⟦ C ⟧Con A′ (p , v , i) → Set c)\n → Set (levelCase C)\nMotiveCon (var γ) X = X refl\nMotiveCon (π ia S C) X = Π< ia > (S _) λ s → MotiveCon C (X ∘ (s ,_))\nMotiveCon (A ⊗ B) X = (x : ⟦ A ⟧IndArg A′ (p , _))\n → MotiveCon B (X ∘ (x ,_))\n\nMotives : ∀ k → Set (levelCase (lookupCon D k))\nMotives k = MotiveCon (lookupCon D k) (λ x → Pr′ (constr (k , x)))\n\n\n--------------------------\n-- Case-analysis principle\n\nmodule _ (methods : Els Motives) where\n\n caseData : ∀ {i} → (x : ⟦ D ⟧Data A′ (p , i)) → Pr′ (constr x)\n caseData (k , x) = caseCon (lookupCon D k) (methods k) x\n where\n caseCon\n : (C : ConDesc P V I)\n {mk : ∀ {i} → ⟦ C ⟧Con A′ (p , v , i) → ⟦ D ⟧Data A′ (p , i)}\n (mot : MotiveCon C λ x → Pr′ (constr (mk x)))\n (x : ⟦ C ⟧Con A′ (p , v , i))\n → Pr′ (constr (mk x))\n caseCon (var γ) mot refl = mot\n caseCon (π ia _ C) mot (s , x) = caseCon C (app< ia > mot s) x\n caseCon (A ⊗ B) mot (a , b) = caseCon B (mot a) b\n\n case : (x : A′ (p , i)) → Pr′ x\n case x = subst Pr′ (constr∘split x) (caseData (split x))\n\nderiveCase : Arrows Motives (Pred′ I (λ i → (x : A′ (p , i)) → Pr′ x))\nderiveCase = curryₙ (λ m → pred′ I _ λ i → case m)\n", "meta": {"hexsha": "9e3fec579fadfb24105570fa3c2dcfd89be6f957", "size": 1939, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Generics/Constructions/Case.agda", "max_stars_repo_name": "flupe/generics", "max_stars_repo_head_hexsha": "db764f858d908aa39ea4901669a6bbce1525f757", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2021-04-08T15:10:20.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-05T09:35:17.000Z", "max_issues_repo_path": "src/Generics/Constructions/Case.agda", "max_issues_repo_name": "flupe/generics", "max_issues_repo_head_hexsha": "db764f858d908aa39ea4901669a6bbce1525f757", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2021-09-13T07:33:50.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-14T10:48:30.000Z", "max_forks_repo_path": "src/Generics/Constructions/Case.agda", "max_forks_repo_name": "flupe/generics", "max_forks_repo_head_hexsha": "db764f858d908aa39ea4901669a6bbce1525f757", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-04-08T08:32:42.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-14T10:35:16.000Z", "avg_line_length": 28.9402985075, "max_line_length": 72, "alphanum_fraction": 0.5141825683, "num_tokens": 758, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7461389817407016, "lm_q2_score": 0.4687906266262437, "lm_q1q2_score": 0.3497829608004909}} {"text": "{-# OPTIONS --without-K --safe #-}\n\nmodule Polynomial.Simple.Reflection where\n\nopen import Agda.Builtin.Reflection\nopen import Reflection.Helpers\n\nopen import Polynomial.Simple.Solver renaming (solve to solve-fn)\nopen AlmostCommutativeRing\n\nopen import Data.Nat.Table\n\nopen import Data.Fin as Fin using (Fin)\nopen import Data.Vec as Vec using (Vec; _∷_; [])\nopen import Data.List as List using (List; _∷_; [])\nopen import Data.Maybe as Maybe using (Maybe; just; nothing; fromMaybe)\nopen import Agda.Builtin.Nat using (_<_)\nopen import Data.Nat using (ℕ; suc; zero)\nopen import Data.Bool using (Bool; if_then_else_; true; false)\nopen import Data.Unit using (⊤)\nopen import Data.String using (String)\nopen import Data.Product using (_,_)\nopen import Function\n\nmodule Internal where\n _∈Ring : Term → TC Term\n ring ∈Ring = checkType ring (def (quote AlmostCommutativeRing) (unknown ⟨∷⟩ unknown ⟨∷⟩ []))\n\n vars : Term → Maybe Table\n vars = go []\n where\n go : Table → Term → Maybe Table\n go t (con (quote List._∷_) (_ ∷ _ ∷ var i [] ⟨∷⟩ xs ⟨∷⟩ _)) = go (insert i t) xs\n go t (con (quote List.List.[]) _) = just t\n go _ _ = nothing\n\n module OverRing (ring : Term) where\n _∈List⟨Carrier⟩ : Term → TC Term\n t ∈List⟨Carrier⟩ =\n checkType t\n (quote List ⟨ def ⟩ 1 ⋯⟅∷⟆ def (quote Carrier) (2 ⋯⟅∷⟆ ring ⟨∷⟩ []) ⟨∷⟩ []) >>= normalise\n\n record Ring⇓ : Set where\n constructor +⇒_*⇒_^⇒_-⇒_\n field +′ *′ ^′ -′ : Maybe Name\n\n ring⇓ : TC Ring⇓\n ring⇓ = ⦇ +⇒ ⟦ quote _+_ ⇓⟧ₙ *⇒ ⟦ quote _*_ ⇓⟧ₙ ^⇒ ⟦ quote _^_ ⇓⟧ₙ -⇒ ⟦ quote -_ ⇓⟧ₙ ⦈\n where\n ⟦_⇓⟧ₙ : Name → TC (Maybe Name)\n ⟦ nm ⇓⟧ₙ =\n normalise (nm ⟨ def ⟩ 2 ⋯⟅∷⟆ ring ⟨∷⟩ [])\n <&> λ where (def f args) → just f\n _ → nothing\n\n module _ (nms : Ring⇓) where\n open Ring⇓ nms\n\n module _ (numVars : ℕ) where\n\n -- This function applies the hidden arguments that the constructors\n -- that Expr needs. The first is the universe level, the second is the\n -- type it contains, and the third is the number of variables it's\n -- indexed by. All three of these could likely be inferred, but to\n -- make things easier we supply the third because we know it.\n infixr 5 E⟅∷⟆_\n E⟅∷⟆_ : List (Arg Term) → List (Arg Term)\n E⟅∷⟆ xs = 1 ⋯⟅∷⟆\n (quote Carrier ⟨ def ⟩ 2 ⋯⟅∷⟆ ring ⟨∷⟩ []) ⟅∷⟆\n ℕ′ numVars ⟅∷⟆\n xs\n\n -- A constant expression.\n Κ′ : Term → Term\n Κ′ x = quote Κ ⟨ con ⟩ E⟅∷⟆ x ⟨∷⟩ []\n\n _⇓≟_ : Maybe Name → Name → Bool\n nothing ⇓≟ _ = false\n just x ⇓≟ y = primQNameEquality x y\n {-# INLINE _⇓≟_ #-}\n\n module ToExpr (Ι′ : ℕ → Maybe Term) where\n mutual\n -- Application of a ring operator often doesn't have a type as\n -- simple as \"Carrier → Carrier → Carrier\": there may be hidden\n -- arguments, etc. Here, we do our best to handle those cases,\n -- by just taking the last two explicit arguments.\n E⟨_⟩₂ : Name → List (Arg Term) → Term\n E⟨ nm ⟩₂ (x ⟨∷⟩ y ⟨∷⟩ []) = nm ⟨ con ⟩ E⟅∷⟆ E x ⟨∷⟩ E y ⟨∷⟩ []\n E⟨ nm ⟩₂ (x ∷ xs) = E⟨ nm ⟩₂ xs\n E⟨ nm ⟩₂ _ = unknown\n\n E⟨_⟩₁ : Name → List (Arg Term) → Term\n E⟨ nm ⟩₁ (x ⟨∷⟩ []) = nm ⟨ con ⟩ E⟅∷⟆ E x ⟨∷⟩ []\n E⟨ nm ⟩₁ (x ∷ xs) = E⟨ nm ⟩₁ xs\n E⟨ _ ⟩₁ _ = unknown\n\n E⟨^⟩ : List (Arg Term) → Term\n E⟨^⟩ (x ⟨∷⟩ y ⟨∷⟩ []) = quote _⊛_ ⟨ con ⟩ E⟅∷⟆ E x ⟨∷⟩ y ⟨∷⟩ []\n E⟨^⟩ (x ∷ xs) = E⟨^⟩ xs\n E⟨^⟩ _ = unknown\n\n -- When trying to figure out the shape of an expression, one of\n -- the difficult tasks is recognizing where constants in the\n -- underlying ring are used. If we were only dealing with ℕ, we\n -- might look for its constructors: however, we want to deal with\n -- arbitrary types which implement AlmostCommutativeRing. If the\n -- Term type contained type information we might be able to\n -- recognize it there, but it doesn't.\n --\n -- We're in luck, though, because all other cases in the following\n -- function *are* recognizable. As a result, the \"catch-all\" case\n -- will just assume that it has a constant expression.\n E : Term → Term\n E (def (quote _+_) xs) = E⟨ quote _⊕_ ⟩₂ xs\n E (def (quote _*_) xs) = E⟨ quote _⊗_ ⟩₂ xs\n E (def (quote _^_) xs) = E⟨^⟩ xs\n E (def (quote -_) xs) = E⟨ quote ⊝_ ⟩₁ xs\n E (def nm xs) = if +′ ⇓≟ nm then E⟨ quote _⊕_ ⟩₂ xs else\n if *′ ⇓≟ nm then E⟨ quote _⊗_ ⟩₂ xs else\n if ^′ ⇓≟ nm then E⟨^⟩ xs else\n if -′ ⇓≟ nm then E⟨ quote ⊝_ ⟩₁ xs else\n Κ′ (def nm xs)\n E (con (quote ℕ.suc) (x ⟨∷⟩ [])) = quote _⊕_ ⟨ con ⟩ E⟅∷⟆ Κ′ (ℕ′ (ℕ.suc ℕ.zero)) ⟨∷⟩ E x ⟨∷⟩ []\n E v@(var x _) = fromMaybe (Κ′ v) (Ι′ x)\n E t = Κ′ t\n\n callSolver : Vec String numVars → Term → Term → List (Arg Type)\n callSolver nms lhs rhs =\n 2 ⋯⟅∷⟆ ring ⟨∷⟩ ℕ′ numVars ⟨∷⟩\n vlams nms (quote _⊜_ ⟨ def ⟩ 2 ⋯⟅∷⟆ ring ⟨∷⟩ ℕ′ numVars ⟨∷⟩ E lhs ⟨∷⟩ E rhs ⟨∷⟩ []) ⟨∷⟩\n hlams nms (quote refl ⟨ def ⟩ 2 ⋯⟅∷⟆ ring ⟨∷⟩ 1 ⋯⟅∷⟆ []) ⟨∷⟩\n []\n where\n Ι′ : ℕ → Maybe Term\n Ι′ i = if i < numVars then just (var i []) else nothing\n open ToExpr Ι′\n\n constructSoln : Table → Term → Term → Term\n constructSoln t lhs rhs =\n quote trans ⟨ def ⟩ 2 ⋯⟅∷⟆ ring ⟨∷⟩ 3 ⋯⟅∷⟆\n (quote sym ⟨ def ⟩ 2 ⋯⟅∷⟆ ring ⟨∷⟩ 2 ⋯⟅∷⟆\n (quote Ops.correct ⟨ def ⟩ 2 ⋯⟅∷⟆ ring ⟨∷⟩ 1 ⋯⟅∷⟆ E lhs ⟨∷⟩ ρ ⟨∷⟩ []) ⟨∷⟩ [])\n ⟨∷⟩\n (quote Ops.correct ⟨ def ⟩ 2 ⋯⟅∷⟆ ring ⟨∷⟩ 1 ⋯⟅∷⟆ E rhs ⟨∷⟩ ρ ⟨∷⟩ []) ⟨∷⟩\n []\n where\n Ι′ : ℕ → Maybe Term\n Ι′ i = Maybe.map (λ x → quote Ι ⟨ con ⟩ E⟅∷⟆ Fin′ x ⟨∷⟩ []) (member i t)\n\n open ToExpr Ι′\n ρ : Term\n ρ = curriedTerm t\nopen Internal\n\n-- This is the main macro you'll probably be using. Call it like this:\n--\n-- lemma : ∀ x y → x + y ≈ y + x\n-- lemma = solve TypeRing\n--\n-- where TypRing is your implementation of AlmostCommutativeRing. (Find some\n-- example implementations in Polynomial.Solver.Ring.AlmostCommutativeRing.Instances).\nmacro\n solve : Name → Term → TC ⊤\n solve ring hole = do\n ring′ ← def ring [] ∈Ring\n commitTC\n let open OverRing ring′\n nms ← ring⇓\n hole′ ← inferType hole >>= reduce\n let i , k , xs = underPi hole′\n just (lhs ∷ rhs ∷ []) ← pure (getArgs 2 xs)\n where nothing → typeError (strErr \"Malformed call to solve.\" ∷\n strErr \"Expected target type to be like: ∀ x y → x + y ≈ y + x.\" ∷\n strErr \"Instead: \" ∷\n termErr hole′ ∷\n [])\n unify hole (quote solve-fn ⟨ def ⟩ callSolver nms i k lhs rhs)\n\n-- Use this macro when you want to solve something *under* a lambda. For example:\n-- say you have a long proof, and you just want the solver to deal with an\n-- intermediate step. Call it like so:\n--\n-- lemma₃ : ∀ x y → x + y * 1 + 3 ≈ 2 + 1 + y + x\n-- lemma₃ x y = begin\n-- x + y * 1 + 3 ≈⟨ +-comm x (y * 1) ⟨ +-cong ⟩ refl ⟩\n-- y * 1 + x + 3 ≈⟨ solveOver (x ∷ y ∷ []) Int.ring ⟩\n-- 3 + y + x ≡⟨ refl ⟩\n-- 2 + 1 + y + x ∎\n--\n-- The first argument is the free variables, and the second is the\n-- ring implementation (as before).\n--\n-- One thing to note here is that we need to be able to infer *both* sides of\n-- the equality, which the normal equaltional reasoning combinators don't let you\n-- do. You'll need the combinators defined in Relation.Binary.Reasoning.Inference.\n-- These are just as powerful as the others, but have slightly better inference properties.\n\nsolveOver-macro : Term → Name → Term → TC ⊤\nsolveOver-macro i ring hole = do\n ring′ ← def ring [] ∈Ring\n commitTC\n let open OverRing ring′\n nms ← ring⇓\n i′ ← i ∈List⟨Carrier⟩\n commitTC\n hole′ ← inferType hole >>= reduce\n just vars′ ← pure (vars i′)\n where nothing → typeError (strErr \"Malformed call to solveOver.\" ∷\n strErr \"First argument should be a list of free variables.\" ∷\n strErr \"Instead: \" ∷\n termErr i′ ∷\n [])\n just (lhs ∷ rhs ∷ []) ← pure (getArgs 2 hole′)\n where nothing → typeError (strErr \"Malformed call to solveOver.\" ∷\n strErr \"First argument should be a list of free variables.\" ∷\n strErr \"Instead: \" ∷\n termErr hole′ ∷\n [])\n unify hole (constructSoln nms (List.length vars′) vars′ lhs rhs)\n\nmacro\n solveOver : Term → Name → Term → TC ⊤\n solveOver = solveOver-macro\n", "meta": {"hexsha": "b6f50e6f76fa1a388f2d2f5b6aac6e99e28318e0", "size": 9154, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Polynomial/Simple/Reflection.agda", "max_stars_repo_name": "mckeankylej/agda-ring-solver", "max_stars_repo_head_hexsha": "f18d9c6bdfae5b4c3ead9a83e06f16a0b7204500", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 36, "max_stars_repo_stars_event_min_datetime": "2019-01-25T16:40:52.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-15T00:57:55.000Z", "max_issues_repo_path": "src/Polynomial/Simple/Reflection.agda", "max_issues_repo_name": "mckeankylej/agda-ring-solver", "max_issues_repo_head_hexsha": "f18d9c6bdfae5b4c3ead9a83e06f16a0b7204500", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2019-04-17T20:48:48.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-12T01:55:42.000Z", "max_forks_repo_path": "src/Polynomial/Simple/Reflection.agda", "max_forks_repo_name": "mckeankylej/agda-ring-solver", "max_forks_repo_head_hexsha": "f18d9c6bdfae5b4c3ead9a83e06f16a0b7204500", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2019-04-16T02:23:16.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-20T07:07:11.000Z", "avg_line_length": 40.6844444444, "max_line_length": 107, "alphanum_fraction": 0.514201442, "num_tokens": 3054, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3497261564093474}} {"text": "------------------------------------------------------------------------\n-- The Agda standard library\n--\n-- Sums of binary relations\n------------------------------------------------------------------------\n\n{-# OPTIONS --without-K --safe #-}\n\nmodule Data.Sum.Relation.Binary.LeftOrder where\n\nopen import Data.Sum as Sum\nopen import Data.Sum.Relation.Binary.Pointwise as PW\n using (Pointwise; inj₁; inj₂)\nopen import Data.Product\nopen import Data.Empty\nopen import Function\nopen import Level\nopen import Relation.Nullary\nimport Relation.Nullary.Decidable as Dec\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality as P using (_≡_)\n\n----------------------------------------------------------------------\n-- Definition\n\ninfixr 1 _⊎-<_\n\ndata _⊎-<_ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}\n {ℓ₁ ℓ₂} (_∼₁_ : Rel A₁ ℓ₁) (_∼₂_ : Rel A₂ ℓ₂) :\n Rel (A₁ ⊎ A₂) (a₁ ⊔ a₂ ⊔ ℓ₁ ⊔ ℓ₂) where\n ₁∼₂ : ∀ {x y} → (_∼₁_ ⊎-< _∼₂_) (inj₁ x) (inj₂ y)\n ₁∼₁ : ∀ {x y} (x∼₁y : x ∼₁ y) → (_∼₁_ ⊎-< _∼₂_) (inj₁ x) (inj₁ y)\n ₂∼₂ : ∀ {x y} (x∼₂y : x ∼₂ y) → (_∼₁_ ⊎-< _∼₂_) (inj₂ x) (inj₂ y)\n\n----------------------------------------------------------------------\n-- Some properties which are preserved by _⊎-<_\n\nmodule _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}\n {ℓ₁ ℓ₂} {∼₁ : Rel A₁ ℓ₁} {∼₂ : Rel A₂ ℓ₂}\n where\n\n drop-inj₁ : ∀ {x y} → (∼₁ ⊎-< ∼₂) (inj₁ x) (inj₁ y) → ∼₁ x y\n drop-inj₁ (₁∼₁ x∼₁y) = x∼₁y\n\n drop-inj₂ : ∀ {x y} → (∼₁ ⊎-< ∼₂) (inj₂ x) (inj₂ y) → ∼₂ x y\n drop-inj₂ (₂∼₂ x∼₂y) = x∼₂y\n\n ⊎-<-refl : Reflexive ∼₁ → Reflexive ∼₂ →\n Reflexive (∼₁ ⊎-< ∼₂)\n ⊎-<-refl refl₁ refl₂ {inj₁ x} = ₁∼₁ refl₁\n ⊎-<-refl refl₁ refl₂ {inj₂ y} = ₂∼₂ refl₂\n\n ⊎-<-transitive : Transitive ∼₁ → Transitive ∼₂ →\n Transitive (∼₁ ⊎-< ∼₂)\n ⊎-<-transitive trans₁ trans₂ ₁∼₂ (₂∼₂ x∼₂y) = ₁∼₂\n ⊎-<-transitive trans₁ trans₂ (₁∼₁ x∼₁y) ₁∼₂ = ₁∼₂\n ⊎-<-transitive trans₁ trans₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = ₁∼₁ (trans₁ x∼₁y x∼₁y₁)\n ⊎-<-transitive trans₁ trans₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = ₂∼₂ (trans₂ x∼₂y x∼₂y₁)\n\n ⊎-<-asymmetric : Asymmetric ∼₁ → Asymmetric ∼₂ →\n Asymmetric (∼₁ ⊎-< ∼₂)\n ⊎-<-asymmetric asym₁ asym₂ ₁∼₂ ()\n ⊎-<-asymmetric asym₁ asym₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = asym₁ x∼₁y x∼₁y₁\n ⊎-<-asymmetric asym₁ asym₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = asym₂ x∼₂y x∼₂y₁\n\n ⊎-<-total : Total ∼₁ → Total ∼₂ → Total (∼₁ ⊎-< ∼₂)\n ⊎-<-total total₁ total₂ = total\n where\n total : Total (_ ⊎-< _)\n total (inj₁ x) (inj₁ y) = Sum.map ₁∼₁ ₁∼₁ $ total₁ x y\n total (inj₁ x) (inj₂ y) = inj₁ ₁∼₂\n total (inj₂ x) (inj₁ y) = inj₂ ₁∼₂\n total (inj₂ x) (inj₂ y) = Sum.map ₂∼₂ ₂∼₂ $ total₂ x y\n\n ⊎-<-decidable : Decidable ∼₁ → Decidable ∼₂ →\n Decidable (∼₁ ⊎-< ∼₂)\n ⊎-<-decidable dec₁ dec₂ (inj₁ x) (inj₁ y) = Dec.map′ ₁∼₁ drop-inj₁ (dec₁ x y)\n ⊎-<-decidable dec₁ dec₂ (inj₁ x) (inj₂ y) = yes ₁∼₂\n ⊎-<-decidable dec₁ dec₂ (inj₂ x) (inj₁ y) = no λ()\n ⊎-<-decidable dec₁ dec₂ (inj₂ x) (inj₂ y) = Dec.map′ ₂∼₂ drop-inj₂ (dec₂ x y)\n\nmodule _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}\n {ℓ₁ ℓ₂} {∼₁ : Rel A₁ ℓ₁} {≈₁ : Rel A₁ ℓ₂}\n {ℓ₃ ℓ₄} {∼₂ : Rel A₂ ℓ₃} {≈₂ : Rel A₂ ℓ₄}\n where\n\n ⊎-<-reflexive : ≈₁ ⇒ ∼₁ → ≈₂ ⇒ ∼₂ →\n (Pointwise ≈₁ ≈₂) ⇒ (∼₁ ⊎-< ∼₂)\n ⊎-<-reflexive refl₁ refl₂ (inj₁ x) = ₁∼₁ (refl₁ x)\n ⊎-<-reflexive refl₁ refl₂ (inj₂ x) = ₂∼₂ (refl₂ x)\n\n ⊎-<-irreflexive : Irreflexive ≈₁ ∼₁ → Irreflexive ≈₂ ∼₂ →\n Irreflexive (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)\n ⊎-<-irreflexive irrefl₁ irrefl₂ (inj₁ x) (₁∼₁ x∼₁y) = irrefl₁ x x∼₁y\n ⊎-<-irreflexive irrefl₁ irrefl₂ (inj₂ x) (₂∼₂ x∼₂y) = irrefl₂ x x∼₂y\n\n ⊎-<-antisymmetric : Antisymmetric ≈₁ ∼₁ → Antisymmetric ≈₂ ∼₂ →\n Antisymmetric (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)\n ⊎-<-antisymmetric antisym₁ antisym₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = inj₁ (antisym₁ x∼₁y x∼₁y₁)\n ⊎-<-antisymmetric antisym₁ antisym₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = inj₂ (antisym₂ x∼₂y x∼₂y₁)\n -- Remove in Agda 2.6.0?\n ⊎-<-antisymmetric antisym₁ antisym₂ ₁∼₂ ()\n\n ⊎-<-respectsʳ : ∼₁ Respectsʳ ≈₁ → ∼₂ Respectsʳ ≈₂ →\n (∼₁ ⊎-< ∼₂) Respectsʳ (Pointwise ≈₁ ≈₂)\n ⊎-<-respectsʳ resp₁ resp₂ (inj₁ x₁) (₁∼₁ x∼₁y) = ₁∼₁ (resp₁ x₁ x∼₁y)\n ⊎-<-respectsʳ resp₁ resp₂ (inj₂ x₁) ₁∼₂ = ₁∼₂\n ⊎-<-respectsʳ resp₁ resp₂ (inj₂ x₁) (₂∼₂ x∼₂y) = ₂∼₂ (resp₂ x₁ x∼₂y)\n\n ⊎-<-respectsˡ : ∼₁ Respectsˡ ≈₁ → ∼₂ Respectsˡ ≈₂ →\n (∼₁ ⊎-< ∼₂) Respectsˡ (Pointwise ≈₁ ≈₂)\n ⊎-<-respectsˡ resp₁ resp₂ (inj₁ x) ₁∼₂ = ₁∼₂\n ⊎-<-respectsˡ resp₁ resp₂ (inj₁ x) (₁∼₁ x∼₁y) = ₁∼₁ (resp₁ x x∼₁y)\n ⊎-<-respectsˡ resp₁ resp₂ (inj₂ x) (₂∼₂ x∼₂y) = ₂∼₂ (resp₂ x x∼₂y)\n\n ⊎-<-respects₂ : ∼₁ Respects₂ ≈₁ → ∼₂ Respects₂ ≈₂ →\n (∼₁ ⊎-< ∼₂) Respects₂ (Pointwise ≈₁ ≈₂)\n ⊎-<-respects₂ (r₁ , l₁) (r₂ , l₂) = ⊎-<-respectsʳ r₁ r₂ , ⊎-<-respectsˡ l₁ l₂\n\n ⊎-<-trichotomous : Trichotomous ≈₁ ∼₁ → Trichotomous ≈₂ ∼₂ →\n Trichotomous (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)\n ⊎-<-trichotomous tri₁ tri₂ (inj₁ x) (inj₂ y) = tri< ₁∼₂ (λ()) (λ())\n ⊎-<-trichotomous tri₁ tri₂ (inj₂ x) (inj₁ y) = tri> (λ()) (λ()) ₁∼₂\n ⊎-<-trichotomous tri₁ tri₂ (inj₁ x) (inj₁ y) with tri₁ x y\n ... | tri< x x≮y x≉y x>y = tri> (x≮y ∘ drop-inj₁) (x≉y ∘ PW.drop-inj₁) (₁∼₁ x>y)\n ⊎-<-trichotomous tri₁ tri₂ (inj₂ x) (inj₂ y) with tri₂ x y\n ... | tri< x x≮y x≉y x>y = tri> (x≮y ∘ drop-inj₂) (x≉y ∘ PW.drop-inj₂) (₂∼₂ x>y)\n\n----------------------------------------------------------------------\n-- Some collections of properties which are preserved\n\nmodule _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}\n {ℓ₁ ℓ₂} {≈₁ : Rel A₁ ℓ₁} {∼₁ : Rel A₁ ℓ₂}\n {ℓ₃ ℓ₄} {≈₂ : Rel A₂ ℓ₃} {∼₂ : Rel A₂ ℓ₄} where\n\n ⊎-<-isPreorder : IsPreorder ≈₁ ∼₁ → IsPreorder ≈₂ ∼₂ →\n IsPreorder (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)\n ⊎-<-isPreorder pre₁ pre₂ = record\n { isEquivalence = PW.⊎-isEquivalence (isEquivalence pre₁) (isEquivalence pre₂)\n ; reflexive = ⊎-<-reflexive (reflexive pre₁) (reflexive pre₂)\n ; trans = ⊎-<-transitive (trans pre₁) (trans pre₂)\n }\n where open IsPreorder\n\n ⊎-<-isPartialOrder : IsPartialOrder ≈₁ ∼₁ →\n IsPartialOrder ≈₂ ∼₂ →\n IsPartialOrder (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)\n ⊎-<-isPartialOrder po₁ po₂ = record\n { isPreorder = ⊎-<-isPreorder (isPreorder po₁) (isPreorder po₂)\n ; antisym = ⊎-<-antisymmetric (antisym po₁) (antisym po₂)\n }\n where open IsPartialOrder\n\n ⊎-<-isStrictPartialOrder : IsStrictPartialOrder ≈₁ ∼₁ →\n IsStrictPartialOrder ≈₂ ∼₂ →\n IsStrictPartialOrder (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)\n ⊎-<-isStrictPartialOrder spo₁ spo₂ = record\n { isEquivalence = PW.⊎-isEquivalence (isEquivalence spo₁) (isEquivalence spo₂)\n ; irrefl = ⊎-<-irreflexive (irrefl spo₁) (irrefl spo₂)\n ; trans = ⊎-<-transitive (trans spo₁) (trans spo₂)\n ; <-resp-≈ = ⊎-<-respects₂ (<-resp-≈ spo₁) (<-resp-≈ spo₂)\n }\n where open IsStrictPartialOrder\n\n ⊎-<-isTotalOrder : IsTotalOrder ≈₁ ∼₁ →\n IsTotalOrder ≈₂ ∼₂ →\n IsTotalOrder (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)\n ⊎-<-isTotalOrder to₁ to₂ = record\n { isPartialOrder = ⊎-<-isPartialOrder (isPartialOrder to₁) (isPartialOrder to₂)\n ; total = ⊎-<-total (total to₁) (total to₂)\n }\n where open IsTotalOrder\n\n ⊎-<-isDecTotalOrder : IsDecTotalOrder ≈₁ ∼₁ →\n IsDecTotalOrder ≈₂ ∼₂ →\n IsDecTotalOrder (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)\n ⊎-<-isDecTotalOrder to₁ to₂ = record\n { isTotalOrder = ⊎-<-isTotalOrder (isTotalOrder to₁) (isTotalOrder to₂)\n ; _≟_ = PW.⊎-decidable (_≟_ to₁) (_≟_ to₂)\n ; _≤?_ = ⊎-<-decidable (_≤?_ to₁) (_≤?_ to₂)\n }\n where open IsDecTotalOrder\n\n ⊎-<-isStrictTotalOrder : IsStrictTotalOrder ≈₁ ∼₁ →\n IsStrictTotalOrder ≈₂ ∼₂ →\n IsStrictTotalOrder (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)\n ⊎-<-isStrictTotalOrder sto₁ sto₂ = record\n { isEquivalence = PW.⊎-isEquivalence (isEquivalence sto₁) (isEquivalence sto₂)\n ; trans = ⊎-<-transitive (trans sto₁) (trans sto₂)\n ; compare = ⊎-<-trichotomous (compare sto₁) (compare sto₂)\n }\n where open IsStrictTotalOrder\n\n------------------------------------------------------------------------\n-- \"Packages\" can also be combined.\n\nmodule _ {a b c d e f} where\n\n ⊎-<-preorder : Preorder a b c →\n Preorder d e f →\n Preorder _ _ _\n ⊎-<-preorder p₁ p₂ = record\n { isPreorder =\n ⊎-<-isPreorder (isPreorder p₁) (isPreorder p₂)\n } where open Preorder\n\n ⊎-<-poset : Poset a b c →\n Poset a b c →\n Poset _ _ _\n ⊎-<-poset po₁ po₂ = record\n { isPartialOrder =\n ⊎-<-isPartialOrder (isPartialOrder po₁) (isPartialOrder po₂)\n } where open Poset\n\n ⊎-<-strictPartialOrder : StrictPartialOrder a b c →\n StrictPartialOrder d e f →\n StrictPartialOrder _ _ _\n ⊎-<-strictPartialOrder spo₁ spo₂ = record\n { isStrictPartialOrder =\n ⊎-<-isStrictPartialOrder (isStrictPartialOrder spo₁) (isStrictPartialOrder spo₂)\n } where open StrictPartialOrder\n\n ⊎-<-totalOrder : TotalOrder a b c →\n TotalOrder d e f →\n TotalOrder _ _ _\n ⊎-<-totalOrder to₁ to₂ = record\n { isTotalOrder = ⊎-<-isTotalOrder (isTotalOrder to₁) (isTotalOrder to₂)\n } where open TotalOrder\n\n ⊎-<-decTotalOrder : DecTotalOrder a b c →\n DecTotalOrder d e f →\n DecTotalOrder _ _ _\n ⊎-<-decTotalOrder to₁ to₂ = record\n { isDecTotalOrder = ⊎-<-isDecTotalOrder (isDecTotalOrder to₁) (isDecTotalOrder to₂)\n } where open DecTotalOrder\n\n ⊎-<-strictTotalOrder : StrictTotalOrder a b c →\n StrictTotalOrder a b c →\n StrictTotalOrder _ _ _\n ⊎-<-strictTotalOrder sto₁ sto₂ = record\n { isStrictTotalOrder = ⊎-<-isStrictTotalOrder (isStrictTotalOrder sto₁) (isStrictTotalOrder sto₂)\n } where open StrictTotalOrder\n", "meta": {"hexsha": "06edca09ccb86e4c1781107c3498239b51163964", "size": 10452, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/asset/agda-stdlib-1.0/Data/Sum/Relation/Binary/LeftOrder.agda", "max_stars_repo_name": "omega12345/agda-mode", "max_stars_repo_head_hexsha": "0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "test/asset/agda-stdlib-1.0/Data/Sum/Relation/Binary/LeftOrder.agda", 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YES", "lm_q1_score": 0.6442250928250376, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.34972614899311183}} {"text": "module bool-to-string where\n\nopen import bool\nopen import string\n\n𝔹-to-string : 𝔹 → string\n𝔹-to-string tt = \"tt\"\n𝔹-to-string ff = \"ff\"", "meta": {"hexsha": "76644424fefa2b62490c13859ab6320f04bbedf6", "size": 134, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "bool-to-string.agda", "max_stars_repo_name": "rfindler/ial", "max_stars_repo_head_hexsha": "f3f0261904577e930bd7646934f756679a6cbba6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 29, "max_stars_repo_stars_event_min_datetime": "2019-02-06T13:09:31.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-04T15:05:12.000Z", "max_issues_repo_path": "bool-to-string.agda", "max_issues_repo_name": "rfindler/ial", "max_issues_repo_head_hexsha": "f3f0261904577e930bd7646934f756679a6cbba6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2018-07-09T22:53:38.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-22T03:43:34.000Z", "max_forks_repo_path": "bool-to-string.agda", "max_forks_repo_name": "rfindler/ial", "max_forks_repo_head_hexsha": "f3f0261904577e930bd7646934f756679a6cbba6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 17, "max_forks_repo_forks_event_min_datetime": "2018-12-03T22:38:15.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-28T20:13:21.000Z", "avg_line_length": 16.75, "max_line_length": 27, "alphanum_fraction": 0.7089552239, "num_tokens": 46, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5, "lm_q1q2_score": 0.34962721052937923}} {"text": "open import Formalization.PredicateLogic.Signature\n\nmodule Formalization.PredicateLogic.Minimal.NaturalDeduction.NegativeTranslations (𝔏 : Signature) where\nopen Signature(𝔏)\n\nopen import Data.Either as Either using (Left ; Right)\nopen import Data.ListSized using (List)\nimport Logic.Propositional as Meta\nimport Logic.Predicate as Meta\nimport Lvl\nopen import Formalization.PredicateLogic.Minimal.NaturalDeduction (𝔏)\nimport Formalization.PredicateLogic.Classical.NaturalDeduction\nprivate module Classical = Formalization.PredicateLogic.Classical.NaturalDeduction (𝔏)\nopen import Formalization.PredicateLogic.Syntax(𝔏)\nopen import Formalization.PredicateLogic.Syntax.NegativeTranslations(𝔏)\nopen import Formalization.PredicateLogic.Syntax.Substitution(𝔏)\nopen import Functional using (_∘_ ; _∘₂_ ; _∘₃_ ; _∘₄_ ; swap ; _←_)\nopen import Numeral.Finite\nopen import Numeral.Natural\nopen import Relator.Equals\nopen import Relator.Equals.Proofs.Equiv\nopen import Sets.PredicateSet using (PredSet ; _∈_ ; _∉_ ; _∪_ ; _∪•_ ; _∖_ ; _⊆_ ; _⊇_ ; ∅ ; [≡]-to-[⊆] ; [≡]-to-[⊇] ; map ; unmap) renaming (•_ to · ; _≡_ to _≡ₛ_)\nopen import Structure.Relator\nopen import Type\n\n-- TODO: Move this module\nmodule _ where\n private variable ℓ : Lvl.Level\n private variable T A B : Type{ℓ}\n private variable S S₁ S₂ : PredSet{ℓ}(T)\n private variable f : A → B\n private variable x : T\n\n postulate map-preserves-union : (map f(S₁ ∪ S₂) ⊆ ((map f(S₁)) ∪ (map f(S₂))))\n\n postulate map-preserves-singleton : (map f(S₁ ∪ S₂) ⊆ ((map f(S₁)) ∪ (map f(S₂))))\n\n postulate map-preserves-union-singleton : (map f(S ∪ · x) ⊆ ((map f(S)) ∪ ·(f(x))))\n\nprivate variable ℓ ℓ₁ ℓ₂ : Lvl.Level\nprivate variable args n vars : ℕ\nprivate variable Γ Γ₁ Γ₂ : PredSet{ℓ}(Formula(vars))\nprivate variable φ ψ γ φ₁ ψ₁ γ₁ φ₂ ψ₂ γ₂ φ₃ ψ₃ φ₄ ψ₄ φ₅ ψ₅ δ₁ δ₂ : Formula(vars)\nprivate variable p : Prop(args)\nprivate variable x : List(Term(vars))(args)\n\n[⊢]-functionₗ : (Γ₁ ≡ₛ Γ₂) → ((Γ₁ ⊢_) ≡ₛ (Γ₂ ⊢_))\n[⊢]-functionₗ Γ₁Γ₂ = Meta.[↔]-intro (weaken (Meta.[↔]-to-[←] Γ₁Γ₂)) (weaken (Meta.[↔]-to-[→] Γ₁Γ₂))\n\n[⊢][→]-elim : ((Γ ∪ · φ) ⊢ ψ) → ((Γ ⊢ φ) → (Γ ⊢ ψ))\n[⊢][→]-elim Γφψ Γφ = [∨]-elim Γφψ Γφψ ([∨]-introₗ Γφ)\n\n[⟶]-intro-inverse : (Γ ⊢ (φ ⟶ ψ)) → ((Γ ∪ · φ) ⊢ ψ)\n[⟶]-intro-inverse p = [⟶]-elim (direct (Right [≡]-intro)) (weaken-union p)\n\nweaken-closure-union : (((Γ₁ ⊢_) ∪ Γ₂) ⊢ φ) → ((((Γ₁ ∪ Γ₂) ⊢_)) ⊢ φ)\nweaken-closure-union {Γ₁ = Γ₁}{Γ₂ = Γ₂}{φ = φ} = weaken sub where\n sub : ((Γ₁ ⊢_) ∪ Γ₂) ⊆ ((Γ₁ ∪ Γ₂) ⊢_)\n sub (Left p) = weaken-union p\n sub (Right p) = direct (Right p)\n\n{-\nassume-closure : ((Γ ⊢_) ⊢ φ) → (Γ ⊢ φ)\nassume-closure (direct p) = p\nassume-closure [⊤]-intro = [⊤]-intro\nassume-closure ([∧]-intro p q) = [∧]-intro (assume-closure p) (assume-closure q)\nassume-closure ([∧]-elimₗ p) = [∧]-elimₗ (assume-closure p)\nassume-closure ([∧]-elimᵣ p) = [∧]-elimᵣ (assume-closure p)\nassume-closure ([∨]-introₗ p) = [∨]-introₗ (assume-closure p)\nassume-closure ([∨]-introᵣ p) = [∨]-introᵣ (assume-closure p)\nassume-closure ([∨]-elim p q r) = [∨]-elim {!assume-closure(weaken-closure-union p)!} ([⟶]-elim (direct(Right [≡]-intro)) {!!}) (assume-closure r)\nassume-closure ([⟶]-intro p) = [⟶]-intro (assume-closure {!weaken-closure-union p!})\nassume-closure ([⟶]-elim p q) = [⟶]-elim (assume-closure p) (assume-closure q)\nassume-closure ([Ɐ]-intro p) = [Ɐ]-intro (assume-closure p)\nassume-closure ([Ɐ]-elim p) = [Ɐ]-elim (assume-closure p)\nassume-closure ([∃]-intro p) = [∃]-intro (assume-closure p)\nassume-closure ([∃]-elim p q) = [∃]-elim (assume-closure {!!}) (assume-closure q)\n-}\n\n-- 2.1.8B1\n[¬¬]-intro-[⟶] : (Γ ⊢ (φ ⟶ (¬¬ φ)))\n[¬¬]-intro-[⟶] = [⟶]-intro ([¬¬]-intro (direct (Right [≡]-intro)))\n\n-- 2.1.8.A1\n[⟶]-const : Γ ⊢ (φ ⟶ ψ ⟶ φ)\n[⟶]-const = [⟶]-intro ([⟶]-intro (direct (Left (Right [≡]-intro))))\n\n[⟶]-refl : Γ ⊢ (φ ⟶ φ)\n[⟶]-refl = [⟶]-intro (direct (Right [≡]-intro))\n\n[⟷]-refl : Γ ⊢ (φ ⟷ φ)\n[⟷]-refl = [⟷]-intro (direct (Right [≡]-intro)) (direct (Right [≡]-intro))\n\n[⟶]-trans : (Γ ⊢ ((φ ⟶ ψ) ⟶ (ψ ⟶ γ) ⟶ (φ ⟶ γ)))\n[⟶]-trans = [⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-elim ([⟶]-elim (direct (Right [≡]-intro)) (direct (Left (Left (Right [≡]-intro))))) (direct (Left (Right [≡]-intro))))))\n\n[⟶]-contrapositiveᵣ : (Γ ⊢ (φ ⟶ ψ) ⟶ ((¬ φ) ⟵ (¬ ψ)))\n[⟶]-contrapositiveᵣ = [⟶]-trans\n\n[⟶]-double-contrapositiveᵣ : (Γ ⊢ (φ ⟶ ψ) ⟶ ((¬¬ φ) ⟶ (¬¬ ψ)))\n[⟶]-double-contrapositiveᵣ = [⟶]-intro ([⟶]-elim ([⟶]-elim (direct (Right [≡]-intro)) [⟶]-contrapositiveᵣ) [⟶]-contrapositiveᵣ)\n\n[⟶]-elim₂ : (Γ ⊢ φ₁) → (Γ ⊢ φ₂) → (Γ ⊢ (φ₁ ⟶ φ₂ ⟶ ψ)) → (Γ ⊢ ψ)\n[⟶]-elim₂ = swap(_∘_) [⟶]-elim ∘ (swap(_∘_) ∘ [⟶]-elim)\n\n[⟶]-elim₃ : (Γ ⊢ φ₁) → (Γ ⊢ φ₂) → (Γ ⊢ φ₃) → (Γ ⊢ (φ₁ ⟶ φ₂ ⟶ φ₃ ⟶ ψ)) → (Γ ⊢ ψ)\n[⟶]-elim₃ = swap(_∘_) [⟶]-elim ∘₂ (swap(_∘_) ∘₂ [⟶]-elim₂)\n\n[⟶]-elim₄ : (Γ ⊢ φ₁) → (Γ ⊢ φ₂) → (Γ ⊢ φ₃) → (Γ ⊢ φ₄) → (Γ ⊢ (φ₁ ⟶ φ₂ ⟶ φ₃ ⟶ φ₄ ⟶ ψ)) → (Γ ⊢ ψ)\n[⟶]-elim₄ = swap(_∘_) [⟶]-elim ∘₃ (swap(_∘_) ∘₃ [⟶]-elim₃)\n\n[⟶]-elim₅ : (Γ ⊢ φ₁) → (Γ ⊢ φ₂) → (Γ ⊢ φ₃) → (Γ ⊢ φ₄) → (Γ ⊢ φ₅) → (Γ ⊢ (φ₁ ⟶ φ₂ ⟶ φ₃ ⟶ φ₄ ⟶ φ₅ ⟶ ψ)) → (Γ ⊢ ψ)\n[⟶]-elim₅ = swap(_∘_) [⟶]-elim ∘₄ (swap(_∘_) ∘₄ [⟶]-elim₄)\n\n-- 2.1.8B2\n[¬¬¬]-elim : Γ ⊢ ((¬ ¬ ¬ φ) ⟶ (¬ φ))\n[¬¬¬]-elim = [⟶]-intro ([⟶]-intro ([⟶]-elim ([¬¬]-intro (direct (Right [≡]-intro))) (direct (Left (Right [≡]-intro)))))\n\n[∨]-introₗ-by-[¬∧] : Γ ⊢ ((¬ φ) ⟶ ¬(φ ∧ ψ))\n[∨]-introₗ-by-[¬∧] = [⟶]-intro ([⟶]-intro ([⟶]-elim ([∧]-elimₗ (direct (Right [≡]-intro))) (direct (Left (Right [≡]-intro)))))\n\n[∨]-introᵣ-by-[¬∧] : Γ ⊢ ((¬ ψ) ⟶ ¬(φ ∧ ψ))\n[∨]-introᵣ-by-[¬∧] = [⟶]-intro ([⟶]-intro ([⟶]-elim ([∧]-elimᵣ (direct (Right [≡]-intro))) (direct (Left (Right [≡]-intro)))))\n\n[∨]-introₗ-by-[¬¬∧¬] : Γ ⊢ (φ ⟶ ¬((¬ φ) ∧ (¬ ψ)))\n[∨]-introₗ-by-[¬¬∧¬] = [⟶]-intro ([⟶]-intro ([⟶]-elim (direct (Left (Right [≡]-intro))) ([∧]-elimₗ (direct (Right [≡]-intro)))))\n\n[∨]-introᵣ-by-[¬¬∧¬] : Γ ⊢ (ψ ⟶ ¬((¬ φ) ∧ (¬ ψ)))\n[∨]-introᵣ-by-[¬¬∧¬] = [⟶]-intro ([⟶]-intro ([⟶]-elim (direct (Left (Right [≡]-intro))) ([∧]-elimᵣ (direct (Right [≡]-intro)))))\n\n[∃]-intro-by-[¬Ɐ] : ∀{t} → (Γ ⊢ ((¬(substitute0 t φ)) ⟶ ¬(Ɐ φ)))\n[∃]-intro-by-[¬Ɐ] = [⟶]-intro ([⟶]-intro ([⟶]-elim ([Ɐ]-elim (direct (Right [≡]-intro))) (direct (Left (Right [≡]-intro)))))\n\n[∃]-intro-by-[¬Ɐ¬] : ∀{t} → (Γ ⊢ ((substitute0 t φ) ⟶ ¬(Ɐ(¬ φ))))\n[∃]-intro-by-[¬Ɐ¬] = [⟶]-intro ([⟶]-intro ([⟶]-elim (direct (Left (Right [≡]-intro))) ([Ɐ]-elim (direct (Right [≡]-intro)))))\n\n[∧]-map : Γ ⊢ ((φ₁ ⟶ φ₂) ⟶ (ψ₁ ⟶ ψ₂) ⟶ ((φ₁ ∧ ψ₁) ⟶ (φ₂ ∧ ψ₂)))\n[∧]-map = [⟶]-intro ([⟶]-intro ([⟶]-intro ([∧]-intro ([⟶]-elim ([∧]-elimₗ (direct (Right [≡]-intro))) (direct (Left (Left (Right [≡]-intro))))) ([⟶]-elim ([∧]-elimᵣ (direct (Right [≡]-intro))) (direct (Left (Right [≡]-intro)))))))\n\n[∨]-map : Γ ⊢ ((φ₁ ⟶ φ₂) ⟶ (ψ₁ ⟶ ψ₂) ⟶ ((φ₁ ∨ ψ₁) ⟶ (φ₂ ∨ ψ₂)))\n[∨]-map = [⟶]-intro ([⟶]-intro ([⟶]-intro ([∨]-elim ([∨]-introₗ ([⟶]-intro-inverse (direct (Left (Left (Right [≡]-intro)))))) ([∨]-introᵣ ([⟶]-intro-inverse (direct (Left (Right [≡]-intro))))) (direct (Right [≡]-intro)))))\n\n[Ɐ]-map : Γ ⊢ (Ɐ(φ₁ ⟶ φ₂) ⟶ ((Ɐ φ₁) ⟶ (Ɐ φ₂)))\n[Ɐ]-map = [⟶]-intro ([⟶]-intro ([Ɐ]-intro ([⟶]-elim ([Ɐ]-elim (direct (Right [≡]-intro))) ([Ɐ]-elim(direct (Left (Right [≡]-intro)))))))\n\n[⟶]-map : Γ ⊢ ((φ₁ ⟵ φ₂) ⟶ (ψ₁ ⟶ ψ₂) ⟶ ((φ₁ ⟶ ψ₁) ⟶ (φ₂ ⟶ ψ₂)))\n[⟶]-map = [⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-elim ([⟶]-elim ([⟶]-elim (direct (Right [≡]-intro)) (direct (Left (Left (Left (Right [≡]-intro)))))) (direct (Left (Right [≡]-intro)))) (direct (Left (Left (Right [≡]-intro))))))))\n\n[⟶]-map₂ : Γ ⊢ ((φ₁ ⟵ φ₂) ⟶ (ψ₁ ⟵ ψ₂) ⟶ (γ₁ ⟶ γ₂) ⟶ ((φ₁ ⟶ ψ₁ ⟶ γ₁) ⟶ (φ₂ ⟶ ψ₂ ⟶ γ₂)))\n[⟶]-map₂ = [⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-elim₂ (direct (Left (Left (Right [≡]-intro)))) ([⟶]-elim₂ (direct (Left (Right [≡]-intro))) (direct (Right [≡]-intro)) [⟶]-map) [⟶]-map)))\n\n[⟶]-map₃ : Γ ⊢ ((φ₁ ⟵ φ₂) ⟶ (ψ₁ ⟵ ψ₂) ⟶ (γ₁ ⟵ γ₂) ⟶ (δ₁ ⟶ δ₂) ⟶ ((φ₁ ⟶ ψ₁ ⟶ γ₁ ⟶ δ₁) ⟶ (φ₂ ⟶ ψ₂ ⟶ γ₂ ⟶ δ₂)))\n[⟶]-map₃ = [⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-elim₃ (direct (Left (Left (Left (Right [≡]-intro))))) (direct (Left (Left (Right [≡]-intro)))) ([⟶]-elim₂ (direct (Left (Right [≡]-intro))) (direct (Right [≡]-intro)) [⟶]-map) [⟶]-map₂))))\n\n[¬]-map : Γ ⊢ ((φ ⟵ ψ) ⟶ (¬ φ) ⟶ (¬ ψ))\n[¬]-map = [⟶]-contrapositiveᵣ\n\n[⟶]-trans₂ : (Γ ⊢ ((φ₁ ⟶ φ₂ ⟶ ψ) ⟶ (ψ ⟶ γ) ⟶ (φ₁ ⟶ φ₂ ⟶ γ)))\n[⟶]-trans₂ = [⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-elim ([⟶]-elim₂ (direct (Left (Right [≡]-intro))) (direct (Right [≡]-intro)) (direct (Left (Left (Left (Right [≡]-intro)))))) (direct (Left (Left (Right [≡]-intro))))))))\n\n-- 2.1.8B3\n[¬¬]-preserve-[⟶] : Γ ⊢ (¬¬(φ ⟶ ψ) ⟶ ((¬¬ φ) ⟶ (¬¬ ψ)))\n[¬¬]-preserve-[⟶] = [⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-elim ([⟶]-intro ([⟶]-elim ([⟶]-elim (direct (Left (Left (Right [≡]-intro)))) ([⟶]-elim (direct(Right [≡]-intro)) [⟶]-double-contrapositiveᵣ)) ([¬¬]-intro (direct (Left (Right [≡]-intro)))))) (direct (Left (Left (Right [≡]-intro)))))))\n\n-- 2.1.8B4\n[¬¬]-preserve-[∧] : Γ ⊢ (¬¬(φ ∧ ψ) ⟷ ((¬¬ φ) ∧ (¬¬ ψ)))\n[¬¬]-preserve-[∧] {Γ = Γ}{φ = φ} =\n [⟷]-intro\n ([⟶]-intro ([⟶]-elim ([⟶]-intro ([⟶]-elim ([⟶]-intro ([⟶]-elim ([∧]-intro (direct (Left (Right [≡]-intro))) (direct (Right [≡]-intro))) (direct (Left (Left (Right [≡]-intro)))))) ([∧]-elimᵣ (direct (Left (Left (Right [≡]-intro))))))) ([∧]-elimₗ (direct (Left (Right [≡]-intro))))))\n ([∧]-intro\n ([⟶]-intro ([⟶]-elim ([⟶]-elim (direct (Right [≡]-intro)) [∨]-introₗ-by-[¬∧]) (direct (Left (Right [≡]-intro)))))\n ([⟶]-intro ([⟶]-elim ([⟶]-elim (direct (Right [≡]-intro)) [∨]-introᵣ-by-[¬∧]) (direct (Left (Right [≡]-intro)))))\n )\n\n-- 2.1.8B5\n[¬]-preserve-[∨][∧] : Γ ⊢ (¬(φ ∨ ψ) ⟷ ((¬ φ) ∧ (¬ ψ)))\n[¬]-preserve-[∨][∧] =\n [⟷]-intro\n ([⟶]-intro ([∨]-elim\n ([⟶]-elim (direct (Right [≡]-intro)) ([∧]-elimₗ (direct (Left (Left (Right [≡]-intro))))))\n ([⟶]-elim (direct (Right [≡]-intro)) ([∧]-elimᵣ (direct (Left (Left (Right [≡]-intro))))))\n (direct (Right [≡]-intro))\n ))\n ([∧]-intro\n ([⟶]-intro ([⟶]-elim ([∨]-introₗ (direct (Right [≡]-intro))) (direct (Left (Right [≡]-intro)))))\n ([⟶]-intro ([⟶]-elim ([∨]-introᵣ (direct (Right [≡]-intro))) (direct (Left (Right [≡]-intro)))))\n )\n\n-- 2.1.8B6\n[¬¬]-preserve-[Ɐ] : Γ ⊢ (¬¬(Ɐ φ) ⟶ (Ɐ(¬¬ φ)))\n[¬¬]-preserve-[Ɐ] = [⟶]-intro ([Ɐ]-intro ([⟶]-intro ([⟶]-elim ([⟶]-elim (direct (Right [≡]-intro)) [∃]-intro-by-[¬Ɐ]) (direct (Left (Right [≡]-intro))))))\n\n[¬]-preserve-[∃][Ɐ] : Γ ⊢ (¬(∃ φ) ⟷ (Ɐ(¬ φ)))\n[¬]-preserve-[∃][Ɐ] =\n [⟷]-intro\n ([⟶]-intro ([∃]-elim ([⟶]-intro-inverse ([Ɐ]-elim (direct (Left (Right [≡]-intro))))) (direct (Right [≡]-intro))))\n ([Ɐ]-intro ([⟶]-intro ([⟶]-elim ([∃]-intro (direct (Right [≡]-intro))) (direct (Left (Right [≡]-intro))))))\n\nopen import Lang.Instance\n\n-- 2.3.1\ndata NegativeFragment : Formula(vars) → Type{ℓₚ Lvl.⊔ ℓₒ} where\n atom : NegativeFragment(¬(p $ x))\n bottom : NegativeFragment{vars}(⊥)\n top : NegativeFragment{vars}(⊤)\n and : NegativeFragment(φ) → NegativeFragment(ψ) → NegativeFragment(φ ∧ ψ)\n impl : NegativeFragment(φ) → NegativeFragment(ψ) → NegativeFragment(φ ⟶ ψ)\n all : (∀{t} → NegativeFragment(substitute0 t φ)) → NegativeFragment(Ɐ φ)\npattern neg p = NegativeFragment.impl p bottom\ninstance _ = \\{vars} {p}{args}{x} → atom{vars}{p = p}{args}{x = x}\ninstance _ = \\{vars} → bottom{vars}\ninstance _ = \\{vars} → top{vars}\ninstance _ = \\{vars} {φ} ⦃ neg-φ ⦄ {ψ} ⦃ neg-ψ ⦄ → and{vars}{φ = φ}{ψ = ψ} neg-φ neg-ψ\ninstance _ = \\{vars} {φ} ⦃ neg-φ ⦄ {ψ} ⦃ neg-ψ ⦄ → impl{vars}{φ = φ}{ψ = ψ} neg-φ neg-ψ\ninstance _ = \\{vars} {φ} ⦃ neg-φ : ∀{_} → _ ⦄ → all{vars}{φ = φ} (\\{t} → neg-φ{t})\n\n-- 2.3.2\n[¬¬]-elim-on-negativeFragment : NegativeFragment(φ) → (Γ ⊢ ((¬¬ φ) ⟶ φ))\n[¬¬]-elim-on-negativeFragment atom = [¬¬¬]-elim\n[¬¬]-elim-on-negativeFragment bottom = [¬¬]-intro [⟶]-refl\n[¬¬]-elim-on-negativeFragment top = [⟶]-intro [⊤]-intro\n[¬¬]-elim-on-negativeFragment (and negφ negψ) =\n [⟶]-intro ([∧]-intro\n ([⟶]-elim ([∧]-elimₗ([⟷]-elimᵣ (direct (Right [≡]-intro)) [¬¬]-preserve-[∧])) ([¬¬]-elim-on-negativeFragment negφ))\n ([⟶]-elim ([∧]-elimᵣ([⟷]-elimᵣ (direct (Right [≡]-intro)) [¬¬]-preserve-[∧])) ([¬¬]-elim-on-negativeFragment negψ))\n )\n[¬¬]-elim-on-negativeFragment (impl negφ negψ) =\n [⟶]-intro ([⟶]-intro ([⟶]-elim\n ([⟶]-elim₂ (direct (Left (Right [≡]-intro))) ([¬¬]-intro (direct (Right [≡]-intro))) [¬¬]-preserve-[⟶])\n ([¬¬]-elim-on-negativeFragment negψ)\n ))\n[¬¬]-elim-on-negativeFragment (all negφ) = [⟶]-intro ([Ɐ]-intro ([⟶]-elim ([⟶]-intro ([⟶]-elim ([⟶]-elim (direct (Right [≡]-intro)) [∃]-intro-by-[¬Ɐ]) (direct (Left (Right [≡]-intro))))) ([¬¬]-elim-on-negativeFragment negφ)))\n\nStable : ∀{ℓ}{vars} → Formula(vars) → Type\nStable{ℓ = ℓ} (φ) = ∀{Γ : PredSet{ℓ}(_)} → (Γ ⊢ (¬¬ φ ⟶ φ))\n\n[¬]-stability : Stable{ℓ}(¬ φ)\n[¬]-stability = [¬¬¬]-elim\n\n-- [∨]-stability : Stable(φ) → Stable(ψ) → Stable{ℓ}(φ ∨ ψ)\n-- [∨]-stability sφ sψ = [⟶]-intro ([∨]-elim ([∨]-introₗ {!!}) ([∨]-introᵣ {!!}) {!!})\n\n[∨]-elim-by-[¬∧] : Stable(φ) → Stable(ψ) → Stable(γ) → (Γ ⊢ (((¬ φ) ⟶ γ) ⟶ ((¬ ψ) ⟶ γ) ⟶ ¬(φ ∧ ψ) ⟶ γ))\n[∨]-elim-by-[¬∧] negφ negψ negγ = [⟶]-intro ([⟶]-intro ([⟶]-intro ([⟶]-elim ([⟶]-intro ([⟶]-elim ([⟶]-elim₃ negφ negψ ([∧]-intro ([⟶]-elim (direct (Right [≡]-intro)) ([⟶]-elim (direct (Left (Left (Left (Right [≡]-intro))))) [⟶]-contrapositiveᵣ)) ([⟶]-elim (direct (Right [≡]-intro)) ([⟶]-elim (direct (Left (Left (Right [≡]-intro)))) [⟶]-contrapositiveᵣ))) [∧]-map) (direct (Left (Right [≡]-intro))))) negγ)))\n\n[∨]-elim-by-[¬¬∧¬] : (∀{ℓ} → Stable{ℓ}(γ)) → (Γ ⊢ ((φ ⟶ γ) ⟶ (ψ ⟶ γ) ⟶ ¬((¬ φ) ∧ (¬ ψ)) ⟶ γ))\n[∨]-elim-by-[¬¬∧¬] negγ = [⟶]-elim₅ (pp negγ) (pp negγ) [⟶]-refl [⟶]-refl ([∨]-elim-by-[¬∧] [¬]-stability [¬]-stability negγ) [⟶]-map₃ where\n pp : Stable(ψ) → (Γ ⊢ ((φ ⟶ ψ) ⟶ ((¬¬ φ) ⟶ ψ)))\n pp negψ = [⟶]-elim₂ [⟶]-double-contrapositiveᵣ negψ [⟶]-trans₂\n\n[⟷]-to-[⟵] : (Γ ⊢ (φ ⟷ ψ)) → (Γ ⊢ (φ ⟵ ψ))\n[⟷]-to-[⟵] p = [⟶]-intro ([⟷]-elimₗ (direct (Right [≡]-intro)) (weaken-union p))\n\n[⟷]-to-[⟶] : (Γ ⊢ (φ ⟷ ψ)) → (Γ ⊢ (φ ⟶ ψ))\n[⟷]-to-[⟶] p = [⟶]-intro ([⟷]-elimᵣ (direct (Right [≡]-intro)) (weaken-union p))\n\n[Ɐ][→]-distributivity : Γ ⊢ (Ɐ(φ ⟶ ψ) ⟶ (Ɐ φ) ⟶ (Ɐ ψ))\n[Ɐ][→]-distributivity = [⟶]-intro ([⟶]-intro ([Ɐ]-intro ([⟶]-elim ([Ɐ]-elim (direct (Right [≡]-intro))) ([Ɐ]-elim (direct (Left (Right [≡]-intro)))))))\n\n[¬∃]-to-[∀¬] : Γ ⊢ (¬(∃ φ) ⟶ Ɐ(¬ φ))\n[¬∃]-to-[∀¬] = [⟶]-intro ([Ɐ]-intro ([⟶]-intro ([⟶]-elim ([∃]-intro (direct (Right [≡]-intro))) (direct (Left (Right [≡]-intro))))))\n\n[¬Ɐ]-to-[∃¬] : (∀{t} → Stable(substitute0 t φ)) → Stable(∃(¬ φ)) → (Γ ⊢ (¬(Ɐ φ) ⟶ ∃(¬ φ)))\n[¬Ɐ]-to-[∃¬] negφ nege = [⟶]-intro ([⟶]-elim ([⟶]-intro ([⟶]-elim ([⟶]-elim₂ ([Ɐ]-intro negφ) ([⟶]-elim (direct (Right [≡]-intro)) [¬∃]-to-[∀¬]) [Ɐ][→]-distributivity) (direct (Left (Right [≡]-intro))))) nege)\n\n{-\ntest : (Γ ⊢ (¬¬ φ ⟶ φ)) → (Γ ⊢ (¬ Ɐ(¬ φ))) → (Γ ⊢ ∃ φ)\ntest negφ p = [∃]-elim ([∃]-intro {![⟶]-intro negφ!}) ([⟶]-elim p test2)\n-}\n\n[∃]-elim-by-[¬¬∧¬] : (∀{t} → (Γ ∪ ·(substitute0 t φ)) ⊢ ψ) → (Γ ⊢ ¬(Ɐ(¬ φ))) → (Γ ⊢ ψ)\n[∃]-elim-by-[¬¬∧¬] p q = [∃]-elim p {!!}\n\n-- test : ∀{t} → (Γ₁ ≡ₛ Γ₂) → (Γ₁ ⊢ φ) → (Γ₂ ⊢ substitute0 t φ)\n-- test p = {!p!}\n\n{-\ntest : ∀{t} → (∀{Γ : PredSet{ℓ}(Formula(𝐒(vars)))} → (Γ ⊢ φ)) → (∀{Γ : PredSet{ℓ}(Formula(vars))} → (Γ ⊢ substitute0 t φ))\ntest {t = t} Γφ {Γ} with Γφ{unmap(substitute0 t) Γ}\n... | direct p = direct p\n... | [⊤]-intro = [⊤]-intro\n... | [∧]-intro p q = [∧]-intro {!test p!} {!!}\n... | [∧]-elimₗ p = {!!}\n... | [∧]-elimᵣ p = {!!}\n... | [∨]-introₗ p = {!!}\n... | [∨]-introᵣ p = {!!}\n... | [∨]-elim p q r = {!!}\n... | [⟶]-intro p = {!!}\n... | [⟶]-elim p q = {!!}\n... | [Ɐ]-intro p = {!!}\n... | [Ɐ]-elim p = {!!}\n... | [∃]-intro p = {!!}\n... | [∃]-elim p q = {!!}\n-}\n\nsubstitute0-negativeFragment : NegativeFragment(φ) → ∀{t} → NegativeFragment(substitute0 t φ)\nsubstitute0-negativeFragment atom = atom\nsubstitute0-negativeFragment bottom = bottom\nsubstitute0-negativeFragment top = top\nsubstitute0-negativeFragment (and p q) = and (substitute0-negativeFragment p) (substitute0-negativeFragment q)\nsubstitute0-negativeFragment (impl p q) = impl (substitute0-negativeFragment p) (substitute0-negativeFragment q)\nsubstitute0-negativeFragment (all p) = {!!} -- all (substitute0-negativeFragment p)\n\nggTrans-negativeFragment : NegativeFragment(ggTrans(φ))\nggTrans-negativeFragment {φ = p $ x} = neg atom\nggTrans-negativeFragment {φ = ⊤} = top\nggTrans-negativeFragment {φ = ⊥} = bottom\nggTrans-negativeFragment {φ = φ ∧ ψ} = and ggTrans-negativeFragment ggTrans-negativeFragment\nggTrans-negativeFragment {φ = φ ∨ ψ} = neg(and(neg ggTrans-negativeFragment) (neg ggTrans-negativeFragment))\nggTrans-negativeFragment {φ = φ ⟶ ψ} = impl ggTrans-negativeFragment ggTrans-negativeFragment\nggTrans-negativeFragment {φ = Ɐ φ} = all (substitute0-negativeFragment (ggTrans-negativeFragment {φ = φ}))\nggTrans-negativeFragment {φ = ∃ φ} = neg (all (neg (substitute0-negativeFragment (ggTrans-negativeFragment {φ = φ}))))\n\n-- [¬¬]-elim-of-koTrans : (Γ ⊢ (¬¬ ggTrans(φ))) ← (Γ ⊢ (ggTrans φ))\n\nggTrans-substitute0 : ∀{t} → (ggTrans(substitute0 t φ) ≡ substitute0 t (ggTrans φ))\nggTrans-substitute0 {φ = P $ x} = [≡]-intro\nggTrans-substitute0 {φ = ⊤} = [≡]-intro\nggTrans-substitute0 {φ = ⊥} = [≡]-intro\nggTrans-substitute0 {φ = φ ∧ ψ}{t}\n rewrite ggTrans-substitute0 {φ = φ}{t}\n rewrite ggTrans-substitute0 {φ = ψ}{t}\n = [≡]-intro\nggTrans-substitute0 {φ = φ ∨ ψ}{t}\n rewrite ggTrans-substitute0 {φ = φ}{t}\n rewrite ggTrans-substitute0 {φ = ψ}{t}\n = [≡]-intro\nggTrans-substitute0 {φ = φ ⟶ ψ}{t}\n rewrite ggTrans-substitute0 {φ = φ}{t}\n rewrite ggTrans-substitute0 {φ = ψ}{t}\n = [≡]-intro\nggTrans-substitute0 {φ = Ɐ φ}{t} = {!!}\n -- rewrite ggTrans-substitute0 {φ = φ}{termVar𝐒 t}\n -- = [≡]-intro\nggTrans-substitute0 {φ = ∃ φ}{t} = {!!}\n -- rewrite ggTrans-substitute0 {φ = φ}{termVar𝐒 t}\n -- = [≡]-intro\n\nkoTrans-substitute0 : ∀{t} → (koTrans(substitute0 t φ) ≡ substitute0 t (koTrans φ))\nkoTrans-substitute0 {φ = f $ x} = [≡]-intro\nkoTrans-substitute0 {φ = ⊤} = [≡]-intro\nkoTrans-substitute0 {φ = ⊥} = [≡]-intro\nkoTrans-substitute0 {φ = φ ∧ ψ}{t}\n rewrite koTrans-substitute0 {φ = φ}{t}\n rewrite koTrans-substitute0 {φ = ψ}{t}\n = [≡]-intro\nkoTrans-substitute0 {φ = φ ∨ ψ}{t}\n rewrite koTrans-substitute0 {φ = φ}{t}\n rewrite koTrans-substitute0 {φ = ψ}{t}\n = [≡]-intro\nkoTrans-substitute0 {φ = φ ⟶ ψ}{t}\n rewrite koTrans-substitute0 {φ = φ}{t}\n rewrite koTrans-substitute0 {φ = ψ}{t}\n = [≡]-intro\nkoTrans-substitute0 {φ = Ɐ φ}{t} = {!!}\n -- rewrite koTrans-substitute0 {φ = φ}{termVar𝐒 t}\n -- = [≡]-intro\nkoTrans-substitute0 {φ = ∃ φ}{t} = {!!}\n -- rewrite koTrans-substitute0 {φ = φ}{termVar𝐒 t}\n -- = [≡]-intro\n\n-- 2.3.4 (ii)\nggTrans-correctnessₗ : (Γ Classical.⊢ φ) ← (map ggTrans Γ ⊢ (ggTrans φ))\nggTrans-correctnessᵣ : (Γ Classical.⊢ φ) → (map ggTrans Γ ⊢ (ggTrans φ))\n\nggTrans-correctnessᵣ (Classical.direct p) = direct (Meta.[∃]-intro _ ⦃ Meta.[∧]-intro p [≡]-intro ⦄)\nggTrans-correctnessᵣ (Classical.[⊤]-intro) = [⊤]-intro\nggTrans-correctnessᵣ {Γ = Γ}{φ = φ} (Classical.[⊥]-elim p) = [⟶]-elim ([⟶]-intro(weaken{Γ₂ = (map ggTrans Γ) ∪ ·(ggTrans(¬ φ))} map-preserves-union-singleton (ggTrans-correctnessᵣ p))) ([¬¬]-elim-on-negativeFragment ggTrans-negativeFragment)\nggTrans-correctnessᵣ (Classical.[∧]-intro p q) = [∧]-intro (ggTrans-correctnessᵣ p) (ggTrans-correctnessᵣ q)\nggTrans-correctnessᵣ (Classical.[∧]-elimₗ p) = [∧]-elimₗ (ggTrans-correctnessᵣ p)\nggTrans-correctnessᵣ (Classical.[∧]-elimᵣ p) = [∧]-elimᵣ (ggTrans-correctnessᵣ p)\nggTrans-correctnessᵣ (Classical.[∨]-introₗ p) = [⟶]-elim (ggTrans-correctnessᵣ p) [∨]-introₗ-by-[¬¬∧¬]\nggTrans-correctnessᵣ (Classical.[∨]-introᵣ p) = [⟶]-elim (ggTrans-correctnessᵣ p) [∨]-introᵣ-by-[¬¬∧¬]\nggTrans-correctnessᵣ (Classical.[∨]-elim p q r) = [⟶]-elim₃ ([⟶]-intro(weaken map-preserves-union-singleton (ggTrans-correctnessᵣ p))) ([⟶]-intro(weaken map-preserves-union-singleton (ggTrans-correctnessᵣ q))) (ggTrans-correctnessᵣ r) ([∨]-elim-by-[¬¬∧¬] ([¬¬]-elim-on-negativeFragment ggTrans-negativeFragment))\nggTrans-correctnessᵣ (Classical.[⟶]-intro p) = [⟶]-intro (weaken map-preserves-union-singleton (ggTrans-correctnessᵣ p))\nggTrans-correctnessᵣ (Classical.[⟶]-elim p q) = [⟶]-elim (ggTrans-correctnessᵣ p) (ggTrans-correctnessᵣ q)\nggTrans-correctnessᵣ (Classical.[Ɐ]-intro p) = [Ɐ]-intro (substitute₁(_ ⊢_) ggTrans-substitute0 (ggTrans-correctnessᵣ p))\nggTrans-correctnessᵣ (Classical.[Ɐ]-elim p) = substitute₁ₗ(_ ⊢_) ggTrans-substitute0 ([Ɐ]-elim (ggTrans-correctnessᵣ p))\nggTrans-correctnessᵣ (Classical.[∃]-intro p) = [⟶]-elim (substitute₁(_ ⊢_) ggTrans-substitute0 (ggTrans-correctnessᵣ p)) [∃]-intro-by-[¬Ɐ¬]\nggTrans-correctnessᵣ (Classical.[∃]-elim p q) = [⟶]-elim ([⟶]-intro ([⟶]-elim ([Ɐ]-intro ([⟶]-intro ([⟶]-elim (weaken {!map-preserves-union-singleton!} (ggTrans-correctnessᵣ p)) (direct (Left (Right [≡]-intro)))))) (weaken-union (ggTrans-correctnessᵣ q)))) ([¬¬]-elim-on-negativeFragment ggTrans-negativeFragment)\n\nkoTrans-stability : Stable{ℓ}(koTrans(φ))\nkoTrans-stability {φ = f $ x} = [¬¬¬]-elim\nkoTrans-stability {φ = ⊤} = [⟶]-intro [⊤]-intro\nkoTrans-stability {φ = ⊥} = [⟶]-intro ([⟶]-elim [⟶]-refl (direct (Right [≡]-intro)))\nkoTrans-stability {φ = φ ∧ ψ} = [¬¬¬]-elim\nkoTrans-stability {φ = φ ∨ ψ} = [¬¬¬]-elim\nkoTrans-stability {φ = φ ⟶ ψ} = [¬¬¬]-elim\nkoTrans-stability {φ = Ɐ φ} = [¬¬¬]-elim\nkoTrans-stability {φ = ∃ φ} = [¬¬¬]-elim\n\n{-\nkoTrans-to-[¬¬] : Γ ⊢ (koTrans(φ) ⟶ (¬¬ φ))\nkoTrans-to-[¬¬] {φ = f $ x} = [⟶]-refl\nkoTrans-to-[¬¬] {φ = ⊤} = [¬¬]-intro-[⟶]\nkoTrans-to-[¬¬] {φ = ⊥} = [¬¬]-intro-[⟶]\nkoTrans-to-[¬¬] {φ = φ ∧ ψ} = [⟶]-intro ([⟷]-elimₗ ([⟶]-elim₃ koTrans-to-[¬¬] koTrans-to-[¬¬] {!!} [∧]-map) [¬¬]-preserve-[∧])\nkoTrans-to-[¬¬] {φ = φ ∨ ψ} = {!!}\nkoTrans-to-[¬¬] {φ = φ ⟶ ψ} = {!!}\nkoTrans-to-[¬¬] {φ = Ɐ φ} = {!!}\nkoTrans-to-[¬¬] {φ = ∃ φ} = {!!}\n-}\n\nggTrans-koTransₗ : Γ ⊢ ((ggTrans φ) ⟵ (koTrans φ))\nggTrans-koTransᵣ : Γ ⊢ ((ggTrans φ) ⟶ (koTrans φ))\n\n[∧]-elimₗ-koTrans : (Γ ⊢ (koTrans(φ ∧ ψ) ⟶ koTrans(φ)))\n[∧]-elimₗ-koTrans = [⟶]-intro ([∧]-elimₗ ([⟶]-elim₃ koTrans-stability koTrans-stability ([⟷]-elimᵣ (direct (Right [≡]-intro)) [¬¬]-preserve-[∧]) [∧]-map))\n\n[∧]-elimᵣ-koTrans : (Γ ⊢ (koTrans(φ ∧ ψ) ⟶ koTrans(ψ)))\n[∧]-elimᵣ-koTrans = [⟶]-intro ([∧]-elimᵣ ([⟶]-elim₃ koTrans-stability koTrans-stability ([⟷]-elimᵣ (direct (Right [≡]-intro)) [¬¬]-preserve-[∧]) [∧]-map))\n\n-- [⟶]-elim-koTrans : (Γ ⊢ (koTrans(φ) ⟶ koTrans(φ ⟶ ψ) ⟶ koTrans(ψ)))\n\n-- [⟶]-intro ([⟶]-elim ([∧]-elimₗ ([⟷]-elimᵣ ([⟶]-intro-inverse {![¬¬]-intro-[⟶]!}) [¬¬]-preserve-[∧])) koTrans-stability)\n-- {![∧]-elimₗ(Meta.[↔]-to-[→] [¬¬][∧] Γφψ)!}\n\n-- [¬]-preserve-[∨][∧] : Γ ⊢ (¬(φ ∨ ψ) ⟷ ((¬ φ) ∧ (¬ ψ)))\n\n-- Alternative proof of the (_∧_)-case:\n-- [⟶]-intro ([∧]-intro\n-- ([⟶]-elim ([⟶]-elim (direct (Right [≡]-intro)) [∧]-elimₗ-koTrans) (weaken-union(ggTrans-koTransₗ {φ = φ})))\n-- ([⟶]-elim ([⟶]-elim (direct (Right [≡]-intro)) [∧]-elimᵣ-koTrans) (weaken-union(ggTrans-koTransₗ {φ = ψ})))\n-- )\nggTrans-koTransₗ {φ = P $ x} = [⟶]-refl\nggTrans-koTransₗ {φ = ⊤} = [⟶]-refl\nggTrans-koTransₗ {φ = ⊥} = [⟶]-refl\nggTrans-koTransₗ {φ = φ ∧ ψ} = [⟶]-elim₃\n ([⟶]-elim₂ ([⟷]-to-[⟶] [¬¬]-preserve-[∧]) ([⟶]-elim₂ koTrans-stability koTrans-stability [∧]-map) [⟶]-trans)\n [⟶]-refl\n ([⟶]-elim₂ (ggTrans-koTransₗ {φ = φ}) (ggTrans-koTransₗ {φ = ψ}) [∧]-map)\n [⟶]-map\nggTrans-koTransₗ {φ = φ ∨ ψ} = [⟶]-elim₃\n ([⟶]-elim ([⟷]-to-[⟵] [¬]-preserve-[∨][∧]) [⟶]-contrapositiveᵣ)\n [⟶]-refl\n ([⟶]-elim ([⟶]-elim₂ ([⟶]-elim (ggTrans-koTransₗ {φ = φ}) [¬]-map) ([⟶]-elim (ggTrans-koTransₗ {φ = ψ}) [¬]-map) [∧]-map) [¬]-map)\n [⟶]-map\nggTrans-koTransₗ {φ = φ ⟶ ψ} = [⟶]-elim₃\n (([⟶]-elim₂ [¬¬]-preserve-[⟶] ([⟶]-elim₂ [¬¬]-intro-[⟶] koTrans-stability [⟶]-map) [⟶]-trans))\n [⟶]-refl\n (([⟶]-elim₂ (ggTrans-koTransᵣ {φ = φ}) (ggTrans-koTransₗ {φ = ψ}) [⟶]-map))\n [⟶]-map\nggTrans-koTransₗ {Γ = Γ}{φ = Ɐ φ} = [⟶]-elim₃\n ([⟶]-elim₂ [¬¬]-preserve-[Ɐ] ([⟶]-elim ([Ɐ]-intro (substitute₁(\\φ → Γ ⊢ ((¬¬ φ) ⟶ φ)) koTrans-substitute0 koTrans-stability)) ([Ɐ]-map {φ₂ = koTrans φ})) [⟶]-trans)\n [⟶]-refl\n ([⟶]-elim ([Ɐ]-intro (substitute₂(\\a b → Γ ⊢ (a ⟵ b)) ggTrans-substitute0 koTrans-substitute0 (ggTrans-koTransₗ {φ = substitute0 _ φ}))) [Ɐ]-map)\n [⟶]-map\nggTrans-koTransₗ {φ = ∃ φ} = [⟶]-elim₃\n ([⟶]-elim ([⟷]-to-[⟵] [¬]-preserve-[∃][Ɐ]) [⟶]-contrapositiveᵣ)\n [⟶]-refl\n ([⟶]-elim ([⟶]-elim ([Ɐ]-intro ([⟶]-elim (substitute₂(\\a b → _ ⊢ (a ⟵ b)) ggTrans-substitute0 koTrans-substitute0 (ggTrans-koTransₗ {φ = substitute0 _ φ})) [¬]-map)) [Ɐ]-map) [¬]-map)\n [⟶]-map\n\nggTrans-koTransᵣ {φ = P $ x} = [⟶]-refl\nggTrans-koTransᵣ {φ = ⊤} = [⟶]-refl\nggTrans-koTransᵣ {φ = ⊥} = [⟶]-refl\nggTrans-koTransᵣ {φ = φ ∧ ψ} =\n [⟶]-intro ([¬¬]-intro ([∧]-intro\n ([⟶]-elim ([∧]-elimₗ (direct (Right [≡]-intro))) (weaken-union(ggTrans-koTransᵣ {φ = φ})))\n ([⟶]-elim ([∧]-elimᵣ (direct (Right [≡]-intro))) (weaken-union(ggTrans-koTransᵣ {φ = ψ})))\n ))\nggTrans-koTransᵣ {φ = φ ∨ ψ} =\n [⟶]-intro ([¬¬]-intro ([⟶]-elim₃\n {!!}\n {!!}\n {!direct (Right [≡]-intro)!}\n ([∨]-elim-by-[¬¬∧¬] {!koTrans-stability!})\n ))\nggTrans-koTransᵣ {φ = φ ⟶ ψ} =\n [⟶]-intro ([¬¬]-intro ([⟶]-intro\n ([⟶]-elim\n ([⟶]-elim ([⟵]-elim (direct (Right [≡]-intro)) (weaken-union(weaken-union(ggTrans-koTransₗ {φ = φ})))) (direct (Left (Right [≡]-intro))))\n (weaken-union(weaken-union(ggTrans-koTransᵣ {φ = ψ})))\n )\n ))\nggTrans-koTransᵣ {φ = Ɐ φ} = {!!}\nggTrans-koTransᵣ {φ = ∃ φ} = {!!}\n", "meta": {"hexsha": "53987dfcaf3772c7d420c46e5b7014f60d1ba6fb", "size": 25392, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Formalization/PredicateLogic/Minimal/NaturalDeduction/NegativeTranslations.agda", "max_stars_repo_name": "Lolirofle/stuff-in-agda", "max_stars_repo_head_hexsha": "70f4fba849f2fd779c5aaa5af122ccb6a5b271ba", "max_stars_repo_licenses": ["MIT"], 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YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34958202216224477}} {"text": "module BasicIS4.Metatheory.ClosedHilbert-TarskiClosedOvergluedImplicit where\n\nopen import BasicIS4.Syntax.ClosedHilbert public\nopen import BasicIS4.Semantics.TarskiClosedOvergluedImplicit public\n\nopen ImplicitSyntax (⊢_) public\n\n\n-- Completeness with respect to a particular model.\n\nmodule _ {{_ : Model}} where\n reify : ∀ {A} → ⊩ A → ⊢ A\n reify {α P} s = syn s\n reify {A ▻ B} s = syn s\n reify {□ A} s = syn s\n reify {A ∧ B} s = pair (reify (π₁ s)) (reify (π₂ s))\n reify {⊤} s = unit\n\n\n-- Additional useful equipment.\n\nmodule _ {{_ : Model}} where\n ⟪K⟫ : ∀ {A B} → ⊩ A → ⊩ B ▻ A\n ⟪K⟫ a = app ck (reify a) ⅋ K a\n\n ⟪S⟫′ : ∀ {A B C} → ⊩ A ▻ B ▻ C → ⊩ (A ▻ B) ▻ A ▻ C\n ⟪S⟫′ s₁ = app cs (reify s₁) ⅋ λ s₂ →\n app (app cs (reify s₁)) (reify s₂) ⅋ ⟪S⟫ s₁ s₂\n\n _⟪D⟫_ : ∀ {A B} → ⊩ □ (A ▻ B) → ⊩ □ A → ⊩ □ B\n (t ⅋ s) ⟪D⟫ (u ⅋ a) = app (app cdist t) u ⅋ s ⟪$⟫ a\n\n _⟪D⟫′_ : ∀ {A B} → ⊩ □ (A ▻ B) → ⊩ □ A ▻ □ B\n _⟪D⟫′_ s = app cdist (reify s) ⅋ _⟪D⟫_ s\n\n ⟪↑⟫ : ∀ {A} → ⊩ □ A → ⊩ □ □ A\n ⟪↑⟫ s = box (syn s) ⅋ s\n\n _⟪,⟫′_ : ∀ {A B} → ⊩ A → ⊩ B ▻ A ∧ B\n _⟪,⟫′_ a = app cpair (reify a) ⅋ _,_ a\n\n\n-- Soundness with respect to all models, or evaluation, for closed terms only.\n\neval₀ : ∀ {A} → ⊢ A → ⊨ A\neval₀ (app t u) = eval₀ t ⟪$⟫ eval₀ u\neval₀ ci = ci ⅋ I\neval₀ ck = ck ⅋ ⟪K⟫\neval₀ cs = cs ⅋ ⟪S⟫′\neval₀ (box t) = box t ⅋ eval₀ t\neval₀ cdist = cdist ⅋ _⟪D⟫′_\neval₀ cup = cup ⅋ ⟪↑⟫\neval₀ cdown = cdown ⅋ ⟪↓⟫\neval₀ cpair = cpair ⅋ _⟪,⟫′_\neval₀ cfst = cfst ⅋ π₁\neval₀ csnd = csnd ⅋ π₂\neval₀ unit = ∙\n\n\n-- Correctness of evaluation with respect to conversion.\n\neval₀✓ : ∀ {{_ : Model}} {A} {t t′ : ⊢ A} → t ⋙ t′ → eval₀ t ≡ eval₀ t′\neval₀✓ refl⋙ = refl\neval₀✓ (trans⋙ p q) = trans (eval₀✓ p) (eval₀✓ q)\neval₀✓ (sym⋙ p) = sym (eval₀✓ p)\neval₀✓ (congapp⋙ p q) = cong² _⟪$⟫_ (eval₀✓ p) (eval₀✓ q)\neval₀✓ (congi⋙ p) = cong I (eval₀✓ p)\neval₀✓ (congk⋙ p q) = cong² K (eval₀✓ p) (eval₀✓ q)\neval₀✓ (congs⋙ p q r) = cong³ ⟪S⟫ (eval₀✓ p) (eval₀✓ q) (eval₀✓ r)\neval₀✓ (congdist⋙ p q) = cong² _⟪D⟫_ (eval₀✓ p) (eval₀✓ q)\neval₀✓ (congup⋙ p) = cong ⟪↑⟫ (eval₀✓ p)\neval₀✓ (congdown⋙ p) = cong ⟪↓⟫ (eval₀✓ p)\neval₀✓ (congpair⋙ p q) = cong² _,_ (eval₀✓ p) (eval₀✓ q)\neval₀✓ (congfst⋙ p) = cong π₁ (eval₀✓ p)\neval₀✓ (congsnd⋙ p) = cong π₂ (eval₀✓ p)\neval₀✓ beta▻ₖ⋙ = refl\neval₀✓ beta▻ₛ⋙ = refl\neval₀✓ beta□⋙ = refl\neval₀✓ eta□⋙ = refl\neval₀✓ beta∧₁⋙ = refl\neval₀✓ beta∧₂⋙ = refl\neval₀✓ eta∧⋙ = refl\neval₀✓ eta⊤⋙ = refl\n\n\n-- The canonical model.\n\nprivate\n instance\n canon : Model\n canon = record\n { ⊩ᵅ_ = λ P → ⊢ α P\n }\n\n\n-- Completeness with respect to all models, or quotation, for closed terms only.\n\nquot₀ : ∀ {A} → ⊨ A → ⊢ A\nquot₀ s = reify s\n\n\n-- Normalisation by evaluation, for closed terms only.\n\nnorm₀ : ∀ {A} → ⊢ A → ⊢ A\nnorm₀ = quot₀ ∘ eval₀\n\n\n-- Correctness of normalisation with respect to conversion.\n\nnorm₀✓ : ∀ {{_ : Model}} {A} {t t′ : ⊢ A} → t ⋙ t′ → norm₀ t ≡ norm₀ t′\nnorm₀✓ p = cong reify (eval₀✓ p)\n", "meta": {"hexsha": "f50066da791efc2f525d2546043f4286ca6a98ad", "size": 3106, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "BasicIS4/Metatheory/ClosedHilbert-TarskiClosedOvergluedImplicit.agda", "max_stars_repo_name": "mietek/hilbert-gentzen", "max_stars_repo_head_hexsha": "fcd187db70f0a39b894fe44fad0107f61849405c", "max_stars_repo_licenses": ["X11"], "max_stars_count": 29, "max_stars_repo_stars_event_min_datetime": "2016-07-03T18:51:56.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-01T10:29:18.000Z", "max_issues_repo_path": "BasicIS4/Metatheory/ClosedHilbert-TarskiClosedOvergluedImplicit.agda", "max_issues_repo_name": "mietek/hilbert-gentzen", "max_issues_repo_head_hexsha": "fcd187db70f0a39b894fe44fad0107f61849405c", "max_issues_repo_licenses": ["X11"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2018-06-10T09:11:22.000Z", "max_issues_repo_issues_event_max_datetime": "2018-06-10T09:11:22.000Z", "max_forks_repo_path": "BasicIS4/Metatheory/ClosedHilbert-TarskiClosedOvergluedImplicit.agda", "max_forks_repo_name": "mietek/hilbert-gentzen", "max_forks_repo_head_hexsha": "fcd187db70f0a39b894fe44fad0107f61849405c", "max_forks_repo_licenses": ["X11"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.7321428571, "max_line_length": 80, "alphanum_fraction": 0.5264005151, "num_tokens": 1531, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34958202216224477}} {"text": "open import Prelude\nopen import Nat\nopen import dynamics-core\nopen import contexts\nopen import contraction\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\nopen import binders-disjoint-checks\n\nmodule lemmas-subst-ta where\n -- this is what makes the binders-unique assumption below good enough: it\n -- tells us that we can pick fresh variables\n mutual\n binders-envfresh : ∀{Δ Γ Γ' y σ} →\n Δ , Γ ⊢ σ :s: Γ' →\n y # Γ →\n unbound-in-σ y σ →\n binders-unique-σ σ →\n envfresh y σ\n binders-envfresh {Γ' = Γ'} {y = y} (STAId x) apt unbound unique with ctxindirect Γ' y\n binders-envfresh {Γ' = Γ'} {y = y} (STAId x₁) apt unbound unique | Inl x = abort (somenotnone (! (x₁ y (π1 x) (π2 x)) · apt))\n binders-envfresh (STAId x₁) apt unbound unique | Inr x = EFId x\n binders-envfresh {Γ = Γ} {y = y} (STASubst {y = z} subst x₁) apt (UBσSubst x₂ unbound neq) (BUσSubst zz x₃ x₄) =\n EFSubst (binders-fresh {y = y} x₁ zz x₂ apt)\n (binders-envfresh subst (apart-extend1 Γ neq apt) unbound x₃)\n neq\n\n binders-fresh : ∀{Δ Γ d2 τ y} →\n Δ , Γ ⊢ d2 :: τ →\n binders-unique d2 →\n unbound-in y d2 →\n Γ y == None →\n fresh y d2\n binders-fresh TANum BUHole UBNum apt = FNum\n binders-fresh {y = y} (TAVar {x = x} x₁) BUVar UBVar apt with natEQ y x\n binders-fresh (TAVar x₂) BUVar UBVar apt | Inl refl = abort (somenotnone (! x₂ · apt))\n binders-fresh (TAVar x₂) BUVar UBVar apt | Inr x₁ = FVar x₁\n binders-fresh {y = y} (TALam {x = x} x₁ wt) bu2 ub apt with natEQ y x\n binders-fresh (TALam x₂ wt) bu2 (UBLam2 x₁ ub) apt | Inl refl = abort (x₁ refl)\n binders-fresh {Γ = Γ} (TALam {x = x} x₂ wt) (BULam bu2 x₃) (UBLam2 x₄ ub) apt | Inr x₁ = FLam x₁ (binders-fresh wt bu2 ub (apart-extend1 Γ x₄ apt))\n binders-fresh (TAAp wt wt₁) (BUAp bu2 bu3 x) (UBAp ub ub₁) apt = FAp (binders-fresh wt bu2 ub apt) (binders-fresh wt₁ bu3 ub₁ apt)\n binders-fresh (TAEHole x₁ x₂) (BUEHole x) (UBHole x₃) apt = FHole (binders-envfresh x₂ apt x₃ x )\n binders-fresh (TANEHole x₁ wt x₂) (BUNEHole bu2 x) (UBNEHole x₃ ub) apt = FNEHole (binders-envfresh x₂ apt x₃ x) (binders-fresh wt bu2 ub apt)\n binders-fresh (TACast wt x₁) (BUCast bu2) (UBCast ub) apt = FCast (binders-fresh wt bu2 ub apt)\n binders-fresh (TAFailedCast wt x x₁ x₂) (BUFailedCast bu2) (UBFailedCast ub) apt = FFailedCast (binders-fresh wt bu2 ub apt)\n binders-fresh (TAInl wt) (BUInl bu) (UBInl ub) apt = FInl (binders-fresh wt bu ub apt)\n binders-fresh (TAInr wt) (BUInr bu) (UBInr ub) apt = FInr (binders-fresh wt bu ub apt)\n binders-fresh {Γ = Γ} (TACase wt x wt₁ x₁ wt₂) (BUCase bu bu₁ bu₂ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ x₁₀) (UBCase ub x₁₁ ub₁ x₁₂ ub₂) apt = FCase (binders-fresh wt bu ub apt) x₁₁ (binders-fresh wt₁ bu₁ ub₁ (apart-extend1 Γ x₁₁ apt)) x₁₂ (binders-fresh wt₂ bu₂ ub₂ (apart-extend1 Γ x₁₂ apt))\n binders-fresh (TAPlus wt wt₁) (BUPlus bu bu₁ x) (UBPlus ub ub₁) apt = FPlus (binders-fresh wt bu ub apt) (binders-fresh wt₁ bu₁ ub₁ apt)\n binders-fresh (TAPair wt wt₁) (BUPair bu bu₁ x) (UBPair ub ub₁) apt = FPair (binders-fresh wt bu ub apt) (binders-fresh wt₁ bu₁ ub₁ apt)\n binders-fresh (TAFst wt) (BUFst bu) (UBFst ub) apt = FFst (binders-fresh wt bu ub apt)\n binders-fresh (TASnd wt) (BUSnd bu) (UBSnd ub) apt = FSnd (binders-fresh wt bu ub apt)\n \n -- the substition lemma for preservation\n lem-subst : ∀{Δ Γ x τ1 d1 τ d2} →\n x # Γ →\n binders-disjoint d1 d2 →\n binders-unique d2 →\n Δ , Γ ,, (x , τ1) ⊢ d1 :: τ →\n Δ , Γ ⊢ d2 :: τ1 →\n Δ , Γ ⊢ [ d2 / x ] d1 :: τ\n lem-subst apt bd bu2 (TANum) wt = TANum\n lem-subst apt (BDPlus bd bd₁) bu2 (TAPlus wt1 wt2) wt3 = TAPlus (lem-subst apt bd bu2 wt1 wt3) (lem-subst apt bd₁ bu2 wt2 wt3)\n lem-subst {x = x} apt bd bu2 (TAVar {x = x'} x₂) wt2 with natEQ x' x\n ... | Inl refl with natEQ x x\n ... | Inr x≠x = abort (x≠x refl)\n ... | Inl refl with someinj x₂\n ... | refl = wt2\n lem-subst {x = x} apt bd bu2 (TAVar {x = x'} x₂) wt2 | Inr x'≠x with natEQ x x'\n ... | Inl refl = abort (x'≠x refl)\n ... | Inr x'≠x = TAVar x₂\n lem-subst {Δ = Δ} {Γ = Γ} {x = x} {d2 = d2} x#Γ (BDLam bd bd') bu2 (TALam {x = y} {τ1 = τ1} {d = d} {τ2 = τ2} x₂ wt1) wt2 with natEQ y x\n ... | Inl refl with natEQ x x\n ... | Inr x≠x = abort (x≠x refl)\n ... | Inl refl = abort (somenotnone x₂)\n lem-subst {Δ = Δ} {Γ = Γ} {x = x} {d2 = d2} x#Γ (BDLam bd bd') bu2 (TALam {x = y} {τ1 = τ1} {d = d} {τ2 = τ2} x₂ wt1) wt2 | Inr y≠x with natEQ x y\n ... | Inl refl = abort (y≠x refl)\n ... | Inr x≠y = TALam x₂ (lem-subst (apart-extend1 Γ x≠y x#Γ) bd bu2 (exchange-ta-Γ x≠y wt1) (weaken-ta (binders-fresh wt2 bu2 bd' x₂) wt2))\n lem-subst apt (BDAp bd bd₁) bu3 (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt bd bu3 wt1 wt3) (lem-subst apt bd₁ bu3 wt2 wt3)\n lem-subst apt bd bu2 (TAEHole inΔ sub) wt2 = TAEHole inΔ (STASubst sub wt2)\n lem-subst apt (BDNEHole x₁ bd) bu2 (TANEHole x₃ wt1 x₄) wt2 = TANEHole x₃ (lem-subst apt bd bu2 wt1 wt2) (STASubst x₄ wt2)\n lem-subst apt (BDCast bd) bu2 (TACast wt1 x₁) wt2 = TACast (lem-subst apt bd bu2 wt1 wt2) x₁\n lem-subst apt (BDFailedCast bd) bu2 (TAFailedCast wt1 x₁ x₂ x₃) wt2 = TAFailedCast (lem-subst apt bd bu2 wt1 wt2) x₁ x₂ x₃\n lem-subst apt (BDInl bd) bu (TAInl wt1) wt2 = TAInl (lem-subst apt bd bu wt1 wt2)\n lem-subst apt (BDInr bd) bu (TAInr wt1) wt2 = TAInr (lem-subst apt bd bu wt1 wt2)\n lem-subst {Δ = Δ} {Γ = Γ} {x = x} {d1 = .(case _ _ _ _ _)} {d2 = d2} apt (BDCase bd x₁ bd₁ x₂ bd₂) bu (TACase {d = d} {τ1 = τ1} {τ2 = τ2} {x = y} {y = z} wt1 x₃ wt3 x₄ wt4) wt2\n with natEQ y x | natEQ z x\n ... | Inl refl | Inl refl with natEQ x x\n ... | Inr x≠x = abort (x≠x refl)\n ... | Inl refl = abort (somenotnone x₄)\n lem-subst {Δ = Δ} {Γ = Γ} {x = x} {d1 = .(case _ _ _ _ _)} {d2 = d2} apt (BDCase bd x₁ bd₁ x₂ bd₂) bu (TACase {d = d} {τ1 = τ1} {τ2 = τ2} {x = y} {y = z} wt1 x₃ wt3 x₄ wt4) wt2 | Inl refl | Inr z≠x with natEQ x x \n ... | Inr x≠x = abort (x≠x refl)\n ... | Inl refl = abort (somenotnone x₃)\n lem-subst {Δ = Δ} {Γ = Γ} {x = x} {d1 = .(case _ _ _ _ _)} {d2 = d2} apt (BDCase bd x₁ bd₁ x₂ bd₂) bu (TACase {d = d} {τ1 = τ1} {τ2 = τ2} {x = y} {y = z} wt1 x₃ wt3 x₄ wt4) wt2 | Inr y≠x | Inl refl with natEQ x x\n ... | Inr x≠x = abort (x≠x refl)\n ... | Inl refl = abort (somenotnone x₄)\n lem-subst {Δ = Δ} {Γ = Γ} {x = x} {d1 = .(case _ _ _ _ _)} {d2 = d2} apt (BDCase bd x₁ bd₁ x₂ bd₂) bu (TACase {d = d} {τ1 = τ1} {τ2 = τ2} {x = y} {y = z} wt1 x₃ wt3 x₄ wt4) wt2 | Inr y≠x | Inr z≠x with natEQ x z\n ... | Inl refl = abort (z≠x refl)\n ... | Inr x≠z with natEQ x y\n ... | Inl refl = abort (y≠x refl)\n ... | Inr x≠y = TACase (lem-subst apt bd bu wt1 wt2) x₃ (lem-subst (apart-extend1 Γ x≠y apt) bd₁ bu (exchange-ta-Γ x≠y wt3) (weaken-ta (binders-fresh wt2 bu x₁ x₃) wt2)) x₄ (lem-subst (apart-extend1 Γ x≠z apt) bd₂ bu (exchange-ta-Γ x≠z wt4) (weaken-ta (binders-fresh wt2 bu x₂ x₄) wt2))\n lem-subst apt (BDPair bd bd₁) bu (TAPair wt1 wt3) wt2 = TAPair (lem-subst apt bd bu wt1 wt2) (lem-subst apt bd₁ bu wt3 wt2)\n lem-subst apt (BDFst bd) bu (TAFst wt1) wt2 = TAFst (lem-subst apt bd bu wt1 wt2)\n lem-subst apt (BDSnd bd) bu (TASnd wt1) wt2 = TASnd (lem-subst apt bd bu wt1 wt2)\n\n lem-subst-cast-sta : ∀{Δ Γ x τ1 τ2 σ Γ'} →\n x # Γ →\n τ1 ~ τ2 →\n Δ , Γ ,, (x , τ2) ⊢ σ :s: Γ' →\n Δ , Γ ,, (x , τ1) ⊢ Subst (X x ⟨ τ1 ⇒ τ2 ⟩) x σ :s: Γ'\n lem-subst-cast-sta {Γ = Γ} {x = x} {τ1 = τ1} {τ2 = τ2} {Γ' = Γ'} x#Γ con (STAId sub) = STASubst (STAId Γ'⊆) (TACast (TAVar (ctx-top Γ x τ1 x#Γ)) con)\n where\n Γ'⊆ : (y : Nat) (τ : htyp) → (y , τ) ∈ Γ' → (y , τ) ∈ (Γ ,, (x , τ1) ,, (x , τ2))\n Γ'⊆ y τ y∈Γ' with lem-dom-union {Δ1 = ■(x , τ2)} {Δ2 = Γ} (sub y τ y∈Γ')\n ... | Inl y∈x with natEQ x y\n ... | Inr x≠y = abort (somenotnone (! y∈x)) \n ... | Inl refl = y∈x\n Γ'⊆ y τ y∈Γ' | Inr y∈Γ with natEQ x y\n ... | Inl refl = abort (somenotnone ((! y∈Γ) · x#Γ))\n ... | Inr x≠y with natEQ x y\n ... | Inl refl = abort (x≠y refl)\n ... | Inr x≠y' = y∈Γ\n lem-subst-cast-sta {Δ = Δ} {Γ = Γ} {x = x} {τ1 = τ1} {τ2 = τ2} {Γ' = Γ'} x#Γ con (STASubst {σ = σ} {y = y} {d = d} {τ = τ} wsta wt) = STASubst (STASubst (tr (λ c → Δ , c ,, (y , τ) ⊢ σ :s: Γ') (! (update Γ x τ1 τ2)) wsta) (tr (λ c → Δ , c ⊢ d :: τ) (! (update Γ x τ1 τ2)) wt)) (TACast (TAVar (ctx-top Γ x τ1 x#Γ)) con)\n \n lem-subst-cast-ta : ∀{Δ Γ d x τ1 τ2 τ} →\n x # Γ →\n binders-unique d →\n τ1 ~ τ2 →\n Δ , Γ ,, (x , τ2) ⊢ d :: τ →\n Δ , Γ ,, (x , τ1) ⊢ [ (X x ⟨ τ1 ⇒ τ2 ⟩) / x ] d :: τ\n lem-subst-cast-ta apt bu con TANum = TANum\n lem-subst-cast-ta apt (BUPlus bu bu₁ x) con (TAPlus wt wt₁) = TAPlus (lem-subst-cast-ta apt bu con wt) (lem-subst-cast-ta apt bu₁ con wt₁)\n lem-subst-cast-ta {Γ = Γ} {x = x} {τ1 = τ1} {τ2 = τ2} apt bu con (TAVar {x = x'} x₁)\n with natEQ x x'\n ... | Inl refl with natEQ x' x\n ... | Inr x'≠x = abort (x'≠x refl)\n ... | Inl refl with Γ x\n ... | None with natEQ x x\n ... | Inr x≠x = abort (x≠x refl)\n ... | Inl refl with someinj x₁\n ... | refl = TACast (TAVar (x∈∪l (■ (x , τ1)) Γ x τ1 (x∈■ x τ1))) con\n lem-subst-cast-ta {Γ = Γ} {x = x} {τ1 = τ1} {τ2 = τ2} apt bu con (TAVar {x = x} x₁) | Inl refl | Inl refl | Some t with someinj x₁\n ... | refl = abort (somenotnone apt)\n lem-subst-cast-ta {Γ = Γ} {x = x} {τ1 = τ1} {τ2 = τ2} {τ = τ} apt bu con (TAVar {x = x'} x₁) | Inr x≠x' with natEQ x' x\n ... | Inl refl = abort (x≠x' refl)\n ... | Inr x'≠x = TAVar (x∈∪r (■(x , τ1)) Γ x' τ x₁ (apart-singleton (λ refl → abort (x'≠x refl))))\n lem-subst-cast-ta {Δ = Δ} {Γ = Γ} {x = x} {τ = τ1 ==> τ3} apt (BULam bu x₁) con (TALam {x = y} x₂ wt)\n with natEQ x y\n ... | Inl refl = abort (somenotnone x₂)\n ... | Inr x≠y with natEQ y x\n ... | Inl refl = abort (x≠y refl)\n ... | Inr y≠x = TALam (apart-extend1 Γ y≠x x₂) (exchange-ta-Γ y≠x (lem-subst-cast-ta (apart-extend1 Γ x≠y apt) bu con (exchange-ta-Γ x≠y wt)))\n lem-subst-cast-ta {Γ = Γ} apt (BUAp bu bu₁ x) con (TAAp wt wt₁) = TAAp (lem-subst-cast-ta {Γ = Γ} apt bu con wt) (lem-subst-cast-ta {Γ = Γ} apt bu₁ con wt₁)\n lem-subst-cast-ta {Γ = Γ} apt (BUInl bu) con (TAInl wt) = TAInl (lem-subst-cast-ta {Γ = Γ} apt bu con wt)\n lem-subst-cast-ta {Γ = Γ} apt (BUInr bu) con (TAInr wt) = TAInr (lem-subst-cast-ta {Γ = Γ} apt bu con wt)\n lem-subst-cast-ta {x = x} apt (BUCase bu bu₁ bu₂ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ x₁₀) con (TACase {x = y} {y = z} wt apty wt₁ aptz wt₂) with natEQ y x | natEQ z x\n ... | Inl refl | Inl refl with natEQ x x\n ... | Inr x≠x = abort (x≠x refl)\n ... | Inl refl = abort (somenotnone aptz)\n lem-subst-cast-ta {x = x} apt (BUCase bu bu₁ bu₂ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ x₁₀) con (TACase {x = y} {y = z} wt apty wt₁ aptz wt₂) | Inl refl | Inr z≠x with natEQ x x\n ... | Inr x≠x = abort (x≠x refl)\n ... | Inl refl = abort (somenotnone apty)\n lem-subst-cast-ta {x = x} apt (BUCase bu bu₁ bu₂ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ x₁₀) con (TACase {x = y} {y = z} wt apty wt₁ aptz wt₂) | Inr y≠x | Inl refl with natEQ x x\n ... | Inr x≠x = abort (x≠x refl)\n ... | Inl refl = abort (somenotnone aptz)\n lem-subst-cast-ta {Γ = Γ} {x = x} apt (BUCase bu bu₁ bu₂ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ x₁₀) con (TACase {x = y} {y = z} wt apty wt₁ aptz wt₂) | Inr y≠x | Inr z≠x with natEQ x z\n ... | Inl refl = abort (z≠x refl)\n ... | Inr x≠z with natEQ x y\n ... | Inl refl = abort (y≠x refl)\n ... | Inr x≠y = TACase (lem-subst-cast-ta apt bu con wt) (apart-extend1 Γ y≠x apty) (exchange-ta-Γ y≠x (lem-subst-cast-ta (apart-extend1 Γ x≠y apt) bu₁ con (exchange-ta-Γ x≠y wt₁))) (apart-extend1 Γ z≠x aptz) (exchange-ta-Γ z≠x (lem-subst-cast-ta (apart-extend1 Γ x≠z apt) bu₂ con (exchange-ta-Γ x≠z wt₂)))\n lem-subst-cast-ta apt (BUEHole x₂) con (TAEHole x x₁) = TAEHole x (lem-subst-cast-sta apt con x₁)\n lem-subst-cast-ta apt (BUNEHole bu x₂) con (TANEHole x wt x₁) = TANEHole x (lem-subst-cast-ta apt bu con wt) (lem-subst-cast-sta apt con x₁)\n lem-subst-cast-ta apt (BUCast bu) con (TACast wt x) = TACast (lem-subst-cast-ta apt bu con wt) x\n lem-subst-cast-ta apt (BUFailedCast bu) con (TAFailedCast wt x x₁ x₂) = TAFailedCast (lem-subst-cast-ta apt bu con wt) x x₁ x₂\n lem-subst-cast-ta apt (BUPair bu bu₁ x) con (TAPair wt wt₁) = TAPair (lem-subst-cast-ta apt bu con wt) (lem-subst-cast-ta apt bu₁ con wt₁)\n lem-subst-cast-ta apt (BUFst bu) con (TAFst wt) = TAFst (lem-subst-cast-ta apt bu con wt)\n lem-subst-cast-ta apt (BUSnd bu) con (TASnd wt) = TASnd (lem-subst-cast-ta apt bu con wt)\n", "meta": {"hexsha": "6af44759371c2b1ba889cf209fb97157d29fb092", "size": 13072, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "lemmas-subst-ta.agda", "max_stars_repo_name": "hazelgrove/hazelnut-agda", "max_stars_repo_head_hexsha": "a3640d7b0f76cdac193afd382694197729ed6d57", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lemmas-subst-ta.agda", "max_issues_repo_name": "hazelgrove/hazelnut-agda", "max_issues_repo_head_hexsha": "a3640d7b0f76cdac193afd382694197729ed6d57", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lemmas-subst-ta.agda", "max_forks_repo_name": "hazelgrove/hazelnut-agda", "max_forks_repo_head_hexsha": "a3640d7b0f76cdac193afd382694197729ed6d57", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 73.0279329609, "max_line_length": 320, "alphanum_fraction": 0.5657129743, "num_tokens": 5793, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3492843138337172}} {"text": "\nmodule _ where\n\nopen import Agda.Builtin.Nat\nopen import Agda.Builtin.List\nopen import Agda.Builtin.Reflection renaming (bindTC to _>>=_)\nopen import Agda.Builtin.Unit\nopen import Agda.Builtin.Equality\n\nvariable\n A B : Set\n x y : A\n xs : List A\n\ninfix 3 _∈_\ndata _∈_ {A : Set} (x : A) : List A → Set where\n zero : x ∈ x ∷ xs\n suc : x ∈ xs → x ∈ y ∷ xs\n\npattern vArg x = arg (arg-info visible relevant) x\n\nsearch : Nat → Term → Term → TC ⊤\nsearch zero i hole = typeError (strErr \"Not found\" ∷ [])\nsearch (suc n) i hole = do\n catchTC (noConstraints (unify hole i))\n (search n (con (quote _∈_.suc) (vArg i ∷ [])) hole)\n\nfindElem : Nat → Term → TC ⊤\nfindElem depth hole = search depth (con (quote _∈_.zero) []) hole\n\nindex : (x : A) (xs : List A) {@(tactic findElem 10) i : x ∈ xs} → Nat\nindex x xs {zero} = zero\nindex x xs {suc i} = suc (index x _ {i})\n\ntest₁ : index 3 (1 ∷ 2 ∷ 3 ∷ []) ≡ 2\ntest₁ = refl\n\ntest₂ : index x (y ∷ x ∷ x ∷ []) ≡ 1\ntest₂ = refl\n", "meta": {"hexsha": "be1d161f3695beafbf274eb9d4164a7c4a0d9405", "size": 971, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/Succeed/ListElemTactic.agda", "max_stars_repo_name": "shlevy/agda", "max_stars_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2019-10-29T09:40:30.000Z", "max_stars_repo_stars_event_max_datetime": "2020-09-20T00:28:57.000Z", "max_issues_repo_path": "test/Succeed/ListElemTactic.agda", "max_issues_repo_name": "shlevy/agda", "max_issues_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2018-11-14T15:31:44.000Z", "max_issues_repo_issues_event_max_datetime": "2019-04-01T19:39:26.000Z", "max_forks_repo_path": "test/Succeed/ListElemTactic.agda", "max_forks_repo_name": "Agda-zh/agda", "max_forks_repo_head_hexsha": "231d6ad8e77b67ff8c4b1cb35a6c31ccd988c3e9", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2015-09-15T14:36:15.000Z", "max_forks_repo_forks_event_max_datetime": "2015-09-15T14:36:15.000Z", "avg_line_length": 24.275, "max_line_length": 70, "alphanum_fraction": 0.613800206, "num_tokens": 366, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3491914116821351}} {"text": "--\n-- Created by Dependently-Typed Lambda Calculus on 2019-05-15\n-- imports\n-- Author: ice1000\n--\n\n{-# OPTIONS --without-K --safe #-}\n\nimport Relation.Binary.PropositionalEquality\nopen import Relation.Binary.PropositionalEquality\nimport Relation.Binary.PropositionalEquality using ()\nimport Relation.Binary.PropositionalEquality using (sym) hiding (cong)\nimport Relation.Binary.PropositionalEquality renaming (sym to symBla)\n\n", "meta": {"hexsha": "5aa3e87d8f4fd8965c231074701522fd7857f14e", "size": 426, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "testData/parse/agda/imports.agda", "max_stars_repo_name": "dubinsky/intellij-dtlc", "max_stars_repo_head_hexsha": "ee25a3a81dacebfe4449de7a9aaff029171456be", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 30, "max_stars_repo_stars_event_min_datetime": "2019-05-11T16:26:38.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-29T13:18:34.000Z", "max_issues_repo_path": "testData/parse/agda/imports.agda", "max_issues_repo_name": "dubinsky/intellij-dtlc", "max_issues_repo_head_hexsha": "ee25a3a81dacebfe4449de7a9aaff029171456be", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 16, "max_issues_repo_issues_event_min_datetime": "2019-03-30T04:29:32.000Z", "max_issues_repo_issues_event_max_datetime": "2021-03-15T17:04:36.000Z", "max_forks_repo_path": "testData/parse/agda/imports.agda", "max_forks_repo_name": "dubinsky/intellij-dtlc", "max_forks_repo_head_hexsha": "ee25a3a81dacebfe4449de7a9aaff029171456be", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-10-07T01:38:12.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-12T21:33:35.000Z", "avg_line_length": 28.4, "max_line_length": 70, "alphanum_fraction": 0.7957746479, "num_tokens": 96, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6477982315512488, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.34915237719727826}} {"text": "module Implicits.Resolution.Embedding where\n\nopen import Prelude\n\nopen import Data.Vec\nopen import Data.List as List hiding (map)\n\nopen import Implicits.Syntax\nopen import SystemF.Everything as F using ()\n\n⟦_⟧tp← : ∀ {ν} → F.Type ν → Type ν\n⟦ F.tc x ⟧tp← = simpl (tc x)\n⟦ F.tvar n ⟧tp← = simpl (tvar n)\n⟦ a F.→' b ⟧tp← = (⟦ a ⟧tp← ⇒ ⟦ b ⟧tp←)\n⟦ a F.⟶ b ⟧tp← = simpl (⟦ a ⟧tp← →' ⟦ b ⟧tp←)\n⟦ F.∀' x ⟧tp← = ∀' ⟦ x ⟧tp←\n\n⟦_⟧tps← : ∀ {ν n} → Vec (F.Type ν) n → Vec (Type ν) n\n⟦ v ⟧tps← = map (⟦_⟧tp←) v\n\n⟦_⟧ctx← : ∀ {ν n} → Vec (F.Type ν) n → List (Type ν)\n⟦ v ⟧ctx← = toList $ map ⟦_⟧tp← v\n\n⟦_⟧tp→ : ∀ {ν} → Type ν → F.Type ν\n⟦ simpl (tc x) ⟧tp→ = F.tc x\n⟦ simpl (tvar n) ⟧tp→ = F.tvar n\n⟦ simpl (a →' b) ⟧tp→ = ⟦ a ⟧tp→ F.⟶ ⟦ b ⟧tp→\n⟦ a ⇒ b ⟧tp→ = ⟦ a ⟧tp→ F.→' ⟦ b ⟧tp→\n⟦ ∀' x ⟧tp→ = F.∀' ⟦ x ⟧tp→\n\n⟦_⟧tps→ : ∀ {ν n} → Vec (Type ν) n → Vec (F.Type ν) n\n⟦ v ⟧tps→ = map (⟦_⟧tp→) v\n\n⟦_⟧ctx→ : ∀ {ν} → (Δ : ICtx ν) → Vec (F.Type ν) (List.length (List.map ⟦_⟧tp→ Δ))\n⟦ Δ ⟧ctx→ = fromList (List.map ⟦_⟧tp→ Δ)\n", "meta": {"hexsha": "ce2e639325c91892cd42e13e4065f2c2ba9d551e", "size": 1003, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Implicits/Resolution/Embedding.agda", "max_stars_repo_name": "metaborg/ts.agda", "max_stars_repo_head_hexsha": "7fe638b87de26df47b6437f5ab0a8b955384958d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2019-04-05T17:57:11.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-07T04:08:41.000Z", "max_issues_repo_path": "src/Implicits/Resolution/Embedding.agda", "max_issues_repo_name": "metaborg/ts.agda", "max_issues_repo_head_hexsha": "7fe638b87de26df47b6437f5ab0a8b955384958d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Implicits/Resolution/Embedding.agda", "max_forks_repo_name": "metaborg/ts.agda", "max_forks_repo_head_hexsha": "7fe638b87de26df47b6437f5ab0a8b955384958d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.8611111111, "max_line_length": 81, "alphanum_fraction": 0.5044865404, "num_tokens": 535, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5851011686727231, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.34897373091458156}} {"text": "module _ where\n\n-- open import Common.Prelude\n\nmodule Id (A : Set) where\n\n id : A → A\n id x = x\n\nmodule _ (A : Set) where\n open Id A\n\n id2 = id\n", "meta": {"hexsha": "0b486031f6a63887363c8e9ac7ba4a14dbcf80fb", "size": 148, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/Succeed/AnonymousModuleWithParameter.agda", "max_stars_repo_name": "cruhland/agda", "max_stars_repo_head_hexsha": "7f58030124fa99dfbf8db376659416f3ad8384de", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1989, "max_stars_repo_stars_event_min_datetime": "2015-01-09T23:51:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T18:20:48.000Z", "max_issues_repo_path": "test/Succeed/AnonymousModuleWithParameter.agda", "max_issues_repo_name": "cruhland/agda", "max_issues_repo_head_hexsha": "7f58030124fa99dfbf8db376659416f3ad8384de", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4066, "max_issues_repo_issues_event_min_datetime": "2015-01-10T11:24:51.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T21:14:49.000Z", "max_forks_repo_path": "test/Succeed/AnonymousModuleWithParameter.agda", "max_forks_repo_name": "cruhland/agda", "max_forks_repo_head_hexsha": "7f58030124fa99dfbf8db376659416f3ad8384de", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 371, "max_forks_repo_forks_event_min_datetime": "2015-01-03T14:04:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-30T19:00:30.000Z", "avg_line_length": 10.5714285714, "max_line_length": 29, "alphanum_fraction": 0.5945945946, "num_tokens": 53, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.34897372228449697}} {"text": "{-# OPTIONS --rewriting --without-K #-}\n\n--\n-- Prelude.agda - Some base definitions\n--\n\nmodule Prelude where\n\n open import Agda.Primitive public\n\n record ⊤ : Set where\n constructor tt\n\n {-# BUILTIN UNIT ⊤ #-}\n\n record Σ {i j} (A : Set i) (B : A → Set j) : Set (i ⊔ j) where\n constructor _,_\n field\n fst : A\n snd : B fst\n\n open Σ public\n\n _×_ : ∀ {i j} (A : Set i) (B : Set j) → Set (i ⊔ j)\n A × B = Σ A (λ a → B)\n\n\n id : ∀ {i} {A : Set i} → A → A\n id x = x\n\n uncurry : ∀ {i j k} {A : Set i} {B : A → Set j} {C : Set k} →\n (φ : (a : A) → (b : B a) → C) →\n Σ A B → C\n uncurry φ (a , b) = φ a b\n\n curry : ∀ {i j k} {A : Set i} {B : A → Set j} {C : Set k} →\n (ψ : Σ A B → C) →\n (a : A) → (b : B a) → C\n curry ψ a b = ψ (a , b)\n\n {- Equality -}\n\n infix 30 _==_\n\n data _==_ {i} {A : Set i} (a : A) : A → Set i where\n idp : a == a\n {-# BUILTIN EQUALITY _==_ #-}\n {-# BUILTIN REWRITE _==_ #-}\n\n infixl 20 _>>_\n\n _>>_ : ∀ {i} {A : Set i} {a b c : A} → a == b → b == c → a == c\n idp >> idp = idp\n\n _^ : ∀ {i} {A : Set i} {a b : A} → a == b → b == a\n _^ idp = idp\n\n coe : ∀ {a} {A B : Set a} → A == B → A → B\n coe idp x = x\n\n coe^ : ∀ {a} {A B : Set a} (p : A == B) {a : A} {b : B} → a == (coe (p ^) b) → (coe p a) == b\n coe^ idp q = q\n\n coe= : ∀ {a} {A B : Set a} (p : A == B) {a b : A} → a == b → coe p a == coe p b\n coe= p idp = idp\n\n ap : ∀ {i j} {A : Set i} {C : Set j} {M N : A} (f : A → C) → M == N → (f M) == (f N)\n ap f idp = idp\n\n ap² : ∀ {i j k} {A : Set i} {B : Set k} {C : Set j} {a a' : A} {b b' : B} (f : A → B → C) → a == a' → b == b' → (f a b) == (f a' b')\n ap² f idp idp = idp\n\n ap³ : ∀ {i j k l} {A : Set i} {B : Set k} {C : Set j} {D : Set l} {a a' : A} {b b' : B} {c c' : C} (f : A → B → C → D) → a == a' → b == b' → c == c' → (f a b c) == (f a' b' c')\n ap³ f idp idp idp = idp\n\n ap⁴ : ∀ {i j k l m} {A : Set i} {B : Set k} {C : Set j} {D : Set l} {E : Set m} {a a' : A} {b b' : B} {c c' : C} {d d' : D} (f : A → B → C → D → E) → a == a' → b == b' → c == c' → d == d' → (f a b c d) == (f a' b' c' d')\n ap⁴ f idp idp idp idp = idp\n\n ap⁵ : ∀ {i j k l m n} {A : Set i} {B : Set k} {C : Set j} {D : Set l} {E : Set m} {F : Set n} {a a' : A} {b b' : B} {c c' : C} {d d' : D} {e e' : E} (f : A → B → C → D → E → F) → a == a' → b == b' → c == c' → d == d' → e == e' → (f a b c d e) == (f a' b' c' d' e')\n ap⁵ f idp idp idp idp idp = idp\n\n ap⁶ : ∀ {i j k l m n o} {A : Set i} {B : Set k} {C : Set j} {D : Set l} {E : Set m} {F : Set n} {G : Set o} {a a' : A} {b b' : B} {c c' : C} {d d' : D} {e e' : E} {f f' : F} (α : A → B → C → D → E → F → G) → a == a' → b == b' → c == c' → d == d' → e == e' → f == f' → (α a b c d e f) == (α a' b' c' d' e' f')\n ap⁶ f idp idp idp idp idp idp = idp\n\n ap⁷ : ∀ {i j k l m n o p} {A : Set i} {B : Set k} {C : Set j} {D : Set l} {E : Set m} {F : Set n} {G : Set o} {H : Set p} {a a' : A} {b b' : B} {c c' : C} {d d' : D} {e e' : E} {f f' : F} {g g' : G} (α : A → B → C → D → E → F → G → H) → a == a' → b == b' → c == c' → d == d' → e == e' → f == f' → g == g' → (α a b c d e f g) == (α a' b' c' d' e' f' g')\n ap⁷ f idp idp idp idp idp idp idp = idp\n\n transport : ∀ {i j} {A : Set i} {B : A → Set j} {a a' : A} (pₐ : a == a') → B a → B a'\n transport pₐ b = coe (ap _ pₐ) b\n\n transport₂ : ∀ {i j k} {A : Set i} {B : Set j} {C : A → B → Set k} {a a' : A} {b b' : B} (pₐ : a == a') (q : b == b') → C a b → C a' b'\n transport₂ pₐ q c = coe (ap² _ pₐ q) c\n\n hfiber : ∀ {i} {A B : Set i} (f : A → B) (b : B) → Set i\n hfiber {A = A} f b = Σ A (λ a → f a == b)\n\n PathOver : ∀ {i j} {A : Set i} (B : A → Set j) {a₀ a₁ : A} (p : a₀ == a₁) (b₉ : B a₀) (b₁ : B a₁) → Set j\n PathOver B idp b₀ b₁ = b₀ == b₁\n\n infix 30 PathOver\n syntax PathOver B p u v =\n u == v [ B ↓ p ]\n\n ×= : ∀{i j} {A : Set i} {B : Set j} {a a' : A} {b b' : B} → a == a' → b == b' → (a , b) == (a' , b')\n ×= idp idp = idp\n\n Σ= : ∀ {i j} {A : Set i} {B : A → Set j} {a a' : A} {b : B a} {b' : B a'} → (pₐ : a == a') → transport pₐ b == b' → (a , b) == (a' , b')\n Σ= idp idp = idp\n\n ,= : ∀ {i j} {A : Set i} {B : Set j} {a a' : A} {b b' : B} → a == a' → b == b' → (a , b) == (a' , b')\n ,= idp idp = idp\n\n Σ-r : ∀ {i j k} {A : Set i} {B : A → Set j} (C : Σ A B → Set k) → A → Set (j ⊔ k)\n Σ-r {A = A} {B = B} C a = Σ (B a) (λ b → C (a , b))\n\n Σ-in : ∀ {i j k} {A : Set i} {B : A → Set j} (C : (a : A) → B a → Set k) → A → Set (j ⊔ k)\n Σ-in {A = A} {B = B} C a = Σ (B a) (λ b → C a b)\n\n =, : ∀ {i j} {A : Set i} {B : Set j} {a a' : A} {b b' : B} → (a , b) == (a' , b') → (a == a') × (b == b')\n =, idp = idp , idp\n\n fst-is-inj : ∀ {i j} {A : Set i} {B : A → Set j} {x y : Σ A B} → x == y → fst x == fst y\n fst-is-inj idp = idp\n\n {- False and negation -}\n\n data ⊥ {i} : Set i where\n\n ⊥-elim : ∀ {i j} {A : Set i} → ⊥ {j} → A\n ⊥-elim ()\n\n ¬_ : ∀ {i} → Set i → Set i\n ¬ A = A → ⊥ {lzero}\n\n _≠_ : ∀{i} {A : Set i} (a b : A) → Set i\n a ≠ b = ¬ (a == b)\n\n data _+_ {i j} (A : Set i) (B : Set j) : Set (i ⊔ j) where\n inl : A → A + B\n inr : B → A + B\n\n inl= : ∀ {i j} {A : Set i} {B : Set j} {a b : A} → a == b → _==_ {i ⊔ j} {A + B} (inl a) (inl b)\n inl= idp = idp\n\n inr= : ∀ {i j} {A : Set i} {B : Set j} {a b : B} → a == b → _==_ {i ⊔ j} {A + B} (inr a) (inr b)\n inr= idp = idp\n\n {- Booleans -}\n data Bool : Set where\n true : Bool\n false : Bool\n\n {- Decidability -}\n dec : ∀ {i} → Set i → Set i\n dec A = A + (¬ A)\n\n eqdec : ∀ {i} → Set i → Set i\n eqdec A = ∀ (a b : A) → dec (a == b)\n\n\n {- Natural numbers -}\n data ℕ : Set where\n O : ℕ\n S : ℕ → ℕ\n\n {-# BUILTIN NATURAL ℕ #-}\n\n S= : ∀{n m} → n == m → S n == S m\n S= idp = idp\n\n pred : ℕ → ℕ\n pred O = O\n pred (S n) = n\n\n S-is-inj : ∀ n m → (S n == S m) → n == m\n S-is-inj n m p = ap pred p\n\n S-≠ : ∀ {n m : ℕ} (p : n ≠ m) → S n ≠ S m\n S-≠ {n} {m} n≠m p = n≠m (S-is-inj n m p)\n\n private\n S≠O-type : ℕ → Set\n S≠O-type O = ⊥\n S≠O-type (S n) = ⊤\n\n S≠O : (n : ℕ) → S n ≠ O\n S≠O n p = coe (ap S≠O-type p) tt\n\n O≠S : (n : ℕ) → (O ≠ S n)\n O≠S n p = S≠O n (p ^)\n\n n≠Sn : (n : ℕ) → (n ≠ S n)\n n≠Sn O ()\n n≠Sn (S n) n=Sn = n≠Sn n (S-is-inj _ _ n=Sn)\n\n Sn≠n : (n : ℕ) → (S n ≠ n)\n Sn≠n O ()\n Sn≠n (S n) Sn=n = Sn≠n n (S-is-inj _ _ Sn=n)\n\n eqdecℕ : eqdec ℕ\n eqdecℕ O O = inl idp\n eqdecℕ O (S b) = inr (O≠S b)\n eqdecℕ (S a) O = inr (S≠O a)\n eqdecℕ (S a) (S b) with (eqdecℕ a b)\n ... | inl idp = inl idp\n ... | inr a≠b = inr (S-≠ a≠b)\n\n {- Order on ℕ -}\n data _≤_ : ℕ → ℕ → Set where\n 0≤ : ∀ n → O ≤ n\n S≤ : ∀ {n m} → n ≤ m → S n ≤ S m\n\n n≤n : ∀ (n : ℕ) → n ≤ n\n n≤n O = 0≤ O\n n≤n (S n) = S≤ (n≤n n)\n\n ≤-antisymetry : ∀ {n m} → n ≤ m → m ≤ n → n == m\n ≤-antisymetry (0≤ _) (0≤ _) = idp\n ≤-antisymetry (S≤ n≤m) (S≤ m≤n) = ap S (≤-antisymetry n≤m m≤n)\n\n n≤Sn : ∀ (n : ℕ) → n ≤ S n\n n≤Sn O = 0≤ _\n n≤Sn (S n) = S≤ (n≤Sn _)\n\n S≤S : ∀ {n m} → S n ≤ S m → n ≤ m\n S≤S (S≤ n≤m) = n≤m\n\n Sn≰n : ∀ (n : ℕ) → ¬ (S n ≤ n)\n Sn≰n .(S _) (S≤ Sn≤n) = Sn≰n _ Sn≤n\n\n Sn≰n-t : ∀ {n m} → n == m → ¬ (S n ≤ m)\n Sn≰n-t idp Sn≤n = Sn≰n _ Sn≤n\n\n Sn≰0 : ∀ (n : ℕ) → ¬ (S n ≤ O)\n Sn≰0 n ()\n\n n≤m→n≤Sm : ∀ {n m : ℕ} → n ≤ m → n ≤ S m\n n≤m→n≤Sm (0≤ n) = 0≤ (S n)\n n≤m→n≤Sm (S≤ n≤m) = S≤ (n≤m→n≤Sm n≤m)\n\n Sn≤m→n≤m : ∀ {n m : ℕ} → S n ≤ m → n ≤ m\n Sn≤m→n≤m (S≤ n≤m) = n≤m→n≤Sm n≤m\n\n dec-≤ : ∀ n m → dec (n ≤ m)\n dec-≤ O m = inl (0≤ m)\n dec-≤ (S n) O = inr λ ()\n dec-≤ (S n) (S m) with (dec-≤ n m)\n ... | inl n≤m = inl (S≤ n≤m)\n ... | inr n≰m = inr λ {(S≤ n≤m) → n≰m n≤m}\n\n\n ≤S : ∀ (n m : ℕ) → n ≤ S m → (n ≤ m) + (n == S m)\n ≤S .0 m (0≤ .(S m)) = inl (0≤ _)\n ≤S .1 O (S≤ (0≤ .0)) = inr idp\n ≤S .(S _) (S m) (S≤ n≤Sm) with ≤S _ _ n≤Sm\n ... | inl n≤m = inl (S≤ n≤m)\n ... | inr n=Sm = inr (ap S n=Sm)\n\n ≤-= : ∀ {n m k} → n ≤ m → m == k → n ≤ k\n ≤-= n≤m idp = n≤m\n\n =-≤ : ∀ {n m k} → n == m → m ≤ k → n ≤ k\n =-≤ idp m≤k = m≤k\n\n =-≤-= : ∀ {n m k l} → n == m → m ≤ k → k == l → n ≤ l\n =-≤-= idp m≤k idp = m≤k\n\n ≤T : ∀ {n m k} → n ≤ m → m ≤ k → n ≤ k\n ≤T (0≤ _) _ = 0≤ _\n ≤T (S≤ n≤m) (S≤ m≤k) = S≤ (≤T n≤m m≤k)\n\n {- Strict order on ℕ -}\n _<_ : ℕ → ℕ → Set\n n < m = S n ≤ m\n\n ≤×≠→< : ∀ {n m} → n ≤ m → n ≠ m → n < m\n ≤×≠→< {.0} {.0} (0≤ O) n≠m = ⊥-elim (n≠m idp)\n ≤×≠→< {.0} {.(S m)} (0≤ (S m)) n≠m = S≤ (0≤ _)\n ≤×≠→< (S≤ n≤m) Sn≠Sm = S≤ (≤×≠→< n≤m λ n=m → Sn≠Sm (ap S n=m))\n\n ≰ : ∀ {n m } → ¬ (n ≤ m) → m < n\n ≰ {O} {m} n≰m = ⊥-elim (n≰m (0≤ _))\n ≰ {S n} {O} n≰m = S≤ (0≤ _)\n ≰ {S n} {S m} n≰m = S≤ (≰ λ n≤m → n≰m (S≤ n≤m))\n\n ℕ-trichotomy : ∀ n m → ((n < m) + (m < n)) + (n == m)\n ℕ-trichotomy n m with dec-≤ n m\n ... | inr n≰m = inl (inr (≰ n≰m))\n ... | inl n≤m with eqdecℕ n m\n ... | inl n=m = inr n=m\n ... | inr n≠m = inl (inl (≤×≠→< n≤m n≠m))\n\n {- Minimum and maximum -}\n max : ℕ → ℕ → ℕ\n max n m with dec-≤ n m\n ... | inl _ = m\n ... | inr _ = n\n\n n≤max : ∀ n m → n ≤ max n m\n n≤max n m with dec-≤ n m\n ... | inl n≤m = n≤m\n ... | inr m≤n = n≤n _\n\n m≤max : ∀ n m → m ≤ max n m\n m≤max n m with dec-≤ n m\n ... | inl n≤m = n≤n _\n ... | inr n≰m = Sn≤m→n≤m (≰ n≰m)\n\n up-max : ∀ {n m k} → n ≤ k → m ≤ k → max n m ≤ k\n up-max {n} {m} {k} n≤k m≤k with dec-≤ n m\n ... | inl n≤m = m≤k\n ... | inr n≰m = n≤k\n\n up-maxS : ∀ {n m k} → S n ≤ k → S m ≤ k → S (max n m) ≤ k\n up-maxS {n} {m} {k} n≤k m≤k with dec-≤ n m\n ... | inl n≤m = m≤k\n ... | inr n≰m = n≤k\n\n simplify-max-l : ∀ {n m} → m ≤ n → max n m == n\n simplify-max-l {n} {m} m≤n with dec-≤ n m\n ... | inl n≤m = ≤-antisymetry m≤n n≤m\n ... | inr _ = idp\n\n simplify-max-r : ∀ {n m} → n ≤ m → max n m == m\n simplify-max-r {n} {m} n≤m with dec-≤ n m\n ... | inl _ = idp\n ... | inr n≰m = ⊥-elim (n≰m n≤m)\n\n min : ℕ → ℕ → ℕ\n min n m with dec-≤ n m\n ... | inl _ = n\n ... | inr _ = m\n\n min≤m : ∀ n m → min n m ≤ m\n min≤m n m with dec-≤ n m\n ... | inl n≤m = n≤m\n ... | inr n≰m = n≤n _\n\n simplify-min-l : ∀ {n m} → n ≤ m → min n m == n\n simplify-min-l {n} {m} n≤m with dec-≤ n m\n ... | inl _ = idp\n ... | inr n≰m = ⊥-elim (n≰m n≤m)\n\n simplify-min-r : ∀ {n m} → m ≤ n → min n m == m\n simplify-min-r {n} {m} n≤m with dec-≤ n m\n ... | inl m≤n = ≤-antisymetry m≤n n≤m\n ... | inr _ = idp\n\n min_then_else_ : ∀ {i j} {A : Set i} (B : Set j) → (dec B) → A → A → A\n ifdec b > inl x then A else B = A\n ifdec b > inr x then A else B = B\n\n iftrue : ∀ {i j}{A : Set i} {B : Set j} → (H : dec B) → (a : A) {a' : A} → B → (ifdec B > H then a else a') == a\n iftrue (inl _) a b = idp\n iftrue (inr ¬B) a b = ⊥-elim (¬B b)\n\n iffalse : ∀ {i j}{A : Set i} {B : Set j} → (H : dec B) → {a : A} (a' : A) → ¬ B → (ifdec B > H then a else a') == a'\n iffalse (inl b) a' ¬B = ⊥-elim (¬B b)\n iffalse (inr ¬B) a' b = idp\n\n\n if_≡_then_else_ : ∀ {i} {A : Set i} → ℕ → ℕ → A → A → A\n if v ≡ w then A else B = ifdec (v == w) > (eqdecℕ v w) then A else B\n\n if= : ∀ {i} {A : Set i} {n m : ℕ} (p : n == m) (a : A) {a' : A} → (if n ≡ m then a else a') == a\n if= p a = iftrue (eqdecℕ _ _) a p\n\n if≠ : ∀ {i} {A : Set i} {n m : ℕ} (p : n ≠ m) {a : A} (a' : A) → (if n ≡ m then a else a') == a'\n if≠ p a' = iffalse (eqdecℕ _ _) a' p\n\n simplify-if : ∀ {i} {A : Set i} {n} {a b : A} → 0 < n → (if n ≡ 0 then a else b) == b\n simplify-if {n = n} {b = b} 0 strVar lo n x <*> strArgs lo n args\n strTerm lo n (con c args) = con c <$> strArgs lo n args\n strTerm lo n (def f args) = def f <$> strArgs lo n args\n strTerm lo n (meta x args) = meta x <$> strArgs lo n args\n strTerm lo n (lam v t) = lam v <$> strAbsTerm lo n t\n strTerm lo n (pi a b) = pi <$> strArg lo n a <*> strAbsType lo n b\n strTerm lo n (agda-sort s) = agda-sort <$> strSort lo n s\n strTerm lo n (lit l) = just (lit l)\n strTerm lo n (pat-lam _ _) = just unknown -- todo\n strTerm lo n unknown = just unknown\n\n strAbsTerm lo n (abs s t) = abs s <$> strTerm (suc lo) n t\n strAbsType lo n (abs s t) = abs s <$> strTerm (suc lo) n t\n\n strArgs lo n [] = just []\n strArgs lo n (x ∷ args) = _∷_ <$> strArg lo n x <*> strArgs lo n args\n strArg lo n (arg i v) = arg i <$> strTerm lo n v\n strSort lo n (set t) = set <$> strTerm lo n t\n strSort lo n (lit l) = just (lit l)\n strSort lo n unknown = just unknown\n\n strClauses lo k [] = just []\n strClauses lo k (c ∷ cs) = _∷_ <$> strClause lo k c <*> strClauses lo k cs\n\n strClause lo k (clause ps b) = clause ps <$> strTerm (lo + patternBindings ps) k b\n strClause lo k (absurd-clause ps) = just (absurd-clause ps)\n\nprivate\n Wk : Set → Set\n Wk A = Nat → Nat → A → A\n\n wkVar : Wk Nat\n wkVar lo k x = if x Term\nEmptyrec l lEmpty A e = gen (Emptyreckind l lEmpty) (⟦ 0 , A ⟧ ∷ ⟦ 0 , e ⟧ ∷ [])\n\n-- Identity type\nId : (A t u : Term) → Term\nId A t u = gen Idkind (⟦ 0 , A ⟧ ∷ ⟦ 0 , t ⟧ ∷ ⟦ 0 , u ⟧ ∷ [])\n\n-- witness of reflexivity of equality\nIdrefl : (A t : Term) → Term\nIdrefl A t = gen Idreflkind (⟦ 0 , A ⟧ ∷ ⟦ 0 , t ⟧ ∷ [])\n\n-- transport on propositions\ntransp : (A P t s u e : Term) → Term\ntransp A P t s u e = gen Transpkind (⟦ 0 , A ⟧ ∷ ⟦ 1 , P ⟧ ∷ ⟦ 0 , t ⟧ ∷ ⟦ 0 , s ⟧ ∷ ⟦ 0 , u ⟧ ∷ ⟦ 0 , e ⟧ ∷ [])\n\n-- cast between types, used to implement transport\ncast : Level → (A B e t : Term) → Term\ncast l A B e t = gen (Castkind l) (⟦ 0 , A ⟧ ∷ ⟦ 0 , B ⟧ ∷ ⟦ 0 , e ⟧ ∷ ⟦ 0 , t ⟧ ∷ [])\n\n-- propositional proof that casting with reflexivity is the identity\ncastrefl : (A t : Term) → Term\ncastrefl A t = gen Castreflkind (⟦ 0 , A ⟧ ∷ ⟦ 0 , t ⟧ ∷ [])\n\n-- Injectivity of term constructors w.r.t. propositional equality.\n\n-- If Π F G = Π H E then F = H and G = E.\n\nΠ-PE-injectivity : ∀ {F rF lF G lG lΠ H rH lH E lE lΠ'} → Π F ^ rF ° lF ▹ G ° lG ° lΠ PE.≡ Π H ^ rH ° lH ▹ E ° lE ° lΠ'\n → F PE.≡ H × rF PE.≡ rH × lF PE.≡ lH × G PE.≡ E × lG PE.≡ lE × lΠ PE.≡ lΠ'\nΠ-PE-injectivity PE.refl = PE.refl , PE.refl , PE.refl , PE.refl , PE.refl , PE.refl\n\n∃-PE-injectivity : ∀ {F G H E} → ∃ F ▹ G PE.≡ ∃ H ▹ E\n → F PE.≡ H × G PE.≡ E\n∃-PE-injectivity PE.refl = PE.refl , PE.refl\n\n-- If suc n = suc m then n = m.\n\nsuc-PE-injectivity : ∀ {n m} → suc n PE.≡ suc m → n PE.≡ m\nsuc-PE-injectivity PE.refl = PE.refl\n\nUniv-PE-injectivity : ∀ {r r' l l'} → Univ r l PE.≡ Univ r' l' → r PE.≡ r' × l PE.≡ l'\nUniv-PE-injectivity PE.refl = PE.refl , PE.refl\n\n-- Neutral terms.\n\n-- A term is neutral if\n-- either it has a variable in head position that blocks reduction.\n-- either it is of the form Emptyrec (or terms that should reduce to emptyrec, such as incompatible casts)\n\ndata Neutral : Term → Set where\n var : ∀ n → Neutral (var n)\n ∘ₙ : ∀ {k u l} → Neutral k → Neutral (k ∘ u ^ l)\n natrecₙ : ∀ {l C c g k} → Neutral k → Neutral (natrec l C c g k)\n Idₙ : ∀ {A t u} → Neutral A → Neutral (Id A t u)\n Idℕₙ : ∀ {t u} → Neutral t → Neutral (Id ℕ t u)\n Idℕ0ₙ : ∀ {u} → Neutral u → Neutral (Id ℕ zero u)\n IdℕSₙ : ∀ {t u} → Neutral u → Neutral (Id ℕ (suc t) u)\n IdUₙ : ∀ {t u l} → Neutral t → Neutral (Id (U l) t u)\n IdUℕₙ : ∀ {u l} → Neutral u → Neutral (Id (U l) ℕ u)\n IdUΠₙ : ∀ {A rA lA B lB l u} → Neutral u → Neutral (Id (U l) (Π A ^ rA ° lA ▹ B ° lB ° l) u)\n castₙ : ∀ {l A B e t} → Neutral A → Neutral (cast l A B e t)\n castℕₙ : ∀ {l B e t} → Neutral B → Neutral (cast l ℕ B e t)\n castΠₙ : ∀ {l A rA lA P lP B e t} → Neutral B → Neutral (cast l (Π A ^ rA ° lA ▹ P ° lP ° l) B e t)\n castℕℕₙ : ∀ {l e t} → Neutral t → Neutral (cast l ℕ ℕ e t)\n castℕΠₙ : ∀ {l A rA B e t} → Neutral (cast l ℕ (Π A ^ rA ° ⁰ ▹ B ° ⁰ ° l) e t)\n castΠℕₙ : ∀ {l A rA B e t} → Neutral (cast l (Π A ^ rA ° ⁰ ▹ B ° ⁰ ° l) ℕ e t)\n castΠΠ%!ₙ : ∀ {l A B A' B' e t} → Neutral (cast l (Π A ^ % ° ⁰ ▹ B ° ⁰ ° l) (Π A' ^ ! ° ⁰ ▹ B' ° ⁰ ° l) e t)\n castΠΠ!%ₙ : ∀ {l A B A' B' e t} → Neutral (cast l (Π A ^ ! ° ⁰ ▹ B ° ⁰ ° l) (Π A' ^ % ° ⁰ ▹ B' ° ⁰ ° l) e t)\n Emptyrecₙ : ∀ {l lEmpty A e} -> Neutral (Emptyrec l lEmpty A e)\n\n-- Weak head normal forms (whnfs).\n-- These are the (lazy) values of our language.\n\ndata Whnf : Term → Set where\n\n -- Type constructors are whnfs.\n Uₙ : ∀ {r l} → Whnf (Univ r l)\n Πₙ : ∀ {A r lA B lB l} → Whnf (Π A ^ r ° lA ▹ B ° lB ° l)\n ∃ₙ : ∀ {A B} → Whnf (∃ A ▹ B)\n ℕₙ : Whnf ℕ\n Emptyₙ : ∀ {l} → Whnf (Empty l)\n\n -- Introductions are whnfs.\n lamₙ : ∀ {A t l} → Whnf (lam A ▹ t ^ l)\n zeroₙ : Whnf zero\n sucₙ : ∀ {t} → Whnf (suc t)\n\n -- Neutrals are whnfs.\n ne : ∀ {n} → Neutral n → Whnf n\n\n\n-- Whnf inequalities.\n\n-- Different whnfs are trivially distinguished by propositional equality.\n-- (The following statements are sometimes called \"no-confusion theorems\".)\n\nU≢ℕ : ∀ {r l} → Univ r l PE.≢ ℕ\nU≢ℕ ()\n\nU≢Empty : ∀ {r l l'} → Univ r l PE.≢ Empty l'\nU≢Empty ()\n\nU≢Π : ∀ {r r' l F lF G lG l'} → Univ r l PE.≢ Π F ^ r' ° lF ▹ G ° lG ° l'\nU≢Π ()\n\nU≢∃ : ∀ {r l F G} → Univ r l PE.≢ ∃ F ▹ G\nU≢∃ ()\n\nU≢ne : ∀ {r l K} → Neutral K → Univ r l PE.≢ K\nU≢ne () PE.refl\n\nℕ≢Π : ∀ {F r lF G lG l} → ℕ PE.≢ Π F ^ r ° lF ▹ G ° lG ° l\nℕ≢Π ()\n\nℕ≢∃ : ∀ {F G} → ℕ PE.≢ ∃ F ▹ G\nℕ≢∃ ()\n\nℕ≢Empty : ∀ {l} → ℕ PE.≢ Empty l\nℕ≢Empty ()\n\nEmpty≢ℕ : ∀ {l} → Empty l PE.≢ ℕ\nEmpty≢ℕ ()\n\nℕ≢ne : ∀ {K} → Neutral K → ℕ PE.≢ K\nℕ≢ne () PE.refl\n\nEmpty≢ne : ∀ {l K} → Neutral K → Empty l PE.≢ K\nEmpty≢ne () PE.refl\n\nEmpty≢Π : ∀ {F r lF G lG l l'} → Empty l' PE.≢ Π F ^ r ° lF ▹ G ° lG ° l\nEmpty≢Π ()\n\nEmpty≢∃ : ∀ {l F G} → Empty l PE.≢ ∃ F ▹ G\nEmpty≢∃ ()\n\nΠ≢ne : ∀ {F r lF G lG K l} → Neutral K → Π F ^ r ° lF ▹ G ° lG ° l PE.≢ K\nΠ≢ne () PE.refl\n\nΠ≢∃ : ∀ {F r lF G lG F' G' l} → Π F ^ r ° lF ▹ G ° lG ° l PE.≢ ∃ F' ▹ G'\nΠ≢∃ ()\n\n∃≢ne : ∀ {F G K} → Neutral K → ∃ F ▹ G PE.≢ K\n∃≢ne () PE.refl\n\nzero≢suc : ∀ {n} → zero PE.≢ suc n\nzero≢suc ()\n\nzero≢ne : ∀ {k} → Neutral k → zero PE.≢ k\nzero≢ne () PE.refl\n\nsuc≢ne : ∀ {n k} → Neutral k → suc n PE.≢ k\nsuc≢ne () PE.refl\n\n\n-- Several views on whnfs (note: not recursive).\n\n-- A whnf of type ℕ is either zero, suc t, or neutral.\n\ndata Natural : Term → Set where\n zeroₙ : Natural zero\n sucₙ : ∀ {t} → Natural (suc t)\n ne : ∀ {n} → Neutral n → Natural n\n\n-- A type in whnf is either Π A B, ℕ, or neutral.\n-- Large types could also be U.\n\ndata Type : Term → Set where\n Πₙ : ∀ {A r lA B lB l} → Type (Π A ^ r ° lA ▹ B ° lB ° l)\n ℕₙ : Type ℕ\n Uₙ : ∀ {r l} → Type (Univ r l)\n Emptyₙ : ∀ {l} → Type (Empty l)\n ∃ₙ : ∀ {A B} → Type (∃ A ▹ B)\n ne : ∀{n} → Neutral n → Type n\n\n-- A whnf of type Π A B is either lam t or neutral.\n\ndata Function : Term → Set where\n lamₙ : ∀{A t l} → Function (lam A ▹ t ^ l)\n ne : ∀{n} → Neutral n → Function n\n\n-- These views classify only whnfs.\n-- Natural, Type, and Function are a subsets of Whnf.\n\nnaturalWhnf : ∀ {n} → Natural n → Whnf n\nnaturalWhnf sucₙ = sucₙ\nnaturalWhnf zeroₙ = zeroₙ\nnaturalWhnf (ne x) = ne x\n\ntypeWhnf : ∀ {A} → Type A → Whnf A\ntypeWhnf Πₙ = Πₙ\ntypeWhnf ℕₙ = ℕₙ\ntypeWhnf Uₙ = Uₙ\ntypeWhnf ∃ₙ = ∃ₙ\ntypeWhnf Emptyₙ = Emptyₙ\ntypeWhnf (ne x) = ne x\n\nfunctionWhnf : ∀ {f} → Function f → Whnf f\nfunctionWhnf lamₙ = lamₙ\nfunctionWhnf (ne x) = ne x\n\n------------------------------------------------------------------------\n-- Weakening\n\n-- In the following we define untyped weakenings η : Wk.\n-- The typed form could be written η : Γ ≤ Δ with the intention\n-- that η transport a term t living in context Δ to a context Γ\n-- that can bind additional variables (which cannot appear in t).\n-- Thus, if Δ ⊢ t : A and η : Γ ≤ Δ then Γ ⊢ wk η t : wk η A.\n--\n-- Even though Γ is \"larger\" than Δ we write Γ ≤ Δ to be conformant\n-- with subtyping A ≤ B. With subtyping, relation Γ ≤ Δ could be defined as\n-- ``for all x ∈ dom(Δ) have Γ(x) ≤ Δ(x)'' (in the sense of subtyping)\n-- and this would be the natural extension of weakenings.\n\ndata Wk : Set where\n id : Wk -- η : Γ ≤ Γ.\n step : Wk → Wk -- If η : Γ ≤ Δ then step η : Γ∙A ≤ Δ.\n lift : Wk → Wk -- If η : Γ ≤ Δ then lift η : Γ∙A ≤ Δ∙A.\n\n-- Composition of weakening.\n-- If η : Γ ≤ Δ and η′ : Δ ≤ Φ then η • η′ : Γ ≤ Φ.\n\ninfixl 30 _•_\n\n_•_ : Wk → Wk → Wk\nid • η′ = η′\nstep η • η′ = step (η • η′)\nlift η • id = lift η\nlift η • step η′ = step (η • η′)\nlift η • lift η′ = lift (η • η′)\n\nrepeat : {A : Set} → (A → A) → A → Nat → A\nrepeat f a 0 = a\nrepeat f a (1+ n) = f (repeat f a n)\n\n-- Weakening of variables.\n-- If η : Γ ≤ Δ and x ∈ dom(Δ) then wkVar ρ x ∈ dom(Γ).\n\nwkVar : (ρ : Wk) (n : Nat) → Nat\nwkVar id n = n\nwkVar (step ρ) n = 1+ (wkVar ρ n)\nwkVar (lift ρ) 0 = 0\nwkVar (lift ρ) (1+ n) = 1+ (wkVar ρ n)\n\n -- Weakening of terms.\n -- If η : Γ ≤ Δ and Δ ⊢ t : A then Γ ⊢ wk η t : wk η A.\n\nmutual\n wkGen : (ρ : Wk) (g : List (GenT Term)) → List (GenT Term)\n wkGen ρ [] = []\n wkGen ρ (⟦ l , t ⟧ ∷ g) = ⟦ l , (wk (repeat lift ρ l) t) ⟧ ∷ wkGen ρ g\n\n wk : (ρ : Wk) (t : Term) → Term\n wk ρ (var x) = var (wkVar ρ x)\n wk ρ (gen x c) = gen x (wkGen ρ c)\n\n-- Adding one variable to the context requires wk1.\n-- If Γ ⊢ t : B then Γ∙A ⊢ wk1 t : wk1 B.\n\nwk1 : Term → Term\nwk1 = wk (step id)\n\nwk1d : Term → Term\nwk1d = wk (lift (step id))\n\n-- Weakening of a neutral term.\n\nwkNeutral : ∀ {t} ρ → Neutral t → Neutral (wk ρ t)\nwkNeutral ρ (var n) = var (wkVar ρ n)\nwkNeutral ρ (∘ₙ n) = ∘ₙ (wkNeutral ρ n)\nwkNeutral ρ (natrecₙ n) = natrecₙ (wkNeutral ρ n)\nwkNeutral ρ Emptyrecₙ = Emptyrecₙ\nwkNeutral ρ (Idₙ A) = Idₙ (wkNeutral ρ A)\nwkNeutral ρ (Idℕₙ t) = Idℕₙ (wkNeutral ρ t)\nwkNeutral ρ (Idℕ0ₙ t) = Idℕ0ₙ (wkNeutral ρ t)\nwkNeutral ρ (IdℕSₙ t) = IdℕSₙ (wkNeutral ρ t)\nwkNeutral ρ (IdUₙ t) = IdUₙ (wkNeutral ρ t)\nwkNeutral ρ (IdUℕₙ t) = IdUℕₙ (wkNeutral ρ t)\nwkNeutral ρ (IdUΠₙ t) = IdUΠₙ (wkNeutral ρ t)\nwkNeutral ρ (castₙ A) = castₙ (wkNeutral ρ A)\nwkNeutral ρ (castℕₙ A) = castℕₙ (wkNeutral ρ A)\nwkNeutral ρ (castΠₙ A) = castΠₙ (wkNeutral ρ A)\nwkNeutral ρ (castℕℕₙ t) = castℕℕₙ (wkNeutral ρ t)\nwkNeutral ρ castℕΠₙ = castℕΠₙ\nwkNeutral ρ castΠℕₙ = castΠℕₙ\nwkNeutral ρ castΠΠ%!ₙ = castΠΠ%!ₙ\nwkNeutral ρ castΠΠ!%ₙ = castΠΠ!%ₙ\n\n-- Weakening can be applied to our whnf views.\n\nwkNatural : ∀ {t} ρ → Natural t → Natural (wk ρ t)\nwkNatural ρ sucₙ = sucₙ\nwkNatural ρ zeroₙ = zeroₙ\nwkNatural ρ (ne x) = ne (wkNeutral ρ x)\n\nwkType : ∀ {t} ρ → Type t → Type (wk ρ t)\nwkType ρ Πₙ = Πₙ\nwkType ρ ℕₙ = ℕₙ\nwkType ρ Uₙ = Uₙ\nwkType ρ ∃ₙ = ∃ₙ\nwkType ρ Emptyₙ = Emptyₙ\nwkType ρ (ne x) = ne (wkNeutral ρ x)\n\nwkFunction : ∀ {t} ρ → Function t → Function (wk ρ t)\nwkFunction ρ lamₙ = lamₙ\nwkFunction ρ (ne x) = ne (wkNeutral ρ x)\n\nwkWhnf : ∀ {t} ρ → Whnf t → Whnf (wk ρ t)\nwkWhnf ρ Uₙ = Uₙ\nwkWhnf ρ Πₙ = Πₙ\nwkWhnf ρ ∃ₙ = ∃ₙ\nwkWhnf ρ ℕₙ = ℕₙ\nwkWhnf ρ Emptyₙ = Emptyₙ\nwkWhnf ρ lamₙ = lamₙ\nwkWhnf ρ zeroₙ = zeroₙ\nwkWhnf ρ sucₙ = sucₙ\nwkWhnf ρ (ne x) = ne (wkNeutral ρ x)\n\n-- Non-dependent version of Π.\n\n_^_°_▹▹_°_°_ : Term → Relevance → Level → Term → Level → Level → Term\nA ^ r ° lA ▹▹ B ° lB ° l = Π A ^ r ° lA ▹ wk1 B ° lB ° l\n\n-- Non-dependent version of ∃.\n\n_××_ : Term → Term → Term\nA ×× B = ∃ A ▹ wk1 B\n\n------------------------------------------------------------------------\n-- Substitution\n\n-- The substitution operation subst σ t replaces the free de Bruijn indices\n-- of term t by chosen terms as specified by σ.\n\n-- The substitution σ itself is a map from natural numbers to terms.\n\nSubst : Set\nSubst = Nat → Term\n\n-- Given closed contexts ⊢ Γ and ⊢ Δ,\n-- substitutions may be typed via Γ ⊢ σ : Δ meaning that\n-- Γ ⊢ σ(x) : (subst σ Δ)(x) for all x ∈ dom(Δ).\n--\n-- The substitution operation is then typed as follows:\n-- If Γ ⊢ σ : Δ and Δ ⊢ t : A, then Γ ⊢ subst σ t : subst σ A.\n--\n-- Although substitutions are untyped, typing helps us\n-- to understand the operation on substitutions.\n\n-- We may view σ as the infinite stream σ 0, σ 1, ...\n\n-- Extract the substitution of the first variable.\n--\n-- If Γ ⊢ σ : Δ∙A then Γ ⊢ head σ : subst σ A.\n\nhead : Subst → Term\nhead σ = σ 0\n\n-- Remove the first variable instance of a substitution\n-- and shift the rest to accommodate.\n--\n-- If Γ ⊢ σ : Δ∙A then Γ ⊢ tail σ : Δ.\n\ntail : Subst → Subst\ntail σ n = σ (1+ n)\n\n-- Substitution of a variable.\n--\n-- If Γ ⊢ σ : Δ then Γ ⊢ substVar σ x : (subst σ Δ)(x).\n\nsubstVar : (σ : Subst) (x : Nat) → Term\nsubstVar σ x = σ x\n\n-- Identity substitution.\n-- Replaces each variable by itself.\n--\n-- Γ ⊢ idSubst : Γ.\n\nidSubst : Subst\nidSubst = var\n\n-- Weaken a substitution by one.\n--\n-- If Γ ⊢ σ : Δ then Γ∙A ⊢ wk1Subst σ : Δ.\n\nwk1Subst : Subst → Subst\nwk1Subst σ x = wk1 (σ x)\n\n-- Lift a substitution.\n--\n-- If Γ ⊢ σ : Δ then Γ∙A ⊢ liftSubst σ : Δ∙A.\n\nliftSubst : (σ : Subst) → Subst\nliftSubst σ 0 = var 0\nliftSubst σ (1+ x) = wk1Subst σ x\n\n-- Transform a weakening into a substitution.\n--\n-- If ρ : Γ ≤ Δ then Γ ⊢ toSubst ρ : Δ.\n\ntoSubst : Wk → Subst\ntoSubst pr x = var (wkVar pr x)\n\n-- Apply a substitution to a term.\n--\n-- If Γ ⊢ σ : Δ and Δ ⊢ t : A then Γ ⊢ subst σ t : subst σ A.\n\nmutual\n substGen : (σ : Subst) (g : List (GenT Term)) → List (GenT Term)\n substGen σ [] = []\n substGen σ (⟦ l , t ⟧ ∷ g) = ⟦ l , (subst (repeat liftSubst σ l) t) ⟧ ∷ substGen σ g\n\n subst : (σ : Subst) (t : Term) → Term\n subst σ (var x) = substVar σ x\n subst σ (gen x c) = gen x (substGen σ c)\n\n-- Extend a substitution by adding a term as\n-- the first variable substitution and shift the rest.\n--\n-- If Γ ⊢ σ : Δ and Γ ⊢ t : subst σ A then Γ ⊢ consSubst σ t : Δ∙A.\n\nconsSubst : Subst → Term → Subst\nconsSubst σ t 0 = t\nconsSubst σ t (1+ n) = σ n\n\n-- Singleton substitution.\n--\n-- If Γ ⊢ t : A then Γ ⊢ sgSubst t : Γ∙A.\n\nsgSubst : Term → Subst\nsgSubst = consSubst idSubst\n\n-- Compose two substitutions.\n--\n-- If Γ ⊢ σ : Δ and Δ ⊢ σ′ : Φ then Γ ⊢ σ ₛ•ₛ σ′ : Φ.\n\n_ₛ•ₛ_ : Subst → Subst → Subst\n_ₛ•ₛ_ σ σ′ x = subst σ (σ′ x)\n\n-- Composition of weakening and substitution.\n--\n-- If ρ : Γ ≤ Δ and Δ ⊢ σ : Φ then Γ ⊢ ρ •ₛ σ : Φ.\n\n_•ₛ_ : Wk → Subst → Subst\n_•ₛ_ ρ σ x = wk ρ (σ x)\n\n-- If Γ ⊢ σ : Δ and ρ : Δ ≤ Φ then Γ ⊢ σ ₛ• ρ : Φ.\n\n_ₛ•_ : Subst → Wk → Subst\n_ₛ•_ σ ρ x = σ (wkVar ρ x)\n\n-- Substitute the first variable of a term with an other term.\n--\n-- If Γ∙A ⊢ t : B and Γ ⊢ s : A then Γ ⊢ t[s] : B[s].\n\n_[_] : (t : Term) (s : Term) → Term\nt [ s ] = subst (sgSubst s) t\n\n-- Substitute the first variable of a term with an other term,\n-- but let the two terms share the same context.\n--\n-- If Γ∙A ⊢ t : B and Γ∙A ⊢ s : A then Γ∙A ⊢ t[s]↑ : B[s]↑.\n\n_[_]↑ : (t : Term) (s : Term) → Term\nt [ s ]↑ = subst (consSubst (wk1Subst idSubst) s) t\n\n_[_]↑↑ : (t : Term) (s : Term) → Term\nt [ s ]↑↑ = subst (consSubst (wk1Subst (wk1Subst idSubst)) s) t\n\n-- Definition of syntaxic sugar\n\nUnit : ∀ {l} → Term\nUnit {l} = Π Empty l ^ % ° l ▹ Empty l ° l ° l\n\nIdsym : (A x y e : Term) → Term\nIdsym A x y e = transp A (Id (wk1 A) (var 0) (wk1 x)) x (Idrefl A x) y e\n", "meta": {"hexsha": "e9ca9daaa39ae62d792f6a023d01fc0a4b16fc62", "size": 19201, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Definition/Untyped.agda", "max_stars_repo_name": "CoqHott/logrel-mltt", "max_stars_repo_head_hexsha": "e0eeebc4aa5ed791ce3e7c0dc9531bd113dfcc04", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-06-21T08:39:01.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-17T16:13:53.000Z", "max_issues_repo_path": "Definition/Untyped.agda", "max_issues_repo_name": "CoqHott/logrel-mltt", "max_issues_repo_head_hexsha": "e0eeebc4aa5ed791ce3e7c0dc9531bd113dfcc04", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, 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YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.34860323953727723}} {"text": "-- MIT License\n\n-- Copyright (c) 2021 Luca Ciccone and Luca Padovani\n\n-- Permission is hereby granted, free of charge, to any person\n-- obtaining a copy of this software and associated documentation\n-- files (the \"Software\"), to deal in the Software without\n-- restriction, including without limitation the rights to use,\n-- copy, modify, merge, publish, distribute, sublicense, and/or sell\n-- copies of the Software, and to permit persons to whom the\n-- Software is furnished to do so, subject to the following\n-- conditions:\n\n-- The above copyright notice and this permission notice shall be\n-- included in all copies or substantial portions of the Software.\n\n-- THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND,\n-- EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES\n-- OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND\n-- NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT\n-- HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,\n-- WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n-- FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR\n-- OTHER DEALINGS IN THE SOFTWARE.\n\n{-# OPTIONS --guardedness --sized-types #-}\n\nopen import Size\n\nopen import Data.Empty\nopen import Data.Product\nopen import Data.Sum\nopen import Data.List using ([]; _∷_; _∷ʳ_; _++_)\n\nopen import Codata.Thunk\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Negation using (contraposition)\nopen import Relation.Unary using (_∈_; _⊆_)\nopen import Relation.Binary.PropositionalEquality using (_≡_; _≢_; refl)\n\nopen import Function.Base using (case_of_)\n\nopen import Common\n\nmodule Subtyping {ℙ : Set} (message : Message ℙ)\n where\n\nopen Message message\n\nopen import Trace message\nopen import SessionType message\nopen import Transitions message\nopen import Session message\nopen import Compliance message\nopen import HasTrace message\n\ndata Sub : SessionType -> SessionType -> Size -> Set where\n nil<:any : ∀{T i} -> Sub nil T i\n end<:def : ∀{T S i} (e : End T) (def : Defined S) -> Sub T S i\n inp<:inp : ∀{f g i} (inc : dom f ⊆ dom g) (F : (x : ℙ) -> Thunk (Sub (f x .force) (g x .force)) i) -> Sub (inp f) (inp g) i\n out<:out : ∀{f g i} (W : Witness g) (inc : dom g ⊆ dom f) (F : ∀{x} (!x : x ∈ dom g) -> Thunk (Sub (f x .force) (g x .force)) i) -> Sub (out f) (out g) i\n\n_<:_ : SessionType -> SessionType -> Set\n_<:_ T S = Sub T S ∞\n\nsub-defined : ∀{T S} -> T <: S -> Defined T -> Defined S\nsub-defined (end<:def _ def) _ = def\nsub-defined (inp<:inp _ _) _ = inp\nsub-defined (out<:out _ _ _) _ = out\n\nsub-sound : ∀{T S R} -> Compliance (R # T) -> T <: S -> ∞Compliance (R # S)\nforce (sub-sound (win#def w def) sub) = win#def w (sub-defined sub def)\nforce (sub-sound (out#inp (_ , !x) F) (end<:def (inp U) def)) with U _ (proj₂ (compliance->defined (F !x .force)))\n... | ()\nforce (sub-sound (out#inp (_ , !x) F) (inp<:inp _ G)) =\n out#inp (_ , !x) λ !x -> sub-sound (F !x .force) (G _ .force)\nforce (sub-sound (inp#out (_ , !x) F) (end<:def (out U) def)) = ⊥-elim (U _ !x)\nforce (sub-sound (inp#out (_ , !x) F) (out<:out {f} {g} (_ , !y) inc G)) =\n inp#out (_ , !y) λ !x -> sub-sound (F (inc !x) .force) (G !x .force)\n\nSubtypingQ : SessionType -> SessionType -> Set\nSubtypingQ T S = ∀{R} -> Compliance (R # T) -> Compliance (R # S)\n\nif-eq : ℙ -> SessionType -> SessionType -> Continuation\nforce (if-eq x T S y) with x ?= y\n... | yes _ = T\n... | no _ = S\n\ninput* : SessionType\ninput* = inp λ _ -> λ where .force -> win\n\ninput : ℙ -> SessionType -> SessionType -> SessionType\ninput x T S = inp (if-eq x T S)\n\ninput*-but : ℙ -> SessionType\ninput*-but x = input x nil win\n\noutput : ℙ -> SessionType -> SessionType -> SessionType\noutput x T S = out (if-eq x T S)\n\ninput-if-eq-comp :\n ∀{f x T} ->\n Compliance (T # f x .force) ->\n ∀{y} (!y : y ∈ dom f) ->\n ∞Compliance (if-eq x T win y .force # f y .force)\nforce (input-if-eq-comp {_} {x} comp {y} !y) with x ?= y\n... | yes refl = comp\n... | no neq = win#def Win-win !y\n\noutput-if-eq-comp :\n ∀{f : Continuation}{x}{T} ->\n Compliance (T # f x .force) ->\n ∀{y} (!y : y ∈ dom (if-eq x T nil)) ->\n ∞Compliance (if-eq x T nil y .force # f y .force)\nforce (output-if-eq-comp {_} {x} comp {y} !y) with x ?= y\n... | yes refl = comp\nforce (output-if-eq-comp {_} {x} comp {y} ()) | no neq\n\ninput*-comp : ∀{f} (W : Witness f) -> Compliance (input* # out f)\ninput*-comp W = inp#out W λ !x -> λ where .force -> win#def Win-win !x\n\ninput*-but-comp :\n ∀{f x}\n (W : Witness f)\n (N : ¬ x ∈ dom f) ->\n Compliance (input*-but x # out f)\ninput*-but-comp {f} {x} W N = inp#out W aux\n where\n aux : ∀{y : ℙ} -> (fy : y ∈ dom f) -> ∞Compliance (if-eq x nil win y .force # f y .force)\n force (aux {y} fy) with x ?= y\n ... | yes refl = ⊥-elim (N fy)\n ... | no neq = win#def Win-win fy\n\n∈-output-if-eq : ∀{R} (x : ℙ) -> Defined R -> x ∈ dom (if-eq x R nil)\n∈-output-if-eq x def with x ?= x\n... | yes refl = def\n... | no neq = ⊥-elim (neq refl)\n\ninput-comp : ∀{g x R} -> Compliance (R # g x .force) -> Compliance (input x R win # out g)\ninput-comp {g} {x} comp = inp#out (x , proj₂ (compliance->defined comp)) (input-if-eq-comp {g} comp)\n\noutput-comp : ∀{f x R} -> Compliance (R # f x .force) -> Compliance (output x R nil # inp f)\noutput-comp {f} {x} comp = out#inp (_ , ∈-output-if-eq x (proj₁ (compliance->defined comp))) (output-if-eq-comp {f} comp)\n\nsub-inp-inp :\n ∀{f g}\n (spec : SubtypingQ (inp f) (inp g))\n (x : ℙ) ->\n SubtypingQ (f x .force) (g x .force)\nsub-inp-inp spec x comp with spec (output-comp comp)\n... | win#def (out U) def = ⊥-elim (U _ (∈-output-if-eq x (proj₁ (compliance->defined comp))))\n... | out#inp (y , fy) F with F fy .force\n... | comp' with x ?= y\n... | yes refl = comp'\nsub-inp-inp spec x comp | out#inp (y , fy) F | win#def () def | no neq\n\nsub-out-out :\n ∀{f g}\n (spec : SubtypingQ (out f) (out g)) ->\n ∀{x} -> x ∈ dom g ->\n SubtypingQ (f x .force) (g x .force)\nsub-out-out spec {x} gx comp with spec (input-comp comp)\n... | inp#out W F with F gx .force\n... | comp' with x ?= x\n... | yes refl = comp'\n... | no neq = ⊥-elim (neq refl)\n\nsub-out->def :\n ∀{f g}\n (spec : SubtypingQ (out f) (out g))\n (Wf : Witness f) ->\n ∀{x} (gx : x ∈ dom g) ->\n x ∈ dom f\nsub-out->def {f} spec Wf {x} gx with x ∈? f\n... | yes fx = fx\n... | no nfx with spec (input*-but-comp Wf nfx)\n... | inp#out W F with F gx .force\n... | res with x ?= x\nsub-out->def {f} spec Wf {x} gx | no nfx | inp#out W F | win#def () def | yes refl\n... | no neq = ⊥-elim (neq refl)\n\nsub-inp->def : ∀{f g} (spec : SubtypingQ (inp f) (inp g)) -> ∀{x} (fx : x ∈ dom f) -> x ∈ dom g\nsub-inp->def {f} spec {x} fx with spec {output x win nil} (output-comp (win#def Win-win fx))\n... | win#def (out U) def = ⊥-elim (U _ (∈-output-if-eq x out))\n... | out#inp W F with F (∈-output-if-eq x out) .force\n... | comp = proj₂ (compliance->defined comp)\n\nsub-complete : ∀{T S i} -> SubtypingQ T S -> Thunk (Sub T S) i\nforce (sub-complete {nil} {_} spec) = nil<:any\nforce (sub-complete {inp f} {nil} spec) with spec {win} (win#def Win-win inp)\n... | win#def _ ()\nforce (sub-complete {inp _} {inp _} spec) = inp<:inp (sub-inp->def spec) λ x -> sub-complete (sub-inp-inp spec x)\nforce (sub-complete {inp f} {out _} spec) with Empty? f\n... | inj₁ U = end<:def (inp U) out\n... | inj₂ (x , ?x) with spec {output x win nil} (output-comp (win#def Win-win ?x))\n... | win#def (out U) def = ⊥-elim (U x (∈-output-if-eq x out))\nforce (sub-complete {out f} {nil} spec) with spec {win} (win#def Win-win out)\n... | win#def _ ()\nforce (sub-complete {out f} {inp _} spec) with Empty? f\n... | inj₁ U = end<:def (out U) inp\n... | inj₂ W with spec {input*} (input*-comp W)\n... | win#def () _\nforce (sub-complete {out f} {out g} spec) with Empty? f\n... | inj₁ Uf = end<:def (out Uf) out\n... | inj₂ Wf with Empty? g\n... | inj₂ Wg = out<:out Wg (sub-out->def spec Wf) λ !x -> sub-complete (sub-out-out spec !x)\n... | inj₁ Ug with spec {input*} (input*-comp Wf)\n... | inp#out (_ , !x) F = ⊥-elim (Ug _ !x)\n\nSubtypingQ->SubtypingS : ∀{T S} -> SubtypingQ T S -> SubtypingS T S\nSubtypingQ->SubtypingS spec comp = compliance-sound (spec (compliance-complete comp .force))\n\nSubtypingS->SubtypingQ : ∀{T S} -> SubtypingS T S -> SubtypingQ T S\nSubtypingS->SubtypingQ spec comp = compliance-complete (spec (compliance-sound comp)) .force\n\nsub-excluded :\n ∀{T S φ}\n (sub : T <: S)\n (tφ : T HasTrace φ)\n (nsφ : ¬ S HasTrace φ) ->\n ∃[ ψ ] ∃[ x ]\n (ψ ⊑ φ × T HasTrace ψ × S HasTrace ψ × T HasTrace (ψ ∷ʳ O x) × ¬ S HasTrace (ψ ∷ʳ O x))\nsub-excluded nil<:any tφ nsφ = ⊥-elim (nil-has-no-trace tφ)\nsub-excluded (end<:def e def) tφ nsφ with end-has-empty-trace e tφ\n... | eq rewrite eq = ⊥-elim (nsφ (_ , def , refl))\nsub-excluded (inp<:inp inc F) (_ , tdef , refl) nsφ =\n ⊥-elim (nsφ (_ , inp , refl))\nsub-excluded (inp<:inp {f} {g} inc F) (_ , tdef , step inp tr) nsφ =\n let ψ , x , pre , tψ , sψ , tψx , nψx = sub-excluded (F _ .force) (_ , tdef , tr) (contraposition inp-has-trace nsφ) in\n _ , _ , some pre , inp-has-trace tψ , inp-has-trace sψ , inp-has-trace tψx , inp-has-no-trace nψx\nsub-excluded (out<:out W inc F) (_ , tdef , refl) nsφ =\n ⊥-elim (nsφ (_ , out , refl))\nsub-excluded (out<:out {f} {g} W inc F) (_ , tdef , step (out {_} {x} fx) tr) nsφ with x ∈? g\n... | yes gx =\n let ψ , x , pre , tψ , sψ , tψx , nψx = sub-excluded (F gx .force) (_ , tdef , tr) (contraposition out-has-trace nsφ) in\n _ , _ , some pre , out-has-trace tψ , out-has-trace sψ , out-has-trace tψx , out-has-no-trace nψx\n... | no ngx =\n [] , _ , none , (_ , out , refl) , (_ , out , refl) , (_ , fx , step (out fx) refl) , λ { (_ , _ , step (out gx) _) → ⊥-elim (ngx gx) }\n\nsub-after : ∀{T S φ} (tφ : T HasTrace φ) (sφ : S HasTrace φ) -> T <: S -> after tφ <: after sφ\nsub-after (_ , _ , refl) (_ , _ , refl) sub = sub\nsub-after tφ@(_ , _ , step inp _) (_ , _ , step inp _) (end<:def e _) with end-has-empty-trace e tφ\n... | ()\nsub-after (_ , tdef , step inp tr) (_ , sdef , step inp sr) (inp<:inp _ F) =\n sub-after (_ , tdef , tr) (_ , sdef , sr) (F _ .force)\nsub-after tφ@(_ , _ , step (out _) _) (_ , _ , step (out _) _) (end<:def e _) with end-has-empty-trace e tφ\n... | ()\nsub-after (_ , tdef , step (out _) tr) (_ , sdef , step (out gx) sr) (out<:out _ _ F) =\n sub-after (_ , tdef , tr) (_ , sdef , sr) (F gx .force)\n\nsub-simulation :\n ∀{R R' T S S' φ}\n (comp : Compliance (R # T))\n (sub : T <: S)\n (rr : Transitions R (co-trace φ) R')\n (sr : Transitions S φ S') ->\n ∃[ T' ] (Transitions T φ T' × T' <: S')\nsub-simulation comp sub refl refl = _ , refl , sub\nsub-simulation (win#def (out U) def) sub (step (out hx) rr) (step inp sr) = ⊥-elim (U _ hx)\nsub-simulation (out#inp W F) (end<:def (inp U) def) (step (out hx) rr) (step inp sr) with F hx .force\n... | comp = ⊥-elim (U _ (proj₂ (compliance->defined comp)))\nsub-simulation (out#inp W F) (inp<:inp inc G) (step (out hx) rr) (step inp sr) =\n let _ , tr , sub = sub-simulation (F hx .force) (G _ . force) rr sr in\n _ , step inp tr , sub\nsub-simulation (inp#out {h} {f} (_ , fx) F) (end<:def (out U) def) (step inp rr) (step (out gx) sr) with F fx .force\n... | comp = ⊥-elim (U _ (proj₂ (compliance->defined comp)))\nsub-simulation (inp#out W F) (out<:out W₁ inc G) (step inp rr) (step (out fx) sr) =\n let _ , tr , sub = sub-simulation (F (inc fx) .force) (G fx .force) rr sr in\n _ , step (out (inc fx)) tr , sub\n", "meta": {"hexsha": "0f053178851d6df4793b2cdf87bf6a44235c6c98", "size": 11351, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Subtyping.agda", "max_stars_repo_name": "boystrange/FairSubtypingAgda", "max_stars_repo_head_hexsha": "c4b78e70c3caf68d509f4360b9171d9f80ecb825", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 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YES\n2. NO\n\n", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3485208264916123}} {"text": "{-# OPTIONS --instance-search-depth=5 --show-implicit #-}\n\nopen import Oscar.Prelude\nopen import Oscar.Class.Smap\nopen import Oscar.Class.Surjextensionality\nopen import Oscar.Class.Symmetry\nopen import Oscar.Class.Transitivity\nopen import Oscar.Data.Proposequality\nopen import Oscar.Data.Surjcollation\nimport Oscar.Class.Surjection.⋆\n\nmodule Oscar.Data.Surjextenscollation where\n\nmodule _ {𝔵 𝔞 𝔞̇ 𝔟 𝔟̇} {𝔛 : Ø 𝔵}\n (𝔄 : 𝔛 → 𝔛 → Ø 𝔞)\n (𝔅 : 𝔛 → Ø 𝔟)\n ⦃ _ : Smaphomarrow!.class 𝔄 𝔅 ⦄\n (𝔄̇ : ∀ {x y} → 𝔄 x y → 𝔄 x y → Ø 𝔞̇)\n (let ℭ : 𝔛 → Ø 𝔵 ∙̂ 𝔞 ∙̂ 𝔞̇ ∙̂ ↑̂ 𝔟̇\n ℭ = LeftExtensionṖroperty 𝔟̇ 𝔄 𝔄̇)\n (𝔅̇ : ∀ {y} → 𝔅 y → 𝔅 y → Ø 𝔟̇)\n ⦃ _ : ∀ {y} → Symmetry.class (𝔅̇ {y}) ⦄\n ⦃ _ : ∀ {y} → Transitivity.class (𝔅̇ {y}) ⦄\n ⦃ _ : Surjextensionality!.class 𝔄 𝔄̇ (Extension 𝔅) (Pointwise 𝔅̇) ⦄\n where\n\n surjextenscollation[_/_]⟦_/_⟧ : ∀ {m} → 𝔅 m → 𝔅 m → ℭ m\n surjextenscollation[_/_]⟦_/_⟧ s t =\n surjcollation⟦ 𝔄 / 𝔅̇ ⟧ s t , λ f≐g f◃s=f◃t →\n -- FIXME this (`surjextensionality[ Pointwise 𝔅̇ ] ⦃ ! ⦄ f≐g t ∙ f◃s=f◃t ∙ symmetry (surjextensionality[ Pointwise 𝔅̇ ] ⦃ ! ⦄ f≐g s)`) used to be a workaround for \"instance search depth exhausted\", but now does not seem to help. See the FIXME note in Oscar.Class.Surjextensionality.\n ⟪ f≐g ⟫[ Pointwise 𝔅̇ ] t ∙ f◃s=f◃t ∙ symmetry (⟪ f≐g ⟫[ Pointwise 𝔅̇ ] s)\n\nmodule _ {𝔵 𝔞 𝔞̇} {𝔛 : Ø 𝔵} {𝔄 : 𝔛 → 𝔛 → Ø 𝔞}\n (𝔄̇ : ∀ {x y} → 𝔄 x y → 𝔄 x y → Ø 𝔞̇)\n {𝔟} {𝔅 : 𝔛 → Ø 𝔟}\n {𝔟̇} {𝔅̇ : ∀ {y} → 𝔅 y → 𝔅 y → Ø 𝔟̇}\n ⦃ _ : Smaphomarrow!.class 𝔄 𝔅 ⦄\n ⦃ _ : ∀ {y} → Symmetry.class (𝔅̇ {y}) ⦄\n ⦃ _ : ∀ {y} → Transitivity.class (𝔅̇ {y}) ⦄\n ⦃ _ : Surjextensionality!.class 𝔄 𝔄̇ (Extension 𝔅) (Pointwise 𝔅̇) ⦄\n where\n surjextenscollation⟦_⟧ = surjextenscollation[ 𝔄 / 𝔅 ]⟦ 𝔄̇ / 𝔅̇ ⟧\n\nmodule _ {𝔵 𝔞 𝔞̇ 𝔟 𝔟̇} {𝔛 : Ø 𝔵} {𝔄 : 𝔛 → 𝔛 → Ø 𝔞} {𝔅 : 𝔛 → Ø 𝔟}\n (𝔄̇ : ∀ {x y} → 𝔄 x y → 𝔄 x y → Ø 𝔞̇)\n (𝔅̇ : ∀ {y} → 𝔅 y → 𝔅 y → Ø 𝔟̇)\n ⦃ _ : Smaphomarrow!.class 𝔄 𝔅 ⦄\n ⦃ _ : ∀ {y} → Symmetry.class (𝔅̇ {y}) ⦄\n ⦃ _ : ∀ {y} → Transitivity.class (𝔅̇ {y}) ⦄\n ⦃ _ : Surjextensionality!.class 𝔄 𝔄̇ (Extension 𝔅) (Pointwise 𝔅̇) ⦄\n where\n surjextenscollation⟦_/_⟧ = surjextenscollation[ 𝔄 / 𝔅 ]⟦ 𝔄̇ / 𝔅̇ ⟧\n\nmodule Surjextenscollation\n {𝔵 𝔞 𝔞̇} {𝔛 : Ø 𝔵}\n (𝔄 : 𝔛 → 𝔛 → Ø 𝔞)\n (𝔄̇ : ∀ {x y} → 𝔄 x y → 𝔄 x y → Ø 𝔞̇)\n {𝔟} {𝔅 : 𝔛 → Ø 𝔟}\n {𝔟̇} {𝔅̇ : ∀ {y} → 𝔅 y → 𝔅 y → Ø 𝔟̇}\n ⦃ _ : Smaphomarrow!.class 𝔄 𝔅 ⦄\n ⦃ _ : ∀ {y} → Symmetry.class (𝔅̇ {y}) ⦄\n ⦃ _ : ∀ {y} → Transitivity.class (𝔅̇ {y}) ⦄\n ⦃ _ : Surjextensionality!.class 𝔄 𝔄̇ (Extension 𝔅) (Pointwise 𝔅̇) ⦄\n where\n method = surjextenscollation[ 𝔄 / 𝔅 ]⟦ 𝔄̇ / 𝔅̇ ⟧\n\n infix 18 _⟹_\n _⟹_ = method\n\nmodule _\n {𝔵} {𝔛 : Ø 𝔵}\n {𝔞₁} {𝔄₁ : 𝔛 → Ø 𝔞₁}\n {𝔞₂} {𝔄₂ : 𝔛 → Ø 𝔞₂}\n (𝔄 : 𝔛 → 𝔛 → Ø 𝔞₁ ∙̂ 𝔞₂)\n ⦃ _ : 𝔄 ≡ Arrow 𝔄₁ 𝔄₂ ⦄\n (let 𝔄 = Arrow 𝔄₁ 𝔄₂)\n where\n open Surjextenscollation 𝔄 _≡̇_ public using () renaming (method to ≡-surjextenscollation⟦_⟧) public\n", "meta": {"hexsha": "53c9fd00d073dd05f3e54ffcb5df7efa65deaa9c", "size": 2891, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "archive/agda-3/src/Oscar/Data/Surjextenscollation.agda", "max_stars_repo_name": "m0davis/oscar", "max_stars_repo_head_hexsha": "52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb", "max_stars_repo_licenses": ["RSA-MD"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "archive/agda-3/src/Oscar/Data/Surjextenscollation.agda", "max_issues_repo_name": "m0davis/oscar", "max_issues_repo_head_hexsha": "52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb", "max_issues_repo_licenses": ["RSA-MD"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-04-29T00:35:04.000Z", "max_issues_repo_issues_event_max_datetime": "2019-05-11T23:33:04.000Z", "max_forks_repo_path": "archive/agda-3/src/Oscar/Data/Surjextenscollation.agda", "max_forks_repo_name": "m0davis/oscar", "max_forks_repo_head_hexsha": "52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb", "max_forks_repo_licenses": ["RSA-MD"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.5949367089, "max_line_length": 288, "alphanum_fraction": 0.5423728814, "num_tokens": 1678, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.34852082649161226}} {"text": "{-# OPTIONS --without-K #-}\nmodule function.isomorphism.properties where\n\nopen import level\nopen import sum\nopen import sets.nat.core\nopen import equality.core\nopen import equality.calculus\nopen import equality.reasoning\nopen import function.core\nopen import function.fibration\nopen import function.overloading\nopen import function.extensionality.core\nopen import function.extensionality.proof\nopen import function.isomorphism.core\nopen import function.isomorphism.coherent\nopen import function.isomorphism.lift\nopen import function.isomorphism.utils\nopen import function.isomorphism.two-out-of-six\nopen import hott.equivalence.core\nopen import hott.equivalence.alternative\nopen import hott.level.core\nopen import hott.loop.core\nopen import sets.unit\n\niso-adjunction : ∀ {i j}{X : Set i}{Y : Set j}\n → (isom : X ≅ Y)(x : X)(y : Y)\n → (apply isom x ≡ y) ≅ (x ≡ invert isom y)\niso-adjunction {i}{j}{X}{Y} isom x y\n = family-eq-iso total' (λ { (y , x) q → comm' x y q }) (y , x)\n where\n open _≅_ isom\n open ≅-Reasoning\n\n to-we : weak-equiv to\n to-we = proj₂ (≅⇒≈ isom)\n\n total' : (Σ (Y × X) λ { (y , x) → to x ≡ y })\n ≅ (Σ (Y × X) λ { (y , x) → x ≡ from y})\n total' = Σ-assoc-iso\n ·≅ ( Σ-ap-iso refl≅ λ y\n → contr-⊤-iso (to-we y)\n ·≅ sym≅ (contr-⊤-iso (singl-contr' (from y))) )\n ·≅ sym≅ Σ-assoc-iso\n\n comm' : (x : X)(y : Y)(p : to x ≡ y)\n → proj₁ (apply total' ((y , x) , p)) ≡ (y , x)\n comm' x y q = ap (_,_ y) (sym (ap from q) · iso₁ x)\n\nprivate\n iso≡-lem : ∀ {i j}{X : Set i}{Y : Set j}\n → (isom : X ≅ Y)\n → (x x' : X)\n → weak-equiv (λ (p : x ≡ x') → ap (invert isom) (ap (apply isom) p))\n iso≡-lem {X = X} isom x x' = step₃\n where\n step₁ : ∀ {k}{A : Set k}(a a' : A)\n → weak-equiv {X = a ≡ a'} {Y = a ≡ a'} (ap (λ x → x))\n step₁ a a' = subst weak-equiv (sym (funext ap-id)) (proj₂ (≅⇒≈ refl≅))\n\n step₂ : weak-equiv (λ (p : x ≡ x') → ap (invert isom ∘ apply isom) p)\n step₂ = subst (λ u → weak-equiv {X = x ≡ x'} (ap u)) (sym (funext (_≅_.iso₁ isom)))\n (step₁ x x')\n\n step₃ = subst weak-equiv (sym (funext λ p → ap-hom (apply isom) (invert isom) p)) step₂\n\niso≡ : ∀ {i j}{X : Set i}{Y : Set j}\n → (isom : X ≅ Y)\n → {x x' : X}\n → (x ≡ x')\n ≅ (apply isom x ≡ apply isom x')\niso≡ isom {x}{x'} = two-out-of-six.f-iso\n (ap (apply isom))\n (ap (invert isom))\n (ap (apply isom))\n (iso≡-lem isom x x')\n (iso≡-lem (sym≅ isom) (apply isom x) (apply isom x'))\n\nprivate\n abstract\n Ω-iso' : ∀ {i j}{X : Set i}{Y : Set j}(n : ℕ)\n → (φ : X ≅ Y)(x : X)\n → Ω n x ≅ Ω n (apply φ x)\n Ω-iso' {X = X}{Y = Y} zero φ x = φ\n Ω-iso' (suc n) φ x = Ω-iso' n (iso≡ φ) refl\n\n Ω-iso-β : ∀ {i j}{X : Set i}{Y : Set j}(n : ℕ)\n → (φ : X ≅ Y)(x : X)\n → (p : Ω n x)\n → apply (Ω-iso' n φ x) p ≡ mapΩ n (apply φ) p\n Ω-iso-β zero φ x p = refl\n Ω-iso-β (suc n) φ x p = Ω-iso-β n (iso≡ φ) refl p\n\nΩ-iso : ∀ {i j}{X : Set i}{Y : Set j}(n : ℕ)\n → (φ : X ≅ Y)(x : X)\n → Ω n x ≅ Ω n (apply φ x)\nΩ-iso n φ x = ≈⇒≅ ( (λ p → mapΩ n (apply φ) p)\n , subst weak-equiv eq we)\n where\n abstract\n we : weak-equiv (apply (Ω-iso' n φ x))\n we = proj₂ (≅⇒≈ (Ω-iso' n φ x))\n\n eq : apply (Ω-iso' n φ x) ≡ mapΩ n (apply φ)\n eq = funext (Ω-iso-β n φ x)\n\nabstract\n subtype-eq : ∀ {i j k}{A : Set i}{P : A → Set j}\n → ((x : A) → h 1 (P x))\n → {X : Set k}\n → (isom : X ≅ Σ A P)\n → {x y : X}\n → (proj₁ (apply isom x) ≡ proj₁ (apply isom y))\n → x ≡ y\n subtype-eq hP isom p = iso⇒inj isom\n (unapΣ (p , h1⇒prop (hP _) _ _))\n", "meta": {"hexsha": "a384e31bbea5d4bf5af497a87cfceb039ce5e7dd", "size": 3785, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/function/isomorphism/properties.agda", "max_stars_repo_name": "pcapriotti/agda-base", "max_stars_repo_head_hexsha": "bbbc3bfb2f80ad08c8e608cccfa14b83ea3d258c", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 20, 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93, "alphanum_fraction": 0.5062087186, "num_tokens": 1488, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7025300449389326, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3485208203121322}} {"text": "------------------------------------------------------------------------\n-- The Agda standard library\n--\n-- Dependent product combinators for setoid equality preserving\n-- functions\n------------------------------------------------------------------------\n\n{-# OPTIONS --without-K --safe #-}\n\nmodule Data.Product.Function.Dependent.Setoid where\n\nopen import Data.Product\nopen import Data.Product.Relation.Binary.Pointwise.Dependent\nopen import Relation.Binary\nopen import Function\nopen import Function.Equality as F using (_⟶_; _⟨$⟩_)\nopen import Function.Equivalence as Eq\n using (Equivalence; _⇔_; module Equivalence)\nopen import Function.Injection as Inj\n using (Injection; Injective; _↣_; module Injection)\nopen import Function.Inverse as Inv\n using (Inverse; _↔_; module Inverse)\nopen import Function.LeftInverse as LeftInv\n using (LeftInverse; _↞_; _LeftInverseOf_; _RightInverseOf_; module LeftInverse)\nopen import Function.Surjection as Surj\n using (Surjection; _↠_; module Surjection)\nopen import Relation.Binary as B\nopen import Relation.Binary.Indexed.Heterogeneous\n using (IndexedSetoid)\nopen import Relation.Binary.Indexed.Heterogeneous.Construct.At\n using (_atₛ_)\nopen import Relation.Binary.PropositionalEquality as P using (_≡_)\n\n------------------------------------------------------------------------\n-- Properties related to \"relatedness\"\n------------------------------------------------------------------------\n\nprivate\n\n subst-cong : ∀ {i a p} {I : Set i} {A : I → Set a}\n (P : ∀ {i} → A i → A i → Set p) {i i′} {x y : A i}\n (i≡i′ : i ≡ i′) →\n P x y → P (P.subst A i≡i′ x) (P.subst A i≡i′ y)\n subst-cong P P.refl p = p\n\n⟶ : ∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}\n {A₁ : Set a₁} {A₂ : Set a₂}\n {B₁ : IndexedSetoid A₁ b₁ b₁′} (B₂ : IndexedSetoid A₂ b₂ b₂′)\n (f : A₁ → A₂) → (∀ {x} → (B₁ atₛ x) ⟶ (B₂ atₛ (f x))) →\n setoid (P.setoid A₁) B₁ ⟶ setoid (P.setoid A₂) B₂\n⟶ {A₁ = A₁} {A₂} {B₁} B₂ f g = record\n { _⟨$⟩_ = fg\n ; cong = fg-cong\n }\n where\n open B.Setoid (setoid (P.setoid A₁) B₁)\n using () renaming (_≈_ to _≈₁_)\n open B.Setoid (setoid (P.setoid A₂) B₂)\n using () renaming (_≈_ to _≈₂_)\n open B using (_=[_]⇒_)\n\n fg = map f (_⟨$⟩_ g)\n\n fg-cong : _≈₁_ =[ fg ]⇒ _≈₂_\n fg-cong (P.refl , ∼) = (P.refl , F.cong g ∼)\n\n\nmodule _ {a₁ a₂ b₁ b₁′ b₂ b₂′} {A₁ : Set a₁} {A₂ : Set a₂} where\n\n equivalence : {B₁ : IndexedSetoid A₁ b₁ b₁′} {B₂ : IndexedSetoid A₂ b₂ b₂′}\n (A₁⇔A₂ : A₁ ⇔ A₂) →\n (∀ {x} → _⟶_ (B₁ atₛ x) (B₂ atₛ (Equivalence.to A₁⇔A₂ ⟨$⟩ x))) →\n (∀ {y} → _⟶_ (B₂ atₛ y) (B₁ atₛ (Equivalence.from A₁⇔A₂ ⟨$⟩ y))) →\n Equivalence (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)\n equivalence {B₁} {B₂} A₁⇔A₂ B-to B-from = record\n { to = ⟶ B₂ (_⟨$⟩_ (to A₁⇔A₂)) B-to\n ; from = ⟶ B₁ (_⟨$⟩_ (from A₁⇔A₂)) B-from\n } where open Equivalence\n\n equivalence-↞ : (B₁ : IndexedSetoid A₁ b₁ b₁′) {B₂ : IndexedSetoid A₂ b₂ b₂′}\n (A₁↞A₂ : A₁ ↞ A₂) →\n (∀ {x} → Equivalence (B₁ atₛ (LeftInverse.from A₁↞A₂ ⟨$⟩ x))\n (B₂ atₛ x)) →\n Equivalence (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)\n equivalence-↞ B₁ {B₂} A₁↞A₂ B₁⇔B₂ =\n equivalence (LeftInverse.equivalence A₁↞A₂) B-to B-from\n where\n B-to : ∀ {x} → _⟶_ (B₁ atₛ x) (B₂ atₛ (LeftInverse.to A₁↞A₂ ⟨$⟩ x))\n B-to = record\n { _⟨$⟩_ = λ x → Equivalence.to B₁⇔B₂ ⟨$⟩\n P.subst (IndexedSetoid.Carrier B₁)\n (P.sym $ LeftInverse.left-inverse-of A₁↞A₂ _)\n x\n ; cong = F.cong (Equivalence.to B₁⇔B₂) ∘\n subst-cong (λ {x} → IndexedSetoid._≈_ B₁ {x} {x})\n (P.sym (LeftInverse.left-inverse-of A₁↞A₂ _))\n }\n\n B-from : ∀ {y} → _⟶_ (B₂ atₛ y) (B₁ atₛ (LeftInverse.from A₁↞A₂ ⟨$⟩ y))\n B-from = Equivalence.from B₁⇔B₂\n\n equivalence-↠ : {B₁ : IndexedSetoid A₁ b₁ b₁′} (B₂ : IndexedSetoid A₂ b₂ b₂′)\n (A₁↠A₂ : A₁ ↠ A₂) →\n (∀ {x} → Equivalence (B₁ atₛ x) (B₂ atₛ (Surjection.to A₁↠A₂ ⟨$⟩ x))) →\n Equivalence (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)\n equivalence-↠ {B₁ = B₁} B₂ A₁↠A₂ B₁⇔B₂ =\n equivalence (Surjection.equivalence A₁↠A₂) B-to B-from\n where\n B-to : ∀ {x} → _⟶_ (B₁ atₛ x) (B₂ atₛ (Surjection.to A₁↠A₂ ⟨$⟩ x))\n B-to = Equivalence.to B₁⇔B₂\n\n B-from : ∀ {y} → _⟶_ (B₂ atₛ y) (B₁ atₛ (Surjection.from A₁↠A₂ ⟨$⟩ y))\n B-from = record\n { _⟨$⟩_ = λ x → Equivalence.from B₁⇔B₂ ⟨$⟩\n P.subst (IndexedSetoid.Carrier B₂)\n (P.sym $ Surjection.right-inverse-of A₁↠A₂ _)\n x\n ; cong = F.cong (Equivalence.from B₁⇔B₂) ∘\n subst-cong (λ {x} → IndexedSetoid._≈_ B₂ {x} {x})\n (P.sym (Surjection.right-inverse-of A₁↠A₂ _))\n }\n\n injection : {B₁ : IndexedSetoid A₁ b₁ b₁′} (B₂ : IndexedSetoid A₂ b₂ b₂′) →\n (A₁↣A₂ : A₁ ↣ A₂) →\n (∀ {x} → Injection (B₁ atₛ x) (B₂ atₛ (Injection.to A₁↣A₂ ⟨$⟩ x))) →\n Injection (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)\n injection {B₁ = B₁} B₂ A₁↣A₂ B₁↣B₂ = record\n { to = to\n ; injective = inj\n }\n where\n to = ⟶ B₂ (Injection.to A₁↣A₂ ⟨$⟩_) (Injection.to B₁↣B₂)\n\n inj : Injective to\n inj (x , y) =\n Injection.injective A₁↣A₂ x ,\n lemma (Injection.injective A₁↣A₂ x) y\n where\n lemma :\n ∀ {x x′}\n {y : IndexedSetoid.Carrier B₁ x} {y′ : IndexedSetoid.Carrier B₁ x′} →\n x ≡ x′ →\n (eq : IndexedSetoid._≈_ B₂ (Injection.to B₁↣B₂ ⟨$⟩ y)\n (Injection.to B₁↣B₂ ⟨$⟩ y′)) →\n IndexedSetoid._≈_ B₁ y y′\n lemma P.refl = Injection.injective B₁↣B₂\n\n left-inverse : (B₁ : IndexedSetoid A₁ b₁ b₁′) {B₂ : IndexedSetoid A₂ b₂ b₂′} →\n (A₁↞A₂ : A₁ ↞ A₂) →\n (∀ {x} → LeftInverse (B₁ atₛ (LeftInverse.from A₁↞A₂ ⟨$⟩ x))\n (B₂ atₛ x)) →\n LeftInverse (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)\n left-inverse B₁ {B₂} A₁↞A₂ B₁↞B₂ = record\n { to = Equivalence.to eq\n ; from = Equivalence.from eq\n ; left-inverse-of = left\n }\n where\n eq = equivalence-↞ B₁ A₁↞A₂ (LeftInverse.equivalence B₁↞B₂)\n\n left : Equivalence.from eq LeftInverseOf Equivalence.to eq\n left (x , y) =\n LeftInverse.left-inverse-of A₁↞A₂ x ,\n IndexedSetoid.trans B₁\n (LeftInverse.left-inverse-of B₁↞B₂ _)\n (lemma (P.sym (LeftInverse.left-inverse-of A₁↞A₂ x)))\n where\n lemma :\n ∀ {x x′ y} (eq : x ≡ x′) →\n IndexedSetoid._≈_ B₁ (P.subst (IndexedSetoid.Carrier B₁) eq y) y\n lemma P.refl = IndexedSetoid.refl B₁\n\n surjection : {B₁ : IndexedSetoid A₁ b₁ b₁′} (B₂ : IndexedSetoid A₂ b₂ b₂′) →\n (A₁↠A₂ : A₁ ↠ A₂) →\n (∀ {x} → Surjection (B₁ atₛ x) (B₂ atₛ (Surjection.to A₁↠A₂ ⟨$⟩ x))) →\n Surjection (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)\n surjection B₂ A₁↠A₂ B₁↠B₂ = record\n { to = Equivalence.to eq\n ; surjective = record\n { from = Equivalence.from eq\n ; right-inverse-of = right\n }\n }\n where\n eq = equivalence-↠ B₂ A₁↠A₂ (Surjection.equivalence B₁↠B₂)\n\n right : Equivalence.from eq RightInverseOf Equivalence.to eq\n right (x , y) =\n Surjection.right-inverse-of A₁↠A₂ x ,\n IndexedSetoid.trans B₂\n (Surjection.right-inverse-of B₁↠B₂ _)\n (lemma (P.sym $ Surjection.right-inverse-of A₁↠A₂ x))\n where\n lemma : ∀ {x x′ y} (eq : x ≡ x′) →\n IndexedSetoid._≈_ B₂ (P.subst (IndexedSetoid.Carrier B₂) eq y) y\n lemma P.refl = IndexedSetoid.refl B₂\n\n -- See also Data.Product.Function.Dependent.Setoid.WithK.inverse.\n", "meta": {"hexsha": "54f5eaf5d8f9ad4366a1d6ad8f8ac63a10f9f408", "size": 7650, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/asset/agda-stdlib-1.0/Data/Product/Function/Dependent/Setoid.agda", "max_stars_repo_name": "omega12345/agda-mode", "max_stars_repo_head_hexsha": "0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "test/asset/agda-stdlib-1.0/Data/Product/Function/Dependent/Setoid.agda", "max_issues_repo_name": "omega12345/agda-mode", "max_issues_repo_head_hexsha": "0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "test/asset/agda-stdlib-1.0/Data/Product/Function/Dependent/Setoid.agda", "max_forks_repo_name": "omega12345/agda-mode", "max_forks_repo_head_hexsha": "0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.4422110553, "max_line_length": 81, "alphanum_fraction": 0.5533333333, "num_tokens": 2914, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190477, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3484376866260387}} {"text": "{-# OPTIONS --without-K --rewriting #-}\n\nmodule Sharp where\n open import Basics\n open import Flat\n\n open import lib.NType2\n open import lib.Equivalence2\n open import lib.types.Modality\n\n\n -- --------------------------------------------------------------------------------------\n -- Postulating the ♯ Modality\n -- --------------------------------------------------------------------------------------\n postulate\n -- Here, I've tried to implement the rules for ♯ as in Shulman as closely as possible\n -- to the sequents in Figure 3 of Section 3.\n\n -- I'm pretty convinced now that this approach will work\n -- However, as currently (02/18/19) rewrite rules do not see the ♭ modality,\n -- it is not fully operational (and, clearly, has not been cleaned up).\n\n -- The litmus is proving the lexness of ♯ along the lines of Shulman 3.7\n\n {-\n We implement the ♯ typing operations in two ways.\n\n First, naively, as follows:\n -}\n\n -- Rule 1/5: Given a type A, we may form ♯ A\n ♯ : {i : ULevel} → Type i → Type i\n\n -- Rule 2/5: Given an a : A, we may form a^♯ : ♯ A\n _^♯ : {i : ULevel} {A : Type i} → A → ♯ A\n\n -- Rule 3/5: Given a crisp a :: ♯ A, we may form a ↓♯ : A\n _↓♯ : {@♭ i : ULevel} {@♭ A : Type i} → ♯ A ::→ A\n\n {-\n Rule 4/5\n Given a :: A, we have that (a ^♯) ↓♯ ≡ a.\n\n Δ | ∙ ⊢ a : A\n -------------------------\n Δ | Γ ⊢ (a ^♯) ↓♯ ≡ a : A\n\n A and a are crisp, but we can have any other context Γ appear\n which is fine, we just don't mention it.\n -}\n ^↓♯ : {@♭ i : ULevel} {@♭ A : Type i} (@♭ a : A)\n → ((a ^♯) ↓♯) ↦ a\n {-# REWRITE ^↓♯ #-}\n\n {-\n Rule 5/5\n Given a : ♯ A in any context, we have (a ↓♯) ^♯ ≡ a\n\n Δ | Γ ⊢ a : ♯ A\n ---------------\n Δ | Γ ⊢ (a ↓♯) ^♯ ≡ a : ♯ A\n\n Here we can do this in any context, but for agda that might mean\n \"factoring through\" the ptwise operations first...\n\n [WARNING] This implementation needs A to be crisp, and so doesn't\n express the above sequent (I think...)\n -}\n ↓^♯ : {@♭ i : ULevel} {@♭ A : Type i} (@♭ a : ♯ A)\n → ((a ↓♯) ^♯) ↦ a\n {-# REWRITE ↓^♯ #-}\n\n {-\n Now, we will postulate \"pointwise\" versions of the above operations which \n take in an additional crisp \"context\" Γ or Δ. The name Γ or Δ is chosen to match\n the role of the explicit \"context\" with its analogy in Shulman.\n\n In each of these, the context either becomes cohesive or not, according to the\n rules from Shulman (which are reproduced here for comparison).\n -}\n \n {-\n Rule 1/5, pointwise version\n If one has a crisp context Γ and a type A which depends crisply on Γ,\n Then one may form ♯ A which depends cohesively on Γ.\n\n Or, read backwards, to construct ♯ A in the context Γ, it suffices to assume\n That every variable x : Γ appearing in A is crisp.\n\n Δ, Γ | ∙ ⊢ A : Type\n -------------------\n Δ | Γ ⊢ ♯ A : Type\n -}\n -- compare to ♯ : {i : ULevel} → Type i → Type i\n ♯-ptwise : {@♭ i : ULevel} {@♭ Γ : Type i} {j : ULevel}\n (A : Γ ::→ Type j)\n → (Γ → Type j)\n\n {-\n We relate the pointwise # with the naive one in the obvious way.\n\n That is, if we apply a pointwise operation to a crisp variable of the \"context\",\n this is the same as applying the naive operation pointwise to that variable.\n \n We take this sameness to be a judgemental equality, implemented as a rewrite rule.\n -}\n\n ♯-law : {@♭ i : ULevel} {@♭ Γ : Type i} {j : ULevel}\n (A : Γ ::→ Type j) (@♭ x : Γ)\n → (♯-ptwise A) x ↦ ♯ (A x)\n {-# REWRITE ♯-law #-}\n\n {-\n Rule 2/5, pointwise version\n If one has a : A in a crisp context Γ, \n one may form a ^♯ : ♯ A in the cohesive context Γ.\n\n Δ, Γ | ∙ ⊢ a : A\n -------------------\n Δ | Γ ⊢ a ^♯ : ♯ A\n -}\n ^♯-ptwise : {@♭ i : ULevel} {@♭ Γ : Type i} {j : ULevel}\n {A : Γ ::→ Type j} (a : (@♭ x : Γ) → A x)\n → (x : Γ) → (♯-ptwise A) x\n\n {-\n Relate the pointwise ^# with the naive one\n -} \n -- ^♯-law : {@♭ i : ULevel} {@♭ Γ : Type i} {j : ULevel}\n -- {A : Γ ::→ Type j} (a : (@♭ x : Γ) → A x)\n -- (@♭ x : Γ) → (^♯-ptwise a) x ↦ ((a x) ^♯)\n -- {-# REWRITE ^♯-law #-}\n\n {-\n Rule 3/5, pointwise version\n If one has a :: ♯ A in a crisp context Γ, \n one may form a ↓♯ : A, also in a crisp context Γ\n\n Δ | ∙ ⊢ a : ♯ A\n ---------------\n Δ | Γ ⊢ a ↓♯ : A\n\n It seems to me that to have a crisp variable of ♯ A requires A to be crisp...\n But this pointwise definition one doesn't. It just requires that the context \n is crisp.\n -}\n ↓♯-ptwise : {@♭ i : ULevel} {@♭ Δ : Type i} {j : ULevel} {A : Δ ::→ Type j}\n (a : (@♭ x : Δ) → (♯-ptwise A) x)\n → (@♭ x : Δ) → A x\n\n {-\n Rule 4/5, pointwise version\n -}\n ^↓♯-ptwise : {@♭ i : ULevel} {@♭ Γ : Type i} {j : ULevel}\n {A : Γ ::→ Type j} (a : (@♭ x : Γ) → A x) (@♭ x : Γ)\n → (((^♯-ptwise a) x) ↓♯) ↦ a x\n\n {-\n Rule 5/5, pointwise version\n I'm having a difficulty implementing the ptwise version of ↓^♯ that will fire\n whether or not A and a are crisp. \n I think this is because if they are, it is a special case of the ↓^♯ rule,\n But it is not strictly more general, since it keeps track of the \"explicit context\" Δ\n\n Or, if I take everything in ↓♯-ptwise to be crisp, then it doesn't typecheck at this \n generality.\n -}\n ↓^♯-ptwise : {@♭ i : ULevel} {j : ULevel} {@♭ Δ : Type i} {A : Δ ::→ Type j}\n (a : (@♭ x : Δ) → (♯-ptwise A) x) (@♭ x : Δ)\n → (((↓♯-ptwise a) x) ^♯) ↦ a x\n -- ^^^ : (♯-ptwise A) x\n -- ^^^^^^^^^^^ : (@♭ x : Δ) → A x\n -- ^^^^^^^^^^^^^^^^ : A x\n -- ^^^^^^^^^^^^^^^^^^^^^ : # (A x)\n {-# REWRITE ↓^♯-ptwise #-} \n \n -- [WARNING] When normalizing λ A x → (^♯-ptwise a) x, the rewrite ^♯-law will fire\n -- turning it into ((a x) ^♯), which is ill typed on cohesive x : Γ (and the typechecker \n -- complains)\n\n ↓♯-law : {@♭ i : ULevel} {@♭ Δ : Type i} {@♭ j : ULevel} {@♭ A : Δ ::→ Type j}\n (@♭ a : (@♭ x : Δ) → (♯-ptwise A) x)\n (@♭ x : Δ) → (↓♯-ptwise a) x ↦ ((a x) ↓♯)\n {-# REWRITE ↓♯-law #-}\n\n\n {-\n Finally, we define some convenient notation for the ptwise operations.\n \n To see how these work in practice, see below.\n -}\n syntax ♯-ptwise (λ γ → A) ctx = let♯ γ ::= ctx in♯-♯ A\n\n syntax ^♯-ptwise (λ γ → a) ctx = let♯ γ ::= ctx in♯ a ^^♯\n\n ^♯-ptwise-explicit : {@♭ i : ULevel} {@♭ Γ : Type i} {j : ULevel}\n (A : Γ ::→ Type j) (a : (@♭ x : Γ) → A x)\n → (x : Γ) → (♯-ptwise A) x\n ^♯-ptwise-explicit A = ^♯-ptwise {A = A}\n syntax ^♯-ptwise-explicit A (λ γ → t) ctx = let♯ γ ::= ctx in♯ t ^^♯-in-family A\n\n \n syntax ↓♯-ptwise (λ γ → a) ctx = let♯ γ ::= ctx in♯ a ↓↓♯\n\n -- ----------------------------------------------------------------------------------------------\n -- End of Postulates\n -- ----------------------------------------------------------------------------------------------\n\n\n -- We have to leave the universe levels out and assume they are crisp,\n -- otherwise the \"context record\" becomes large.\n -- It shouldn't matter tho, since ULevel is discrete.\n module _ {@♭ i j : ULevel} where\n private \n record Γ : Type (lsucc (lmax i j)) where\n constructor ctx\n field\n ᶜA : Type i\n ᶜB : Type j\n ᶜf : ᶜA → ᶜB\n ᶜa : ♯ ᶜA\n open Γ\n\n -- Functoriality of ♯\n ♯→ : {A : Type i} {B : Type j}\n (f : A → B) → (♯ A) → (♯ B)\n ♯→ {A} {B} f a =\n let♯ γ ::= (ctx A B f a) in♯\n (ᶜf γ (ᶜa γ ↓♯)) ^^♯ -- (f (a ↓♯)) ^♯\n\n -- The naturality square of the unit (judgemental!)\n ♯→-nat : {A : Type i} {B : Type j} (f : A → B)\n (a : A) → ((f a) ^♯) == ((♯→ f) (a ^♯))\n ♯→-nat {A} {B} f a = refl\n\n -- ♯-elmination (Shulman Theorem 3.4)\n {-\n The general form of these definitions is:\n Take the context (or the part of the context) you want to make crisp,\n and make a private record Γ with fields ᶜx for every variable x in the context.\n Then use the let♯ notation to crispify in the context.\n -}\n module _ {@♭ i j : ULevel} where\n private\n record Γ : Type (lsucc (lmax i j)) where\n constructor ctx\n field\n ᶜA : Type i\n ᶜB : (♯ ᶜA) → Type j\n ᶜf : (a : ᶜA) → ♯ (ᶜB (a ^♯))\n ᶜa : ♯ ᶜA\n open Γ\n \n ♯-elim : {A : Type i} (B : ♯ A → Type j)\n (f : (a : A) → ♯ (B (a ^♯)))\n → ((a : ♯ A) → ♯ (B a))\n ♯-elim {A} B f a =\n let♯ γ ::= (ctx A B f a) in♯\n (ᶜf γ (ᶜa γ ↓♯)) ↓♯ ^^♯\n syntax ♯-elim B (λ x → t) a = let♯ x ^♯:= a in♯ t in-family B\n\n -- Elimination with implicit family,\n ♯-elim' : {A : Type i} {B : ♯ A → Type j}\n (f : (a : A) → ♯ (B (a ^♯)))\n → ((a : ♯ A) → ♯ (B a))\n ♯-elim' {A} {B} f a = ♯-elim {A} B f a \n syntax ♯-elim' (λ x → t) a = let♯ x ^♯:= a in♯ t\n\n -- Crisp eliminators\n ♯-elim-crisp : {@♭ A : Type i} (@♭ B : ♯ A → Type j)\n (f : (@♭ a : A) → ♯ (B (a ^♯)))\n → (@♭ a : ♯ A) → ♯ (B a)\n ♯-elim-crisp B f a =\n let♯ ᶜf ::= f in♯ ((ᶜf (a ↓♯)) ↓♯) ^^♯\n syntax ♯-elim-crisp B (λ x → t) a = let♯ x ^♯::= a in♯ t in-family B\n \n -- β holds judgementally :)\n ♯-elim-β : {A : Type i} {B : ♯ A → Type j}\n (f : (a : A) → ♯ (B (a ^♯))) (a : A)\n → (♯-elim B f (a ^♯)) == (f a)\n ♯-elim-β f a = refl\n\n -- ♯-elim is inverse to precomposition by _^♯\n -- This proves that ♯ is a uniquely eliminating modality\n ♯-universal : {A : Type i} (B : ♯ A → Type j)\n → ((a : ♯ A) → ♯ (B a)) ≃ ((a : A) → ♯ (B (a ^♯)))\n ♯-universal {A} B = equiv to fro to-fro fro-to\n where\n to : (f : (a : ♯ A) → ♯ (B a))\n → (a : A) → ♯ (B (a ^♯))\n to f = f ∘ _^♯\n\n fro : ((a : A) → ♯ (B (a ^♯))) → ((a : ♯ A) → ♯ (B a))\n fro = ♯-elim B\n\n to-fro : (f : (a : A) → ♯ (B (a ^♯))) → to (fro f) == f\n to-fro f = refl\n\n fro-to : (f : (a : ♯ A) → ♯ (B a)) → fro (to f) == f\n fro-to f = refl\n\n -- A type is codiscrete if the inclusion a ↦ a ^♯ is an equivalence.\n _is-codiscrete : {i : ULevel} (A : Type i) → Type i\n A is-codiscrete = (_^♯ {A = A}) is-an-equiv\n\n codisc-eq : {i : ULevel} {A : Type i} (p : A is-codiscrete) → A ≃ (♯ A)\n codisc-eq = _^♯ ,_\n\n un♯ : {i : ULevel} {A : Type i} (p : A is-codiscrete) → ♯ A → A\n un♯ p = <– (codisc-eq p)\n\n _is-codisc-is-a-prop : {i : ULevel} (A : Type i) → (A is-codiscrete) is-a-prop\n A is-codisc-is-a-prop = is-equiv-is-prop\n\n -- Shulman Theorem 3.5\n -- ♯ A is codiscrete.\n module _ {@♭ i : ULevel} where\n private\n record Γ : Type (lsucc i) where\n constructor ctx\n field\n ᶜA : Type i\n ᶜa : ♯ (♯ ᶜA)\n open Γ\n \n ♯-is-codiscrete : (A : Type i) → (♯ A) is-codiscrete\n ♯-is-codiscrete = λ A → \n (_^♯ {A = ♯ A}) is-an-equivalence-because fro is-inverse-by to-fro and fro-to\n where\n fro : {A : Type i} → ♯ (♯ A) → ♯ A\n fro {A} a = let♯ γ ::= (ctx A a) in♯ ((ᶜa γ ↓♯) ↓♯) ^^♯ \n\n to-fro : {A : Type i} → (a : ♯ (♯ A)) → ((fro a) ^♯) == a\n to-fro a = refl\n\n fro-to : {A : Type i} → (a : ♯ A) → fro (a ^♯) == a\n fro-to a = refl\n\n {-\n module _ {@♭ i j : ULevel} {@♭ i j : Type i} where\n private\n record Γ : Type (lsucc (lmax i j)) where\n constructor ctx\n field\n ᶜA : Δ ::→ Type j\n ᶜx : Δ\n open Γ\n ♯-ptwise-is-codiscrete : (A : Δ ::→ Type j) (x : Δ)\n → ((♯-ptwise A) x) is-codiscrete\n ♯-ptwise-is-codiscrete = λ A x → \n {!(_^\\# {A = (♯-ptwise A) x}) is-an-equivalence-because fro is-inverse-by to-fro and fro-to!}\n where\n module _ {A : Δ ::→ Type j} {x : Δ} where\n fro : ♯ ((♯-ptwise A) x) → (♯-ptwise A) x\n fro = {!!}\n \n -- to-fro : (a : ♯ ((♯-ptwise A) x)) → ((fro a) ^♯) == a\n -- to-fro = {!!}\n\n -- fro-to : ∀ a → fro (a ^♯) == a\n -- fro-to = {!!}\n -}\n\n module _ {@♭ i j : ULevel} where\n record CTX-uncrisp {@♭ A : Type i} : Type (lsucc (lmax i j)) where\n constructor ctx\n field\n ᶜB : A → Type j\n ᶜf : (@♭ a : A) → ♯ (ᶜB a)\n ᶜa : A\n \n uncrisp : {@♭ A : Type i} (B : A → Type j)\n → ((@♭ a : A) → ♯ (B a))\n → ((a : A) → ♯ (B a))\n uncrisp B f a =\n let♯ γ ::= (ctx B f a) in♯\n (((ᶜf γ) (ᶜa γ)) ↓♯) ^^♯\n where open CTX-uncrisp\n\n Π-codisc : {@♭ i j : ULevel} {A : Type i} (B : A → Type j)\n → ((a : A) → ♯ (B a)) is-codiscrete\n Π-codisc {A = A} B =\n _^♯ is-an-equivalence-because\n (λ f a → let♯ g ^♯:= f in♯ (g a)) is-inverse-by\n (λ _ → refl) and (λ _ → refl)\n\n -- The map ♯ (x == y) → (x == y) for x y : ♯ A, following RSS Lemma 1.25\n module _ {@♭ i : ULevel} {A : Type i} {x y : ♯ A} where\n private\n constx : ♯ (x == y) → ♯ A\n constx _ = x\n \n consty : ♯ (x == y) → ♯ A\n consty _ = y\n\n lemma₀ : (constx ∘ _^♯) == (consty ∘ _^♯)\n lemma₀ = λ= (λ p → p)\n\n lemma₁ : constx == consty\n lemma₁ = -- constx == consty because they are equalized by _^♯, via the universal prop of ♯\n –>-is-inj (♯-universal (λ (_ : ♯ (x == y)) → A)) constx consty lemma₀\n\n ♯-=-retract : ♯ (x == y) → x == y\n ♯-=-retract p = app= lemma₁ p\n\n -- To prove a type is an equivalence, it suffices to give a retract of _^♯\n _is-codiscrete-because_is-retract-by_ : {@♭ i : ULevel} (A : Type i)\n (r : ♯ A → A) (p : (a : A) → r (a ^♯) == a)\n → A is-codiscrete\n A is-codiscrete-because r is-retract-by p =\n (_^♯ {A = A}) is-an-equivalence-because r is-inverse-by\n (λ a → -- Given an a : ♯ A, we will show ♯ ((r a)^♯ == a)\n (let♯ b ^♯:= a in♯ -- We suppose a is b ^♯\n ((ap _^♯ (p b)) ^♯) -- apply p to b to get r (b ^♯) == b,\n -- then apply _^♯ to get (r b^♯)^♯ == b^♯\n -- then hit it with _^♯ to get ♯ ((r b^♯)^♯ == b^♯)\n in-family (λ (a : ♯ A) → ((r a) ^♯) == a) -- which is ♯ ((r a)^♯ == a) by our hypothesis,\n ) -- and we can strip the ♯ becacuse equality types in ♯ are codiscrete.\n |> (♯-=-retract {x = (r a) ^♯} {a})\n )\n and p\n\n -- We follow RSS Lemma 1.25\n =-is-codiscrete : {@♭ i : ULevel} {A : Type i} (x y : ♯ A)\n → (x == y) is-codiscrete\n =-is-codiscrete {A = A} x y =\n (x == y) is-codiscrete-because ♯-=-retract is-retract-by proof\n where\n abstract -- UNFINISHED\n proof : (p : x == y) → (♯-=-retract (p ^♯)) == p\n proof = trust-me\n where postulate trust-me : (p : x == y) → (♯-=-retract (p ^♯)) == p\n\n ♯-modality : {@♭ i : ULevel} → Modality i\n ♯-modality {i} = record\n { is-local = _is-codiscrete\n ; is-local-is-prop = λ {A} → A is-codisc-is-a-prop\n ; ◯ = ♯\n ; ◯-is-local = λ {A} → ♯-is-codiscrete A\n ; η = _^♯\n ; ◯-elim = λ {A} {B} p f a → un♯ (p a) (♯-elim B (λ a → (f a) ^♯) a)\n ; ◯-elim-β = λ {A} {B} p f a → <–-inv-l (codisc-eq (p (a ^♯))) (f a) \n ; ◯-=-is-local = =-is-codiscrete\n }\n\n _is-infinitesimal : {@♭ i : ULevel} → Type i → Type i\n _is-infinitesimal = Modality.is-◯-connected ♯-modality\n\n ♯→e : {@♭ i : ULevel} {A B : Type i} → A ≃ B → (♯ A) ≃ (♯ B)\n ♯→e = Modality.◯-emap ♯-modality\n\n ♯-Σ : ∀ {@♭ i} {A : Type i} (B : A → Type i)\n → A is-codiscrete → ((a : A) → (B a) is-codiscrete)\n → (Σ A B) is-codiscrete\n ♯-Σ {i} = (Modality.Σ-is-local {i}) ♯-modality\n\n ♯-Π : ∀ {@♭ i} {A : Type i} {B : A → Type i} (w : (a : A) → (B a) is-codiscrete)\n → (Π A B) is-codiscrete\n ♯-Π {i} = (Modality.Π-is-local {i}) ♯-modality\n\n -- Theorem 6.22 of Shulman\n -- Points of ♯ A are the points of A, and ♯ of the points of A is ♯ A.\n ♭♯-eq : {@♭ i : ULevel} {@♭ A : Type i} → ♭ (♯ A) ≃ ♭ A\n ♭♯-eq {A = A} = equiv to fro to-fro fro-to\n where\n to : ♭ (♯ A) → ♭ A\n to (a ^♭) = (a ↓♯) ^♭\n\n fro : ♭ A → ♭ (♯ A)\n fro (a ^♭) = (a ^♯) ^♭\n\n to-fro : (a : ♭ A) → to (fro a) == a\n to-fro (a ^♭) = refl\n\n fro-to : (a : ♭ (♯ A)) → fro (to a) == a\n fro-to (a ^♭) = ♭-ap _^♭ refl\n \n ♯♭-eq : {@♭ i : ULevel} {@♭ A : Type i} → ♯ (♭ A) ≃ ♯ A\n ♯♭-eq {A = A} = equiv to fro to-fro fro-to\n where\n to : ♯ (♭ A) → ♯ A\n to a =\n let♯ u ^♯:= a in♯\n let♭ v ^♭:= u in♭ (v ^♯)\n\n fro : ♯ A → ♯ (♭ A)\n fro a =\n let♯ u ^♯:= a in♯ \n let♯ v ::= u in♯ (v ^♭) ^^♯\n \n abstract \n to-fro : (a : ♯ A) → to (fro a) == a\n to-fro a = -- It suffices to show ♯ (to fro a == a),\n (let♯ u ^♯:= a in♯ -- which lets us assume a = u^♯\n refl ^♯ -- so that the equality follows judgementally.\n in-family (λ a → to (fro a) == a))\n |> ♯-=-retract\n\n fro-to : (a : ♯ (♭ A)) → fro (to a) == a\n fro-to a = -- It suffices to show ♯ (fro to a == a),\n (let♯ u ^♯:= a in♯ -- which lets us assume a = u^♯ with u : ♭ A,\n (let♭ v ^♭:= u in♭ -- so we can assume u = v^♭,\n refl ^♯ -- so that the equality follows judgementally.\n in-family (λ u → ♯ (fro (to (u ^♯)) == (u ^♯))))\n in-family (λ a → fro (to a) == a))\n |> ♯-=-retract\n\n -- Theorem 6.27 of Shulman\n -- The adjunction between ♭ and ♯\n ♭♯-adjoint : {@♭ i j : ULevel} {@♭ A : Type i} {@♭ B : A → Type j}\n → ♭ ((a : ♭ A) → B (a ↓♭)) ≃ ♭ ((a : A) → ♯ (B a))\n ♭♯-adjoint {A = A} {B = B} = equiv to fro to-fro fro-to\n where\n to : ♭ ((a : ♭ A) → B (a ↓♭)) → ♭ ((a : A) → ♯ (B a))\n to (f ^♭) = (λ a → let♯ u ::= a in♯ f (u ^♭) ^^♯) ^♭\n\n fro : ♭ ((a : A) → ♯ (B a)) → ♭ ((a : ♭ A) → B (a ↓♭))\n fro (f ^♭) = (λ a → let♭ u ^♭:= a in♭ ((f u) ↓♯) in-family (λ a → B (a ↓♭))) ^♭\n\n to-fro : ∀ f → to (fro f) == f\n to-fro (f ^♭) = refl\n\n fro-to : ∀ f → fro (to f) == f\n fro-to (f ^♭) =\n ♭-ap _^♭ -- We can strip ^♭ from both sides,\n ( λ= (λ a → let♭ u ^♭:= a in♭ -- then, working at a crisp argument u,\n refl -- we find both sides are judgementally the same.\n in-family (λ a → ♭-elim (λ a₁ → B (a₁ ↓♭)) (λ (@♭ u : _) → f (u ^♭)) a == f a) )\n ) -- The \"in family\" gibberish just reminds agda what we are trying to prove. \n\n -- Shulman Theorem 3.7\n -- ♯ is left exact, in that x^♯ == y^♯ is ♯ (x == y)\n module _ {@♭ i : ULevel} where\n private\n record CTX-code : Type (lsucc i) where\n constructor ctx\n field\n ᶜA : Type i\n ᶜx : ♯ ᶜA\n ᶜy : ♯ ᶜA\n\n code : {A : Type i} → ♯ A → ♯ A → Type i\n code {A} x y =\n let♯ γ ::= (ctx A x y) in♯-♯ (((ᶜx γ) ↓♯) == ((ᶜy γ) ↓♯))\n where open CTX-code\n\n private\n record CTX-r : Type (lsucc i) where\n constructor ctx\n field\n ᶜA : Type i\n ᶜx : ♯ ᶜA\n \n r : {A : Type i} (x : ♯ A) → code x x\n r {A} x =\n let♯ γ ::= (ctx A x) in♯ (idp {a = (ᶜx γ) ↓♯}) ^^♯\n where open CTX-r\n -- for some reason, I can't use refl here, I need idp???\n \n encode : {A : Type i} {a b : ♯ A} → (a == b) → code a b\n encode {A} {a} {b} p = transport (λ y → code a y) p (r a)\n\n decode : {A : Type i} {a b : ♯ A} → code a b → (a == b)\n decode {A} {a} {b} = -- It suffices to give ♯ (code a b → a == b) by lemma.\n lemma a b (decode' a b)\n where\n lemma : {A : Type i} (a b : ♯ A)\n → ♯ (code a b → (a == b))\n → code a b → (a == b)\n lemma a b p e =\n ♯-=-retract $ -- it suffices to show ♯ (a == b)\n let♯ q ^♯:= p in♯ ((q e) ^♯) -- so we can assume p is of the form q ^♯.\n \n decode' : {A : Type i} (a b : ♯ A) → ♯ (code a b → (a == b))\n decode' {A} a b =\n let♯ u ^♯:= a in♯ -- By ♯-elim, we can assume a and b are of the form\n let♯ v ^♯:= b in♯ -- u ^♯ and v ^♯, and we'll give\n -- code (u ^♯) (v ^♯) → (u ^♯) == (v ^♯).\n (λ p → ♯-=-retract -- Assuming a code p, it suffices to give\n -- ♯ (u ^♯ == v ^♯).\n (let♯ q ^♯:= p in♯ ((ap _^♯ q)^♯)) )^♯ -- So, we let p be q^♯\n -- with q : u == v, and then\n -- push this through the unit.\n in-family (λ b' → code (u ^♯) b' → (u ^♯) == b')\n in-family (λ a' → code a' b → a' == b)\n\n \n\n private -- context for encode-decode'\n record CTX-encode-decode : Type (lsucc i) where\n constructor ctx\n field\n ᶜA : Type i\n ᶜa : ♯ ᶜA\n ᶜb : ♯ ᶜA\n -- ᶜp : code ᶜa ᶜb\n open CTX-encode-decode\n {-\n encode-decode' : {A : Type i} {a b : ♯ A} → ♯ ((encode {a = a}{b = b})∘ decode == (idf (code a b)))\n encode-decode' {A} {a} {b} =\n let♯ γ ::= (ctx A a b) in♯\n {!let♯ u ^♯:= (ᶜa γ) in♯\n ?\n in-family (λ a' → (encode {a = a'}{b = ᶜb γ})∘ decode == (idf (code a' (ᶜb γ))))!}\n ^^♯-in-family (λ γ' → (encode {a = ᶜa γ'}{b = ᶜb γ'})∘ decode == (idf (code (ᶜa γ') (ᶜb γ'))))\n \n -- [WARNING] I get an error here complaining about A in (a ↓♯) == (b ↓♯) if doing this at\n -- p : code a b (I believe this to be because rewrite rules do not see the ♭ modality)\n encode-decode : {A : Type i} {a b : ♯ A} → (encode {a = a}{b = b})∘ decode == (idf (code a b))\n encode-decode {A} {a} {b} =\n λ= (λ p → {!!})\n\n -}\n \n\n -- For now, we'll just postulate it\n ♯-=-compare : {@♭ i : ULevel} {A : Type i} {x y : A}\n → ♯ (x == y) → (x ^♯) == (y ^♯)\n ♯-=-compare {x = x} {y = y} p =\n ♯-=-retract $ -- it suffices to give ♯ (x ^♯ == y ^♯)\n let♯ q ^♯:= p in♯ -- in which case we may assume p = q ^♯ with q : x == y\n ((ap _^♯ q) ^♯) -- and so we can push this through.\n module _ {@♭ i : ULevel} where\n private\n record CTX-♯-lex : Type (lsucc i) where\n constructor ctx\n field\n ᶜA : Type i\n ᶜx : ᶜA\n ᶜy : ᶜA\n postulate\n ♯-lex : {A : Type i} {x y : A}\n → (♯-=-compare {x = x} {y = y}) is-an-equiv\n\n ♯-lex-eq : {A : Type i} {x y : A}\n → (♯ (x == y)) ≃ ((x ^♯) == (y ^♯))\n ♯-lex-eq {A}{x}{y} = ♯-=-compare , ♯-lex\n\n test : {A : Type i} {x y : ♯ A}\n → ♯ ((x ^♯) == (y ^♯) → ♯ (x == y))\n test {A} {x} {y} =\n let♯ γ ::= (ctx (♯ A) x y) in♯\n (λ p → let♯ q ::= p in♯\n (♭-ap _↓♯ q)\n ^^♯-in-family (λ _ → (ᶜx γ == ᶜy γ)))\n ^^♯-in-family (λ γ → ((ᶜx γ) ^♯) == ((ᶜy γ) ^♯) → ♯ ((ᶜx γ) == (ᶜy γ)))\n where open CTX-♯-lex\n\n ♯-has-level-is-codisc : {@♭ i : ULevel} {A : Type i}\n {n : ℕ₋₂} → (has-level n (♯ A)) is-codiscrete\n ♯-has-level-is-codisc {i} {A} {n} = replete helper (has-level-def-eq ⁻¹)\n where\n replete = (Modality.local-is-replete {i}) ♯-modality\n\n helper : {A : Type i}\n {n : ℕ₋₂} → (has-level-aux n (♯ A)) is-codiscrete\n helper {A} {⟨-2⟩} =\n ♯-Σ (λ x → (y : ♯ A) → x == y) (♯-is-codiscrete A) $\n λ x → ♯-Π (λ y → =-is-codiscrete x y) \n\n helper {A} {n = S n} =\n ♯-Π (λ x →\n ♯-Π (λ y →\n replete (replete (helper {A = (x == y)} {n}) (has-level-def-eq ⁻¹))\n (≃-preserves-level-eq ((codisc-eq $ =-is-codiscrete x y) ⁻¹))))\n\n \n ♯-preserves-level : {@♭ i : ULevel} {A : Type i}\n {n : ℕ₋₂} (p : has-level n A)\n → has-level n (♯ A)\n ♯-preserves-level {i} {A} {⟨-2⟩} p =\n has-level-in ( ((contr-center p)^♯) ,\n (λ y → ♯-=-retract $ -- It suffices to prove ♯ (center == y)\n let♯ u ^♯:= y in♯ -- so we can assume y is u ^♯\n ap _^♯ (contr-path p u) ^♯ -- and then apply _^♯ to the contractibility of A.\n in-family (λ y → ((contr-center p) ^♯) == y))\n )\n ♯-preserves-level {i} {A} {S n} p =\n has-level-in\n (λ x y → -- Given x y : ♯ A, we need to show that x == y has level n.\n ≃-preserves-level ((codisc-eq (=-is-codiscrete x y))⁻¹) lemma \n ) -- since x == y is ♯ (x == y), it will suffice to show that has level n.\n where\n lemma : {x y : ♯ A} → has-level n (♯ (x == y))\n lemma {x} {y} = -- Since has-level is codiscrete on codiscretes,\n un♯ ♯-has-level-is-codisc $ -- we can let x and y be u^♯ and v^♯.\n let♯ u ^♯:= x in♯ -- Then, we use the lex-ness of ♯ and a bit of jiggling\n let♯ v ^♯:= y in♯ -- to show that we might as well prove the result of\n (≃-preserves-level (e u v) -- ♯ (u == v), which we can do by recursing.\n (♯-preserves-level (has-level-apply p u v)))^♯\n in-family (λ y → has-level n (♯ ((u ^♯) == y)))\n in-family (λ x → has-level n (♯ (x == y)))\n where\n e : (u v : A) → ♯ (u == v) ≃ ♯ ((u ^♯) == (v ^♯))\n e u v = ♯ (u == v)\n ≃⟨ codisc-eq (♯-is-codiscrete (u == v)) ⟩\n ♯ (♯ (u == v))\n ≃⟨ ♯→e ♯-lex-eq ⟩\n ♯ ((u ^♯) == (v ^♯))\n ≃∎\n\n ♯ₙ : {@♭ i : ULevel} {n : ℕ₋₂}\n → (A : n -Type i) → (n -Type i)\n ♯ₙ A = (♯ (fst A)) , ♯-preserves-level (snd A)\n", "meta": {"hexsha": "fc83b3365b4cb67214ec806cf65cd6d06521bd6a", "size": 26043, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "cohesion/david_jaz_261/Sharp.agda", "max_stars_repo_name": "glangmead/formalization", "max_stars_repo_head_hexsha": "497e720a1ddaa2ec713c060f999f4b3ee2fe5e8a", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-10-06T17:39:22.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-13T05:51:12.000Z", "max_issues_repo_path": "cohesion/david_jaz_261/Sharp.agda", "max_issues_repo_name": 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YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3484376866260385}} {"text": "------------------------------------------------------------------------\n-- The Agda standard library\n--\n-- Properties of operations on the Stream type\n------------------------------------------------------------------------\n\n{-# OPTIONS --without-K --safe --sized-types #-}\n\nmodule Codata.Stream.Properties where\n\nopen import Level using (Level)\nopen import Size\nopen import Codata.Thunk as Thunk using (Thunk; force)\nopen import Codata.Stream\nopen import Codata.Stream.Bisimilarity\n\nopen import Data.Nat.Base\nopen import Data.Nat.GeneralisedArithmetic using (fold; fold-pull)\n\nopen import Data.List.Base as List using ([]; _∷_)\nopen import Data.List.NonEmpty as List⁺ using (List⁺; _∷_)\nimport Data.List.Relation.Binary.Equality.Propositional as Eq\nopen import Data.Product as Prod using (_,_)\nopen import Data.Vec.Base as Vec using (_∷_)\n\nopen import Function\nopen import Relation.Binary.PropositionalEquality as P using (_≡_; _≢_)\n\nprivate\n variable\n a b c : Level\n A : Set a\n B : Set b\n C : Set c\n i : Size\n\n------------------------------------------------------------------------\n-- repeat\n\nlookup-repeat-identity : (n : ℕ) (a : A) → lookup n (repeat a) ≡ a\nlookup-repeat-identity zero a = P.refl\nlookup-repeat-identity (suc n) a = lookup-repeat-identity n a\n\ntake-repeat-identity : (n : ℕ) (a : A) → take n (repeat a) ≡ Vec.replicate a\ntake-repeat-identity zero a = P.refl\ntake-repeat-identity (suc n) a = P.cong (a Vec.∷_) (take-repeat-identity n a)\n\nsplitAt-repeat-identity : (n : ℕ) (a : A) → splitAt n (repeat a) ≡ (Vec.replicate a , repeat a)\nsplitAt-repeat-identity zero a = P.refl\nsplitAt-repeat-identity (suc n) a = P.cong (Prod.map₁ (a ∷_)) (splitAt-repeat-identity n a)\n\nreplicate-repeat : ∀ {i} (n : ℕ) (a : A) → i ⊢ List.replicate n a ++ repeat a ≈ repeat a\nreplicate-repeat zero a = refl\nreplicate-repeat (suc n) a = P.refl ∷ λ where .force → replicate-repeat n a\n\ncycle-replicate : ∀ {i} (n : ℕ) (n≢0 : n ≢ 0) (a : A) → i ⊢ cycle (List⁺.replicate n n≢0 a) ≈ repeat a\ncycle-replicate {i} n n≢0 a = let as = List⁺.replicate n n≢0 a in begin\n cycle as ≡⟨⟩\n as ⁺++ _ ≈⟨ ⁺++⁺ Eq.≋-refl (λ where .force → cycle-replicate n n≢0 a) ⟩\n as ⁺++ (λ where .force → repeat a) ≈⟨ P.refl ∷ (λ where .force → replicate-repeat (pred n) a) ⟩\n repeat a ∎ where open ≈-Reasoning\n\nmodule _ {a b} {A : Set a} {B : Set b} where\n\n map-repeat : ∀ (f : A → B) a {i} → i ⊢ map f (repeat a) ≈ repeat (f a)\n map-repeat f a = P.refl ∷ λ where .force → map-repeat f a\n\n ap-repeat : ∀ (f : A → B) a {i} → i ⊢ ap (repeat f) (repeat a) ≈ repeat (f a)\n ap-repeat f a = P.refl ∷ λ where .force → ap-repeat f a\n\n ap-repeatˡ : ∀ (f : A → B) as {i} → i ⊢ ap (repeat f) as ≈ map f as\n ap-repeatˡ f (a ∷ as) = P.refl ∷ λ where .force → ap-repeatˡ f (as .force)\n\n ap-repeatʳ : ∀ (fs : Stream (A → B) ∞) (a : A) {i} → i ⊢ ap fs (repeat a) ≈ map (_$ a) fs\n ap-repeatʳ (f ∷ fs) a = P.refl ∷ λ where .force → ap-repeatʳ (fs .force) a\n\n map-++ : ∀ {i} (f : A → B) as xs → i ⊢ map f (as ++ xs) ≈ List.map f as ++ map f xs\n map-++ f [] xs = refl\n map-++ f (a ∷ as) xs = P.refl ∷ λ where .force → map-++ f as xs\n\n map-⁺++ : ∀ {i} (f : A → B) as xs → i ⊢ map f (as ⁺++ xs) ≈ List⁺.map f as ⁺++ Thunk.map (map f) xs\n map-⁺++ f (a ∷ as) xs = P.refl ∷ (λ where .force → map-++ f as (xs .force))\n\n map-cycle : ∀ {i} (f : A → B) as → i ⊢ map f (cycle as) ≈ cycle (List⁺.map f as)\n map-cycle f as = begin\n map f (cycle as) ≈⟨ map-⁺++ f as _ ⟩\n List⁺.map f as ⁺++ _ ≈⟨ ⁺++⁺ Eq.≋-refl (λ where .force → map-cycle f as) ⟩\n cycle (List⁺.map f as) ∎ where open ≈-Reasoning\n\n------------------------------------------------------------------------\n-- Functor laws\n\nmap-identity : ∀ (as : Stream A ∞) → i ⊢ map id as ≈ as\nmap-identity (a ∷ as) = P.refl ∷ λ where .force → map-identity (as .force)\n\nmap-map-fusion : ∀ (f : A → B) (g : B → C) as → i ⊢ map g (map f as) ≈ map (g ∘ f) as\nmap-map-fusion f g (a ∷ as) = P.refl ∷ λ where .force → map-map-fusion f g (as .force)\n\n\n------------------------------------------------------------------------\n-- splitAt\n\nsplitAt-map : ∀ n (f : A → B) xs →\n splitAt n (map f xs) ≡ Prod.map (Vec.map f) (map f) (splitAt n xs)\nsplitAt-map zero f xs = P.refl\nsplitAt-map (suc n) f (x ∷ xs) =\n P.cong (Prod.map₁ (f x Vec.∷_)) (splitAt-map n f (xs .force))\n\n------------------------------------------------------------------------\n-- iterate\n\nlookup-iterate-identity : ∀ n f (a : A) → lookup n (iterate f a) ≡ fold a f n\nlookup-iterate-identity zero f a = P.refl\nlookup-iterate-identity (suc n) f a = begin\n lookup (suc n) (iterate f a) ≡⟨⟩\n lookup n (iterate f (f a)) ≡⟨ lookup-iterate-identity n f (f a) ⟩\n fold (f a) f n ≡⟨ fold-pull (const ∘′ f) (f a) P.refl (λ _ → P.refl) n ⟩\n f (fold a f n) ≡⟨⟩\n fold a f (suc n) ∎ where open P.≡-Reasoning\n\n------------------------------------------------------------------------\n-- DEPRECATED NAMES\n------------------------------------------------------------------------\n-- Please use the new names as continuing support for the old names is\n-- not guaranteed.\n\n-- Version 1.1\n\nrepeat-ap-identity = ap-repeatˡ\n{-# WARNING_ON_USAGE repeat-ap-identity\n\"Warning: repeat-ap-identity was deprecated in v1.1.\nPlease use ap-repeatˡ instead.\"\n#-}\n\nap-repeat-identity = ap-repeatʳ\n{-# WARNING_ON_USAGE ap-repeat-identity\n\"Warning: ap-repeat-identity was deprecated in v1.1.\nPlease use ap-repeatʳ instead.\"\n#-}\n\nap-repeat-commute = ap-repeat\n{-# WARNING_ON_USAGE ap-repeat-commute\n\"Warning: ap-repeat-commute was deprecated in v1.1.\nPlease use ap-repeat instead.\"\n#-}\n\nmap-repeat-commute = map-repeat\n{-# WARNING_ON_USAGE map-repeat-commute\n\"Warning: map-repeat-commute was deprecated in v1.1.\nPlease use map-repeat instead.\"\n#-}\n", "meta": {"hexsha": 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YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.34839763850849137}} {"text": "module PiFrac.Properties where\nopen import Data.Empty\nopen import Data.Unit hiding (_≟_)\nopen import Data.Sum\nopen import Data.Product\nopen import Relation.Binary.Core\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Maybe\nopen import PiFrac.Syntax\nopen import PiFrac.Opsem\nopen import PiFrac.AuxLemmas\nopen import PiFrac.NoRepeat\nopen import PiFrac.Eval\nopen import PiFrac.Interp\nopen import PiFrac.Invariants\n\n-- Forward evaluator is reversible\nevalIsRev : ∀ {A B} → (c : A ↔ B) (v₁ : ⟦ A ⟧) (v₂ : ⟦ B ⟧)\n → eval c v₁ ≡ (just v₂) → evalᵣₑᵥ c v₂ ≡ (just v₁)\nevalIsRev c v₁ v₂ eq with ev {κ = ☐} c v₁\nevalIsRev c v₁ v₂ refl | inj₁ (v₂ , rs) with evᵣₑᵥ {κ = ☐} c v₂\n... | inj₁ (v₁' , rs') with deterministicᵣₑᵥ* rs' (Rev↦ rs) (λ ()) (λ ()) (λ ())\n... | refl = refl\nevalIsRev c v₁ v₂ refl | inj₁ (v₂ , rs) | inj₂ (_ , v , _ , _ , neq , rs') with deterministicᵣₑᵥ* rs' (Rev↦ rs) (λ ()) (Lemma₆ neq) (λ ())\n... | ()\n\n-- Backward evaluator is reversible\nevalᵣₑᵥIsRev : ∀ {A B} → (c : A ↔ B) (v₁ : ⟦ B ⟧) (v₂ : ⟦ A ⟧)\n → evalᵣₑᵥ c v₁ ≡ (just v₂) → eval c v₂ ≡ (just v₁)\nevalᵣₑᵥIsRev c v₁ v₂ eq with evᵣₑᵥ {κ = ☐} c v₁\nevalᵣₑᵥIsRev c v₁ v₂ refl | inj₁ (v₂ , rs) with ev {κ = ☐} c v₂\n... | inj₁ (_ , rs') with deterministic* rs' (Rev↦ᵣₑᵥ rs) (λ ()) (λ ())\n... | refl = refl\nevalᵣₑᵥIsRev c v₁ v₂ refl | inj₁ (v₂ , rs) | inj₂ rs' with deterministic* rs' (Rev↦ᵣₑᵥ rs) (λ ()) (λ ())\n... | ()\n\n-- The abstract machine semantics is equivalent to the big-step semantics\neval≡interp : ∀ {A B} → (c : A ↔ B) → (v : ⟦ A ⟧) → eval c v ≡ interp c v\neval≡interp uniti₊l v = refl\neval≡interp unite₊l (inj₂ v) = refl\neval≡interp swap₊ (inj₁ x) = refl\neval≡interp swap₊ (inj₂ y) = refl\neval≡interp assocl₊ (inj₁ x) = refl\neval≡interp assocl₊ (inj₂ (inj₁ y)) = refl\neval≡interp assocl₊ (inj₂ (inj₂ z)) = refl\neval≡interp assocr₊ (inj₁ (inj₁ x)) = refl\neval≡interp assocr₊ (inj₁ (inj₂ y)) = refl\neval≡interp assocr₊ (inj₂ z) = refl\neval≡interp unite⋆l (tt , v) = refl\neval≡interp uniti⋆l v = refl\neval≡interp swap⋆ (x , y) = refl\neval≡interp assocl⋆ (x , (y , z)) = refl\neval≡interp assocr⋆ ((x , y) , z) = refl\neval≡interp dist (inj₁ x , z) = refl\neval≡interp dist (inj₂ y , z) = refl\neval≡interp factor (inj₁ (x , z)) = refl\neval≡interp factor (inj₂ (y , z)) = refl\neval≡interp id↔ v = refl\neval≡interp (c₁ ⨾ c₂) v with ev {κ = ☐} c₁ v | inspect (ev {κ = ☐} c₁) v\neval≡interp (c₁ ⨾ c₂) v | inj₁ (v' , rs) | [ eq ] with ev {κ = ☐} c₂ v' | inspect (ev {κ = ☐} c₂) v'\neval≡interp (c₁ ⨾ c₂) v | inj₁ (v' , rs) | [ eq ] | inj₁ (v'' , rs') | [ eq' ] with ev {κ = ☐} (c₁ ⨾ c₂) v | inspect (ev {κ = ☐} (c₁ ⨾ c₂)) v\neval≡interp (c₁ ⨾ c₂) v | inj₁ (v' , rs) | [ eq ] | inj₁ (v'' , rs') | [ eq' ] | inj₁ (u , rs'') | [ eq'' ] rewrite (sym (eval≡interp c₁ v)) | eq | (sym (eval≡interp c₂ v')) | eq' with deterministic* ((↦₃ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (☐⨾ c₂ • ☐) ++↦ (↦₇ ∷ ◾) ++↦ appendκ↦* rs' (λ ()) (λ ()) refl (c₁ ⨾☐• ☐) ++↦ (↦₁₀ ∷ ◾)) rs'' (λ ()) (λ ())\n... | refl = refl\neval≡interp (c₁ ⨾ c₂) v | inj₁ (v' , rs) | [ eq ] | inj₁ (v'' , rs') | [ eq' ] | inj₂ rs'' | [ eq'' ] rewrite (sym (eval≡interp c₁ v)) | eq | (sym (eval≡interp c₂ v')) | eq' with deterministic* ((↦₃ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (☐⨾ c₂ • ☐) ++↦ (↦₇ ∷ ◾) ++↦ appendκ↦* rs' (λ ()) (λ ()) refl (c₁ ⨾☐• ☐) ++↦ (↦₁₀ ∷ ◾)) rs'' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⨾ c₂) v | inj₁ (v' , rs) | [ eq ] | inj₂ rs' | [ eq' ] with ev {κ = ☐} (c₁ ⨾ c₂) v | inspect (ev {κ = ☐} (c₁ ⨾ c₂)) v\neval≡interp (c₁ ⨾ c₂) v | inj₁ (v' , rs) | [ eq ] | inj₂ rs' | [ eq' ] | inj₁ (u , rs'') | [ eq'' ] rewrite (sym (eval≡interp c₁ v)) | eq | (sym (eval≡interp c₂ v')) | eq' with deterministic* ((↦₃ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (☐⨾ c₂ • ☐) ++↦ (↦₇ ∷ ◾) ++↦ appendκ↦*⊠ rs' (λ ()) (c₁ ⨾☐• ☐)) rs'' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⨾ c₂) v | inj₁ (v' , rs) | [ eq ] | inj₂ rs' | [ eq' ] | inj₂ rs'' | [ eq'' ] rewrite (sym (eval≡interp c₁ v)) | eq | (sym (eval≡interp c₂ v')) | eq' = refl\neval≡interp (c₁ ⨾ c₂) v | inj₂ rs | [ eq ] with ev {κ = ☐} (c₁ ⨾ c₂) v | inspect (ev {κ = ☐} (c₁ ⨾ c₂)) v\neval≡interp (c₁ ⨾ c₂) v | inj₂ rs | [ eq ] | inj₁ (v'' , rs') | [ eq' ] with deterministic* ((↦₃ ∷ ◾) ++↦ appendκ↦*⊠ rs (λ ()) (☐⨾ c₂ • ☐)) rs' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⨾ c₂) v | inj₂ rs | [ eq ] | inj₂ rs' | [ eq' ] rewrite (sym (eval≡interp c₁ v)) | eq = refl\neval≡interp (c₁ ⊕ c₂) (inj₁ x) with ev {κ = ☐} c₁ x | inspect (ev {κ = ☐} c₁) x\neval≡interp (c₁ ⊕ c₂) (inj₁ x) | inj₁ (x' , rs) | [ eq ] with ev {κ = ☐} (c₁ ⊕ c₂) (inj₁ x) | inspect (ev {κ = ☐} (c₁ ⊕ c₂)) (inj₁ x)\neval≡interp (c₁ ⊕ c₂) (inj₁ x) | inj₁ (x' , rs) | [ eq ] | inj₁ (x'' , rs') | [ eq' ] rewrite (sym (eval≡interp c₁ x)) | eq with deterministic* ((↦₄ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (☐⊕ c₂ • ☐) ++↦ (↦₁₁ ∷ ◾)) rs' (λ ()) (λ ())\n... | refl = refl\neval≡interp (c₁ ⊕ c₂) (inj₁ x) | inj₁ (x' , rs) | [ eq ] | inj₂ rs' | [ eq' ] rewrite (sym (eval≡interp c₁ x)) | eq with deterministic* ((↦₄ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (☐⊕ c₂ • ☐) ++↦ (↦₁₁ ∷ ◾)) rs' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⊕ c₂) (inj₁ x) | inj₂ rs | [ eq ] with ev {κ = ☐} (c₁ ⊕ c₂) (inj₁ x) | inspect (ev {κ = ☐} (c₁ ⊕ c₂)) (inj₁ x)\neval≡interp (c₁ ⊕ c₂) (inj₁ x) | inj₂ rs | [ eq ] | inj₁ (x'' , rs') | [ eq' ] rewrite (sym (eval≡interp c₁ x)) | eq with deterministic* ((↦₄ ∷ ◾) ++↦ appendκ↦*⊠ rs (λ ()) (☐⊕ c₂ • ☐)) rs' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⊕ c₂) (inj₁ x) | inj₂ rs | [ eq ] | inj₂ rs' | [ eq' ] rewrite (sym (eval≡interp c₁ x)) | eq = refl\neval≡interp (c₁ ⊕ c₂) (inj₂ y) with ev {κ = ☐} c₂ y | inspect (ev {κ = ☐} c₂) y\neval≡interp (c₁ ⊕ c₂) (inj₂ y) | inj₁ (y' , rs) | [ eq ] with ev {κ = ☐} (c₁ ⊕ c₂) (inj₂ y) | inspect (ev {κ = ☐} (c₁ ⊕ c₂)) (inj₂ y)\neval≡interp (c₁ ⊕ c₂) (inj₂ y) | inj₁ (y' , rs) | [ eq ] | inj₁ (y'' , rs') | [ eq' ] rewrite (sym (eval≡interp c₂ y)) | eq with deterministic* ((↦₅ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (c₁ ⊕☐• ☐) ++↦ (↦₁₂ ∷ ◾)) rs' (λ ()) (λ ())\n... | refl = refl\neval≡interp (c₁ ⊕ c₂) (inj₂ y) | inj₁ (y' , rs) | [ eq ] | inj₂ rs' | [ eq' ] rewrite (sym (eval≡interp c₂ y)) | eq with deterministic* ((↦₅ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (c₁ ⊕☐• ☐) ++↦ (↦₁₂ ∷ ◾)) rs' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⊕ c₂) (inj₂ y) | inj₂ rs | [ eq ] with ev {κ = ☐} (c₁ ⊕ c₂) (inj₂ y) | inspect (ev {κ = ☐} (c₁ ⊕ c₂)) (inj₂ y)\neval≡interp (c₁ ⊕ c₂) (inj₂ y) | inj₂ rs | [ eq ] | inj₁ (y'' , rs') | [ eq' ] rewrite (sym (eval≡interp c₂ y)) | eq with deterministic* ((↦₅ ∷ ◾) ++↦ appendκ↦*⊠ rs (λ ()) (c₁ ⊕☐• ☐)) rs' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⊕ c₂) (inj₂ y) | inj₂ rs | [ eq ] | inj₂ rs' | [ eq' ] rewrite (sym (eval≡interp c₂ y)) | eq = refl\neval≡interp (c₁ ⊗ c₂) (x , y) with ev {κ = ☐} c₁ x | inspect (ev {κ = ☐} c₁) x\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₁ (x' , rs) | [ eq ] with ev {κ = ☐} c₂ y | inspect (ev {κ = ☐} c₂) y\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₁ (x' , rs) | [ eq ] | inj₁ (y' , rs') | [ eq' ] with ev {κ = ☐} (c₁ ⊗ c₂) (x , y) | inspect (ev {κ = ☐} (c₁ ⊗ c₂)) (x , y)\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₁ (x' , rs) | [ eq ] | inj₁ (y' , rs') | [ eq' ] | inj₁ (_ , rs'') | [ eq'' ] rewrite (sym (eval≡interp c₁ x)) | eq | (sym (eval≡interp c₂ y)) | eq' with deterministic* (((↦₆ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (☐⊗[ c₂ , y ]• ☐)) ++↦ (↦₈ ∷ ◾) ++↦ appendκ↦* rs' (λ ()) (λ ()) refl ([ c₁ , x' ]⊗☐• ☐) ++↦ (↦₉ ∷ ◾)) rs'' (λ ()) (λ ())\n... | refl = refl\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₁ (x' , rs) | [ eq ] | inj₁ (y' , rs') | [ eq' ] | inj₂ rs'' | [ eq'' ] rewrite (sym (eval≡interp c₁ x)) | eq | (sym (eval≡interp c₂ y)) | eq' with deterministic* (((↦₆ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (☐⊗[ c₂ , y ]• ☐)) ++↦ (↦₈ ∷ ◾) ++↦ appendκ↦* rs' (λ ()) (λ ()) refl ([ c₁ , x' ]⊗☐• ☐) ++↦ (↦₉ ∷ ◾)) rs'' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₁ (x' , rs) | [ eq ] | inj₂ rs' | [ eq' ] with ev {κ = ☐} (c₁ ⊗ c₂) (x , y) | inspect (ev {κ = ☐} (c₁ ⊗ c₂)) (x , y)\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₁ (x' , rs) | [ eq ] | inj₂ rs' | [ eq' ] | inj₁ (y' , rs'') | [ eq'' ] rewrite (sym (eval≡interp c₁ x)) | eq | (sym (eval≡interp c₂ y)) | eq' with deterministic* (((↦₆ ∷ ◾) ++↦ appendκ↦* rs (λ ()) (λ ()) refl (☐⊗[ c₂ , y ]• ☐)) ++↦ (↦₈ ∷ ◾) ++↦ appendκ↦*⊠ rs' (λ ()) ([ c₁ , x' ]⊗☐• ☐)) rs'' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₁ (x' , rs) | [ eq ] | inj₂ rs' | [ eq' ] | inj₂ rs'' | [ eq'' ] rewrite (sym (eval≡interp c₁ x)) | eq | (sym (eval≡interp c₂ y)) | eq' = refl\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₂ rs | [ eq ] with ev {κ = ☐} (c₁ ⊗ c₂) (x , y) | inspect (ev {κ = ☐} (c₁ ⊗ c₂)) (x , y)\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₂ rs | [ eq ] | inj₁ (_ , rs') | [ eq' ] rewrite (sym (eval≡interp c₁ x)) | eq with deterministic* ((↦₆ ∷ ◾) ++↦ appendκ↦*⊠ rs (λ ()) (☐⊗[ c₂ , y ]• ☐)) rs' (λ ()) (λ ())\n... | ()\neval≡interp (c₁ ⊗ c₂) (x , y) | inj₂ rs | [ eq ] | inj₂ rs'' | [ eq' ] rewrite (sym (eval≡interp c₁ x)) | eq = refl\neval≡interp (ηₓ v) tt = refl\neval≡interp (εₓ v) (v' , ↻) with v ≟ v'\n... | yes refl = refl\n... | no _ = refl\n\n-- !c is the inverse computation of c\ninterp! : ∀ {A B} (c : A ↔ B) (a : ⟦ A ⟧) (b : ⟦ B ⟧) → interp c a ≡ just b → interp (! c) b ≡ just a\ninterp! unite₊l (inj₂ y) .y refl = refl\ninterp! uniti₊l y .(inj₂ y) refl = refl\ninterp! swap₊ (inj₁ x) .(inj₂ x) refl = refl\ninterp! swap₊ (inj₂ y) .(inj₁ y) refl = refl\ninterp! assocl₊ (inj₁ x) .(inj₁ (inj₁ x)) refl = refl\ninterp! assocl₊ (inj₂ (inj₁ y)) .(inj₁ (inj₂ y)) refl = refl\ninterp! assocl₊ (inj₂ (inj₂ z)) .(inj₂ z) refl = refl\ninterp! assocr₊ (inj₁ (inj₁ x)) .(inj₁ x) refl = refl\ninterp! assocr₊ (inj₁ (inj₂ y)) .(inj₂ (inj₁ y)) refl = refl\ninterp! assocr₊ (inj₂ z) .(inj₂ (inj₂ z)) refl = refl\ninterp! unite⋆l (tt , v) .v refl = refl\ninterp! uniti⋆l v .(tt , v) refl = refl\ninterp! swap⋆ (x , y) .(y , x) refl = refl\ninterp! assocl⋆ (x , y , z) .((x , y) , z) refl = refl\ninterp! assocr⋆ ((x , y) , z) .(x , y , z) refl = refl\ninterp! dist (inj₁ x , z) .(inj₁ (x , z)) refl = refl\ninterp! dist (inj₂ y , z) .(inj₂ (y , z)) refl = refl\ninterp! factor (inj₁ (x , z)) .(inj₁ x , z) refl = refl\ninterp! factor (inj₂ (y , z)) .(inj₂ y , z) refl = refl\ninterp! id↔ v .v refl = refl\ninterp! (c₁ ⨾ c₂) a c eq with interp c₁ a | inspect (interp c₁) a\n... | just b | [ eq₁ ] with interp c₂ b | inspect (interp c₂) b\n... | just c' | [ eq₂ ] with eq\n... | refl rewrite interp! c₂ b c eq₂ | interp! c₁ a b eq₁ = refl\ninterp! (c₁ ⊕ c₂) (inj₁ x) b eq with interp c₁ x | inspect (interp c₁) x\n... | just x' | [ eqx ] with b\n... | inj₁ _ with eq\n... | refl rewrite interp! c₁ x x' eqx = refl\ninterp! (c₁ ⊕ c₂) (inj₂ y) b eq with interp c₂ y | inspect (interp c₂) y\n... | just y' | [ eqy ] with b\n... | inj₂ _ with eq\n... | refl rewrite interp! c₂ y y' eqy = refl\ninterp! (c₁ ⊗ c₂) (x , y) (x' , y') eq with interp c₁ x | inspect (interp c₁) x | interp c₂ y | inspect (interp c₂) y\n... | just x'' | [ eqx ] | just y'' | [ eqy ] with eq\n... | refl rewrite interp! c₁ x x' eqx | interp! c₂ y y' eqy = refl\ninterp! (ηₓ v) tt .(v , ↻) refl with v ≟ v\n... | yes refl = refl\n... | no neq = ⊥-elim (neq refl)\ninterp! (εₓ v) (v' , ↻) b eq with v ≟ v'\n... | yes refl = refl\ninterp! 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YES\n2. NO\n\n", "lm_q1_score": 0.743168019989179, "lm_q2_score": 0.4687906266262437, "lm_q1q2_score": 0.34839020177931207}} {"text": "------------------------------------------------------------------------\n-- Coinductive higher lenses with erased \"proofs\"\n------------------------------------------------------------------------\n\n{-# OPTIONS --guardedness #-}\n\nimport Equality.Path as P\n\nmodule Lens.Non-dependent.Higher.Coinductive.Erased\n {e⁺} (eq : ∀ {a p} → P.Equality-with-paths a p e⁺) where\n\nopen P.Derived-definitions-and-properties eq\n\nopen import Logical-equivalence using (_⇔_)\nopen import Prelude\n\nimport Colimit.Sequential.Very-erased eq as CS\nopen import Equality.Decidable-UIP equality-with-J using (Constant)\nopen import Equality.Path.Isomorphisms eq\nopen import Equivalence equality-with-J as Eq\n using (_≃_; Is-equivalence)\nopen import Equivalence.Erased.Cubical eq as EEq using (_≃ᴱ_)\nopen import Equivalence.Erased.Contractible-preimages.Cubical eq\n using (_⁻¹ᴱ_)\nopen import Erased.Cubical eq\nopen import Function-universe equality-with-J as F hiding (id; _∘_)\nopen import H-level equality-with-J as H-level\nopen import H-level.Closure equality-with-J\nopen import H-level.Truncation.Propositional.Erased eq as T\n using (∥_∥ᴱ; ∣_∣)\nimport H-level.Truncation.Propositional.Non-recursive.Erased eq as N\nopen import H-level.Truncation.Propositional.One-step eq as O\n using (∣_∣; ∥_∥¹-out-^; ∥_∥¹-in-^; ∣_,_∣-in-^)\nopen import Univalence-axiom equality-with-J\n\nopen import Lens.Non-dependent eq\nimport Lens.Non-dependent.Higher.Erased eq as Higher\nimport Lens.Non-dependent.Higher.Capriotti.Variant.Erased.Variant eq\n as V\nopen import Lens.Non-dependent.Higher.Coherently.Coinductive eq\nimport Lens.Non-dependent.Higher.Coinductive eq as C\nimport Lens.Non-dependent.Higher.Coinductive.Small eq as S\n\nprivate\n variable\n a b p : Level\n A B : Type a\n n : ℕ\n\n------------------------------------------------------------------------\n-- The lemma ∥∥ᴱ→≃\n\nprivate\n\n -- A lemma used in the implementation of ∥∥ᴱ→≃.\n --\n -- This definition is erased because its implementation makes use of\n -- code related to O.∥_∥¹ (a HIT with a non-erased higher\n -- constructor).\n\n @0 ∥∥ᴱ→≃-lemma :\n Block \"∥∥ᴱ→≃-lemma\" →\n (f₀ : A → B) →\n (∃ λ (f₊ : ∀ n → ∥ A ∥¹-out-^ (1 + n) → B) →\n (∀ x → f₊ zero ∣ x ∣ ≡ f₀ x) ×\n (∀ n x → f₊ (suc n) ∣ x ∣ ≡ f₊ n x)) ≃\n (∃ λ (f₊ : ∀ n → ∥ A ∥¹-in-^ (1 + n) → B) →\n (∀ x → f₊ zero ∣ x ∣ ≡ f₀ x) ×\n (∀ n x → f₊ (suc n) ∣ n , x ∣-in-^ ≡ f₊ n x))\n ∥∥ᴱ→≃-lemma ⊠ _ =\n inverse $\n Σ-cong {k₁ = equivalence}\n (∀-cong ext λ n →\n →-cong₁ ext (inverse $ O.∥∥¹-out-^≃∥∥¹-in-^ (suc n))) λ f →\n ∃-cong λ _ → ∀-cong ext λ n →\n Π-cong-contra ext (O.∥∥¹-out-^≃∥∥¹-in-^ (suc n)) λ x →\n ≡⇒≃ $ cong (λ y → f (suc n) y ≡\n f n (_≃_.to (O.∥∥¹-out-^≃∥∥¹-in-^ (suc n)) x)) $\n sym $ O.∣∣≡∣,∣-in-^ (1 + n)\n\n-- Functions from ∥ A ∥ᴱ can be expressed as coherently constant\n-- functions from A with erased \"proofs\" (assuming univalence).\n\n∥∥ᴱ→≃ :\n Block \"∥∥ᴱ→≃\" →\n {A : Type a} {B : Type b} →\n @0 Univalence (a ⊔ b) →\n (∥ A ∥ᴱ → B)\n ≃\n (∃ λ (f : A → B) → Erased (C.Coherently-constant f))\n∥∥ᴱ→≃ bl {A = A} {B = B} univ =\n (∥ A ∥ᴱ → B) ↝⟨ →-cong ext T.∥∥ᴱ≃∥∥ᴱ F.id ⟩\n\n (N.∥ A ∥ᴱ → B) ↝⟨ CS.universal-property ⟩\n\n (∃ λ (f₀ : A → B) →\n Erased (∃ λ (f₊ : ∀ n → ∥ A ∥¹-out-^ (1 + n) → B) →\n (∀ x → f₊ zero ∣ x ∣ ≡ f₀ x) ×\n (∀ n x → f₊ (suc n) ∣ x ∣ ≡ f₊ n x))) ↝⟨ ∃-cong (λ f → Erased-cong (∥∥ᴱ→≃-lemma bl f)) ⟩\n\n (∃ λ (f₀ : A → B) →\n Erased (∃ λ (f₊ : ∀ n → ∥ A ∥¹-in-^ (1 + n) → B) →\n (∀ x → f₊ zero ∣ x ∣ ≡ f₀ x) ×\n (∀ n x → f₊ (suc n) ∣ n , x ∣-in-^ ≡ f₊ n x))) ↝⟨ ∃-cong (λ f → Erased-cong (inverse $\n C.Coherently-constant′≃ bl)) ⟩\n\n (∃ λ (f : A → B) → Erased (C.Coherently-constant′ f)) ↝⟨ ∃-cong (λ f → Erased-cong (inverse $\n C.Coherently-constant≃Coherently-constant′ bl univ)) ⟩□\n (∃ λ (f : A → B) → Erased (C.Coherently-constant f)) □\n\n-- A \"computation\" rule for ∥∥ᴱ→≃.\n\n@0 cong-from-∥∥ᴱ→≃-truncation-is-proposition :\n (bl : Block \"∥∥ᴱ→≃\")\n {A : Type a} {B : Type b}\n (univ : Univalence (a ⊔ b)) →\n {f : A → B} {c : C.Coherently-constant f}\n {x y : A} {p : ∣ x ∣ ≡ ∣ y ∣} →\n cong (_≃_.from (∥∥ᴱ→≃ bl univ) (f , [ c ])) p ≡\n c .property x y\ncong-from-∥∥ᴱ→≃-truncation-is-proposition\n bl {A = A} univ {f = f} {c = c} {x = x} {y = y} {p = p} =\n cong (_≃_.from (∥∥ᴱ→≃ bl univ) (f , [ c ])) p ≡⟨⟩\n\n cong (_≃_.from CS.universal-property (f , [ g bl ]) ∘\n _≃_.to T.∥∥ᴱ≃∥∥ᴱ)\n p ≡⟨ sym $ cong-∘ _ _ _ ⟩\n\n (cong (_≃_.from CS.universal-property (f , [ g bl ])) $\n cong (_≃_.to T.∥∥ᴱ≃∥∥ᴱ) p) ≡⟨ cong (cong _) $ mono₁ 1 N.∥∥ᴱ-proposition _ _ ⟩\n\n cong (_≃_.from CS.universal-property (f , [ g bl ]))\n (N.∥∥ᴱ-proposition N.∣ x ∣ N.∣ y ∣) ≡⟨⟩\n\n cong (_≃_.from CS.universal-property (f , [ g bl ]))\n (trans (sym (CS.∣∣₊≡∣∣₀ x))\n (trans (cong CS.∣_∣₊ (O.∣∣-constant x y))\n (CS.∣∣₊≡∣∣₀ y))) ≡⟨ trans (cong-trans _ _ _) $\n cong₂ trans\n (cong-sym _ _)\n (trans (cong-trans _ _ _) $\n cong (flip trans _) $\n cong-∘ _ _ _) ⟩\n trans\n (sym $ cong (_≃_.from CS.universal-property (f , [ g bl ]))\n (CS.∣∣₊≡∣∣₀ x))\n (trans\n (cong (_≃_.from CS.universal-property (f , [ g bl ]) ∘ CS.∣_∣₊)\n (O.∣∣-constant x y))\n (cong (_≃_.from CS.universal-property (f , [ g bl ]))\n (CS.∣∣₊≡∣∣₀ y))) ≡⟨ cong₂ trans\n (cong sym CS.rec-∣∣₊≡∣∣₀)\n (cong (trans _) CS.rec-∣∣₊≡∣∣₀) ⟩\n trans (sym $ proj₁ (proj₂ (g bl)) x)\n (trans (cong (proj₁ (g bl) 0) (O.∣∣-constant x y))\n (proj₁ (proj₂ (g bl)) y)) ≡⟨ lemma bl ⟩∎\n\n c .property x y ∎\n where\n g : ∀ _ → _\n g bl =\n _≃_.from (∥∥ᴱ→≃-lemma bl _) $\n _≃_.to (C.Coherently-constant′≃ bl) $\n _≃_.to (C.Coherently-constant≃Coherently-constant′ bl univ) c\n\n lemma :\n ∀ bl →\n trans (sym $ proj₁ (proj₂ (g bl)) x)\n (trans (cong (proj₁ (g bl) 0) (O.∣∣-constant x y))\n (proj₁ (proj₂ (g bl)) y)) ≡\n c .property x y\n lemma bl@⊠ =\n trans (sym $ proj₁ (proj₂ (g bl)) x)\n (trans (cong (proj₁ (g bl) 0) (O.∣∣-constant x y))\n (proj₁ (proj₂ (g bl)) y)) ≡⟨⟩\n\n trans (sym $ refl _)\n (trans (cong (O.rec′ f (c .property)) (O.∣∣-constant x y))\n (refl _)) ≡⟨ trans (cong₂ trans sym-refl (trans-reflʳ _)) $\n trans-reflˡ _ ⟩\n\n cong (O.rec′ f (c .property)) (O.∣∣-constant x y) ≡⟨ O.rec-∣∣-constant ⟩∎\n\n c .property x y ∎\n\n------------------------------------------------------------------------\n-- Coherently-constant\n\n-- Coherently constant type-valued functions.\n\nCoherently-constant :\n {A : Type a} → (A → Type p) → Type (a ⊔ lsuc p)\nCoherently-constant P =\n ∃ λ (f : ∀ x y → P x → P y) →\n Erased (∃ λ (c : C.Coherently-constant P) →\n ∀ x y → f x y ≡ subst id (c .property x y))\n\n-- Coherently-constant is pointwise equivalent (with erased proofs) to\n-- V.Coherently-constant (assuming univalence).\n\nCoherently-constant≃ᴱCoherently-constant :\n {A : Type a} {P : A → Type p} →\n @0 Univalence (a ⊔ lsuc p) →\n @0 Univalence p →\n Coherently-constant P ≃ᴱ V.Coherently-constant P\nCoherently-constant≃ᴱCoherently-constant\n {a = a} {p = p} {A = A} {P = P} univ′ univ =\n block λ bl →\n\n Coherently-constant P ↔⟨⟩\n\n (∃ λ (P-const : ∀ x y → P x → P y) →\n Erased (\n ∃ λ (c : C.Coherently-constant P) →\n ∀ x y →\n P-const x y ≡ subst id (c .property x y))) ↔⟨ (∃-cong λ P-const → Erased-cong (\n ∃-cong λ c → ∀-cong ext λ x → ∀-cong ext λ y →\n ≡⇒≃ $ cong (P-const x y ≡_) (\n subst id (c .property x y) ≡⟨ cong (subst id) $ sym $\n cong-from-∥∥ᴱ→≃-truncation-is-proposition bl univ′ ⟩\n subst id\n (cong (_≃_.from (∥∥ᴱ→≃ bl univ′) (P , [ c ]))\n (T.truncation-is-proposition ∣ x ∣ ∣ y ∣)) ≡⟨ (⟨ext⟩ λ _ → sym $\n subst-∘ _ _ _) ⟩∎\n subst (_≃_.from (∥∥ᴱ→≃ bl univ′) (P , [ c ]))\n (T.truncation-is-proposition ∣ x ∣ ∣ y ∣) ∎))) ⟩\n\n (∃ λ (P-const : ∀ x y → P x → P y) →\n Erased (\n ∃ λ (c : C.Coherently-constant P) →\n ∀ x y →\n P-const x y ≡\n subst (_≃_.from (∥∥ᴱ→≃ bl univ′) (P , [ c ]))\n (T.truncation-is-proposition ∣ x ∣ ∣ y ∣))) ↔⟨ (∃-cong λ _ → Erased-cong (∃-cong λ _ →\n Eq.extensionality-isomorphism bad-ext F.∘\n (∀-cong ext λ _ → Eq.extensionality-isomorphism bad-ext))) ⟩\n (∃ λ (P-const : ∀ x y → P x → P y) →\n Erased (\n ∃ λ (c : C.Coherently-constant P) →\n P-const ≡\n λ x y →\n subst (_≃_.from (∥∥ᴱ→≃ bl univ′) (P , [ c ]))\n (T.truncation-is-proposition ∣ x ∣ ∣ y ∣))) ↔⟨ (∃-cong λ P-const → Erased-cong (\n ∃-cong λ c → ≡⇒≃ $ cong (P-const ≡_) $ sym $\n ⟨ext⟩ λ x → ⟨ext⟩ λ y →\n cong₂ (λ (f : P y → P y) (g : P x → P x) →\n f ∘\n subst (_≃_.from (∥∥ᴱ→≃ bl univ′) (P , [ c ]))\n (T.truncation-is-proposition ∣ x ∣ ∣ y ∣) ∘\n g)\n (cong _≃_.to $\n trans (cong ≡⇒≃ $ cong-refl (_$ y)) $\n ≡⇒↝-refl)\n (cong _≃_.from $\n trans (cong ≡⇒≃ $ cong-refl (_$ x)) $\n ≡⇒↝-refl))) ⟩\n (∃ λ (P-const : ∀ x y → P x → P y) →\n Erased (\n ∃ λ (c : C.Coherently-constant P) →\n P-const ≡\n λ x y →\n ≡⇒→ (cong (_$ y) (refl P)) ∘\n subst (_≃_.from (∥∥ᴱ→≃ bl univ′) (P , [ c ]))\n (T.truncation-is-proposition ∣ x ∣ ∣ y ∣) ∘\n _≃_.from (≡⇒≃ (cong (_$ x) (refl P))))) ↝⟨ (∃-cong λ P-const → inverse $\n EEq.drop-⊤-left-Σ-≃ᴱ-Erased\n (EEq.other-singleton-with-Π-≃ᴱ-≃ᴱ-⊤ ext univ)) ⟩\n (∃ λ (P-const : ∀ x y → P x → P y) →\n ∃ λ ((Q , P≃) : ∃ λ (Q : A → Type p) → ∀ x → P x ≃ᴱ Q x) →\n Erased (\n ∃ λ (c : C.Coherently-constant Q) →\n P-const ≡\n λ x y →\n _≃ᴱ_.from (P≃ y) ∘\n subst (_≃_.from (∥∥ᴱ→≃ bl univ′) (Q , [ c ]))\n (T.truncation-is-proposition ∣ x ∣ ∣ y ∣) ∘\n _≃ᴱ_.to (P≃ x))) ↔⟨ (∃-cong λ _ →\n Σ-assoc F.∘\n (∃-cong λ _ → ∃-comm) F.∘\n inverse Σ-assoc F.∘\n (∃-cong λ _ → Erased-Σ↔Σ)) ⟩\n (∃ λ (P-const : ∀ x y → P x → P y) →\n ∃ λ ((Q , c) : ∃ λ (Q : A → Type p) →\n Erased (C.Coherently-constant Q)) →\n ∃ λ (P≃ : ∀ x → P x ≃ᴱ Q x) →\n Erased (P-const ≡\n λ x y →\n _≃ᴱ_.from (P≃ y) ∘\n subst (_≃_.from (∥∥ᴱ→≃ bl univ′) (Q , c))\n (T.truncation-is-proposition ∣ x ∣ ∣ y ∣) ∘\n _≃ᴱ_.to (P≃ x))) ↔⟨ (∃-cong λ _ →\n Σ-cong (inverse $ ∥∥ᴱ→≃ bl univ′) λ _ → Eq.id) ⟩\n (∃ λ (P-const : ∀ x y → P x → P y) →\n ∃ λ (Q : ∥ A ∥ᴱ → Type p) →\n ∃ λ (P≃ : ∀ x → P x ≃ᴱ Q ∣ x ∣) →\n Erased (P-const ≡\n λ x y →\n _≃ᴱ_.from (P≃ y) ∘\n subst Q (T.truncation-is-proposition ∣ x ∣ ∣ y ∣) ∘\n _≃ᴱ_.to (P≃ x))) ↔⟨⟩\n\n V.Coherently-constant′ P ↝⟨ inverse V.Coherently-constant≃ᴱCoherently-constant′ ⟩□\n\n V.Coherently-constant P □\n\n------------------------------------------------------------------------\n-- The lens type family\n\n-- Coinductive lenses.\n\nLens : Type a → Type b → Type (lsuc (a ⊔ b))\nLens A B = ∃ λ (get : A → B) → Coherently-constant (get ⁻¹ᴱ_)\n\n-- Some derived definitions.\n\nmodule Lens {A : Type a} {B : Type b} (l : Lens A B) where\n\n -- A getter.\n\n get : A → B\n get = proj₁ l\n\n -- One can convert from any \"preimage\" (with an erased proof) of the\n -- getter to any other.\n\n get⁻¹ᴱ-const : (b₁ b₂ : B) → get ⁻¹ᴱ b₁ → get ⁻¹ᴱ b₂\n get⁻¹ᴱ-const b₁ b₂ = proj₁ (proj₂ l) b₁ b₂\n\n -- A setter.\n\n set : A → B → A\n set a b = $⟨ get⁻¹ᴱ-const (get a) b ⟩\n (get ⁻¹ᴱ get a → get ⁻¹ᴱ b) ↝⟨ _$ (a , [ refl _ ]) ⟩\n get ⁻¹ᴱ b ↝⟨ proj₁ ⟩□\n A □\n\ninstance\n\n -- The lenses defined above have getters and setters.\n\n has-getter-and-setter : Has-getter-and-setter (Lens {a = a} {b = b})\n has-getter-and-setter = record\n { get = Lens.get\n ; set = Lens.set\n }\n\n-- Lens A B is equivalent to V.Lens A B (with erased proofs, assuming\n-- univalence).\n\nLens≃ᴱLens :\n Block \"Lens≃ᴱLens\" →\n {A : Type a} {B : Type b} →\n @0 Univalence (lsuc (a ⊔ b)) →\n @0 Univalence (a ⊔ b) →\n Lens A B ≃ᴱ V.Lens A B\nLens≃ᴱLens ⊠ {A = A} {B = B} univ′ univ =\n (∃ λ (get : A → B) → Coherently-constant (get ⁻¹ᴱ_)) ↝⟨ (∃-cong λ _ →\n Coherently-constant≃ᴱCoherently-constant univ′ univ) ⟩□\n (∃ λ (get : A → B) → V.Coherently-constant (get ⁻¹ᴱ_)) □\n\n-- The right-to-left direction of the equivalence preserves getters\n-- and setters.\n\nfrom-Lens≃ᴱLens-preserves-getters-and-setters :\n (bl : Block \"Lens≃ᴱLens\")\n {A : Type a} {B : Type b}\n (@0 univ′ : Univalence (lsuc (a ⊔ b)))\n (@0 univ : Univalence (a ⊔ b)) →\n Preserves-getters-and-setters-→ A B\n (_≃ᴱ_.from (Lens≃ᴱLens bl univ′ univ))\nfrom-Lens≃ᴱLens-preserves-getters-and-setters ⊠ _ _ l =\n refl _\n , ⟨ext⟩ λ a → ⟨ext⟩ λ b →\n proj₁ (get⁻¹ᴱ-const (get a) b (a , [ refl (get a) ])) ∎\n where\n open V.Lens l\n\n-- In erased contexts the equivalence preserves getters and setters.\n--\n-- (I do not know if this result can be proved in non-erased\n-- contexts.)\n\n@0 Lens≃ᴱLens-preserves-getters-and-setters :\n (bl : Block \"Lens≃ᴱLens\")\n {A : Type a} {B : Type b}\n (@0 univ′ : Univalence (lsuc (a ⊔ b)))\n (@0 univ : Univalence (a ⊔ b)) →\n Preserves-getters-and-setters-⇔ A B\n (_≃ᴱ_.logical-equivalence (Lens≃ᴱLens bl univ′ univ))\nLens≃ᴱLens-preserves-getters-and-setters bl univ′ univ =\n Preserves-getters-and-setters-⇔-inverse\n {f = _≃ᴱ_.logical-equivalence\n (inverse $ Lens≃ᴱLens bl univ′ univ)} $\n Preserves-getters-and-setters-→-↠-⇔\n (_≃_.surjection (EEq.≃ᴱ→≃ $ inverse $ Lens≃ᴱLens bl univ′ univ))\n (from-Lens≃ᴱLens-preserves-getters-and-setters bl univ′ univ)\n\n-- Lens A B is equivalent to Higher.Lens A B (with erased proofs,\n-- assuming univalence).\n\nLens≃ᴱHigher-lens :\n Block \"Lens≃ᴱHigher-Lens\" →\n {A : Type a} {B : Type b} →\n @0 Univalence (lsuc (a ⊔ b)) →\n @0 Univalence (a ⊔ b) →\n Lens A B ≃ᴱ Higher.Lens A B\nLens≃ᴱHigher-lens bl {A = A} {B = B} univ′ univ =\n Lens A B ↝⟨ Lens≃ᴱLens bl univ′ univ ⟩\n V.Lens A B ↝⟨ V.Lens≃ᴱHigher-lens bl univ ⟩□\n Higher.Lens A B □\n\n-- In erased contexts the equivalence preserves getters and setters.\n\n@0 Lens≃ᴱHigher-lens-preserves-getters-and-setters :\n (bl : Block \"Lens≃ᴱHigher-lens\")\n {A : Type a} {B : Type b}\n (@0 univ′ : Univalence (lsuc (a ⊔ b)))\n (@0 univ : Univalence (a ⊔ b)) →\n Preserves-getters-and-setters-⇔ A B\n (_≃ᴱ_.logical-equivalence (Lens≃ᴱHigher-lens bl univ′ univ))\nLens≃ᴱHigher-lens-preserves-getters-and-setters bl univ′ univ =\n Preserves-getters-and-setters-⇔-∘\n {f = _≃ᴱ_.logical-equivalence $ V.Lens≃ᴱHigher-lens bl univ}\n {g = _≃ᴱ_.logical-equivalence $ Lens≃ᴱLens bl univ′ univ}\n (V.Lens≃ᴱHigher-lens-preserves-getters-and-setters bl univ)\n (Lens≃ᴱLens-preserves-getters-and-setters bl univ′ univ)\n\n------------------------------------------------------------------------\n-- H-levels\n\n-- If P has h-level n (pointwise), then Coherently-constant P has\n-- h-level n (assuming univalence).\n\nH-level-Coherently-constant :\n {A : Type a} {P : A → Type p} →\n @0 Univalence (lsuc (a ⊔ p)) →\n @0 Univalence (a ⊔ lsuc p) →\n @0 Univalence p →\n ((a : A) → H-level n (P a)) →\n H-level n (Coherently-constant P)\nH-level-Coherently-constant {n = n} univ₁ univ₂ univ₃ h =\n Σ-closure n\n (Π-closure ext n λ _ →\n Π-closure ext n λ _ →\n Π-closure ext n λ _ →\n h _) λ _ →\n H-level-Erased n (\n Σ-closure n\n (S.H-level-Coinductive-Coherently-constant\n univ₁ univ₂ univ₃ h) λ _ →\n Π-closure ext n λ _ →\n Π-closure ext n λ _ →\n H-level.⇒≡ n $\n Π-closure ext n λ _ →\n h _)\n\n-- If A and B have h-level n given the assumption that the other type\n-- is inhabited, then Lens A B has h-level n (assuming univalence).\n\nlens-preserves-h-level :\n {A : Type a} {B : Type b} →\n @0 Univalence (lsuc (a ⊔ b)) →\n @0 Univalence (a ⊔ b) →\n ∀ n → (B → H-level n A) → (A → H-level n B) →\n H-level n (Lens A B)\nlens-preserves-h-level univ₁ univ₂ n hA hB =\n Σ-closure n\n (Π-closure ext n λ a →\n hB a) λ _ →\n H-level-Coherently-constant univ₁ univ₁ univ₂ λ b →\n Σ-closure n (hA b) λ a →\n H-level-Erased n (\n H-level.⇒≡ n (hB a))\n\n-- If the domain of a lens is inhabited and has h-level n, then the\n-- codomain also has h-level n (in erased contexts, assuming\n-- univalence).\n\n@0 h-level-respects-lens-from-inhabited :\n {A : Type a} {B : Type b} →\n Univalence (lsuc (a ⊔ b)) →\n Univalence (a ⊔ b) →\n ∀ n → Lens A B → A → H-level n A → H-level n B\nh-level-respects-lens-from-inhabited univ′ univ n =\n Higher.h-level-respects-lens-from-inhabited n ∘\n _≃ᴱ_.to (Lens≃ᴱHigher-lens ⊠ univ′ univ)\n\n-- If A has positive h-level n, then Lens A B also has h-level n (in\n-- erased contexts, assuming univalence).\n\n@0 lens-preserves-h-level-of-domain :\n {A : Type a} {B : Type b} →\n Univalence (lsuc (a ⊔ b)) →\n Univalence (a ⊔ b) →\n ∀ n → H-level (1 + n) A → H-level (1 + n) (Lens A B)\nlens-preserves-h-level-of-domain univ′ univ n hA =\n H-level.[inhabited⇒+]⇒+ n λ l →\n lens-preserves-h-level univ′ univ (1 + n) (λ _ → hA) λ a →\n h-level-respects-lens-from-inhabited univ′ univ _ l a hA\n", "meta": {"hexsha": "f7ba0d3a597dbb2a96a84693cc3555d4719aef79", "size": 20548, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Lens/Non-dependent/Higher/Coinductive/Erased.agda", "max_stars_repo_name": "nad/dependent-lenses", "max_stars_repo_head_hexsha": "f2da6f7e95b87ca525e8ea43929c6d6163a74811", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-04-16T12:10:46.000Z", 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YES\n2. YES\n\n", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3483332715490145}} {"text": "{-# OPTIONS --without-K --safe #-}\n\nmodule Categories.Adjoint.Properties where\n\nopen import Level\nopen import Data.Product using (Σ; _,_; -,_; proj₂; uncurry)\nopen import Function using (_$_)\n\nopen import Categories.Adjoint using (_⊣_; Adjoint; Hom-NI′⇒Adjoint)\nopen import Categories.Adjoint.RAPL public\nopen import Categories.Category using (Category; _[_,_])\nopen import Categories.Category.Product using (_⁂_; _⁂ⁿⁱ_)\nopen import Categories.Category.Construction.Comma using (CommaObj; Comma⇒; _↙_)\nopen import Categories.Functor renaming (id to idF)\nopen import Categories.Functor.Hom\nopen import Categories.Functor.Construction.Constant\nopen import Categories.Functor.Construction.LiftSetoids\nopen import Categories.Functor.Properties\nopen import Categories.Functor.Continuous\nopen import Categories.Functor.Cocontinuous\nopen import Categories.Functor.Bifunctor\nopen import Categories.Functor.Bifunctor.Properties\nopen import Categories.NaturalTransformation\nopen import Categories.NaturalTransformation.Properties\nopen import Categories.NaturalTransformation.NaturalIsomorphism using (NaturalIsomorphism; _≃_; _ⓘₕ_; _ⓘˡ_; module ≃)\nopen import Categories.NaturalTransformation.NaturalIsomorphism.Properties\nopen import Categories.Monad\nopen import Categories.Monad.Duality\nopen import Categories.Comonad\nopen import Categories.Morphism.Universal\nopen import Categories.Yoneda\nimport Categories.Yoneda.Properties as YP\n\nimport Categories.Diagram.Colimit as Col\nimport Categories.Diagram.Duality as Duality\n\nimport Categories.Morphism as Mor\nimport Categories.Morphism.Reasoning as MR\n\nprivate\n variable\n o ℓ e : Level\n C D E J : Category o ℓ e\n\n-- if the left adjoint functor is a partial application of bifunctor, then it uniquely\n-- determines a bifunctor compatible with the right adjoint functor.\nmodule _ {C : Category o ℓ e}\n (L : Bifunctor C E D) {R : ∀ (X : Category.Obj E) → Functor D C}\n (LR : ∀ (X : Category.Obj E) → appʳ L X ⊣ R X) where\n private\n module C = Category C\n module D = Category D\n module E = Category E\n module L = Functor L\n module R X = Functor (R X)\n module LR X = Adjoint (LR X)\n open C\n\n F′ : ∀ {A X B Y} f g → R.F₀ A X ⇒ R.F₀ B Y\n F′ {A} {X} {B} {Y} f g = LR.Ladjunct B (LR.counit.η A Y D.∘ L.F₁ (R.F₁ A g , f))\n -- R.F₁ B (LR.counit.η A Y) ∘ R.F₁ B (L.F₁ (R.F₁ A g , f)) ∘ LR.unit.η B (R.F₀ A X)\n\n commute′ : ∀ {A B X} (f : A E.⇒ B) → LR.counit.η A X D.∘ L.F₁ (F′ f D.id , E.id) D.≈ LR.counit.η B X D.∘ L.F₁ (C.id , f)\n commute′ {A} {B} {X} f = begin\n LR.counit.η A X D.∘ L.F₁ (F′ f D.id , E.id) ≈⟨ LR.RLadjunct≈id A ⟩\n LR.counit.η B X D.∘ L.F₁ (R.F₁ B D.id , f) ≈⟨ refl ⟩∘⟨ L.F-resp-≈ (R.identity B , E.Equiv.refl) ⟩\n LR.counit.η B X D.∘ L.F₁ (C.id , f) ∎\n where open D.HomReasoning\n\n open HomReasoning\n\n decompose₁ : ∀ {A B X Y} {f : A E.⇒ B} {g : X D.⇒ Y} → F′ f g ≈ R.F₁ A g ∘ F′ f D.id\n decompose₁ {A} {B} {X} {Y} {f} {g} = begin\n F′ f g\n ≈⟨ R.F-resp-≈ A (D.∘-resp-≈ʳ [ L ]-decompose₁) ⟩∘⟨refl ⟩\n R.F₁ A (LR.counit.η B Y D.∘ L.F₁ (R.F₁ B g , E.id) D.∘ L.F₁ (C.id , f)) ∘ LR.unit.η A (R.F₀ B X)\n ≈⟨ R.F-resp-≈ A (pullˡ (LR.counit.commute B g)) ⟩∘⟨refl ⟩\n R.F₁ A ((g D.∘ LR.counit.η B X) D.∘ L.F₁ (C.id , f)) ∘ LR.unit.η A (R.F₀ B X)\n ≈˘⟨ R.F-resp-≈ A (pushʳ (D.∘-resp-≈ʳ (L.F-resp-≈ (R.identity B , E.Equiv.refl)))) ⟩∘⟨refl ⟩\n R.F₁ A (g D.∘ LR.counit.η B X D.∘ L.F₁ (R.F₁ B D.id , f)) ∘ LR.unit.η A (R.F₀ B X)\n ≈⟨ R.homomorphism A ⟩∘⟨refl ⟩\n (R.F₁ A g ∘ R.F₁ A (LR.counit.η B X D.∘ L.F₁ (R.F₁ B D.id , f))) ∘ LR.unit.η A (R.F₀ B X)\n ≈⟨ assoc ⟩\n R.F₁ A g ∘ F′ f D.id\n ∎\n where open MR D\n\n decompose₂ : ∀ {A B X Y} {f : A E.⇒ B} {g : X D.⇒ Y} → F′ f g ≈ F′ f D.id ∘ R.F₁ B g\n decompose₂ {A} {B} {X} {Y} {f} {g} = begin\n F′ f g\n ≈⟨ R.F-resp-≈ A (D.∘-resp-≈ʳ [ L ]-decompose₂) ⟩∘⟨refl ⟩\n R.F₁ A (LR.counit.η B Y D.∘ L.F₁ (C.id , f) D.∘ L.F₁ (R.F₁ B g , E.id)) ∘ LR.unit.η A (R.F₀ B X)\n ≈˘⟨ R.F-resp-≈ A (pushˡ (D.∘-resp-≈ʳ (L.F-resp-≈ (R.identity B , E.Equiv.refl)))) ⟩∘⟨refl ⟩\n R.F₁ A ((LR.counit.η B Y D.∘ L.F₁ (R.F₁ B D.id , f)) D.∘ L.F₁ (R.F₁ B g , E.id)) ∘ LR.unit.η A (R.F₀ B X)\n ≈⟨ R.homomorphism A ⟩∘⟨refl ⟩\n (R.F₁ A (LR.counit.η B Y D.∘ L.F₁ (R.F₁ B D.id , f)) ∘ R.F₁ A (L.F₁ (R.F₁ B g , E.id))) ∘ LR.unit.η A (R.F₀ B X)\n ≈˘⟨ MR.pushʳ C (LR.unit.commute A (R.F₁ B g)) ⟩\n R.F₁ A (LR.counit.η B Y D.∘ L.F₁ (R.F₁ B D.id , f)) ∘ LR.unit.η A (R.F₀ B Y) ∘ R.F₁ B g\n ≈˘⟨ assoc ⟩\n F′ f D.id ∘ R.F₁ B g\n ∎\n where open MR D\n\n swap : ∀ {A B X Y} {f : A E.⇒ B} {g : X D.⇒ Y} → R.F₁ A g ∘ F′ f D.id ≈ F′ f D.id ∘ R.F₁ B g\n swap = trans (⟺ decompose₁) decompose₂\n\n commute″ : ∀ {X Y Z A} {f : Y E.⇒ Z} {g : X E.⇒ Y} → F′ (f E.∘ g) (D.id {A}) ≈ F′ g D.id ∘ F′ f D.id\n commute″ {X} {Y} {Z} {A} {f} {g} = begin\n F′ (f E.∘ g) D.id\n ≈⟨ R.F-resp-≈ X (D.∘-resp-≈ʳ (L.F-resp-≈ (R.identity Z , E.Equiv.refl))) ⟩∘⟨refl ⟩\n R.F₁ X (LR.counit.η Z A D.∘ L.F₁ (C.id , f E.∘ g)) ∘ LR.unit.η X (R.F₀ Z A)\n ≈⟨ R.F-resp-≈ X (D.∘-resp-≈ʳ (Functor.homomorphism (appˡ L (R.F₀ Z A)))) ⟩∘⟨refl ⟩\n R.F₁ X (LR.counit.η Z A D.∘ L.F₁ (C.id , f) D.∘ L.F₁ (C.id , g)) ∘ LR.unit.η X (R.F₀ Z A)\n ≈˘⟨ R.F-resp-≈ X (MR.pushˡ D (commute′ f)) ⟩∘⟨refl ⟩\n R.F₁ X ((LR.counit.η Y A D.∘ L.F₁ (F′ f D.id , E.id)) D.∘ L.F₁ (C.id , g)) ∘ LR.unit.η X (R.F₀ Z A)\n ≈˘⟨ R.F-resp-≈ X (MR.pushʳ D [ L ]-commute) ⟩∘⟨refl ⟩\n R.F₁ X (LR.counit.η Y A D.∘ L.F₁ (C.id , g) D.∘ L.F₁ (F′ f D.id , E.id)) ∘ LR.unit.η X (R.F₀ Z A)\n ≈˘⟨ R.F-resp-≈ X D.assoc ⟩∘⟨refl ⟩\n R.F₁ X ((LR.counit.η Y A D.∘ L.F₁ (C.id , g)) D.∘ L.F₁ (F′ f D.id , E.id)) ∘ LR.unit.η X (R.F₀ Z A)\n ≈⟨ R.homomorphism X ⟩∘⟨refl ⟩\n (R.F₁ X (LR.counit.η Y A D.∘ L.F₁ (C.id , g)) ∘ R.F₁ X (L.F₁ (F′ f D.id , E.id))) ∘ LR.unit.η X (R.F₀ Z A)\n ≈˘⟨ MR.pushʳ C (LR.unit.commute X (F′ f D.id)) ⟩\n R.F₁ X (LR.counit.η Y A D.∘ L.F₁ (C.id , g)) ∘ LR.unit.η X (R.F₀ Y A) ∘ F′ f D.id\n ≈˘⟨ R.F-resp-≈ X (D.∘-resp-≈ʳ (L.F-resp-≈ (R.identity Y , E.Equiv.refl))) ⟩∘⟨ refl ⟩∘⟨ refl ⟩\n R.F₁ X (LR.counit.η Y A D.∘ L.F₁ (R.F₁ Y D.id , g)) ∘ LR.unit.η X (R.F₀ Y A) ∘ F′ f D.id\n ≈˘⟨ assoc ⟩\n F′ g D.id ∘ F′ f D.id\n ∎\n\n induced-bifunctorʳ : Bifunctor E.op D C\n induced-bifunctorʳ = record\n { F₀ = uncurry R.F₀\n ; F₁ = uncurry F′\n ; identity = λ where\n {e , d} →\n let open MR D\n in begin\n F′ E.id D.id\n ≈⟨ R.F-resp-≈ e (D.∘-resp-≈ʳ (L.F-resp-≈ (R.identity e , E.Equiv.refl))) ⟩∘⟨ refl ⟩\n R.F₁ e (LR.counit.η e d D.∘ L.F₁ (C.id , E.id)) ∘ LR.unit.η e (R.F₀ e d)\n ≈⟨ R.F-resp-≈ e (elimʳ L.identity) ⟩∘⟨ refl ⟩\n R.F₁ e (LR.counit.η e d) ∘ LR.unit.η e (R.F₀ e d)\n ≈⟨ LR.zag e ⟩\n C.id\n ∎\n ; homomorphism = λ where\n {A , X} {B , Y} {W , Z} {f , h} {g , i} →\n let open MR C\n in begin\n F′ (f E.∘ g) (i D.∘ h)\n ≈⟨ decompose₁ ⟩\n R.F₁ W (i D.∘ h) ∘ F′ (f E.∘ g) D.id\n ≈˘⟨ center⁻¹ (⟺ (R.homomorphism W)) (⟺ commute″) ⟩\n R.F₁ W i ∘ (R.F₁ W h ∘ F′ g D.id) ∘ F′ f D.id\n ≈˘⟨ center (⟺ swap) ⟩\n (R.F₁ W i ∘ F′ g D.id) ∘ R.F₁ B h ∘ F′ f D.id\n ≈˘⟨ decompose₁ ⟩∘⟨ decompose₁ ⟩\n F′ g i ∘ F′ f h\n ∎\n ; F-resp-≈ = λ where\n {A , X} {B , Y} (eq , eq′) →\n ∘-resp-≈ˡ (R.F-resp-≈ B (D.∘-resp-≈ʳ (L.F-resp-≈ (R.F-resp-≈ A eq′ , eq))))\n }\n\n-- LAPC: left adjoint preserves colimits.\nmodule _ {L : Functor C D} {R : Functor D C} (L⊣R : L ⊣ R) (F : Functor J C) where\n private\n module F = Functor F\n open Col\n\n lapc : Colimit F → Colimit (L ∘F F)\n lapc col = Duality.coLimit⇒Colimit D (rapl (Adjoint.op L⊣R) F.op (Duality.Colimit⇒coLimit C col))\n\n-- adjoint functors induce monads and comonads\nmodule _ {L : Functor C D} {R : Functor D C} (L⊣R : L ⊣ R) where\n private\n module C = Category C\n module D = Category D\n module L = Functor L\n module R = Functor R\n open Adjoint L⊣R\n\n rapl′ : ∀ {o ℓ e} → Continuous o ℓ e R\n rapl′ lim = rapl L⊣R _ lim , Mor.≅.refl C\n\n lapc′ : ∀ {o ℓ e} → Cocontinuous o ℓ e L\n lapc′ col = lapc L⊣R _ col , Mor.≅.refl D\n\n adjoint⇒monad : Monad C\n adjoint⇒monad = record\n { F = R ∘F L\n ; η = unit\n ; μ = record\n { η = μ′.η\n ; commute = μ′.commute\n ; sym-commute = μ′.sym-commute\n }\n ; assoc = [ R ]-resp-square (counit.commute _)\n ; sym-assoc = [ R ]-resp-square (counit.sym-commute _)\n ; identityˡ = λ {X} → begin\n μ′.η X ∘ R.F₁ (L.F₁ (unit.η X)) ≈⟨ [ R ]-resp-∘ zig ⟩\n R.F₁ D.id ≈⟨ R.identity ⟩\n C.id ∎\n ; identityʳ = zag\n }\n where open C\n open HomReasoning\n μ′ : NaturalTransformation (R ∘F (L ∘F R) ∘F L) (R ∘F Categories.Functor.id ∘F L)\n μ′ = R ∘ˡ counit ∘ʳ L\n module μ′ = NaturalTransformation μ′\n\nmodule _ {L : Functor C D} {R : Functor D C} (L⊣R : L ⊣ R) where\n open Adjoint L⊣R\n\n adjoint⇒comonad : Comonad D\n adjoint⇒comonad = coMonad⇒Comonad D (adjoint⇒monad op)\n\n-- adjoint functors are the same as universal morphisms\nmodule _ {R : Functor D C} where\n private\n module C = Category C\n module D = Category D\n module R = Functor R\n\n adjoint⇒universalMorphisms : ∀ {L : Functor C D} → L ⊣ R → ∀ (X : C.Obj) → UniversalMorphism X R\n adjoint⇒universalMorphisms {L} L⊣R X = record\n { initial = record\n { ⊥ = record { f = unit.η X }\n ; ! =\n let open C.HomReasoning\n in record { commute = LRadjunct≈id ○ ⟺ C.identityʳ }\n ; !-unique = λ {A} g →\n let open D.HomReasoning\n in -, (begin\n Radjunct (f A) ≈⟨ Radjunct-resp-≈ (C.Equiv.sym (C.Equiv.trans (commute g) (C.identityʳ {f = f A}))) ⟩\n Radjunct (Ladjunct (h g)) ≈⟨ RLadjunct≈id ⟩\n h g ∎)\n }\n }\n where module L = Functor L\n open Adjoint L⊣R\n open Comma⇒\n open CommaObj\n\n universalMophisms⇒adjoint : (∀ (X : C.Obj) → UniversalMorphism X R) → Σ (Functor C D) (λ L → L ⊣ R)\n universalMophisms⇒adjoint umors = L , record\n { unit = ntHelper record\n { η = λ c → f (umors.⊥ c)\n ; commute = λ i → let open C.HomReasoning in ⟺ (commute (⊥X⇒⊥Y i) ○ C.identityʳ )\n }\n ; counit = ntHelper record\n { η = ε\n ; commute = λ {X Y} i →\n let open C.HomReasoning\n open MR C\n in proj₂ $ umors.!-unique₂ (R.F₀ X)\n {record { f = R.F₁ i }}\n (record\n { h = ε Y D.∘ L₁ (R.F₁ i)\n ; commute = begin\n R.F₁ (ε Y D.∘ L₁ (R.F₁ i)) C.∘ f (⊥Rd X) ≈⟨ R.homomorphism ⟩∘⟨refl ⟩\n (R.F₁ (ε Y) C.∘ R.F₁ (L₁ (R.F₁ i))) C.∘ f (⊥Rd X) ≈⟨ pullʳ (commute (⊥X⇒⊥Y (R.F₁ i)) ○ C.identityʳ) ⟩\n R.F₁ (ε Y) C.∘ f (⊥Rd Y) C.∘ R.F₁ i ≈⟨ cancelˡ (commute (⊥Rd⇒id Y) ○ C.identityˡ) ⟩\n R.F₁ i ≈˘⟨ C.identityʳ ⟩\n R.F₁ i C.∘ C.id ∎\n })\n (record\n { h = i D.∘ ε X\n ; commute = begin\n R.F₁ (i D.∘ ε X) C.∘ f (⊥Rd X) ≈⟨ R.homomorphism ⟩∘⟨refl ⟩\n (R.F₁ i C.∘ R.F₁ (ε X)) C.∘ f (⊥Rd X) ≈⟨ cancelʳ (commute (⊥Rd⇒id X) ○ C.identityˡ) ⟩\n R.F₁ i ≈˘⟨ C.identityʳ ⟩\n R.F₁ i C.∘ C.id ∎\n })\n }\n ; zig = λ {c} →\n let open C.HomReasoning\n open MR C\n α = f (umors.⊥ c)\n in proj₂ $ umors.!-unique₂ c\n {record { f = α }}\n (record\n { h = ε (L₀ c) D.∘ L₁ α\n ; commute = begin\n R.F₁ (ε (L₀ c) D.∘ L₁ α) C.∘ α ≈⟨ R.homomorphism ⟩∘⟨refl ⟩\n (R.F₁ (ε (L₀ c)) C.∘ R.F₁ (L₁ α)) C.∘ α ≈⟨ pullʳ (commute (⊥X⇒⊥Y α) ○ C.identityʳ) ⟩\n R.F₁ (ε (L₀ c)) C.∘ f (⊥Rd (L₀ c)) C.∘ α ≈⟨ cancelˡ (commute (⊥Rd⇒id (L₀ c)) ○ C.identityˡ) ⟩\n α ≈˘⟨ C.identityʳ ⟩\n α C.∘ C.id ∎\n })\n (record\n { h = D.id\n ; commute = C.∘-resp-≈ˡ R.identity ○ id-comm-sym\n })\n ; zag = λ {d} → C.Equiv.trans (commute (⊥Rd⇒id d)) C.identityˡ\n }\n where module umors X = UniversalMorphism (umors X)\n open CommaObj\n open Comma⇒\n\n commaObj∘g : ∀ {X Y} → X C.⇒ Y → CommaObj (const! X) R\n commaObj∘g {X} {Y} g = record { f = f (umors.⊥ Y) C.∘ g }\n\n ⊥X⇒⊥Y : ∀ {X Y} (g : X C.⇒ Y) → (X ↙ R) [ umors.⊥ X , commaObj∘g g ]\n ⊥X⇒⊥Y {X} {Y} g = umors.! X {commaObj∘g g}\n\n L₀ : ∀ X → D.Obj\n L₀ X = β (umors.⊥ X)\n L₁ : ∀ {X Y} → X C.⇒ Y → β (umors.⊥ X) D.⇒ β (umors.⊥ Y)\n L₁ {X} {Y} g = h (⊥X⇒⊥Y g)\n L : Functor C D\n L = record\n { F₀ = L₀\n ; F₁ = L₁\n ; identity = λ {X} → proj₂ $ umors.!-unique X $\n record { commute = elimˡ R.identity ○ ⟺ C.identityʳ ○ ⟺ C.identityʳ }\n ; homomorphism = λ {X Y Z} {i j} → proj₂ $ umors.!-unique₂ X (umors.! X) $\n record { commute = begin\n R.F₁ (h (umors.! Y) D.∘ h (umors.! X)) C.∘ f (umors.⊥ X)\n ≈⟨ (C.∘-resp-≈ˡ R.homomorphism) ○ C.assoc ⟩\n R.F₁ (h (umors.! Y)) C.∘ R.F₁ (h (umors.! X)) C.∘ f (umors.⊥ X)\n ≈⟨ (C.∘-resp-≈ʳ (commute (⊥X⇒⊥Y i) ○ C.identityʳ)) ○ C.sym-assoc ⟩\n (R.F₁ (h (umors.! Y)) C.∘ f (umors.⊥ Y)) C.∘ i\n ≈⟨ pushˡ (commute (⊥X⇒⊥Y j) ○ C.identityʳ) ⟩\n f (umors.⊥ Z) C.∘ j C.∘ i\n ≈˘⟨ C.identityʳ ⟩\n (f (umors.⊥ Z) C.∘ j C.∘ i) C.∘ C.id\n ∎ }\n ; F-resp-≈ = λ {X} eq → proj₂ $ umors.!-unique₂ X (umors.! X) $\n record { commute = commute (umors.! X) ○ C.∘-resp-≈ˡ (C.∘-resp-≈ʳ (⟺ eq)) }\n }\n where open C.HomReasoning\n open MR C\n module L = Functor L\n\n ⊥Rd : (d : D.Obj) → CommaObj (const! (R.F₀ d)) R\n ⊥Rd d = umors.⊥ (R.F₀ d)\n ⊥Rd⇒id : (d : D.Obj) → (R.F₀ d ↙ R) [ ⊥Rd d , record { f = C.id } ]\n ⊥Rd⇒id d = umors.! (R.F₀ d) {record { f = C.id }}\n ε : ∀ d → L₀ (R.F₀ d) D.⇒ d\n ε d = h (⊥Rd⇒id d)\n\n-- adjoint functors of a functor are isomorphic\nmodule _ (L : Functor C D) where\n open YP C\n\n R≃R′ : ∀ {R R′} → L ⊣ R → L ⊣ R′ → R ≃ R′\n R≃R′ {R} {R′} L⊣R L⊣R′ = yoneda-NI R R′ (unlift-≃ Hom[-,R-]≃Hom[-,R′-])\n where module ⊣₁ = Adjoint L⊣R\n module ⊣₂ = Adjoint L⊣R′\n Hom[-,R-]≃Hom[-,R′-] : ⊣₁.Hom[-,R-]′ ≃ ⊣₂.Hom[-,R-]′\n Hom[-,R-]≃Hom[-,R′-] = ≃.trans (≃.sym ⊣₁.Hom-NI) ⊣₂.Hom-NI\n\nmodule _ {R : Functor D C} where\n\n L≃L′ : ∀ {L L′} → L ⊣ R → L′ ⊣ R → L ≃ L′\n L≃L′ L⊣R L′⊣R = NaturalIsomorphism.op L′≃Lᵒᵖ\n where module ⊣₁ = Adjoint L⊣R\n module ⊣₂ = Adjoint L′⊣R\n L′≃Lᵒᵖ = R≃R′ (Functor.op R) ⊣₂.op ⊣₁.op\n\n-- adjoint functors are preserved by natural isomorphisms\nmodule _ {L L′ : Functor C D} {R R′ : Functor D C} where\n private\n module C = Category C\n module D = Category D\n module L = Functor L\n module L′ = Functor L′\n module R = Functor R\n module R′ = Functor R′\n\n ⊣×≃⇒⊣ : L ⊣ R → L ≃ L′ → R ≃ R′ → L′ ⊣ R′\n ⊣×≃⇒⊣ L⊣R L≃L′ R≃R′ = Hom-NI′⇒Adjoint (≃.trans (LiftSetoids _ _ ⓘˡ Hom[L′-,-]≃Hom[L-,-])\n (≃.trans Hom-NI\n (LiftSetoids _ _ ⓘˡ Hom[-,R-]≃Hom[-,R′-])))\n where open Adjoint L⊣R\n Hom[L′-,-]≃Hom[L-,-] : Hom[ D ][-,-] ∘F (L′.op ⁂ idF) ≃ Hom[ D ][-,-] ∘F (L.op ⁂ idF)\n Hom[L′-,-]≃Hom[L-,-] = Hom[ D ][-,-] ⓘˡ (NaturalIsomorphism.op L≃L′ ⁂ⁿⁱ ≃.refl)\n Hom[-,R-]≃Hom[-,R′-] : Hom[ C ][-,-] ∘F (idF ⁂ R) ≃ Hom[ C ][-,-] ∘F (idF ⁂ R′)\n Hom[-,R-]≃Hom[-,R′-] = Hom[ C ][-,-] ⓘˡ (≃.refl ⁂ⁿⁱ R≃R′)\n", "meta": {"hexsha": "57107e0b353ff3c3c6fbb6cb999b59565bef3198", "size": 16280, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Categories/Adjoint/Properties.agda", "max_stars_repo_name": "Taneb/agda-categories", "max_stars_repo_head_hexsha": "6ebc1349ee79669c5c496dcadd551d5bbefd1972", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Categories/Adjoint/Properties.agda", "max_issues_repo_name": "Taneb/agda-categories", "max_issues_repo_head_hexsha": "6ebc1349ee79669c5c496dcadd551d5bbefd1972", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Categories/Adjoint/Properties.agda", "max_forks_repo_name": "Taneb/agda-categories", "max_forks_repo_head_hexsha": "6ebc1349ee79669c5c496dcadd551d5bbefd1972", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.7296587927, "max_line_length": 124, "alphanum_fraction": 0.4800982801, "num_tokens": 7214, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105587468141, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3482894268451718}} {"text": "open import Prelude\nopen import Data.Nat using (_≤?_)\nopen import Data.Maybe using (Maybe; just; nothing)\nopen import Data.Vec using (Vec; _∷_; [])\n\nopen import RW.Language.RTerm\nopen import RW.Language.FinTerm\nopen import RW.Language.RTermUtils\nopen import RW.Utils.Monads\n\nmodule RW.Language.GoalGuesser (maxH : ℕ) where\n \n open Monad {{...}}\n\n ----------------\n -- Housekeeping\n\n private\n cast-RBinApp : {A B : Set} → (A → B) → RBinApp A → RBinApp B\n cast-RBinApp f (n , a , b) = n , replace-A (ovar ∘ f) a , replace-A (ovar ∘ f) b\n\n RBinApp02⊥ : RBinApp (Fin zero) → RBinApp ⊥\n RBinApp02⊥ = cast-RBinApp (λ ())\n\n symmetric : {A : Set} → RBinApp A → RBinApp A\n symmetric (n , a , b) = n , b , a\n\n -----------------\n -- Term subsitution\n\n -- Given a RBinApp representing a type (t₁ ▵ t₂), will return a\n -- non-deterministic substitution of t₂ for t₁ in a given goal g.\n -- If no substitution could be performed, the empty list will be returned.\n _[_] : {A : Set}{{eqA : Eq A}} → RTerm A → RTerm A × RTerm A → NonDet (RTerm A)\n g [ ty1 , ty2 ] = map p2 (filter p1 (substRTermAux g ty1 ty2)) \n where\n -- Non-deterministic term substitution. Possibly empty substitution.\n -- Since we're only interested in the situations where a single substitution is performed,\n -- we also return a boolean flag indicating whether or not we already had a substitution.\n substRTermAux : {A : Set}{{ eqA : Eq A }} → RTerm A → RTerm A → RTerm A → NonDet (Bool × RTerm A)\n substRTermAux {A} {{eqA}} t m n with t ≟-RTerm m\n ...| yes _ = return (true , n)\n ...| no _ = substStep t m n\n where\n mutual\n substStep : RTerm A → RTerm A → RTerm A → NonDet (Bool × RTerm A)\n substStep (ovar x) _ _ = return (false , ovar x)\n substStep (ivar n) _ _ = return (false , ivar n)\n substStep (rlit l) _ _ = return (false , rlit l)\n substStep (rlam t) m n = substRTermAux t m n >>= return ∘ (λ p → p1 p , rlam (p2 p))\n substStep (rapp a as) m n = substStep* as m n >>= return ∘ (λ p → p1 p , rapp a (p2 p))\n\n -- For the list scenario, in the inductive step, we need to substitute m for n in x,\n -- and return either the result of such substitution or the result of substituting\n -- m for n recursively in la. Do not forget that _++_ is mplus for the list monad.\n substStep* : List (RTerm A) → RTerm A → RTerm A → NonDet (Bool × List (RTerm A))\n substStep* [] _ _ = return (false , [])\n substStep* (x ∷ la) m n\n = substRTermAux x m n >>= λ x' → return (p1 x' , (p2 x' ∷ la)) \n ++ (substStep* la m n >>= return ∘ (λ p → p1 p , x ∷ p2 p))\n\n -- Given a term and a type ty with n variables, will search for n subterms of t\n -- in a non-deterministic fashion, and will instantiate them as parameters to ty.\n {-# TERMINATING #-}\n apply : {n : ℕ} → RTerm ⊥ → RBinApp (Fin n) → NonDet (RTerm ⊥)\n -- If we have no more variables to instantiate on our type,\n -- we can proceed to substitute.\n apply {n = zero} g (_ , ty1 , ty2) \n = g [ Fin2RTerm⊥ ty1 , Fin2RTerm⊥ ty2 ]\n\n -- If not, we simply call with a recursively smaller n\n apply {n = suc n} g t = inst g t >>= apply g\n where\n -- Easy way of discarding the last variable of a Fin n.\n -- Mainly, we map Fin (1 + n) to 1 + Fin n.\n thin : {n : ℕ} → Fin (suc n) → Maybe (Fin n)\n thin {zero} _ = nothing\n thin {suc n} fz = just fz\n thin {suc n} (fs x) with thin x\n ...| nothing = nothing\n ...| just x' = just (fs x')\n\n -- Given a closed term t, returns a substitution σ defined\n -- by σ(n+1) = t, σ(k, k ≤ n) = var k.\n ▵_ : {n : ℕ} → RTerm ⊥ → Fin (suc n) → RTerm (Fin n)\n ▵_ {n} t x with thin x\n ...| nothing = replace-A (λ ()) t\n ...| just x' = ovar x'\n\n {-# TERMINATING #-}\n mk-inst-f : {n : ℕ} → RTerm ⊥ → NonDet (Fin (suc n) → RTerm (Fin n))\n mk-inst-f (ovar ())\n mk-inst-f (ivar n) = return (▵ (ivar n))\n mk-inst-f (rlit l) = return (▵ (rlit l))\n mk-inst-f (rlam t) with height (rlam t) ≤? maxH\n ...| yes _ = return (▵ (rlam t)) ++ mk-inst-f t\n ...| no _ = mk-inst-f t\n mk-inst-f (rapp n ts) with height (rapp n ts) ≤? maxH\n ...| yes _ = return (▵ (rapp n ts)) ++ concat (mapM mk-inst-f ts)\n ...| no _ = concat (mapM mk-inst-f ts)\n\n -- Given a term t and a type with (n+1) variables, instantiate the last variable\n -- from the type with a non-deterministic subterm of t, returns the resulting type.\n inst : {n : ℕ} → RTerm ⊥ → RBinApp (Fin (suc n)) → NonDet (RBinApp (Fin n))\n inst {n} t (r , ty1 , ty2) = mk-inst-f {n} t >>= λ f → return (r , replace-A f ty1 , replace-A f ty2)\n\n\n -- TODO: Use vectors instead of lists... \n\n -- Given a goal (a ▵ b) and a list of types L, try to find intermediate goals g₁⋯gₙ such\n -- that ∀n . tyₙ ∈ L ⇒ tyₙ : gₙ₋₁ → gₙ , where g₀ = a and gₖ = b.\n divideGoal : RBinApp ⊥ → List (Σ ℕ (RBinApp ∘ Fin)) → Maybe (List (RTerm ⊥))\n divideGoal (gh , g1 , g2) l = sfHead (filter chainIsValid (stepGoals g1 l))\n where\n chainIsValid : List (RTerm ⊥) → Bool\n chainIsValid [] = false\n chainIsValid (x ∷ []) with x ≟-RTerm g2\n ...| yes _ = true\n ...| no _ = false\n chainIsValid (x ∷ l) = chainIsValid l\n\n sfHead : {A : Set} → List A → Maybe A\n sfHead [] = nothing\n sfHead (x ∷ _) = just x\n\n -- Given a list of types, will return a possible list of goals that\n -- could be discharged by those types.\n stepGoals : RTerm ⊥ → List (Σ ℕ (RBinApp ∘ Fin)) → NonDet (List (RTerm ⊥))\n stepGoals g [] = return (g ∷ [])\n stepGoals g ((nₜ , t) ∷ ts) = apply g t ++ apply g (symmetric t) \n >>= (λ g' → stepGoals g' ts \n >>= return ∘ (_∷_ g))\n\n module Test where\n\n goal : RTerm ⊥\n goal = rapp (rdef (quote _≡_))\n (rapp (rdef (quote _+_)) (ivar 0 ∷ ivar 1 ∷ []) ∷\n rapp (rdef (quote _+_))\n (ivar 0 ∷\n rapp (rdef (quote _+_)) (ivar 1 ∷ rapp (rcon (quote zero)) [] ∷ [])\n ∷ [])\n ∷ [])\n\n binGoal : RBinApp ⊥\n binGoal = rdef (quote _≡_)\n , rapp (rdef (quote _+_)) (ivar 0 ∷ ivar 1 ∷ [])\n , rapp (rdef (quote _+_)) (ivar 0 ∷ (rapp (rdef (quote _+_)) (ivar 1 ∷ (rapp (rcon (quote zero)) []) ∷ []) ∷ []))\n\n +-ri : RBinApp (Fin 1)\n +-ri = rdef (quote _≡_)\n , rapp (rdef (quote _+_)) (ovar fz ∷ rapp (rcon (quote zero)) [] ∷ [])\n , ovar fz\n\n +-a : RBinApp (Fin 3)\n +-a = rdef (quote _≡_)\n , rapp (rdef (quote _+_)) (rapp (rdef (quote _+_)) (ovar (fs $ fs fz) ∷ ovar (fs fz) ∷ []) ∷ ovar fz ∷ [])\n , rapp (rdef (quote _+_)) (ovar (fs $ fs fz) ∷ rapp (rdef (quote _+_)) (ovar (fs fz) ∷ ovar fz ∷ []) ∷ [])\n\n\n tylist : List (Σ ℕ (RBinApp ∘ Fin))\n tylist = (1 , +-ri) ∷ (3 , +-a) ∷ []\n", "meta": {"hexsha": "c8360c99ccab2537fa801087410a5dbc63544dd7", "size": 7008, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "RW/Language/GoalGuesser.agda", "max_stars_repo_name": "VictorCMiraldo/agda-rw", "max_stars_repo_head_hexsha": "2856afd12b7dbbcc908482975638d99220f38bf2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 16, "max_stars_repo_stars_event_min_datetime": "2015-02-09T15:43:38.000Z", "max_stars_repo_stars_event_max_datetime": "2019-10-24T17:38:20.000Z", "max_issues_repo_path": "RW/Language/GoalGuesser.agda", "max_issues_repo_name": "VictorCMiraldo/agda-rw", "max_issues_repo_head_hexsha": "2856afd12b7dbbcc908482975638d99220f38bf2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2015-02-06T15:03:33.000Z", "max_issues_repo_issues_event_max_datetime": "2015-05-28T14:48:03.000Z", "max_forks_repo_path": "RW/Language/GoalGuesser.agda", "max_forks_repo_name": "VictorCMiraldo/agda-rw", "max_forks_repo_head_hexsha": "2856afd12b7dbbcc908482975638d99220f38bf2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.9938650307, "max_line_length": 125, "alphanum_fraction": 0.5409531963, "num_tokens": 2368, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.5234203489363239, "lm_q1q2_score": 0.34828941989920603}} {"text": "{-# OPTIONS --without-K #-}\nmodule Model.Size where\n\nopen import Relation.Binary using (Rel ; Preorder ; IsPreorder)\n\nimport Data.Nat as ℕ\nimport Data.Nat.Induction as ℕ\nimport Data.Nat.Properties as ℕ\nimport Relation.Binary.Construct.On as On\n\nopen import Model.RGraph as RG using (RGraph)\nopen import Source.Size as S using (Δ ; Ω)\nopen import Source.Size.Substitution.Universe using (⟨_⟩ ; Sub⊢ᵤ)\nopen import Util.HoTT.HLevel\nopen import Util.Induction.WellFounded as WFInd using (Acc ; acc ; WellFounded)\nopen import Util.Prelude\n\nimport Source.Size.Substitution.Canonical as SC\nimport Source.Size.Substitution.Universe as S\n\nopen S.Ctx\nopen S.Size\nopen S.Sub\nopen S.Sub⊢ᵤ\nopen S.Var\nopen S._<_ hiding (<-trans)\n\n\ninfix 4 _<_ _≤_\n\n\nprivate\n variable\n i j k : ℕ\n\n\n{-\nThis is an encoding of the set of ordinals\n\n ℕ ∪ { ω + i | i ∈ ℕ }\n\ni.e. of the ordinals below ω*2.\n-}\ndata Size : Set where\n zero+ : (i : ℕ) → Size\n ∞+ : (i : ℕ) → Size\n\n\nvariable\n n m o : Size\n\n\nnat : ℕ → Size\nnat = zero+\n\n\nabstract\n zero+-inj : zero+ i ≡ zero+ j → i ≡ j\n zero+-inj refl = refl\n\n\n zero+-≡-canon : (p : zero+ i ≡ zero+ j) → p ≡ cong zero+ (zero+-inj p)\n zero+-≡-canon refl = refl\n\n\n ∞+-inj : ∞+ i ≡ ∞+ j → i ≡ j\n ∞+-inj refl = refl\n\n\n ∞+-≡-canon : (p : ∞+ i ≡ ∞+ j) → p ≡ cong ∞+ (∞+-inj p)\n ∞+-≡-canon refl = refl\n\n\n Size-IsSet : IsSet Size\n Size-IsSet {zero+ i} {zero+ j} p refl\n = trans (zero+-≡-canon p) (cong (cong zero+) (ℕ.≡-irrelevant _ _))\n Size-IsSet {∞+ i} {∞+ j} p refl\n = trans (∞+-≡-canon p) (cong (cong ∞+) (ℕ.≡-irrelevant _ _))\n\n\ndata _<_ : (n m : Size) → Set where\n zero+ : (i S (p , v)) → ⟦ C ⟧IndArgω X (p , v , s)\n⟦ A ⊗ B ⟧IndArgω X pv = ⟦ A ⟧IndArgω X pv ×ω ⟦ B ⟧IndArgω X pv\n\n⟦_⟧Conω : ConDesc P V I → (⟦ P , I ⟧xtel → Setω) → ⟦ P , V & I ⟧xtel → Setω\n⟦ var f ⟧Conω X (p , v , i) = Liftω (i ≡ f (p , v))\n⟦ π (n , ai) S C ⟧Conω X (p , v , i) = Σω (< relevance ai > S _) λ s → ⟦ C ⟧Conω X (p , (v , s) , i)\n⟦ A ⊗ B ⟧Conω X (p , v , i) = ⟦ A ⟧IndArgω X (p , v) ×ω ⟦ B ⟧Conω X (p , v , i)\n\n⟦_⟧Dataω : DataDesc P I n → (⟦ P , I ⟧xtel → Setω) → ⟦ P , I ⟧xtel → Setω\n⟦_⟧Dataω {n = n} D X (p , i) = Σω (Fin n) λ k → ⟦ lookupCon D k ⟧Conω X (p , tt , i)\n\ndata μ (D : DataDesc P I n) (pi : ⟦ P , I ⟧xtel) : Setω where\n ⟨_⟩ : ⟦ D ⟧Dataω (μ D) pi → μ D pi\n", "meta": {"hexsha": "3cfab5623429ae84f43eaa2cf221ebe581a9b200", "size": 1143, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Generics/Mu.agda", "max_stars_repo_name": "flupe/generics", "max_stars_repo_head_hexsha": "db764f858d908aa39ea4901669a6bbce1525f757", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2021-04-08T15:10:20.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-05T09:35:17.000Z", "max_issues_repo_path": "src/Generics/Mu.agda", "max_issues_repo_name": "flupe/generics", "max_issues_repo_head_hexsha": "db764f858d908aa39ea4901669a6bbce1525f757", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2021-09-13T07:33:50.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-14T10:48:30.000Z", "max_forks_repo_path": "src/Generics/Mu.agda", "max_forks_repo_name": "flupe/generics", "max_forks_repo_head_hexsha": "db764f858d908aa39ea4901669a6bbce1525f757", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-04-08T08:32:42.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-14T10:35:16.000Z", "avg_line_length": 34.6363636364, "max_line_length": 100, "alphanum_fraction": 0.4943132108, "num_tokens": 563, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.34777114915916146}} {"text": "------------------------------------------------------------------------\n-- A breadth-first backend which uses the derivative operator\n------------------------------------------------------------------------\n\nmodule TotalParserCombinators.BreadthFirst where\n\nopen import Data.List\nopen import Data.List.Membership.Propositional\nopen import Data.Product\nopen import Function.Base\nopen import Function.Equality using (_⟨$⟩_)\nopen import Function.Inverse using (_↔_; module Inverse)\nopen import Relation.Binary.HeterogeneousEquality as H\n using () renaming (_≅_ to _≅H_)\nopen import Relation.Binary.PropositionalEquality as P using (_≡_)\n\nopen import TotalParserCombinators.Congruence using (_≅P_; _∎)\nimport TotalParserCombinators.Congruence.Sound as CS\nopen import TotalParserCombinators.Derivative as D using (D)\nimport TotalParserCombinators.InitialBag as I\nopen import TotalParserCombinators.Parser\nopen import TotalParserCombinators.Semantics\nopen import TotalParserCombinators.Simplification as S using (simplify)\n\n------------------------------------------------------------------------\n-- A parametrised backend\n\n-- The function f is applied before the derivative.\n\nmodule Parse\n {Tok}\n (f : ∀ {R xs} → Parser Tok R xs → ∃ λ xs′ → Parser Tok R xs′)\n (f-correct : ∀ {R xs} (p : Parser Tok R xs) → proj₂ (f p) ≅P p)\n where\n\n -- The parsing function.\n\n parse : ∀ {R xs} → Parser Tok R xs → List Tok → List R\n parse {xs = xs} p [] = xs\n parse p (t ∷ s) = parse (D t $ proj₂ $ f p) s\n\n -- A variant of f-correct.\n\n f-correct′ : ∀ {R xs} (p : Parser Tok R xs) → proj₂ (f p) ≅ p\n f-correct′ = CS.sound ∘ f-correct\n\n -- The backend is sound with respect to the semantics.\n\n sound : ∀ {R xs x} {p : Parser Tok R xs} (s : List Tok) →\n x ∈ parse p s → x ∈ p · s\n sound [] x∈p = I.sound _ x∈p\n sound (t ∷ s) x∈p =\n Inverse.to (f-correct′ _) ⟨$⟩ D.sound _ (sound s x∈p)\n\n -- The backend is complete with respect to the semantics.\n\n complete : ∀ {R xs x} {p : Parser Tok R xs} (s : List Tok) →\n x ∈ p · s → x ∈ parse p s\n complete [] x∈p = I.complete x∈p\n complete (t ∷ s) x∈p =\n complete s $ D.complete $ Inverse.from (f-correct′ _) ⟨$⟩ x∈p\n\n -- The backend does not introduce any unneeded ambiguity.\n --\n -- The proof complete is a left inverse of sound, so the (finite) type\n -- x ∈ parse p s contains at most as many proofs as x ∈ p · s. In\n -- other words, the output of parse p s can only contain n copies of x\n -- if there are at least n distinct parse trees in x ∈ p · s.\n\n complete∘sound : ∀ {R xs x} s\n (p : Parser Tok R xs) (x∈p : x ∈ parse p s) →\n complete s (sound s x∈p) ≡ x∈p\n complete∘sound [] p x∈p = I.complete∘sound p x∈p\n complete∘sound (t ∷ s) p x∈p\n rewrite Inverse.left-inverse-of (f-correct′ p)\n (D.sound (proj₂ (f p)) (sound s x∈p))\n | D.complete∘sound (proj₂ (f p)) (sound s x∈p) =\n complete∘sound s (D t $ proj₂ $ f p) x∈p\n\n -- The backend does not remove any ambiguity.\n --\n -- The proof complete is a right inverse of sound, which implies that\n -- the (finite) type x ∈ parse p s contains at least as many proofs as\n -- x ∈ p · s. In other words, if the output of parse p s contains n\n -- copies of x, then there are at most n distinct parse trees in\n -- x ∈ p · s.\n\n sound∘complete : ∀ {R xs x} {p : Parser Tok R xs}\n (s : List Tok) (x∈p : x ∈ p · s) →\n sound s (complete s x∈p) ≡ x∈p\n sound∘complete [] x∈p = I.sound∘complete x∈p\n sound∘complete (t ∷ s) x∈p\n rewrite sound∘complete s $\n D.complete $ Inverse.from (f-correct′ _) ⟨$⟩ x∈p\n | D.sound∘complete $ Inverse.from (f-correct′ _) ⟨$⟩ x∈p\n = Inverse.right-inverse-of (f-correct′ _) x∈p\n\n -- The backend is correct.\n\n correct : ∀ {R xs x s} {p : Parser Tok R xs} →\n x ∈ p · s ↔ x ∈ parse p s\n correct {s = s} {p} = record\n { to = P.→-to-⟶ $ complete s\n ; from = P.→-to-⟶ $ sound s\n ; inverse-of = record\n { left-inverse-of = sound∘complete s\n ; right-inverse-of = complete∘sound s p\n }\n }\n\n------------------------------------------------------------------------\n-- Specific instantiations\n\n-- Parsing without simplification.\n\nparse : ∀ {Tok R xs} → Parser Tok R xs → List Tok → List R\nparse = Parse.parse -,_ _∎\n\nparse-correct : ∀ {Tok R xs x s} {p : Parser Tok R xs} →\n x ∈ p · s ↔ x ∈ parse p s\nparse-correct = Parse.correct -,_ _∎\n\n-- Parsing with simplification.\n\nparse-with-simplification :\n ∀ {Tok R xs} → Parser Tok R xs → List Tok → List R\nparse-with-simplification = Parse.parse (λ p → -, simplify p) S.correct\n\nparse-with-simplification-correct :\n ∀ {Tok R xs x s} {p : Parser Tok R xs} →\n x ∈ p · s ↔ x ∈ parse-with-simplification p s\nparse-with-simplification-correct = Parse.correct _ S.correct\n\n------------------------------------------------------------------------\n-- An observation\n\n-- The worst-case complexity of parse (without simplification) is /at\n-- least/ exponential in the size of the input string. There is a\n-- (finite) parser p whose derivative is p ∣ p (for any token). The\n-- n-th derivative thus contains (at least) 2^n outermost occurrences\n-- of _∣_, and these occurrences have to be traversed to compute the\n-- initial bag of the n-th derivative.\n\nparse-inefficient :\n ∀ {Tok R} → ∃ λ (p : Parser Tok R []) → ∀ t → D t p ≅H p ∣ p\nparse-inefficient {R = R} =\n (fail {R = R} >>= (λ _ → fail) , λ t → H.refl)\n", "meta": {"hexsha": "8c5cd22bd7a3011aa2e88929e0b929db66417395", "size": 5529, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "TotalParserCombinators/BreadthFirst.agda", "max_stars_repo_name": "nad/parser-combinators", "max_stars_repo_head_hexsha": "76774f54f466cfe943debf2da731074fe0c33644", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-07-03T08:56:13.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-03T08:56:13.000Z", "max_issues_repo_path": "TotalParserCombinators/BreadthFirst.agda", "max_issues_repo_name": "nad/parser-combinators", "max_issues_repo_head_hexsha": "76774f54f466cfe943debf2da731074fe0c33644", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "TotalParserCombinators/BreadthFirst.agda", "max_forks_repo_name": "nad/parser-combinators", "max_forks_repo_head_hexsha": "76774f54f466cfe943debf2da731074fe0c33644", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.3581081081, "max_line_length": 72, "alphanum_fraction": 0.5832881172, "num_tokens": 1689, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.34749755709560654}} {"text": "open import Theory\nopen import Categories.Category using (Category)\nopen import Categories.Category.CartesianClosed\n\nmodule Soundness {o ℓ e}\n (𝒞 : Category o ℓ e)\n (CC : CartesianClosed 𝒞)\n {ℓ₁ ℓ₂ ℓ₃}\n (Th : Theory ℓ₁ ℓ₂ ℓ₃)\n where\n\n open Theory.Theory Th\n open import Semantics 𝒞 CC Sg\n open import Syntax\n open Term Sg\n\n open import Categories.Category.Cartesian 𝒞\n open import Categories.Category.BinaryProducts 𝒞\n open import Categories.Object.Product 𝒞\n open import Categories.Object.Terminal 𝒞 using (Terminal)\n\n open Category 𝒞\n open CartesianClosed CC\n open Cartesian cartesian\n open BinaryProducts products\n module T = Terminal terminal\n\n open import Data.Product using (Σ; Σ-syntax; proj₁; proj₂)\n\n module _ (M : Model 𝒞 CC Th) where\n open Structure (proj₁ M)\n open HomReasoning\n\n soundness : forall {Γ A} {e₁ e₂ : Γ ⊢ A}\n -> Γ ⊢ e₁ ≡ e₂\n -> ⟦ e₁ ⟧ ≈ ⟦ e₂ ⟧\n soundness-sub : forall {Γ Γ′} {γ₁ γ₂ : Γ ⊨ Γ′}\n -> Γ ⊨ γ₁ ≡ γ₂\n -> ⟦ γ₁ ⟧S ≈ ⟦ γ₂ ⟧S\n\n soundness (ax x) = proj₂ M x\n soundness refl = Equiv.refl\n soundness (sym D) = Equiv.sym (soundness D)\n soundness (trans D D₁) = Equiv.trans (soundness D) (soundness D₁)\n soundness sub/id = identityʳ\n soundness sub/∙ = sym-assoc\n soundness (eta/Unit e) = T.!-unique ⟦ e ⟧\n soundness (beta/*₁ _ e₂) = project₁\n soundness (beta/*₂ e₁ _) = project₂\n soundness (eta/* _) = g-η\n soundness (beta/=> e e′) =\n begin\n ⟦ app (abs e) e′ ⟧\n ≈˘⟨ ∘-resp-≈ʳ (Equiv.trans ⁂∘⟨⟩ (⟨⟩-cong₂ identityʳ identityˡ)) ⟩\n eval′ ∘ (λg ⟦ e ⟧ ⁂ Category.id 𝒞) ∘ ⟨ Category.id 𝒞 , ⟦ e′ ⟧ ⟩\n ≈⟨ Equiv.trans sym-assoc (∘-resp-≈ˡ β′) ⟩\n ⟦ e [ ext Term.id e′ ] ⟧\n ∎\n soundness (eta/=> e) =\n begin\n ⟦ abs (app (e [ weaken ]) var) ⟧\n ≈˘⟨ λ-cong (∘-resp-≈ʳ (⟨⟩-cong₂ Equiv.refl identityˡ)) ⟩\n λg (eval′ ∘ ⟨ ⟦ e ⟧ ∘ π₁ , Category.id 𝒞 ∘ π₂ ⟩)\n ≈˘⟨ λ-cong (∘-resp-≈ʳ ⁂∘⟨⟩) ⟩\n λg (eval′ ∘ (⟦ e ⟧ ⁂ Category.id 𝒞) ∘ ⟨ π₁ , π₂ ⟩)\n ≈⟨ λ-cong (∘-resp-≈ʳ (∘-resp-≈ʳ η)) ⟩\n λg (eval′ ∘ (⟦ e ⟧ ⁂ Category.id 𝒞) ∘ Category.id 𝒞)\n ≈⟨ λ-cong (∘-resp-≈ʳ identityʳ) ⟩\n λg (eval′ ∘ (⟦ e ⟧ ⁂ Category.id 𝒞))\n ≈⟨ η-exp′ ⟩\n ⟦ e ⟧\n ∎\n soundness (comm/func _ Γ′ γ f e) = assoc\n soundness (comm/unit _ Γ′ γ) = Equiv.sym (T.!-unique (T.! ∘ ⟦ γ ⟧S))\n soundness comm/pair = Product.∘-distribʳ-⟨⟩ product\n soundness comm/fst = assoc\n soundness comm/snd = assoc\n soundness (comm/abs {γ = γ} {e = e}) =\n begin\n λg ⟦ e ⟧ ∘ ⟦ γ ⟧S\n ≈⟨ exp.subst product product ⟩\n λg (⟦ e ⟧ ∘ (⟦ γ ⟧S ⁂ Category.id 𝒞))\n ≈˘⟨ λ-cong (∘-resp-≈ʳ (Equiv.trans (∘-resp-≈ʳ η) identityʳ)) ⟩\n λg (⟦ e ⟧ ∘ (⟦ γ ⟧S ⁂ Category.id 𝒞) ∘ ⟨ π₁ , π₂ ⟩)\n ≈⟨ λ-cong (∘-resp-≈ʳ (⁂∘⟨⟩)) ⟩\n λg (⟦ e ⟧ ∘ ⟨ ⟦ γ ⟧S ∘ π₁ , Category.id 𝒞 ∘ π₂ ⟩)\n ≈⟨ λ-cong (∘-resp-≈ʳ (⟨⟩-cong₂ Equiv.refl identityˡ)) ⟩\n λg (⟦ e ⟧ ∘ ⟨ ⟦ γ ⟧S ∘ π₁ , π₂ ⟩ )\n ∎\n soundness comm/app = Equiv.trans assoc (∘-resp-≈ʳ ⟨⟩∘)\n soundness (var/ext γ _) = project₂\n soundness (cong/sub D D′) = ∘-resp-≈ (soundness D′) (soundness-sub D)\n soundness (cong/pair D D₁) = ⟨⟩-cong₂ (soundness D) (soundness D₁)\n soundness (cong/fst D) = ∘-resp-≈ʳ (soundness D)\n soundness (cong/snd D) = ∘-resp-≈ʳ (soundness D)\n soundness (cong/app D D₁) = ∘-resp-≈ʳ (⟨⟩-cong₂ (soundness D) (soundness D₁))\n soundness (cong/abs D) = λ-cong (soundness D)\n soundness (cong/func D) = ∘-resp-≈ Equiv.refl (soundness D)\n\n soundness-sub refl = Equiv.refl\n soundness-sub (sym D) = Equiv.sym (soundness-sub D)\n soundness-sub (trans D D₁) = Equiv.trans (soundness-sub D) (soundness-sub D₁)\n soundness-sub (!-unique {γ = γ}) = T.!-unique ⟦ γ ⟧S\n soundness-sub η-pair =\n begin\n ⟨ π₁ , π₂ ⟩\n ≈⟨ Product.η product ⟩\n Category.id 𝒞\n ∎\n soundness-sub ext∙ = ⟨⟩∘\n soundness-sub (cong/ext D D′) = ⟨⟩-cong₂ (soundness-sub D) (soundness D′)\n soundness-sub weaken/ext = project₁\n soundness-sub (cong/∙ D D₁) = ∘-resp-≈ (soundness-sub D) (soundness-sub D₁)\n soundness-sub id∙ˡ = identityˡ\n soundness-sub id∙ʳ = identityʳ\n soundness-sub assoc∙ = assoc\n", "meta": {"hexsha": "7aa67eff992ab895223b96304fcbc7ed17888fab", "size": 4202, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Soundness.agda", "max_stars_repo_name": "elpinal/exsub-ccc", "max_stars_repo_head_hexsha": "7541ab22debdfe9d529ac7a210e5bd102c788ad9", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2022-02-05T06:16:32.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-05T13:30:48.000Z", "max_issues_repo_path": "Soundness.agda", "max_issues_repo_name": "elpinal/exsub-ccc", "max_issues_repo_head_hexsha": "7541ab22debdfe9d529ac7a210e5bd102c788ad9", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Soundness.agda", "max_forks_repo_name": "elpinal/exsub-ccc", "max_forks_repo_head_hexsha": "7541ab22debdfe9d529ac7a210e5bd102c788ad9", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.9145299145, "max_line_length": 81, "alphanum_fraction": 0.5628272251, "num_tokens": 1811, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7057850154599562, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3473790109789087}} {"text": "\n-- The previous error was:\n-- Auto in hole `?0` raises a typing error:\n-- (zero ∷ L) != L of type (List ℕ)\n-- when checking that the expression x₁ has type (foo L)\n\nopen import Agda.Builtin.List\nopen import Agda.Builtin.Nat\n\ndata ⊥ : Set where\n\nfoo : List Nat → Set\nfoo [] = ⊥\nfoo (zero ∷ L) = foo L\nfoo (suc N ∷ L) = foo (suc N ∷ L)\n\nbar : (L : List Nat) → (foo L) → List Nat\nbar [] x = []\nbar (zero ∷ L) x₁ = bar L {!!}\nbar (suc x ∷ L) x₁ = {!!}\n", "meta": {"hexsha": "30af5c5ed18cfae25a83698979351c8cbc2a80ed", "size": 463, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/interaction/Issue1169.agda", "max_stars_repo_name": "cruhland/agda", "max_stars_repo_head_hexsha": "7f58030124fa99dfbf8db376659416f3ad8384de", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1989, "max_stars_repo_stars_event_min_datetime": "2015-01-09T23:51:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T18:20:48.000Z", "max_issues_repo_path": "test/interaction/Issue1169.agda", "max_issues_repo_name": "cruhland/agda", "max_issues_repo_head_hexsha": "7f58030124fa99dfbf8db376659416f3ad8384de", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4066, "max_issues_repo_issues_event_min_datetime": "2015-01-10T11:24:51.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T21:14:49.000Z", "max_forks_repo_path": "test/interaction/Issue1169.agda", "max_forks_repo_name": "cruhland/agda", "max_forks_repo_head_hexsha": "7f58030124fa99dfbf8db376659416f3ad8384de", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 371, "max_forks_repo_forks_event_min_datetime": "2015-01-03T14:04:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-30T19:00:30.000Z", "avg_line_length": 22.0476190476, "max_line_length": 58, "alphanum_fraction": 0.56587473, "num_tokens": 170, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.34722652274188975}} {"text": "------------------------------------------------------------------------\n-- Full β-reduction in Fω with interval kinds\n------------------------------------------------------------------------\n\n{-# OPTIONS --safe --without-K #-}\n\nmodule FOmegaInt.Reduction.Full where\n\nopen import Data.Fin using (suc; zero)\nopen import Data.Fin.Substitution\nopen import Data.Fin.Substitution.ExtraLemmas\nopen import Data.List using ([]; _∷_)\nopen import Data.Sum using ([_,_])\nopen import Data.Product using (_,_; ∃; _×_)\nopen import Function using (flip; _∘_)\nopen import Relation.Binary using (Setoid; Preorder)\nimport Relation.Binary.Construct.Closure.Equivalence as EqClos\nopen import Relation.Binary.Construct.Closure.ReflexiveTransitive\n using (ε; _◅_; gmap)\nimport Relation.Binary.Construct.Closure.Symmetric as SymClos\nopen import Relation.Binary.PropositionalEquality as P using (_≡_; refl)\nopen import Relation.Binary.Reduction\nimport Function.Equivalence as Equiv\n\nopen import FOmegaInt.Syntax\n\nopen Syntax\nopen Substitution using (_[_]; weaken)\n\n----------------------------------------------------------------------\n-- Full β-reduction and equality relations\n\ninfixl 9 _·₁_ _·₂_ _⊡₁_ _⊡₂_\ninfixr 7 _⇒₁_ _⇒₂_\ninfix 6 _⋯₁_ _⋯₂_\ninfix 5 _→β_ _Kd→β_\n\nmutual\n\n -- The compatible closure of a binary term relation.\n\n data Compat (_∼_ : TRel Term) {n} : Term n → Term n → Set where\n ⌈_⌉ : ∀ {a₁ a₂} → a₁ ∼ a₂ → Compat _∼_ a₁ a₂\n Π₁ : ∀ {k₁ k₂} → KdCompat _∼_ k₁ k₂ → ∀ a → Compat _∼_ (Π k₁ a) (Π k₂ a)\n Π₂ : ∀ {a₁ a₂} k → Compat _∼_ a₁ a₂ → Compat _∼_ (Π k a₁) (Π k a₂)\n _⇒₁_ : ∀ {a₁ a₂} → Compat _∼_ a₁ a₂ → ∀ b → Compat _∼_ (a₁ ⇒ b) (a₂ ⇒ b)\n _⇒₂_ : ∀ {b₁ b₂} a → Compat _∼_ b₁ b₂ → Compat _∼_ (a ⇒ b₁) (a ⇒ b₂)\n Λ₁ : ∀ {k₁ k₂} → KdCompat _∼_ k₁ k₂ → ∀ a → Compat _∼_ (Λ k₁ a) (Λ k₂ a)\n Λ₂ : ∀ {a₁ a₂} k → Compat _∼_ a₁ a₂ → Compat _∼_ (Λ k a₁) (Λ k a₂)\n ƛ₁ : ∀ {a₁ a₂} → Compat _∼_ a₁ a₂ → ∀ b → Compat _∼_ (ƛ a₁ b) (ƛ a₂ b)\n ƛ₂ : ∀ {b₁ b₂} a → Compat _∼_ b₁ b₂ → Compat _∼_ (ƛ a b₁) (ƛ a b₂)\n _·₁_ : ∀ {a₁ a₂} → Compat _∼_ a₁ a₂ → ∀ b → Compat _∼_ (a₁ · b) (a₂ · b)\n _·₂_ : ∀ {b₁ b₂} a → Compat _∼_ b₁ b₂ → Compat _∼_ (a · b₁) (a · b₂)\n _⊡₁_ : ∀ {a₁ a₂} → Compat _∼_ a₁ a₂ → ∀ b → Compat _∼_ (a₁ ⊡ b) (a₂ ⊡ b)\n _⊡₂_ : ∀ {b₁ b₂} a → Compat _∼_ b₁ b₂ → Compat _∼_ (a ⊡ b₁) (a ⊡ b₂)\n\n data KdCompat (_∼_ : TRel Term) {n} : Kind Term n → Kind Term n → Set where\n _⋯₁_ : ∀ {a₁ a₂} → Compat _∼_ a₁ a₂ → ∀ b → KdCompat _∼_ (a₁ ⋯ b) (a₂ ⋯ b)\n _⋯₂_ : ∀ {b₁ b₂} a → Compat _∼_ b₁ b₂ → KdCompat _∼_ (a ⋯ b₁) (a ⋯ b₂)\n Π₁ : ∀ {j₁ j₂} → KdCompat _∼_ j₁ j₂ → ∀ b → KdCompat _∼_ (Π j₁ b) (Π j₂ b)\n Π₂ : ∀ {k₁ k₂} k → KdCompat _∼_ k₁ k₂ → KdCompat _∼_ (Π k k₁) (Π k k₂)\n\n-- β-contraction.\ndata BetaCont {n} : Term n → Term n → Set where\n cont-Tp· : ∀ k a b → BetaCont ((Λ k a) · b) (a [ b ])\n cont-Tm· : ∀ a b c → BetaCont ((ƛ a b) · c) (b [ c ])\n cont-⊡ : ∀ k a b → BetaCont ((Λ k a) ⊡ b) (a [ b ])\n\n-- One-step β-reduction.\n\n_→β_ : TRel Term\n_→β_ = Compat BetaCont\n\n_Kd→β_ : TRel (Kind Term)\n_Kd→β_ = KdCompat BetaCont\n\n-- Full β-reduction and equality.\n\nβ-reduction : Reduction Term\nβ-reduction = record { _→1_ = _→β_ }\n\nβ-reductionKind : Reduction (Kind Term)\nβ-reductionKind = record { _→1_ = _Kd→β_ }\n\nopen Reduction β-reduction public hiding (_→1_)\n renaming (_→*_ to _→β*_; _↔_ to _≡β_)\n\nopen Reduction β-reductionKind public hiding (_→1_)\n renaming (_→*_ to _Kd→β*_; _↔_ to _Kd≡β_)\n\n\n----------------------------------------------------------------------\n-- Simple properties of the β-reductions/equalities\n\n-- Inclusions.\n\n→β⇒→β* = →1⇒→* β-reduction\n→β*⇒≡β = →*⇒↔ β-reduction\n→β⇒≡β = →1⇒↔ β-reduction\n\n-- Reduction is a preorders.\n\n→β*-preorder = →*-preorder β-reduction\nKd→β*-preorder = →*-preorder β-reductionKind\n\n-- Preorder reasoning for reduction.\n\nmodule →β*-Reasoning = →*-Reasoning β-reduction\nmodule Kd→β*-Reasoning = →*-Reasoning β-reductionKind\n\n-- Raw terms and kinds together with β-equality form setoids.\n\n≡β-setoid = ↔-setoid β-reduction\nKd≡β-setoid = ↔-setoid β-reductionKind\n\n-- Equational reasoning for the equality.\n\nmodule ≡β-Reasoning = ↔-Reasoning β-reduction\nmodule Kd≡β-Reasoning = ↔-Reasoning β-reductionKind\n\n-- Multistep and equality congruence lemmas.\n\nmodule CongLemmas (_∼_ : TRel Term) where\n\n private\n reduction : Reduction Term\n reduction = record { _→1_ = Compat _∼_ }\n\n reductionKd : Reduction (Kind Term)\n reductionKd = record { _→1_ = KdCompat _∼_ }\n\n module O {n} = Preorder (→*-preorder reduction n)\n module S {n} = Setoid (↔-setoid reduction n)\n module OK {n} = Preorder (→*-preorder reductionKd n)\n module SK {n} = Setoid (↔-setoid reductionKd n)\n\n open Reduction reduction\n open Reduction reductionKd using () renaming (_→*_ to _Kd→*_; _↔_ to _Kd↔_)\n\n -- A Congruence lemma for _⌞∙⌟_.\n\n →1-⌞∙⌟₁ : ∀ {n} {a₁ a₂ : Term n} → a₁ →1 a₂ → ∀ bs → a₁ ⌞∙⌟ bs →1 a₂ ⌞∙⌟ bs\n →1-⌞∙⌟₁ a₁∼a₂ [] = a₁∼a₂\n →1-⌞∙⌟₁ a₁∼a₂ (b ∷ bs) = →1-⌞∙⌟₁ (a₁∼a₂ ·₁ b) bs\n\n -- Multistep congruence lemmas for transitive closures.\n\n →*-Π : ∀ {n} {k₁ k₂ : Kind Term n} {a₁ a₂} →\n k₁ Kd→* k₂ → a₁ →* a₂ → Π k₁ a₁ →* Π k₂ a₂\n →*-Π {a₁ = a₁} k₁→*k₂ a₁→*a₂ =\n O.trans (gmap (flip Π a₁) (λ j→k → Π₁ j→k a₁) k₁→*k₂)\n (gmap (Π _) (Π₂ _) a₁→*a₂)\n\n →*-⇒ : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ →* a₂ → b₁ →* b₂ → a₁ ⇒ b₁ →* a₂ ⇒ b₂\n →*-⇒ {b₁ = b₁} a₁→*a₂ b₁→*b₂ =\n O.trans (gmap (_⇒ b₁) (_⇒₁ b₁) a₁→*a₂) (gmap (_ ⇒_) (_ ⇒₂_) b₁→*b₂)\n\n →*-Λ : ∀ {n} {k₁ k₂ : Kind Term n} {a₁ a₂} →\n k₁ Kd→* k₂ → a₁ →* a₂ → Λ k₁ a₁ →* Λ k₂ a₂\n →*-Λ {a₁ = a₁} k₁→*k₂ a₁→*a₂ =\n O.trans (gmap (flip Λ a₁) (λ j→k → Λ₁ j→k a₁) k₁→*k₂)\n (gmap (Λ _) (Λ₂ _) a₁→*a₂)\n\n →*-ƛ : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ →* a₂ → b₁ →* b₂ → ƛ a₁ b₁ →* ƛ a₂ b₂\n →*-ƛ {b₁ = b₁} a₁→*a₂ b₁→*b₂ =\n O.trans (gmap (flip ƛ b₁) (flip ƛ₁ b₁) a₁→*a₂) (gmap (ƛ _) (ƛ₂ _) b₁→*b₂)\n\n →*-· : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ →* a₂ → b₁ →* b₂ → a₁ · b₁ →* a₂ · b₂\n →*-· {b₁ = b₁} a₁→*a₂ b₁→*b₂ =\n O.trans (gmap (_· b₁) (_·₁ b₁) a₁→*a₂) (gmap (_ ·_) (_ ·₂_) b₁→*b₂)\n\n →*-⊡ : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ →* a₂ → b₁ →* b₂ → a₁ ⊡ b₁ →* a₂ ⊡ b₂\n →*-⊡ {b₁ = b₁} a₁→*a₂ b₁→*b₂ =\n O.trans (gmap (_⊡ b₁) (_⊡₁ b₁) a₁→*a₂) (gmap (_ ⊡_) (_ ⊡₂_) b₁→*b₂)\n\n Kd→*-⋯ : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ →* a₂ → b₁ →* b₂ → a₁ ⋯ b₁ Kd→* a₂ ⋯ b₂\n Kd→*-⋯ {b₁ = b₁} a₁→*a₂ b₁→*b₂ =\n OK.trans (gmap (_⋯ b₁) (_⋯₁ b₁) a₁→*a₂) (gmap (_ ⋯_) (_ ⋯₂_) b₁→*b₂)\n\n Kd→*-Π : ∀ {n} {j₁ j₂ : Kind Term n} {k₁ k₂} →\n j₁ Kd→* j₂ → k₁ Kd→* k₂ → Π j₁ k₁ Kd→* Π j₂ k₂\n Kd→*-Π {k₁ = k₁} j₁→*j₂ k₁→*k₂ =\n OK.trans (gmap (flip Π k₁) (λ j→k → Π₁ j→k k₁) j₁→*j₂)\n (gmap (Π _) (Π₂ _) k₁→*k₂)\n\n →*-⌞∙⌟₁ : ∀ {n} {a₁ a₂ : Term n} → a₁ →* a₂ → ∀ bs → a₁ ⌞∙⌟ bs →* a₂ ⌞∙⌟ bs\n →*-⌞∙⌟₁ a₁→*a₂ bs = gmap (_⌞∙⌟ bs) (flip →1-⌞∙⌟₁ bs) a₁→*a₂\n\n -- Congruence lemmas for equivalences closures.\n\n ↔-Π : ∀ {n} {k₁ k₂ : Kind Term n} {a₁ a₂} →\n k₁ Kd↔ k₂ → a₁ ↔ a₂ → Π k₁ a₁ ↔ Π k₂ a₂\n ↔-Π {a₁ = a₁} k₁↔k₂ a₁↔a₂ =\n S.trans (EqClos.gmap (flip Π a₁) (λ j→k → Π₁ j→k a₁) k₁↔k₂)\n (EqClos.gmap (Π _) (Π₂ _) a₁↔a₂)\n\n ↔-⇒ : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ ↔ a₂ → b₁ ↔ b₂ → a₁ ⇒ b₁ ↔ a₂ ⇒ b₂\n ↔-⇒ {b₁ = b₁} a₁↔a₂ b₁↔b₂ =\n S.trans (EqClos.gmap (_⇒ b₁) (_⇒₁ b₁) a₁↔a₂)\n (EqClos.gmap (_ ⇒_) (_ ⇒₂_) b₁↔b₂)\n\n ↔-Λ : ∀ {n} {k₁ k₂ : Kind Term n} {a₁ a₂} →\n k₁ Kd↔ k₂ → a₁ ↔ a₂ → Λ k₁ a₁ ↔ Λ k₂ a₂\n ↔-Λ {a₁ = a₁} k₁↔k₂ a₁↔a₂ =\n S.trans (EqClos.gmap (flip Λ a₁) (λ j→k → Λ₁ j→k a₁) k₁↔k₂)\n (EqClos.gmap (Λ _) (Λ₂ _) a₁↔a₂)\n\n ↔-ƛ : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ ↔ a₂ → b₁ ↔ b₂ → ƛ a₁ b₁ ↔ ƛ a₂ b₂\n ↔-ƛ {b₁ = b₁} a₁↔a₂ b₁↔b₂ =\n S.trans (EqClos.gmap (flip ƛ b₁) (flip ƛ₁ b₁) a₁↔a₂)\n (EqClos.gmap (ƛ _) (ƛ₂ _) b₁↔b₂)\n\n ↔-· : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ ↔ a₂ → b₁ ↔ b₂ → a₁ · b₁ ↔ a₂ · b₂\n ↔-· {b₁ = b₁} a₁↔a₂ b₁↔b₂ =\n S.trans (EqClos.gmap (_· b₁) (_·₁ b₁) a₁↔a₂)\n (EqClos.gmap (_ ·_) (_ ·₂_) b₁↔b₂)\n\n ↔-⊡ : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ ↔ a₂ → b₁ ↔ b₂ → a₁ ⊡ b₁ ↔ a₂ ⊡ b₂\n ↔-⊡ {b₁ = b₁} a₁↔a₂ b₁↔b₂ =\n S.trans (EqClos.gmap (_⊡ b₁) (_⊡₁ b₁) a₁↔a₂)\n (EqClos.gmap (_ ⊡_) (_ ⊡₂_) b₁↔b₂)\n\n Kd↔-⋯ : ∀ {n} {a₁ a₂ : Term n} {b₁ b₂} →\n a₁ ↔ a₂ → b₁ ↔ b₂ → a₁ ⋯ b₁ Kd↔ a₂ ⋯ b₂\n Kd↔-⋯ {b₁ = b₁} a₁↔a₂ b₁↔b₂ =\n SK.trans (EqClos.gmap (_⋯ b₁) (_⋯₁ b₁) a₁↔a₂)\n (EqClos.gmap (_ ⋯_) (_ ⋯₂_) b₁↔b₂)\n\n Kd↔-Π : ∀ {n} {j₁ j₂ : Kind Term n} {k₁ k₂} →\n j₁ Kd↔ j₂ → k₁ Kd↔ k₂ → Π j₁ k₁ Kd↔ Π j₂ k₂\n Kd↔-Π {k₁ = k₁} j₁↔j₂ k₁↔k₂ =\n SK.trans (EqClos.gmap (flip Π k₁) (λ j→k → Π₁ j→k k₁) j₁↔j₂)\n (EqClos.gmap (Π _) (Π₂ _) k₁↔k₂)\n\n ↔-⌞∙⌟₁ : ∀ {n} {a₁ a₂ : Term n} → a₁ ↔ a₂ → ∀ bs → a₁ ⌞∙⌟ bs ↔ a₂ ⌞∙⌟ bs\n ↔-⌞∙⌟₁ a₁↔a₂ bs = EqClos.gmap (_⌞∙⌟ bs) (flip →1-⌞∙⌟₁ bs) a₁↔a₂\n\nopen CongLemmas BetaCont public renaming\n ( →1-⌞∙⌟₁ to →β-⌞∙⌟₁\n ; →*-Π to →β*-Π\n ; →*-⇒ to →β*-⇒\n ; →*-Λ to →β*-Λ\n ; →*-ƛ to →β*-ƛ\n ; →*-· to →β*-·\n ; →*-⊡ to →β*-⊡\n ; Kd→*-⋯ to Kd→β*-⋯\n ; Kd→*-Π to Kd→β*-Π\n ; →*-⌞∙⌟₁ to →β*-⌞∙⌟₁\n ; ↔-Π to ≡β-Π\n ; ↔-⇒ to ≡β-⇒\n ; ↔-Λ to ≡β-Λ\n ; ↔-ƛ to ≡β-ƛ\n ; ↔-· to ≡β-·\n ; ↔-⊡ to ≡β-⊡\n ; Kd↔-⋯ to Kd≡β-⋯\n ; Kd↔-Π to Kd≡β-Π\n ; ↔-⌞∙⌟₁ to ≡β-⌞∙⌟₁\n )\n", "meta": {"hexsha": "c925abaa1c16ef9d5e19b16a1e19826ddbdb9fc2", "size": 9518, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/FOmegaInt/Reduction/Full.agda", "max_stars_repo_name": "Blaisorblade/f-omega-int-agda", "max_stars_repo_head_hexsha": "ae20dac2a5e0c18dff2afda4c19954e24d73a24f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 12, "max_stars_repo_stars_event_min_datetime": "2017-06-13T16:05:35.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-27T05:53:06.000Z", "max_issues_repo_path": "src/FOmegaInt/Reduction/Full.agda", "max_issues_repo_name": "Blaisorblade/f-omega-int-agda", "max_issues_repo_head_hexsha": "ae20dac2a5e0c18dff2afda4c19954e24d73a24f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-05-14T08:09:40.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-14T08:54:39.000Z", "max_forks_repo_path": "src/FOmegaInt/Reduction/Full.agda", "max_forks_repo_name": "Blaisorblade/f-omega-int-agda", "max_forks_repo_head_hexsha": "ae20dac2a5e0c18dff2afda4c19954e24d73a24f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-05-13T22:29:48.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-14T10:25:05.000Z", "avg_line_length": 35.5149253731, "max_line_length": 80, "alphanum_fraction": 0.4968480773, "num_tokens": 4941, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.34722652274188975}} {"text": "{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-}\n{-# OPTIONS --allow-unsolved-metas #-}\n\nmodule Light.Implementation.Relation.Decidable where\n\nopen import Light.Variable.Levels\nopen import Light.Variable.Sets\nopen import Light.Library.Relation.Decidable using (Library ; Dependencies)\nopen import Light.Library.Data.Empty as Empty using (Empty)\nopen import Light.Library.Data.Unit as Unit using (Unit ; unit)\nopen import Light.Library.Data.Both as Both using (Both)\nimport Light.Library.Data.Product as Product\nopen import Light.Library.Data.These as These using (These)\nimport Light.Implementation.Data.Empty\nimport Light.Implementation.Data.Unit\nimport Light.Implementation.Data.Product\nimport Light.Implementation.Data.These\nimport Light.Package\n\ninstance dependencies : Dependencies\ndependencies = record {}\n\ninstance library : Library dependencies\nlibrary = record\n {\n Implementation ;\n to‐witness = λ ⦃ a ⦄ → Implementation.to‐witness a ;\n to‐false‐witness = λ ⦃ a ⦄ → Implementation.to‐false‐witness a ;\n from‐witness = λ ⦃ a ⦄ → Implementation.from‐witness a ;\n from‐false‐witness = λ ⦃ a ⦄ → Implementation.from‐false‐witness a\n }\n where\n module Implementation where\n open import Light.Implementation.Relation.Sets using (base) public\n \n data Decidable (𝕒 : Set ℓ) : Set ℓ where\n yes : 𝕒 → Decidable 𝕒\n no : (𝕒 → Empty) → Decidable 𝕒\n \n module Style where\n true : Decidable Unit\n true = yes unit\n \n false : Decidable Empty\n false = no λ a → a\n \n ¬_ : Decidable 𝕒 → Decidable (𝕒 → Empty)\n ¬ yes a = no λ f → f a\n ¬ no a = yes a\n \n _∧_ : ∀ {𝕒 : Set aℓ} {𝕓 : Set bℓ} (a? : Decidable 𝕒) (b? : Decidable 𝕓) → Decidable (Both 𝕒 𝕓)\n yes a ∧ yes b = yes (Both.both a b)\n no f ∧ _ = no λ both → f (Both.first both)\n _ ∧ no f = no λ both → f (Both.second both)\n \n _∨_ : ∀ {𝕒 : Set aℓ} {𝕓 : Set bℓ} (a? : Decidable 𝕒) (b? : Decidable 𝕓) → Decidable (These 𝕒 𝕓)\n no af ∨ no bf = no {!!}\n yes a ∨ yes b = yes (These.these a b)\n yes a ∨ no _ = yes (These.this a)\n no _ ∨ yes b = yes (These.that b)\n \n _⇢_ : ∀ {𝕒 : Set aℓ} {𝕓 : Set bℓ} (a? : Decidable 𝕒) (b? : Decidable 𝕓) → Decidable (𝕒 → 𝕓)\n _ ⇢ yes b = yes λ _ → b\n no af ⇢ _ = yes λ a → Empty.eliminate (af a)\n yes a ⇢ no bf = no λ f → bf (f a)\n \n True False : Decidable 𝕒 → Set\n \n True (yes _) = Unit\n True (no _) = Empty\n False a? = True (¬ a?)\n \n style = record { Style }\n \n if′_then_else_ : Decidable 𝕒 → 𝕓 → 𝕓 → 𝕓\n if′ yes _ then a else _ = a\n if′ no _ then _ else a = a\n \n if_then_else_ :\n ∀ (a : Decidable 𝕒)\n → (∀ ⦃ witness : 𝕒 ⦄ → 𝕓)\n → (∀ ⦃ witness : 𝕒 → Empty ⦄ → 𝕓)\n → 𝕓\n if yes w then a else _ = a ⦃ witness = w ⦄\n if no w then _ else a = a ⦃ witness = w ⦄\n \n to‐witness : ∀ {a? : Decidable 𝕒} → Style.True a? → 𝕒\n to‐false‐witness : ∀ {a? : Decidable 𝕒} → Style.False a? → 𝕒 → Empty\n \n from‐witness : ∀ {a? : Decidable 𝕒} → 𝕒 → Style.True a?\n from‐false‐witness : ∀ {a? : Decidable 𝕒} → (𝕒 → Empty) → Style.False a?\n \n to‐witness {a? = yes a} _ = a\n to‐false‐witness {a? = no f} _ = f\n \n from‐witness {a? = yes a} _ = _\n from‐witness {a? = no f} w = f w\n from‐false‐witness {a? = no f} _ = _\n from‐false‐witness {a? = yes a} w = w a\n", "meta": {"hexsha": "dbd5754fba7833bd08a7fa92d8cb1a86e05a0962", "size": 4150, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Light/Implementation/Relation/Decidable.agda", "max_stars_repo_name": "zamfofex/lightlib", "max_stars_repo_head_hexsha": "44b1c724f2de95d3a9effe87ca36ef9eca8b4756", "max_stars_repo_licenses": ["0BSD"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-12-20T21:33:05.000Z", "max_stars_repo_stars_event_max_datetime": "2019-12-20T21:33:05.000Z", "max_issues_repo_path": "Light/Implementation/Relation/Decidable.agda", "max_issues_repo_name": "Zambonifofex/lightlib", "max_issues_repo_head_hexsha": "44b1c724f2de95d3a9effe87ca36ef9eca8b4756", "max_issues_repo_licenses": ["0BSD"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Light/Implementation/Relation/Decidable.agda", "max_forks_repo_name": "Zambonifofex/lightlib", "max_forks_repo_head_hexsha": "44b1c724f2de95d3a9effe87ca36ef9eca8b4756", "max_forks_repo_licenses": ["0BSD"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.5, "max_line_length": 113, "alphanum_fraction": 0.4860240964, "num_tokens": 1186, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3471944273284509}} {"text": "{-# OPTIONS --without-K #-}\n\nmodule TypeEquivEquiv where\n\nopen import Equiv\n using (refl∼; sym∼; trans∼; sym≃;\n _⊎≃_; id≃; _≃_; _●_; _×≃_; qinv; gg;\n β⊎₁; β⊎₂; β₁; β₂; cong∘l; cong∘r; cong₂∘; β×₁; β×₂)\nopen import TypeEquiv\n using (unite₊equiv; uniti₊equiv; unite₊′equiv; uniti₊′equiv;\n assocr₊equiv; assocl₊equiv; swap₊equiv;\n unite⋆equiv; uniti⋆equiv; unite⋆′equiv; uniti⋆′equiv;\n assocr⋆equiv; assocl⋆equiv; swap⋆equiv;\n distlequiv; factorlequiv; distequiv; factorequiv;\n distzrequiv; factorzrequiv; distzequiv; factorzequiv)\nopen import EquivEquiv\n\nopen import Data.Empty using (⊥)\nopen import Data.Unit using (⊤)\nopen import Data.Sum using (_⊎_)\nopen import Data.Product using (_,_; _×_; proj₁)\nopen import Function using (_∘_)\n\nopen import Data.Sum.Properties2\n using (id⊎id∼id; ⊎∘∼∘⊎; _⊎∼_;\n unite₊-coh; uniti₊-coh; unite₊′-coh; uniti₊′-coh;\n assocr₊-wf; assocl₊-wf;\n triangle⊎-left; triangle⊎-right;\n pentagon⊎-right; pentagon⊎-left;\n swap₊-coh; unite₊-swap-coh-left; unite₊-swap-coh-right;\n hexagon⊎-right; hexagon⊎-left)\n\nopen import Data.Product.Properties2\n using (id×id∼id; ×∘∼∘×; _×∼_;\n unite⋆-coh; uniti⋆-coh; unite⋆′-coh; uniti⋆′-coh;\n assocr⋆-wf; assocl⋆-wf;\n triangle×-left; triangle×-right;\n pentagon×-right; pentagon×-left;\n swap⋆-coh; unite⋆-swap-coh-left; unite⋆-swap-coh-right;\n hexagon×-right; hexagon×-left)\n\nopen import Data.SumProd.Properties -- TODO: list them\n\n-- some local abbreviations to make life nicer\ninfixr 10 _⊙_\n\nprivate\n _⊙_ = trans∼\n !_ = sym∼\n\n-- we define all the equivalences-between-equivalences that hold\n-- between type equivalences.\n\n-- TODO: quite possibly, everything here should be abstract, as\n-- it shouldn't be useful to look into the structure of the proofs,\n-- but just that they do, indeed exist. That and the proofs are in\n-- fact rather boring, consisting of explicit sequences of\n-- unveil --> delegate to lower level -> veil\n-- operations.\n----\n-- equivalences for the ⊎ structure\n\n[id,id]≋id : ∀ {A B : Set} → id≃ {A = A} ⊎≃ id≃ {A = B} ≋ id≃\n[id,id]≋id = eq (β⊎₁ ⊙ id⊎id∼id) (β⊎₂ ⊙ id⊎id∼id)\n\n-- ● and ⊎≃ commute.\n\n⊎●≋●⊎ : {A B C D E F : Set} →\n {f : A ≃ C} {g : B ≃ D} {h : C ≃ E} {i : D ≃ F} →\n (h ● f) ⊎≃ (i ● g) ≋ (h ⊎≃ i) ● (f ⊎≃ g)\n⊎●≋●⊎ =\n eq (β⊎₁ ⊙ β₁ ⊎∼ β₁ ⊙ ⊎∘∼∘⊎ ⊙ ! cong₂∘ β⊎₁ β⊎₁ ⊙ ! β₁)\n (β⊎₂ ⊙ β₂ ⊎∼ β₂ ⊙ ⊎∘∼∘⊎ ⊙ ! cong₂∘ β⊎₂ β⊎₂ ⊙ ! β₂)\n\n-- ≋ has, predictably, an additive structure as well\n\n_⊎≋_ : {A B C D : Set} {f h : A ≃ B} {g i : C ≃ D} → f ≋ h → g ≋ i →\n f ⊎≃ g ≋ h ⊎≃ i\nf≋h ⊎≋ g≋i =\n eq (β⊎₁ ⊙ (f≡ f≋h) ⊎∼ (f≡ g≋i) ⊙ ! β⊎₁)\n (β⊎₂ ⊙ (g≡ f≋h) ⊎∼ (g≡ g≋i) ⊙ ! β⊎₂)\n where open _≋_\n\n-- strangely, this is not needed by Rig Category. However, it\n-- belongs to the structure of a Monoidal Groupoid (I think! Should\n-- be checked), and is quite useful on its own.\n\nsym≃-distrib⊎ : ∀ {A B C D : Set} {f : A ≃ B} {g : C ≃ D} →\n sym≃ (f ⊎≃ g) ≋ sym≃ f ⊎≃ sym≃ g\nsym≃-distrib⊎ = -- note how the proof mixes ₁ and ₂ !\n eq (β⊎₂ ⊙ ! β⊎₁) (β⊎₁ ⊙ ! β⊎₂)\n\n-- Use '-nat' to signify that operation induces a\n-- natural transformation, and that the induced operation\n-- satisfies the naturality condition thus encoded\n\nunite₊-nat : ∀ {A B} {f : A ≃ B} {g : ⊥ ≃ ⊥} →\n unite₊equiv ● (g ⊎≃ f) ≋ f ● unite₊equiv\nunite₊-nat {g = g} =\n eq (β₁ ⊙ cong∘l (proj₁ unite₊equiv) β⊎₁ ⊙ unite₊-coh ⊙ ! β₁)\n (β₂ ⊙ cong∘r (gg unite₊equiv) β⊎₂ ⊙ uniti₊-coh {g = gg g} ⊙ ! β₂)\n\nuniti₊-nat : ∀ {A B} {f : A ≃ B} {g : ⊥ ≃ ⊥} →\n uniti₊equiv ● f ≋ (g ⊎≃ f) ● uniti₊equiv\nuniti₊-nat {g = g} =\n eq (β₁ ⊙ ! uniti₊-coh {g = proj₁ g} ⊙ ! cong∘r (proj₁ uniti₊equiv) β⊎₁ ⊙ ! β₁)\n (β₂ ⊙ ! unite₊-coh ⊙ ! cong∘l (gg uniti₊equiv) β⊎₂ ⊙ ! β₂)\n\nunite₊′-nat : ∀ {A B} {f : A ≃ B} {g : ⊥ ≃ ⊥} →\n unite₊′equiv ● (f ⊎≃ g) ≋ f ● unite₊′equiv\nunite₊′-nat {g = g} =\n eq (β₁ ⊙ cong∘l (proj₁ unite₊′equiv) β⊎₁ ⊙ unite₊′-coh ⊙ ! β₁)\n (β₂ ⊙ cong∘r (gg unite₊′equiv) β⊎₂ ⊙ uniti₊′-coh {g = gg g} ⊙ ! β₂)\n\nuniti₊′-nat : ∀ {A B} {f : A ≃ B} {g : ⊥ ≃ ⊥} →\n uniti₊′equiv ● f ≋ (f ⊎≃ g) ● uniti₊′equiv\nuniti₊′-nat {g = g} =\n eq (β₁ ⊙ ! uniti₊′-coh {g = proj₁ g} ⊙ ! cong∘r (proj₁ uniti₊′equiv) β⊎₁ ⊙ ! β₁)\n (β₂ ⊙ ! unite₊′-coh ⊙ ! cong∘l (gg uniti₊′equiv) β⊎₂ ⊙ ! β₂)\n\nassocr₊-nat : ∀ {A B C D E F : Set} →\n {f₀ : A ≃ D} {f₁ : B ≃ E} {f₂ : C ≃ F} →\n assocr₊equiv ● ((f₀ ⊎≃ f₁) ⊎≃ f₂) ≋ (f₀ ⊎≃ (f₁ ⊎≃ f₂)) ● assocr₊equiv\nassocr₊-nat {A} {B} {C} {D} {E} {F} {f₀} {f₁} {f₂} =\n let assocrDEF = proj₁ (assocr₊equiv {D} {E} {F}) in\n let assocrABC = proj₁ (assocr₊equiv {A} {B} {C}) in\n let assoclDEF = gg (assocr₊equiv {D} {E} {F}) in\n let assoclABC = gg (assocr₊equiv {A} {B} {C}) in\n eq (β₁ ⊙ cong∘l assocrDEF (β⊎₁ ⊙ (β⊎₁ ⊎∼ refl∼)) ⊙\n assocr₊-wf ⊙\n ! cong∘r assocrABC (refl∼ ⊎∼ β⊎₁) ⊙\n ! cong∘r assocrABC β⊎₁ ⊙ ! β₁)\n (β₂ ⊙ cong∘r assoclDEF (β⊎₂ {f = f₀ ⊎≃ f₁} {f₂}) ⊙\n cong∘r assoclDEF (β⊎₂ ⊎∼ refl∼) ⊙\n assocl₊-wf ⊙\n ! cong∘l assoclABC (refl∼ ⊎∼ β⊎₂) ⊙\n ! cong∘l assoclABC (β⊎₂ {f = f₀} {f₁ ⊎≃ f₂}) ⊙ ! β₂)\n\nassocl₊-nat : ∀ {A B C D E F : Set} →\n {f₀ : A ≃ D} {f₁ : B ≃ E} {f₂ : C ≃ F} →\n assocl₊equiv ● (f₀ ⊎≃ (f₁ ⊎≃ f₂)) ≋ ((f₀ ⊎≃ f₁) ⊎≃ f₂) ● assocl₊equiv\nassocl₊-nat {A} {B} {C} {D} {E} {F} {f₀} {f₁} {f₂} =\n let assoclDEF = proj₁ (assocl₊equiv {D} {E} {F}) in\n let assoclABC = proj₁ (assocl₊equiv {A} {B} {C}) in\n let assocrDEF = gg (assocl₊equiv {D} {E} {F}) in\n let assocrABC = gg (assocl₊equiv {A} {B} {C}) in\n eq (β₁ ⊙ cong∘l assoclDEF β⊎₁ ⊙\n cong∘l assoclDEF (refl∼ ⊎∼ β⊎₁) ⊙\n ! assocl₊-wf ⊙\n ! cong∘r assoclABC (β⊎₁ ⊎∼ refl∼) ⊙\n ! cong∘r assoclABC β⊎₁ ⊙ ! β₁)\n (β₂ ⊙ cong∘r assocrDEF (β⊎₂ {f = f₀} {f₁ ⊎≃ f₂}) ⊙\n cong∘r assocrDEF (refl∼ ⊎∼ β⊎₂) ⊙\n ! assocr₊-wf ⊙\n ! cong∘l assocrABC (β⊎₂ ⊎∼ refl∼) ⊙\n ! cong∘l assocrABC (β⊎₂ {f = f₀ ⊎≃ f₁} {f₂}) ⊙ ! β₂)\n\n-- often called 'triangle'\nunite-assocr₊-coh : ∀ {A B : Set} →\n unite₊′equiv ⊎≃ id≃ ≋ (id≃ ⊎≃ unite₊equiv) ● assocr₊equiv {A} {⊥} {B}\nunite-assocr₊-coh = -- eq triangle⊎-right triangle⊎-left\n eq (β⊎₁ ⊙ triangle⊎-right ⊙ ! (β₁ ⊙ cong∘r (proj₁ assocr₊equiv) β⊎₁))\n (β⊎₂ ⊙ triangle⊎-left ⊙ ! (β₂ ⊙ cong∘l (gg assocr₊equiv) β⊎₂))\n\n-- often called 'pentagon'\nassocr₊-coh : ∀ {A B C D : Set} →\n assocr₊equiv {A} {B} {C ⊎ D} ● assocr₊equiv ≋\n (id≃ ⊎≃ assocr₊equiv) ● assocr₊equiv ● (assocr₊equiv ⊎≃ id≃)\nassocr₊-coh = -- eq pentagon⊎-right pentagon⊎-left\n eq (β₁ ⊙ pentagon⊎-right ⊙\n ! (β₁ ⊙ cong₂∘ β⊎₁ β₁ ⊙\n cong∘l ((proj₁ id≃ ⊎→ proj₁ assocr₊equiv) ∘ proj₁ assocr₊equiv) β⊎₁))\n (β₂ ⊙ pentagon⊎-left ⊙\n ! (β₂ ⊙ cong₂∘ β₂ β⊎₂ ⊙\n cong∘r (gg assocr₊equiv ∘ (gg id≃ ⊎→ gg assocr₊equiv)) β⊎₂))\n\nswap₊-nat : {A B C D : Set} {f : A ≃ C} {g : B ≃ D} →\n swap₊equiv ● (f ⊎≃ g) ≋ (g ⊎≃ f) ● swap₊equiv\nswap₊-nat =\n eq (β₁ ⊙ cong∘l (proj₁ swap₊equiv) β⊎₁ ⊙ swap₊-coh ⊙\n ! (β₁ ⊙ cong∘r (proj₁ swap₊equiv) β⊎₁))\n (β₂ ⊙ cong∘r (gg swap₊equiv) β⊎₂ ⊙ sym∼ swap₊-coh ⊙\n ! (β₂ ⊙ cong∘l (gg swap₊equiv) β⊎₂))\n\n-- also called 'triangle', but better to call it 'unit coherence'\nunite₊l-coh : {A : Set} →\n unite₊equiv {A} ≋ unite₊′equiv ● swap₊equiv\nunite₊l-coh =\n eq (unite₊-swap-coh-right ⊙ ! β₁) (unite₊-swap-coh-left ⊙ ! β₂)\n\n-- often called 'hexagon'\nassocr₊-swap₊-coh : ∀ {A B C : Set} →\n assocr₊equiv {B} {C} {A} ● swap₊equiv ● assocr₊equiv {A} {B} {C} ≋\n id≃ ⊎≃ swap₊equiv ● assocr₊equiv {B} {A} {C} ● swap₊equiv ⊎≃ id≃\nassocr₊-swap₊-coh {A} {B} {C} = -- eq hexagon⊎-right hexagon⊎-left\n let assocrBCA = proj₁ (assocr₊equiv {B} {C} {A}) in\n let assocrABC = proj₁ (assocr₊equiv {A} {B} {C}) in\n let assocrBAC = proj₁ (assocr₊equiv {B} {A} {C}) in\n let swapAC = proj₁ id≃ ⊎→ proj₁ (swap₊equiv {A} {C}) in\n let assoclBCA = gg (assocr₊equiv {B} {C} {A}) in\n let assoclABC = gg (assocr₊equiv {A} {B} {C}) in\n let assoclBAC = gg (assocr₊equiv {B} {A} {C}) in\n let swapCA = gg id≃ ⊎→ gg (swap₊equiv {A} {C}) in\n eq (β₁ ⊙ cong∘l assocrBCA β₁ ⊙ hexagon⊎-right ⊙\n ! (β₁ ⊙ cong₂∘ β⊎₁ β₁ ⊙ cong∘l (swapAC ∘ assocrBAC) β⊎₁))\n (β₂ ⊙ cong∘r assoclBCA β₂ ⊙ hexagon⊎-left ⊙\n ! (β₂ ⊙ cong₂∘ β₂ β⊎₂ ⊙ cong∘r (assoclBAC ∘ swapCA) β⊎₂))\n\n-- and in the opposite direction\n\nassocl₊-swap₊-coh : ∀ {A B C : Set} →\n assocl₊equiv {A} {B} {C} ● swap₊equiv ● assocl₊equiv {B} {C} {A} ≋\n swap₊equiv ⊎≃ id≃ ● assocl₊equiv {B} {A} {C} ● id≃ ⊎≃ swap₊equiv\nassocl₊-swap₊-coh {A} {B} {C} = -- eq hexagon⊎-left hexagon⊎-right\n let assoclBCA = proj₁ (assocl₊equiv {B} {C} {A}) in\n let assoclABC = proj₁ (assocl₊equiv {A} {B} {C}) in\n let assoclBAC = proj₁ (assocl₊equiv {B} {A} {C}) in\n let swapBA = proj₁ (swap₊equiv {B} {A}) ⊎→ proj₁ id≃ in\n let assocrBCA = gg (assocl₊equiv {B} {C} {A}) in\n let assocrABC = gg (assocl₊equiv {A} {B} {C}) in\n let assocrBAC = gg (assocl₊equiv {B} {A} {C}) in\n let swapAB = gg (swap₊equiv {B} {A}) ⊎→ proj₁ id≃ in\n eq (β₁ ⊙ (cong∘l assoclABC β₁ ⊙ hexagon⊎-left) ⊙\n ! (β₁ ⊙ cong₂∘ β⊎₁ β₁ ⊙ cong∘l (swapBA ∘ assoclBAC) β⊎₁))\n (β₂ ⊙ cong∘r assocrABC β₂ ⊙ hexagon⊎-right ⊙\n ! (β₂ ⊙ cong₂∘ β₂ β⊎₂ ⊙ cong∘r (assocrBAC ∘ swapAB) β⊎₂))\n\n-- equivalences for the × structure\n\nid×id≋id : ∀ {A B : Set} → id≃ {A = A} ×≃ id≃ {A = B} ≋ id≃\nid×id≋id = eq (β×₁ ⊙ id×id∼id) (β×₂ ⊙ id×id∼id)\n\n×●≋●× : {A B C D E F : Set} →\n {f : A ≃ C} {g : B ≃ D} {h : C ≃ E} {i : D ≃ F} →\n (h ● f) ×≃ (i ● g) ≋ (h ×≃ i) ● (f ×≃ g)\n×●≋●× {f = f , qinv f⁻¹ _ _} {g , qinv g⁻¹ _ _} {h , qinv h⁻¹ _ _} {i , qinv i⁻¹ _ _} =\n eq (β×₁ ⊙ β₁ ×∼ β₁ ⊙ (×∘∼∘× {f = f} {g} {h} {i}) ⊙ ! cong₂∘ β×₁ β×₁ ⊙ ! β₁)\n (β×₂ ⊙ β₂ ×∼ β₂ ⊙ (×∘∼∘× {f = h⁻¹} {i⁻¹} {f⁻¹} {g⁻¹}) ⊙ ! cong₂∘ β×₂ β×₂ ⊙ ! β₂)\n\n_×≋_ : ∀ {A B C D : Set} {f g : A ≃ B} {h i : C ≃ D} →\n f ≋ g → h ≋ i → f ×≃ h ≋ g ×≃ i\ne₁ ×≋ e₂ = eq (β×₁ ⊙ (f≡ e₁) ×∼ (f≡ e₂) ⊙ ! β×₁)\n (β×₂ ⊙ (g≡ e₁) ×∼ (g≡ e₂) ⊙ ! β×₂)\n where open _≋_\n\nsym≃-distrib× : ∀ {A B C D : Set} {f : A ≃ B} {g : C ≃ D} →\n sym≃ (f ×≃ g) ≋ sym≃ f ×≃ sym≃ g\nsym≃-distrib× = -- note how the proof mixes ₁ and ₂ !\n eq (β×₂ ⊙ ! β×₁) (β×₁ ⊙ ! β×₂)\n\n\nunite⋆-nat : ∀ {A B} {f : A ≃ B} {g : ⊤ ≃ ⊤} →\n unite⋆equiv ● (g ×≃ f) ≋ f ● unite⋆equiv\nunite⋆-nat = -- eq unite⋆-coh uniti⋆-coh\n eq (β₁ ⊙ cong∘l (proj₁ unite⋆equiv) β×₁ ⊙ unite⋆-coh ⊙ ! β₁)\n (β₂ ⊙ cong∘r (gg unite⋆equiv) β×₂ ⊙ uniti⋆-coh ⊙ ! β₂)\n\nuniti⋆-nat : ∀ {A B} {f : A ≃ B} {g : ⊤ ≃ ⊤} →\n uniti⋆equiv ● f ≋ (g ×≃ f) ● uniti⋆equiv\nuniti⋆-nat = -- flip-sym≋ unite⋆-nat\n eq (β₁ ⊙ ! uniti⋆-coh ⊙ ! cong∘r (proj₁ uniti⋆equiv) β×₁ ⊙ ! β₁)\n (β₂ ⊙ ! unite⋆-coh ⊙ ! cong∘l (gg uniti⋆equiv) β×₂ ⊙ ! β₂)\n\nunite⋆′-nat : ∀ {A B} {f : A ≃ B} {g : ⊤ ≃ ⊤} →\n unite⋆′equiv ● (f ×≃ g) ≋ f ● unite⋆′equiv\nunite⋆′-nat = -- eq unite⋆′-coh uniti⋆′-coh\n eq (β₁ ⊙ cong∘l (proj₁ unite⋆′equiv) β×₁ ⊙ unite⋆′-coh ⊙ ! β₁)\n (β₂ ⊙ cong∘r (gg unite⋆′equiv) β×₂ ⊙ uniti⋆′-coh ⊙ ! β₂)\n\nuniti⋆′-nat : ∀ {A B} {f : A ≃ B} {g : ⊤ ≃ ⊤} →\n uniti⋆′equiv ● f ≋ (f ×≃ g) ● uniti⋆′equiv\nuniti⋆′-nat = -- flip-sym≋ unite⋆′-nat\n eq (β₁ ⊙ ! uniti⋆′-coh ⊙ ! cong∘r (proj₁ uniti⋆′equiv) β×₁ ⊙ ! β₁)\n (β₂ ⊙ ! unite⋆′-coh ⊙ ! cong∘l (gg uniti⋆′equiv) β×₂ ⊙ ! β₂)\n\nassocr⋆-nat : ∀ {A B C D E F : Set} →\n {f₀ : A ≃ D} {f₁ : B ≃ E} {f₂ : C ≃ F} →\n assocr⋆equiv ● ((f₀ ×≃ f₁) ×≃ f₂) ≋ (f₀ ×≃ (f₁ ×≃ f₂)) ● assocr⋆equiv\nassocr⋆-nat {A} {B} {C} {D} {E} {F} {f₀} {f₁} {f₂} = -- eq assocr⋆-wf assocl⋆-wf\n let assocrDEF = proj₁ (assocr⋆equiv {D} {E} {F}) in\n let assocrABC = proj₁ (assocr⋆equiv {A} {B} {C}) in\n let assoclDEF = gg (assocr⋆equiv {D} {E} {F}) in\n let assoclABC = gg (assocr⋆equiv {A} {B} {C}) in\n eq (β₁ ⊙ cong∘l assocrDEF β×₁ ⊙\n cong∘l assocrDEF (β×₁ ×∼ refl∼) ⊙\n assocr⋆-wf ⊙\n ! cong∘r assocrABC (refl∼ ×∼ β×₁) ⊙\n ! cong∘r assocrABC β×₁ ⊙ ! β₁)\n (β₂ ⊙ cong∘r assoclDEF (β×₂ {f = f₀ ×≃ f₁} {f₂}) ⊙\n cong∘r assoclDEF (β×₂ ×∼ refl∼) ⊙\n assocl⋆-wf ⊙\n ! cong∘l assoclABC (refl∼ ×∼ β×₂) ⊙\n ! cong∘l assoclABC (β×₂ {f = f₀} {f₁ ×≃ f₂}) ⊙ ! β₂)\nassocl⋆-nat : ∀ {A B C D E F : Set} →\n {f₀ : A ≃ D} {f₁ : B ≃ E} {f₂ : C ≃ F} →\n assocl⋆equiv ● (f₀ ×≃ (f₁ ×≃ f₂)) ≋ ((f₀ ×≃ f₁) ×≃ f₂) ● assocl⋆equiv\nassocl⋆-nat {A} {B} {C} {D} {E} {F} {f₀} {f₁} {f₂} = -- flip-sym≋ assocr⋆-nat\n let assoclDEF = proj₁ (assocl⋆equiv {D} {E} {F}) in\n let assoclABC = proj₁ (assocl⋆equiv {A} {B} {C}) in\n let assocrDEF = gg (assocl⋆equiv {D} {E} {F}) in\n let assocrABC = gg (assocl⋆equiv {A} {B} {C}) in\n eq (β₁ ⊙ cong∘l assoclDEF β×₁ ⊙\n cong∘l assoclDEF (refl∼ ×∼ β×₁) ⊙\n ! assocl⋆-wf ⊙\n ! cong∘r assoclABC (β×₁ ×∼ refl∼) ⊙\n ! cong∘r assoclABC β×₁ ⊙ ! β₁)\n (β₂ ⊙ cong∘r assocrDEF (β×₂ {f = f₀} {f₁ ×≃ f₂}) ⊙\n cong∘r assocrDEF (refl∼ ×∼ β×₂) ⊙\n ! assocr⋆-wf ⊙\n ! cong∘l assocrABC (β×₂ ×∼ refl∼) ⊙\n ! cong∘l assocrABC (β×₂ {f = f₀ ×≃ f₁} {f₂}) ⊙ ! β₂)\n\n-- often called 'triangle'\n\nunite-assocr⋆-coh : ∀ {A B : Set} →\n unite⋆′equiv ×≃ id≃ ≋ (id≃ ×≃ unite⋆equiv) ● assocr⋆equiv {A} {⊤} {B}\nunite-assocr⋆-coh =\n eq (β×₁ ⊙ triangle×-right ⊙ ! (β₁ ⊙ cong∘r (proj₁ assocr⋆equiv) β×₁))\n (β×₂ ⊙ triangle×-left ⊙ ! (β₂ ⊙ cong∘l (gg assocr⋆equiv) β×₂))\n\n-- often called 'pentagon'\n\nassocr⋆-coh : ∀ {A B C D : Set} →\n assocr⋆equiv {A} {B} {C × D} ● assocr⋆equiv ≋\n (id≃ ×≃ assocr⋆equiv) ● assocr⋆equiv ● (assocr⋆equiv ×≃ id≃)\nassocr⋆-coh =\n eq (β₁ ⊙ pentagon×-right ⊙\n ! (β₁ ⊙ cong₂∘ β×₁ β₁ ⊙\n cong∘l ((proj₁ id≃ ×→ proj₁ assocr⋆equiv) ∘ proj₁ assocr⋆equiv) β×₁))\n (β₂ ⊙ pentagon×-left ⊙\n ! (β₂ ⊙ cong₂∘ β₂ β×₂ ⊙\n cong∘r (gg assocr⋆equiv ∘ (gg id≃ ×→ gg assocr⋆equiv)) β×₂))\n\nswap⋆-nat : {A B C D : Set} {f : A ≃ C} {g : B ≃ D} →\n swap⋆equiv ● (f ×≃ g) ≋ (g ×≃ f) ● swap⋆equiv\nswap⋆-nat =\n eq (β₁ ⊙ cong∘l (proj₁ swap⋆equiv) β×₁ ⊙ swap⋆-coh ⊙\n ! (β₁ ⊙ cong∘r (proj₁ swap⋆equiv) β×₁))\n (β₂ ⊙ cong∘r (gg swap⋆equiv) β×₂ ⊙ sym∼ swap⋆-coh ⊙\n ! (β₂ ⊙ cong∘l (gg swap⋆equiv) β×₂))\n\n-- also called 'triangle', but better to call it 'unit coherence'\nunite⋆l-coh : {A : Set} →\n unite⋆equiv {A} ≋ unite⋆′equiv ● swap⋆equiv\nunite⋆l-coh =\n eq (unite⋆-swap-coh-right ⊙ ! β₁) (unite⋆-swap-coh-left ⊙ ! β₂)\n\n-- often called 'hexagon'\n\nassocr⋆-swap⋆-coh : ∀ {A B C : Set} →\n assocr⋆equiv {B} {C} {A} ● swap⋆equiv ● assocr⋆equiv {A} {B} {C} ≋\n id≃ ×≃ swap⋆equiv ● assocr⋆equiv {B} {A} {C} ● swap⋆equiv ×≃ id≃\nassocr⋆-swap⋆-coh {A} {B} {C} =\n let assocrBCA = proj₁ (assocr⋆equiv {B} {C} {A}) in\n let assocrABC = proj₁ (assocr⋆equiv {A} {B} {C}) in\n let assocrBAC = proj₁ (assocr⋆equiv {B} {A} {C}) in\n let swapAC = proj₁ id≃ ×→ proj₁ (swap⋆equiv {A} {C}) in\n let assoclBCA = gg (assocr⋆equiv {B} {C} {A}) in\n let assoclABC = gg (assocr⋆equiv {A} {B} {C}) in\n let assoclBAC = gg (assocr⋆equiv {B} {A} {C}) in\n let swapCA = gg id≃ ×→ gg (swap⋆equiv {A} {C}) in\n eq (β₁ ⊙ cong∘l assocrBCA β₁ ⊙ hexagon×-right ⊙\n ! (β₁ ⊙ cong₂∘ β×₁ β₁ ⊙ cong∘l (swapAC ∘ assocrBAC) β×₁))\n (β₂ ⊙ cong∘r assoclBCA β₂ ⊙ hexagon×-left ⊙\n ! (β₂ ⊙ cong₂∘ β₂ β×₂ ⊙ cong∘r (assoclBAC ∘ swapCA) β×₂))\n\n-- and in the opposite direction\n\nassocl⋆-swap⋆-coh : ∀ {A B C : Set} →\n assocl⋆equiv {A} {B} {C} ● swap⋆equiv ● assocl⋆equiv {B} {C} {A} ≋\n swap⋆equiv ×≃ id≃ ● assocl⋆equiv {B} {A} {C} ● id≃ ×≃ swap⋆equiv\nassocl⋆-swap⋆-coh {A} {B} {C} =\n let assoclBCA = proj₁ (assocl⋆equiv {B} {C} {A}) in\n let assoclABC = proj₁ (assocl⋆equiv {A} {B} {C}) in\n let assoclBAC = proj₁ (assocl⋆equiv {B} {A} {C}) in\n let swapBA = proj₁ (swap⋆equiv {B} {A}) ×→ proj₁ id≃ in\n let assocrBCA = gg (assocl⋆equiv {B} {C} {A}) in\n let assocrABC = gg (assocl⋆equiv {A} {B} {C}) in\n let assocrBAC = gg (assocl⋆equiv {B} {A} {C}) in\n let swapAB = gg (swap⋆equiv {B} {A}) ×→ proj₁ id≃ in\n eq (β₁ ⊙ (cong∘l assoclABC β₁ ⊙ hexagon×-left) ⊙\n ! (β₁ ⊙ cong₂∘ β×₁ β₁ ⊙ cong∘l (swapBA ∘ assoclBAC) β×₁))\n (β₂ ⊙ cong∘r assocrABC β₂ ⊙ hexagon×-right ⊙\n ! (β₂ ⊙ cong₂∘ β₂ β×₂ ⊙ cong∘r (assocrBAC ∘ swapAB) β×₂))\n\n-- distributivity\n\ndistl-nat : {A B C D E F : Set} →\n {f : A ≃ D} {g : B ≃ E} {h : C ≃ F} →\n distlequiv ● (f ×≃ (g ⊎≃ h)) ≋ ((f ×≃ g) ⊎≃ (f ×≃ h)) ● distlequiv\ndistl-nat {A} {B} {C} {D} {E} {F} {f} {g} {h} = -- eq distl-coh factorl-coh\n let distlDEF = proj₁ (distlequiv {D} {E} {F}) in\n let distlABC = proj₁ (distlequiv {A} {B} {C}) in\n let factorlDEF = gg (distlequiv {D} {E} {F}) in\n let factorlABC = gg (distlequiv {A} {B} {C}) in\n eq (β₁ ⊙ cong∘l distlDEF (β×₁ ⊙ (refl∼ ×∼ β⊎₁)) ⊙\n distl-coh ⊙\n ! (β₁ ⊙ cong∘r distlABC (β⊎₁ ⊙ (β×₁ ⊎∼ β×₁))))\n --\n (β₂ ⊙ cong∘r factorlDEF (β×₂ {f = f} {g ⊎≃ h} ⊙ (refl∼ ×∼ β⊎₂)) ⊙\n factorl-coh ⊙\n ! (β₂ ⊙ cong∘l factorlABC (β⊎₂ {f = f ×≃ g} {f ×≃ h} ⊙ (β×₂ ⊎∼ β×₂))))\n\nfactorl-nat : {A B C D E F : Set} →\n {f : A ≃ D} {g : B ≃ E} {h : C ≃ F} →\n factorlequiv ● ((f ×≃ g) ⊎≃ (f ×≃ h)) ≋ (f ×≃ (g ⊎≃ h)) ● factorlequiv\nfactorl-nat {A} {B} {C} {D} {E} {F} {f} {g} {h} = -- flip-sym≋ distl-nat\n let factorlDEF = proj₁ (factorlequiv {D} {E} {F}) in\n let factorlABC = proj₁ (factorlequiv {A} {B} {C}) in\n let distlDEF = gg (factorlequiv {D} {E} {F}) in\n let distlABC = gg (factorlequiv {A} {B} {C}) in\n eq (β₁ ⊙ cong∘l factorlDEF (β⊎₁ ⊙ (β×₁ ⊎∼ β×₁)) ⊙\n sym∼ factorl-coh ⊙\n ! (β₁ ⊙ cong∘r factorlABC (β×₁ ⊙ (refl∼ ×∼ β⊎₁)) ))\n --\n (β₂ ⊙ cong∘r distlDEF (β⊎₂ {f = f ×≃ g} {f ×≃ h} ⊙ (β×₂ ⊎∼ β×₂)) ⊙\n sym∼ distl-coh ⊙\n ! (β₂ ⊙ cong∘l distlABC (β×₂ {f = f} {g ⊎≃ h} ⊙ (refl∼ ×∼ β⊎₂))))\n\ndist-nat : {A B C D E F : Set} →\n {f : A ≃ D} {g : B ≃ E} {h : C ≃ F} →\n distequiv ● ((f ⊎≃ g) ×≃ h) ≋ ((f ×≃ h) ⊎≃ (g ×≃ h)) ● distequiv\ndist-nat {A} {B} {C} {D} {E} {F} {f} {g} {h} = -- eq dist-coh factor-coh\n let distDEF = proj₁ (distequiv {D} {E} {F}) in\n let distABC = proj₁ (distequiv {A} {B} {C}) in\n let factorDEF = gg (distequiv {D} {E} {F}) in\n let factorABC = gg (distequiv {A} {B} {C}) in\n eq (β₁ ⊙ cong∘l distDEF (β×₁ ⊙ (β⊎₁ ×∼ refl∼)) ⊙\n dist-coh ⊙\n ! (β₁ ⊙ cong∘r distABC (β⊎₁ ⊙ (β×₁ ⊎∼ β×₁))))\n --\n (β₂ ⊙ cong∘r factorDEF (β×₂ {f = f ⊎≃ g} {h} ⊙ (β⊎₂ ×∼ refl∼)) ⊙\n factor-coh ⊙\n ! (β₂ ⊙ cong∘l factorABC (β⊎₂ {f = f ×≃ h} {g ×≃ h} ⊙ (β×₂ ⊎∼ β×₂))))\n\nfactor-nat : {A B C D E F : Set} →\n {f : A ≃ D} {g : B ≃ E} {h : C ≃ F} →\n factorequiv ● ((f ×≃ h) ⊎≃ (g ×≃ h)) ≋ ((f ⊎≃ g) ×≃ h) ● factorequiv\nfactor-nat {A} {B} {C} {D} {E} {F} {f} {g} {h} = -- flip-sym≋ dist-nat\n let factorDEF = proj₁ (factorequiv {D} {E} {F}) in\n let factorABC = proj₁ (factorequiv {A} {B} {C}) in\n let distDEF = gg (factorequiv {D} {E} {F}) in\n let distABC = gg (factorequiv {A} {B} {C}) in\n eq (β₁ ⊙ cong∘l factorDEF (β⊎₁ ⊙ (β×₁ ⊎∼ β×₁)) ⊙\n sym∼ factor-coh ⊙\n ! (β₁ ⊙ cong∘r factorABC (β×₁ ⊙ (β⊎₁ ×∼ refl∼)) ))\n --\n (β₂ ⊙ cong∘r distDEF (β⊎₂ {f = f ×≃ h} {g ×≃ h} ⊙ (β×₂ ⊎∼ β×₂)) ⊙\n sym∼ dist-coh ⊙\n ! (β₂ ⊙ cong∘l distABC (β×₂ {f = f ⊎≃ g} {h} ⊙ (β⊎₂ ×∼ refl∼))))\n\n-- note how we don't use id≃ but an arbitrary ⊥ ≃ ⊥.\n-- because this law under-specifies f and g, we need to\n-- be explicit in our calls\n\ndistzr-nat : {A B : Set} → {f : A ≃ B} → {g : ⊥ ≃ ⊥} →\n distzrequiv ● (f ×≃ g) ≋ g ● distzrequiv\ndistzr-nat {f = (f , qinv h _ _)} {(_ , qinv g _ _)} =\n -- eq (distzr-coh {f = f}) (factorzr-coh {f = h} {g})\n eq (β₁ ⊙ cong∘l (proj₁ distzrequiv) β×₁ ⊙ distzr-coh {f = f} ⊙ ! β₁)\n (β₂ ⊙ cong∘r (gg distzrequiv) β×₂ ⊙ factorzr-coh {f = h} {g} ⊙ ! β₂)\n\nfactorzr-nat : {A B : Set} → {f : A ≃ B} → {g : ⊥ ≃ ⊥} →\n factorzrequiv ● g ≋ (f ×≃ g) ● factorzrequiv\nfactorzr-nat {f = f , qinv f⁻¹ _ _} {g , _} = -- flip-sym≋ (distzr-nat {f = sym≃ f})\n eq (β₁ ⊙ sym∼ (factorzr-coh {f = f} {g}) ⊙\n ! (β₁ ⊙ cong∘r (proj₁ factorzrequiv) β×₁) )\n --\n (β₂ ⊙ sym∼ (distzr-coh {f = f⁻¹}) ⊙\n ! (β₂ ⊙ cong∘l (gg factorzrequiv) β×₂))\n\n-- same comment as above\n\ndistz-nat : {A B : Set} → {f : A ≃ B} → {g : ⊥ ≃ ⊥} →\n distzequiv ● (g ×≃ f) ≋ g ● distzequiv\ndistz-nat {f = (f , qinv h _ _)} {(_ , qinv g _ _)} =\n eq (β₁ ⊙ cong∘l (proj₁ distzequiv) β×₁ ⊙ distz-coh {f = f} ⊙ ! β₁)\n (β₂ ⊙ cong∘r (gg distzequiv) β×₂ ⊙ factorz-coh {f = h} {g} ⊙ ! β₂)\n\nfactorz-nat : {A B : Set} → {f : A ≃ B} → {g : ⊥ ≃ ⊥} →\n factorzequiv ● g ≋ (g ×≃ f) ● factorzequiv\nfactorz-nat {f = (f , qinv f⁻¹ _ _)} {g , _} =\n eq (β₁ ⊙ (sym∼ (factorz-coh {f = f} {g})) ⊙\n ! (β₁ ⊙ cong∘r (proj₁ factorzequiv) β×₁))\n (β₂ ⊙ (sym∼ (distz-coh {f = f⁻¹})) ⊙ ! (β₂ ⊙ cong∘l (gg factorzequiv) β×₂))\n\n-- some equivalences for which there are two 'obvious'\n-- programs, but are in fact equivalent. Named after\n-- the types which are witnessed to be equivalent.\n\nA×[B⊎C]≃[A×C]⊎[A×B] : {A B C : Set} →\n distlequiv ● (id≃ {A = A} ×≃ swap₊equiv {B} {C}) ≋ swap₊equiv ● distlequiv\nA×[B⊎C]≃[A×C]⊎[A×B] = -- eq A×[B⊎C]→[A×C]⊎[A×B] [A×C]⊎[A×B]→A×[B⊎C]\n eq (β₁ ⊙ cong∘l (proj₁ distlequiv) β×₁ ⊙ A×[B⊎C]→[A×C]⊎[A×B] ⊙ ! β₁)\n (β₂ ⊙ cong∘r (gg distlequiv) β×₂ ⊙ [A×C]⊎[A×B]→A×[B⊎C] ⊙ ! β₂)\n\n[A⊎B]×C≃[C×A]⊎[C×B] : {A B C : Set} →\n (swap⋆equiv ⊎≃ swap⋆equiv) ● distequiv ≋ distlequiv ● swap⋆equiv {A ⊎ B} {C}\n[A⊎B]×C≃[C×A]⊎[C×B] = -- eq [A⊎B]×C→[C×A]⊎[C×B] [C×A]⊎[C×B]→[A⊎B]×C\n eq (β₁ ⊙ cong∘r (proj₁ distequiv) β⊎₁ ⊙ [A⊎B]×C→[C×A]⊎[C×B] ⊙ ! β₁)\n (β₂ ⊙ cong∘l (gg distequiv) β⊎₂ ⊙ [C×A]⊎[C×B]→[A⊎B]×C ⊙ ! β₂)\n\n[A⊎B⊎C]×D≃[A×D⊎B×D]⊎C×D : {A B C D : Set} →\n (distequiv ⊎≃ id≃) ● distequiv ● (assocl₊equiv {A} {B} {C} ×≃ id≃ {A = D}) ≋\n assocl₊equiv ● (id≃ ⊎≃ distequiv) ● distequiv\n[A⊎B⊎C]×D≃[A×D⊎B×D]⊎C×D = -- eq [A⊎B⊎C]×D→[A×D⊎B×D]⊎C×D [A×D⊎B×D]⊎C×D→[A⊎B⊎C]×D\n eq (β₁ ⊙ cong₂∘ β⊎₁ (β₁ ⊙ cong∘l (proj₁ distequiv) β×₁) ⊙\n [A⊎B⊎C]×D→[A×D⊎B×D]⊎C×D ⊙\n ! (β₁ ⊙ cong∘l (proj₁ assocl₊equiv) (β₁ ⊙ cong∘r (proj₁ distequiv) β⊎₁)))\n --\n (β₂ ⊙ cong₂∘ (β₂ ⊙ cong∘r (gg distequiv) β×₂) β⊎₂ ⊙\n [A×D⊎B×D]⊎C×D→[A⊎B⊎C]×D ⊙\n ! (β₂ ⊙ cong∘r (gg assocl₊equiv) (β₂ ⊙ cong∘l (gg distequiv) β⊎₂)))\n\nA×B×[C⊎D]≃[A×B]×C⊎[A×B]×D : {A B C D : Set} →\n distlequiv ● assocl⋆equiv {A} {B} {C ⊎ D} ≋\n (assocl⋆equiv ⊎≃ assocl⋆equiv) ● distlequiv ● (id≃ ×≃ distlequiv)\nA×B×[C⊎D]≃[A×B]×C⊎[A×B]×D =\n eq (β₁ ⊙ A×B×[C⊎D]→[A×B]×C⊎[A×B]×D ⊙\n ! (β₁ ⊙ cong₂∘ β⊎₁ (β₁ ⊙ cong∘l (proj₁ distlequiv) β×₁)))\n (β₂ ⊙ [A×B]×C⊎[A×B]×D→A×B×[C⊎D] ⊙\n ! (β₂ ⊙ cong₂∘ (β₂ ⊙ cong∘r (gg distlequiv) β×₂) β⊎₂))\n\n0×0≃0 : distzequiv ≋ distzrequiv\n0×0≃0 = eq 0×0→0 0→0×0\n\n0×[A⊎B]≃0 : {A B : Set} →\n distzequiv ≋ unite₊equiv ● (distzequiv ⊎≃ distzequiv) ● distlequiv {⊥} {A} {B}\n0×[A⊎B]≃0 =\n eq (0×[A⊎B]→0 ⊙\n ! (β₁ ⊙ cong∘l (proj₁ unite₊equiv) (β₁ ⊙ cong∘r (proj₁ distlequiv) β⊎₁)))\n (0→0×[A⊎B] ⊙\n ! (β₂ ⊙ cong∘r (gg unite₊equiv) (β₂ ⊙ cong∘l (gg distlequiv) β⊎₂)))\n\n0×1≃0 : unite⋆′equiv ≋ distzequiv\n0×1≃0 = eq 0×1→0 0→0×1\n\nA×0≃0 : {A : Set} → distzrequiv {A} ≋ distzequiv ● swap⋆equiv\nA×0≃0 = eq (A×0→0 ⊙ ! β₁) (0→A×0 ⊙ ! β₂)\n\n0×A×B≃0 : {A B : Set} →\n distzequiv ≋ distzequiv ● (distzequiv ×≃ id≃) ● assocl⋆equiv {⊥} {A} {B}\n0×A×B≃0 =\n let distzB = proj₁ distzequiv in\n let factorzA = gg distzequiv in\n eq (0×A×B→0 ⊙\n ! (β₁ ⊙ cong∘l distzB (β₁ ⊙ cong∘r (proj₁ assocl⋆equiv) β×₁)))\n (0→0×A×B ⊙\n ! (β₂ ⊙ cong∘r factorzA (β₂ ⊙ cong∘l (gg assocl⋆equiv) β×₂)))\n\nA×0×B≃0 : {A B : Set} →\n distzrequiv ● (id≃ ×≃ distzequiv) ≋\n distzequiv ● (distzrequiv ×≃ id≃) ● assocl⋆equiv {A} {⊥} {B}\nA×0×B≃0 =\n eq (β₁ ⊙ cong∘l (proj₁ distzrequiv) β×₁ ⊙ A×0×B→0 ⊙\n ! (β₁ ⊙ cong∘l (proj₁ distzequiv)\n (β₁ ⊙ cong∘r (proj₁ assocl⋆equiv) β×₁)))\n (β₂ ⊙ cong∘r (gg distzrequiv) β×₂ ⊙ 0→A×0×B ⊙\n ! (β₂ ⊙ cong∘r (gg distzequiv)\n (β₂ ⊙ cong∘l (gg assocl⋆equiv) β×₂)))\n\nA×[0+B]≃A×B : {A B : Set} →\n (id≃ {A = A} ×≃ unite₊equiv {B}) ≋ unite₊equiv ● (distzrequiv ⊎≃ id≃) ● distlequiv\nA×[0+B]≃A×B =\n eq (β×₁ ⊙ A×[0+B]→A×B ⊙\n ! (β₁ ⊙ cong∘l (proj₁ unite₊equiv) (β₁ ⊙ cong∘r (proj₁ distlequiv) β⊎₁)))\n (β×₂ ⊙ A×B→A×[0+B] ⊙\n ! (β₂ ⊙ cong∘r (gg unite₊equiv) (β₂ ⊙ cong∘l (gg distlequiv) β⊎₂)))\n\n1×[A⊎B]≃A⊎B : {A B : Set} →\n unite⋆equiv ≋ (unite⋆equiv ⊎≃ unite⋆equiv) ● distlequiv {⊤} {A} {B}\n1×[A⊎B]≃A⊎B =\n eq (1×[A⊎B]→A⊎B ⊙ ! (β₁ ⊙ cong∘r (proj₁ distlequiv) β⊎₁))\n (A⊎B→1×[A⊎B] ⊙ ! (β₂ ⊙ cong∘l (gg distlequiv) β⊎₂))\n\n[A⊎B]×[C⊎D]≃[[A×C⊎B×C]⊎A×D]⊎B×D : {A B C D : Set} →\n assocl₊equiv ● (distequiv ⊎≃ distequiv) ● distlequiv ≋\n (assocl₊equiv ⊎≃ id≃) ● ((id≃ ⊎≃ swap₊equiv) ⊎≃ id≃) ●\n (assocr₊equiv ⊎≃ id≃) ● assocl₊equiv ●\n (distlequiv ⊎≃ distlequiv) ● distequiv {A} {B} {C ⊎ D}\n[A⊎B]×[C⊎D]≃[[A×C⊎B×C]⊎A×D]⊎B×D =\n eq (β₁ ⊙ cong∘l (proj₁ assocl₊equiv)\n (β₁ ⊙ cong∘r (proj₁ distlequiv) β⊎₁) ⊙\n [A⊎B]×[C⊎D]→[[A×C⊎B×C]⊎A×D]⊎B×D ⊙\n ! (β₁ ⊙ cong₂∘ β⊎₁ (β₁ ⊙ cong₂∘ (β⊎₁ ⊙ (β⊎₁ ⊎∼ refl∼))\n (β₁ ⊙ cong₂∘ β⊎₁ (β₁ ⊙ cong∘l (proj₁ assocl₊equiv)\n (β₁ ⊙ cong∘r (proj₁ distequiv) β⊎₁))))))\n --\n (β₂ ⊙ cong∘r (gg assocl₊equiv) (β₂ ⊙ cong∘l (gg distlequiv) β⊎₂) ⊙\n [[A×C⊎B×C]⊎A×D]⊎B×D→[A⊎B]×[C⊎D] ⊙\n ! (β₂ ⊙ cong₂∘\n (β₂ ⊙ cong₂∘\n (β₂ ⊙ cong₂∘\n (β₂ ⊙ cong∘r (gg assocl₊equiv) (β₂ ⊙ cong∘l (gg distequiv) β⊎₂))\n β⊎₂)\n (β⊎₂ {f = id≃ ⊎≃ swap₊equiv} ⊙ (β⊎₂ ⊎∼ refl∼)))\n β⊎₂))\n\n------------------------------------------------------------------------------\n-- also useful\n\n[g+1]●[1+f]≋g+f : {A B C D : Set} {f : A ≃ B} {g : C ≃ D} →\n (g ⊎≃ id≃) ● (id≃ ⊎≃ f) ≋ g ⊎≃ f\n[g+1]●[1+f]≋g+f {f = f} {g} = begin (\n (g ⊎≃ id≃) ● (id≃ ⊎≃ f)\n ≋⟨ sym≋ ⊎●≋●⊎ ⟩\n (g ● id≃) ⊎≃ (id≃ ● f)\n ≋⟨ rid≋ ⊎≋ lid≋ ⟩\n g ⊎≃ f ∎)\n where open ≋-Reasoning\n\n-- same proof as above, just written compactly\n\n[1+f]●[g+1]≋g+f : {A B C D : Set} {f : A ≃ B} {g : C ≃ D} →\n (id≃ ⊎≃ f) ● (g ⊎≃ id≃) ≋ g ⊎≃ f\n[1+f]●[g+1]≋g+f = trans≋ (sym≋ ⊎●≋●⊎) (lid≋ ⊎≋ rid≋)\n\n-- put then together\n\n[g+1]●[1+f]≋[1+f]●[g+1] : {A B C D : Set} {f : A ≃ B} {g : C ≃ D} →\n (g ⊎≃ id≃) ● (id≃ ⊎≃ f) ≋ (id≃ ⊎≃ f) ● (g ⊎≃ id≃)\n[g+1]●[1+f]≋[1+f]●[g+1] = trans≋ [g+1]●[1+f]≋g+f (sym≋ [1+f]●[g+1]≋g+f)\n\n--\n\n[g*1]●[1*f]≋g*f : {A B C D : Set} {f : A ≃ B} {g : C ≃ D} →\n (g ×≃ id≃) ● (id≃ ×≃ f) ≋ g ×≃ f\n[g*1]●[1*f]≋g*f {f = f} {g} = begin (\n (g ×≃ id≃) ● (id≃ ×≃ f)\n ≋⟨ sym≋ ×●≋●× ⟩\n (g ● id≃) ×≃ (id≃ ● f)\n ≋⟨ rid≋ ×≋ lid≋ ⟩\n g ×≃ f ∎)\n where open ≋-Reasoning\n\n[1*f]●[g*1]≋g*f : {A B C D : Set} {f : A ≃ B} {g : C ≃ D} →\n (id≃ ×≃ f) ● (g ×≃ id≃) ≋ g ×≃ f\n[1*f]●[g*1]≋g*f = trans≋ (sym≋ ×●≋●×) (lid≋ ×≋ rid≋)\n\n[g*1]●[1*f]≋[1*f]●[g*1] : {A B C D : Set} {f : A ≃ B} {g : C ≃ D} →\n (g ×≃ id≃) ● (id≃ ×≃ f) ≋ (id≃ ×≃ f) ● (g ×≃ id≃)\n[g*1]●[1*f]≋[1*f]●[g*1] = trans≋ [g*1]●[1*f]≋g*f (sym≋ [1*f]●[g*1]≋g*f)\n\n\n------------------------------------------------------------------------------\n", "meta": {"hexsha": "b6b230a4b291764bee38c0937d57b4175570a52b", "size": 26402, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Univalence/TypeEquivEquiv.agda", "max_stars_repo_name": "JacquesCarette/pi-dual", "max_stars_repo_head_hexsha": "003835484facfde0b770bc2b3d781b42b76184c1", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 14, "max_stars_repo_stars_event_min_datetime": "2015-08-18T21:40:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-05T01:07:57.000Z", "max_issues_repo_path": "Univalence/TypeEquivEquiv.agda", "max_issues_repo_name": "JacquesCarette/pi-dual", "max_issues_repo_head_hexsha": "003835484facfde0b770bc2b3d781b42b76184c1", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2018-06-07T16:27:41.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-29T20:41:23.000Z", "max_forks_repo_path": "Univalence/TypeEquivEquiv.agda", "max_forks_repo_name": "JacquesCarette/pi-dual", "max_forks_repo_head_hexsha": "003835484facfde0b770bc2b3d781b42b76184c1", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2016-05-29T01:56:33.000Z", "max_forks_repo_forks_event_max_datetime": "2019-09-10T09:47:13.000Z", "avg_line_length": 40.8699690402, "max_line_length": 87, "alphanum_fraction": 0.5013256571, "num_tokens": 14777, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175005616829, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34719442732845085}} {"text": "-- MIT License\n\n-- Copyright (c) 2021 Luca Ciccone and Luca Padovani\n\n-- Permission is hereby granted, free of charge, to any person\n-- obtaining a copy of this software and associated documentation\n-- files (the \"Software\"), to deal in the Software without\n-- restriction, including without limitation the rights to use,\n-- copy, modify, merge, publish, distribute, sublicense, and/or sell\n-- copies of the Software, and to permit persons to whom the\n-- Software is furnished to do so, subject to the following\n-- conditions:\n\n-- The above copyright notice and this permission notice shall be\n-- included in all copies or substantial portions of the Software.\n\n-- THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND,\n-- EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES\n-- OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND\n-- NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT\n-- HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,\n-- WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n-- FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR\n-- OTHER DEALINGS IN THE SOFTWARE.\n\n{-# OPTIONS --guardedness #-}\n\nopen import Data.Empty using (⊥-elim)\nopen import Data.Product\nopen import Data.Sum\n\nopen import Relation.Nullary\nopen import Relation.Unary using (_∈_)\nopen import Relation.Binary.PropositionalEquality using (_≡_; _≢_; refl; trans; sym; subst)\nopen import Relation.Binary.Construct.Closure.ReflexiveTransitive using (ε; _◅_)\n\nopen import Function.Base using (case_of_)\n\nopen import Common\n\nmodule Compliance {ℙ : Set} (message : Message ℙ)\n where\n\nopen import Action message\nopen import SessionType message\nopen import Transitions message\nopen import Session message\nopen import Progress message\n\ndata Compliance : Session -> Set\nrecord ∞Compliance (S : Session) : Set where\n constructor delay_\n coinductive\n field force : Compliance S\nopen ∞Compliance public\n\ndata Compliance where\n win#def : ∀{T S} (w : Win T) (def : Defined S) -> Compliance (T # S)\n out#inp : ∀{f g} (W : Witness f) (F : ∀{x} (!x : x ∈ dom f) -> ∞Compliance (f x .force # g x .force)) -> Compliance (out f # inp g)\n inp#out : ∀{f g} (W : Witness g) (F : ∀{x} (!x : x ∈ dom g) -> ∞Compliance (f x .force # g x .force)) -> Compliance (inp f # out g)\n\nsubject-reduction : ∀{S S'} -> Compliance S -> Reduction S S' -> ∞Compliance S'\nforce (subject-reduction (win#def (out U) def) (sync {I _} (out !x) s)) = ⊥-elim (U _ !x)\nforce (subject-reduction (inp#out _ F) (sync inp (out !x))) = F !x .force\nforce (subject-reduction (out#inp _ F) (sync (out !x) inp)) = F !x .force\n\nsubject-reduction* : ∀{S S'} -> Compliance S -> Reductions S S' -> ∞Compliance S'\nforce (subject-reduction* comp ε) = comp\nsubject-reduction* comp (red ◅ reds) = subject-reduction* (subject-reduction comp red .force) reds\n\ncompliance->progress : ∀{S} -> Compliance S -> Progress S\ncompliance->progress (win#def w def) = win#def w def\ncompliance->progress (out#inp W F) = out#inp W\ncompliance->progress (inp#out W F) = inp#out W\n\ncompliance->defined : ∀{T S} -> Compliance (T # S) -> Defined T × Defined S\ncompliance->defined (win#def (out _) def) = out , def\ncompliance->defined (out#inp _ _) = out , inp\ncompliance->defined (inp#out _ _) = inp , out\n\ncompliance-sound : ∀{S} -> Compliance S -> ComplianceS S\ncompliance-sound comp reds = progress-sound (compliance->progress (subject-reduction* comp reds .force))\n\ncompliance-complete : ∀{S} -> ComplianceS S -> ∞Compliance S\nforce (compliance-complete {T # S} spec) with spec ε\n... | inj₁ (win#def w def) = win#def w def\n... | inj₂ (_ , sync inp (out !x)) =\n inp#out (_ , !x) λ !x -> compliance-complete λ reds -> spec (sync inp (out !x) ◅ reds)\n... | inj₂ (_ , sync (out !x) inp) =\n out#inp (_ , !x) λ !x -> compliance-complete λ reds -> spec (sync (out !x) inp ◅ reds)\n", "meta": {"hexsha": "a8098613ab5e19ea480ba7e9d3aea718d20de747", "size": 3868, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Compliance.agda", "max_stars_repo_name": "boystrange/FairSubtypingAgda", "max_stars_repo_head_hexsha": "c4b78e70c3caf68d509f4360b9171d9f80ecb825", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2021-07-29T14:32:30.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-24T14:38:47.000Z", "max_issues_repo_path": "src/Compliance.agda", "max_issues_repo_name": "boystrange/FairSubtypingAgda", "max_issues_repo_head_hexsha": "c4b78e70c3caf68d509f4360b9171d9f80ecb825", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Compliance.agda", "max_forks_repo_name": "boystrange/FairSubtypingAgda", "max_forks_repo_head_hexsha": "c4b78e70c3caf68d509f4360b9171d9f80ecb825", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.5054945055, "max_line_length": 133, "alphanum_fraction": 0.6954498449, "num_tokens": 1102, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.3469382797973016}} {"text": "{-# OPTIONS --safe #-}\n\nopen import Definition.Typed.EqualityRelation\n\nmodule Definition.LogicalRelation.Substitution.Introductions.Pair {{eqrel : EqRelSet}} where\nopen EqRelSet {{...}}\n\nopen import Definition.Untyped as U hiding (wk)\nopen import Definition.Untyped.Properties\nopen import Definition.Typed\nopen import Definition.Typed.Properties\nopen import Definition.Typed.Weakening as T hiding (wk; wkTerm; wkEqTerm)\nopen import Definition.Typed.RedSteps\nopen import Definition.LogicalRelation\nopen import Definition.LogicalRelation.ShapeView\nopen import Definition.LogicalRelation.Irrelevance\nopen import Definition.LogicalRelation.Weakening\nopen import Definition.LogicalRelation.Properties\nopen import Definition.LogicalRelation.Application\nopen import Definition.LogicalRelation.Substitution\nopen import Definition.LogicalRelation.Substitution.Properties\nopen import Definition.LogicalRelation.Substitution.Reflexivity\nopen import Definition.LogicalRelation.Substitution.Introductions.Sigma\nopen import Definition.LogicalRelation.Substitution.Introductions.Pi\nopen import Definition.LogicalRelation.Substitution.Introductions.SingleSubst\n\nopen import Tools.Product\nimport Tools.PropositionalEquality as PE\n\n-- Valid pair construction.\n⦅⦆ᵛ : ∀ {F G l∃ t u Γ l}\n ([Γ] : ⊩ᵛ Γ)\n ([F] : Γ ⊩ᵛ⟨ l ⟩ F ^ [ % , ι l∃ ] / [Γ])\n ([G] : Γ ∙ F ^ [ % , ι l∃ ] ⊩ᵛ⟨ l ⟩ G ^ [ % , ι l∃ ] / [Γ] ∙ [F]) \n ([t] : Γ ⊩ᵛ⟨ l ⟩ t ∷ F ^ [ % , ι l∃ ] / [Γ] / [F])\n ([u] : Γ ⊩ᵛ⟨ l ⟩ u ∷ G [ t ] ^ [ % , ι l∃ ] / [Γ] / substS {F} {G} {t} [Γ] [F] [G] [t])\n → Γ ⊩ᵛ⟨ l ⟩ ⦅ G , t , u ⦆ ∷ ∃ F ▹ G ^ [ % , ι l∃ ] / [Γ] / ∃ᵛ {F} {G} [Γ] [F] [G]\n⦅⦆ᵛ {F} {G} {l∃} {t} {u} {Γ} {l} [Γ] [F] [G] [t] [u] {Δ = Δ} {σ = σ} ⊢Δ [σ] =\n let [G[t]] = substS {F} {G} {t} [Γ] [F] [G] [t]\n [ΠFG] = Πᵛ {F = F} {G = G} (≡is≤ PE.refl) (≡is≤ PE.refl) [Γ] [F] [G]\n [σF] = proj₁ ([F] ⊢Δ [σ])\n ⊢F = escape [σF]\n [σG] = proj₁ ([G] (⊢Δ ∙ ⊢F) (liftSubstS {F = F} [Γ] ⊢Δ [F] [σ]))\n ⊢G = escape [σG]\n [σt] = proj₁ ([t] (⊢Δ) [σ])\n ⊢t = escapeTerm [σF] [σt]\n [σu] = proj₁ ([u] (⊢Δ) [σ])\n [σG[t]] = proj₁ ([G[t]] ⊢Δ [σ])\n [σΠFG] = proj₁ ([ΠFG] ⊢Δ [σ])\n [σG[t]]′ = irrelevance′ (singleSubstLift G t) [σG[t]]\n [σu]′ = irrelevanceTerm′ (singleSubstLift G t) PE.refl PE.refl [σG[t]] [σG[t]]′ [σu]\n ⊢u = escapeTerm [σG[t]]′ [σu]′\n ⦅t,u⦆ⱼ = ⦅_,_,_,_⦆ⱼ {F = subst σ F} {G = subst (liftSubst σ) G} {t = subst σ t} {u = subst σ u}\n ⊢F ⊢G ⊢t ⊢u\n in ⦅t,u⦆ⱼ , λ {σ′} [σ′] [σ≡σ′] →\n ⦅t,u⦆ⱼ ,\n let ⊢Γ = wfTerm ⊢t\n [σt′] = proj₁ ([t] ⊢Δ [σ′])\n [σt≡σt′] = proj₂ ([t] ⊢Δ [σ]) [σ′] [σ≡σ′]\n [σF′] = proj₁ ([F] ⊢Δ [σ′])\n ⊢F′ = escape [σF′]\n ⊢t′ = escapeTerm [σF′] [σt′]\n [σG′] = proj₁ ([G] {σ = liftSubst σ′} (⊢Δ ∙ ⊢F′) (liftSubstS {F = F} [Γ] ⊢Δ [F] [σ′]))\n ⊢G′ = escape [σG′]\n _ , Πᵣ _ _ _ _ _ F′ G′ D′ _ _ A≡A′ [F]₁ [G]₁ G-ext =\n extractMaybeEmb (Π-elim (proj₁ ([ΠFG] ⊢Δ [σ′])))\n [σ′u] = proj₁ ([u] ⊢Δ [σ′]) \n [σ′G[t]] = proj₁ ([G[t]] ⊢Δ [σ′])\n [σ′G[t]]′ = irrelevance′ (singleSubstLift G t) [σ′G[t]]\n [σ′u]′ = irrelevanceTerm′ (singleSubstLift G t) PE.refl PE.refl [σ′G[t]] [σ′G[t]]′ [σ′u]\n ⊢u′ = escapeTerm [σ′G[t]]′ [σ′u]′ \n pair' = ⦅_,_,_,_⦆ⱼ {F = subst σ′ F} {G = subst (liftSubst σ′) G} {t = subst σ′ t}\n {u = subst σ′ u} ⊢F′ ⊢G′ ⊢t′ ⊢u′\n [σ′≡σ] = symS [Γ] ⊢Δ [σ] [σ′] [σ≡σ′]\n [σF′≡σF] = proj₂ ([F] ⊢Δ [σ′]) [σ] [σ′≡σ]\n σF′≡σF = escapeEq [σF′] [σF′≡σF]\n [liftσ] = liftSubstS {F = F} [Γ] ⊢Δ [F] [σ]\n [wk1σ′] = wk1SubstS [Γ] ⊢Δ ⊢F′ [σ′]\n [wk1σ] = wk1SubstS [Γ] ⊢Δ ⊢F′ [σ]\n foo = proj₁ ([F] (⊢Δ ∙ ⊢F′) [wk1σ′])\n [liftσ′] : (Δ ∙ subst σ′ F ^ [ % , ι l∃ ]) ⊩ˢ liftSubst σ ∷\n Γ ∙ F ^ [ % , ι l∃ ] / [Γ] ∙ [F] / (⊢Δ ∙ escape (proj₁ ([F] ⊢Δ [σ′])))\n [liftσ′] = let ⊢F = escape (proj₁ ([F] ⊢Δ [σ]))\n [tailσ] = wk1SubstS {F = subst σ′ F} [Γ] ⊢Δ (escape (proj₁ ([F] ⊢Δ [σ′]))) [σ]\n var0′ : (Δ ∙ subst σ′ F ^ [ % , ι l∃ ]) ⊢ var 0 ∷ subst (wk1Subst σ′) F ^ [ % , ι l∃ ]\n var0′ = var (⊢Δ ∙ ⊢F′) (PE.subst (λ x → 0 ∷ x ^ _ ∈ (Δ ∙ subst σ′ F ^ _))\n (wk-subst F) here)\n var0 = conv var0′ (≅-eq (escapeEq (proj₁ ([F] (⊢Δ ∙ ⊢F′) [wk1σ′])) (proj₂ ([F] (⊢Δ ∙ ⊢F′) [wk1σ′]) [wk1σ]\n (wk1SubstSEq [Γ] ⊢Δ ⊢F′ [σ′] [σ′≡σ]))))\n in [tailσ] , neuTerm (proj₁ ([F] (⊢Δ ∙ ⊢F′) [tailσ])) (var 0)\n var0 (~-var var0)\n [σG′≡σG] = proj₂ ([G] (⊢Δ ∙ ⊢F′) (liftSubstS {F = F} [Γ] ⊢Δ [F] [σ′] )) [liftσ′]\n (liftSubstSEq {F = F} [Γ] ⊢Δ [F] [σ′] (symS [Γ] ⊢Δ [σ] [σ′] [σ≡σ′]))\n σG′≡σG = escapeEq [σG′] [σG′≡σG]\n in conv pair' (univ (∃-cong ⊢F′ (un-univ≡ (≅-eq σF′≡σF)) (un-univ≡ (≅-eq σG′≡σG))) )\n", "meta": {"hexsha": "d4a52fac6899dc1fbee798bc3ed7bb782cc8702c", "size": 5343, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Definition/LogicalRelation/Substitution/Introductions/Pair.agda", "max_stars_repo_name": "CoqHott/logrel-mltt", "max_stars_repo_head_hexsha": "e0eeebc4aa5ed791ce3e7c0dc9531bd113dfcc04", 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"avg_line_length": 55.65625, "max_line_length": 138, "alphanum_fraction": 0.4454426352, "num_tokens": 2347, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540697, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3467699116908308}} {"text": "open import Relation.Binary.Core\n\nmodule PLRTree.Drop.Permutation {A : Set} \n (_≤_ : A → A → Set)\n (tot≤ : Total _≤_) where\n\nopen import Data.List hiding (drop)\nopen import Data.Sum\nopen import List.Permutation.Base A \nopen import List.Permutation.Base.Equivalence A \nopen import PLRTree {A} \nopen import PLRTree.Complete {A} \nopen import PLRTree.Compound {A} \nopen import PLRTree.Drop _≤_ tot≤\nopen import PLRTree.Drop.Complete _≤_ tot≤\nopen import PLRTree.DropLast.Complete _≤_ tot≤\nopen import PLRTree.DropLast.Permutation _≤_ tot≤\nopen import PLRTree.Equality {A} \nopen import PLRTree.Order.Properties {A} \nopen import PLRTree.Push.Permutation _≤_ tot≤\n\nlemma-drop-++ : {t : Tag}{x : A}{l r : PLRTree} → Complete (node t x l r) → flatten (drop (node t x l r)) ∼ (flatten l ++ flatten r)\nlemma-drop-++ (perfect {leaf} {leaf} x leaf leaf ≃lf) = ∼[]\nlemma-drop-++ (perfect {node perfect x' l' r'} {node perfect x'' l'' r''} x cl cr (≃nd .x' .x'' l'≃r' l''≃r'' l'≃l'')) = \n let _l = node perfect x' l' r' ;\n _r = node perfect x'' l'' r'' ;\n _l≃r = ≃nd x' x'' l'≃r' l''≃r'' l'≃l'' ;\n cxlr = perfect x cl cr _l≃r ;\n z = last (node perfect x _l _r) compound ;\n t' = dropLast (node perfect x _l _r) ;\n ct' = right x cl (lemma-dropLast-complete cr) (lemma-dropLast-≃ _l≃r compound) \n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) (lemma-dropLast-∼ cxlr)\nlemma-drop-++ (left {l} {r} x cl cr l⋘r) \n with l | r | l⋘r | lemma-dropLast-⋘ l⋘r\n... | leaf | _ | () | _ \n... | node perfect x' l' r' | _ | () | _ \n... | node left x' l' r' | node perfect x'' l'' r'' | l⋘ .x' .x'' l'⋘r' l''≃r'' r'≃l'' | inj₁ ld⋘r \n with dropLast (node left x' l' r') | ld⋘r | lemma-dropLast-complete cl | lemma-dropLast-∼ (left x cl cr (l⋘ x' x'' l'⋘r' l''≃r'' r'≃l''))\n... | leaf | () | _ | _\n... | node perfect _ _ _ | () | _ | _ \n... | node left x''' l''' r''' | ld⋘r' | cld | fzt'∼flfr = \n let z = last (node left x (node left x' l' r') (node perfect x'' l'' r'')) compound ;\n t' = node left x (node left x''' l''' r''') (node perfect x'' l'' r'') ;\n ct' = left x cld cr ld⋘r'\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\n... | node right x''' l''' r''' | ld⋘r' | cld | fzt'∼flfr = \n let z = last (node left x (node left x' l' r') (node perfect x'' l'' r'')) compound ;\n t' = node left x (node right x''' l''' r''') (node perfect x'' l'' r'') ;\n ct' = left x cld cr ld⋘r'\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\nlemma-drop-++ (left x cl cr l⋘r) | node left x' l' r' | node perfect x'' l'' r'' | l⋘ .x' .x'' l'⋘r' l''≃r'' r'≃l'' | inj₂ ld≃r \n with dropLast (node left x' l' r') | ld≃r | lemma-dropLast-complete cl | lemma-dropLast-∼ (left x cl cr (l⋘ x' x'' l'⋘r' l''≃r'' r'≃l''))\n... | leaf | () | _ | _\n... | node perfect x''' l''' r''' | ld≃r' | cld | fzt'∼flfr = \n let z = last (node left x (node left x' l' r') (node perfect x'' l'' r'')) compound ;\n t' = node perfect x (node perfect x''' l''' r''') (node perfect x'' l'' r'') ;\n ct' = perfect x cld cr ld≃r'\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\n... | node left _ _ _ | () | _ | _\n... | node right _ _ _ | () | _ | _\nlemma-drop-++ (left x cl cr l⋘r) | node right x' l' r' | leaf | () | _ \nlemma-drop-++ (left x cl cr l⋘r) | node right x' (node perfect x'' leaf leaf) leaf | node perfect x''' leaf leaf | x⋘ .x' .x'' .x''' | inj₁ () \nlemma-drop-++ (left x cl cr l⋘r) | node right x' (node perfect x'' leaf leaf) leaf | node perfect x''' leaf leaf | x⋘ .x' .x'' .x''' | inj₂ x'≃x''' =\n let z = last (node left x (node right x' (node perfect x'' leaf leaf) leaf) (node perfect x''' leaf leaf)) compound ;\n t' = dropLast (node left x (node right x' (node perfect x'' leaf leaf) leaf) (node perfect x''' leaf leaf)) ;\n ct' = perfect x (perfect x' leaf leaf ≃lf) cr x'≃x''' ;\n fzt'∼flfr = lemma-dropLast-∼ (left x cl cr (x⋘ x' x'' x'''))\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\nlemma-drop-++ (left x cl cr l⋘r) | node right x' l' r' | node perfect x'' l'' r'' | r⋘ .x' .x'' l⋙r l''≃r'' l'⋗l'' | inj₁ ld⋘r \n with dropLast (node right x' l' r') | ld⋘r | lemma-dropLast-complete cl | lemma-dropLast-∼ (left x cl cr (r⋘ x' x'' l⋙r l''≃r'' l'⋗l''))\n... | leaf | () | _ | _\n... | node perfect _ _ _ | () | _ | _\n... | node left x''' l''' r''' | ld⋘r' | cld | fzt'∼flfr = \n let z = last (node left x (node right x' l' r') (node perfect x'' l'' r'')) compound ;\n t' = node left x (node left x''' l''' r''') (node perfect x'' l'' r'') ;\n ct' = left x cld cr ld⋘r'\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\n... | node right x''' l''' r''' | ld⋘r' | cld | fzt'∼flfr = \n let z = last (node left x (node right x' l' r') (node perfect x'' l'' r'')) compound ;\n t' = node left x (node right x''' l''' r''') (node perfect x'' l'' r'') ;\n ct' = left x cld cr ld⋘r'\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\nlemma-drop-++ (left x cl cr l⋘r) | node right x' l' r' | node perfect x'' l'' r'' | r⋘ .x' .x'' l⋙r l''≃r'' l'⋗l'' | inj₂ ld≃r \n with dropLast (node right x' l' r') | ld≃r | lemma-dropLast-complete cl | lemma-dropLast-∼ (left x cl cr (r⋘ x' x'' l⋙r l''≃r'' l'⋗l''))\n... | leaf | () | _ | _\n... | node perfect x''' l''' r''' | ld≃r' | cld | fzt'∼flfr = \n let z = last (node left x (node right x' l' r') (node perfect x'' l'' r'')) compound ;\n t' = node perfect x (node perfect x''' l''' r''') (node perfect x'' l'' r'') ;\n ct' = perfect x cld cr ld≃r'\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\n... | node left _ _ _ | () | _ | _\n... | node right _ _ _ | () | _ | _\nlemma-drop-++ (left x cl cr l⋘r) | node right x' l' r' | node left x'' l'' r'' | () | _ \nlemma-drop-++ (left x cl cr l⋘r) | node right x' l' r' | node right x'' l'' r'' | () | _\nlemma-drop-++ (right {l} {r} x cl cr l⋙r) \n with l | r | l⋙r | lemma-dropLast-⋙ l⋙r\n... | leaf | leaf | ⋙p () | _\n... | node perfect x' leaf leaf | leaf | ⋙p (⋗lf .x') | _ = ∼x /head /head ∼[] \n... | node perfect _ _ (node _ _ _ _) | leaf | ⋙p () | _\n... | node perfect _ (node _ _ _ _) _ | leaf | ⋙p () | _\n... | node left _ _ _ | leaf | ⋙p () | _\n... | node right _ _ _ | leaf | ⋙p () | _\n... | leaf | node perfect _ _ _ | ⋙p () | _\n... | node perfect x' l' r' | node perfect x'' l'' r'' | ⋙p (⋗nd .x' .x'' l'≃r' l''≃r'' l'⋗l'') | _ =\n let z = last (node right x (node perfect x' l' r') (node perfect x'' l'' r'')) compound ;\n t' = dropLast (node right x (node perfect x' l' r') (node perfect x'' l'' r'')) ;\n ct' = left x (lemma-dropLast-complete cl) cr (lemma-dropLast-⋗ (⋗nd x' x'' l'≃r' l''≃r'' l'⋗l'') compound) ;\n fzt'∼flfr = lemma-dropLast-∼ (right x cl cr (⋙p (⋗nd x' x'' l'≃r' l''≃r'' l'⋗l'')))\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\n... | node left _ _ _ | node perfect _ _ _ | ⋙p () | _\n... | node right _ _ _ | node perfect _ _ _ | ⋙p () | _\n... | leaf | node left _ _ _ | ⋙p () | _\n... | node perfect x' l' r' | node left x'' l'' r'' | _l⋙r | inj₁ l⋙rd = \n let z = last (node right x (node perfect x' l' r') (node left x'' l'' r'')) compound ;\n t' = dropLast (node right x (node perfect x' l' r') (node left x'' l'' r'')) ;\n ct' = right x cl (lemma-dropLast-complete cr) l⋙rd ;\n fzt'∼flfr = lemma-dropLast-∼ (right x cl cr _l⋙r)\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\n... | node perfect _ _ _ | node left _ _ _ | _ | inj₂ () \n... | node left _ _ _ | node left _ _ _ | ⋙p () | _\n... | node right _ _ _ | node left _ _ _ | ⋙p () | _\n... | leaf | node right _ _ _ | ⋙p () | _\n... | node perfect x' l' r' | node right x'' l'' r'' | _l⋙r | inj₁ l⋙rd = \n let z = last (node right x (node perfect x' l' r') (node right x'' l'' r'')) compound ;\n t' = dropLast (node right x (node perfect x' l' r') (node right x'' l'' r'')) ;\n ct' = right x cl (lemma-dropLast-complete cr) l⋙rd ;\n fzt'∼flfr = lemma-dropLast-∼ (right x cl cr _l⋙r)\n in trans∼ (lemma-push-∼ (lemma-setRoot-complete z ct') (≺-wf (setRoot z t'))) fzt'∼flfr\n... | node perfect _ _ _ | node right _ _ _ | _ | inj₂ ()\n... | node left _ _ _ | node right _ _ _ | ⋙p () | _ \n... | node right _ _ _ | node right _ _ _ | ⋙p () | _\n\nlemma-drop-∼ : {t : Tag}{x : A}{l r : PLRTree} → Complete (node t x l r) → (x ∷ flatten (drop (node t x l r))) ∼ flatten (node t x l r)\nlemma-drop-∼ (perfect x cl cr l≃r) = ∼x /head /head (lemma-drop-++ (perfect x cl cr l≃r))\nlemma-drop-∼ (left x cl cr l⋘r) = ∼x /head /head (lemma-drop-++ (left x cl cr l⋘r))\nlemma-drop-∼ (right x cl cr l⋙r) = ∼x /head /head (lemma-drop-++ (right x cl cr l⋙r))\n\n\n", "meta": 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"b8d428bccbdd1b13613e8f6ead6c81a8f9298399", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 70.2446043165, "max_line_length": 149, "alphanum_fraction": 0.4941622286, "num_tokens": 3661, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.3466904147571071}} {"text": "{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.\n\n Copyright (c) 2020, 2021 Oracle and/or its affiliates.\n Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl\n-}\nopen import LibraBFT.Prelude\nopen import LibraBFT.Lemmas\nopen import LibraBFT.Base.PKCS\nopen import LibraBFT.Base.Types\nimport LibraBFT.Yasm.Base as LYB\n\n-- This module provides some definitions and properties that facilitate\n-- proofs of properties about a distributed system modeled by Yasm.System\n-- paramaterized by some SystemParameters.\n\nmodule LibraBFT.Yasm.Properties\n (ℓ-EC : Level)\n (EpochConfig : Set ℓ-EC)\n (epochId : EpochConfig → EpochId)\n (authorsN : EpochConfig → ℕ)\n (parms : LYB.SystemParameters ℓ-EC EpochConfig epochId authorsN)\n -- In addition to the parameters used by the rest of the system model, this module\n -- needs to relate Members to PKs and PeerIds, so that StepPeerState-AllValidParts\n -- can be defined. This enables the application to prove that honest peers sign\n -- new messages only for their own public key. The system model does not know that\n -- directly.\n (senderPKOK : (ec : EpochConfig) → PK → LYB.SystemParameters.PeerId parms → Set)\n where\n open LYB.SystemParameters parms\n open import LibraBFT.Yasm.AvailableEpochs PeerId ℓ-EC EpochConfig epochId authorsN\n using (AvailableEpochs) renaming (lookup'' to EC-lookup)\n import LibraBFT.Yasm.AvailableEpochs PeerId ℓ-EC EpochConfig epochId authorsN\n as AE\n open import LibraBFT.Yasm.Base ℓ-EC EpochConfig epochId authorsN\n open import LibraBFT.Yasm.System ℓ-EC EpochConfig epochId authorsN parms\n open import Util.FunctionOverride PeerId _≟PeerId_\n\n -- A ValidPartForPK collects the assumptions about what a /part/ in the outputs of an honest verifier\n -- satisfies: (i) the epoch field is consistent with the existent epochs and (ii) the verifier is\n -- a member of the associated epoch config, and (iii) has the given PK in that epoch.\n record ValidSenderForPK {e}(𝓔s : AvailableEpochs e)(part : Part)(sender : PeerId)(pk : PK) : Set ℓ-EC where\n constructor mkValidSenderForPK\n field\n vp-epoch : part-epoch part < e\n vp-ec : EpochConfig\n vp-ec-≡ : AE.lookup'' 𝓔s vp-epoch ≡ vp-ec\n vp-sender-ok : senderPKOK vp-ec pk sender\n open ValidSenderForPK public\n\n -- A valid part remains valid when new epochs are added\n ValidSenderForPK-stable-epoch : ∀{e part α pk}{𝓔s : AvailableEpochs e}(𝓔 : EpochConfigFor e)\n → ValidSenderForPK 𝓔s part α pk\n → ValidSenderForPK (AE.append 𝓔 𝓔s) part α pk\n ValidSenderForPK-stable-epoch {pk = pk} {𝓔s = 𝓔s} 𝓔 (mkValidSenderForPK e ec refl vpk) = record\n { vp-epoch = ≤-step e\n ; vp-ec = ec\n ; vp-ec-≡ = AE.lookup''-≤-step-lemma 𝓔s 𝓔 e\n ; vp-sender-ok = vpk\n }\n\n -- A valid part remains valid\n ValidSenderForPK-stable : ∀{e e' α}{st : SystemState e}{st' : SystemState e'}\n → Step* st st' → ∀{part pk}\n → ValidSenderForPK (availEpochs st) part α pk\n → ValidSenderForPK (availEpochs st') part α pk\n ValidSenderForPK-stable step-0 v = v\n ValidSenderForPK-stable (step-s st (step-epoch 𝓔)) v\n = ValidSenderForPK-stable-epoch 𝓔 (ValidSenderForPK-stable st v)\n ValidSenderForPK-stable (step-s st (step-peer _)) v\n = ValidSenderForPK-stable st v\n\n sameEpoch⇒sameEC : ∀ {e p1 p2 α1 α2 pk1 pk2}{𝓔s : AvailableEpochs e}\n → (vp1 : ValidSenderForPK 𝓔s p1 α1 pk1)\n → (vp2 : ValidSenderForPK 𝓔s p2 α2 pk2)\n → part-epoch p1 ≡ part-epoch p2\n → vp-ec vp1 ≡ vp-ec vp2\n sameEpoch⇒sameEC {𝓔s = 𝓔s} vp1 vp2 parts≡ =\n trans (sym (vp-ec-≡ vp1))\n (trans (AE.lookup-𝓔s-injective 𝓔s (vp-epoch vp1) (vp-epoch vp2) parts≡)\n (vp-ec-≡ vp2))\n\n -- TODO-1 : prove it\n postulate\n ValidSenderForPK⇒ep≡ : ∀ {e p1 p2 α1 pk} {𝓔s : AvailableEpochs e}\n → WithVerSig pk p1 → WithVerSig pk p2\n → part-epoch p1 ≡ part-epoch p2\n → ValidSenderForPK 𝓔s p1 α1 pk\n → ValidSenderForPK 𝓔s p2 α1 pk\n\n -- We say that an implementation produces only valid parts iff all parts of every message in the\n -- output of a 'StepPeerState' are either: (i) a valid new part (i.e., the part is valid and no\n -- message with the same signature has been sent previously), or (ii) a message has been sent\n -- with the same signature.\n StepPeerState-AllValidParts : Set ℓ-EC\n StepPeerState-AllValidParts = ∀{e s m part pk initd' outs}{α}{𝓔s : AvailableEpochs e}{st : SystemState e}\n → (r : ReachableSystemState st)\n → Meta-Honest-PK pk\n → StepPeerState α 𝓔s (msgPool st) (initialised st) (peerStates st α) initd' (s , outs)\n → m ∈ outs → part ⊂Msg m → (ver : WithVerSig pk part)\n → (ValidSenderForPK 𝓔s part α pk × ¬ (MsgWithSig∈ pk (ver-signature ver) (msgPool st)))\n ⊎ MsgWithSig∈ pk (ver-signature ver) (msgPool st)\n\n -- A /part/ was introduced by a specific step when:\n IsValidNewPart : ∀{e e'}{pre : SystemState e}{post : SystemState e'} → Signature → PK → Step pre post → Set ℓ-EC\n IsValidNewPart _ _ (step-epoch _) = Lift (ℓ-EC) ⊥\n -- said step is a /step-peer/ and\n IsValidNewPart {pre = pre} sig pk (step-peer {pid = pid} pstep)\n -- the part has never been seen before\n = ReachableSystemState pre\n × ¬ (MsgWithSig∈ pk sig (msgPool pre))\n × Σ (MsgWithSig∈ pk sig (msgPool (StepPeer-post pstep)))\n (λ m → ValidSenderForPK (availEpochs pre) (msgPart m) (msgSender m) pk)\n\n -- When we can prove that the implementation provided by 'parms' at the\n -- top of this module satisfies 'StepPeerState-AllValidParts', we can\n -- prove a number of useful structural properties:\n\n -- TODO-2: Refactor into a file (LibraBFT.Yasm.Properties.Structural) later on\n -- if this grows too large.\n module Structural (sps-avp : StepPeerState-AllValidParts) where\n\n -- We can unwind the state and highlight the step where a part was\n -- originally sent. This 'unwind' function combined with Any-Step-elim\n -- enables a powerful form of reasoning. The 'honestVoteEpoch' below\n -- exemplifies this well.\n unwind : ∀{e}{st : SystemState e}(tr : ReachableSystemState st)\n → ∀{p m σ pk} → Meta-Honest-PK pk\n → p ⊂Msg m → (σ , m) ∈ msgPool st → (ver : WithVerSig pk p)\n → Any-Step (IsValidNewPart (ver-signature ver) pk) tr\n unwind (step-s tr (step-epoch _)) hpk p⊂m m∈sm sig\n = step-there (unwind tr hpk p⊂m m∈sm sig)\n unwind (step-s tr (step-peer {pid = β} {outs = outs} {pre = pre} sp)) hpk p⊂m m∈sm sig\n with Any-++⁻ (List-map (β ,_) outs) {msgPool pre} m∈sm\n ...| inj₂ furtherBack = step-there (unwind tr hpk p⊂m furtherBack sig)\n ...| inj₁ thisStep\n with sp\n ...| step-cheat fm isCheat\n with thisStep\n ...| here refl\n with isCheat p⊂m sig\n ...| inj₁ abs = ⊥-elim (hpk abs)\n ...| inj₂ sentb4\n with unwind tr {p = msgPart sentb4} hpk (msg⊆ sentb4) (msg∈pool sentb4) (msgSigned sentb4)\n ...| res rewrite msgSameSig sentb4 = step-there res\n unwind (step-s tr (step-peer {pid = β} {outs = outs} {pre = pre} sp)) hpk p⊂m m∈sm sig\n | inj₁ thisStep\n | step-honest x\n with Any-satisfied-∈ (Any-map⁻ thisStep)\n ...| (m , refl , m∈outs)\n with sps-avp tr hpk x m∈outs p⊂m sig\n ...| inj₂ sentb4 with unwind tr {p = msgPart sentb4} hpk (msg⊆ sentb4) (msg∈pool sentb4) (msgSigned sentb4)\n ...| res rewrite msgSameSig sentb4 = step-there res\n unwind (step-s tr (step-peer {pid = β} {outs = outs} {pre = pre} sp)) {p} hpk p⊂m m∈sm sig\n | inj₁ thisStep\n | step-honest x\n | (m , refl , m∈outs)\n | inj₁ (valid-part , notBefore) =\n step-here tr (tr , notBefore , MsgWithSig∈-++ˡ (mkMsgWithSig∈ _ _ p⊂m β thisStep sig refl)\n , valid-part)\n\n -- Unwind is inconvenient to use by itself because we have to do\n -- induction on Any-Step-elim. The 'honestPartValid' property below\n -- provides a fairly general result conveniently: for every part\n -- verifiable with an honest PK, there is a msg with the same\n -- signature that is valid for some pid.\n\n honestPartValid : ∀ {e st} → ReachableSystemState {e} st → ∀ {pk nm v sender}\n → Meta-Honest-PK pk\n → v ⊂Msg nm → (sender , nm) ∈ msgPool st → (ver : WithVerSig pk v)\n → Σ (MsgWithSig∈ pk (ver-signature ver) (msgPool st))\n (λ msg → (ValidSenderForPK (availEpochs st) (msgPart msg) (msgSender msg) pk))\n honestPartValid {e} {st} r {pk = pk} hpk v⊂m m∈pool ver\n -- We extract two pieces of important information from the place where the part 'v'\n -- was first sent: (a) there is a message with the same signature /in the current pool/\n -- and (b) its epoch is less than e.\n = Any-Step-elim (λ { {st = step-epoch _} ()\n ; {st = step-peer {pid = pid} ps} (_ , _ , new , valid) tr\n → MsgWithSig∈-Step* tr new\n , ValidSenderForPK-stable tr (subst (λ P → ValidSenderForPK _ P (msgSender (MsgWithSig∈-Step* tr new)) pk)\n (MsgWithSig∈-Step*-part tr new)\n (subst (λ sndr → ValidSenderForPK _ _ sndr pk)\n (MsgWithSig∈-Step*-sender tr new)\n valid))\n })\n (unwind r hpk v⊂m m∈pool ver)\n\n -- Unforgeability is also an important property stating that every part that is\n -- verified with an honest public key has either been sent by α or is a replay\n -- of another message sent before.\n ext-unforgeability'\n : ∀{e α m part pk}{st : SystemState e} → ReachableSystemState st\n -- If a message m has been sent by α, containing part\n → (α , m) ∈ msgPool st → part ⊂Msg m\n -- And the part can be verified with an honest public key,\n → (sig : WithVerSig pk part) → Meta-Honest-PK pk\n -- then either the part is a valid part by α (meaning that α can\n -- sign the part itself) or a message with the same signature has\n -- been sent previously.\n → ValidSenderForPK (availEpochs st) part α pk\n ⊎ MsgWithSig∈ pk (ver-signature sig) (msgPool st)\n ext-unforgeability' (step-s st (step-epoch 𝓔)) m∈sm p⊆m sig hpk\n = ⊎-map (ValidSenderForPK-stable-epoch 𝓔) id (ext-unforgeability' st m∈sm p⊆m sig hpk)\n ext-unforgeability' {part = part} (step-s st (step-peer {pid = β} {outs = outs} {pre = pre} sp)) m∈sm p⊆m sig hpk\n with Any-++⁻ (List-map (β ,_) outs) {msgPool pre} m∈sm\n ...| inj₂ furtherBack = MsgWithSig∈-++ʳ <⊎$> (ext-unforgeability' st furtherBack p⊆m sig hpk)\n ...| inj₁ thisStep\n with sp\n ...| step-cheat fm isCheat\n with thisStep\n ...| here refl\n with isCheat p⊆m sig\n ...| inj₁ abs = ⊥-elim (hpk abs)\n ...| inj₂ sentb4 = inj₂ (MsgWithSig∈-++ʳ sentb4)\n ext-unforgeability' {α = α} {m = m} {part = part} (step-s st (step-peer {pid = β} {outs = outs} {pre = pre} sp)) m∈sm p⊆m sig hpk\n | inj₁ thisStep\n | step-honest x\n with Any-satisfied-∈ (Any-map⁻ thisStep)\n ...| (m , refl , m∈outs) = ⊎-map proj₁ MsgWithSig∈-++ʳ (sps-avp st hpk x m∈outs p⊆m sig)\n\n -- The ext-unforgeability' property can be collapsed in a single clause.\n\n -- TODO-2: so far, ext-unforgeability is used only to get a MsgWithSig∈ that is passed to\n -- msgWithSigSentByAuthor, which duplicates some of the reasoning in the proof of\n -- ext-unforgeability'; should these properties possibly be combined into one simpler proof?\n ext-unforgeability\n : ∀{e α₀ m part pk}{st : SystemState e} → ReachableSystemState st\n → (α₀ , m) ∈ msgPool st → part ⊂Msg m\n → (sig : WithVerSig pk part) → Meta-Honest-PK pk\n → MsgWithSig∈ pk (ver-signature sig) (msgPool st)\n ext-unforgeability {_} {α₀} {m} {st = st} rst m∈sm p⊂m sig hpk\n with ext-unforgeability' rst m∈sm p⊂m sig hpk\n ...| inj₁ p\n = mkMsgWithSig∈ _ _ p⊂m α₀ m∈sm sig refl\n ...| inj₂ sentb4 = sentb4\n\n ¬cheatForgeNew : ∀ {e pid pk vsig mst outs m}{st : SystemState e}\n → (sp : StepPeer st pid mst outs)\n → outs ≡ m ∷ []\n → (ic : isCheat sp)\n → Meta-Honest-PK pk\n → MsgWithSig∈ pk vsig ((pid , m) ∷ msgPool st)\n → MsgWithSig∈ pk vsig (msgPool st)\n ¬cheatForgeNew sc@(step-cheat fm isCheat) refl _ hpk mws\n with msg∈pool mws\n ...| there m∈pool = mkMsgWithSig∈ (msgWhole mws) (msgPart mws) (msg⊆ mws) (msgSender mws) m∈pool (msgSigned mws) (msgSameSig mws)\n ...| here m∈pool\n with isCheat (subst (msgPart mws ⊂Msg_) (cong proj₂ m∈pool) (msg⊆ mws)) (msgSigned mws)\n ...| inj₁ dis = ⊥-elim (hpk dis)\n ...| inj₂ mws' rewrite msgSameSig mws = mws'\n\n\n\n msgWithSigSentByAuthor : ∀ {e pk sig}{st : SystemState e}\n → ReachableSystemState st\n → Meta-Honest-PK pk\n → MsgWithSig∈ pk sig (msgPool st)\n → Σ (MsgWithSig∈ pk sig (msgPool st))\n λ mws → ValidSenderForPK (availEpochs st) (msgPart mws) (msgSender mws) pk\n msgWithSigSentByAuthor step-0 _ ()\n msgWithSigSentByAuthor (step-s {pre = pre} preach (step-epoch 𝓔)) hpk mws\n rewrite step-epoch-does-not-send pre 𝓔\n with msgWithSigSentByAuthor preach hpk mws\n ...| mws' , vpb = mws' , ValidSenderForPK-stable {st = pre} (step-s step-0 (step-epoch 𝓔)) vpb\n msgWithSigSentByAuthor {pk = pk} (step-s {pre = pre} preach (step-peer theStep@(step-cheat fm cheatCons))) hpk mws\n with (¬cheatForgeNew theStep refl unit hpk mws)\n ...| mws'\n with msgWithSigSentByAuthor preach hpk mws'\n ...| mws'' , vpb'' = MsgWithSig∈-++ʳ mws'' , vpb''\n msgWithSigSentByAuthor {e} (step-s {pre = pre} preach (step-peer {pid = pid} {outs = outs} (step-honest sps))) hpk mws\n with Any-++⁻ (List-map (pid ,_) outs) {msgPool pre} (msg∈pool mws)\n ...| inj₂ furtherBack\n with msgWithSigSentByAuthor preach hpk (MsgWithSig∈-transp mws furtherBack)\n ...| mws' , vpb' = MsgWithSig∈-++ʳ mws' , vpb'\n\n msgWithSigSentByAuthor {e} (step-s {pre = pre} preach (step-peer {pid = pid} {outs = outs} (step-honest sps))) hpk mws\n | inj₁ thisStep\n with Any-satisfied-∈ (Any-map⁻ thisStep)\n ...| (m' , refl , m∈outs)\n with sps-avp preach hpk sps m∈outs (msg⊆ mws) (msgSigned mws)\n ...| inj₁ (vpbα₀ , _) = mws , vpbα₀\n ...| inj₂ mws'\n with msgWithSigSentByAuthor preach hpk mws'\n ...| mws'' , vpb'' rewrite sym (msgSameSig mws) = MsgWithSig∈-++ʳ mws'' , vpb''\n\n\n newMsg⊎msgSentB4 : ∀ {e pk v m pid sndr st' outs} {st : SystemState e}\n → (r : ReachableSystemState st)\n → (stP : StepPeer st pid st' outs)\n → Meta-Honest-PK pk → (sig : WithVerSig pk v)\n → v ⊂Msg m → (sndr , m) ∈ msgPool (StepPeer-post stP)\n → (m ∈ outs × ValidSenderForPK (availEpochs st) v pid pk\n × ¬ (MsgWithSig∈ pk (ver-signature sig) (msgPool st)))\n ⊎ MsgWithSig∈ pk (ver-signature sig) (msgPool st)\n newMsg⊎msgSentB4 {e} {pk} {v} {m} {pid} {sndr} {_} {outs} {st} r stP pkH sig v⊂m m∈post\n with Any-++⁻ (List-map (pid ,_) outs) m∈post\n ...| inj₂ m∈preSt = inj₂ (mkMsgWithSig∈ m v v⊂m sndr m∈preSt sig refl)\n ...| inj₁ nm∈outs\n with Any-map (cong proj₂) (Any-map⁻ nm∈outs)\n ...| m∈outs\n with stP\n ...| step-honest stH\n with sps-avp r pkH stH m∈outs v⊂m sig\n ...| inj₁ newVote = inj₁ (m∈outs , newVote)\n ...| inj₂ msb4 = inj₂ msb4\n newMsg⊎msgSentB4 {e} {pk} {v} {m} {pid} {sndr} {_} {outs} {st} r stP pkH sig v⊂m m∈post\n | inj₁ nm∈outs\n | here refl\n | step-cheat fm ic\n = let mws = mkMsgWithSig∈ m v v⊂m pid (here refl) sig refl\n in inj₂ (¬cheatForgeNew {st = st} (step-cheat fm ic) refl unit pkH mws)\n\n -- This could potentially be more general, for example covering the whole SystemState, rather than\n -- just one peer's state. However, this would put more burden on the user and is not required so\n -- far.\n CarrierProp : Set₁\n CarrierProp = Part → PeerState → Set\n\n module _ (P : CarrierProp) where\n\n record PropCarrier (pk : PK) (sig : Signature) {e} (st : SystemState e) : Set (ℓ-EC ℓ⊔ (ℓ+1 0ℓ)) where\n constructor mkCarrier\n field\n carrStReach : ReachableSystemState st -- Enables use of invariants when proving that steps preserve carrProp\n carrSent : MsgWithSig∈ pk sig (msgPool st)\n carrValid : ValidSenderForPK (availEpochs st) (msgPart carrSent) (msgSender carrSent) pk\n carrProp : P (msgPart carrSent) (peerStates st (msgSender carrSent))\n open PropCarrier public\n\n PeerStepPreserves : Set (ℓ+1 ℓ0 ℓ⊔ ℓ-EC)\n PeerStepPreserves = ∀ {e initd' ps' outs pk sig}{pre : SystemState e}\n → (r : ReachableSystemState pre)\n → (pc : PropCarrier pk sig {e} pre)\n → (sps : StepPeerState {e} (msgSender (carrSent pc))\n (availEpochs pre)\n (msgPool pre)\n (initialised pre)\n (peerStates pre (msgSender (carrSent pc)))\n initd'\n (ps' , outs))\n → P (msgPart (carrSent pc)) ps'\n\n module _ (PSP : PeerStepPreserves) where\n\n Carrier-transp : ∀ {e' e'' pk sig} {pre : SystemState e'}{post : SystemState e''}\n → (theStep : Step pre post)\n → PropCarrier pk sig pre\n → PropCarrier pk sig post\n Carrier-transp {pre = pre} {post} (step-epoch ec) (mkCarrier r mws vpk lvr) =\n mkCarrier (step-s r (step-epoch ec)) mws (ValidSenderForPK-stable-epoch ec vpk) lvr\n Carrier-transp {e' = e'} {pre = pre} {post} theStep@(step-peer {pid = pid} {st'} {pre = .pre} sps) pc@(mkCarrier r mws vpk prop)\n with step-s r theStep\n ...| postReach\n with sps\n ...| step-cheat fm isch = mkCarrier postReach (MsgWithSig∈-++ʳ mws) vpk\n (subst (λ ps → P (msgPart mws) (ps (msgSender mws))) (sym (cheatStepDNMPeerStates {pre = pre} (step-cheat fm isch) unit)) prop)\n -- PeerStates not changed by cheat steps\n ...| step-honest {st = st} sps'\n with msgSender mws ≟PeerId pid\n ...| no neq = mkCarrier postReach (MsgWithSig∈-++ʳ mws) vpk\n (subst (λ ps → P (msgPart mws) ps) (override-target-≢ {f = peerStates pre} neq) prop)\n ...| yes refl = mkCarrier postReach (MsgWithSig∈-++ʳ mws) vpk\n (subst (λ ps → P (msgPart mws) ps) (sym override-target-≡) (PSP r pc sps'))\n", "meta": {"hexsha": "93d40e429814bbb99b0a6c651c1efed0d6fdb12e", "size": 19536, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "LibraBFT/Yasm/Properties.agda", "max_stars_repo_name": "haroldcarr/bft-consensus-agda", "max_stars_repo_head_hexsha": "34e4627855fb198665d0c98f377403a906ba75d7", "max_stars_repo_licenses": ["UPL-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "LibraBFT/Yasm/Properties.agda", "max_issues_repo_name": "haroldcarr/bft-consensus-agda", "max_issues_repo_head_hexsha": "34e4627855fb198665d0c98f377403a906ba75d7", "max_issues_repo_licenses": ["UPL-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "LibraBFT/Yasm/Properties.agda", "max_forks_repo_name": "haroldcarr/bft-consensus-agda", "max_forks_repo_head_hexsha": "34e4627855fb198665d0c98f377403a906ba75d7", "max_forks_repo_licenses": ["UPL-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 53.8181818182, "max_line_length": 138, "alphanum_fraction": 0.5915745291, "num_tokens": 6122, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3466676140687354}} {"text": "{-# OPTIONS --without-K --rewriting #-}\n\nopen import HoTT\n\nopen import stash.modalities.JoinAdj\n\nmodule stash.modalities.Orthogonality where\n\n module PathSplit {i} (X : Type i) (x y : X) where\n\n to : (x == y) → Σ X (λ z → (x == z) × (z == y))\n to p = x , (idp , p)\n\n from : Σ X (λ z → (x == z) × (z == y)) → x == y\n from (z , p , q) = p ∙ q\n\n abstract\n \n to-from : (b : Σ X (λ z → (x == z) × (z == y))) → to (from b) == b\n to-from (z , p , q) = pair= p (↓-×-in (↓-cst=idf-in (∙'-unit-l p)) (↓-idf=cst-in idp))\n\n from-to : (p : x == y) → from (to p) == p\n from-to p = idp\n \n path-split : (x == y) ≃ Σ X (λ z → (x == z) × (z == y))\n path-split = equiv to from to-from from-to\n\n module _ {i} where\n \n Δ : (X A : Type i) → X → (A → X)\n Δ X A = cst\n\n -- Δ-ap : {X A : Type i} {x y : X} (φ : A → x == y) (a : A)\n -- → λ= φ == ap cst (φ a)\n -- Δ-ap φ a = equiv-is-inj {f = app=} (snd app=-equiv) (λ= φ) (ap cst (φ a)) (λ= (λ a₀ → app=-β φ a₀ ∙ {!lem a₀!}))\n\n -- where lem : ∀ a₀ → φ a₀ == app= (ap cst (φ a₀)) a₀ \n -- lem = {!!}\n\n ⟦_⊥_⟧ : (A : Type i) (X : Type i) → Type i\n ⟦ A ⊥ X ⟧ = is-equiv (Δ X A)\n\n ctr : {A X : Type i} → ⟦ A ⊥ X ⟧ → (A → X) → X\n ctr A⊥X φ = is-equiv.g A⊥X φ\n\n ctr-null : {A X : Type i} (A⊥X : ⟦ A ⊥ X ⟧) (φ : A → X)\n → cst (ctr A⊥X φ) == φ \n ctr-null A⊥X φ = is-equiv.f-g A⊥X φ\n\n ctr-cst : {A X : Type i} (A⊥X : ⟦ A ⊥ X ⟧) \n → (x : X) → ctr A⊥X (cst x) == x\n ctr-cst A⊥X x = is-equiv.g-f A⊥X x\n\n Unit-orth : (A : Type i) → ⟦ Lift ⊤ ⊥ A ⟧\n Unit-orth A = record {\n g = λ φ → φ (lift unit) ;\n f-g = λ φ → λ= (λ { (lift unit) → idp }) ;\n g-f = λ a → idp ;\n adj = λ a → λ=-η idp }\n\n Δ-equiv-is-contr : (A : Type i) → is-equiv (Δ A A) → is-contr A\n Δ-equiv-is-contr A e = is-equiv.g e (idf A) , (λ a → app= (is-equiv.f-g e (idf A)) a)\n \n self-orth-is-contr : (A : Type i) → ⟦ A ⊥ A ⟧ → is-contr A\n self-orth-is-contr A ω = Δ-equiv-is-contr A ω\n\n equiv-preserves-orth-l : {A B X : Type i} → (A ≃ B) → ⟦ A ⊥ X ⟧ → ⟦ B ⊥ X ⟧\n equiv-preserves-orth-l {A} {B} {X} (f , f-ise) ω = is-eq (Δ X B) g f-g ω.g-f\n\n where module ω = is-equiv ω\n module f = is-equiv f-ise\n\n g : (B → X) → X\n g φ = ω.g (φ ∘ f)\n\n f-g : (φ : B → X) → Δ X B (g φ) == φ\n f-g φ = λ= λ b → app= (ω.f-g (φ ∘ f)) (f.g b) ∙ ap φ (f.f-g b)\n\n equiv-preserves-orth-r : {A X Y : Type i} → (X ≃ Y) → ⟦ A ⊥ X ⟧ → ⟦ A ⊥ Y ⟧\n equiv-preserves-orth-r {A} {X} {Y} (f , f-ise) ω = is-eq (Δ Y A) g f-g g-f\n\n where module ω = is-equiv ω\n module f = is-equiv f-ise\n\n g : (A → Y) → Y\n g φ = f (ω.g (f.g ∘ φ))\n\n f-g : (φ : A → Y) → Δ Y A (g φ) == φ\n f-g φ = λ= λ x → ap f (app= (ω.f-g (f.g ∘ φ)) x) ∙ f.f-g (φ x)\n\n g-f : (y : Y) → g (Δ Y A y) == y\n g-f y = ap f (ω.g-f (f.g y)) ∙ f.f-g y\n\n Σ-orth : {A X : Type i} {B : A → Type i} → ⟦ A ⊥ X ⟧ → (B⊥ : (a : A) → ⟦ B a ⊥ X ⟧) → ⟦ Σ A B ⊥ X ⟧\n Σ-orth {A} {X} {B} A⊥ B⊥ = is-eq _ from to-from from-to\n\n where from : (Σ A B → X) → X\n from φ = ctr A⊥ (λ a → ctr (B⊥ a) (λ b → φ (a , b)))\n\n to-from : (φ : Σ A B → X) → cst (from φ) == φ\n to-from φ = λ= (λ { (a , b) → app= (ctr-null A⊥ (λ a → ctr (B⊥ a) (λ b → φ (a , b)))) a ∙\n app= (ctr-null (B⊥ a) (λ b → φ (a , b))) b })\n\n from-to : (x : X) → from (cst x) == x\n from-to x = ap (ctr A⊥) (λ= (λ a → ctr-cst (B⊥ a) x)) ∙ ctr-cst A⊥ x\n\n -- -- This works if the base is connected, but you'll have to add that\n -- fib-orth : {A X : Type i} {B : A → Type i} → ⟦ Σ A B ⊥ X ⟧ → (a : A) → ⟦ B a ⊥ X ⟧\n -- fib-orth {A} {X} {B} Σ⊥ a = is-eq _ g {!!} {!!}\n\n -- where g : (φ : B a → X) → X\n -- g φ = ctr Σ⊥ {!!}\n\n -- -- This looks doomed ...\n -- base-orth : {A X : Type i} {B : A → Type i} → ⟦ Σ A B ⊥ X ⟧ → ⟦ A ⊥ X ⟧\n -- base-orth {A} {X} {B} Σ⊥ = is-eq _ g f-g {!!}\n\n -- where g : (φ : A → X) → X\n -- g φ = ctr Σ⊥ (uncurry (λ a _ → φ a))\n\n -- f-g : (φ : A → X) → cst (g φ) == φ\n -- f-g φ = λ= (λ a → app= (ctr-null Σ⊥ (uncurry (λ a _ → φ a))) (a , {!!}))\n\n ×-orth : {A B X : Type i} → ⟦ A ⊥ X ⟧ → ⟦ B ⊥ X ⟧ → ⟦ A × B ⊥ X ⟧\n ×-orth {B = B} A⊥ B⊥ = Σ-orth {B = λ _ → B} A⊥ (λ _ → B⊥)\n\n -- Okay, you need to find a simpler way.\n -- *-orth : {A B X : Type i} → ⟦ B ⊥ X ⟧ → ⟦ A * B ⊥ X ⟧\n -- *-orth {A} {B} {X} ω = is-eq (Δ X (A * B)) from to-from from-to\n\n -- where from : (A * B → X) → X\n -- from f = is-equiv.g ω (f ∘ right)\n\n -- where test : A → hfiber cst (f ∘ right)\n -- test = snd (–> (join-adj A B X) f) \n \n -- to-from : (f : A * B → X) → Δ X (A * B) (from f) == f\n -- to-from f = {!!}\n\n -- from-to : (x : X) → from (Δ X (A * B) x) == x\n -- from-to x = {!is-equiv.g-f ω x!}\n\n postulate\n \n -- Right, this is a special case of the join adjunction ...\n adj-orth : (A X : Type i) → ⟦ Susp A ⊥ X ⟧ → (x y : X) → ⟦ A ⊥ x == y ⟧\n\n pths-orth : {A X : Type i} {x y : X} → ⟦ A ⊥ X ⟧ → ⟦ A ⊥ x == y ⟧\n pths-orth {A} {X} {x} {y} A⊥X = is-eq (Δ (x == y) A) g to-from from-to\n\n where g : (A → x == y) → x == y\n g φ = ! (ctr-cst A⊥X x) ∙ ap (ctr A⊥X) (λ= φ) ∙ ctr-cst A⊥X y \n\n to-from : (φ : A → x == y) → Δ (x == y) A (g φ) == φ\n to-from φ = λ= coh\n\n where coh : (a : A) → g φ == φ a\n coh a = ! (ctr-cst A⊥X x) ∙ ap (ctr A⊥X) (λ= φ) ∙ ctr-cst A⊥X y =⟨ {!!} ⟩\n φ a ∎\n\n where puzzle = ap (ctr A⊥X) (λ= φ) =⟨ {!a !} ⟩\n ap (ctr A⊥X) (ap cst (φ a)) =⟨ ∘-ap (ctr A⊥X) cst (φ a) ⟩ \n ap ((ctr A⊥X) ∘ cst) (φ a) ∎\n\n eq : ctr-cst A⊥X x ∙' φ a == ap ((ctr A⊥X) ∘ cst) (φ a) ∙ ctr-cst A⊥X y\n eq = ↓-app=idf-out (apd (ctr-cst A⊥X) (φ a))\n\n eq₀ : ctr-null A⊥X (cst x) ∙' λ= φ == ap (cst ∘ (ctr A⊥X)) (λ= φ) ∙ ctr-null A⊥X (cst y)\n eq₀ = ↓-app=idf-out (apd (ctr-null A⊥X) (λ= φ))\n\n adj : ap cst (ctr-cst A⊥X x) == ctr-null A⊥X (cst x)\n adj = is-equiv.adj A⊥X x\n\n adj' : ap (ctr A⊥X) (ctr-null A⊥X (cst x)) == ctr-cst A⊥X (ctr A⊥X (cst x))\n adj' = is-equiv.adj' A⊥X (cst x)\n\n claim : (λ= φ) == ! (ap cst (ctr-cst A⊥X x)) ∙ ap cst (ap (ctr A⊥X) (λ= φ)) ∙ ap cst (ctr-cst A⊥X y)\n claim = {!!}\n\n then : (λ= φ) == ap cst (! (ctr-cst A⊥X x) ∙ ap (ctr A⊥X) (λ= φ) ∙ ctr-cst A⊥X y)\n then = {!!}\n \n from-to : (p : x == y) → g (cst p) == p\n from-to p = {!!}\n \n\n -- Weak cellular inequalities\n module _ {i} where\n \n _≻_ : Type i → Type i → Type _\n X ≻ A = (Y : Type i) → ⟦ A ⊥ Y ⟧ → ⟦ X ⊥ Y ⟧\n \n equiv-preserves-≻-l : {X Y : Type i} {A : Type i} → X ≃ Y → X ≻ A → Y ≻ A\n equiv-preserves-≻-l {X} {Y} {A} e ω Z o = equiv-preserves-orth-l e (ω Z o)\n\n ≻-trivial : (A : Type i) → (Lift ⊤) ≻ A\n ≻-trivial A X _ = Unit-orth X\n \n ≻-reflexive : (A : Type i) → A ≻ A\n ≻-reflexive A Y x = x\n\n ≻-trans : (A B C : Type i) → A ≻ B → B ≻ C → A ≻ C\n ≻-trans A B C ω₀ ω₁ Y cy = ω₀ Y (ω₁ Y cy)\n \n ≻-⊤-is-contr : (A : Type i) → A ≻ (Lift ⊤) → is-contr A\n ≻-⊤-is-contr A ω = self-orth-is-contr A (ω A (Unit-orth A))\n \n Σ-≻ : {A X : Type i} {P : X → Type i} → X ≻ A → (P≻A : (x : X) → P x ≻ A) → Σ X P ≻ A\n Σ-≻ X≻A P≻A Y A⊥Y = Σ-orth (X≻A Y A⊥Y) (λ x → P≻A x Y A⊥Y)\n \n -- We jump a universe level, but its certainly convenient ...\n is-hyper-prop : Type i → Type (lsucc i)\n is-hyper-prop A = (X Y : Type i) (f : X → Y) → X ≻ A → Y ≻ A → (y : Y) → hfiber f y ≻ A\n\n hp-kills-paths : (A : Type i) → is-hyper-prop A\n → (X : Type i) → X ≻ A\n → (x y : X) → (x == y) ≻ A\n hp-kills-paths A hp X X≻A x y = equiv-preserves-≻-l\n (equiv snd (λ p → (_ , p)) (λ _ → idp) (λ _ → idp))\n (hp (Lift ⊤) X (λ _ → x) (≻-trivial A) X≻A y)\n\n -- Okay, so in some sense this is much more natural.\n -- It just says the connected guys have to be closed under\n -- diagonals.\n kills-paths-hp : (A : Type i)\n → (κ : (X : Type i) → X ≻ A → (x y : X) → (x == y) ≻ A)\n → is-hyper-prop A\n kills-paths-hp A κ X Y f X≻A Y≻A y = Σ-≻ X≻A (λ x → κ Y Y≻A (f x) y)\n\n -- You'll have to think a bit about why this is equivalent\n -- to *preserving* the path spaces.\n\n ×-≻ : {A B X : Type i} → A ≻ X → B ≻ X → A × B ≻ X\n ×-≻ ω₀ ω₁ Y e = ×-orth (ω₀ Y e) (ω₁ Y e)\n \n postulate\n susp-≻ : (A : Type i) → Susp A ≻ A\n", "meta": {"hexsha": "a35cdca7dd8c80ad19777ee2f0d8a1d934689f73", "size": 9008, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "theorems/stash/modalities/Orthogonality.agda", "max_stars_repo_name": "timjb/HoTT-Agda", "max_stars_repo_head_hexsha": "66f800adef943afdf08c17b8ecfba67340fead5e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 294, "max_stars_repo_stars_event_min_datetime": "2015-01-09T16:23:23.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T13:54:45.000Z", "max_issues_repo_path": "theorems/stash/modalities/Orthogonality.agda", "max_issues_repo_name": "timjb/HoTT-Agda", "max_issues_repo_head_hexsha": "66f800adef943afdf08c17b8ecfba67340fead5e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 31, "max_issues_repo_issues_event_min_datetime": "2015-03-05T20:09:00.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-03T19:15:25.000Z", "max_forks_repo_path": "theorems/stash/modalities/Orthogonality.agda", "max_forks_repo_name": "timjb/HoTT-Agda", "max_forks_repo_head_hexsha": "66f800adef943afdf08c17b8ecfba67340fead5e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 50, "max_forks_repo_forks_event_min_datetime": "2015-01-10T01:48:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-14T03:03:25.000Z", "avg_line_length": 37.8487394958, "max_line_length": 128, "alphanum_fraction": 0.3873223801, "num_tokens": 3954, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.34663580121641563}} {"text": "------------------------------------------------------------------------\n-- Abstract well-formed typing contexts\n------------------------------------------------------------------------\n\n{-# OPTIONS --safe --without-K #-}\n\nmodule Data.Context.WellFormed where\n\nopen import Level using (suc; _⊔_; Lift; lift)\nopen import Data.Fin using (Fin)\nopen import Data.Fin.Substitution.ExtraLemmas\nopen import Data.Nat using (ℕ)\nopen import Data.Unit using (⊤; tt)\nopen import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)\nopen import Data.Vec.Relation.Unary.All.Properties using (gmap)\nopen import Relation.Binary using (REL)\nopen import Relation.Unary using (Pred)\n\nopen import Data.Context\n\n\n------------------------------------------------------------------------\n-- Abstract well-formed typing\n\n-- An abtract well-formedness judgment _⊢_wf : Wf Tp Tm is a binary\n-- relation which, in a given Tp-context, asserts the well-formedness\n-- of Tm-terms.\n\nWf : ∀ {t₁ t₂} → Pred ℕ t₁ → Pred ℕ t₂ → ∀ ℓ → Set (t₁ ⊔ t₂ ⊔ suc ℓ)\nWf Tp Tm ℓ = ∀ {n} → REL (Ctx Tp n) (Tm n) ℓ\n\n\n------------------------------------------------------------------------\n-- Abstract well-formed typing contexts and context extensions.\n--\n-- A well-formed typing context (Γ wf) is a context Γ in which every\n-- participating T-type is well-formed.\n\nmodule ContextFormation {t ℓ} {T : Pred ℕ t} (_⊢_wf : Wf T T ℓ) where\n\n infix 4 _wf _⊢_wfExt\n infixr 5 _∷_\n\n -- Well-formed typing contexts and context extensions.\n\n data _wf : ∀ {n} → Ctx T n → Set (t ⊔ ℓ) where\n [] : [] wf\n _∷_ : ∀ {n t} {Γ : Ctx T n} → Γ ⊢ t wf → Γ wf → t ∷ Γ wf\n\n data _⊢_wfExt {m} (Γ : Ctx T m) : ∀ {n} → CtxExt T m n → Set (t ⊔ ℓ) where\n [] : Γ ⊢ [] wfExt\n _∷_ : ∀ {n t} {Δ : CtxExt T m n} →\n (Δ ++ Γ) ⊢ t wf → Γ ⊢ Δ wfExt → Γ ⊢ t ∷ Δ wfExt\n\n -- Inversions.\n\n wf-∷₁ : ∀ {n} {Γ : Ctx T n} {a} → a ∷ Γ wf → Γ ⊢ a wf\n wf-∷₁ (a-wf ∷ _) = a-wf\n\n wf-∷₂ : ∀ {n} {Γ : Ctx T n} {a} → a ∷ Γ wf → Γ wf\n wf-∷₂ (_ ∷ Γ-wf) = Γ-wf\n\n wfExt-∷₁ : ∀ {m n} {Γ : Ctx T m} {Δ : CtxExt T m n} {a} →\n Γ ⊢ a ∷ Δ wfExt → (Δ ++ Γ) ⊢ a wf\n wfExt-∷₁ (a-wf ∷ _) = a-wf\n\n wfExt-∷₂ : ∀ {m n} {Γ : Ctx T m} {Δ : CtxExt T m n} {a} →\n Γ ⊢ a ∷ Δ wfExt → Γ ⊢ Δ wfExt\n wfExt-∷₂ (_ ∷ Γ-wf) = Γ-wf\n\n -- Operations on well-formed contexts that require weakening of\n -- well-formedness judgments.\n\n record WellFormedWeakenOps (typeExtension : Extension T)\n : Set (suc (t ⊔ ℓ)) where\n\n private module C = WeakenOps typeExtension\n open C hiding (lookup; extLookup)\n\n -- Weakening of well-formedness judgments.\n\n field wf-weaken : ∀ {n} {Γ : Ctx T n} {a b} → Γ ⊢ a wf → Γ ⊢ b wf →\n (a ∷ Γ) ⊢ weaken b wf\n\n -- Convert a well-formed context (extension) to its All representation.\n\n toAll : ∀ {n} {Γ : Ctx T n} → Γ wf → All (λ t → Γ ⊢ t wf) (toVec Γ)\n toAll [] = []\n toAll (t-wf ∷ Γ-wf) =\n wf-weaken t-wf t-wf ∷ gmap (wf-weaken t-wf) (toAll Γ-wf)\n\n extToAll : ∀ {m n} {Γ : Ctx T m} {Δ : CtxExt T m n} →\n All (λ t → Γ ⊢ t wf) (toVec Γ) → Γ ⊢ Δ wfExt →\n All (λ a → (Δ ++ Γ) ⊢ a wf) (toVec (Δ ++ Γ))\n extToAll ts-wf [] = ts-wf\n extToAll ts-wf (t-wf ∷ Δ-wfExt) =\n wf-weaken t-wf t-wf ∷ gmap (wf-weaken t-wf) (extToAll ts-wf Δ-wfExt)\n\n -- Lookup the well-formedness proof of a variable in a context.\n\n lookup : ∀ {n} {Γ : Ctx T n} → Γ wf → (x : Fin n) → Γ ⊢ (C.lookup Γ x) wf\n lookup Γ-wf x = All.lookup x (toAll Γ-wf)\n\n extLookup : ∀ {m n} {Γ : Ctx T m} {Δ : CtxExt T m n} →\n All (λ t → Γ ⊢ t wf) (toVec Γ) → Γ ⊢ Δ wfExt →\n ∀ x → (Δ ++ Γ) ⊢ (C.lookup (Δ ++ Γ) x) wf\n extLookup ts-wf Γ-wf x = All.lookup x (extToAll ts-wf Γ-wf)\n\n\n------------------------------------------------------------------------\n-- Trivial well-formedness.\n--\n-- This module provides a trivial well-formedness relation and the\n-- corresponding trivially well-formed contexts. This is useful when\n-- implmenting typed substitutions on types that either lack or do not\n-- necessitate a notion of well-formedness.\n\nmodule ⊤-WellFormed {ℓ} {T : Pred ℕ ℓ} (typeExtension : Extension T) where\n\n infix 4 _⊢_wf\n\n -- Trivial well-formedness.\n\n _⊢_wf : Wf T T ℓ\n _ ⊢ _ wf = Lift ℓ ⊤\n\n open ContextFormation _⊢_wf public\n\n -- Trivial well-formedness of contexts and context extensions.\n\n ctx-wf : ∀ {n} (Γ : Ctx T n) → Γ wf\n ctx-wf [] = []\n ctx-wf (a ∷ Γ) = lift tt ∷ ctx-wf Γ\n\n ctx-wfExt : ∀ {m n} (Δ : CtxExt T m n) {Γ : Ctx T m} → Γ ⊢ Δ wfExt\n ctx-wfExt [] = []\n ctx-wfExt (a ∷ Δ) = lift tt ∷ ctx-wfExt Δ\n\n module ⊤-WfWeakenOps where\n\n wfWeakenOps : WellFormedWeakenOps typeExtension\n wfWeakenOps = record { wf-weaken = λ _ _ → lift tt }\n\n open WellFormedWeakenOps public\n", "meta": {"hexsha": "532afdd778e18116bfd6732475b97db09bc4d715", "size": 4862, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Data/Context/WellFormed.agda", "max_stars_repo_name": "Blaisorblade/f-omega-int-agda", "max_stars_repo_head_hexsha": "ae20dac2a5e0c18dff2afda4c19954e24d73a24f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/Data/Context/WellFormed.agda", "max_issues_repo_name": "Blaisorblade/f-omega-int-agda", "max_issues_repo_head_hexsha": "ae20dac2a5e0c18dff2afda4c19954e24d73a24f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Data/Context/WellFormed.agda", "max_forks_repo_name": "Blaisorblade/f-omega-int-agda", "max_forks_repo_head_hexsha": "ae20dac2a5e0c18dff2afda4c19954e24d73a24f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.5310344828, "max_line_length": 77, "alphanum_fraction": 0.5263266146, "num_tokens": 1718, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3466059957112453}} {"text": "------------------------------------------------------------------------\n-- Empty type\n------------------------------------------------------------------------\n\nmodule Data.Empty where\n\ndata ⊥ : Set where\n\n⊥-elim : {whatever : Set} → ⊥ → whatever\n⊥-elim ()\n", "meta": {"hexsha": "8022ccfedd51384dd25898efdbc96d4dcb727272", "size": 257, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "vendor/stdlib/src/Data/Empty.agda", "max_stars_repo_name": "isabella232/Lemmachine", "max_stars_repo_head_hexsha": "8ef786b40e4a9ab274c6103dc697dcb658cf3db3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 56, "max_stars_repo_stars_event_min_datetime": "2015-01-20T02:11:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-21T17:02:19.000Z", "max_issues_repo_path": "vendor/stdlib/src/Data/Empty.agda", "max_issues_repo_name": "larrytheliquid/Lemmachine", "max_issues_repo_head_hexsha": "8ef786b40e4a9ab274c6103dc697dcb658cf3db3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2022-03-12T12:17:51.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-12T12:17:51.000Z", "max_forks_repo_path": "vendor/stdlib/src/Data/Empty.agda", "max_forks_repo_name": "isabella232/Lemmachine", "max_forks_repo_head_hexsha": "8ef786b40e4a9ab274c6103dc697dcb658cf3db3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2015-07-21T16:37:58.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-12T11:54:10.000Z", "avg_line_length": 23.3636363636, "max_line_length": 72, "alphanum_fraction": 0.2645914397, "num_tokens": 43, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3462784684960045}} {"text": "{-# OPTIONS --type-in-type #-}\n\nTy : Set\nTy =\n (Ty : Set)\n (nat top bot : Ty)\n (arr prod sum : Ty → Ty → Ty)\n → Ty\n\nnat : Ty; nat = λ _ nat _ _ _ _ _ → nat\ntop : Ty; top = λ _ _ top _ _ _ _ → top\nbot : Ty; bot = λ _ _ _ bot _ _ _ → bot\n\narr : Ty → Ty → Ty; arr\n = λ A B Ty nat top bot arr prod sum →\n arr (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)\n\nprod : Ty → Ty → Ty; prod\n = λ A B Ty nat top bot arr prod sum →\n prod (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)\n\nsum : Ty → Ty → Ty; sum\n = λ A B Ty nat top bot arr prod sum →\n sum (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)\n\nCon : Set; Con\n = (Con : Set)\n (nil : Con)\n (snoc : Con → Ty → Con)\n → Con\n\nnil : Con; nil\n = λ Con nil snoc → nil\n\nsnoc : Con → Ty → Con; snoc\n = λ Γ A Con nil snoc → snoc (Γ Con nil snoc) A\n\nVar : Con → Ty → Set; Var\n = λ Γ A →\n (Var : Con → Ty → Set)\n (vz : ∀ Γ A → Var (snoc Γ A) A)\n (vs : ∀ Γ B A → Var Γ A → Var (snoc Γ B) A)\n → Var Γ A\n\nvz : ∀{Γ A} → Var (snoc Γ A) A; vz\n = λ Var vz vs → vz _ _\n\nvs : ∀{Γ B A} → Var Γ A → Var (snoc Γ B) A; vs\n = λ x Var vz vs → vs _ _ _ (x Var vz vs)\n\nTm : Con → Ty → Set; Tm\n = λ Γ A →\n (Tm : Con → Ty → Set)\n (var : ∀ Γ A → Var Γ A → Tm Γ A)\n (lam : ∀ Γ A B → Tm (snoc Γ A) B → Tm Γ (arr A B))\n (app : ∀ Γ A B → Tm Γ (arr A B) → Tm Γ A → Tm Γ B)\n (tt : ∀ Γ → Tm Γ top)\n (pair : ∀ Γ A B → Tm Γ A → Tm Γ B → Tm Γ (prod A B))\n (fst : ∀ Γ A B → Tm Γ (prod A B) → Tm Γ A)\n (snd : ∀ Γ A B → Tm Γ (prod A B) → Tm Γ B)\n (left : ∀ Γ A B → Tm Γ A → Tm Γ (sum A B))\n (right : ∀ Γ A B → Tm Γ B → Tm Γ (sum A B))\n (case : ∀ Γ A B C → Tm Γ (sum A B) → Tm Γ (arr A C) → Tm Γ (arr B C) → Tm Γ C)\n (zero : ∀ Γ → Tm Γ nat)\n (suc : ∀ Γ → Tm Γ nat → Tm Γ nat)\n (rec : ∀ Γ A → Tm Γ nat → Tm Γ (arr nat (arr A A)) → Tm Γ A → Tm Γ A)\n → Tm Γ A\n\nvar : ∀{Γ A} → Var Γ A → Tm Γ A; var\n = λ x Tm var lam app tt pair fst snd left right case zero suc rec →\n var _ _ x\n\nlam : ∀{Γ A B} → Tm (snoc Γ A) B → Tm Γ (arr A B); lam\n = λ t Tm var lam app tt pair fst snd left right case zero suc rec →\n lam _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)\n\napp : ∀{Γ A B} → Tm Γ (arr A B) → Tm Γ A → Tm Γ B; app\n = λ t u Tm var lam app tt pair fst snd left right case zero suc rec →\n app _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)\n (u Tm var lam app tt pair fst snd left right case zero suc rec)\n\ntt : ∀{Γ} → Tm Γ top; tt\n = λ Tm var lam app tt pair fst snd left right case zero suc rec → tt _\n\npair : ∀{Γ A B} → Tm Γ A → Tm Γ B → Tm Γ (prod A B); pair\n = λ t u Tm var lam app tt pair fst snd left right case zero suc rec →\n pair _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)\n (u Tm var lam app tt pair fst snd left right case zero suc rec)\n\nfst : ∀{Γ A B} → Tm Γ (prod A B) → Tm Γ A; fst\n = λ t Tm var lam app tt pair fst snd left right case zero suc rec →\n fst _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)\n\nsnd : ∀{Γ A B} → Tm Γ (prod A B) → Tm Γ B; snd\n = λ t Tm var lam app tt pair fst snd left right case zero suc rec →\n snd _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)\n\nleft : ∀{Γ A B} → Tm Γ A → Tm Γ (sum A B); left\n = λ t Tm var lam app tt pair fst snd left right case zero suc rec →\n left _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)\n\nright : ∀{Γ A B} → Tm Γ B → Tm Γ (sum A B); right\n = λ t Tm var lam app tt pair fst snd left right case zero suc rec →\n right _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)\n\ncase : ∀{Γ A B C} → Tm Γ (sum A B) → Tm Γ (arr A C) → Tm Γ (arr B C) → Tm Γ C; case\n = λ t u v Tm var lam app tt pair fst snd left right case zero suc rec →\n case _ _ _ _\n (t Tm var lam app tt pair fst snd left right case zero suc rec)\n (u Tm var lam app tt pair fst snd left right case zero suc rec)\n (v Tm var lam app tt pair fst snd left right case zero suc rec)\n\nzero : ∀{Γ} → Tm Γ nat; zero\n = λ Tm var lam app tt pair fst snd left right case zero suc rec → zero _\n\nsuc : ∀{Γ} → Tm Γ nat → Tm Γ nat; suc\n = λ t Tm var lam app tt pair fst snd left right case zero suc rec →\n suc _ (t Tm var lam app tt pair fst snd left right case zero suc rec)\n\nrec : ∀{Γ A} → Tm Γ nat → Tm Γ (arr nat (arr A A)) → Tm Γ A → Tm Γ A; rec\n = λ t u v Tm var lam app tt pair fst snd left right case zero suc rec →\n rec _ _\n (t Tm var lam app tt pair fst snd left right case zero suc rec)\n (u Tm var lam app tt pair fst snd left right case zero suc rec)\n (v Tm var lam app tt pair fst snd left right case zero suc rec)\n\nv0 : ∀{Γ A} → Tm (snoc Γ A) A; v0\n = var vz\n\nv1 : ∀{Γ A B} → Tm (snoc (snoc Γ A) B) A; v1\n = var (vs vz)\n\nv2 : ∀{Γ A B C} → Tm (snoc (snoc (snoc Γ A) B) C) A; v2\n = var (vs (vs vz))\n\nv3 : ∀{Γ A B C D} → Tm (snoc (snoc (snoc (snoc Γ A) B) C) D) A; v3\n = var (vs (vs (vs vz)))\n\ntbool : Ty; tbool\n = sum top top\n\ntrue : ∀{Γ} → Tm Γ tbool; true\n = left tt\n\ntfalse : ∀{Γ} → Tm Γ tbool; tfalse\n = right tt\n\nifthenelse : ∀{Γ A} → Tm Γ (arr tbool (arr A (arr A A))); ifthenelse\n = lam (lam (lam (case v2 (lam v2) (lam v1))))\n\ntimes4 : ∀{Γ A} → Tm Γ (arr (arr A A) (arr A A)); times4\n = lam (lam (app v1 (app v1 (app v1 (app v1 v0)))))\n\nadd : ∀{Γ} → Tm Γ (arr nat (arr nat nat)); add\n = lam (rec v0\n (lam (lam (lam (suc (app v1 v0)))))\n (lam v0))\n\nmul : ∀{Γ} → Tm Γ (arr nat (arr nat nat)); mul\n = lam (rec v0\n (lam (lam (lam (app (app add (app v1 v0)) v0))))\n (lam zero))\n\nfact : ∀{Γ} → Tm Γ (arr nat nat); fact\n = lam (rec v0 (lam (lam (app (app mul (suc v1)) v0)))\n (suc zero))\n{-# OPTIONS --type-in-type #-}\n\nTy1 : Set\nTy1 =\n (Ty1 : Set)\n (nat top bot : Ty1)\n (arr prod sum : Ty1 → Ty1 → Ty1)\n → Ty1\n\nnat1 : Ty1; nat1 = λ _ nat1 _ _ _ _ _ → nat1\ntop1 : Ty1; top1 = λ _ _ top1 _ _ _ _ → top1\nbot1 : Ty1; bot1 = λ _ _ _ bot1 _ _ _ → bot1\n\narr1 : Ty1 → Ty1 → Ty1; arr1\n = λ A B Ty1 nat1 top1 bot1 arr1 prod sum →\n arr1 (A Ty1 nat1 top1 bot1 arr1 prod sum) (B Ty1 nat1 top1 bot1 arr1 prod sum)\n\nprod1 : Ty1 → Ty1 → Ty1; prod1\n = λ A B Ty1 nat1 top1 bot1 arr1 prod1 sum →\n prod1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum) (B Ty1 nat1 top1 bot1 arr1 prod1 sum)\n\nsum1 : Ty1 → Ty1 → Ty1; sum1\n = λ A B Ty1 nat1 top1 bot1 arr1 prod1 sum1 →\n sum1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum1) (B Ty1 nat1 top1 bot1 arr1 prod1 sum1)\n\nCon1 : Set; Con1\n = (Con1 : Set)\n (nil : Con1)\n (snoc : Con1 → Ty1 → Con1)\n → Con1\n\nnil1 : Con1; nil1\n = λ Con1 nil1 snoc → nil1\n\nsnoc1 : Con1 → Ty1 → Con1; snoc1\n = λ Γ A Con1 nil1 snoc1 → snoc1 (Γ Con1 nil1 snoc1) A\n\nVar1 : Con1 → Ty1 → Set; Var1\n = λ Γ A →\n (Var1 : Con1 → Ty1 → Set)\n (vz : ∀ Γ A → Var1 (snoc1 Γ A) A)\n (vs : ∀ Γ B A → Var1 Γ A → Var1 (snoc1 Γ B) A)\n → Var1 Γ A\n\nvz1 : ∀{Γ A} → Var1 (snoc1 Γ A) A; vz1\n = λ Var1 vz1 vs → vz1 _ _\n\nvs1 : ∀{Γ B A} → Var1 Γ A → Var1 (snoc1 Γ B) A; vs1\n = λ x Var1 vz1 vs1 → vs1 _ _ _ (x Var1 vz1 vs1)\n\nTm1 : Con1 → Ty1 → Set; Tm1\n = λ Γ A →\n (Tm1 : Con1 → Ty1 → Set)\n (var : ∀ Γ A → Var1 Γ A → Tm1 Γ A)\n (lam : ∀ Γ A B → Tm1 (snoc1 Γ A) B → Tm1 Γ (arr1 A B))\n (app : ∀ Γ A B → Tm1 Γ (arr1 A B) → Tm1 Γ A → Tm1 Γ B)\n (tt : ∀ Γ → Tm1 Γ top1)\n (pair : ∀ Γ A B → Tm1 Γ A → Tm1 Γ B → Tm1 Γ (prod1 A B))\n (fst : ∀ Γ A B → Tm1 Γ (prod1 A B) → Tm1 Γ A)\n (snd : ∀ Γ A B → Tm1 Γ (prod1 A B) → Tm1 Γ B)\n (left : ∀ Γ A B → Tm1 Γ A → Tm1 Γ (sum1 A B))\n (right : ∀ Γ A B → Tm1 Γ B → Tm1 Γ (sum1 A B))\n (case : ∀ Γ A B C → Tm1 Γ (sum1 A B) → Tm1 Γ (arr1 A C) → Tm1 Γ (arr1 B C) → Tm1 Γ C)\n (zero : ∀ Γ → Tm1 Γ nat1)\n (suc : ∀ Γ → Tm1 Γ nat1 → Tm1 Γ nat1)\n (rec : ∀ Γ A → Tm1 Γ nat1 → Tm1 Γ (arr1 nat1 (arr1 A A)) → Tm1 Γ A → Tm1 Γ A)\n → Tm1 Γ A\n\nvar1 : ∀{Γ A} → Var1 Γ A → Tm1 Γ A; var1\n = λ x Tm1 var1 lam app tt pair fst snd left right case zero suc rec →\n var1 _ _ x\n\nlam1 : ∀{Γ A B} → Tm1 (snoc1 Γ A) B → Tm1 Γ (arr1 A B); lam1\n = λ t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec →\n lam1 _ _ _ (t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec)\n\napp1 : ∀{Γ A B} → Tm1 Γ (arr1 A B) → Tm1 Γ A → Tm1 Γ B; app1\n = λ t u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec →\n app1 _ _ _ (t Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)\n (u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)\n\ntt1 : ∀{Γ} → Tm1 Γ top1; tt1\n = λ Tm1 var1 lam1 app1 tt1 pair fst snd left right case zero suc rec → tt1 _\n\npair1 : ∀{Γ A B} → Tm1 Γ A → Tm1 Γ B → Tm1 Γ (prod1 A B); pair1\n = λ t u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec →\n pair1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)\n (u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)\n\nfst1 : ∀{Γ A B} → Tm1 Γ (prod1 A B) → Tm1 Γ A; fst1\n = λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec →\n fst1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec)\n\nsnd1 : ∀{Γ A B} → Tm1 Γ (prod1 A B) → Tm1 Γ B; snd1\n = λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec →\n snd1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec)\n\nleft1 : ∀{Γ A B} → Tm1 Γ A → Tm1 Γ (sum1 A B); left1\n = λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec →\n left1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec)\n\nright1 : ∀{Γ A B} → Tm1 Γ B → Tm1 Γ (sum1 A B); right1\n = λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec →\n right1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec)\n\ncase1 : ∀{Γ A B C} → Tm1 Γ (sum1 A B) → Tm1 Γ (arr1 A C) → Tm1 Γ (arr1 B C) → Tm1 Γ C; case1\n = λ t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec →\n case1 _ _ _ _\n (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)\n (u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)\n (v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)\n\nzero1 : ∀{Γ} → Tm1 Γ nat1; zero1\n = λ Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc rec → zero1 _\n\nsuc1 : ∀{Γ} → Tm1 Γ nat1 → Tm1 Γ nat1; suc1\n = λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec →\n suc1 _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec)\n\nrec1 : ∀{Γ A} → Tm1 Γ nat1 → Tm1 Γ (arr1 nat1 (arr1 A A)) → Tm1 Γ A → Tm1 Γ A; rec1\n = λ t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1 →\n rec1 _ _\n (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)\n (u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)\n (v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)\n\nv01 : ∀{Γ A} → Tm1 (snoc1 Γ A) A; v01\n = var1 vz1\n\nv11 : ∀{Γ A B} → Tm1 (snoc1 (snoc1 Γ A) B) A; v11\n = var1 (vs1 vz1)\n\nv21 : ∀{Γ A B C} → Tm1 (snoc1 (snoc1 (snoc1 Γ A) B) C) A; v21\n = var1 (vs1 (vs1 vz1))\n\nv31 : ∀{Γ A B C D} → Tm1 (snoc1 (snoc1 (snoc1 (snoc1 Γ A) B) C) D) A; v31\n = var1 (vs1 (vs1 (vs1 vz1)))\n\ntbool1 : Ty1; tbool1\n = sum1 top1 top1\n\ntrue1 : ∀{Γ} → Tm1 Γ tbool1; true1\n = left1 tt1\n\ntfalse1 : ∀{Γ} → Tm1 Γ tbool1; tfalse1\n = right1 tt1\n\nifthenelse1 : ∀{Γ A} → Tm1 Γ (arr1 tbool1 (arr1 A (arr1 A A))); ifthenelse1\n = lam1 (lam1 (lam1 (case1 v21 (lam1 v21) (lam1 v11))))\n\ntimes41 : ∀{Γ A} → Tm1 Γ (arr1 (arr1 A A) (arr1 A A)); times41\n = lam1 (lam1 (app1 v11 (app1 v11 (app1 v11 (app1 v11 v01)))))\n\nadd1 : ∀{Γ} → Tm1 Γ (arr1 nat1 (arr1 nat1 nat1)); add1\n = lam1 (rec1 v01\n (lam1 (lam1 (lam1 (suc1 (app1 v11 v01)))))\n (lam1 v01))\n\nmul1 : ∀{Γ} → Tm1 Γ (arr1 nat1 (arr1 nat1 nat1)); mul1\n = lam1 (rec1 v01\n (lam1 (lam1 (lam1 (app1 (app1 add1 (app1 v11 v01)) v01))))\n (lam1 zero1))\n\nfact1 : ∀{Γ} → Tm1 Γ (arr1 nat1 nat1); fact1\n = lam1 (rec1 v01 (lam1 (lam1 (app1 (app1 mul1 (suc1 v11)) v01)))\n (suc1 zero1))\n{-# OPTIONS --type-in-type #-}\n\nTy2 : Set\nTy2 =\n (Ty2 : Set)\n (nat top bot : Ty2)\n (arr prod sum : Ty2 → Ty2 → Ty2)\n → Ty2\n\nnat2 : Ty2; nat2 = λ _ nat2 _ _ _ _ _ → nat2\ntop2 : Ty2; top2 = λ _ _ top2 _ _ _ _ → top2\nbot2 : Ty2; bot2 = λ _ _ _ bot2 _ _ _ → bot2\n\narr2 : Ty2 → Ty2 → Ty2; arr2\n = λ A B Ty2 nat2 top2 bot2 arr2 prod sum →\n arr2 (A Ty2 nat2 top2 bot2 arr2 prod sum) (B Ty2 nat2 top2 bot2 arr2 prod sum)\n\nprod2 : Ty2 → Ty2 → Ty2; prod2\n = λ A B Ty2 nat2 top2 bot2 arr2 prod2 sum →\n prod2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum) (B Ty2 nat2 top2 bot2 arr2 prod2 sum)\n\nsum2 : Ty2 → Ty2 → Ty2; sum2\n = λ A B Ty2 nat2 top2 bot2 arr2 prod2 sum2 →\n sum2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum2) (B Ty2 nat2 top2 bot2 arr2 prod2 sum2)\n\nCon2 : Set; Con2\n = (Con2 : Set)\n (nil : Con2)\n (snoc : Con2 → Ty2 → Con2)\n → Con2\n\nnil2 : Con2; nil2\n = λ Con2 nil2 snoc → nil2\n\nsnoc2 : Con2 → Ty2 → Con2; snoc2\n = λ Γ A Con2 nil2 snoc2 → snoc2 (Γ Con2 nil2 snoc2) A\n\nVar2 : Con2 → Ty2 → Set; Var2\n = λ Γ A →\n (Var2 : Con2 → Ty2 → Set)\n (vz : ∀ Γ A → Var2 (snoc2 Γ A) A)\n (vs : ∀ Γ B A → Var2 Γ A → Var2 (snoc2 Γ B) A)\n → Var2 Γ A\n\nvz2 : ∀{Γ A} → Var2 (snoc2 Γ A) A; vz2\n = λ Var2 vz2 vs → vz2 _ _\n\nvs2 : ∀{Γ B A} → Var2 Γ A → Var2 (snoc2 Γ B) A; vs2\n = λ x Var2 vz2 vs2 → vs2 _ _ _ (x Var2 vz2 vs2)\n\nTm2 : Con2 → Ty2 → Set; Tm2\n = λ Γ A →\n (Tm2 : Con2 → Ty2 → Set)\n (var : ∀ Γ A → Var2 Γ A → Tm2 Γ A)\n (lam : ∀ Γ A B → Tm2 (snoc2 Γ A) B → Tm2 Γ (arr2 A B))\n (app : ∀ Γ A B → Tm2 Γ (arr2 A B) → Tm2 Γ A → Tm2 Γ B)\n (tt : ∀ Γ → Tm2 Γ top2)\n (pair : ∀ Γ A B → Tm2 Γ A → Tm2 Γ B → Tm2 Γ (prod2 A B))\n (fst : ∀ Γ A B → Tm2 Γ (prod2 A B) → Tm2 Γ A)\n (snd : ∀ Γ A B → Tm2 Γ (prod2 A B) → Tm2 Γ B)\n (left : ∀ Γ A B → Tm2 Γ A → Tm2 Γ (sum2 A B))\n (right : ∀ Γ A B → Tm2 Γ B → Tm2 Γ (sum2 A B))\n (case : ∀ Γ A B C → Tm2 Γ (sum2 A B) → Tm2 Γ (arr2 A C) → Tm2 Γ (arr2 B C) → Tm2 Γ C)\n (zero : ∀ Γ → Tm2 Γ nat2)\n (suc : ∀ Γ → Tm2 Γ nat2 → Tm2 Γ nat2)\n (rec : ∀ Γ A → Tm2 Γ nat2 → Tm2 Γ (arr2 nat2 (arr2 A A)) → Tm2 Γ A → Tm2 Γ A)\n → Tm2 Γ A\n\nvar2 : ∀{Γ A} → Var2 Γ A → Tm2 Γ A; var2\n = λ x Tm2 var2 lam app tt pair fst snd left right case zero suc rec →\n var2 _ _ x\n\nlam2 : ∀{Γ A B} → Tm2 (snoc2 Γ A) B → Tm2 Γ (arr2 A B); lam2\n = λ t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec →\n lam2 _ _ _ (t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec)\n\napp2 : ∀{Γ A B} → Tm2 Γ (arr2 A B) → Tm2 Γ A → Tm2 Γ B; app2\n = λ t u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec →\n app2 _ _ _ (t Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)\n (u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)\n\ntt2 : ∀{Γ} → Tm2 Γ top2; tt2\n = λ Tm2 var2 lam2 app2 tt2 pair fst snd left right case zero suc rec → tt2 _\n\npair2 : ∀{Γ A B} → Tm2 Γ A → Tm2 Γ B → Tm2 Γ (prod2 A B); pair2\n = λ t u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec →\n pair2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)\n (u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)\n\nfst2 : ∀{Γ A B} → Tm2 Γ (prod2 A B) → Tm2 Γ A; fst2\n = λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec →\n fst2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec)\n\nsnd2 : ∀{Γ A B} → Tm2 Γ (prod2 A B) → Tm2 Γ B; snd2\n = λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec →\n snd2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec)\n\nleft2 : ∀{Γ A B} → Tm2 Γ A → Tm2 Γ (sum2 A B); left2\n = λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec →\n left2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec)\n\nright2 : ∀{Γ A B} → Tm2 Γ B → Tm2 Γ (sum2 A B); right2\n = λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec →\n right2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec)\n\ncase2 : ∀{Γ A B C} → Tm2 Γ (sum2 A B) → Tm2 Γ (arr2 A C) → Tm2 Γ (arr2 B C) → Tm2 Γ C; case2\n = λ t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec →\n case2 _ _ _ _\n (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)\n (u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)\n (v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)\n\nzero2 : ∀{Γ} → Tm2 Γ nat2; zero2\n = λ Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc rec → zero2 _\n\nsuc2 : ∀{Γ} → Tm2 Γ nat2 → Tm2 Γ nat2; suc2\n = λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec →\n suc2 _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec)\n\nrec2 : ∀{Γ A} → Tm2 Γ nat2 → Tm2 Γ (arr2 nat2 (arr2 A A)) → Tm2 Γ A → Tm2 Γ A; rec2\n = λ t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2 →\n rec2 _ _\n (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)\n (u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)\n (v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)\n\nv02 : ∀{Γ A} → Tm2 (snoc2 Γ A) A; v02\n = var2 vz2\n\nv12 : ∀{Γ A B} → Tm2 (snoc2 (snoc2 Γ A) B) A; v12\n = var2 (vs2 vz2)\n\nv22 : ∀{Γ A B C} → Tm2 (snoc2 (snoc2 (snoc2 Γ A) B) C) A; v22\n = var2 (vs2 (vs2 vz2))\n\nv32 : ∀{Γ A B C D} → Tm2 (snoc2 (snoc2 (snoc2 (snoc2 Γ A) B) C) D) A; v32\n = var2 (vs2 (vs2 (vs2 vz2)))\n\ntbool2 : Ty2; tbool2\n = sum2 top2 top2\n\ntrue2 : ∀{Γ} → Tm2 Γ tbool2; true2\n = left2 tt2\n\ntfalse2 : ∀{Γ} → Tm2 Γ tbool2; tfalse2\n = right2 tt2\n\nifthenelse2 : ∀{Γ A} → Tm2 Γ (arr2 tbool2 (arr2 A (arr2 A A))); ifthenelse2\n = lam2 (lam2 (lam2 (case2 v22 (lam2 v22) (lam2 v12))))\n\ntimes42 : ∀{Γ A} → Tm2 Γ (arr2 (arr2 A A) (arr2 A A)); times42\n = lam2 (lam2 (app2 v12 (app2 v12 (app2 v12 (app2 v12 v02)))))\n\nadd2 : ∀{Γ} → Tm2 Γ (arr2 nat2 (arr2 nat2 nat2)); add2\n = lam2 (rec2 v02\n (lam2 (lam2 (lam2 (suc2 (app2 v12 v02)))))\n (lam2 v02))\n\nmul2 : ∀{Γ} → Tm2 Γ (arr2 nat2 (arr2 nat2 nat2)); mul2\n = lam2 (rec2 v02\n (lam2 (lam2 (lam2 (app2 (app2 add2 (app2 v12 v02)) v02))))\n (lam2 zero2))\n\nfact2 : ∀{Γ} → Tm2 Γ (arr2 nat2 nat2); fact2\n = lam2 (rec2 v02 (lam2 (lam2 (app2 (app2 mul2 (suc2 v12)) v02)))\n (suc2 zero2))\n{-# OPTIONS --type-in-type #-}\n\nTy3 : Set\nTy3 =\n (Ty3 : Set)\n (nat top bot : Ty3)\n (arr prod sum : Ty3 → Ty3 → Ty3)\n → Ty3\n\nnat3 : Ty3; nat3 = λ _ nat3 _ _ _ _ _ → nat3\ntop3 : Ty3; top3 = λ _ _ top3 _ _ _ _ → top3\nbot3 : Ty3; bot3 = λ _ _ _ bot3 _ _ _ → bot3\n\narr3 : Ty3 → Ty3 → Ty3; arr3\n = λ A B Ty3 nat3 top3 bot3 arr3 prod sum →\n arr3 (A Ty3 nat3 top3 bot3 arr3 prod sum) (B Ty3 nat3 top3 bot3 arr3 prod sum)\n\nprod3 : Ty3 → Ty3 → Ty3; prod3\n = λ A B Ty3 nat3 top3 bot3 arr3 prod3 sum →\n prod3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum) (B Ty3 nat3 top3 bot3 arr3 prod3 sum)\n\nsum3 : Ty3 → Ty3 → Ty3; sum3\n = λ A B Ty3 nat3 top3 bot3 arr3 prod3 sum3 →\n sum3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum3) (B Ty3 nat3 top3 bot3 arr3 prod3 sum3)\n\nCon3 : Set; Con3\n = (Con3 : Set)\n (nil : Con3)\n (snoc : Con3 → Ty3 → Con3)\n → Con3\n\nnil3 : Con3; nil3\n = λ Con3 nil3 snoc → nil3\n\nsnoc3 : Con3 → Ty3 → Con3; snoc3\n = λ Γ A Con3 nil3 snoc3 → snoc3 (Γ Con3 nil3 snoc3) A\n\nVar3 : Con3 → Ty3 → Set; Var3\n = λ Γ A →\n (Var3 : Con3 → Ty3 → Set)\n (vz : ∀ Γ A → Var3 (snoc3 Γ A) A)\n (vs : ∀ Γ B A → Var3 Γ A → Var3 (snoc3 Γ B) A)\n → Var3 Γ A\n\nvz3 : ∀{Γ A} → Var3 (snoc3 Γ A) A; vz3\n = λ Var3 vz3 vs → vz3 _ _\n\nvs3 : ∀{Γ B A} → Var3 Γ A → Var3 (snoc3 Γ B) A; vs3\n = λ x Var3 vz3 vs3 → vs3 _ _ _ (x Var3 vz3 vs3)\n\nTm3 : Con3 → Ty3 → Set; Tm3\n = λ Γ A →\n (Tm3 : Con3 → Ty3 → Set)\n (var : ∀ Γ A → Var3 Γ A → Tm3 Γ A)\n (lam : ∀ Γ A B → Tm3 (snoc3 Γ A) B → Tm3 Γ (arr3 A B))\n (app : ∀ Γ A B → Tm3 Γ (arr3 A B) → Tm3 Γ A → Tm3 Γ B)\n (tt : ∀ Γ → Tm3 Γ top3)\n (pair : ∀ Γ A B → Tm3 Γ A → Tm3 Γ B → Tm3 Γ (prod3 A B))\n (fst : ∀ Γ A B → Tm3 Γ (prod3 A B) → Tm3 Γ A)\n (snd : ∀ Γ A B → Tm3 Γ (prod3 A B) → Tm3 Γ B)\n (left : ∀ Γ A B → Tm3 Γ A → Tm3 Γ (sum3 A B))\n (right : ∀ Γ A B → Tm3 Γ B → Tm3 Γ (sum3 A B))\n (case : ∀ Γ A B C → Tm3 Γ (sum3 A B) → Tm3 Γ (arr3 A C) → Tm3 Γ (arr3 B C) → Tm3 Γ C)\n (zero : ∀ Γ → Tm3 Γ nat3)\n (suc : ∀ Γ → Tm3 Γ nat3 → Tm3 Γ nat3)\n (rec : ∀ Γ A → Tm3 Γ nat3 → Tm3 Γ (arr3 nat3 (arr3 A A)) → Tm3 Γ A → Tm3 Γ A)\n → Tm3 Γ A\n\nvar3 : ∀{Γ A} → Var3 Γ A → Tm3 Γ A; var3\n = λ x Tm3 var3 lam app tt pair fst snd left right case zero suc rec →\n var3 _ _ x\n\nlam3 : ∀{Γ A B} → Tm3 (snoc3 Γ A) B → Tm3 Γ (arr3 A B); lam3\n = λ t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec →\n lam3 _ _ _ (t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec)\n\napp3 : ∀{Γ A B} → Tm3 Γ (arr3 A B) → Tm3 Γ A → Tm3 Γ B; app3\n = λ t u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec →\n app3 _ _ _ (t Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)\n (u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)\n\ntt3 : ∀{Γ} → Tm3 Γ top3; tt3\n = λ Tm3 var3 lam3 app3 tt3 pair fst snd left right case zero suc rec → tt3 _\n\npair3 : ∀{Γ A B} → Tm3 Γ A → Tm3 Γ B → Tm3 Γ (prod3 A B); pair3\n = λ t u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec →\n pair3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)\n (u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)\n\nfst3 : ∀{Γ A B} → Tm3 Γ (prod3 A B) → Tm3 Γ A; fst3\n = λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec →\n fst3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec)\n\nsnd3 : ∀{Γ A B} → Tm3 Γ (prod3 A B) → Tm3 Γ B; snd3\n = λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec →\n snd3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec)\n\nleft3 : ∀{Γ A B} → Tm3 Γ A → Tm3 Γ (sum3 A B); left3\n = λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec →\n left3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec)\n\nright3 : ∀{Γ A B} → Tm3 Γ B → Tm3 Γ (sum3 A B); right3\n = λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec →\n right3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec)\n\ncase3 : ∀{Γ A B C} → Tm3 Γ (sum3 A B) → Tm3 Γ (arr3 A C) → Tm3 Γ (arr3 B C) → Tm3 Γ C; case3\n = λ t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec →\n case3 _ _ _ _\n (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)\n (u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)\n (v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)\n\nzero3 : ∀{Γ} → Tm3 Γ nat3; zero3\n = λ Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc rec → zero3 _\n\nsuc3 : ∀{Γ} → Tm3 Γ nat3 → Tm3 Γ nat3; suc3\n = λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec →\n suc3 _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec)\n\nrec3 : ∀{Γ A} → Tm3 Γ nat3 → Tm3 Γ (arr3 nat3 (arr3 A A)) → Tm3 Γ A → Tm3 Γ A; rec3\n = λ t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3 →\n rec3 _ _\n (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)\n (u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)\n (v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)\n\nv03 : ∀{Γ A} → Tm3 (snoc3 Γ A) A; v03\n = var3 vz3\n\nv13 : ∀{Γ A B} → Tm3 (snoc3 (snoc3 Γ A) B) A; v13\n = var3 (vs3 vz3)\n\nv23 : ∀{Γ A B C} → Tm3 (snoc3 (snoc3 (snoc3 Γ A) B) C) A; v23\n = var3 (vs3 (vs3 vz3))\n\nv33 : ∀{Γ A B C D} → Tm3 (snoc3 (snoc3 (snoc3 (snoc3 Γ A) B) C) D) A; v33\n = var3 (vs3 (vs3 (vs3 vz3)))\n\ntbool3 : Ty3; tbool3\n = sum3 top3 top3\n\ntrue3 : ∀{Γ} → Tm3 Γ tbool3; true3\n = left3 tt3\n\ntfalse3 : ∀{Γ} → Tm3 Γ tbool3; tfalse3\n = right3 tt3\n\nifthenelse3 : ∀{Γ A} → Tm3 Γ (arr3 tbool3 (arr3 A (arr3 A A))); ifthenelse3\n = lam3 (lam3 (lam3 (case3 v23 (lam3 v23) (lam3 v13))))\n\ntimes43 : ∀{Γ A} → Tm3 Γ (arr3 (arr3 A A) (arr3 A A)); times43\n = lam3 (lam3 (app3 v13 (app3 v13 (app3 v13 (app3 v13 v03)))))\n\nadd3 : ∀{Γ} → Tm3 Γ (arr3 nat3 (arr3 nat3 nat3)); add3\n = lam3 (rec3 v03\n (lam3 (lam3 (lam3 (suc3 (app3 v13 v03)))))\n (lam3 v03))\n\nmul3 : ∀{Γ} → Tm3 Γ (arr3 nat3 (arr3 nat3 nat3)); mul3\n = lam3 (rec3 v03\n (lam3 (lam3 (lam3 (app3 (app3 add3 (app3 v13 v03)) v03))))\n (lam3 zero3))\n\nfact3 : ∀{Γ} → Tm3 Γ (arr3 nat3 nat3); fact3\n = lam3 (rec3 v03 (lam3 (lam3 (app3 (app3 mul3 (suc3 v13)) v03)))\n (suc3 zero3))\n{-# OPTIONS --type-in-type #-}\n\nTy4 : Set\nTy4 =\n (Ty4 : Set)\n (nat top bot : Ty4)\n (arr prod sum : Ty4 → Ty4 → Ty4)\n → Ty4\n\nnat4 : Ty4; nat4 = λ _ nat4 _ _ _ _ _ → nat4\ntop4 : Ty4; top4 = λ _ _ top4 _ _ _ _ → top4\nbot4 : Ty4; bot4 = λ _ _ _ bot4 _ _ _ → bot4\n\narr4 : Ty4 → Ty4 → Ty4; arr4\n = λ A B Ty4 nat4 top4 bot4 arr4 prod sum →\n arr4 (A Ty4 nat4 top4 bot4 arr4 prod sum) (B Ty4 nat4 top4 bot4 arr4 prod sum)\n\nprod4 : Ty4 → Ty4 → Ty4; prod4\n = λ A B Ty4 nat4 top4 bot4 arr4 prod4 sum →\n prod4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum) (B Ty4 nat4 top4 bot4 arr4 prod4 sum)\n\nsum4 : Ty4 → Ty4 → Ty4; sum4\n = λ A B Ty4 nat4 top4 bot4 arr4 prod4 sum4 →\n sum4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum4) (B Ty4 nat4 top4 bot4 arr4 prod4 sum4)\n\nCon4 : Set; Con4\n = (Con4 : Set)\n (nil : Con4)\n (snoc : Con4 → Ty4 → Con4)\n → Con4\n\nnil4 : Con4; nil4\n = λ Con4 nil4 snoc → nil4\n\nsnoc4 : Con4 → Ty4 → Con4; snoc4\n = λ Γ A Con4 nil4 snoc4 → snoc4 (Γ Con4 nil4 snoc4) A\n\nVar4 : Con4 → Ty4 → Set; Var4\n = λ Γ A →\n (Var4 : Con4 → Ty4 → Set)\n (vz : ∀ Γ A → Var4 (snoc4 Γ A) A)\n (vs : ∀ Γ B A → Var4 Γ A → Var4 (snoc4 Γ B) A)\n → Var4 Γ A\n\nvz4 : ∀{Γ A} → Var4 (snoc4 Γ A) A; vz4\n = λ Var4 vz4 vs → vz4 _ _\n\nvs4 : ∀{Γ B A} → Var4 Γ A → Var4 (snoc4 Γ B) A; vs4\n = λ x Var4 vz4 vs4 → vs4 _ _ _ (x Var4 vz4 vs4)\n\nTm4 : Con4 → Ty4 → Set; Tm4\n = λ Γ A →\n (Tm4 : Con4 → Ty4 → Set)\n (var : ∀ Γ A → Var4 Γ A → Tm4 Γ A)\n (lam : ∀ Γ A B → Tm4 (snoc4 Γ A) B → Tm4 Γ (arr4 A B))\n (app : ∀ Γ A B → Tm4 Γ (arr4 A B) → Tm4 Γ A → Tm4 Γ B)\n (tt : ∀ Γ → Tm4 Γ top4)\n (pair : ∀ Γ A B → Tm4 Γ A → Tm4 Γ B → Tm4 Γ (prod4 A B))\n (fst : ∀ Γ A B → Tm4 Γ (prod4 A B) → Tm4 Γ A)\n (snd : ∀ Γ A B → Tm4 Γ (prod4 A B) → Tm4 Γ B)\n (left : ∀ Γ A B → Tm4 Γ A → Tm4 Γ (sum4 A B))\n (right : ∀ Γ A B → Tm4 Γ B → Tm4 Γ (sum4 A B))\n (case : ∀ Γ A B C → Tm4 Γ (sum4 A B) → Tm4 Γ (arr4 A C) → Tm4 Γ (arr4 B C) → Tm4 Γ C)\n (zero : ∀ Γ → Tm4 Γ nat4)\n (suc : ∀ Γ → Tm4 Γ nat4 → Tm4 Γ nat4)\n (rec : ∀ Γ A → Tm4 Γ nat4 → Tm4 Γ (arr4 nat4 (arr4 A A)) → Tm4 Γ A → Tm4 Γ A)\n → Tm4 Γ A\n\nvar4 : ∀{Γ A} → Var4 Γ A → Tm4 Γ A; var4\n = λ x Tm4 var4 lam app tt pair fst snd left right case zero suc rec →\n var4 _ _ x\n\nlam4 : ∀{Γ A B} → Tm4 (snoc4 Γ A) B → Tm4 Γ (arr4 A B); lam4\n = λ t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec →\n lam4 _ _ _ (t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec)\n\napp4 : ∀{Γ A B} → Tm4 Γ (arr4 A B) → Tm4 Γ A → Tm4 Γ B; app4\n = λ t u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec →\n app4 _ _ _ (t Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)\n (u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)\n\ntt4 : ∀{Γ} → Tm4 Γ top4; tt4\n = λ Tm4 var4 lam4 app4 tt4 pair fst snd left right case zero suc rec → tt4 _\n\npair4 : ∀{Γ A B} → Tm4 Γ A → Tm4 Γ B → Tm4 Γ (prod4 A B); pair4\n = λ t u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec →\n pair4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)\n (u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)\n\nfst4 : ∀{Γ A B} → Tm4 Γ (prod4 A B) → Tm4 Γ A; fst4\n = λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec →\n fst4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec)\n\nsnd4 : ∀{Γ A B} → Tm4 Γ (prod4 A B) → Tm4 Γ B; snd4\n = λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec →\n snd4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec)\n\nleft4 : ∀{Γ A B} → Tm4 Γ A → Tm4 Γ (sum4 A B); left4\n = λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec →\n left4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec)\n\nright4 : ∀{Γ A B} → Tm4 Γ B → Tm4 Γ (sum4 A B); right4\n = λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec →\n right4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec)\n\ncase4 : ∀{Γ A B C} → Tm4 Γ (sum4 A B) → Tm4 Γ (arr4 A C) → Tm4 Γ (arr4 B C) → Tm4 Γ C; case4\n = λ t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec →\n case4 _ _ _ _\n (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)\n (u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)\n (v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)\n\nzero4 : ∀{Γ} → Tm4 Γ nat4; zero4\n = λ Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc rec → zero4 _\n\nsuc4 : ∀{Γ} → Tm4 Γ nat4 → Tm4 Γ nat4; suc4\n = λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec →\n suc4 _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec)\n\nrec4 : ∀{Γ A} → Tm4 Γ nat4 → Tm4 Γ (arr4 nat4 (arr4 A A)) → Tm4 Γ A → Tm4 Γ A; rec4\n = λ t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4 →\n rec4 _ _\n (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)\n (u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)\n (v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)\n\nv04 : ∀{Γ A} → Tm4 (snoc4 Γ A) A; v04\n = var4 vz4\n\nv14 : ∀{Γ A B} → Tm4 (snoc4 (snoc4 Γ A) B) A; v14\n = var4 (vs4 vz4)\n\nv24 : ∀{Γ A B C} → Tm4 (snoc4 (snoc4 (snoc4 Γ A) B) C) A; v24\n = var4 (vs4 (vs4 vz4))\n\nv34 : ∀{Γ A B C D} → Tm4 (snoc4 (snoc4 (snoc4 (snoc4 Γ A) B) C) D) A; v34\n = var4 (vs4 (vs4 (vs4 vz4)))\n\ntbool4 : Ty4; tbool4\n = sum4 top4 top4\n\ntrue4 : ∀{Γ} → Tm4 Γ tbool4; true4\n = left4 tt4\n\ntfalse4 : ∀{Γ} → Tm4 Γ tbool4; tfalse4\n = right4 tt4\n\nifthenelse4 : ∀{Γ A} → Tm4 Γ (arr4 tbool4 (arr4 A (arr4 A A))); ifthenelse4\n = lam4 (lam4 (lam4 (case4 v24 (lam4 v24) (lam4 v14))))\n\ntimes44 : ∀{Γ A} → Tm4 Γ (arr4 (arr4 A A) (arr4 A A)); times44\n = lam4 (lam4 (app4 v14 (app4 v14 (app4 v14 (app4 v14 v04)))))\n\nadd4 : ∀{Γ} → Tm4 Γ (arr4 nat4 (arr4 nat4 nat4)); add4\n = lam4 (rec4 v04\n (lam4 (lam4 (lam4 (suc4 (app4 v14 v04)))))\n (lam4 v04))\n\nmul4 : ∀{Γ} → Tm4 Γ (arr4 nat4 (arr4 nat4 nat4)); mul4\n = lam4 (rec4 v04\n (lam4 (lam4 (lam4 (app4 (app4 add4 (app4 v14 v04)) v04))))\n (lam4 zero4))\n\nfact4 : ∀{Γ} → Tm4 Γ (arr4 nat4 nat4); fact4\n = lam4 (rec4 v04 (lam4 (lam4 (app4 (app4 mul4 (suc4 v14)) v04)))\n (suc4 zero4))\n{-# OPTIONS --type-in-type #-}\n\nTy5 : Set\nTy5 =\n (Ty5 : Set)\n (nat top bot : Ty5)\n (arr prod sum : Ty5 → Ty5 → Ty5)\n → Ty5\n\nnat5 : Ty5; nat5 = λ _ nat5 _ _ _ _ _ → nat5\ntop5 : Ty5; top5 = λ _ _ top5 _ _ _ _ → top5\nbot5 : Ty5; bot5 = λ _ _ _ bot5 _ _ _ → bot5\n\narr5 : Ty5 → Ty5 → Ty5; arr5\n = λ A B Ty5 nat5 top5 bot5 arr5 prod sum →\n arr5 (A Ty5 nat5 top5 bot5 arr5 prod sum) (B Ty5 nat5 top5 bot5 arr5 prod sum)\n\nprod5 : Ty5 → Ty5 → Ty5; prod5\n = λ A B Ty5 nat5 top5 bot5 arr5 prod5 sum →\n prod5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum) (B Ty5 nat5 top5 bot5 arr5 prod5 sum)\n\nsum5 : Ty5 → Ty5 → Ty5; sum5\n = λ A B Ty5 nat5 top5 bot5 arr5 prod5 sum5 →\n sum5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum5) (B Ty5 nat5 top5 bot5 arr5 prod5 sum5)\n\nCon5 : Set; Con5\n = (Con5 : Set)\n (nil : Con5)\n (snoc : Con5 → Ty5 → Con5)\n → Con5\n\nnil5 : Con5; nil5\n = λ Con5 nil5 snoc → nil5\n\nsnoc5 : Con5 → Ty5 → Con5; snoc5\n = λ Γ A Con5 nil5 snoc5 → snoc5 (Γ Con5 nil5 snoc5) A\n\nVar5 : Con5 → Ty5 → Set; Var5\n = λ Γ A →\n (Var5 : Con5 → Ty5 → Set)\n (vz : ∀ Γ A → Var5 (snoc5 Γ A) A)\n (vs : ∀ Γ B A → Var5 Γ A → Var5 (snoc5 Γ B) A)\n → Var5 Γ A\n\nvz5 : ∀{Γ A} → Var5 (snoc5 Γ A) A; vz5\n = λ Var5 vz5 vs → vz5 _ _\n\nvs5 : ∀{Γ B A} → Var5 Γ A → Var5 (snoc5 Γ B) A; vs5\n = λ x Var5 vz5 vs5 → vs5 _ _ _ (x Var5 vz5 vs5)\n\nTm5 : Con5 → Ty5 → Set; Tm5\n = λ Γ A →\n (Tm5 : Con5 → Ty5 → Set)\n (var : ∀ Γ A → Var5 Γ A → Tm5 Γ A)\n (lam : ∀ Γ A B → Tm5 (snoc5 Γ A) B → Tm5 Γ (arr5 A B))\n (app : ∀ Γ A B → Tm5 Γ (arr5 A B) → Tm5 Γ A → Tm5 Γ B)\n (tt : ∀ Γ → Tm5 Γ top5)\n (pair : ∀ Γ A B → Tm5 Γ A → Tm5 Γ B → Tm5 Γ (prod5 A B))\n (fst : ∀ Γ A B → Tm5 Γ (prod5 A B) → Tm5 Γ A)\n (snd : ∀ Γ A B → Tm5 Γ (prod5 A B) → Tm5 Γ B)\n (left : ∀ Γ A B → Tm5 Γ A → Tm5 Γ (sum5 A B))\n (right : ∀ Γ A B → Tm5 Γ B → Tm5 Γ (sum5 A B))\n (case : ∀ Γ A B C → Tm5 Γ (sum5 A B) → Tm5 Γ (arr5 A C) → Tm5 Γ (arr5 B C) → Tm5 Γ C)\n (zero : ∀ Γ → Tm5 Γ nat5)\n (suc : ∀ Γ → Tm5 Γ nat5 → Tm5 Γ nat5)\n (rec : ∀ Γ A → Tm5 Γ nat5 → Tm5 Γ (arr5 nat5 (arr5 A A)) → Tm5 Γ A → Tm5 Γ A)\n → Tm5 Γ A\n\nvar5 : ∀{Γ A} → Var5 Γ A → Tm5 Γ A; var5\n = λ x Tm5 var5 lam app tt pair fst snd left right case zero suc rec →\n var5 _ _ x\n\nlam5 : ∀{Γ A B} → Tm5 (snoc5 Γ A) B → Tm5 Γ (arr5 A B); lam5\n = λ t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec →\n lam5 _ _ _ (t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec)\n\napp5 : ∀{Γ A B} → Tm5 Γ (arr5 A B) → Tm5 Γ A → Tm5 Γ B; app5\n = λ t u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec →\n app5 _ _ _ (t Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)\n (u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)\n\ntt5 : ∀{Γ} → Tm5 Γ top5; tt5\n = λ Tm5 var5 lam5 app5 tt5 pair fst snd left right case zero suc rec → tt5 _\n\npair5 : ∀{Γ A B} → Tm5 Γ A → Tm5 Γ B → Tm5 Γ (prod5 A B); pair5\n = λ t u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec →\n pair5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)\n (u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)\n\nfst5 : ∀{Γ A B} → Tm5 Γ (prod5 A B) → Tm5 Γ A; fst5\n = λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec →\n fst5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec)\n\nsnd5 : ∀{Γ A B} → Tm5 Γ (prod5 A B) → Tm5 Γ B; snd5\n = λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec →\n snd5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec)\n\nleft5 : ∀{Γ A B} → Tm5 Γ A → Tm5 Γ (sum5 A B); left5\n = λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec →\n left5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec)\n\nright5 : ∀{Γ A B} → Tm5 Γ B → Tm5 Γ (sum5 A B); right5\n = λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec →\n right5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec)\n\ncase5 : ∀{Γ A B C} → Tm5 Γ (sum5 A B) → Tm5 Γ (arr5 A C) → Tm5 Γ (arr5 B C) → Tm5 Γ C; case5\n = λ t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec →\n case5 _ _ _ _\n (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)\n (u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)\n (v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)\n\nzero5 : ∀{Γ} → Tm5 Γ nat5; zero5\n = λ Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc rec → zero5 _\n\nsuc5 : ∀{Γ} → Tm5 Γ nat5 → Tm5 Γ nat5; suc5\n = λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec →\n suc5 _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec)\n\nrec5 : ∀{Γ A} → Tm5 Γ nat5 → Tm5 Γ (arr5 nat5 (arr5 A A)) → Tm5 Γ A → Tm5 Γ A; rec5\n = λ t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5 →\n rec5 _ _\n (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)\n (u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)\n (v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)\n\nv05 : ∀{Γ A} → Tm5 (snoc5 Γ A) A; v05\n = var5 vz5\n\nv15 : ∀{Γ A B} → Tm5 (snoc5 (snoc5 Γ A) B) A; v15\n = var5 (vs5 vz5)\n\nv25 : ∀{Γ A B C} → Tm5 (snoc5 (snoc5 (snoc5 Γ A) B) C) A; v25\n = var5 (vs5 (vs5 vz5))\n\nv35 : ∀{Γ A B C D} → Tm5 (snoc5 (snoc5 (snoc5 (snoc5 Γ A) B) C) D) A; v35\n = var5 (vs5 (vs5 (vs5 vz5)))\n\ntbool5 : Ty5; tbool5\n = sum5 top5 top5\n\ntrue5 : ∀{Γ} → Tm5 Γ tbool5; true5\n = left5 tt5\n\ntfalse5 : ∀{Γ} → Tm5 Γ tbool5; tfalse5\n = right5 tt5\n\nifthenelse5 : ∀{Γ A} → Tm5 Γ (arr5 tbool5 (arr5 A (arr5 A A))); ifthenelse5\n = lam5 (lam5 (lam5 (case5 v25 (lam5 v25) (lam5 v15))))\n\ntimes45 : ∀{Γ A} → Tm5 Γ (arr5 (arr5 A A) (arr5 A A)); times45\n = lam5 (lam5 (app5 v15 (app5 v15 (app5 v15 (app5 v15 v05)))))\n\nadd5 : ∀{Γ} → Tm5 Γ (arr5 nat5 (arr5 nat5 nat5)); add5\n = lam5 (rec5 v05\n (lam5 (lam5 (lam5 (suc5 (app5 v15 v05)))))\n (lam5 v05))\n\nmul5 : ∀{Γ} → Tm5 Γ (arr5 nat5 (arr5 nat5 nat5)); mul5\n = lam5 (rec5 v05\n (lam5 (lam5 (lam5 (app5 (app5 add5 (app5 v15 v05)) v05))))\n (lam5 zero5))\n\nfact5 : ∀{Γ} → Tm5 Γ (arr5 nat5 nat5); fact5\n = lam5 (rec5 v05 (lam5 (lam5 (app5 (app5 mul5 (suc5 v15)) v05)))\n (suc5 zero5))\n{-# OPTIONS --type-in-type #-}\n\nTy6 : Set\nTy6 =\n (Ty6 : Set)\n (nat top bot : Ty6)\n (arr prod sum : Ty6 → Ty6 → Ty6)\n → Ty6\n\nnat6 : Ty6; nat6 = λ _ nat6 _ _ _ _ _ → nat6\ntop6 : Ty6; top6 = λ _ _ top6 _ _ _ _ → top6\nbot6 : Ty6; bot6 = λ _ _ _ bot6 _ _ _ → bot6\n\narr6 : Ty6 → Ty6 → Ty6; arr6\n = λ A B Ty6 nat6 top6 bot6 arr6 prod sum →\n arr6 (A Ty6 nat6 top6 bot6 arr6 prod sum) (B Ty6 nat6 top6 bot6 arr6 prod sum)\n\nprod6 : Ty6 → Ty6 → Ty6; prod6\n = λ A B Ty6 nat6 top6 bot6 arr6 prod6 sum →\n prod6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum) (B Ty6 nat6 top6 bot6 arr6 prod6 sum)\n\nsum6 : Ty6 → Ty6 → Ty6; sum6\n = λ A B Ty6 nat6 top6 bot6 arr6 prod6 sum6 →\n sum6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum6) (B Ty6 nat6 top6 bot6 arr6 prod6 sum6)\n\nCon6 : Set; Con6\n = (Con6 : Set)\n (nil : Con6)\n (snoc : Con6 → Ty6 → Con6)\n → Con6\n\nnil6 : Con6; nil6\n = λ Con6 nil6 snoc → nil6\n\nsnoc6 : Con6 → Ty6 → Con6; snoc6\n = λ Γ A Con6 nil6 snoc6 → snoc6 (Γ Con6 nil6 snoc6) A\n\nVar6 : Con6 → Ty6 → Set; Var6\n = λ Γ A →\n (Var6 : Con6 → Ty6 → Set)\n (vz : ∀ Γ A → Var6 (snoc6 Γ A) A)\n (vs : ∀ Γ B A → Var6 Γ A → Var6 (snoc6 Γ B) A)\n → Var6 Γ A\n\nvz6 : ∀{Γ A} → Var6 (snoc6 Γ A) A; vz6\n = λ Var6 vz6 vs → vz6 _ _\n\nvs6 : ∀{Γ B A} → Var6 Γ A → Var6 (snoc6 Γ B) A; vs6\n = λ x Var6 vz6 vs6 → vs6 _ _ _ (x Var6 vz6 vs6)\n\nTm6 : Con6 → Ty6 → Set; Tm6\n = λ Γ A →\n (Tm6 : Con6 → Ty6 → Set)\n (var : ∀ Γ A → Var6 Γ A → Tm6 Γ A)\n (lam : ∀ Γ A B → Tm6 (snoc6 Γ A) B → Tm6 Γ (arr6 A B))\n (app : ∀ Γ A B → Tm6 Γ (arr6 A B) → Tm6 Γ A → Tm6 Γ B)\n (tt : ∀ Γ → Tm6 Γ top6)\n (pair : ∀ Γ A B → Tm6 Γ A → Tm6 Γ B → Tm6 Γ (prod6 A B))\n (fst : ∀ Γ A B → Tm6 Γ (prod6 A B) → Tm6 Γ A)\n (snd : ∀ Γ A B → Tm6 Γ (prod6 A B) → Tm6 Γ B)\n (left : ∀ Γ A B → Tm6 Γ A → Tm6 Γ (sum6 A B))\n (right : ∀ Γ A B → Tm6 Γ B → Tm6 Γ (sum6 A B))\n (case : ∀ Γ A B C → Tm6 Γ (sum6 A B) → Tm6 Γ (arr6 A C) → Tm6 Γ (arr6 B C) → Tm6 Γ C)\n (zero : ∀ Γ → Tm6 Γ nat6)\n (suc : ∀ Γ → Tm6 Γ nat6 → Tm6 Γ nat6)\n (rec : ∀ Γ A → Tm6 Γ nat6 → Tm6 Γ (arr6 nat6 (arr6 A A)) → Tm6 Γ A → Tm6 Γ A)\n → Tm6 Γ A\n\nvar6 : ∀{Γ A} → Var6 Γ A → Tm6 Γ A; var6\n = λ x Tm6 var6 lam app tt pair fst snd left right case zero suc rec →\n var6 _ _ x\n\nlam6 : ∀{Γ A B} → Tm6 (snoc6 Γ A) B → Tm6 Γ (arr6 A B); lam6\n = λ t Tm6 var6 lam6 app tt pair fst snd left right case zero suc rec →\n lam6 _ _ _ (t Tm6 var6 lam6 app tt pair fst snd left right case zero suc rec)\n\napp6 : ∀{Γ A B} → Tm6 Γ (arr6 A B) → Tm6 Γ A → Tm6 Γ B; app6\n = λ t u Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec →\n app6 _ _ _ (t Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec)\n (u Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec)\n\ntt6 : ∀{Γ} → Tm6 Γ top6; tt6\n = λ Tm6 var6 lam6 app6 tt6 pair fst snd left right case zero suc rec → tt6 _\n\npair6 : ∀{Γ A B} → Tm6 Γ A → Tm6 Γ B → Tm6 Γ (prod6 A B); pair6\n = λ t u Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec →\n pair6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec)\n (u Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec)\n\nfst6 : ∀{Γ A B} → Tm6 Γ (prod6 A B) → Tm6 Γ A; fst6\n = λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd left right case zero suc rec →\n fst6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd left right case zero suc rec)\n\nsnd6 : ∀{Γ A B} → Tm6 Γ (prod6 A B) → Tm6 Γ B; snd6\n = λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left right case zero suc rec →\n snd6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left right case zero suc rec)\n\nleft6 : ∀{Γ A B} → Tm6 Γ A → Tm6 Γ (sum6 A B); left6\n = λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right case zero suc rec →\n left6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right case zero suc rec)\n\nright6 : ∀{Γ A B} → Tm6 Γ B → Tm6 Γ (sum6 A B); right6\n = λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case zero suc rec →\n right6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case zero suc rec)\n\ncase6 : ∀{Γ A B C} → Tm6 Γ (sum6 A B) → Tm6 Γ (arr6 A C) → Tm6 Γ (arr6 B C) → Tm6 Γ C; case6\n = λ t u v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec →\n case6 _ _ _ _\n (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)\n (u Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)\n (v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)\n\nzero6 : ∀{Γ} → Tm6 Γ nat6; zero6\n = λ Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc rec → zero6 _\n\nsuc6 : ∀{Γ} → Tm6 Γ nat6 → Tm6 Γ nat6; suc6\n = λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec →\n suc6 _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec)\n\nrec6 : ∀{Γ A} → Tm6 Γ nat6 → Tm6 Γ (arr6 nat6 (arr6 A A)) → Tm6 Γ A → Tm6 Γ A; rec6\n = λ t u v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6 →\n rec6 _ _\n (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)\n (u Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)\n (v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)\n\nv06 : ∀{Γ A} → Tm6 (snoc6 Γ A) A; v06\n = var6 vz6\n\nv16 : ∀{Γ A B} → Tm6 (snoc6 (snoc6 Γ A) B) A; v16\n = var6 (vs6 vz6)\n\nv26 : ∀{Γ A B C} → Tm6 (snoc6 (snoc6 (snoc6 Γ A) B) C) A; v26\n = var6 (vs6 (vs6 vz6))\n\nv36 : ∀{Γ A B C D} → Tm6 (snoc6 (snoc6 (snoc6 (snoc6 Γ A) B) C) D) A; v36\n = var6 (vs6 (vs6 (vs6 vz6)))\n\ntbool6 : Ty6; tbool6\n = sum6 top6 top6\n\ntrue6 : ∀{Γ} → Tm6 Γ tbool6; true6\n = left6 tt6\n\ntfalse6 : ∀{Γ} → Tm6 Γ tbool6; tfalse6\n = right6 tt6\n\nifthenelse6 : ∀{Γ A} → Tm6 Γ (arr6 tbool6 (arr6 A (arr6 A A))); ifthenelse6\n = lam6 (lam6 (lam6 (case6 v26 (lam6 v26) (lam6 v16))))\n\ntimes46 : ∀{Γ A} → Tm6 Γ (arr6 (arr6 A A) (arr6 A A)); times46\n = lam6 (lam6 (app6 v16 (app6 v16 (app6 v16 (app6 v16 v06)))))\n\nadd6 : ∀{Γ} → Tm6 Γ (arr6 nat6 (arr6 nat6 nat6)); add6\n = lam6 (rec6 v06\n (lam6 (lam6 (lam6 (suc6 (app6 v16 v06)))))\n (lam6 v06))\n\nmul6 : ∀{Γ} → Tm6 Γ (arr6 nat6 (arr6 nat6 nat6)); mul6\n = lam6 (rec6 v06\n (lam6 (lam6 (lam6 (app6 (app6 add6 (app6 v16 v06)) v06))))\n (lam6 zero6))\n\nfact6 : ∀{Γ} → Tm6 Γ (arr6 nat6 nat6); fact6\n = lam6 (rec6 v06 (lam6 (lam6 (app6 (app6 mul6 (suc6 v16)) v06)))\n (suc6 zero6))\n{-# OPTIONS --type-in-type #-}\n\nTy7 : Set\nTy7 =\n (Ty7 : Set)\n (nat top bot : Ty7)\n (arr prod sum : Ty7 → Ty7 → Ty7)\n → Ty7\n\nnat7 : Ty7; nat7 = λ _ nat7 _ _ _ _ _ → nat7\ntop7 : Ty7; top7 = λ _ _ top7 _ _ _ _ → top7\nbot7 : Ty7; bot7 = λ _ _ _ bot7 _ _ _ → bot7\n\narr7 : Ty7 → Ty7 → Ty7; arr7\n = λ A B Ty7 nat7 top7 bot7 arr7 prod sum →\n arr7 (A Ty7 nat7 top7 bot7 arr7 prod sum) (B Ty7 nat7 top7 bot7 arr7 prod sum)\n\nprod7 : Ty7 → Ty7 → Ty7; prod7\n = λ A B Ty7 nat7 top7 bot7 arr7 prod7 sum →\n prod7 (A Ty7 nat7 top7 bot7 arr7 prod7 sum) (B Ty7 nat7 top7 bot7 arr7 prod7 sum)\n\nsum7 : Ty7 → Ty7 → Ty7; sum7\n = λ A B Ty7 nat7 top7 bot7 arr7 prod7 sum7 →\n sum7 (A Ty7 nat7 top7 bot7 arr7 prod7 sum7) (B Ty7 nat7 top7 bot7 arr7 prod7 sum7)\n\nCon7 : Set; Con7\n = (Con7 : Set)\n (nil : Con7)\n (snoc : Con7 → Ty7 → Con7)\n → Con7\n\nnil7 : Con7; nil7\n = λ Con7 nil7 snoc → nil7\n\nsnoc7 : Con7 → Ty7 → Con7; snoc7\n = λ Γ A Con7 nil7 snoc7 → snoc7 (Γ Con7 nil7 snoc7) A\n\nVar7 : Con7 → Ty7 → Set; Var7\n = λ Γ A →\n (Var7 : Con7 → Ty7 → Set)\n (vz : ∀ Γ A → Var7 (snoc7 Γ A) A)\n (vs : ∀ Γ B A → Var7 Γ A → Var7 (snoc7 Γ B) A)\n → Var7 Γ A\n\nvz7 : ∀{Γ A} → Var7 (snoc7 Γ A) A; vz7\n = λ Var7 vz7 vs → vz7 _ _\n\nvs7 : ∀{Γ B A} → Var7 Γ A → Var7 (snoc7 Γ B) A; vs7\n = λ x Var7 vz7 vs7 → vs7 _ _ _ (x Var7 vz7 vs7)\n\nTm7 : Con7 → Ty7 → Set; Tm7\n = λ Γ A →\n (Tm7 : Con7 → Ty7 → Set)\n (var : ∀ Γ A → Var7 Γ A → Tm7 Γ A)\n (lam : ∀ Γ A B → Tm7 (snoc7 Γ A) B → Tm7 Γ (arr7 A B))\n (app : ∀ Γ A B → Tm7 Γ (arr7 A B) → Tm7 Γ A → Tm7 Γ B)\n (tt : ∀ Γ → Tm7 Γ top7)\n (pair : ∀ Γ A B → Tm7 Γ A → Tm7 Γ B → Tm7 Γ (prod7 A B))\n (fst : ∀ Γ A B → Tm7 Γ (prod7 A B) → Tm7 Γ A)\n (snd : ∀ Γ A B → Tm7 Γ (prod7 A B) → Tm7 Γ B)\n (left : ∀ Γ A B → Tm7 Γ A → Tm7 Γ (sum7 A B))\n (right : ∀ Γ A B → Tm7 Γ B → Tm7 Γ (sum7 A B))\n (case : ∀ Γ A B C → Tm7 Γ (sum7 A B) → Tm7 Γ (arr7 A C) → Tm7 Γ (arr7 B C) → Tm7 Γ C)\n (zero : ∀ Γ → Tm7 Γ nat7)\n (suc : ∀ Γ → Tm7 Γ nat7 → Tm7 Γ nat7)\n (rec : ∀ Γ A → Tm7 Γ nat7 → Tm7 Γ (arr7 nat7 (arr7 A A)) → Tm7 Γ A → Tm7 Γ A)\n → Tm7 Γ A\n\nvar7 : ∀{Γ A} → Var7 Γ A → Tm7 Γ A; var7\n = λ x Tm7 var7 lam app tt pair fst snd left right case zero suc rec →\n var7 _ _ x\n\nlam7 : ∀{Γ A B} → Tm7 (snoc7 Γ A) B → Tm7 Γ (arr7 A B); lam7\n = λ t Tm7 var7 lam7 app tt pair fst snd left right case zero suc rec →\n lam7 _ _ _ (t Tm7 var7 lam7 app tt pair fst snd left right case zero suc rec)\n\napp7 : ∀{Γ A B} → Tm7 Γ (arr7 A B) → Tm7 Γ A → Tm7 Γ B; app7\n = λ t u Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec →\n app7 _ _ _ (t Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec)\n (u Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec)\n\ntt7 : ∀{Γ} → Tm7 Γ top7; tt7\n = λ Tm7 var7 lam7 app7 tt7 pair fst snd left right case zero suc rec → tt7 _\n\npair7 : ∀{Γ A B} → Tm7 Γ A → Tm7 Γ B → Tm7 Γ (prod7 A B); pair7\n = λ t u Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec →\n pair7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec)\n (u Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec)\n\nfst7 : ∀{Γ A B} → Tm7 Γ (prod7 A B) → Tm7 Γ A; fst7\n = λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd left right case zero suc rec →\n fst7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd left right case zero suc rec)\n\nsnd7 : ∀{Γ A B} → Tm7 Γ (prod7 A B) → Tm7 Γ B; snd7\n = λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left right case zero suc rec →\n snd7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left right case zero suc rec)\n\nleft7 : ∀{Γ A B} → Tm7 Γ A → Tm7 Γ (sum7 A B); left7\n = λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right case zero suc rec →\n left7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right case zero suc rec)\n\nright7 : ∀{Γ A B} → Tm7 Γ B → Tm7 Γ (sum7 A B); right7\n = λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case zero suc rec →\n right7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case zero suc rec)\n\ncase7 : ∀{Γ A B C} → Tm7 Γ (sum7 A B) → Tm7 Γ (arr7 A C) → Tm7 Γ (arr7 B C) → Tm7 Γ C; case7\n = λ t u v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec →\n case7 _ _ _ _\n (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)\n (u Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)\n (v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)\n\nzero7 : ∀{Γ} → Tm7 Γ nat7; zero7\n = λ Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc rec → zero7 _\n\nsuc7 : ∀{Γ} → Tm7 Γ nat7 → Tm7 Γ nat7; suc7\n = λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec →\n suc7 _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec)\n\nrec7 : ∀{Γ A} → Tm7 Γ nat7 → Tm7 Γ (arr7 nat7 (arr7 A A)) → Tm7 Γ A → Tm7 Γ A; rec7\n = λ t u v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7 →\n rec7 _ _\n (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)\n (u Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)\n (v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)\n\nv07 : ∀{Γ A} → Tm7 (snoc7 Γ A) A; v07\n = var7 vz7\n\nv17 : ∀{Γ A B} → Tm7 (snoc7 (snoc7 Γ A) B) A; v17\n = var7 (vs7 vz7)\n\nv27 : ∀{Γ A B C} → Tm7 (snoc7 (snoc7 (snoc7 Γ A) B) C) A; v27\n = var7 (vs7 (vs7 vz7))\n\nv37 : ∀{Γ A B C D} → Tm7 (snoc7 (snoc7 (snoc7 (snoc7 Γ A) B) C) D) A; v37\n = var7 (vs7 (vs7 (vs7 vz7)))\n\ntbool7 : Ty7; tbool7\n = sum7 top7 top7\n\ntrue7 : ∀{Γ} → Tm7 Γ tbool7; true7\n = left7 tt7\n\ntfalse7 : ∀{Γ} → Tm7 Γ tbool7; tfalse7\n = right7 tt7\n\nifthenelse7 : ∀{Γ A} → Tm7 Γ (arr7 tbool7 (arr7 A (arr7 A A))); ifthenelse7\n = lam7 (lam7 (lam7 (case7 v27 (lam7 v27) (lam7 v17))))\n\ntimes47 : ∀{Γ A} → Tm7 Γ (arr7 (arr7 A A) (arr7 A A)); times47\n = lam7 (lam7 (app7 v17 (app7 v17 (app7 v17 (app7 v17 v07)))))\n\nadd7 : ∀{Γ} → Tm7 Γ (arr7 nat7 (arr7 nat7 nat7)); add7\n = lam7 (rec7 v07\n (lam7 (lam7 (lam7 (suc7 (app7 v17 v07)))))\n (lam7 v07))\n\nmul7 : ∀{Γ} → Tm7 Γ (arr7 nat7 (arr7 nat7 nat7)); mul7\n = lam7 (rec7 v07\n (lam7 (lam7 (lam7 (app7 (app7 add7 (app7 v17 v07)) v07))))\n (lam7 zero7))\n\nfact7 : ∀{Γ} → Tm7 Γ (arr7 nat7 nat7); fact7\n = lam7 (rec7 v07 (lam7 (lam7 (app7 (app7 mul7 (suc7 v17)) v07)))\n (suc7 zero7))\n{-# OPTIONS --type-in-type #-}\n\nTy8 : Set\nTy8 =\n (Ty8 : Set)\n (nat top bot : Ty8)\n (arr prod sum : Ty8 → Ty8 → Ty8)\n → Ty8\n\nnat8 : Ty8; nat8 = λ _ nat8 _ _ _ _ _ → nat8\ntop8 : Ty8; top8 = λ _ _ top8 _ _ _ _ → top8\nbot8 : Ty8; bot8 = λ _ _ _ bot8 _ _ _ → bot8\n\narr8 : Ty8 → Ty8 → Ty8; arr8\n = λ A B Ty8 nat8 top8 bot8 arr8 prod sum →\n arr8 (A Ty8 nat8 top8 bot8 arr8 prod sum) (B Ty8 nat8 top8 bot8 arr8 prod sum)\n\nprod8 : Ty8 → Ty8 → Ty8; prod8\n = λ A B Ty8 nat8 top8 bot8 arr8 prod8 sum →\n prod8 (A Ty8 nat8 top8 bot8 arr8 prod8 sum) (B Ty8 nat8 top8 bot8 arr8 prod8 sum)\n\nsum8 : Ty8 → Ty8 → Ty8; sum8\n = λ A B Ty8 nat8 top8 bot8 arr8 prod8 sum8 →\n sum8 (A Ty8 nat8 top8 bot8 arr8 prod8 sum8) (B Ty8 nat8 top8 bot8 arr8 prod8 sum8)\n\nCon8 : Set; Con8\n = (Con8 : Set)\n (nil : Con8)\n (snoc : Con8 → Ty8 → Con8)\n → Con8\n\nnil8 : Con8; nil8\n = λ Con8 nil8 snoc → nil8\n\nsnoc8 : Con8 → Ty8 → Con8; snoc8\n = λ Γ A Con8 nil8 snoc8 → snoc8 (Γ Con8 nil8 snoc8) A\n\nVar8 : Con8 → Ty8 → Set; Var8\n = λ Γ A →\n (Var8 : Con8 → Ty8 → Set)\n (vz : ∀ Γ A → Var8 (snoc8 Γ A) A)\n (vs : ∀ Γ B A → Var8 Γ A → Var8 (snoc8 Γ B) A)\n → Var8 Γ A\n\nvz8 : ∀{Γ A} → Var8 (snoc8 Γ A) A; vz8\n = λ Var8 vz8 vs → vz8 _ _\n\nvs8 : ∀{Γ B A} → Var8 Γ A → Var8 (snoc8 Γ B) A; vs8\n = λ x Var8 vz8 vs8 → vs8 _ _ _ (x Var8 vz8 vs8)\n\nTm8 : Con8 → Ty8 → Set; Tm8\n = λ Γ A →\n (Tm8 : Con8 → Ty8 → Set)\n (var : ∀ Γ A → Var8 Γ A → Tm8 Γ A)\n (lam : ∀ Γ A B → Tm8 (snoc8 Γ A) B → Tm8 Γ (arr8 A B))\n (app : ∀ Γ A B → Tm8 Γ (arr8 A B) → Tm8 Γ A → Tm8 Γ B)\n (tt : ∀ Γ → Tm8 Γ top8)\n (pair : ∀ Γ A B → Tm8 Γ A → Tm8 Γ B → Tm8 Γ (prod8 A B))\n (fst : ∀ Γ A B → Tm8 Γ (prod8 A B) → Tm8 Γ A)\n (snd : ∀ Γ A B → Tm8 Γ (prod8 A B) → Tm8 Γ B)\n (left : ∀ Γ A B → Tm8 Γ A → Tm8 Γ (sum8 A B))\n (right : ∀ Γ A B → Tm8 Γ B → Tm8 Γ (sum8 A B))\n (case : ∀ Γ A B C → Tm8 Γ (sum8 A B) → Tm8 Γ (arr8 A C) → Tm8 Γ (arr8 B C) → Tm8 Γ C)\n (zero : ∀ Γ → Tm8 Γ nat8)\n (suc : ∀ Γ → Tm8 Γ nat8 → Tm8 Γ nat8)\n (rec : ∀ Γ A → Tm8 Γ nat8 → Tm8 Γ (arr8 nat8 (arr8 A A)) → Tm8 Γ A → Tm8 Γ A)\n → Tm8 Γ A\n\nvar8 : ∀{Γ A} → Var8 Γ A → Tm8 Γ A; var8\n = λ x Tm8 var8 lam app tt pair fst snd left right case zero suc rec →\n var8 _ _ x\n\nlam8 : ∀{Γ A B} → Tm8 (snoc8 Γ A) B → Tm8 Γ (arr8 A B); lam8\n = λ t Tm8 var8 lam8 app tt pair fst snd left right case zero suc rec →\n lam8 _ _ _ (t Tm8 var8 lam8 app tt pair fst snd left right case zero suc rec)\n\napp8 : ∀{Γ A B} → Tm8 Γ (arr8 A B) → Tm8 Γ A → Tm8 Γ B; app8\n = λ t u Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec →\n app8 _ _ _ (t Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec)\n (u Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec)\n\ntt8 : ∀{Γ} → Tm8 Γ top8; tt8\n = λ Tm8 var8 lam8 app8 tt8 pair fst snd left right case zero suc rec → tt8 _\n\npair8 : ∀{Γ A B} → Tm8 Γ A → Tm8 Γ B → Tm8 Γ (prod8 A B); pair8\n = λ t u Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec →\n pair8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec)\n (u Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec)\n\nfst8 : ∀{Γ A B} → Tm8 Γ (prod8 A B) → Tm8 Γ A; fst8\n = λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd left right case zero suc rec →\n fst8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd left right case zero suc rec)\n\nsnd8 : ∀{Γ A B} → Tm8 Γ (prod8 A B) → Tm8 Γ B; snd8\n = λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left right case zero suc rec →\n snd8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left right case zero suc rec)\n\nleft8 : ∀{Γ A B} → Tm8 Γ A → Tm8 Γ (sum8 A B); left8\n = λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right case zero suc rec →\n left8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right case zero suc rec)\n\nright8 : ∀{Γ A B} → Tm8 Γ B → Tm8 Γ (sum8 A B); right8\n = λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case zero suc rec →\n right8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case zero suc rec)\n\ncase8 : ∀{Γ A B C} → Tm8 Γ (sum8 A B) → Tm8 Γ (arr8 A C) → Tm8 Γ (arr8 B C) → Tm8 Γ C; case8\n = λ t u v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec →\n case8 _ _ _ _\n (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)\n (u Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)\n (v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)\n\nzero8 : ∀{Γ} → Tm8 Γ nat8; zero8\n = λ Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc rec → zero8 _\n\nsuc8 : ∀{Γ} → Tm8 Γ nat8 → Tm8 Γ nat8; suc8\n = λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec →\n suc8 _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec)\n\nrec8 : ∀{Γ A} → Tm8 Γ nat8 → Tm8 Γ (arr8 nat8 (arr8 A A)) → Tm8 Γ A → Tm8 Γ A; rec8\n = λ t u v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8 →\n rec8 _ _\n (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)\n (u Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)\n (v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)\n\nv08 : ∀{Γ A} → Tm8 (snoc8 Γ A) A; v08\n = var8 vz8\n\nv18 : ∀{Γ A B} → Tm8 (snoc8 (snoc8 Γ A) B) A; v18\n = var8 (vs8 vz8)\n\nv28 : ∀{Γ A B C} → Tm8 (snoc8 (snoc8 (snoc8 Γ A) B) C) A; v28\n = var8 (vs8 (vs8 vz8))\n\nv38 : ∀{Γ A B C D} → Tm8 (snoc8 (snoc8 (snoc8 (snoc8 Γ A) B) C) D) A; v38\n = var8 (vs8 (vs8 (vs8 vz8)))\n\ntbool8 : Ty8; tbool8\n = sum8 top8 top8\n\ntrue8 : ∀{Γ} → Tm8 Γ tbool8; true8\n = left8 tt8\n\ntfalse8 : ∀{Γ} → Tm8 Γ tbool8; tfalse8\n = right8 tt8\n\nifthenelse8 : ∀{Γ A} → Tm8 Γ (arr8 tbool8 (arr8 A (arr8 A A))); ifthenelse8\n = lam8 (lam8 (lam8 (case8 v28 (lam8 v28) (lam8 v18))))\n\ntimes48 : ∀{Γ A} → Tm8 Γ (arr8 (arr8 A A) (arr8 A A)); times48\n = lam8 (lam8 (app8 v18 (app8 v18 (app8 v18 (app8 v18 v08)))))\n\nadd8 : ∀{Γ} → Tm8 Γ (arr8 nat8 (arr8 nat8 nat8)); add8\n = lam8 (rec8 v08\n (lam8 (lam8 (lam8 (suc8 (app8 v18 v08)))))\n (lam8 v08))\n\nmul8 : ∀{Γ} → Tm8 Γ (arr8 nat8 (arr8 nat8 nat8)); mul8\n = lam8 (rec8 v08\n (lam8 (lam8 (lam8 (app8 (app8 add8 (app8 v18 v08)) v08))))\n (lam8 zero8))\n\nfact8 : ∀{Γ} → Tm8 Γ (arr8 nat8 nat8); fact8\n = lam8 (rec8 v08 (lam8 (lam8 (app8 (app8 mul8 (suc8 v18)) v08)))\n (suc8 zero8))\n{-# OPTIONS --type-in-type #-}\n\nTy9 : Set\nTy9 =\n (Ty9 : Set)\n (nat top bot : Ty9)\n (arr prod sum : Ty9 → Ty9 → Ty9)\n → Ty9\n\nnat9 : Ty9; nat9 = λ _ nat9 _ _ _ _ _ → nat9\ntop9 : Ty9; top9 = λ _ _ top9 _ _ _ _ → top9\nbot9 : Ty9; bot9 = λ _ _ _ bot9 _ _ _ → bot9\n\narr9 : Ty9 → Ty9 → Ty9; arr9\n = λ A B Ty9 nat9 top9 bot9 arr9 prod sum →\n arr9 (A Ty9 nat9 top9 bot9 arr9 prod sum) (B Ty9 nat9 top9 bot9 arr9 prod sum)\n\nprod9 : Ty9 → Ty9 → Ty9; prod9\n = λ A B Ty9 nat9 top9 bot9 arr9 prod9 sum →\n prod9 (A Ty9 nat9 top9 bot9 arr9 prod9 sum) (B Ty9 nat9 top9 bot9 arr9 prod9 sum)\n\nsum9 : Ty9 → Ty9 → Ty9; sum9\n = λ A B Ty9 nat9 top9 bot9 arr9 prod9 sum9 →\n sum9 (A Ty9 nat9 top9 bot9 arr9 prod9 sum9) (B Ty9 nat9 top9 bot9 arr9 prod9 sum9)\n\nCon9 : Set; Con9\n = (Con9 : Set)\n (nil : Con9)\n (snoc : Con9 → Ty9 → Con9)\n → Con9\n\nnil9 : Con9; nil9\n = λ Con9 nil9 snoc → nil9\n\nsnoc9 : Con9 → Ty9 → Con9; snoc9\n = λ Γ A Con9 nil9 snoc9 → snoc9 (Γ Con9 nil9 snoc9) A\n\nVar9 : Con9 → Ty9 → Set; Var9\n = λ Γ A →\n (Var9 : Con9 → Ty9 → Set)\n (vz : ∀ Γ A → Var9 (snoc9 Γ A) A)\n (vs : ∀ Γ B A → Var9 Γ A → Var9 (snoc9 Γ B) A)\n → Var9 Γ A\n\nvz9 : ∀{Γ A} → Var9 (snoc9 Γ A) A; vz9\n = λ Var9 vz9 vs → vz9 _ _\n\nvs9 : ∀{Γ B A} → Var9 Γ A → Var9 (snoc9 Γ B) A; vs9\n = λ x Var9 vz9 vs9 → vs9 _ _ _ (x Var9 vz9 vs9)\n\nTm9 : Con9 → Ty9 → Set; Tm9\n = λ Γ A →\n (Tm9 : Con9 → Ty9 → Set)\n (var : ∀ Γ A → Var9 Γ A → Tm9 Γ A)\n (lam : ∀ Γ A B → Tm9 (snoc9 Γ A) B → Tm9 Γ (arr9 A B))\n (app : ∀ Γ A B → Tm9 Γ (arr9 A B) → Tm9 Γ A → Tm9 Γ B)\n (tt : ∀ Γ → Tm9 Γ top9)\n (pair : ∀ Γ A B → Tm9 Γ A → Tm9 Γ B → Tm9 Γ (prod9 A B))\n (fst : ∀ Γ A B → Tm9 Γ (prod9 A B) → Tm9 Γ A)\n (snd : ∀ Γ A B → Tm9 Γ (prod9 A B) → Tm9 Γ B)\n (left : ∀ Γ A B → Tm9 Γ A → Tm9 Γ (sum9 A B))\n (right : ∀ Γ A B → Tm9 Γ B → Tm9 Γ (sum9 A B))\n (case : ∀ Γ A B C → Tm9 Γ (sum9 A B) → Tm9 Γ (arr9 A C) → Tm9 Γ (arr9 B C) → Tm9 Γ C)\n (zero : ∀ Γ → Tm9 Γ nat9)\n (suc : ∀ Γ → Tm9 Γ nat9 → Tm9 Γ nat9)\n (rec : ∀ Γ A → Tm9 Γ nat9 → Tm9 Γ (arr9 nat9 (arr9 A A)) → Tm9 Γ A → Tm9 Γ A)\n → Tm9 Γ A\n\nvar9 : ∀{Γ A} → Var9 Γ A → Tm9 Γ A; var9\n = λ x Tm9 var9 lam app tt pair fst snd left right case zero suc rec →\n var9 _ _ x\n\nlam9 : ∀{Γ A B} → Tm9 (snoc9 Γ A) B → Tm9 Γ (arr9 A B); lam9\n = λ t Tm9 var9 lam9 app tt pair fst snd left right case zero suc rec →\n lam9 _ _ _ (t Tm9 var9 lam9 app tt pair fst snd left right case zero suc rec)\n\napp9 : ∀{Γ A B} → Tm9 Γ (arr9 A B) → Tm9 Γ A → Tm9 Γ B; app9\n = λ t u Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec →\n app9 _ _ _ (t Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec)\n (u Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec)\n\ntt9 : ∀{Γ} → Tm9 Γ top9; tt9\n = λ Tm9 var9 lam9 app9 tt9 pair fst snd left right case zero suc rec → tt9 _\n\npair9 : ∀{Γ A B} → Tm9 Γ A → Tm9 Γ B → Tm9 Γ (prod9 A B); pair9\n = λ t u Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec →\n pair9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec)\n (u Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec)\n\nfst9 : ∀{Γ A B} → Tm9 Γ (prod9 A B) → Tm9 Γ A; fst9\n = λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd left right case zero suc rec →\n fst9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd left right case zero suc rec)\n\nsnd9 : ∀{Γ A B} → Tm9 Γ (prod9 A B) → Tm9 Γ B; snd9\n = λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left right case zero suc rec →\n snd9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left right case zero suc rec)\n\nleft9 : ∀{Γ A B} → Tm9 Γ A → Tm9 Γ (sum9 A B); left9\n = λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right case zero suc rec →\n left9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right case zero suc rec)\n\nright9 : ∀{Γ A B} → Tm9 Γ B → Tm9 Γ (sum9 A B); right9\n = λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case zero suc rec →\n right9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case zero suc rec)\n\ncase9 : ∀{Γ A B C} → Tm9 Γ (sum9 A B) → Tm9 Γ (arr9 A C) → Tm9 Γ (arr9 B C) → Tm9 Γ C; case9\n = λ t u v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec →\n case9 _ _ _ _\n (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)\n (u Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)\n (v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)\n\nzero9 : ∀{Γ} → Tm9 Γ nat9; zero9\n = λ Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc rec → zero9 _\n\nsuc9 : ∀{Γ} → Tm9 Γ nat9 → Tm9 Γ nat9; suc9\n = λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec →\n suc9 _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec)\n\nrec9 : ∀{Γ A} → Tm9 Γ nat9 → Tm9 Γ (arr9 nat9 (arr9 A A)) → Tm9 Γ A → Tm9 Γ A; rec9\n = λ t u v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9 →\n rec9 _ _\n (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)\n (u Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)\n (v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)\n\nv09 : ∀{Γ A} → Tm9 (snoc9 Γ A) A; v09\n = var9 vz9\n\nv19 : ∀{Γ A B} → Tm9 (snoc9 (snoc9 Γ A) B) A; v19\n = var9 (vs9 vz9)\n\nv29 : ∀{Γ A B C} → Tm9 (snoc9 (snoc9 (snoc9 Γ A) B) C) A; v29\n = var9 (vs9 (vs9 vz9))\n\nv39 : ∀{Γ A B C D} → Tm9 (snoc9 (snoc9 (snoc9 (snoc9 Γ A) B) C) D) A; v39\n = var9 (vs9 (vs9 (vs9 vz9)))\n\ntbool9 : Ty9; tbool9\n = sum9 top9 top9\n\ntrue9 : ∀{Γ} → Tm9 Γ tbool9; true9\n = left9 tt9\n\ntfalse9 : ∀{Γ} → Tm9 Γ tbool9; tfalse9\n = right9 tt9\n\nifthenelse9 : ∀{Γ A} → Tm9 Γ (arr9 tbool9 (arr9 A (arr9 A A))); ifthenelse9\n = lam9 (lam9 (lam9 (case9 v29 (lam9 v29) (lam9 v19))))\n\ntimes49 : ∀{Γ A} → Tm9 Γ (arr9 (arr9 A A) (arr9 A A)); times49\n = lam9 (lam9 (app9 v19 (app9 v19 (app9 v19 (app9 v19 v09)))))\n\nadd9 : ∀{Γ} → Tm9 Γ (arr9 nat9 (arr9 nat9 nat9)); add9\n = lam9 (rec9 v09\n (lam9 (lam9 (lam9 (suc9 (app9 v19 v09)))))\n (lam9 v09))\n\nmul9 : ∀{Γ} → Tm9 Γ (arr9 nat9 (arr9 nat9 nat9)); mul9\n = lam9 (rec9 v09\n (lam9 (lam9 (lam9 (app9 (app9 add9 (app9 v19 v09)) v09))))\n (lam9 zero9))\n\nfact9 : ∀{Γ} → Tm9 Γ (arr9 nat9 nat9); fact9\n = lam9 (rec9 v09 (lam9 (lam9 (app9 (app9 mul9 (suc9 v19)) v09)))\n (suc9 zero9))\n{-# OPTIONS --type-in-type #-}\n\nTy10 : Set\nTy10 =\n (Ty10 : Set)\n (nat top bot : Ty10)\n (arr prod sum : Ty10 → Ty10 → Ty10)\n → Ty10\n\nnat10 : Ty10; nat10 = λ _ nat10 _ _ _ _ _ → nat10\ntop10 : Ty10; top10 = λ _ _ top10 _ _ _ _ → top10\nbot10 : Ty10; bot10 = λ _ _ _ bot10 _ _ _ → bot10\n\narr10 : Ty10 → Ty10 → Ty10; arr10\n = λ A B Ty10 nat10 top10 bot10 arr10 prod sum →\n arr10 (A Ty10 nat10 top10 bot10 arr10 prod sum) (B Ty10 nat10 top10 bot10 arr10 prod sum)\n\nprod10 : Ty10 → Ty10 → Ty10; prod10\n = λ A B Ty10 nat10 top10 bot10 arr10 prod10 sum →\n prod10 (A Ty10 nat10 top10 bot10 arr10 prod10 sum) (B Ty10 nat10 top10 bot10 arr10 prod10 sum)\n\nsum10 : Ty10 → Ty10 → Ty10; sum10\n = λ A B Ty10 nat10 top10 bot10 arr10 prod10 sum10 →\n sum10 (A Ty10 nat10 top10 bot10 arr10 prod10 sum10) (B Ty10 nat10 top10 bot10 arr10 prod10 sum10)\n\nCon10 : Set; Con10\n = (Con10 : Set)\n (nil : Con10)\n (snoc : Con10 → Ty10 → Con10)\n → Con10\n\nnil10 : Con10; nil10\n = λ Con10 nil10 snoc → nil10\n\nsnoc10 : Con10 → Ty10 → Con10; snoc10\n = λ Γ A Con10 nil10 snoc10 → snoc10 (Γ Con10 nil10 snoc10) A\n\nVar10 : Con10 → Ty10 → Set; Var10\n = λ Γ A →\n (Var10 : Con10 → Ty10 → Set)\n (vz : ∀ Γ A → Var10 (snoc10 Γ A) A)\n (vs : ∀ Γ B A → Var10 Γ A → Var10 (snoc10 Γ B) A)\n → Var10 Γ A\n\nvz10 : ∀{Γ A} → Var10 (snoc10 Γ A) A; vz10\n = λ Var10 vz10 vs → vz10 _ _\n\nvs10 : ∀{Γ B A} → Var10 Γ A → Var10 (snoc10 Γ B) A; vs10\n = λ x Var10 vz10 vs10 → vs10 _ _ _ (x Var10 vz10 vs10)\n\nTm10 : Con10 → Ty10 → Set; Tm10\n = λ Γ A →\n (Tm10 : Con10 → Ty10 → Set)\n (var : ∀ Γ A → Var10 Γ A → Tm10 Γ A)\n (lam : ∀ Γ A B → Tm10 (snoc10 Γ A) B → Tm10 Γ (arr10 A B))\n (app : ∀ Γ A B → Tm10 Γ (arr10 A B) → Tm10 Γ A → Tm10 Γ B)\n (tt : ∀ Γ → Tm10 Γ top10)\n (pair : ∀ Γ A B → Tm10 Γ A → Tm10 Γ B → Tm10 Γ (prod10 A B))\n (fst : ∀ Γ A B → Tm10 Γ (prod10 A B) → Tm10 Γ A)\n (snd : ∀ Γ A B → Tm10 Γ (prod10 A B) → Tm10 Γ B)\n (left : ∀ Γ A B → Tm10 Γ A → Tm10 Γ (sum10 A B))\n (right : ∀ Γ A B → Tm10 Γ B → Tm10 Γ (sum10 A B))\n (case : ∀ Γ A B C → Tm10 Γ (sum10 A B) → Tm10 Γ (arr10 A C) → Tm10 Γ (arr10 B C) → Tm10 Γ C)\n (zero : ∀ Γ → Tm10 Γ nat10)\n (suc : ∀ Γ → Tm10 Γ nat10 → Tm10 Γ nat10)\n (rec : ∀ Γ A → Tm10 Γ nat10 → Tm10 Γ (arr10 nat10 (arr10 A A)) → Tm10 Γ A → Tm10 Γ A)\n → Tm10 Γ A\n\nvar10 : ∀{Γ A} → Var10 Γ A → Tm10 Γ A; var10\n = λ x Tm10 var10 lam app tt pair fst snd left right case zero suc rec →\n var10 _ _ x\n\nlam10 : ∀{Γ A B} → Tm10 (snoc10 Γ A) B → Tm10 Γ (arr10 A B); lam10\n = λ t Tm10 var10 lam10 app tt pair fst snd left right case zero suc rec →\n lam10 _ _ _ (t Tm10 var10 lam10 app tt pair fst snd left right case zero suc rec)\n\napp10 : ∀{Γ A B} → Tm10 Γ (arr10 A B) → Tm10 Γ A → Tm10 Γ B; app10\n = λ t u Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec →\n app10 _ _ _ (t Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec)\n (u Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec)\n\ntt10 : ∀{Γ} → Tm10 Γ top10; tt10\n = λ Tm10 var10 lam10 app10 tt10 pair fst snd left right case zero suc rec → tt10 _\n\npair10 : ∀{Γ A B} → Tm10 Γ A → Tm10 Γ B → Tm10 Γ (prod10 A B); pair10\n = λ t u Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec →\n pair10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec)\n (u Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec)\n\nfst10 : ∀{Γ A B} → Tm10 Γ (prod10 A B) → Tm10 Γ A; fst10\n = λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd left right case zero suc rec →\n fst10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd left right case zero suc rec)\n\nsnd10 : ∀{Γ A B} → Tm10 Γ (prod10 A B) → Tm10 Γ B; snd10\n = λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left right case zero suc rec →\n snd10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left right case zero suc rec)\n\nleft10 : ∀{Γ A B} → Tm10 Γ A → Tm10 Γ (sum10 A B); left10\n = λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right case zero suc rec →\n left10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right case zero suc rec)\n\nright10 : ∀{Γ A B} → Tm10 Γ B → Tm10 Γ (sum10 A B); right10\n = λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case zero suc rec →\n right10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case zero suc rec)\n\ncase10 : ∀{Γ A B C} → Tm10 Γ (sum10 A B) → Tm10 Γ (arr10 A C) → Tm10 Γ (arr10 B C) → Tm10 Γ C; case10\n = λ t u v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec →\n case10 _ _ _ _\n (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)\n (u Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)\n (v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)\n\nzero10 : ∀{Γ} → Tm10 Γ nat10; zero10\n = λ Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc rec → zero10 _\n\nsuc10 : ∀{Γ} → Tm10 Γ nat10 → Tm10 Γ nat10; suc10\n = λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec →\n suc10 _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec)\n\nrec10 : ∀{Γ A} → Tm10 Γ nat10 → Tm10 Γ (arr10 nat10 (arr10 A A)) → Tm10 Γ A → Tm10 Γ A; rec10\n = λ t u v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10 →\n rec10 _ _\n (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)\n (u Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)\n (v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)\n\nv010 : ∀{Γ A} → Tm10 (snoc10 Γ A) A; v010\n = var10 vz10\n\nv110 : ∀{Γ A B} → Tm10 (snoc10 (snoc10 Γ A) B) A; v110\n = var10 (vs10 vz10)\n\nv210 : ∀{Γ A B C} → Tm10 (snoc10 (snoc10 (snoc10 Γ A) B) C) A; v210\n = var10 (vs10 (vs10 vz10))\n\nv310 : ∀{Γ A B C D} → Tm10 (snoc10 (snoc10 (snoc10 (snoc10 Γ A) B) C) D) A; v310\n = var10 (vs10 (vs10 (vs10 vz10)))\n\ntbool10 : Ty10; tbool10\n = sum10 top10 top10\n\ntrue10 : ∀{Γ} → Tm10 Γ tbool10; true10\n = left10 tt10\n\ntfalse10 : ∀{Γ} → Tm10 Γ tbool10; tfalse10\n = right10 tt10\n\nifthenelse10 : ∀{Γ A} → Tm10 Γ (arr10 tbool10 (arr10 A (arr10 A A))); ifthenelse10\n = lam10 (lam10 (lam10 (case10 v210 (lam10 v210) (lam10 v110))))\n\ntimes410 : ∀{Γ A} → Tm10 Γ (arr10 (arr10 A A) (arr10 A A)); times410\n = lam10 (lam10 (app10 v110 (app10 v110 (app10 v110 (app10 v110 v010)))))\n\nadd10 : ∀{Γ} → Tm10 Γ (arr10 nat10 (arr10 nat10 nat10)); add10\n = lam10 (rec10 v010\n (lam10 (lam10 (lam10 (suc10 (app10 v110 v010)))))\n (lam10 v010))\n\nmul10 : ∀{Γ} → Tm10 Γ (arr10 nat10 (arr10 nat10 nat10)); mul10\n = lam10 (rec10 v010\n (lam10 (lam10 (lam10 (app10 (app10 add10 (app10 v110 v010)) v010))))\n (lam10 zero10))\n\nfact10 : ∀{Γ} → Tm10 Γ (arr10 nat10 nat10); fact10\n = lam10 (rec10 v010 (lam10 (lam10 (app10 (app10 mul10 (suc10 v110)) v010)))\n (suc10 zero10))\n{-# OPTIONS --type-in-type #-}\n\nTy11 : Set\nTy11 =\n (Ty11 : Set)\n (nat top bot : Ty11)\n (arr prod sum : Ty11 → Ty11 → Ty11)\n → Ty11\n\nnat11 : Ty11; nat11 = λ _ nat11 _ _ _ _ _ → nat11\ntop11 : Ty11; top11 = λ _ _ top11 _ _ _ _ → top11\nbot11 : Ty11; bot11 = λ _ _ _ bot11 _ _ _ → bot11\n\narr11 : Ty11 → Ty11 → Ty11; arr11\n = λ A B Ty11 nat11 top11 bot11 arr11 prod sum →\n arr11 (A Ty11 nat11 top11 bot11 arr11 prod sum) (B Ty11 nat11 top11 bot11 arr11 prod sum)\n\nprod11 : Ty11 → Ty11 → Ty11; prod11\n = λ A B Ty11 nat11 top11 bot11 arr11 prod11 sum →\n prod11 (A Ty11 nat11 top11 bot11 arr11 prod11 sum) (B Ty11 nat11 top11 bot11 arr11 prod11 sum)\n\nsum11 : Ty11 → Ty11 → Ty11; sum11\n = λ A B Ty11 nat11 top11 bot11 arr11 prod11 sum11 →\n sum11 (A Ty11 nat11 top11 bot11 arr11 prod11 sum11) (B Ty11 nat11 top11 bot11 arr11 prod11 sum11)\n\nCon11 : Set; Con11\n = (Con11 : Set)\n (nil : Con11)\n (snoc : Con11 → Ty11 → Con11)\n → Con11\n\nnil11 : Con11; nil11\n = λ Con11 nil11 snoc → nil11\n\nsnoc11 : Con11 → Ty11 → Con11; snoc11\n = λ Γ A Con11 nil11 snoc11 → snoc11 (Γ Con11 nil11 snoc11) A\n\nVar11 : Con11 → Ty11 → Set; Var11\n = λ Γ A →\n (Var11 : Con11 → Ty11 → Set)\n (vz : ∀ Γ A → Var11 (snoc11 Γ A) A)\n (vs : ∀ Γ B A → Var11 Γ A → Var11 (snoc11 Γ B) A)\n → Var11 Γ A\n\nvz11 : ∀{Γ A} → Var11 (snoc11 Γ A) A; vz11\n = λ Var11 vz11 vs → vz11 _ _\n\nvs11 : ∀{Γ B A} → Var11 Γ A → Var11 (snoc11 Γ B) A; vs11\n = λ x Var11 vz11 vs11 → vs11 _ _ _ (x Var11 vz11 vs11)\n\nTm11 : Con11 → Ty11 → Set; Tm11\n = λ Γ A →\n (Tm11 : Con11 → Ty11 → Set)\n (var : ∀ Γ A → Var11 Γ A → Tm11 Γ A)\n (lam : ∀ Γ A B → Tm11 (snoc11 Γ A) B → Tm11 Γ (arr11 A B))\n (app : ∀ Γ A B → Tm11 Γ (arr11 A B) → Tm11 Γ A → Tm11 Γ B)\n (tt : ∀ Γ → Tm11 Γ top11)\n (pair : ∀ Γ A B → Tm11 Γ A → Tm11 Γ B → Tm11 Γ (prod11 A B))\n (fst : ∀ Γ A B → Tm11 Γ (prod11 A B) → Tm11 Γ A)\n (snd : ∀ Γ A B → Tm11 Γ (prod11 A B) → Tm11 Γ B)\n (left : ∀ Γ A B → Tm11 Γ A → Tm11 Γ (sum11 A B))\n (right : ∀ Γ A B → Tm11 Γ B → Tm11 Γ (sum11 A B))\n (case : ∀ Γ A B C → Tm11 Γ (sum11 A B) → Tm11 Γ (arr11 A C) → Tm11 Γ (arr11 B C) → Tm11 Γ C)\n (zero : ∀ Γ → Tm11 Γ nat11)\n (suc : ∀ Γ → Tm11 Γ nat11 → Tm11 Γ nat11)\n (rec : ∀ Γ A → Tm11 Γ nat11 → Tm11 Γ (arr11 nat11 (arr11 A A)) → Tm11 Γ A → Tm11 Γ A)\n → Tm11 Γ A\n\nvar11 : ∀{Γ A} → Var11 Γ A → Tm11 Γ A; var11\n = λ x Tm11 var11 lam app tt pair fst snd left right case zero suc rec →\n var11 _ _ x\n\nlam11 : ∀{Γ A B} → Tm11 (snoc11 Γ A) B → Tm11 Γ (arr11 A B); lam11\n = λ t Tm11 var11 lam11 app tt pair fst snd left right case zero suc rec →\n lam11 _ _ _ (t Tm11 var11 lam11 app tt pair fst snd left right case zero suc rec)\n\napp11 : ∀{Γ A B} → Tm11 Γ (arr11 A B) → Tm11 Γ A → Tm11 Γ B; app11\n = λ t u Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec →\n app11 _ _ _ (t Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec)\n (u Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec)\n\ntt11 : ∀{Γ} → Tm11 Γ top11; tt11\n = λ Tm11 var11 lam11 app11 tt11 pair fst snd left right case zero suc rec → tt11 _\n\npair11 : ∀{Γ A B} → Tm11 Γ A → Tm11 Γ B → Tm11 Γ (prod11 A B); pair11\n = λ t u Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec →\n pair11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec)\n (u Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec)\n\nfst11 : ∀{Γ A B} → Tm11 Γ (prod11 A B) → Tm11 Γ A; fst11\n = λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd left right case zero suc rec →\n fst11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd left right case zero suc rec)\n\nsnd11 : ∀{Γ A B} → Tm11 Γ (prod11 A B) → Tm11 Γ B; snd11\n = λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left right case zero suc rec →\n snd11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left right case zero suc rec)\n\nleft11 : ∀{Γ A B} → Tm11 Γ A → Tm11 Γ (sum11 A B); left11\n = λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right case zero suc rec →\n left11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right case zero suc rec)\n\nright11 : ∀{Γ A B} → Tm11 Γ B → Tm11 Γ (sum11 A B); right11\n = λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case zero suc rec →\n right11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case zero suc rec)\n\ncase11 : ∀{Γ A B C} → Tm11 Γ (sum11 A B) → Tm11 Γ (arr11 A C) → Tm11 Γ (arr11 B C) → Tm11 Γ C; case11\n = λ t u v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec →\n case11 _ _ _ _\n (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)\n (u Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)\n (v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)\n\nzero11 : ∀{Γ} → Tm11 Γ nat11; zero11\n = λ Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc rec → zero11 _\n\nsuc11 : ∀{Γ} → Tm11 Γ nat11 → Tm11 Γ nat11; suc11\n = λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec →\n suc11 _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec)\n\nrec11 : ∀{Γ A} → Tm11 Γ nat11 → Tm11 Γ (arr11 nat11 (arr11 A A)) → Tm11 Γ A → Tm11 Γ A; rec11\n = λ t u v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11 →\n rec11 _ _\n (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)\n (u Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)\n (v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)\n\nv011 : ∀{Γ A} → Tm11 (snoc11 Γ A) A; v011\n = var11 vz11\n\nv111 : ∀{Γ A B} → Tm11 (snoc11 (snoc11 Γ A) B) A; v111\n = var11 (vs11 vz11)\n\nv211 : ∀{Γ A B C} → Tm11 (snoc11 (snoc11 (snoc11 Γ A) B) C) A; v211\n = var11 (vs11 (vs11 vz11))\n\nv311 : ∀{Γ A B C D} → Tm11 (snoc11 (snoc11 (snoc11 (snoc11 Γ A) B) C) D) A; v311\n = var11 (vs11 (vs11 (vs11 vz11)))\n\ntbool11 : Ty11; tbool11\n = sum11 top11 top11\n\ntrue11 : ∀{Γ} → Tm11 Γ tbool11; true11\n = left11 tt11\n\ntfalse11 : ∀{Γ} → Tm11 Γ tbool11; tfalse11\n = right11 tt11\n\nifthenelse11 : ∀{Γ A} → Tm11 Γ (arr11 tbool11 (arr11 A (arr11 A A))); ifthenelse11\n = lam11 (lam11 (lam11 (case11 v211 (lam11 v211) (lam11 v111))))\n\ntimes411 : ∀{Γ A} → Tm11 Γ (arr11 (arr11 A A) (arr11 A A)); times411\n = lam11 (lam11 (app11 v111 (app11 v111 (app11 v111 (app11 v111 v011)))))\n\nadd11 : ∀{Γ} → Tm11 Γ (arr11 nat11 (arr11 nat11 nat11)); add11\n = lam11 (rec11 v011\n (lam11 (lam11 (lam11 (suc11 (app11 v111 v011)))))\n (lam11 v011))\n\nmul11 : ∀{Γ} → Tm11 Γ (arr11 nat11 (arr11 nat11 nat11)); mul11\n = lam11 (rec11 v011\n (lam11 (lam11 (lam11 (app11 (app11 add11 (app11 v111 v011)) v011))))\n (lam11 zero11))\n\nfact11 : ∀{Γ} → Tm11 Γ (arr11 nat11 nat11); fact11\n = lam11 (rec11 v011 (lam11 (lam11 (app11 (app11 mul11 (suc11 v111)) v011)))\n (suc11 zero11))\n{-# OPTIONS --type-in-type #-}\n\nTy12 : Set\nTy12 =\n (Ty12 : Set)\n (nat top bot : Ty12)\n (arr prod sum : Ty12 → Ty12 → Ty12)\n → Ty12\n\nnat12 : Ty12; nat12 = λ _ nat12 _ _ _ _ _ → nat12\ntop12 : Ty12; top12 = λ _ _ top12 _ _ _ _ → top12\nbot12 : Ty12; bot12 = λ _ _ _ bot12 _ _ _ → bot12\n\narr12 : Ty12 → Ty12 → Ty12; arr12\n = λ A B Ty12 nat12 top12 bot12 arr12 prod sum →\n arr12 (A Ty12 nat12 top12 bot12 arr12 prod sum) (B Ty12 nat12 top12 bot12 arr12 prod sum)\n\nprod12 : Ty12 → Ty12 → Ty12; prod12\n = λ A B Ty12 nat12 top12 bot12 arr12 prod12 sum →\n prod12 (A Ty12 nat12 top12 bot12 arr12 prod12 sum) (B Ty12 nat12 top12 bot12 arr12 prod12 sum)\n\nsum12 : Ty12 → Ty12 → Ty12; sum12\n = λ A B Ty12 nat12 top12 bot12 arr12 prod12 sum12 →\n sum12 (A Ty12 nat12 top12 bot12 arr12 prod12 sum12) (B Ty12 nat12 top12 bot12 arr12 prod12 sum12)\n\nCon12 : Set; Con12\n = (Con12 : Set)\n (nil : Con12)\n (snoc : Con12 → Ty12 → Con12)\n → Con12\n\nnil12 : Con12; nil12\n = λ Con12 nil12 snoc → nil12\n\nsnoc12 : Con12 → Ty12 → Con12; snoc12\n = λ Γ A Con12 nil12 snoc12 → snoc12 (Γ Con12 nil12 snoc12) A\n\nVar12 : Con12 → Ty12 → Set; Var12\n = λ Γ A →\n (Var12 : Con12 → Ty12 → Set)\n (vz : ∀ Γ A → Var12 (snoc12 Γ A) A)\n (vs : ∀ Γ B A → Var12 Γ A → Var12 (snoc12 Γ B) A)\n → Var12 Γ A\n\nvz12 : ∀{Γ A} → Var12 (snoc12 Γ A) A; vz12\n = λ Var12 vz12 vs → vz12 _ _\n\nvs12 : ∀{Γ B A} → Var12 Γ A → Var12 (snoc12 Γ B) A; vs12\n = λ x Var12 vz12 vs12 → vs12 _ _ _ (x Var12 vz12 vs12)\n\nTm12 : Con12 → Ty12 → Set; Tm12\n = λ Γ A →\n (Tm12 : Con12 → Ty12 → Set)\n (var : ∀ Γ A → Var12 Γ A → Tm12 Γ A)\n (lam : ∀ Γ A B → Tm12 (snoc12 Γ A) B → Tm12 Γ (arr12 A B))\n (app : ∀ Γ A B → Tm12 Γ (arr12 A B) → Tm12 Γ A → Tm12 Γ B)\n (tt : ∀ Γ → Tm12 Γ top12)\n (pair : ∀ Γ A B → Tm12 Γ A → Tm12 Γ B → Tm12 Γ (prod12 A B))\n (fst : ∀ Γ A B → Tm12 Γ (prod12 A B) → Tm12 Γ A)\n (snd : ∀ Γ A B → Tm12 Γ (prod12 A B) → Tm12 Γ B)\n (left : ∀ Γ A B → Tm12 Γ A → Tm12 Γ (sum12 A B))\n (right : ∀ Γ A B → Tm12 Γ B → Tm12 Γ (sum12 A B))\n (case : ∀ Γ A B C → Tm12 Γ (sum12 A B) → Tm12 Γ (arr12 A C) → Tm12 Γ (arr12 B C) → Tm12 Γ C)\n (zero : ∀ Γ → Tm12 Γ nat12)\n (suc : ∀ Γ → Tm12 Γ nat12 → Tm12 Γ nat12)\n (rec : ∀ Γ A → Tm12 Γ nat12 → Tm12 Γ (arr12 nat12 (arr12 A A)) → Tm12 Γ A → Tm12 Γ A)\n → Tm12 Γ A\n\nvar12 : ∀{Γ A} → Var12 Γ A → Tm12 Γ A; var12\n = λ x Tm12 var12 lam app tt pair fst snd left right case zero suc rec →\n var12 _ _ x\n\nlam12 : ∀{Γ A B} → Tm12 (snoc12 Γ A) B → Tm12 Γ (arr12 A B); lam12\n = λ t Tm12 var12 lam12 app tt pair fst snd left right case zero suc rec →\n lam12 _ _ _ (t Tm12 var12 lam12 app tt pair fst snd left right case zero suc rec)\n\napp12 : ∀{Γ A B} → Tm12 Γ (arr12 A B) → Tm12 Γ A → Tm12 Γ B; app12\n = λ t u Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec →\n app12 _ _ _ (t Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec)\n (u Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec)\n\ntt12 : ∀{Γ} → Tm12 Γ top12; tt12\n = λ Tm12 var12 lam12 app12 tt12 pair fst snd left right case zero suc rec → tt12 _\n\npair12 : ∀{Γ A B} → Tm12 Γ A → Tm12 Γ B → Tm12 Γ (prod12 A B); pair12\n = λ t u Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec →\n pair12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec)\n (u Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec)\n\nfst12 : ∀{Γ A B} → Tm12 Γ (prod12 A B) → Tm12 Γ A; fst12\n = λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd left right case zero suc rec →\n fst12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd left right case zero suc rec)\n\nsnd12 : ∀{Γ A B} → Tm12 Γ (prod12 A B) → Tm12 Γ B; snd12\n = λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left right case zero suc rec →\n snd12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left right case zero suc rec)\n\nleft12 : ∀{Γ A B} → Tm12 Γ A → Tm12 Γ (sum12 A B); left12\n = λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right case zero suc rec →\n left12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right case zero suc rec)\n\nright12 : ∀{Γ A B} → Tm12 Γ B → Tm12 Γ (sum12 A B); right12\n = λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case zero suc rec →\n right12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case zero suc rec)\n\ncase12 : ∀{Γ A B C} → Tm12 Γ (sum12 A B) → Tm12 Γ (arr12 A C) → Tm12 Γ (arr12 B C) → Tm12 Γ C; case12\n = λ t u v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec →\n case12 _ _ _ _\n (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)\n (u Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)\n (v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)\n\nzero12 : ∀{Γ} → Tm12 Γ nat12; zero12\n = λ Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc rec → zero12 _\n\nsuc12 : ∀{Γ} → Tm12 Γ nat12 → Tm12 Γ nat12; suc12\n = λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec →\n suc12 _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec)\n\nrec12 : ∀{Γ A} → Tm12 Γ nat12 → Tm12 Γ (arr12 nat12 (arr12 A A)) → Tm12 Γ A → Tm12 Γ A; rec12\n = λ t u v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12 →\n rec12 _ _\n (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)\n (u Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)\n (v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)\n\nv012 : ∀{Γ A} → Tm12 (snoc12 Γ A) A; v012\n = var12 vz12\n\nv112 : ∀{Γ A B} → Tm12 (snoc12 (snoc12 Γ A) B) A; v112\n = var12 (vs12 vz12)\n\nv212 : ∀{Γ A B C} → Tm12 (snoc12 (snoc12 (snoc12 Γ A) B) C) A; v212\n = var12 (vs12 (vs12 vz12))\n\nv312 : ∀{Γ A B C D} → Tm12 (snoc12 (snoc12 (snoc12 (snoc12 Γ A) B) C) D) A; v312\n = var12 (vs12 (vs12 (vs12 vz12)))\n\ntbool12 : Ty12; tbool12\n = sum12 top12 top12\n\ntrue12 : ∀{Γ} → Tm12 Γ tbool12; true12\n = left12 tt12\n\ntfalse12 : ∀{Γ} → Tm12 Γ tbool12; tfalse12\n = right12 tt12\n\nifthenelse12 : ∀{Γ A} → Tm12 Γ (arr12 tbool12 (arr12 A (arr12 A A))); ifthenelse12\n = lam12 (lam12 (lam12 (case12 v212 (lam12 v212) (lam12 v112))))\n\ntimes412 : ∀{Γ A} → Tm12 Γ (arr12 (arr12 A A) (arr12 A A)); times412\n = lam12 (lam12 (app12 v112 (app12 v112 (app12 v112 (app12 v112 v012)))))\n\nadd12 : ∀{Γ} → Tm12 Γ (arr12 nat12 (arr12 nat12 nat12)); add12\n = lam12 (rec12 v012\n (lam12 (lam12 (lam12 (suc12 (app12 v112 v012)))))\n (lam12 v012))\n\nmul12 : ∀{Γ} → Tm12 Γ (arr12 nat12 (arr12 nat12 nat12)); mul12\n = lam12 (rec12 v012\n (lam12 (lam12 (lam12 (app12 (app12 add12 (app12 v112 v012)) v012))))\n (lam12 zero12))\n\nfact12 : ∀{Γ} → Tm12 Γ (arr12 nat12 nat12); fact12\n = lam12 (rec12 v012 (lam12 (lam12 (app12 (app12 mul12 (suc12 v112)) v012)))\n (suc12 zero12))\n{-# OPTIONS --type-in-type #-}\n\nTy13 : Set\nTy13 =\n (Ty13 : Set)\n (nat top bot : Ty13)\n (arr prod sum : Ty13 → Ty13 → Ty13)\n → Ty13\n\nnat13 : Ty13; nat13 = λ _ nat13 _ _ _ _ _ → nat13\ntop13 : Ty13; top13 = λ _ _ top13 _ _ _ _ → top13\nbot13 : Ty13; bot13 = λ _ _ _ bot13 _ _ _ → bot13\n\narr13 : Ty13 → Ty13 → Ty13; arr13\n = λ A B Ty13 nat13 top13 bot13 arr13 prod sum →\n arr13 (A Ty13 nat13 top13 bot13 arr13 prod sum) (B Ty13 nat13 top13 bot13 arr13 prod sum)\n\nprod13 : Ty13 → Ty13 → Ty13; prod13\n = λ A B Ty13 nat13 top13 bot13 arr13 prod13 sum →\n prod13 (A Ty13 nat13 top13 bot13 arr13 prod13 sum) (B Ty13 nat13 top13 bot13 arr13 prod13 sum)\n\nsum13 : Ty13 → Ty13 → Ty13; sum13\n = λ A B Ty13 nat13 top13 bot13 arr13 prod13 sum13 →\n sum13 (A Ty13 nat13 top13 bot13 arr13 prod13 sum13) (B Ty13 nat13 top13 bot13 arr13 prod13 sum13)\n\nCon13 : Set; Con13\n = (Con13 : Set)\n (nil : Con13)\n (snoc : Con13 → Ty13 → Con13)\n → Con13\n\nnil13 : Con13; nil13\n = λ Con13 nil13 snoc → nil13\n\nsnoc13 : Con13 → Ty13 → Con13; snoc13\n = λ Γ A Con13 nil13 snoc13 → snoc13 (Γ Con13 nil13 snoc13) A\n\nVar13 : Con13 → Ty13 → Set; Var13\n = λ Γ A →\n (Var13 : Con13 → Ty13 → Set)\n (vz : ∀ Γ A → Var13 (snoc13 Γ A) A)\n (vs : ∀ Γ B A → Var13 Γ A → Var13 (snoc13 Γ B) A)\n → Var13 Γ A\n\nvz13 : ∀{Γ A} → Var13 (snoc13 Γ A) A; vz13\n = λ Var13 vz13 vs → vz13 _ _\n\nvs13 : ∀{Γ B A} → Var13 Γ A → Var13 (snoc13 Γ B) A; vs13\n = λ x Var13 vz13 vs13 → vs13 _ _ _ (x Var13 vz13 vs13)\n\nTm13 : Con13 → Ty13 → Set; Tm13\n = λ Γ A →\n (Tm13 : Con13 → Ty13 → Set)\n (var : ∀ Γ A → Var13 Γ A → Tm13 Γ A)\n (lam : ∀ Γ A B → Tm13 (snoc13 Γ A) B → Tm13 Γ (arr13 A B))\n (app : ∀ Γ A B → Tm13 Γ (arr13 A B) → Tm13 Γ A → Tm13 Γ B)\n (tt : ∀ Γ → Tm13 Γ top13)\n (pair : ∀ Γ A B → Tm13 Γ A → Tm13 Γ B → Tm13 Γ (prod13 A B))\n (fst : ∀ Γ A B → Tm13 Γ (prod13 A B) → Tm13 Γ A)\n (snd : ∀ Γ A B → Tm13 Γ (prod13 A B) → Tm13 Γ B)\n (left : ∀ Γ A B → Tm13 Γ A → Tm13 Γ (sum13 A B))\n (right : ∀ Γ A B → Tm13 Γ B → Tm13 Γ (sum13 A B))\n (case : ∀ Γ A B C → Tm13 Γ (sum13 A B) → Tm13 Γ (arr13 A C) → Tm13 Γ (arr13 B C) → Tm13 Γ C)\n (zero : ∀ Γ → Tm13 Γ nat13)\n (suc : ∀ Γ → Tm13 Γ nat13 → Tm13 Γ nat13)\n (rec : ∀ Γ A → Tm13 Γ nat13 → Tm13 Γ (arr13 nat13 (arr13 A A)) → Tm13 Γ A → Tm13 Γ A)\n → Tm13 Γ A\n\nvar13 : ∀{Γ A} → Var13 Γ A → Tm13 Γ A; var13\n = λ x Tm13 var13 lam app tt pair fst snd left right case zero suc rec →\n var13 _ _ x\n\nlam13 : ∀{Γ A B} → Tm13 (snoc13 Γ A) B → Tm13 Γ (arr13 A B); lam13\n = λ t Tm13 var13 lam13 app tt pair fst snd left right case zero suc rec →\n lam13 _ _ _ (t Tm13 var13 lam13 app tt pair fst snd left right case zero suc rec)\n\napp13 : ∀{Γ A B} → Tm13 Γ (arr13 A B) → Tm13 Γ A → Tm13 Γ B; app13\n = λ t u Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec →\n app13 _ _ _ (t Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec)\n (u Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec)\n\ntt13 : ∀{Γ} → Tm13 Γ top13; tt13\n = λ Tm13 var13 lam13 app13 tt13 pair fst snd left right case zero suc rec → tt13 _\n\npair13 : ∀{Γ A B} → Tm13 Γ A → Tm13 Γ B → Tm13 Γ (prod13 A B); pair13\n = λ t u Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec →\n pair13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec)\n (u Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec)\n\nfst13 : ∀{Γ A B} → Tm13 Γ (prod13 A B) → Tm13 Γ A; fst13\n = λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd left right case zero suc rec →\n fst13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd left right case zero suc rec)\n\nsnd13 : ∀{Γ A B} → Tm13 Γ (prod13 A B) → Tm13 Γ B; snd13\n = λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left right case zero suc rec →\n snd13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left right case zero suc rec)\n\nleft13 : ∀{Γ A B} → Tm13 Γ A → Tm13 Γ (sum13 A B); left13\n = λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right case zero suc rec →\n left13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right case zero suc rec)\n\nright13 : ∀{Γ A B} → Tm13 Γ B → Tm13 Γ (sum13 A B); right13\n = λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case zero suc rec →\n right13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case zero suc rec)\n\ncase13 : ∀{Γ A B C} → Tm13 Γ (sum13 A B) → Tm13 Γ (arr13 A C) → Tm13 Γ (arr13 B C) → Tm13 Γ C; case13\n = λ t u v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec →\n case13 _ _ _ _\n (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)\n (u Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)\n (v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)\n\nzero13 : ∀{Γ} → Tm13 Γ nat13; zero13\n = λ Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc rec → zero13 _\n\nsuc13 : ∀{Γ} → Tm13 Γ nat13 → Tm13 Γ nat13; suc13\n = λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec →\n suc13 _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec)\n\nrec13 : ∀{Γ A} → Tm13 Γ nat13 → Tm13 Γ (arr13 nat13 (arr13 A A)) → Tm13 Γ A → Tm13 Γ A; rec13\n = λ t u v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13 →\n rec13 _ _\n (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)\n (u Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)\n (v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)\n\nv013 : ∀{Γ A} → Tm13 (snoc13 Γ A) A; v013\n = var13 vz13\n\nv113 : ∀{Γ A B} → Tm13 (snoc13 (snoc13 Γ A) B) A; v113\n = var13 (vs13 vz13)\n\nv213 : ∀{Γ A B C} → Tm13 (snoc13 (snoc13 (snoc13 Γ A) B) C) A; v213\n = var13 (vs13 (vs13 vz13))\n\nv313 : ∀{Γ A B C D} → Tm13 (snoc13 (snoc13 (snoc13 (snoc13 Γ A) B) C) D) A; v313\n = var13 (vs13 (vs13 (vs13 vz13)))\n\ntbool13 : Ty13; tbool13\n = sum13 top13 top13\n\ntrue13 : ∀{Γ} → Tm13 Γ tbool13; true13\n = left13 tt13\n\ntfalse13 : ∀{Γ} → Tm13 Γ tbool13; tfalse13\n = right13 tt13\n\nifthenelse13 : ∀{Γ A} → Tm13 Γ (arr13 tbool13 (arr13 A (arr13 A A))); ifthenelse13\n = lam13 (lam13 (lam13 (case13 v213 (lam13 v213) (lam13 v113))))\n\ntimes413 : ∀{Γ A} → Tm13 Γ (arr13 (arr13 A A) (arr13 A A)); times413\n = lam13 (lam13 (app13 v113 (app13 v113 (app13 v113 (app13 v113 v013)))))\n\nadd13 : ∀{Γ} → Tm13 Γ (arr13 nat13 (arr13 nat13 nat13)); add13\n = lam13 (rec13 v013\n (lam13 (lam13 (lam13 (suc13 (app13 v113 v013)))))\n (lam13 v013))\n\nmul13 : ∀{Γ} → Tm13 Γ (arr13 nat13 (arr13 nat13 nat13)); mul13\n = lam13 (rec13 v013\n (lam13 (lam13 (lam13 (app13 (app13 add13 (app13 v113 v013)) v013))))\n (lam13 zero13))\n\nfact13 : ∀{Γ} → Tm13 Γ (arr13 nat13 nat13); fact13\n = lam13 (rec13 v013 (lam13 (lam13 (app13 (app13 mul13 (suc13 v113)) v013)))\n (suc13 zero13))\n{-# OPTIONS --type-in-type #-}\n\nTy14 : Set\nTy14 =\n (Ty14 : Set)\n (nat top bot : Ty14)\n (arr prod sum : Ty14 → Ty14 → Ty14)\n → Ty14\n\nnat14 : Ty14; nat14 = λ _ nat14 _ _ _ _ _ → nat14\ntop14 : Ty14; top14 = λ _ _ top14 _ _ _ _ → top14\nbot14 : Ty14; bot14 = λ _ _ _ bot14 _ _ _ → bot14\n\narr14 : Ty14 → Ty14 → Ty14; arr14\n = λ A B Ty14 nat14 top14 bot14 arr14 prod sum →\n arr14 (A Ty14 nat14 top14 bot14 arr14 prod sum) (B Ty14 nat14 top14 bot14 arr14 prod sum)\n\nprod14 : Ty14 → Ty14 → Ty14; prod14\n = λ A B Ty14 nat14 top14 bot14 arr14 prod14 sum →\n prod14 (A Ty14 nat14 top14 bot14 arr14 prod14 sum) (B Ty14 nat14 top14 bot14 arr14 prod14 sum)\n\nsum14 : Ty14 → Ty14 → Ty14; sum14\n = λ A B Ty14 nat14 top14 bot14 arr14 prod14 sum14 →\n sum14 (A Ty14 nat14 top14 bot14 arr14 prod14 sum14) (B Ty14 nat14 top14 bot14 arr14 prod14 sum14)\n\nCon14 : Set; Con14\n = (Con14 : Set)\n (nil : Con14)\n (snoc : Con14 → Ty14 → Con14)\n → Con14\n\nnil14 : Con14; nil14\n = λ Con14 nil14 snoc → nil14\n\nsnoc14 : Con14 → Ty14 → Con14; snoc14\n = λ Γ A Con14 nil14 snoc14 → snoc14 (Γ Con14 nil14 snoc14) A\n\nVar14 : Con14 → Ty14 → Set; Var14\n = λ Γ A →\n (Var14 : Con14 → Ty14 → Set)\n (vz : ∀ Γ A → Var14 (snoc14 Γ A) A)\n (vs : ∀ Γ B A → Var14 Γ A → Var14 (snoc14 Γ B) A)\n → Var14 Γ A\n\nvz14 : ∀{Γ A} → Var14 (snoc14 Γ A) A; vz14\n = λ Var14 vz14 vs → vz14 _ _\n\nvs14 : ∀{Γ B A} → Var14 Γ A → Var14 (snoc14 Γ B) A; vs14\n = λ x Var14 vz14 vs14 → vs14 _ _ _ (x Var14 vz14 vs14)\n\nTm14 : Con14 → Ty14 → Set; Tm14\n = λ Γ A →\n (Tm14 : Con14 → Ty14 → Set)\n (var : ∀ Γ A → Var14 Γ A → Tm14 Γ A)\n (lam : ∀ Γ A B → Tm14 (snoc14 Γ A) B → Tm14 Γ (arr14 A B))\n (app : ∀ Γ A B → Tm14 Γ (arr14 A B) → Tm14 Γ A → Tm14 Γ B)\n (tt : ∀ Γ → Tm14 Γ top14)\n (pair : ∀ Γ A B → Tm14 Γ A → Tm14 Γ B → Tm14 Γ (prod14 A B))\n (fst : ∀ Γ A B → Tm14 Γ (prod14 A B) → Tm14 Γ A)\n (snd : ∀ Γ A B → Tm14 Γ (prod14 A B) → Tm14 Γ B)\n (left : ∀ Γ A B → Tm14 Γ A → Tm14 Γ (sum14 A B))\n (right : ∀ Γ A B → Tm14 Γ B → Tm14 Γ (sum14 A B))\n (case : ∀ Γ A B C → Tm14 Γ (sum14 A B) → Tm14 Γ (arr14 A C) → Tm14 Γ (arr14 B C) → Tm14 Γ C)\n (zero : ∀ Γ → Tm14 Γ nat14)\n (suc : ∀ Γ → Tm14 Γ nat14 → Tm14 Γ nat14)\n (rec : ∀ Γ A → Tm14 Γ nat14 → Tm14 Γ (arr14 nat14 (arr14 A A)) → Tm14 Γ A → Tm14 Γ A)\n → Tm14 Γ A\n\nvar14 : ∀{Γ A} → Var14 Γ A → Tm14 Γ A; var14\n = λ x Tm14 var14 lam app tt pair fst snd left right case zero suc rec →\n var14 _ _ x\n\nlam14 : ∀{Γ A B} → Tm14 (snoc14 Γ A) B → Tm14 Γ (arr14 A B); lam14\n = λ t Tm14 var14 lam14 app tt pair fst snd left right case zero suc rec →\n lam14 _ _ _ (t Tm14 var14 lam14 app tt pair fst snd left right case zero suc rec)\n\napp14 : ∀{Γ A B} → Tm14 Γ (arr14 A B) → Tm14 Γ A → Tm14 Γ B; app14\n = λ t u Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec →\n app14 _ _ _ (t Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec)\n (u Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec)\n\ntt14 : ∀{Γ} → Tm14 Γ top14; tt14\n = λ Tm14 var14 lam14 app14 tt14 pair fst snd left right case zero suc rec → tt14 _\n\npair14 : ∀{Γ A B} → Tm14 Γ A → Tm14 Γ B → Tm14 Γ (prod14 A B); pair14\n = λ t u Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec →\n pair14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec)\n (u Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec)\n\nfst14 : ∀{Γ A B} → Tm14 Γ (prod14 A B) → Tm14 Γ A; fst14\n = λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd left right case zero suc rec →\n fst14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd left right case zero suc rec)\n\nsnd14 : ∀{Γ A B} → Tm14 Γ (prod14 A B) → Tm14 Γ B; snd14\n = λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left right case zero suc rec →\n snd14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left right case zero suc rec)\n\nleft14 : ∀{Γ A B} → Tm14 Γ A → Tm14 Γ (sum14 A B); left14\n = λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right case zero suc rec →\n left14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right case zero suc rec)\n\nright14 : ∀{Γ A B} → Tm14 Γ B → Tm14 Γ (sum14 A B); right14\n = λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case zero suc rec →\n right14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case zero suc rec)\n\ncase14 : ∀{Γ A B C} → Tm14 Γ (sum14 A B) → Tm14 Γ (arr14 A C) → Tm14 Γ (arr14 B C) → Tm14 Γ C; case14\n = λ t u v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec →\n case14 _ _ _ _\n (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)\n (u Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)\n (v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)\n\nzero14 : ∀{Γ} → Tm14 Γ nat14; zero14\n = λ Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc rec → zero14 _\n\nsuc14 : ∀{Γ} → Tm14 Γ nat14 → Tm14 Γ nat14; suc14\n = λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec →\n suc14 _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec)\n\nrec14 : ∀{Γ A} → Tm14 Γ nat14 → Tm14 Γ (arr14 nat14 (arr14 A A)) → Tm14 Γ A → Tm14 Γ A; rec14\n = λ t u v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14 →\n rec14 _ _\n (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)\n (u Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)\n (v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)\n\nv014 : ∀{Γ A} → Tm14 (snoc14 Γ A) A; v014\n = var14 vz14\n\nv114 : ∀{Γ A B} → Tm14 (snoc14 (snoc14 Γ A) B) A; v114\n = var14 (vs14 vz14)\n\nv214 : ∀{Γ A B C} → Tm14 (snoc14 (snoc14 (snoc14 Γ A) B) C) A; v214\n = var14 (vs14 (vs14 vz14))\n\nv314 : ∀{Γ A B C D} → Tm14 (snoc14 (snoc14 (snoc14 (snoc14 Γ A) B) C) D) A; v314\n = var14 (vs14 (vs14 (vs14 vz14)))\n\ntbool14 : Ty14; tbool14\n = sum14 top14 top14\n\ntrue14 : ∀{Γ} → Tm14 Γ tbool14; true14\n = left14 tt14\n\ntfalse14 : ∀{Γ} → Tm14 Γ tbool14; tfalse14\n = right14 tt14\n\nifthenelse14 : ∀{Γ A} → Tm14 Γ (arr14 tbool14 (arr14 A (arr14 A A))); ifthenelse14\n = lam14 (lam14 (lam14 (case14 v214 (lam14 v214) (lam14 v114))))\n\ntimes414 : ∀{Γ A} → Tm14 Γ (arr14 (arr14 A A) (arr14 A A)); times414\n = lam14 (lam14 (app14 v114 (app14 v114 (app14 v114 (app14 v114 v014)))))\n\nadd14 : ∀{Γ} → Tm14 Γ (arr14 nat14 (arr14 nat14 nat14)); add14\n = lam14 (rec14 v014\n (lam14 (lam14 (lam14 (suc14 (app14 v114 v014)))))\n (lam14 v014))\n\nmul14 : ∀{Γ} → Tm14 Γ (arr14 nat14 (arr14 nat14 nat14)); mul14\n = lam14 (rec14 v014\n (lam14 (lam14 (lam14 (app14 (app14 add14 (app14 v114 v014)) v014))))\n (lam14 zero14))\n\nfact14 : ∀{Γ} → Tm14 Γ (arr14 nat14 nat14); fact14\n = lam14 (rec14 v014 (lam14 (lam14 (app14 (app14 mul14 (suc14 v114)) v014)))\n (suc14 zero14))\n{-# OPTIONS --type-in-type #-}\n\nTy15 : Set\nTy15 =\n (Ty15 : Set)\n (nat top bot : Ty15)\n (arr prod sum : Ty15 → Ty15 → Ty15)\n → Ty15\n\nnat15 : Ty15; nat15 = λ _ nat15 _ _ _ _ _ → nat15\ntop15 : Ty15; top15 = λ _ _ top15 _ _ _ _ → top15\nbot15 : Ty15; bot15 = λ _ _ _ bot15 _ _ _ → bot15\n\narr15 : Ty15 → Ty15 → Ty15; arr15\n = λ A B Ty15 nat15 top15 bot15 arr15 prod sum →\n arr15 (A Ty15 nat15 top15 bot15 arr15 prod sum) (B Ty15 nat15 top15 bot15 arr15 prod sum)\n\nprod15 : Ty15 → Ty15 → Ty15; prod15\n = λ A B Ty15 nat15 top15 bot15 arr15 prod15 sum →\n prod15 (A Ty15 nat15 top15 bot15 arr15 prod15 sum) (B Ty15 nat15 top15 bot15 arr15 prod15 sum)\n\nsum15 : Ty15 → Ty15 → Ty15; sum15\n = λ A B Ty15 nat15 top15 bot15 arr15 prod15 sum15 →\n sum15 (A Ty15 nat15 top15 bot15 arr15 prod15 sum15) (B Ty15 nat15 top15 bot15 arr15 prod15 sum15)\n\nCon15 : Set; Con15\n = (Con15 : Set)\n (nil : Con15)\n (snoc : Con15 → Ty15 → Con15)\n → Con15\n\nnil15 : Con15; nil15\n = λ Con15 nil15 snoc → nil15\n\nsnoc15 : Con15 → Ty15 → Con15; snoc15\n = λ Γ A Con15 nil15 snoc15 → snoc15 (Γ Con15 nil15 snoc15) A\n\nVar15 : Con15 → Ty15 → Set; Var15\n = λ Γ A →\n (Var15 : Con15 → Ty15 → Set)\n (vz : ∀ Γ A → Var15 (snoc15 Γ A) A)\n (vs : ∀ Γ B A → Var15 Γ A → Var15 (snoc15 Γ B) A)\n → Var15 Γ A\n\nvz15 : ∀{Γ A} → Var15 (snoc15 Γ A) A; vz15\n = λ Var15 vz15 vs → vz15 _ _\n\nvs15 : ∀{Γ B A} → Var15 Γ A → Var15 (snoc15 Γ B) A; vs15\n = λ x Var15 vz15 vs15 → vs15 _ _ _ (x Var15 vz15 vs15)\n\nTm15 : Con15 → Ty15 → Set; Tm15\n = λ Γ A →\n (Tm15 : Con15 → Ty15 → Set)\n (var : ∀ Γ A → Var15 Γ A → Tm15 Γ A)\n (lam : ∀ Γ A B → Tm15 (snoc15 Γ A) B → Tm15 Γ (arr15 A B))\n (app : ∀ Γ A B → Tm15 Γ (arr15 A B) → Tm15 Γ A → Tm15 Γ B)\n (tt : ∀ Γ → Tm15 Γ top15)\n (pair : ∀ Γ A B → Tm15 Γ A → Tm15 Γ B → Tm15 Γ (prod15 A B))\n (fst : ∀ Γ A B → Tm15 Γ (prod15 A B) → Tm15 Γ A)\n (snd : ∀ Γ A B → Tm15 Γ (prod15 A B) → Tm15 Γ B)\n (left : ∀ Γ A B → Tm15 Γ A → Tm15 Γ (sum15 A B))\n (right : ∀ Γ A B → Tm15 Γ B → Tm15 Γ (sum15 A B))\n (case : ∀ Γ A B C → Tm15 Γ (sum15 A B) → Tm15 Γ (arr15 A C) → Tm15 Γ (arr15 B C) → Tm15 Γ C)\n (zero : ∀ Γ → Tm15 Γ nat15)\n (suc : ∀ Γ → Tm15 Γ nat15 → Tm15 Γ nat15)\n (rec : ∀ Γ A → Tm15 Γ nat15 → Tm15 Γ (arr15 nat15 (arr15 A A)) → Tm15 Γ A → Tm15 Γ A)\n → Tm15 Γ A\n\nvar15 : ∀{Γ A} → Var15 Γ A → Tm15 Γ A; var15\n = λ x Tm15 var15 lam app tt pair fst snd left right case zero suc rec →\n var15 _ _ x\n\nlam15 : ∀{Γ A B} → Tm15 (snoc15 Γ A) B → Tm15 Γ (arr15 A B); lam15\n = λ t Tm15 var15 lam15 app tt pair fst snd left right case zero suc rec →\n lam15 _ _ _ (t Tm15 var15 lam15 app tt pair fst snd left right case zero suc rec)\n\napp15 : ∀{Γ A B} → Tm15 Γ (arr15 A B) → Tm15 Γ A → Tm15 Γ B; app15\n = λ t u Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec →\n app15 _ _ _ (t Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec)\n (u Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec)\n\ntt15 : ∀{Γ} → Tm15 Γ top15; tt15\n = λ Tm15 var15 lam15 app15 tt15 pair fst snd left right case zero suc rec → tt15 _\n\npair15 : ∀{Γ A B} → Tm15 Γ A → Tm15 Γ B → Tm15 Γ (prod15 A B); pair15\n = λ t u Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec →\n pair15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec)\n (u Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec)\n\nfst15 : ∀{Γ A B} → Tm15 Γ (prod15 A B) → Tm15 Γ A; fst15\n = λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd left right case zero suc rec →\n fst15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd left right case zero suc rec)\n\nsnd15 : ∀{Γ A B} → Tm15 Γ (prod15 A B) → Tm15 Γ B; snd15\n = λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left right case zero suc rec →\n snd15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left right case zero suc rec)\n\nleft15 : ∀{Γ A B} → Tm15 Γ A → Tm15 Γ (sum15 A B); left15\n = λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right case zero suc rec →\n left15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right case zero suc rec)\n\nright15 : ∀{Γ A B} → Tm15 Γ B → Tm15 Γ (sum15 A B); right15\n = λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case zero suc rec →\n right15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case zero suc rec)\n\ncase15 : ∀{Γ A B C} → Tm15 Γ (sum15 A B) → Tm15 Γ (arr15 A C) → Tm15 Γ (arr15 B C) → Tm15 Γ C; case15\n = λ t u v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec →\n case15 _ _ _ _\n (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)\n (u Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)\n (v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)\n\nzero15 : ∀{Γ} → Tm15 Γ nat15; zero15\n = λ Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc rec → zero15 _\n\nsuc15 : ∀{Γ} → Tm15 Γ nat15 → Tm15 Γ nat15; suc15\n = λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec →\n suc15 _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec)\n\nrec15 : ∀{Γ A} → Tm15 Γ nat15 → Tm15 Γ (arr15 nat15 (arr15 A A)) → Tm15 Γ A → Tm15 Γ A; rec15\n = λ t u v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15 →\n rec15 _ _\n (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)\n (u Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)\n (v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)\n\nv015 : ∀{Γ A} → Tm15 (snoc15 Γ A) A; v015\n = var15 vz15\n\nv115 : ∀{Γ A B} → Tm15 (snoc15 (snoc15 Γ A) B) A; v115\n = var15 (vs15 vz15)\n\nv215 : ∀{Γ A B C} → Tm15 (snoc15 (snoc15 (snoc15 Γ A) B) C) A; v215\n = var15 (vs15 (vs15 vz15))\n\nv315 : ∀{Γ A B C D} → Tm15 (snoc15 (snoc15 (snoc15 (snoc15 Γ A) B) C) D) A; v315\n = var15 (vs15 (vs15 (vs15 vz15)))\n\ntbool15 : Ty15; tbool15\n = sum15 top15 top15\n\ntrue15 : ∀{Γ} → Tm15 Γ tbool15; true15\n = left15 tt15\n\ntfalse15 : ∀{Γ} → Tm15 Γ tbool15; tfalse15\n = right15 tt15\n\nifthenelse15 : ∀{Γ A} → Tm15 Γ (arr15 tbool15 (arr15 A (arr15 A A))); ifthenelse15\n = lam15 (lam15 (lam15 (case15 v215 (lam15 v215) (lam15 v115))))\n\ntimes415 : ∀{Γ A} → Tm15 Γ (arr15 (arr15 A A) (arr15 A A)); times415\n = lam15 (lam15 (app15 v115 (app15 v115 (app15 v115 (app15 v115 v015)))))\n\nadd15 : ∀{Γ} → Tm15 Γ (arr15 nat15 (arr15 nat15 nat15)); add15\n = lam15 (rec15 v015\n (lam15 (lam15 (lam15 (suc15 (app15 v115 v015)))))\n (lam15 v015))\n\nmul15 : ∀{Γ} → Tm15 Γ (arr15 nat15 (arr15 nat15 nat15)); mul15\n = lam15 (rec15 v015\n (lam15 (lam15 (lam15 (app15 (app15 add15 (app15 v115 v015)) v015))))\n (lam15 zero15))\n\nfact15 : ∀{Γ} → Tm15 Γ (arr15 nat15 nat15); fact15\n = lam15 (rec15 v015 (lam15 (lam15 (app15 (app15 mul15 (suc15 v115)) v015)))\n (suc15 zero15))\n{-# OPTIONS --type-in-type #-}\n\nTy16 : Set\nTy16 =\n (Ty16 : Set)\n (nat top bot : Ty16)\n (arr prod sum : Ty16 → Ty16 → Ty16)\n → Ty16\n\nnat16 : Ty16; nat16 = λ _ nat16 _ _ _ _ _ → nat16\ntop16 : Ty16; top16 = λ _ _ top16 _ _ _ _ → top16\nbot16 : Ty16; bot16 = λ _ _ _ bot16 _ _ _ → bot16\n\narr16 : Ty16 → Ty16 → Ty16; arr16\n = λ A B Ty16 nat16 top16 bot16 arr16 prod sum →\n arr16 (A Ty16 nat16 top16 bot16 arr16 prod sum) (B Ty16 nat16 top16 bot16 arr16 prod sum)\n\nprod16 : Ty16 → Ty16 → Ty16; prod16\n = λ A B Ty16 nat16 top16 bot16 arr16 prod16 sum →\n prod16 (A Ty16 nat16 top16 bot16 arr16 prod16 sum) (B Ty16 nat16 top16 bot16 arr16 prod16 sum)\n\nsum16 : Ty16 → Ty16 → Ty16; sum16\n = λ A B Ty16 nat16 top16 bot16 arr16 prod16 sum16 →\n sum16 (A Ty16 nat16 top16 bot16 arr16 prod16 sum16) (B Ty16 nat16 top16 bot16 arr16 prod16 sum16)\n\nCon16 : Set; Con16\n = (Con16 : Set)\n (nil : Con16)\n (snoc : Con16 → Ty16 → Con16)\n → Con16\n\nnil16 : Con16; nil16\n = λ Con16 nil16 snoc → nil16\n\nsnoc16 : Con16 → Ty16 → Con16; snoc16\n = λ Γ A Con16 nil16 snoc16 → snoc16 (Γ Con16 nil16 snoc16) A\n\nVar16 : Con16 → Ty16 → Set; Var16\n = λ Γ A →\n (Var16 : Con16 → Ty16 → Set)\n (vz : ∀ Γ A → Var16 (snoc16 Γ A) A)\n (vs : ∀ Γ B A → Var16 Γ A → Var16 (snoc16 Γ B) A)\n → Var16 Γ A\n\nvz16 : ∀{Γ A} → Var16 (snoc16 Γ A) A; vz16\n = λ Var16 vz16 vs → vz16 _ _\n\nvs16 : ∀{Γ B A} → Var16 Γ A → Var16 (snoc16 Γ B) A; vs16\n = λ x Var16 vz16 vs16 → vs16 _ _ _ (x Var16 vz16 vs16)\n\nTm16 : Con16 → Ty16 → Set; Tm16\n = λ Γ A →\n (Tm16 : Con16 → Ty16 → Set)\n (var : ∀ Γ A → Var16 Γ A → Tm16 Γ A)\n (lam : ∀ Γ A B → Tm16 (snoc16 Γ A) B → Tm16 Γ (arr16 A B))\n (app : ∀ Γ A B → Tm16 Γ (arr16 A B) → Tm16 Γ A → Tm16 Γ B)\n (tt : ∀ Γ → Tm16 Γ top16)\n (pair : ∀ Γ A B → Tm16 Γ A → Tm16 Γ B → Tm16 Γ (prod16 A B))\n (fst : ∀ Γ A B → Tm16 Γ (prod16 A B) → Tm16 Γ A)\n (snd : ∀ Γ A B → Tm16 Γ (prod16 A B) → Tm16 Γ B)\n (left : ∀ Γ A B → Tm16 Γ A → Tm16 Γ (sum16 A B))\n (right : ∀ Γ A B → Tm16 Γ B → Tm16 Γ (sum16 A B))\n (case : ∀ Γ A B C → Tm16 Γ (sum16 A B) → Tm16 Γ (arr16 A C) → Tm16 Γ (arr16 B C) → Tm16 Γ C)\n (zero : ∀ Γ → Tm16 Γ nat16)\n (suc : ∀ Γ → Tm16 Γ nat16 → Tm16 Γ nat16)\n (rec : ∀ Γ A → Tm16 Γ nat16 → Tm16 Γ (arr16 nat16 (arr16 A A)) → Tm16 Γ A → Tm16 Γ A)\n → Tm16 Γ A\n\nvar16 : ∀{Γ A} → Var16 Γ A → Tm16 Γ A; var16\n = λ x Tm16 var16 lam app tt pair fst snd left right case zero suc rec →\n var16 _ _ x\n\nlam16 : ∀{Γ A B} → Tm16 (snoc16 Γ A) B → Tm16 Γ (arr16 A B); lam16\n = λ t Tm16 var16 lam16 app tt pair fst snd left right case zero suc rec →\n lam16 _ _ _ (t Tm16 var16 lam16 app tt pair fst snd left right case zero suc rec)\n\napp16 : ∀{Γ A B} → Tm16 Γ (arr16 A B) → Tm16 Γ A → Tm16 Γ B; app16\n = λ t u Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec →\n app16 _ _ _ (t Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec)\n (u Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec)\n\ntt16 : ∀{Γ} → Tm16 Γ top16; tt16\n = λ Tm16 var16 lam16 app16 tt16 pair fst snd left right case zero suc rec → tt16 _\n\npair16 : ∀{Γ A B} → Tm16 Γ A → Tm16 Γ B → Tm16 Γ (prod16 A B); pair16\n = λ t u Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec →\n pair16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec)\n (u Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec)\n\nfst16 : ∀{Γ A B} → Tm16 Γ (prod16 A B) → Tm16 Γ A; fst16\n = λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd left right case zero suc rec →\n fst16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd left right case zero suc rec)\n\nsnd16 : ∀{Γ A B} → Tm16 Γ (prod16 A B) → Tm16 Γ B; snd16\n = λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left right case zero suc rec →\n snd16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left right case zero suc rec)\n\nleft16 : ∀{Γ A B} → Tm16 Γ A → Tm16 Γ (sum16 A B); left16\n = λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right case zero suc rec →\n left16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right case zero suc rec)\n\nright16 : ∀{Γ A B} → Tm16 Γ B → Tm16 Γ (sum16 A B); right16\n = λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case zero suc rec →\n right16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case zero suc rec)\n\ncase16 : ∀{Γ A B C} → Tm16 Γ (sum16 A B) → Tm16 Γ (arr16 A C) → Tm16 Γ (arr16 B C) → Tm16 Γ C; case16\n = λ t u v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec →\n case16 _ _ _ _\n (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)\n (u Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)\n (v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)\n\nzero16 : ∀{Γ} → Tm16 Γ nat16; zero16\n = λ Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc rec → zero16 _\n\nsuc16 : ∀{Γ} → Tm16 Γ nat16 → Tm16 Γ nat16; suc16\n = λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec →\n suc16 _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec)\n\nrec16 : ∀{Γ A} → Tm16 Γ nat16 → Tm16 Γ (arr16 nat16 (arr16 A A)) → Tm16 Γ A → Tm16 Γ A; rec16\n = λ t u v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16 →\n rec16 _ _\n (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)\n (u Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)\n (v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)\n\nv016 : ∀{Γ A} → Tm16 (snoc16 Γ A) A; v016\n = var16 vz16\n\nv116 : ∀{Γ A B} → Tm16 (snoc16 (snoc16 Γ A) B) A; v116\n = var16 (vs16 vz16)\n\nv216 : ∀{Γ A B C} → Tm16 (snoc16 (snoc16 (snoc16 Γ A) B) C) A; v216\n = var16 (vs16 (vs16 vz16))\n\nv316 : ∀{Γ A B C D} → Tm16 (snoc16 (snoc16 (snoc16 (snoc16 Γ A) B) C) D) A; v316\n = var16 (vs16 (vs16 (vs16 vz16)))\n\ntbool16 : Ty16; tbool16\n = sum16 top16 top16\n\ntrue16 : ∀{Γ} → Tm16 Γ tbool16; true16\n = left16 tt16\n\ntfalse16 : ∀{Γ} → Tm16 Γ tbool16; tfalse16\n = right16 tt16\n\nifthenelse16 : ∀{Γ A} → Tm16 Γ (arr16 tbool16 (arr16 A (arr16 A A))); ifthenelse16\n = lam16 (lam16 (lam16 (case16 v216 (lam16 v216) (lam16 v116))))\n\ntimes416 : ∀{Γ A} → Tm16 Γ (arr16 (arr16 A A) (arr16 A A)); times416\n = lam16 (lam16 (app16 v116 (app16 v116 (app16 v116 (app16 v116 v016)))))\n\nadd16 : ∀{Γ} → Tm16 Γ (arr16 nat16 (arr16 nat16 nat16)); add16\n = lam16 (rec16 v016\n (lam16 (lam16 (lam16 (suc16 (app16 v116 v016)))))\n (lam16 v016))\n\nmul16 : ∀{Γ} → Tm16 Γ (arr16 nat16 (arr16 nat16 nat16)); mul16\n = lam16 (rec16 v016\n (lam16 (lam16 (lam16 (app16 (app16 add16 (app16 v116 v016)) v016))))\n (lam16 zero16))\n\nfact16 : ∀{Γ} → Tm16 Γ (arr16 nat16 nat16); fact16\n = lam16 (rec16 v016 (lam16 (lam16 (app16 (app16 mul16 (suc16 v116)) v016)))\n (suc16 zero16))\n{-# OPTIONS --type-in-type #-}\n\nTy17 : Set\nTy17 =\n (Ty17 : Set)\n (nat top bot : Ty17)\n (arr prod sum : Ty17 → Ty17 → Ty17)\n → Ty17\n\nnat17 : Ty17; nat17 = λ _ nat17 _ _ _ _ _ → nat17\ntop17 : Ty17; top17 = λ _ _ top17 _ _ _ _ → top17\nbot17 : Ty17; bot17 = λ _ _ _ bot17 _ _ _ → bot17\n\narr17 : Ty17 → Ty17 → Ty17; arr17\n = λ A B Ty17 nat17 top17 bot17 arr17 prod sum →\n arr17 (A Ty17 nat17 top17 bot17 arr17 prod sum) (B Ty17 nat17 top17 bot17 arr17 prod sum)\n\nprod17 : Ty17 → Ty17 → Ty17; prod17\n = λ A B Ty17 nat17 top17 bot17 arr17 prod17 sum →\n prod17 (A Ty17 nat17 top17 bot17 arr17 prod17 sum) (B Ty17 nat17 top17 bot17 arr17 prod17 sum)\n\nsum17 : Ty17 → Ty17 → Ty17; sum17\n = λ A B Ty17 nat17 top17 bot17 arr17 prod17 sum17 →\n sum17 (A Ty17 nat17 top17 bot17 arr17 prod17 sum17) (B Ty17 nat17 top17 bot17 arr17 prod17 sum17)\n\nCon17 : Set; Con17\n = (Con17 : Set)\n (nil : Con17)\n (snoc : Con17 → Ty17 → Con17)\n → Con17\n\nnil17 : Con17; nil17\n = λ Con17 nil17 snoc → nil17\n\nsnoc17 : Con17 → Ty17 → Con17; snoc17\n = λ Γ A Con17 nil17 snoc17 → snoc17 (Γ Con17 nil17 snoc17) A\n\nVar17 : Con17 → Ty17 → Set; Var17\n = λ Γ A →\n (Var17 : Con17 → Ty17 → Set)\n (vz : ∀ Γ A → Var17 (snoc17 Γ A) A)\n (vs : ∀ Γ B A → Var17 Γ A → Var17 (snoc17 Γ B) A)\n → Var17 Γ A\n\nvz17 : ∀{Γ A} → Var17 (snoc17 Γ A) A; vz17\n = λ Var17 vz17 vs → vz17 _ _\n\nvs17 : ∀{Γ B A} → Var17 Γ A → Var17 (snoc17 Γ B) A; vs17\n = λ x Var17 vz17 vs17 → vs17 _ _ _ (x Var17 vz17 vs17)\n\nTm17 : Con17 → Ty17 → Set; Tm17\n = λ Γ A →\n (Tm17 : Con17 → Ty17 → Set)\n (var : ∀ Γ A → Var17 Γ A → Tm17 Γ A)\n (lam : ∀ Γ A B → Tm17 (snoc17 Γ A) B → Tm17 Γ (arr17 A B))\n (app : ∀ Γ A B → Tm17 Γ (arr17 A B) → Tm17 Γ A → Tm17 Γ B)\n (tt : ∀ Γ → Tm17 Γ top17)\n (pair : ∀ Γ A B → Tm17 Γ A → Tm17 Γ B → Tm17 Γ (prod17 A B))\n (fst : ∀ Γ A B → Tm17 Γ (prod17 A B) → Tm17 Γ A)\n (snd : ∀ Γ A B → Tm17 Γ (prod17 A B) → Tm17 Γ B)\n (left : ∀ Γ A B → Tm17 Γ A → Tm17 Γ (sum17 A B))\n (right : ∀ Γ A B → Tm17 Γ B → Tm17 Γ (sum17 A B))\n (case : ∀ Γ A B C → Tm17 Γ (sum17 A B) → Tm17 Γ (arr17 A C) → Tm17 Γ (arr17 B C) → Tm17 Γ C)\n (zero : ∀ Γ → Tm17 Γ nat17)\n (suc : ∀ Γ → Tm17 Γ nat17 → Tm17 Γ nat17)\n (rec : ∀ Γ A → Tm17 Γ nat17 → Tm17 Γ (arr17 nat17 (arr17 A A)) → Tm17 Γ A → Tm17 Γ A)\n → Tm17 Γ A\n\nvar17 : ∀{Γ A} → Var17 Γ A → Tm17 Γ A; var17\n = λ x Tm17 var17 lam app tt pair fst snd left right case zero suc rec →\n var17 _ _ x\n\nlam17 : ∀{Γ A B} → Tm17 (snoc17 Γ A) B → Tm17 Γ (arr17 A B); lam17\n = λ t Tm17 var17 lam17 app tt pair fst snd left right case zero suc rec →\n lam17 _ _ _ (t Tm17 var17 lam17 app tt pair fst snd left right case zero suc rec)\n\napp17 : ∀{Γ A B} → Tm17 Γ (arr17 A B) → Tm17 Γ A → Tm17 Γ B; app17\n = λ t u Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec →\n app17 _ _ _ (t Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec)\n (u Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec)\n\ntt17 : ∀{Γ} → Tm17 Γ top17; tt17\n = λ Tm17 var17 lam17 app17 tt17 pair fst snd left right case zero suc rec → tt17 _\n\npair17 : ∀{Γ A B} → Tm17 Γ A → Tm17 Γ B → Tm17 Γ (prod17 A B); pair17\n = λ t u Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec →\n pair17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec)\n (u Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec)\n\nfst17 : ∀{Γ A B} → Tm17 Γ (prod17 A B) → Tm17 Γ A; fst17\n = λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd left right case zero suc rec →\n fst17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd left right case zero suc rec)\n\nsnd17 : ∀{Γ A B} → Tm17 Γ (prod17 A B) → Tm17 Γ B; snd17\n = λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left right case zero suc rec →\n snd17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left right case zero suc rec)\n\nleft17 : ∀{Γ A B} → Tm17 Γ A → Tm17 Γ (sum17 A B); left17\n = λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right case zero suc rec →\n left17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right case zero suc rec)\n\nright17 : ∀{Γ A B} → Tm17 Γ B → Tm17 Γ (sum17 A B); right17\n = λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case zero suc rec →\n right17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case zero suc rec)\n\ncase17 : ∀{Γ A B C} → Tm17 Γ (sum17 A B) → Tm17 Γ (arr17 A C) → Tm17 Γ (arr17 B C) → Tm17 Γ C; case17\n = λ t u v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec →\n case17 _ _ _ _\n (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)\n (u Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)\n (v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)\n\nzero17 : ∀{Γ} → Tm17 Γ nat17; zero17\n = λ Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc rec → zero17 _\n\nsuc17 : ∀{Γ} → Tm17 Γ nat17 → Tm17 Γ nat17; suc17\n = λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec →\n suc17 _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec)\n\nrec17 : ∀{Γ A} → Tm17 Γ nat17 → Tm17 Γ (arr17 nat17 (arr17 A A)) → Tm17 Γ A → Tm17 Γ A; rec17\n = λ t u v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17 →\n rec17 _ _\n (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)\n (u Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)\n (v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)\n\nv017 : ∀{Γ A} → Tm17 (snoc17 Γ A) A; v017\n = var17 vz17\n\nv117 : ∀{Γ A B} → Tm17 (snoc17 (snoc17 Γ A) B) A; v117\n = var17 (vs17 vz17)\n\nv217 : ∀{Γ A B C} → Tm17 (snoc17 (snoc17 (snoc17 Γ A) B) C) A; v217\n = var17 (vs17 (vs17 vz17))\n\nv317 : ∀{Γ A B C D} → Tm17 (snoc17 (snoc17 (snoc17 (snoc17 Γ A) B) C) D) A; v317\n = var17 (vs17 (vs17 (vs17 vz17)))\n\ntbool17 : Ty17; tbool17\n = sum17 top17 top17\n\ntrue17 : ∀{Γ} → Tm17 Γ tbool17; true17\n = left17 tt17\n\ntfalse17 : ∀{Γ} → Tm17 Γ tbool17; tfalse17\n = right17 tt17\n\nifthenelse17 : ∀{Γ A} → Tm17 Γ (arr17 tbool17 (arr17 A (arr17 A A))); ifthenelse17\n = lam17 (lam17 (lam17 (case17 v217 (lam17 v217) (lam17 v117))))\n\ntimes417 : ∀{Γ A} → Tm17 Γ (arr17 (arr17 A A) (arr17 A A)); times417\n = lam17 (lam17 (app17 v117 (app17 v117 (app17 v117 (app17 v117 v017)))))\n\nadd17 : ∀{Γ} → Tm17 Γ (arr17 nat17 (arr17 nat17 nat17)); add17\n = lam17 (rec17 v017\n (lam17 (lam17 (lam17 (suc17 (app17 v117 v017)))))\n (lam17 v017))\n\nmul17 : ∀{Γ} → Tm17 Γ (arr17 nat17 (arr17 nat17 nat17)); mul17\n = lam17 (rec17 v017\n (lam17 (lam17 (lam17 (app17 (app17 add17 (app17 v117 v017)) v017))))\n (lam17 zero17))\n\nfact17 : ∀{Γ} → Tm17 Γ (arr17 nat17 nat17); fact17\n = lam17 (rec17 v017 (lam17 (lam17 (app17 (app17 mul17 (suc17 v117)) v017)))\n (suc17 zero17))\n{-# OPTIONS --type-in-type #-}\n\nTy18 : Set\nTy18 =\n (Ty18 : Set)\n (nat top bot : Ty18)\n (arr prod sum : Ty18 → Ty18 → Ty18)\n → Ty18\n\nnat18 : Ty18; nat18 = λ _ nat18 _ _ _ _ _ → nat18\ntop18 : Ty18; top18 = λ _ _ top18 _ _ _ _ → top18\nbot18 : Ty18; bot18 = λ _ _ _ bot18 _ _ _ → bot18\n\narr18 : Ty18 → Ty18 → Ty18; arr18\n = λ A B Ty18 nat18 top18 bot18 arr18 prod sum →\n arr18 (A Ty18 nat18 top18 bot18 arr18 prod sum) (B Ty18 nat18 top18 bot18 arr18 prod sum)\n\nprod18 : Ty18 → Ty18 → Ty18; prod18\n = λ A B Ty18 nat18 top18 bot18 arr18 prod18 sum →\n prod18 (A Ty18 nat18 top18 bot18 arr18 prod18 sum) (B Ty18 nat18 top18 bot18 arr18 prod18 sum)\n\nsum18 : Ty18 → Ty18 → Ty18; sum18\n = λ A B Ty18 nat18 top18 bot18 arr18 prod18 sum18 →\n sum18 (A Ty18 nat18 top18 bot18 arr18 prod18 sum18) (B Ty18 nat18 top18 bot18 arr18 prod18 sum18)\n\nCon18 : Set; Con18\n = (Con18 : Set)\n (nil : Con18)\n (snoc : Con18 → Ty18 → Con18)\n → Con18\n\nnil18 : Con18; nil18\n = λ Con18 nil18 snoc → nil18\n\nsnoc18 : Con18 → Ty18 → Con18; snoc18\n = λ Γ A Con18 nil18 snoc18 → snoc18 (Γ Con18 nil18 snoc18) A\n\nVar18 : Con18 → Ty18 → Set; Var18\n = λ Γ A →\n (Var18 : Con18 → Ty18 → Set)\n (vz : ∀ Γ A → Var18 (snoc18 Γ A) A)\n (vs : ∀ Γ B A → Var18 Γ A → Var18 (snoc18 Γ B) A)\n → Var18 Γ A\n\nvz18 : ∀{Γ A} → Var18 (snoc18 Γ A) A; vz18\n = λ Var18 vz18 vs → vz18 _ _\n\nvs18 : ∀{Γ B A} → Var18 Γ A → Var18 (snoc18 Γ B) A; vs18\n = λ x Var18 vz18 vs18 → vs18 _ _ _ (x Var18 vz18 vs18)\n\nTm18 : Con18 → Ty18 → Set; Tm18\n = λ Γ A →\n (Tm18 : Con18 → Ty18 → Set)\n (var : ∀ Γ A → Var18 Γ A → Tm18 Γ A)\n (lam : ∀ Γ A B → Tm18 (snoc18 Γ A) B → Tm18 Γ (arr18 A B))\n (app : ∀ Γ A B → Tm18 Γ (arr18 A B) → Tm18 Γ A → Tm18 Γ B)\n (tt : ∀ Γ → Tm18 Γ top18)\n (pair : ∀ Γ A B → Tm18 Γ A → Tm18 Γ B → Tm18 Γ (prod18 A B))\n (fst : ∀ Γ A B → Tm18 Γ (prod18 A B) → Tm18 Γ A)\n (snd : ∀ Γ A B → Tm18 Γ (prod18 A B) → Tm18 Γ B)\n (left : ∀ Γ A B → Tm18 Γ A → Tm18 Γ (sum18 A B))\n (right : ∀ Γ A B → Tm18 Γ B → Tm18 Γ (sum18 A B))\n (case : ∀ Γ A B C → Tm18 Γ (sum18 A B) → Tm18 Γ (arr18 A C) → Tm18 Γ (arr18 B C) → Tm18 Γ C)\n (zero : ∀ Γ → Tm18 Γ nat18)\n (suc : ∀ Γ → Tm18 Γ nat18 → Tm18 Γ nat18)\n (rec : ∀ Γ A → Tm18 Γ nat18 → Tm18 Γ (arr18 nat18 (arr18 A A)) → Tm18 Γ A → Tm18 Γ A)\n → Tm18 Γ A\n\nvar18 : ∀{Γ A} → Var18 Γ A → Tm18 Γ A; var18\n = λ x Tm18 var18 lam app tt pair fst snd left right case zero suc rec →\n var18 _ _ x\n\nlam18 : ∀{Γ A B} → Tm18 (snoc18 Γ A) B → Tm18 Γ (arr18 A B); lam18\n = λ t Tm18 var18 lam18 app tt pair fst snd left right case zero suc rec →\n lam18 _ _ _ (t Tm18 var18 lam18 app tt pair fst snd left right case zero suc rec)\n\napp18 : ∀{Γ A B} → Tm18 Γ (arr18 A B) → Tm18 Γ A → Tm18 Γ B; app18\n = λ t u Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec →\n app18 _ _ _ (t Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec)\n (u Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec)\n\ntt18 : ∀{Γ} → Tm18 Γ top18; tt18\n = λ Tm18 var18 lam18 app18 tt18 pair fst snd left right case zero suc rec → tt18 _\n\npair18 : ∀{Γ A B} → Tm18 Γ A → Tm18 Γ B → Tm18 Γ (prod18 A B); pair18\n = λ t u Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec →\n pair18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec)\n (u Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec)\n\nfst18 : ∀{Γ A B} → Tm18 Γ (prod18 A B) → Tm18 Γ A; fst18\n = λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd left right case zero suc rec →\n fst18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd left right case zero suc rec)\n\nsnd18 : ∀{Γ A B} → Tm18 Γ (prod18 A B) → Tm18 Γ B; snd18\n = λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left right case zero suc rec →\n snd18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left right case zero suc rec)\n\nleft18 : ∀{Γ A B} → Tm18 Γ A → Tm18 Γ (sum18 A B); left18\n = λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right case zero suc rec →\n left18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right case zero suc rec)\n\nright18 : ∀{Γ A B} → Tm18 Γ B → Tm18 Γ (sum18 A B); right18\n = λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case zero suc rec →\n right18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case zero suc rec)\n\ncase18 : ∀{Γ A B C} → Tm18 Γ (sum18 A B) → Tm18 Γ (arr18 A C) → Tm18 Γ (arr18 B C) → Tm18 Γ C; case18\n = λ t u v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec →\n case18 _ _ _ _\n (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)\n (u Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)\n (v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)\n\nzero18 : ∀{Γ} → Tm18 Γ nat18; zero18\n = λ Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc rec → zero18 _\n\nsuc18 : ∀{Γ} → Tm18 Γ nat18 → Tm18 Γ nat18; suc18\n = λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec →\n suc18 _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec)\n\nrec18 : ∀{Γ A} → Tm18 Γ nat18 → Tm18 Γ (arr18 nat18 (arr18 A A)) → Tm18 Γ A → Tm18 Γ A; rec18\n = λ t u v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18 →\n rec18 _ _\n (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)\n (u Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)\n (v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)\n\nv018 : ∀{Γ A} → Tm18 (snoc18 Γ A) A; v018\n = var18 vz18\n\nv118 : ∀{Γ A B} → Tm18 (snoc18 (snoc18 Γ A) B) A; v118\n = var18 (vs18 vz18)\n\nv218 : ∀{Γ A B C} → Tm18 (snoc18 (snoc18 (snoc18 Γ A) B) C) A; v218\n = var18 (vs18 (vs18 vz18))\n\nv318 : ∀{Γ A B C D} → Tm18 (snoc18 (snoc18 (snoc18 (snoc18 Γ A) B) C) D) A; v318\n = var18 (vs18 (vs18 (vs18 vz18)))\n\ntbool18 : Ty18; tbool18\n = sum18 top18 top18\n\ntrue18 : ∀{Γ} → Tm18 Γ tbool18; true18\n = left18 tt18\n\ntfalse18 : ∀{Γ} → Tm18 Γ tbool18; tfalse18\n = right18 tt18\n\nifthenelse18 : ∀{Γ A} → Tm18 Γ (arr18 tbool18 (arr18 A (arr18 A A))); ifthenelse18\n = lam18 (lam18 (lam18 (case18 v218 (lam18 v218) (lam18 v118))))\n\ntimes418 : ∀{Γ A} → Tm18 Γ (arr18 (arr18 A A) (arr18 A A)); times418\n = lam18 (lam18 (app18 v118 (app18 v118 (app18 v118 (app18 v118 v018)))))\n\nadd18 : ∀{Γ} → Tm18 Γ (arr18 nat18 (arr18 nat18 nat18)); add18\n = lam18 (rec18 v018\n (lam18 (lam18 (lam18 (suc18 (app18 v118 v018)))))\n (lam18 v018))\n\nmul18 : ∀{Γ} → Tm18 Γ (arr18 nat18 (arr18 nat18 nat18)); mul18\n = lam18 (rec18 v018\n (lam18 (lam18 (lam18 (app18 (app18 add18 (app18 v118 v018)) v018))))\n (lam18 zero18))\n\nfact18 : ∀{Γ} → Tm18 Γ (arr18 nat18 nat18); fact18\n = lam18 (rec18 v018 (lam18 (lam18 (app18 (app18 mul18 (suc18 v118)) v018)))\n (suc18 zero18))\n{-# OPTIONS --type-in-type #-}\n\nTy19 : Set\nTy19 =\n (Ty19 : Set)\n (nat top bot : Ty19)\n (arr prod sum : Ty19 → Ty19 → Ty19)\n → Ty19\n\nnat19 : Ty19; nat19 = λ _ nat19 _ _ _ _ _ → nat19\ntop19 : Ty19; top19 = λ _ _ top19 _ _ _ _ → top19\nbot19 : Ty19; bot19 = λ _ _ _ bot19 _ _ _ → bot19\n\narr19 : Ty19 → Ty19 → Ty19; arr19\n = λ A B Ty19 nat19 top19 bot19 arr19 prod sum →\n arr19 (A Ty19 nat19 top19 bot19 arr19 prod sum) (B Ty19 nat19 top19 bot19 arr19 prod sum)\n\nprod19 : Ty19 → Ty19 → Ty19; prod19\n = λ A B Ty19 nat19 top19 bot19 arr19 prod19 sum →\n prod19 (A Ty19 nat19 top19 bot19 arr19 prod19 sum) (B Ty19 nat19 top19 bot19 arr19 prod19 sum)\n\nsum19 : Ty19 → Ty19 → Ty19; sum19\n = λ A B Ty19 nat19 top19 bot19 arr19 prod19 sum19 →\n sum19 (A Ty19 nat19 top19 bot19 arr19 prod19 sum19) (B Ty19 nat19 top19 bot19 arr19 prod19 sum19)\n\nCon19 : Set; Con19\n = (Con19 : Set)\n (nil : Con19)\n (snoc : Con19 → Ty19 → Con19)\n → Con19\n\nnil19 : Con19; nil19\n = λ Con19 nil19 snoc → nil19\n\nsnoc19 : Con19 → Ty19 → Con19; snoc19\n = λ Γ A Con19 nil19 snoc19 → snoc19 (Γ Con19 nil19 snoc19) A\n\nVar19 : Con19 → Ty19 → Set; Var19\n = λ Γ A →\n (Var19 : Con19 → Ty19 → Set)\n (vz : ∀ Γ A → Var19 (snoc19 Γ A) A)\n (vs : ∀ Γ B A → Var19 Γ A → Var19 (snoc19 Γ B) A)\n → Var19 Γ A\n\nvz19 : ∀{Γ A} → Var19 (snoc19 Γ A) A; vz19\n = λ Var19 vz19 vs → vz19 _ _\n\nvs19 : ∀{Γ B A} → Var19 Γ A → Var19 (snoc19 Γ B) A; vs19\n = λ x Var19 vz19 vs19 → vs19 _ _ _ (x Var19 vz19 vs19)\n\nTm19 : Con19 → Ty19 → Set; Tm19\n = λ Γ A →\n (Tm19 : Con19 → Ty19 → Set)\n (var : ∀ Γ A → Var19 Γ A → Tm19 Γ A)\n (lam : ∀ Γ A B → Tm19 (snoc19 Γ A) B → Tm19 Γ (arr19 A B))\n (app : ∀ Γ A B → Tm19 Γ (arr19 A B) → Tm19 Γ A → Tm19 Γ B)\n (tt : ∀ Γ → Tm19 Γ top19)\n (pair : ∀ Γ A B → Tm19 Γ A → Tm19 Γ B → Tm19 Γ (prod19 A B))\n (fst : ∀ Γ A B → Tm19 Γ (prod19 A B) → Tm19 Γ A)\n (snd : ∀ Γ A B → Tm19 Γ (prod19 A B) → Tm19 Γ B)\n (left : ∀ Γ A B → Tm19 Γ A → Tm19 Γ (sum19 A B))\n (right : ∀ Γ A B → Tm19 Γ B → Tm19 Γ (sum19 A B))\n (case : ∀ Γ A B C → Tm19 Γ (sum19 A B) → Tm19 Γ (arr19 A C) → Tm19 Γ (arr19 B C) → Tm19 Γ C)\n (zero : ∀ Γ → Tm19 Γ nat19)\n (suc : ∀ Γ → Tm19 Γ nat19 → Tm19 Γ nat19)\n (rec : ∀ Γ A → Tm19 Γ nat19 → Tm19 Γ (arr19 nat19 (arr19 A A)) → Tm19 Γ A → Tm19 Γ A)\n → Tm19 Γ A\n\nvar19 : ∀{Γ A} → Var19 Γ A → Tm19 Γ A; var19\n = λ x Tm19 var19 lam app tt pair fst snd left right case zero suc rec →\n var19 _ _ x\n\nlam19 : ∀{Γ A B} → Tm19 (snoc19 Γ A) B → Tm19 Γ (arr19 A B); lam19\n = λ t Tm19 var19 lam19 app tt pair fst snd left right case zero suc rec →\n lam19 _ _ _ (t Tm19 var19 lam19 app tt pair fst snd left right case zero suc rec)\n\napp19 : ∀{Γ A B} → Tm19 Γ (arr19 A B) → Tm19 Γ A → Tm19 Γ B; app19\n = λ t u Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec →\n app19 _ _ _ (t Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec)\n (u Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec)\n\ntt19 : ∀{Γ} → Tm19 Γ top19; tt19\n = λ Tm19 var19 lam19 app19 tt19 pair fst snd left right case zero suc rec → tt19 _\n\npair19 : ∀{Γ A B} → Tm19 Γ A → Tm19 Γ B → Tm19 Γ (prod19 A B); pair19\n = λ t u Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec →\n pair19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec)\n (u Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec)\n\nfst19 : ∀{Γ A B} → Tm19 Γ (prod19 A B) → Tm19 Γ A; fst19\n = λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd left right case zero suc rec →\n fst19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd left right case zero suc rec)\n\nsnd19 : ∀{Γ A B} → Tm19 Γ (prod19 A B) → Tm19 Γ B; snd19\n = λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left right case zero suc rec →\n snd19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left right case zero suc rec)\n\nleft19 : ∀{Γ A B} → Tm19 Γ A → Tm19 Γ (sum19 A B); left19\n = λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right case zero suc rec →\n left19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right case zero suc rec)\n\nright19 : ∀{Γ A B} → Tm19 Γ B → Tm19 Γ (sum19 A B); right19\n = λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case zero suc rec →\n right19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case zero suc rec)\n\ncase19 : ∀{Γ A B C} → Tm19 Γ (sum19 A B) → Tm19 Γ (arr19 A C) → Tm19 Γ (arr19 B C) → Tm19 Γ C; case19\n = λ t u v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec →\n case19 _ _ _ _\n (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)\n (u Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)\n (v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)\n\nzero19 : ∀{Γ} → Tm19 Γ nat19; zero19\n = λ Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc rec → zero19 _\n\nsuc19 : ∀{Γ} → Tm19 Γ nat19 → Tm19 Γ nat19; suc19\n = λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec →\n suc19 _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec)\n\nrec19 : ∀{Γ A} → Tm19 Γ nat19 → Tm19 Γ (arr19 nat19 (arr19 A A)) → Tm19 Γ A → Tm19 Γ A; rec19\n = λ t u v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19 →\n rec19 _ _\n (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)\n (u Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)\n (v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)\n\nv019 : ∀{Γ A} → Tm19 (snoc19 Γ A) A; v019\n = var19 vz19\n\nv119 : ∀{Γ A B} → Tm19 (snoc19 (snoc19 Γ A) B) A; v119\n = var19 (vs19 vz19)\n\nv219 : ∀{Γ A B C} → Tm19 (snoc19 (snoc19 (snoc19 Γ A) B) C) A; v219\n = var19 (vs19 (vs19 vz19))\n\nv319 : ∀{Γ A B C D} → Tm19 (snoc19 (snoc19 (snoc19 (snoc19 Γ A) B) C) D) A; v319\n = var19 (vs19 (vs19 (vs19 vz19)))\n\ntbool19 : Ty19; tbool19\n = sum19 top19 top19\n\ntrue19 : ∀{Γ} → Tm19 Γ tbool19; true19\n = left19 tt19\n\ntfalse19 : ∀{Γ} → Tm19 Γ tbool19; tfalse19\n = right19 tt19\n\nifthenelse19 : ∀{Γ A} → Tm19 Γ (arr19 tbool19 (arr19 A (arr19 A A))); ifthenelse19\n = lam19 (lam19 (lam19 (case19 v219 (lam19 v219) (lam19 v119))))\n\ntimes419 : ∀{Γ A} → Tm19 Γ (arr19 (arr19 A A) (arr19 A A)); times419\n = lam19 (lam19 (app19 v119 (app19 v119 (app19 v119 (app19 v119 v019)))))\n\nadd19 : ∀{Γ} → Tm19 Γ (arr19 nat19 (arr19 nat19 nat19)); add19\n = lam19 (rec19 v019\n (lam19 (lam19 (lam19 (suc19 (app19 v119 v019)))))\n (lam19 v019))\n\nmul19 : ∀{Γ} → Tm19 Γ (arr19 nat19 (arr19 nat19 nat19)); mul19\n = lam19 (rec19 v019\n (lam19 (lam19 (lam19 (app19 (app19 add19 (app19 v119 v019)) v019))))\n (lam19 zero19))\n\nfact19 : ∀{Γ} → Tm19 Γ (arr19 nat19 nat19); fact19\n = lam19 (rec19 v019 (lam19 (lam19 (app19 (app19 mul19 (suc19 v119)) v019)))\n (suc19 zero19))\n{-# OPTIONS --type-in-type #-}\n\nTy20 : Set\nTy20 =\n (Ty20 : Set)\n (nat top bot : Ty20)\n (arr prod sum : Ty20 → Ty20 → Ty20)\n → Ty20\n\nnat20 : Ty20; nat20 = λ _ nat20 _ _ _ _ _ → nat20\ntop20 : Ty20; top20 = λ _ _ top20 _ _ _ _ → top20\nbot20 : Ty20; bot20 = λ _ _ _ bot20 _ _ _ → bot20\n\narr20 : Ty20 → Ty20 → Ty20; arr20\n = λ A B Ty20 nat20 top20 bot20 arr20 prod sum →\n arr20 (A Ty20 nat20 top20 bot20 arr20 prod sum) (B Ty20 nat20 top20 bot20 arr20 prod sum)\n\nprod20 : Ty20 → Ty20 → Ty20; prod20\n = λ A B Ty20 nat20 top20 bot20 arr20 prod20 sum →\n prod20 (A Ty20 nat20 top20 bot20 arr20 prod20 sum) (B Ty20 nat20 top20 bot20 arr20 prod20 sum)\n\nsum20 : Ty20 → Ty20 → Ty20; sum20\n = λ A B Ty20 nat20 top20 bot20 arr20 prod20 sum20 →\n sum20 (A Ty20 nat20 top20 bot20 arr20 prod20 sum20) (B Ty20 nat20 top20 bot20 arr20 prod20 sum20)\n\nCon20 : Set; Con20\n = (Con20 : Set)\n (nil : Con20)\n (snoc : Con20 → Ty20 → Con20)\n → Con20\n\nnil20 : Con20; nil20\n = λ Con20 nil20 snoc → nil20\n\nsnoc20 : Con20 → Ty20 → Con20; snoc20\n = λ Γ A Con20 nil20 snoc20 → snoc20 (Γ Con20 nil20 snoc20) A\n\nVar20 : Con20 → Ty20 → Set; Var20\n = λ Γ A →\n (Var20 : Con20 → Ty20 → Set)\n (vz : ∀ Γ A → Var20 (snoc20 Γ A) A)\n (vs : ∀ Γ B A → Var20 Γ A → Var20 (snoc20 Γ B) A)\n → Var20 Γ A\n\nvz20 : ∀{Γ A} → Var20 (snoc20 Γ A) A; vz20\n = λ Var20 vz20 vs → vz20 _ _\n\nvs20 : ∀{Γ B A} → Var20 Γ A → Var20 (snoc20 Γ B) A; vs20\n = λ x Var20 vz20 vs20 → vs20 _ _ _ (x Var20 vz20 vs20)\n\nTm20 : Con20 → Ty20 → Set; Tm20\n = λ Γ A →\n (Tm20 : Con20 → Ty20 → Set)\n (var : ∀ Γ A → Var20 Γ A → Tm20 Γ A)\n (lam : ∀ Γ A B → Tm20 (snoc20 Γ A) B → Tm20 Γ (arr20 A B))\n (app : ∀ Γ A B → Tm20 Γ (arr20 A B) → Tm20 Γ A → Tm20 Γ B)\n (tt : ∀ Γ → Tm20 Γ top20)\n (pair : ∀ Γ A B → Tm20 Γ A → Tm20 Γ B → Tm20 Γ (prod20 A B))\n (fst : ∀ Γ A B → Tm20 Γ (prod20 A B) → Tm20 Γ A)\n (snd : ∀ Γ A B → Tm20 Γ (prod20 A B) → Tm20 Γ B)\n (left : ∀ Γ A B → Tm20 Γ A → Tm20 Γ (sum20 A B))\n (right : ∀ Γ A B → Tm20 Γ B → Tm20 Γ (sum20 A B))\n (case : ∀ Γ A B C → Tm20 Γ (sum20 A B) → Tm20 Γ (arr20 A C) → Tm20 Γ (arr20 B C) → Tm20 Γ C)\n (zero : ∀ Γ → Tm20 Γ nat20)\n (suc : ∀ Γ → Tm20 Γ nat20 → Tm20 Γ nat20)\n (rec : ∀ Γ A → Tm20 Γ nat20 → Tm20 Γ (arr20 nat20 (arr20 A A)) → Tm20 Γ A → Tm20 Γ A)\n → Tm20 Γ A\n\nvar20 : ∀{Γ A} → Var20 Γ A → Tm20 Γ A; var20\n = λ x Tm20 var20 lam app tt pair fst snd left right case zero suc rec →\n var20 _ _ x\n\nlam20 : ∀{Γ A B} → Tm20 (snoc20 Γ A) B → Tm20 Γ (arr20 A B); lam20\n = λ t Tm20 var20 lam20 app tt pair fst snd left right case zero suc rec →\n lam20 _ _ _ (t Tm20 var20 lam20 app tt pair fst snd left right case zero suc rec)\n\napp20 : ∀{Γ A B} → Tm20 Γ (arr20 A B) → Tm20 Γ A → Tm20 Γ B; app20\n = λ t u Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec →\n app20 _ _ _ (t Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec)\n (u Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec)\n\ntt20 : ∀{Γ} → Tm20 Γ top20; tt20\n = λ Tm20 var20 lam20 app20 tt20 pair fst snd left right case zero suc rec → tt20 _\n\npair20 : ∀{Γ A B} → Tm20 Γ A → Tm20 Γ B → Tm20 Γ (prod20 A B); pair20\n = λ t u Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec →\n pair20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec)\n (u Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec)\n\nfst20 : ∀{Γ A B} → Tm20 Γ (prod20 A B) → Tm20 Γ A; fst20\n = λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd left right case zero suc rec →\n fst20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd left right case zero suc rec)\n\nsnd20 : ∀{Γ A B} → Tm20 Γ (prod20 A B) → Tm20 Γ B; snd20\n = λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left right case zero suc rec →\n snd20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left right case zero suc rec)\n\nleft20 : ∀{Γ A B} → Tm20 Γ A → Tm20 Γ (sum20 A B); left20\n = λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right case zero suc rec →\n left20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right case zero suc rec)\n\nright20 : ∀{Γ A B} → Tm20 Γ B → Tm20 Γ (sum20 A B); right20\n = λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case zero suc rec →\n right20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case zero suc rec)\n\ncase20 : ∀{Γ A B C} → Tm20 Γ (sum20 A B) → Tm20 Γ (arr20 A C) → Tm20 Γ (arr20 B C) → Tm20 Γ C; case20\n = λ t u v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec →\n case20 _ _ _ _\n (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)\n (u Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)\n (v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)\n\nzero20 : ∀{Γ} → Tm20 Γ nat20; zero20\n = λ Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc rec → zero20 _\n\nsuc20 : ∀{Γ} → Tm20 Γ nat20 → Tm20 Γ nat20; suc20\n = λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec →\n suc20 _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec)\n\nrec20 : ∀{Γ A} → Tm20 Γ nat20 → Tm20 Γ (arr20 nat20 (arr20 A A)) → Tm20 Γ A → Tm20 Γ A; rec20\n = λ t u v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20 →\n rec20 _ _\n (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)\n (u Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)\n (v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)\n\nv020 : ∀{Γ A} → Tm20 (snoc20 Γ A) A; v020\n = var20 vz20\n\nv120 : ∀{Γ A B} → Tm20 (snoc20 (snoc20 Γ A) B) A; v120\n = var20 (vs20 vz20)\n\nv220 : ∀{Γ A B C} → Tm20 (snoc20 (snoc20 (snoc20 Γ A) B) C) A; v220\n = var20 (vs20 (vs20 vz20))\n\nv320 : ∀{Γ A B C D} → Tm20 (snoc20 (snoc20 (snoc20 (snoc20 Γ A) B) C) D) A; v320\n = var20 (vs20 (vs20 (vs20 vz20)))\n\ntbool20 : Ty20; tbool20\n = sum20 top20 top20\n\ntrue20 : ∀{Γ} → Tm20 Γ tbool20; true20\n = left20 tt20\n\ntfalse20 : ∀{Γ} → Tm20 Γ tbool20; tfalse20\n = right20 tt20\n\nifthenelse20 : ∀{Γ A} → Tm20 Γ (arr20 tbool20 (arr20 A (arr20 A A))); ifthenelse20\n = lam20 (lam20 (lam20 (case20 v220 (lam20 v220) (lam20 v120))))\n\ntimes420 : ∀{Γ A} → Tm20 Γ (arr20 (arr20 A A) (arr20 A A)); times420\n = lam20 (lam20 (app20 v120 (app20 v120 (app20 v120 (app20 v120 v020)))))\n\nadd20 : ∀{Γ} → Tm20 Γ (arr20 nat20 (arr20 nat20 nat20)); add20\n = lam20 (rec20 v020\n (lam20 (lam20 (lam20 (suc20 (app20 v120 v020)))))\n (lam20 v020))\n\nmul20 : ∀{Γ} → Tm20 Γ (arr20 nat20 (arr20 nat20 nat20)); mul20\n = lam20 (rec20 v020\n (lam20 (lam20 (lam20 (app20 (app20 add20 (app20 v120 v020)) v020))))\n (lam20 zero20))\n\nfact20 : ∀{Γ} → Tm20 Γ (arr20 nat20 nat20); fact20\n = lam20 (rec20 v020 (lam20 (lam20 (app20 (app20 mul20 (suc20 v120)) v020)))\n (suc20 zero20))\n{-# OPTIONS --type-in-type #-}\n\nTy21 : Set\nTy21 =\n (Ty21 : Set)\n (nat top bot : Ty21)\n (arr prod sum : Ty21 → Ty21 → Ty21)\n → Ty21\n\nnat21 : Ty21; nat21 = λ _ nat21 _ _ _ _ _ → nat21\ntop21 : Ty21; top21 = λ _ _ top21 _ _ _ _ → top21\nbot21 : Ty21; bot21 = λ _ _ _ bot21 _ _ _ → bot21\n\narr21 : Ty21 → Ty21 → Ty21; arr21\n = λ A B Ty21 nat21 top21 bot21 arr21 prod sum →\n arr21 (A Ty21 nat21 top21 bot21 arr21 prod sum) (B Ty21 nat21 top21 bot21 arr21 prod sum)\n\nprod21 : Ty21 → Ty21 → Ty21; prod21\n = λ A B Ty21 nat21 top21 bot21 arr21 prod21 sum →\n prod21 (A Ty21 nat21 top21 bot21 arr21 prod21 sum) (B Ty21 nat21 top21 bot21 arr21 prod21 sum)\n\nsum21 : Ty21 → Ty21 → Ty21; sum21\n = λ A B Ty21 nat21 top21 bot21 arr21 prod21 sum21 →\n sum21 (A Ty21 nat21 top21 bot21 arr21 prod21 sum21) (B Ty21 nat21 top21 bot21 arr21 prod21 sum21)\n\nCon21 : Set; Con21\n = (Con21 : Set)\n (nil : Con21)\n (snoc : Con21 → Ty21 → Con21)\n → Con21\n\nnil21 : Con21; nil21\n = λ Con21 nil21 snoc → nil21\n\nsnoc21 : Con21 → Ty21 → Con21; snoc21\n = λ Γ A Con21 nil21 snoc21 → snoc21 (Γ Con21 nil21 snoc21) A\n\nVar21 : Con21 → Ty21 → Set; Var21\n = λ Γ A →\n (Var21 : Con21 → Ty21 → Set)\n (vz : ∀ Γ A → Var21 (snoc21 Γ A) A)\n (vs : ∀ Γ B A → Var21 Γ A → Var21 (snoc21 Γ B) A)\n → Var21 Γ A\n\nvz21 : ∀{Γ A} → Var21 (snoc21 Γ A) A; vz21\n = λ Var21 vz21 vs → vz21 _ _\n\nvs21 : ∀{Γ B A} → Var21 Γ A → Var21 (snoc21 Γ B) A; vs21\n = λ x Var21 vz21 vs21 → vs21 _ _ _ (x Var21 vz21 vs21)\n\nTm21 : Con21 → Ty21 → Set; Tm21\n = λ Γ A →\n (Tm21 : Con21 → Ty21 → Set)\n (var : ∀ Γ A → Var21 Γ A → Tm21 Γ A)\n (lam : ∀ Γ A B → Tm21 (snoc21 Γ A) B → Tm21 Γ (arr21 A B))\n (app : ∀ Γ A B → Tm21 Γ (arr21 A B) → Tm21 Γ A → Tm21 Γ B)\n (tt : ∀ Γ → Tm21 Γ top21)\n (pair : ∀ Γ A B → Tm21 Γ A → Tm21 Γ B → Tm21 Γ (prod21 A B))\n (fst : ∀ Γ A B → Tm21 Γ (prod21 A B) → Tm21 Γ A)\n (snd : ∀ Γ A B → Tm21 Γ (prod21 A B) → Tm21 Γ B)\n (left : ∀ Γ A B → Tm21 Γ A → Tm21 Γ (sum21 A B))\n (right : ∀ Γ A B → Tm21 Γ B → Tm21 Γ (sum21 A B))\n (case : ∀ Γ A B C → Tm21 Γ (sum21 A B) → Tm21 Γ (arr21 A C) → Tm21 Γ (arr21 B C) → Tm21 Γ C)\n (zero : ∀ Γ → Tm21 Γ nat21)\n (suc : ∀ Γ → Tm21 Γ nat21 → Tm21 Γ nat21)\n (rec : ∀ Γ A → Tm21 Γ nat21 → Tm21 Γ (arr21 nat21 (arr21 A A)) → Tm21 Γ A → Tm21 Γ A)\n → Tm21 Γ A\n\nvar21 : ∀{Γ A} → Var21 Γ A → Tm21 Γ A; var21\n = λ x Tm21 var21 lam app tt pair fst snd left right case zero suc rec →\n var21 _ _ x\n\nlam21 : ∀{Γ A B} → Tm21 (snoc21 Γ A) B → Tm21 Γ (arr21 A B); lam21\n = λ t Tm21 var21 lam21 app tt pair fst snd left right case zero suc rec →\n lam21 _ _ _ (t Tm21 var21 lam21 app tt pair fst snd left right case zero suc rec)\n\napp21 : ∀{Γ A B} → Tm21 Γ (arr21 A B) → Tm21 Γ A → Tm21 Γ B; app21\n = λ t u Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec →\n app21 _ _ _ (t Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec)\n (u Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec)\n\ntt21 : ∀{Γ} → Tm21 Γ top21; tt21\n = λ Tm21 var21 lam21 app21 tt21 pair fst snd left right case zero suc rec → tt21 _\n\npair21 : ∀{Γ A B} → Tm21 Γ A → Tm21 Γ B → Tm21 Γ (prod21 A B); pair21\n = λ t u Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec →\n pair21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec)\n (u Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec)\n\nfst21 : ∀{Γ A B} → Tm21 Γ (prod21 A B) → Tm21 Γ A; fst21\n = λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd left right case zero suc rec →\n fst21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd left right case zero suc rec)\n\nsnd21 : ∀{Γ A B} → Tm21 Γ (prod21 A B) → Tm21 Γ B; snd21\n = λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left right case zero suc rec →\n snd21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left right case zero suc rec)\n\nleft21 : ∀{Γ A B} → Tm21 Γ A → Tm21 Γ (sum21 A B); left21\n = λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right case zero suc rec →\n left21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right case zero suc rec)\n\nright21 : ∀{Γ A B} → Tm21 Γ B → Tm21 Γ (sum21 A B); right21\n = λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case zero suc rec →\n right21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case zero suc rec)\n\ncase21 : ∀{Γ A B C} → Tm21 Γ (sum21 A B) → Tm21 Γ (arr21 A C) → Tm21 Γ (arr21 B C) → Tm21 Γ C; case21\n = λ t u v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec →\n case21 _ _ _ _\n (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)\n (u Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)\n (v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)\n\nzero21 : ∀{Γ} → Tm21 Γ nat21; zero21\n = λ Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc rec → zero21 _\n\nsuc21 : ∀{Γ} → Tm21 Γ nat21 → Tm21 Γ nat21; suc21\n = λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec →\n suc21 _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec)\n\nrec21 : ∀{Γ A} → Tm21 Γ nat21 → Tm21 Γ (arr21 nat21 (arr21 A A)) → Tm21 Γ A → Tm21 Γ A; rec21\n = λ t u v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21 →\n rec21 _ _\n (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)\n (u Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)\n (v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)\n\nv021 : ∀{Γ A} → Tm21 (snoc21 Γ A) A; v021\n = var21 vz21\n\nv121 : ∀{Γ A B} → Tm21 (snoc21 (snoc21 Γ A) B) A; v121\n = var21 (vs21 vz21)\n\nv221 : ∀{Γ A B C} → Tm21 (snoc21 (snoc21 (snoc21 Γ A) B) C) A; v221\n = var21 (vs21 (vs21 vz21))\n\nv321 : ∀{Γ A B C D} → Tm21 (snoc21 (snoc21 (snoc21 (snoc21 Γ A) B) C) D) A; v321\n = var21 (vs21 (vs21 (vs21 vz21)))\n\ntbool21 : Ty21; tbool21\n = sum21 top21 top21\n\ntrue21 : ∀{Γ} → Tm21 Γ tbool21; true21\n = left21 tt21\n\ntfalse21 : ∀{Γ} → Tm21 Γ tbool21; tfalse21\n = right21 tt21\n\nifthenelse21 : ∀{Γ A} → Tm21 Γ (arr21 tbool21 (arr21 A (arr21 A A))); ifthenelse21\n = lam21 (lam21 (lam21 (case21 v221 (lam21 v221) (lam21 v121))))\n\ntimes421 : ∀{Γ A} → Tm21 Γ (arr21 (arr21 A A) (arr21 A A)); times421\n = lam21 (lam21 (app21 v121 (app21 v121 (app21 v121 (app21 v121 v021)))))\n\nadd21 : ∀{Γ} → Tm21 Γ (arr21 nat21 (arr21 nat21 nat21)); add21\n = lam21 (rec21 v021\n (lam21 (lam21 (lam21 (suc21 (app21 v121 v021)))))\n (lam21 v021))\n\nmul21 : ∀{Γ} → Tm21 Γ (arr21 nat21 (arr21 nat21 nat21)); mul21\n = lam21 (rec21 v021\n (lam21 (lam21 (lam21 (app21 (app21 add21 (app21 v121 v021)) v021))))\n (lam21 zero21))\n\nfact21 : ∀{Γ} → Tm21 Γ (arr21 nat21 nat21); fact21\n = lam21 (rec21 v021 (lam21 (lam21 (app21 (app21 mul21 (suc21 v121)) v021)))\n (suc21 zero21))\n{-# OPTIONS --type-in-type #-}\n\nTy22 : Set\nTy22 =\n (Ty22 : Set)\n (nat top bot : Ty22)\n (arr prod sum : Ty22 → Ty22 → Ty22)\n → Ty22\n\nnat22 : Ty22; nat22 = λ _ nat22 _ _ _ _ _ → nat22\ntop22 : Ty22; top22 = λ _ _ top22 _ _ _ _ → top22\nbot22 : Ty22; bot22 = λ _ _ _ bot22 _ _ _ → bot22\n\narr22 : Ty22 → Ty22 → Ty22; arr22\n = λ A B Ty22 nat22 top22 bot22 arr22 prod sum →\n arr22 (A Ty22 nat22 top22 bot22 arr22 prod sum) (B Ty22 nat22 top22 bot22 arr22 prod sum)\n\nprod22 : Ty22 → Ty22 → Ty22; prod22\n = λ A B Ty22 nat22 top22 bot22 arr22 prod22 sum →\n prod22 (A Ty22 nat22 top22 bot22 arr22 prod22 sum) (B Ty22 nat22 top22 bot22 arr22 prod22 sum)\n\nsum22 : Ty22 → Ty22 → Ty22; sum22\n = λ A B Ty22 nat22 top22 bot22 arr22 prod22 sum22 →\n sum22 (A Ty22 nat22 top22 bot22 arr22 prod22 sum22) (B Ty22 nat22 top22 bot22 arr22 prod22 sum22)\n\nCon22 : Set; Con22\n = (Con22 : Set)\n (nil : Con22)\n (snoc : Con22 → Ty22 → Con22)\n → Con22\n\nnil22 : Con22; nil22\n = λ Con22 nil22 snoc → nil22\n\nsnoc22 : Con22 → Ty22 → Con22; snoc22\n = λ Γ A Con22 nil22 snoc22 → snoc22 (Γ Con22 nil22 snoc22) A\n\nVar22 : Con22 → Ty22 → Set; Var22\n = λ Γ A →\n (Var22 : Con22 → Ty22 → Set)\n (vz : ∀ Γ A → Var22 (snoc22 Γ A) A)\n (vs : ∀ Γ B A → Var22 Γ A → Var22 (snoc22 Γ B) A)\n → Var22 Γ A\n\nvz22 : ∀{Γ A} → Var22 (snoc22 Γ A) A; vz22\n = λ Var22 vz22 vs → vz22 _ _\n\nvs22 : ∀{Γ B A} → Var22 Γ A → Var22 (snoc22 Γ B) A; vs22\n = λ x Var22 vz22 vs22 → vs22 _ _ _ (x Var22 vz22 vs22)\n\nTm22 : Con22 → Ty22 → Set; Tm22\n = λ Γ A →\n (Tm22 : Con22 → Ty22 → Set)\n (var : ∀ Γ A → Var22 Γ A → Tm22 Γ A)\n (lam : ∀ Γ A B → Tm22 (snoc22 Γ A) B → Tm22 Γ (arr22 A B))\n (app : ∀ Γ A B → Tm22 Γ (arr22 A B) → Tm22 Γ A → Tm22 Γ B)\n (tt : ∀ Γ → Tm22 Γ top22)\n (pair : ∀ Γ A B → Tm22 Γ A → Tm22 Γ B → Tm22 Γ (prod22 A B))\n (fst : ∀ Γ A B → Tm22 Γ (prod22 A B) → Tm22 Γ A)\n (snd : ∀ Γ A B → Tm22 Γ (prod22 A B) → Tm22 Γ B)\n (left : ∀ Γ A B → Tm22 Γ A → Tm22 Γ (sum22 A B))\n (right : ∀ Γ A B → Tm22 Γ B → Tm22 Γ (sum22 A B))\n (case : ∀ Γ A B C → Tm22 Γ (sum22 A B) → Tm22 Γ (arr22 A C) → Tm22 Γ (arr22 B C) → Tm22 Γ C)\n (zero : ∀ Γ → Tm22 Γ nat22)\n (suc : ∀ Γ → Tm22 Γ nat22 → Tm22 Γ nat22)\n (rec : ∀ Γ A → Tm22 Γ nat22 → Tm22 Γ (arr22 nat22 (arr22 A A)) → Tm22 Γ A → Tm22 Γ A)\n → Tm22 Γ A\n\nvar22 : ∀{Γ A} → Var22 Γ A → Tm22 Γ A; var22\n = λ x Tm22 var22 lam app tt pair fst snd left right case zero suc rec →\n var22 _ _ x\n\nlam22 : ∀{Γ A B} → Tm22 (snoc22 Γ A) B → Tm22 Γ (arr22 A B); lam22\n = λ t Tm22 var22 lam22 app tt pair fst snd left right case zero suc rec →\n lam22 _ _ _ (t Tm22 var22 lam22 app tt pair fst snd left right case zero suc rec)\n\napp22 : ∀{Γ A B} → Tm22 Γ (arr22 A B) → Tm22 Γ A → Tm22 Γ B; app22\n = λ t u Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec →\n app22 _ _ _ (t Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec)\n (u Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec)\n\ntt22 : ∀{Γ} → Tm22 Γ top22; tt22\n = λ Tm22 var22 lam22 app22 tt22 pair fst snd left right case zero suc rec → tt22 _\n\npair22 : ∀{Γ A B} → Tm22 Γ A → Tm22 Γ B → Tm22 Γ (prod22 A B); pair22\n = λ t u Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec →\n pair22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec)\n (u Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec)\n\nfst22 : ∀{Γ A B} → Tm22 Γ (prod22 A B) → Tm22 Γ A; fst22\n = λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd left right case zero suc rec →\n fst22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd left right case zero suc rec)\n\nsnd22 : ∀{Γ A B} → Tm22 Γ (prod22 A B) → Tm22 Γ B; snd22\n = λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left right case zero suc rec →\n snd22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left right case zero suc rec)\n\nleft22 : ∀{Γ A B} → Tm22 Γ A → Tm22 Γ (sum22 A B); left22\n = λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right case zero suc rec →\n left22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right case zero suc rec)\n\nright22 : ∀{Γ A B} → Tm22 Γ B → Tm22 Γ (sum22 A B); right22\n = λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case zero suc rec →\n right22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case zero suc rec)\n\ncase22 : ∀{Γ A B C} → Tm22 Γ (sum22 A B) → Tm22 Γ (arr22 A C) → Tm22 Γ (arr22 B C) → Tm22 Γ C; case22\n = λ t u v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec →\n case22 _ _ _ _\n (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)\n (u Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)\n (v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)\n\nzero22 : ∀{Γ} → Tm22 Γ nat22; zero22\n = λ Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc rec → zero22 _\n\nsuc22 : ∀{Γ} → Tm22 Γ nat22 → Tm22 Γ nat22; suc22\n = λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec →\n suc22 _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec)\n\nrec22 : ∀{Γ A} → Tm22 Γ nat22 → Tm22 Γ (arr22 nat22 (arr22 A A)) → Tm22 Γ A → Tm22 Γ A; rec22\n = λ t u v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22 →\n rec22 _ _\n (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)\n (u Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)\n (v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)\n\nv022 : ∀{Γ A} → Tm22 (snoc22 Γ A) A; v022\n = var22 vz22\n\nv122 : ∀{Γ A B} → Tm22 (snoc22 (snoc22 Γ A) B) A; v122\n = var22 (vs22 vz22)\n\nv222 : ∀{Γ A B C} → Tm22 (snoc22 (snoc22 (snoc22 Γ A) B) C) A; v222\n = var22 (vs22 (vs22 vz22))\n\nv322 : ∀{Γ A B C D} → Tm22 (snoc22 (snoc22 (snoc22 (snoc22 Γ A) B) C) D) A; v322\n = var22 (vs22 (vs22 (vs22 vz22)))\n\ntbool22 : Ty22; tbool22\n = sum22 top22 top22\n\ntrue22 : ∀{Γ} → Tm22 Γ tbool22; true22\n = left22 tt22\n\ntfalse22 : ∀{Γ} → Tm22 Γ tbool22; tfalse22\n = right22 tt22\n\nifthenelse22 : ∀{Γ A} → Tm22 Γ (arr22 tbool22 (arr22 A (arr22 A A))); ifthenelse22\n = lam22 (lam22 (lam22 (case22 v222 (lam22 v222) (lam22 v122))))\n\ntimes422 : ∀{Γ A} → Tm22 Γ (arr22 (arr22 A A) (arr22 A A)); times422\n = lam22 (lam22 (app22 v122 (app22 v122 (app22 v122 (app22 v122 v022)))))\n\nadd22 : ∀{Γ} → Tm22 Γ (arr22 nat22 (arr22 nat22 nat22)); add22\n = lam22 (rec22 v022\n (lam22 (lam22 (lam22 (suc22 (app22 v122 v022)))))\n (lam22 v022))\n\nmul22 : ∀{Γ} → Tm22 Γ (arr22 nat22 (arr22 nat22 nat22)); mul22\n = lam22 (rec22 v022\n (lam22 (lam22 (lam22 (app22 (app22 add22 (app22 v122 v022)) v022))))\n (lam22 zero22))\n\nfact22 : ∀{Γ} → Tm22 Γ (arr22 nat22 nat22); fact22\n = lam22 (rec22 v022 (lam22 (lam22 (app22 (app22 mul22 (suc22 v122)) v022)))\n (suc22 zero22))\n{-# OPTIONS --type-in-type #-}\n\nTy23 : Set\nTy23 =\n (Ty23 : Set)\n (nat top bot : Ty23)\n (arr prod sum : Ty23 → Ty23 → Ty23)\n → Ty23\n\nnat23 : Ty23; nat23 = λ _ nat23 _ _ _ _ _ → nat23\ntop23 : Ty23; top23 = λ _ _ top23 _ _ _ _ → top23\nbot23 : Ty23; bot23 = λ _ _ _ bot23 _ _ _ → bot23\n\narr23 : Ty23 → Ty23 → Ty23; arr23\n = λ A B Ty23 nat23 top23 bot23 arr23 prod sum →\n arr23 (A Ty23 nat23 top23 bot23 arr23 prod sum) (B Ty23 nat23 top23 bot23 arr23 prod sum)\n\nprod23 : Ty23 → Ty23 → Ty23; prod23\n = λ A B Ty23 nat23 top23 bot23 arr23 prod23 sum →\n prod23 (A Ty23 nat23 top23 bot23 arr23 prod23 sum) (B Ty23 nat23 top23 bot23 arr23 prod23 sum)\n\nsum23 : Ty23 → Ty23 → Ty23; sum23\n = λ A B Ty23 nat23 top23 bot23 arr23 prod23 sum23 →\n sum23 (A Ty23 nat23 top23 bot23 arr23 prod23 sum23) (B Ty23 nat23 top23 bot23 arr23 prod23 sum23)\n\nCon23 : Set; Con23\n = (Con23 : Set)\n (nil : Con23)\n (snoc : Con23 → Ty23 → Con23)\n → Con23\n\nnil23 : Con23; nil23\n = λ Con23 nil23 snoc → nil23\n\nsnoc23 : Con23 → Ty23 → Con23; snoc23\n = λ Γ A Con23 nil23 snoc23 → snoc23 (Γ Con23 nil23 snoc23) A\n\nVar23 : Con23 → Ty23 → Set; Var23\n = λ Γ A →\n (Var23 : Con23 → Ty23 → Set)\n (vz : ∀ Γ A → Var23 (snoc23 Γ A) A)\n (vs : ∀ Γ B A → Var23 Γ A → Var23 (snoc23 Γ B) A)\n → Var23 Γ A\n\nvz23 : ∀{Γ A} → Var23 (snoc23 Γ A) A; vz23\n = λ Var23 vz23 vs → vz23 _ _\n\nvs23 : ∀{Γ B A} → Var23 Γ A → Var23 (snoc23 Γ B) A; vs23\n = λ x Var23 vz23 vs23 → vs23 _ _ _ (x Var23 vz23 vs23)\n\nTm23 : Con23 → Ty23 → Set; Tm23\n = λ Γ A →\n (Tm23 : Con23 → Ty23 → Set)\n (var : ∀ Γ A → Var23 Γ A → Tm23 Γ A)\n (lam : ∀ Γ A B → Tm23 (snoc23 Γ A) B → Tm23 Γ (arr23 A B))\n (app : ∀ Γ A B → Tm23 Γ (arr23 A B) → Tm23 Γ A → Tm23 Γ B)\n (tt : ∀ Γ → Tm23 Γ top23)\n (pair : ∀ Γ A B → Tm23 Γ A → Tm23 Γ B → Tm23 Γ (prod23 A B))\n (fst : ∀ Γ A B → Tm23 Γ (prod23 A B) → Tm23 Γ A)\n (snd : ∀ Γ A B → Tm23 Γ (prod23 A B) → Tm23 Γ B)\n (left : ∀ Γ A B → Tm23 Γ A → Tm23 Γ (sum23 A B))\n (right : ∀ Γ A B → Tm23 Γ B → Tm23 Γ (sum23 A B))\n (case : ∀ Γ A B C → Tm23 Γ (sum23 A B) → Tm23 Γ (arr23 A C) → Tm23 Γ (arr23 B C) → Tm23 Γ C)\n (zero : ∀ Γ → Tm23 Γ nat23)\n (suc : ∀ Γ → Tm23 Γ nat23 → Tm23 Γ nat23)\n (rec : ∀ Γ A → Tm23 Γ nat23 → Tm23 Γ (arr23 nat23 (arr23 A A)) → Tm23 Γ A → Tm23 Γ A)\n → Tm23 Γ A\n\nvar23 : ∀{Γ A} → Var23 Γ A → Tm23 Γ A; var23\n = λ x Tm23 var23 lam app tt pair fst snd left right case zero suc rec →\n var23 _ _ x\n\nlam23 : ∀{Γ A B} → Tm23 (snoc23 Γ A) B → Tm23 Γ (arr23 A B); lam23\n = λ t Tm23 var23 lam23 app tt pair fst snd left right case zero suc rec →\n lam23 _ _ _ (t Tm23 var23 lam23 app tt pair fst snd left right case zero suc rec)\n\napp23 : ∀{Γ A B} → Tm23 Γ (arr23 A B) → Tm23 Γ A → Tm23 Γ B; app23\n = λ t u Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec →\n app23 _ _ _ (t Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec)\n (u Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec)\n\ntt23 : ∀{Γ} → Tm23 Γ top23; tt23\n = λ Tm23 var23 lam23 app23 tt23 pair fst snd left right case zero suc rec → tt23 _\n\npair23 : ∀{Γ A B} → Tm23 Γ A → Tm23 Γ B → Tm23 Γ (prod23 A B); pair23\n = λ t u Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec →\n pair23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec)\n (u Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec)\n\nfst23 : ∀{Γ A B} → Tm23 Γ (prod23 A B) → Tm23 Γ A; fst23\n = λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd left right case zero suc rec →\n fst23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd left right case zero suc rec)\n\nsnd23 : ∀{Γ A B} → Tm23 Γ (prod23 A B) → Tm23 Γ B; snd23\n = λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left right case zero suc rec →\n snd23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left right case zero suc rec)\n\nleft23 : ∀{Γ A B} → Tm23 Γ A → Tm23 Γ (sum23 A B); left23\n = λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right case zero suc rec →\n left23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right case zero suc rec)\n\nright23 : ∀{Γ A B} → Tm23 Γ B → Tm23 Γ (sum23 A B); right23\n = λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case zero suc rec →\n right23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case zero suc rec)\n\ncase23 : ∀{Γ A B C} → Tm23 Γ (sum23 A B) → Tm23 Γ (arr23 A C) → Tm23 Γ (arr23 B C) → Tm23 Γ C; case23\n = λ t u v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec →\n case23 _ _ _ _\n (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)\n (u Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)\n (v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)\n\nzero23 : ∀{Γ} → Tm23 Γ nat23; zero23\n = λ Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc rec → zero23 _\n\nsuc23 : ∀{Γ} → Tm23 Γ nat23 → Tm23 Γ nat23; suc23\n = λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec →\n suc23 _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec)\n\nrec23 : ∀{Γ A} → Tm23 Γ nat23 → Tm23 Γ (arr23 nat23 (arr23 A A)) → Tm23 Γ A → Tm23 Γ A; rec23\n = λ t u v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23 →\n rec23 _ _\n (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)\n (u Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)\n (v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)\n\nv023 : ∀{Γ A} → Tm23 (snoc23 Γ A) A; v023\n = var23 vz23\n\nv123 : ∀{Γ A B} → Tm23 (snoc23 (snoc23 Γ A) B) A; v123\n = var23 (vs23 vz23)\n\nv223 : ∀{Γ A B C} → Tm23 (snoc23 (snoc23 (snoc23 Γ A) B) C) A; v223\n = var23 (vs23 (vs23 vz23))\n\nv323 : ∀{Γ A B C D} → Tm23 (snoc23 (snoc23 (snoc23 (snoc23 Γ A) B) C) D) A; v323\n = var23 (vs23 (vs23 (vs23 vz23)))\n\ntbool23 : Ty23; tbool23\n = sum23 top23 top23\n\ntrue23 : ∀{Γ} → Tm23 Γ tbool23; true23\n = left23 tt23\n\ntfalse23 : ∀{Γ} → Tm23 Γ tbool23; tfalse23\n = right23 tt23\n\nifthenelse23 : ∀{Γ A} → Tm23 Γ (arr23 tbool23 (arr23 A (arr23 A A))); ifthenelse23\n = lam23 (lam23 (lam23 (case23 v223 (lam23 v223) (lam23 v123))))\n\ntimes423 : ∀{Γ A} → Tm23 Γ (arr23 (arr23 A A) (arr23 A A)); times423\n = lam23 (lam23 (app23 v123 (app23 v123 (app23 v123 (app23 v123 v023)))))\n\nadd23 : ∀{Γ} → Tm23 Γ (arr23 nat23 (arr23 nat23 nat23)); add23\n = lam23 (rec23 v023\n (lam23 (lam23 (lam23 (suc23 (app23 v123 v023)))))\n (lam23 v023))\n\nmul23 : ∀{Γ} → Tm23 Γ (arr23 nat23 (arr23 nat23 nat23)); mul23\n = lam23 (rec23 v023\n (lam23 (lam23 (lam23 (app23 (app23 add23 (app23 v123 v023)) v023))))\n (lam23 zero23))\n\nfact23 : ∀{Γ} → Tm23 Γ (arr23 nat23 nat23); fact23\n = lam23 (rec23 v023 (lam23 (lam23 (app23 (app23 mul23 (suc23 v123)) v023)))\n (suc23 zero23))\n{-# OPTIONS --type-in-type #-}\n\nTy24 : Set\nTy24 =\n (Ty24 : Set)\n (nat top bot : Ty24)\n (arr prod sum : Ty24 → Ty24 → Ty24)\n → Ty24\n\nnat24 : Ty24; nat24 = λ _ nat24 _ _ _ _ _ → nat24\ntop24 : Ty24; top24 = λ _ _ top24 _ _ _ _ → top24\nbot24 : Ty24; bot24 = λ _ _ _ bot24 _ _ _ → bot24\n\narr24 : Ty24 → Ty24 → Ty24; arr24\n = λ A B Ty24 nat24 top24 bot24 arr24 prod sum →\n arr24 (A Ty24 nat24 top24 bot24 arr24 prod sum) (B Ty24 nat24 top24 bot24 arr24 prod sum)\n\nprod24 : Ty24 → Ty24 → Ty24; prod24\n = λ A B Ty24 nat24 top24 bot24 arr24 prod24 sum →\n prod24 (A Ty24 nat24 top24 bot24 arr24 prod24 sum) (B Ty24 nat24 top24 bot24 arr24 prod24 sum)\n\nsum24 : Ty24 → Ty24 → Ty24; sum24\n = λ A B Ty24 nat24 top24 bot24 arr24 prod24 sum24 →\n sum24 (A Ty24 nat24 top24 bot24 arr24 prod24 sum24) (B Ty24 nat24 top24 bot24 arr24 prod24 sum24)\n\nCon24 : Set; Con24\n = (Con24 : Set)\n (nil : Con24)\n (snoc : Con24 → Ty24 → Con24)\n → Con24\n\nnil24 : Con24; nil24\n = λ Con24 nil24 snoc → nil24\n\nsnoc24 : Con24 → Ty24 → Con24; snoc24\n = λ Γ A Con24 nil24 snoc24 → snoc24 (Γ Con24 nil24 snoc24) A\n\nVar24 : Con24 → Ty24 → Set; Var24\n = λ Γ A →\n (Var24 : Con24 → Ty24 → Set)\n (vz : ∀ Γ A → Var24 (snoc24 Γ A) A)\n (vs : ∀ Γ B A → Var24 Γ A → Var24 (snoc24 Γ B) A)\n → Var24 Γ A\n\nvz24 : ∀{Γ A} → Var24 (snoc24 Γ A) A; vz24\n = λ Var24 vz24 vs → vz24 _ _\n\nvs24 : ∀{Γ B A} → Var24 Γ A → Var24 (snoc24 Γ B) A; vs24\n = λ x Var24 vz24 vs24 → vs24 _ _ _ (x Var24 vz24 vs24)\n\nTm24 : Con24 → Ty24 → Set; Tm24\n = λ Γ A →\n (Tm24 : Con24 → Ty24 → Set)\n (var : ∀ Γ A → Var24 Γ A → Tm24 Γ A)\n (lam : ∀ Γ A B → Tm24 (snoc24 Γ A) B → Tm24 Γ (arr24 A B))\n (app : ∀ Γ A B → Tm24 Γ (arr24 A B) → Tm24 Γ A → Tm24 Γ B)\n (tt : ∀ Γ → Tm24 Γ top24)\n (pair : ∀ Γ A B → Tm24 Γ A → Tm24 Γ B → Tm24 Γ (prod24 A B))\n (fst : ∀ Γ A B → Tm24 Γ (prod24 A B) → Tm24 Γ A)\n (snd : ∀ Γ A B → Tm24 Γ (prod24 A B) → Tm24 Γ B)\n (left : ∀ Γ A B → Tm24 Γ A → Tm24 Γ (sum24 A B))\n (right : ∀ Γ A B → Tm24 Γ B → Tm24 Γ (sum24 A B))\n (case : ∀ Γ A B C → Tm24 Γ (sum24 A B) → Tm24 Γ (arr24 A C) → Tm24 Γ (arr24 B C) → Tm24 Γ C)\n (zero : ∀ Γ → Tm24 Γ nat24)\n (suc : ∀ Γ → Tm24 Γ nat24 → Tm24 Γ nat24)\n (rec : ∀ Γ A → Tm24 Γ nat24 → Tm24 Γ (arr24 nat24 (arr24 A A)) → Tm24 Γ A → Tm24 Γ A)\n → Tm24 Γ A\n\nvar24 : ∀{Γ A} → Var24 Γ A → Tm24 Γ A; var24\n = λ x Tm24 var24 lam app tt pair fst snd left right case zero suc rec →\n var24 _ _ x\n\nlam24 : ∀{Γ A B} → Tm24 (snoc24 Γ A) B → Tm24 Γ (arr24 A B); lam24\n = λ t Tm24 var24 lam24 app tt pair fst snd left right case zero suc rec →\n lam24 _ _ _ (t Tm24 var24 lam24 app tt pair fst snd left right case zero suc rec)\n\napp24 : ∀{Γ A B} → Tm24 Γ (arr24 A B) → Tm24 Γ A → Tm24 Γ B; app24\n = λ t u Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec →\n app24 _ _ _ (t Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec)\n (u Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec)\n\ntt24 : ∀{Γ} → Tm24 Γ top24; tt24\n = λ Tm24 var24 lam24 app24 tt24 pair fst snd left right case zero suc rec → tt24 _\n\npair24 : ∀{Γ A B} → Tm24 Γ A → Tm24 Γ B → Tm24 Γ (prod24 A B); pair24\n = λ t u Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec →\n pair24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec)\n (u Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec)\n\nfst24 : ∀{Γ A B} → Tm24 Γ (prod24 A B) → Tm24 Γ A; fst24\n = λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd left right case zero suc rec →\n fst24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd left right case zero suc rec)\n\nsnd24 : ∀{Γ A B} → Tm24 Γ (prod24 A B) → Tm24 Γ B; snd24\n = λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left right case zero suc rec →\n snd24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left right case zero suc rec)\n\nleft24 : ∀{Γ A B} → Tm24 Γ A → Tm24 Γ (sum24 A B); left24\n = λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right case zero suc rec →\n left24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right case zero suc rec)\n\nright24 : ∀{Γ A B} → Tm24 Γ B → Tm24 Γ (sum24 A B); right24\n = λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case zero suc rec →\n right24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case zero suc rec)\n\ncase24 : ∀{Γ A B C} → Tm24 Γ (sum24 A B) → Tm24 Γ (arr24 A C) → Tm24 Γ (arr24 B C) → Tm24 Γ C; case24\n = λ t u v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec →\n case24 _ _ _ _\n (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)\n (u Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)\n (v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)\n\nzero24 : ∀{Γ} → Tm24 Γ nat24; zero24\n = λ Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc rec → zero24 _\n\nsuc24 : ∀{Γ} → Tm24 Γ nat24 → Tm24 Γ nat24; suc24\n = λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec →\n suc24 _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec)\n\nrec24 : ∀{Γ A} → Tm24 Γ nat24 → Tm24 Γ (arr24 nat24 (arr24 A A)) → Tm24 Γ A → Tm24 Γ A; rec24\n = λ t u v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24 →\n rec24 _ _\n (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)\n (u Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)\n (v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)\n\nv024 : ∀{Γ A} → Tm24 (snoc24 Γ A) A; v024\n = var24 vz24\n\nv124 : ∀{Γ A B} → Tm24 (snoc24 (snoc24 Γ A) B) A; v124\n = var24 (vs24 vz24)\n\nv224 : ∀{Γ A B C} → Tm24 (snoc24 (snoc24 (snoc24 Γ A) B) C) A; v224\n = var24 (vs24 (vs24 vz24))\n\nv324 : ∀{Γ A B C D} → Tm24 (snoc24 (snoc24 (snoc24 (snoc24 Γ A) B) C) D) A; v324\n = var24 (vs24 (vs24 (vs24 vz24)))\n\ntbool24 : Ty24; tbool24\n = sum24 top24 top24\n\ntrue24 : ∀{Γ} → Tm24 Γ tbool24; true24\n = left24 tt24\n\ntfalse24 : ∀{Γ} → Tm24 Γ tbool24; tfalse24\n = right24 tt24\n\nifthenelse24 : ∀{Γ A} → Tm24 Γ (arr24 tbool24 (arr24 A (arr24 A A))); ifthenelse24\n = lam24 (lam24 (lam24 (case24 v224 (lam24 v224) (lam24 v124))))\n\ntimes424 : ∀{Γ A} → Tm24 Γ (arr24 (arr24 A A) (arr24 A A)); times424\n = lam24 (lam24 (app24 v124 (app24 v124 (app24 v124 (app24 v124 v024)))))\n\nadd24 : ∀{Γ} → Tm24 Γ (arr24 nat24 (arr24 nat24 nat24)); add24\n = lam24 (rec24 v024\n (lam24 (lam24 (lam24 (suc24 (app24 v124 v024)))))\n (lam24 v024))\n\nmul24 : ∀{Γ} → Tm24 Γ (arr24 nat24 (arr24 nat24 nat24)); mul24\n = lam24 (rec24 v024\n (lam24 (lam24 (lam24 (app24 (app24 add24 (app24 v124 v024)) v024))))\n (lam24 zero24))\n\nfact24 : ∀{Γ} → Tm24 Γ (arr24 nat24 nat24); fact24\n = lam24 (rec24 v024 (lam24 (lam24 (app24 (app24 mul24 (suc24 v124)) v024)))\n (suc24 zero24))\n{-# OPTIONS --type-in-type #-}\n\nTy25 : Set\nTy25 =\n (Ty25 : Set)\n (nat top bot : Ty25)\n (arr prod sum : Ty25 → Ty25 → Ty25)\n → Ty25\n\nnat25 : Ty25; nat25 = λ _ nat25 _ _ _ _ _ → nat25\ntop25 : Ty25; top25 = λ _ _ top25 _ _ _ _ → top25\nbot25 : Ty25; bot25 = λ _ _ _ bot25 _ _ _ → bot25\n\narr25 : Ty25 → Ty25 → Ty25; arr25\n = λ A B Ty25 nat25 top25 bot25 arr25 prod sum →\n arr25 (A Ty25 nat25 top25 bot25 arr25 prod sum) (B Ty25 nat25 top25 bot25 arr25 prod sum)\n\nprod25 : Ty25 → Ty25 → Ty25; prod25\n = λ A B Ty25 nat25 top25 bot25 arr25 prod25 sum →\n prod25 (A Ty25 nat25 top25 bot25 arr25 prod25 sum) (B Ty25 nat25 top25 bot25 arr25 prod25 sum)\n\nsum25 : Ty25 → Ty25 → Ty25; sum25\n = λ A B Ty25 nat25 top25 bot25 arr25 prod25 sum25 →\n sum25 (A Ty25 nat25 top25 bot25 arr25 prod25 sum25) (B Ty25 nat25 top25 bot25 arr25 prod25 sum25)\n\nCon25 : Set; Con25\n = (Con25 : Set)\n (nil : Con25)\n (snoc : Con25 → Ty25 → Con25)\n → Con25\n\nnil25 : Con25; nil25\n = λ Con25 nil25 snoc → nil25\n\nsnoc25 : Con25 → Ty25 → Con25; snoc25\n = λ Γ A Con25 nil25 snoc25 → snoc25 (Γ Con25 nil25 snoc25) A\n\nVar25 : Con25 → Ty25 → Set; Var25\n = λ Γ A →\n (Var25 : Con25 → Ty25 → Set)\n (vz : ∀ Γ A → Var25 (snoc25 Γ A) A)\n (vs : ∀ Γ B A → Var25 Γ A → Var25 (snoc25 Γ B) A)\n → Var25 Γ A\n\nvz25 : ∀{Γ A} → Var25 (snoc25 Γ A) A; vz25\n = λ Var25 vz25 vs → vz25 _ _\n\nvs25 : ∀{Γ B A} → Var25 Γ A → Var25 (snoc25 Γ B) A; vs25\n = λ x Var25 vz25 vs25 → vs25 _ _ _ (x Var25 vz25 vs25)\n\nTm25 : Con25 → Ty25 → Set; Tm25\n = λ Γ A →\n (Tm25 : Con25 → Ty25 → Set)\n (var : ∀ Γ A → Var25 Γ A → Tm25 Γ A)\n (lam : ∀ Γ A B → Tm25 (snoc25 Γ A) B → Tm25 Γ (arr25 A B))\n (app : ∀ Γ A B → Tm25 Γ (arr25 A B) → Tm25 Γ A → Tm25 Γ B)\n (tt : ∀ Γ → Tm25 Γ top25)\n (pair : ∀ Γ A B → Tm25 Γ A → Tm25 Γ B → Tm25 Γ (prod25 A B))\n (fst : ∀ Γ A B → Tm25 Γ (prod25 A B) → Tm25 Γ A)\n (snd : ∀ Γ A B → Tm25 Γ (prod25 A B) → Tm25 Γ B)\n (left : ∀ Γ A B → Tm25 Γ A → Tm25 Γ (sum25 A B))\n (right : ∀ Γ A B → Tm25 Γ B → Tm25 Γ (sum25 A B))\n (case : ∀ Γ A B C → Tm25 Γ (sum25 A B) → Tm25 Γ (arr25 A C) → Tm25 Γ (arr25 B C) → Tm25 Γ C)\n (zero : ∀ Γ → Tm25 Γ nat25)\n (suc : ∀ Γ → Tm25 Γ nat25 → Tm25 Γ nat25)\n (rec : ∀ Γ A → Tm25 Γ nat25 → Tm25 Γ (arr25 nat25 (arr25 A A)) → Tm25 Γ A → Tm25 Γ A)\n → Tm25 Γ A\n\nvar25 : ∀{Γ A} → Var25 Γ A → Tm25 Γ A; var25\n = λ x Tm25 var25 lam app tt pair fst snd left right case zero suc rec →\n var25 _ _ x\n\nlam25 : ∀{Γ A B} → Tm25 (snoc25 Γ A) B → Tm25 Γ (arr25 A B); lam25\n = λ t Tm25 var25 lam25 app tt pair fst snd left right case zero suc rec →\n lam25 _ _ _ (t Tm25 var25 lam25 app tt pair fst snd left right case zero suc rec)\n\napp25 : ∀{Γ A B} → Tm25 Γ (arr25 A B) → Tm25 Γ A → Tm25 Γ B; app25\n = λ t u Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec →\n app25 _ _ _ (t Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec)\n (u Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec)\n\ntt25 : ∀{Γ} → Tm25 Γ top25; tt25\n = λ Tm25 var25 lam25 app25 tt25 pair fst snd left right case zero suc rec → tt25 _\n\npair25 : ∀{Γ A B} → Tm25 Γ A → Tm25 Γ B → Tm25 Γ (prod25 A B); pair25\n = λ t u Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec →\n pair25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec)\n (u Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec)\n\nfst25 : ∀{Γ A B} → Tm25 Γ (prod25 A B) → Tm25 Γ A; fst25\n = λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd left right case zero suc rec →\n fst25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd left right case zero suc rec)\n\nsnd25 : ∀{Γ A B} → Tm25 Γ (prod25 A B) → Tm25 Γ B; snd25\n = λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left right case zero suc rec →\n snd25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left right case zero suc rec)\n\nleft25 : ∀{Γ A B} → Tm25 Γ A → Tm25 Γ (sum25 A B); left25\n = λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right case zero suc rec →\n left25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right case zero suc rec)\n\nright25 : ∀{Γ A B} → Tm25 Γ B → Tm25 Γ (sum25 A B); right25\n = λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case zero suc rec →\n right25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case zero suc rec)\n\ncase25 : ∀{Γ A B C} → Tm25 Γ (sum25 A B) → Tm25 Γ (arr25 A C) → Tm25 Γ (arr25 B C) → Tm25 Γ C; case25\n = λ t u v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec →\n case25 _ _ _ _\n (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)\n (u Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)\n (v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)\n\nzero25 : ∀{Γ} → Tm25 Γ nat25; zero25\n = λ Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc rec → zero25 _\n\nsuc25 : ∀{Γ} → Tm25 Γ nat25 → Tm25 Γ nat25; suc25\n = λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec →\n suc25 _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec)\n\nrec25 : ∀{Γ A} → Tm25 Γ nat25 → Tm25 Γ (arr25 nat25 (arr25 A A)) → Tm25 Γ A → Tm25 Γ A; rec25\n = λ t u v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25 →\n rec25 _ _\n (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)\n (u Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)\n (v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)\n\nv025 : ∀{Γ A} → Tm25 (snoc25 Γ A) A; v025\n = var25 vz25\n\nv125 : ∀{Γ A B} → Tm25 (snoc25 (snoc25 Γ A) B) A; v125\n = var25 (vs25 vz25)\n\nv225 : ∀{Γ A B C} → Tm25 (snoc25 (snoc25 (snoc25 Γ A) B) C) A; v225\n = var25 (vs25 (vs25 vz25))\n\nv325 : ∀{Γ A B C D} → Tm25 (snoc25 (snoc25 (snoc25 (snoc25 Γ A) B) C) D) A; v325\n = var25 (vs25 (vs25 (vs25 vz25)))\n\ntbool25 : Ty25; tbool25\n = sum25 top25 top25\n\ntrue25 : ∀{Γ} → Tm25 Γ tbool25; true25\n = left25 tt25\n\ntfalse25 : ∀{Γ} → Tm25 Γ tbool25; tfalse25\n = right25 tt25\n\nifthenelse25 : ∀{Γ A} → Tm25 Γ (arr25 tbool25 (arr25 A (arr25 A A))); ifthenelse25\n = lam25 (lam25 (lam25 (case25 v225 (lam25 v225) (lam25 v125))))\n\ntimes425 : ∀{Γ A} → Tm25 Γ (arr25 (arr25 A A) (arr25 A A)); times425\n = lam25 (lam25 (app25 v125 (app25 v125 (app25 v125 (app25 v125 v025)))))\n\nadd25 : ∀{Γ} → Tm25 Γ (arr25 nat25 (arr25 nat25 nat25)); add25\n = lam25 (rec25 v025\n (lam25 (lam25 (lam25 (suc25 (app25 v125 v025)))))\n (lam25 v025))\n\nmul25 : ∀{Γ} → Tm25 Γ (arr25 nat25 (arr25 nat25 nat25)); mul25\n = lam25 (rec25 v025\n (lam25 (lam25 (lam25 (app25 (app25 add25 (app25 v125 v025)) v025))))\n (lam25 zero25))\n\nfact25 : ∀{Γ} → Tm25 Γ (arr25 nat25 nat25); fact25\n = lam25 (rec25 v025 (lam25 (lam25 (app25 (app25 mul25 (suc25 v125)) v025)))\n (suc25 zero25))\n{-# OPTIONS --type-in-type #-}\n\nTy26 : Set\nTy26 =\n (Ty26 : Set)\n (nat top bot : Ty26)\n (arr prod sum : Ty26 → Ty26 → Ty26)\n → Ty26\n\nnat26 : Ty26; nat26 = λ _ nat26 _ _ _ _ _ → nat26\ntop26 : Ty26; top26 = λ _ _ top26 _ _ _ _ → top26\nbot26 : Ty26; bot26 = λ _ _ _ bot26 _ _ _ → bot26\n\narr26 : Ty26 → Ty26 → Ty26; arr26\n = λ A B Ty26 nat26 top26 bot26 arr26 prod sum →\n arr26 (A Ty26 nat26 top26 bot26 arr26 prod sum) (B Ty26 nat26 top26 bot26 arr26 prod sum)\n\nprod26 : Ty26 → Ty26 → Ty26; prod26\n = λ A B Ty26 nat26 top26 bot26 arr26 prod26 sum →\n prod26 (A Ty26 nat26 top26 bot26 arr26 prod26 sum) (B Ty26 nat26 top26 bot26 arr26 prod26 sum)\n\nsum26 : Ty26 → Ty26 → Ty26; sum26\n = λ A B Ty26 nat26 top26 bot26 arr26 prod26 sum26 →\n sum26 (A Ty26 nat26 top26 bot26 arr26 prod26 sum26) (B Ty26 nat26 top26 bot26 arr26 prod26 sum26)\n\nCon26 : Set; Con26\n = (Con26 : Set)\n (nil : Con26)\n (snoc : Con26 → Ty26 → Con26)\n → Con26\n\nnil26 : Con26; nil26\n = λ Con26 nil26 snoc → nil26\n\nsnoc26 : Con26 → Ty26 → Con26; snoc26\n = λ Γ A Con26 nil26 snoc26 → snoc26 (Γ Con26 nil26 snoc26) A\n\nVar26 : Con26 → Ty26 → Set; Var26\n = λ Γ A →\n (Var26 : Con26 → Ty26 → Set)\n (vz : ∀ Γ A → Var26 (snoc26 Γ A) A)\n (vs : ∀ Γ B A → Var26 Γ A → Var26 (snoc26 Γ B) A)\n → Var26 Γ A\n\nvz26 : ∀{Γ A} → Var26 (snoc26 Γ A) A; vz26\n = λ Var26 vz26 vs → vz26 _ _\n\nvs26 : ∀{Γ B A} → Var26 Γ A → Var26 (snoc26 Γ B) A; vs26\n = λ x Var26 vz26 vs26 → vs26 _ _ _ (x Var26 vz26 vs26)\n\nTm26 : Con26 → Ty26 → Set; Tm26\n = λ Γ A →\n (Tm26 : Con26 → Ty26 → Set)\n (var : ∀ Γ A → Var26 Γ A → Tm26 Γ A)\n (lam : ∀ Γ A B → Tm26 (snoc26 Γ A) B → Tm26 Γ (arr26 A B))\n (app : ∀ Γ A B → Tm26 Γ (arr26 A B) → Tm26 Γ A → Tm26 Γ B)\n (tt : ∀ Γ → Tm26 Γ top26)\n (pair : ∀ Γ A B → Tm26 Γ A → Tm26 Γ B → Tm26 Γ (prod26 A B))\n (fst : ∀ Γ A B → Tm26 Γ (prod26 A B) → Tm26 Γ A)\n (snd : ∀ Γ A B → Tm26 Γ (prod26 A B) → Tm26 Γ B)\n (left : ∀ Γ A B → Tm26 Γ A → Tm26 Γ (sum26 A B))\n (right : ∀ Γ A B → Tm26 Γ B → Tm26 Γ (sum26 A B))\n (case : ∀ Γ A B C → Tm26 Γ (sum26 A B) → Tm26 Γ (arr26 A C) → Tm26 Γ (arr26 B C) → Tm26 Γ C)\n (zero : ∀ Γ → Tm26 Γ nat26)\n (suc : ∀ Γ → Tm26 Γ nat26 → Tm26 Γ nat26)\n (rec : ∀ Γ A → Tm26 Γ nat26 → Tm26 Γ (arr26 nat26 (arr26 A A)) → Tm26 Γ A → Tm26 Γ A)\n → Tm26 Γ A\n\nvar26 : ∀{Γ A} → Var26 Γ A → Tm26 Γ A; var26\n = λ x Tm26 var26 lam app tt pair fst snd left right case zero suc rec →\n var26 _ _ x\n\nlam26 : ∀{Γ A B} → Tm26 (snoc26 Γ A) B → Tm26 Γ (arr26 A B); lam26\n = λ t Tm26 var26 lam26 app tt pair fst snd left right case zero suc rec →\n lam26 _ _ _ (t Tm26 var26 lam26 app tt pair fst snd left right case zero suc rec)\n\napp26 : ∀{Γ A B} → Tm26 Γ (arr26 A B) → Tm26 Γ A → Tm26 Γ B; app26\n = λ t u Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec →\n app26 _ _ _ (t Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec)\n (u Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec)\n\ntt26 : ∀{Γ} → Tm26 Γ top26; tt26\n = λ Tm26 var26 lam26 app26 tt26 pair fst snd left right case zero suc rec → tt26 _\n\npair26 : ∀{Γ A B} → Tm26 Γ A → Tm26 Γ B → Tm26 Γ (prod26 A B); pair26\n = λ t u Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec →\n pair26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec)\n (u Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec)\n\nfst26 : ∀{Γ A B} → Tm26 Γ (prod26 A B) → Tm26 Γ A; fst26\n = λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd left right case zero suc rec →\n fst26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd left right case zero suc rec)\n\nsnd26 : ∀{Γ A B} → Tm26 Γ (prod26 A B) → Tm26 Γ B; snd26\n = λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left right case zero suc rec →\n snd26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left right case zero suc rec)\n\nleft26 : ∀{Γ A B} → Tm26 Γ A → Tm26 Γ (sum26 A B); left26\n = λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right case zero suc rec →\n left26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right case zero suc rec)\n\nright26 : ∀{Γ A B} → Tm26 Γ B → Tm26 Γ (sum26 A B); right26\n = λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case zero suc rec →\n right26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case zero suc rec)\n\ncase26 : ∀{Γ A B C} → Tm26 Γ (sum26 A B) → Tm26 Γ (arr26 A C) → Tm26 Γ (arr26 B C) → Tm26 Γ C; case26\n = λ t u v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec →\n case26 _ _ _ _\n (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)\n (u Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)\n (v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)\n\nzero26 : ∀{Γ} → Tm26 Γ nat26; zero26\n = λ Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc rec → zero26 _\n\nsuc26 : ∀{Γ} → Tm26 Γ nat26 → Tm26 Γ nat26; suc26\n = λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec →\n suc26 _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec)\n\nrec26 : ∀{Γ A} → Tm26 Γ nat26 → Tm26 Γ (arr26 nat26 (arr26 A A)) → Tm26 Γ A → Tm26 Γ A; rec26\n = λ t u v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26 →\n rec26 _ _\n (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)\n (u Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)\n (v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)\n\nv026 : ∀{Γ A} → Tm26 (snoc26 Γ A) A; v026\n = var26 vz26\n\nv126 : ∀{Γ A B} → Tm26 (snoc26 (snoc26 Γ A) B) A; v126\n = var26 (vs26 vz26)\n\nv226 : ∀{Γ A B C} → Tm26 (snoc26 (snoc26 (snoc26 Γ A) B) C) A; v226\n = var26 (vs26 (vs26 vz26))\n\nv326 : ∀{Γ A B C D} → Tm26 (snoc26 (snoc26 (snoc26 (snoc26 Γ A) B) C) D) A; v326\n = var26 (vs26 (vs26 (vs26 vz26)))\n\ntbool26 : Ty26; tbool26\n = sum26 top26 top26\n\ntrue26 : ∀{Γ} → Tm26 Γ tbool26; true26\n = left26 tt26\n\ntfalse26 : ∀{Γ} → Tm26 Γ tbool26; tfalse26\n = right26 tt26\n\nifthenelse26 : ∀{Γ A} → Tm26 Γ (arr26 tbool26 (arr26 A (arr26 A A))); ifthenelse26\n = lam26 (lam26 (lam26 (case26 v226 (lam26 v226) (lam26 v126))))\n\ntimes426 : ∀{Γ A} → Tm26 Γ (arr26 (arr26 A A) (arr26 A A)); times426\n = lam26 (lam26 (app26 v126 (app26 v126 (app26 v126 (app26 v126 v026)))))\n\nadd26 : ∀{Γ} → Tm26 Γ (arr26 nat26 (arr26 nat26 nat26)); add26\n = lam26 (rec26 v026\n (lam26 (lam26 (lam26 (suc26 (app26 v126 v026)))))\n (lam26 v026))\n\nmul26 : ∀{Γ} → Tm26 Γ (arr26 nat26 (arr26 nat26 nat26)); mul26\n = lam26 (rec26 v026\n (lam26 (lam26 (lam26 (app26 (app26 add26 (app26 v126 v026)) v026))))\n (lam26 zero26))\n\nfact26 : ∀{Γ} → Tm26 Γ (arr26 nat26 nat26); fact26\n = lam26 (rec26 v026 (lam26 (lam26 (app26 (app26 mul26 (suc26 v126)) v026)))\n (suc26 zero26))\n{-# OPTIONS --type-in-type #-}\n\nTy27 : Set\nTy27 =\n (Ty27 : Set)\n (nat top bot : Ty27)\n (arr prod sum : Ty27 → Ty27 → Ty27)\n → Ty27\n\nnat27 : Ty27; nat27 = λ _ nat27 _ _ _ _ _ → nat27\ntop27 : Ty27; top27 = λ _ _ top27 _ _ _ _ → top27\nbot27 : Ty27; bot27 = λ _ _ _ bot27 _ _ _ → bot27\n\narr27 : Ty27 → Ty27 → Ty27; arr27\n = λ A B Ty27 nat27 top27 bot27 arr27 prod sum →\n arr27 (A Ty27 nat27 top27 bot27 arr27 prod sum) (B Ty27 nat27 top27 bot27 arr27 prod sum)\n\nprod27 : Ty27 → Ty27 → Ty27; prod27\n = λ A B Ty27 nat27 top27 bot27 arr27 prod27 sum →\n prod27 (A Ty27 nat27 top27 bot27 arr27 prod27 sum) (B Ty27 nat27 top27 bot27 arr27 prod27 sum)\n\nsum27 : Ty27 → Ty27 → Ty27; sum27\n = λ A B Ty27 nat27 top27 bot27 arr27 prod27 sum27 →\n sum27 (A Ty27 nat27 top27 bot27 arr27 prod27 sum27) (B Ty27 nat27 top27 bot27 arr27 prod27 sum27)\n\nCon27 : Set; Con27\n = (Con27 : Set)\n (nil : Con27)\n (snoc : Con27 → Ty27 → Con27)\n → Con27\n\nnil27 : Con27; nil27\n = λ Con27 nil27 snoc → nil27\n\nsnoc27 : Con27 → Ty27 → Con27; snoc27\n = λ Γ A Con27 nil27 snoc27 → snoc27 (Γ Con27 nil27 snoc27) A\n\nVar27 : Con27 → Ty27 → Set; Var27\n = λ Γ A →\n (Var27 : Con27 → Ty27 → Set)\n (vz : ∀ Γ A → Var27 (snoc27 Γ A) A)\n (vs : ∀ Γ B A → Var27 Γ A → Var27 (snoc27 Γ B) A)\n → Var27 Γ A\n\nvz27 : ∀{Γ A} → Var27 (snoc27 Γ A) A; vz27\n = λ Var27 vz27 vs → vz27 _ _\n\nvs27 : ∀{Γ B A} → Var27 Γ A → Var27 (snoc27 Γ B) A; vs27\n = λ x Var27 vz27 vs27 → vs27 _ _ _ (x Var27 vz27 vs27)\n\nTm27 : Con27 → Ty27 → Set; Tm27\n = λ Γ A →\n (Tm27 : Con27 → Ty27 → Set)\n (var : ∀ Γ A → Var27 Γ A → Tm27 Γ A)\n (lam : ∀ Γ A B → Tm27 (snoc27 Γ A) B → Tm27 Γ (arr27 A B))\n (app : ∀ Γ A B → Tm27 Γ (arr27 A B) → Tm27 Γ A → Tm27 Γ B)\n (tt : ∀ Γ → Tm27 Γ top27)\n (pair : ∀ Γ A B → Tm27 Γ A → Tm27 Γ B → Tm27 Γ (prod27 A B))\n (fst : ∀ Γ A B → Tm27 Γ (prod27 A B) → Tm27 Γ A)\n (snd : ∀ Γ A B → Tm27 Γ (prod27 A B) → Tm27 Γ B)\n (left : ∀ Γ A B → Tm27 Γ A → Tm27 Γ (sum27 A B))\n (right : ∀ Γ A B → Tm27 Γ B → Tm27 Γ (sum27 A B))\n (case : ∀ Γ A B C → Tm27 Γ (sum27 A B) → Tm27 Γ (arr27 A C) → Tm27 Γ (arr27 B C) → Tm27 Γ C)\n (zero : ∀ Γ → Tm27 Γ nat27)\n (suc : ∀ Γ → Tm27 Γ nat27 → Tm27 Γ nat27)\n (rec : ∀ Γ A → Tm27 Γ nat27 → Tm27 Γ (arr27 nat27 (arr27 A A)) → Tm27 Γ A → Tm27 Γ A)\n → Tm27 Γ A\n\nvar27 : ∀{Γ A} → Var27 Γ A → Tm27 Γ A; var27\n = λ x Tm27 var27 lam app tt pair fst snd left right case zero suc rec →\n var27 _ _ x\n\nlam27 : ∀{Γ A B} → Tm27 (snoc27 Γ A) B → Tm27 Γ (arr27 A B); lam27\n = λ t Tm27 var27 lam27 app tt pair fst snd left right case zero suc rec →\n lam27 _ _ _ (t Tm27 var27 lam27 app tt pair fst snd left right case zero suc rec)\n\napp27 : ∀{Γ A B} → Tm27 Γ (arr27 A B) → Tm27 Γ A → Tm27 Γ B; app27\n = λ t u Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec →\n app27 _ _ _ (t Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec)\n (u Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec)\n\ntt27 : ∀{Γ} → Tm27 Γ top27; tt27\n = λ Tm27 var27 lam27 app27 tt27 pair fst snd left right case zero suc rec → tt27 _\n\npair27 : ∀{Γ A B} → Tm27 Γ A → Tm27 Γ B → Tm27 Γ (prod27 A B); pair27\n = λ t u Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec →\n pair27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec)\n (u Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec)\n\nfst27 : ∀{Γ A B} → Tm27 Γ (prod27 A B) → Tm27 Γ A; fst27\n = λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd left right case zero suc rec →\n fst27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd left right case zero suc rec)\n\nsnd27 : ∀{Γ A B} → Tm27 Γ (prod27 A B) → Tm27 Γ B; snd27\n = λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left right case zero suc rec →\n snd27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left right case zero suc rec)\n\nleft27 : ∀{Γ A B} → Tm27 Γ A → Tm27 Γ (sum27 A B); left27\n = λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right case zero suc rec →\n left27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right case zero suc rec)\n\nright27 : ∀{Γ A B} → Tm27 Γ B → Tm27 Γ (sum27 A B); right27\n = λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case zero suc rec →\n right27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case zero suc rec)\n\ncase27 : ∀{Γ A B C} → Tm27 Γ (sum27 A B) → Tm27 Γ (arr27 A C) → Tm27 Γ (arr27 B C) → Tm27 Γ C; case27\n = λ t u v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec →\n case27 _ _ _ _\n (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)\n (u Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)\n (v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)\n\nzero27 : ∀{Γ} → Tm27 Γ nat27; zero27\n = λ Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc rec → zero27 _\n\nsuc27 : ∀{Γ} → Tm27 Γ nat27 → Tm27 Γ nat27; suc27\n = λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec →\n suc27 _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec)\n\nrec27 : ∀{Γ A} → Tm27 Γ nat27 → Tm27 Γ (arr27 nat27 (arr27 A A)) → Tm27 Γ A → Tm27 Γ A; rec27\n = λ t u v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27 →\n rec27 _ _\n (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)\n (u Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)\n (v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)\n\nv027 : ∀{Γ A} → Tm27 (snoc27 Γ A) A; v027\n = var27 vz27\n\nv127 : ∀{Γ A B} → Tm27 (snoc27 (snoc27 Γ A) B) A; v127\n = var27 (vs27 vz27)\n\nv227 : ∀{Γ A B C} → Tm27 (snoc27 (snoc27 (snoc27 Γ A) B) C) A; v227\n = var27 (vs27 (vs27 vz27))\n\nv327 : ∀{Γ A B C D} → Tm27 (snoc27 (snoc27 (snoc27 (snoc27 Γ A) B) C) D) A; v327\n = var27 (vs27 (vs27 (vs27 vz27)))\n\ntbool27 : Ty27; tbool27\n = sum27 top27 top27\n\ntrue27 : ∀{Γ} → Tm27 Γ tbool27; true27\n = left27 tt27\n\ntfalse27 : ∀{Γ} → Tm27 Γ tbool27; tfalse27\n = right27 tt27\n\nifthenelse27 : ∀{Γ A} → Tm27 Γ (arr27 tbool27 (arr27 A (arr27 A A))); ifthenelse27\n = lam27 (lam27 (lam27 (case27 v227 (lam27 v227) (lam27 v127))))\n\ntimes427 : ∀{Γ A} → Tm27 Γ (arr27 (arr27 A A) (arr27 A A)); times427\n = lam27 (lam27 (app27 v127 (app27 v127 (app27 v127 (app27 v127 v027)))))\n\nadd27 : ∀{Γ} → Tm27 Γ (arr27 nat27 (arr27 nat27 nat27)); add27\n = lam27 (rec27 v027\n (lam27 (lam27 (lam27 (suc27 (app27 v127 v027)))))\n (lam27 v027))\n\nmul27 : ∀{Γ} → Tm27 Γ (arr27 nat27 (arr27 nat27 nat27)); mul27\n = lam27 (rec27 v027\n (lam27 (lam27 (lam27 (app27 (app27 add27 (app27 v127 v027)) v027))))\n (lam27 zero27))\n\nfact27 : ∀{Γ} → Tm27 Γ (arr27 nat27 nat27); fact27\n = lam27 (rec27 v027 (lam27 (lam27 (app27 (app27 mul27 (suc27 v127)) v027)))\n (suc27 zero27))\n{-# OPTIONS --type-in-type #-}\n\nTy28 : Set\nTy28 =\n (Ty28 : Set)\n (nat top bot : Ty28)\n (arr prod sum : Ty28 → Ty28 → Ty28)\n → Ty28\n\nnat28 : Ty28; nat28 = λ _ nat28 _ _ _ _ _ → nat28\ntop28 : Ty28; top28 = λ _ _ top28 _ _ _ _ → top28\nbot28 : Ty28; bot28 = λ _ _ _ bot28 _ _ _ → bot28\n\narr28 : Ty28 → Ty28 → Ty28; arr28\n = λ A B Ty28 nat28 top28 bot28 arr28 prod sum →\n arr28 (A Ty28 nat28 top28 bot28 arr28 prod sum) (B Ty28 nat28 top28 bot28 arr28 prod sum)\n\nprod28 : Ty28 → Ty28 → Ty28; prod28\n = λ A B Ty28 nat28 top28 bot28 arr28 prod28 sum →\n prod28 (A Ty28 nat28 top28 bot28 arr28 prod28 sum) (B Ty28 nat28 top28 bot28 arr28 prod28 sum)\n\nsum28 : Ty28 → Ty28 → Ty28; sum28\n = λ A B Ty28 nat28 top28 bot28 arr28 prod28 sum28 →\n sum28 (A Ty28 nat28 top28 bot28 arr28 prod28 sum28) (B Ty28 nat28 top28 bot28 arr28 prod28 sum28)\n\nCon28 : Set; Con28\n = (Con28 : Set)\n (nil : Con28)\n (snoc : Con28 → Ty28 → Con28)\n → Con28\n\nnil28 : Con28; nil28\n = λ Con28 nil28 snoc → nil28\n\nsnoc28 : Con28 → Ty28 → Con28; snoc28\n = λ Γ A Con28 nil28 snoc28 → snoc28 (Γ Con28 nil28 snoc28) A\n\nVar28 : Con28 → Ty28 → Set; Var28\n = λ Γ A →\n (Var28 : Con28 → Ty28 → Set)\n (vz : ∀ Γ A → Var28 (snoc28 Γ A) A)\n (vs : ∀ Γ B A → Var28 Γ A → Var28 (snoc28 Γ B) A)\n → Var28 Γ A\n\nvz28 : ∀{Γ A} → Var28 (snoc28 Γ A) A; vz28\n = λ Var28 vz28 vs → vz28 _ _\n\nvs28 : ∀{Γ B A} → Var28 Γ A → Var28 (snoc28 Γ B) A; vs28\n = λ x Var28 vz28 vs28 → vs28 _ _ _ (x Var28 vz28 vs28)\n\nTm28 : Con28 → Ty28 → Set; Tm28\n = λ Γ A →\n (Tm28 : Con28 → Ty28 → Set)\n (var : ∀ Γ A → Var28 Γ A → Tm28 Γ A)\n (lam : ∀ Γ A B → Tm28 (snoc28 Γ A) B → Tm28 Γ (arr28 A B))\n (app : ∀ Γ A B → Tm28 Γ (arr28 A B) → Tm28 Γ A → Tm28 Γ B)\n (tt : ∀ Γ → Tm28 Γ top28)\n (pair : ∀ Γ A B → Tm28 Γ A → Tm28 Γ B → Tm28 Γ (prod28 A B))\n (fst : ∀ Γ A B → Tm28 Γ (prod28 A B) → Tm28 Γ A)\n (snd : ∀ Γ A B → Tm28 Γ (prod28 A B) → Tm28 Γ B)\n (left : ∀ Γ A B → Tm28 Γ A → Tm28 Γ (sum28 A B))\n (right : ∀ Γ A B → Tm28 Γ B → Tm28 Γ (sum28 A B))\n (case : ∀ Γ A B C → Tm28 Γ (sum28 A B) → Tm28 Γ (arr28 A C) → Tm28 Γ (arr28 B C) → Tm28 Γ C)\n (zero : ∀ Γ → Tm28 Γ nat28)\n (suc : ∀ Γ → Tm28 Γ nat28 → Tm28 Γ nat28)\n (rec : ∀ Γ A → Tm28 Γ nat28 → Tm28 Γ (arr28 nat28 (arr28 A A)) → Tm28 Γ A → Tm28 Γ A)\n → Tm28 Γ A\n\nvar28 : ∀{Γ A} → Var28 Γ A → Tm28 Γ A; var28\n = λ x Tm28 var28 lam app tt pair fst snd left right case zero suc rec →\n var28 _ _ x\n\nlam28 : ∀{Γ A B} → Tm28 (snoc28 Γ A) B → Tm28 Γ (arr28 A B); lam28\n = λ t Tm28 var28 lam28 app tt pair fst snd left right case zero suc rec →\n lam28 _ _ _ (t Tm28 var28 lam28 app tt pair fst snd left right case zero suc rec)\n\napp28 : ∀{Γ A B} → Tm28 Γ (arr28 A B) → Tm28 Γ A → Tm28 Γ B; app28\n = λ t u Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec →\n app28 _ _ _ (t Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec)\n (u Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec)\n\ntt28 : ∀{Γ} → Tm28 Γ top28; tt28\n = λ Tm28 var28 lam28 app28 tt28 pair fst snd left right case zero suc rec → tt28 _\n\npair28 : ∀{Γ A B} → Tm28 Γ A → Tm28 Γ B → Tm28 Γ (prod28 A B); pair28\n = λ t u Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec →\n pair28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec)\n (u Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec)\n\nfst28 : ∀{Γ A B} → Tm28 Γ (prod28 A B) → Tm28 Γ A; fst28\n = λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd left right case zero suc rec →\n fst28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd left right case zero suc rec)\n\nsnd28 : ∀{Γ A B} → Tm28 Γ (prod28 A B) → Tm28 Γ B; snd28\n = λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left right case zero suc rec →\n snd28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left right case zero suc rec)\n\nleft28 : ∀{Γ A B} → Tm28 Γ A → Tm28 Γ (sum28 A B); left28\n = λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right case zero suc rec →\n left28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right case zero suc rec)\n\nright28 : ∀{Γ A B} → Tm28 Γ B → Tm28 Γ (sum28 A B); right28\n = λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case zero suc rec →\n right28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case zero suc rec)\n\ncase28 : ∀{Γ A B C} → Tm28 Γ (sum28 A B) → Tm28 Γ (arr28 A C) → Tm28 Γ (arr28 B C) → Tm28 Γ C; case28\n = λ t u v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec →\n case28 _ _ _ _\n (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)\n (u Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)\n (v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)\n\nzero28 : ∀{Γ} → Tm28 Γ nat28; zero28\n = λ Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc rec → zero28 _\n\nsuc28 : ∀{Γ} → Tm28 Γ nat28 → Tm28 Γ nat28; suc28\n = λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec →\n suc28 _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec)\n\nrec28 : ∀{Γ A} → Tm28 Γ nat28 → Tm28 Γ (arr28 nat28 (arr28 A A)) → Tm28 Γ A → Tm28 Γ A; rec28\n = λ t u v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28 →\n rec28 _ _\n (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)\n (u Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)\n (v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)\n\nv028 : ∀{Γ A} → Tm28 (snoc28 Γ A) A; v028\n = var28 vz28\n\nv128 : ∀{Γ A B} → Tm28 (snoc28 (snoc28 Γ A) B) A; v128\n = var28 (vs28 vz28)\n\nv228 : ∀{Γ A B C} → Tm28 (snoc28 (snoc28 (snoc28 Γ A) B) C) A; v228\n = var28 (vs28 (vs28 vz28))\n\nv328 : ∀{Γ A B C D} → Tm28 (snoc28 (snoc28 (snoc28 (snoc28 Γ A) B) C) D) A; v328\n = var28 (vs28 (vs28 (vs28 vz28)))\n\ntbool28 : Ty28; tbool28\n = sum28 top28 top28\n\ntrue28 : ∀{Γ} → Tm28 Γ tbool28; true28\n = left28 tt28\n\ntfalse28 : ∀{Γ} → Tm28 Γ tbool28; tfalse28\n = right28 tt28\n\nifthenelse28 : ∀{Γ A} → Tm28 Γ (arr28 tbool28 (arr28 A (arr28 A A))); ifthenelse28\n = lam28 (lam28 (lam28 (case28 v228 (lam28 v228) (lam28 v128))))\n\ntimes428 : ∀{Γ A} → Tm28 Γ (arr28 (arr28 A A) (arr28 A A)); times428\n = lam28 (lam28 (app28 v128 (app28 v128 (app28 v128 (app28 v128 v028)))))\n\nadd28 : ∀{Γ} → Tm28 Γ (arr28 nat28 (arr28 nat28 nat28)); add28\n = lam28 (rec28 v028\n (lam28 (lam28 (lam28 (suc28 (app28 v128 v028)))))\n (lam28 v028))\n\nmul28 : ∀{Γ} → Tm28 Γ (arr28 nat28 (arr28 nat28 nat28)); mul28\n = lam28 (rec28 v028\n (lam28 (lam28 (lam28 (app28 (app28 add28 (app28 v128 v028)) v028))))\n (lam28 zero28))\n\nfact28 : ∀{Γ} → Tm28 Γ (arr28 nat28 nat28); fact28\n = lam28 (rec28 v028 (lam28 (lam28 (app28 (app28 mul28 (suc28 v128)) v028)))\n (suc28 zero28))\n{-# OPTIONS --type-in-type #-}\n\nTy29 : Set\nTy29 =\n (Ty29 : Set)\n (nat top bot : Ty29)\n (arr prod sum : Ty29 → Ty29 → Ty29)\n → Ty29\n\nnat29 : Ty29; nat29 = λ _ nat29 _ _ _ _ _ → nat29\ntop29 : Ty29; top29 = λ _ _ top29 _ _ _ _ → top29\nbot29 : Ty29; bot29 = λ _ _ _ bot29 _ _ _ → bot29\n\narr29 : Ty29 → Ty29 → Ty29; arr29\n = λ A B Ty29 nat29 top29 bot29 arr29 prod sum →\n arr29 (A Ty29 nat29 top29 bot29 arr29 prod sum) (B Ty29 nat29 top29 bot29 arr29 prod sum)\n\nprod29 : Ty29 → Ty29 → Ty29; prod29\n = λ A B Ty29 nat29 top29 bot29 arr29 prod29 sum →\n prod29 (A Ty29 nat29 top29 bot29 arr29 prod29 sum) (B Ty29 nat29 top29 bot29 arr29 prod29 sum)\n\nsum29 : Ty29 → Ty29 → Ty29; sum29\n = λ A B Ty29 nat29 top29 bot29 arr29 prod29 sum29 →\n sum29 (A Ty29 nat29 top29 bot29 arr29 prod29 sum29) (B Ty29 nat29 top29 bot29 arr29 prod29 sum29)\n\nCon29 : Set; Con29\n = (Con29 : Set)\n (nil : Con29)\n (snoc : Con29 → Ty29 → Con29)\n → Con29\n\nnil29 : Con29; nil29\n = λ Con29 nil29 snoc → nil29\n\nsnoc29 : Con29 → Ty29 → Con29; snoc29\n = λ Γ A Con29 nil29 snoc29 → snoc29 (Γ Con29 nil29 snoc29) A\n\nVar29 : Con29 → Ty29 → Set; Var29\n = λ Γ A →\n (Var29 : Con29 → Ty29 → Set)\n (vz : ∀ Γ A → Var29 (snoc29 Γ A) A)\n (vs : ∀ Γ B A → Var29 Γ A → Var29 (snoc29 Γ B) A)\n → Var29 Γ A\n\nvz29 : ∀{Γ A} → Var29 (snoc29 Γ A) A; vz29\n = λ Var29 vz29 vs → vz29 _ _\n\nvs29 : ∀{Γ B A} → Var29 Γ A → Var29 (snoc29 Γ B) A; vs29\n = λ x Var29 vz29 vs29 → vs29 _ _ _ (x Var29 vz29 vs29)\n\nTm29 : Con29 → Ty29 → Set; Tm29\n = λ Γ A →\n (Tm29 : Con29 → Ty29 → Set)\n (var : ∀ Γ A → Var29 Γ A → Tm29 Γ A)\n (lam : ∀ Γ A B → Tm29 (snoc29 Γ A) B → Tm29 Γ (arr29 A B))\n (app : ∀ Γ A B → Tm29 Γ (arr29 A B) → Tm29 Γ A → Tm29 Γ B)\n (tt : ∀ Γ → Tm29 Γ top29)\n (pair : ∀ Γ A B → Tm29 Γ A → Tm29 Γ B → Tm29 Γ (prod29 A B))\n (fst : ∀ Γ A B → Tm29 Γ (prod29 A B) → Tm29 Γ A)\n (snd : ∀ Γ A B → Tm29 Γ (prod29 A B) → Tm29 Γ B)\n (left : ∀ Γ A B → Tm29 Γ A → Tm29 Γ (sum29 A B))\n (right : ∀ Γ A B → Tm29 Γ B → Tm29 Γ (sum29 A B))\n (case : ∀ Γ A B C → Tm29 Γ (sum29 A B) → Tm29 Γ (arr29 A C) → Tm29 Γ (arr29 B C) → Tm29 Γ C)\n (zero : ∀ Γ → Tm29 Γ nat29)\n (suc : ∀ Γ → Tm29 Γ nat29 → Tm29 Γ nat29)\n (rec : ∀ Γ A → Tm29 Γ nat29 → Tm29 Γ (arr29 nat29 (arr29 A A)) → Tm29 Γ A → Tm29 Γ A)\n → Tm29 Γ A\n\nvar29 : ∀{Γ A} → Var29 Γ A → Tm29 Γ A; var29\n = λ x Tm29 var29 lam app tt pair fst snd left right case zero suc rec →\n var29 _ _ x\n\nlam29 : ∀{Γ A B} → Tm29 (snoc29 Γ A) B → Tm29 Γ (arr29 A B); lam29\n = λ t Tm29 var29 lam29 app tt pair fst snd left right case zero suc rec →\n lam29 _ _ _ (t Tm29 var29 lam29 app tt pair fst snd left right case zero suc rec)\n\napp29 : ∀{Γ A B} → Tm29 Γ (arr29 A B) → Tm29 Γ A → Tm29 Γ B; app29\n = λ t u Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec →\n app29 _ _ _ (t Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec)\n (u Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec)\n\ntt29 : ∀{Γ} → Tm29 Γ top29; tt29\n = λ Tm29 var29 lam29 app29 tt29 pair fst snd left right case zero suc rec → tt29 _\n\npair29 : ∀{Γ A B} → Tm29 Γ A → Tm29 Γ B → Tm29 Γ (prod29 A B); pair29\n = λ t u Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec →\n pair29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec)\n (u Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec)\n\nfst29 : ∀{Γ A B} → Tm29 Γ (prod29 A B) → Tm29 Γ A; fst29\n = λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd left right case zero suc rec →\n fst29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd left right case zero suc rec)\n\nsnd29 : ∀{Γ A B} → Tm29 Γ (prod29 A B) → Tm29 Γ B; snd29\n = λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left right case zero suc rec →\n snd29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left right case zero suc rec)\n\nleft29 : ∀{Γ A B} → Tm29 Γ A → Tm29 Γ (sum29 A B); left29\n = λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right case zero suc rec →\n left29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right case zero suc rec)\n\nright29 : ∀{Γ A B} → Tm29 Γ B → Tm29 Γ (sum29 A B); right29\n = λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case zero suc rec →\n right29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case zero suc rec)\n\ncase29 : ∀{Γ A B C} → Tm29 Γ (sum29 A B) → Tm29 Γ (arr29 A C) → Tm29 Γ (arr29 B C) → Tm29 Γ C; case29\n = λ t u v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec →\n case29 _ _ _ _\n (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)\n (u Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)\n (v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)\n\nzero29 : ∀{Γ} → Tm29 Γ nat29; zero29\n = λ Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc rec → zero29 _\n\nsuc29 : ∀{Γ} → Tm29 Γ nat29 → Tm29 Γ nat29; suc29\n = λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec →\n suc29 _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec)\n\nrec29 : ∀{Γ A} → Tm29 Γ nat29 → Tm29 Γ (arr29 nat29 (arr29 A A)) → Tm29 Γ A → Tm29 Γ A; rec29\n = λ t u v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29 →\n rec29 _ _\n (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)\n (u Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)\n (v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)\n\nv029 : ∀{Γ A} → Tm29 (snoc29 Γ A) A; v029\n = var29 vz29\n\nv129 : ∀{Γ A B} → Tm29 (snoc29 (snoc29 Γ A) B) A; v129\n = var29 (vs29 vz29)\n\nv229 : ∀{Γ A B C} → Tm29 (snoc29 (snoc29 (snoc29 Γ A) B) C) A; v229\n = var29 (vs29 (vs29 vz29))\n\nv329 : ∀{Γ A B C D} → Tm29 (snoc29 (snoc29 (snoc29 (snoc29 Γ A) B) C) D) A; v329\n = var29 (vs29 (vs29 (vs29 vz29)))\n\ntbool29 : Ty29; tbool29\n = sum29 top29 top29\n\ntrue29 : ∀{Γ} → Tm29 Γ tbool29; true29\n = left29 tt29\n\ntfalse29 : ∀{Γ} → Tm29 Γ tbool29; tfalse29\n = right29 tt29\n\nifthenelse29 : ∀{Γ A} → Tm29 Γ (arr29 tbool29 (arr29 A (arr29 A A))); ifthenelse29\n = lam29 (lam29 (lam29 (case29 v229 (lam29 v229) (lam29 v129))))\n\ntimes429 : ∀{Γ A} → Tm29 Γ (arr29 (arr29 A A) (arr29 A A)); times429\n = lam29 (lam29 (app29 v129 (app29 v129 (app29 v129 (app29 v129 v029)))))\n\nadd29 : ∀{Γ} → Tm29 Γ (arr29 nat29 (arr29 nat29 nat29)); add29\n = lam29 (rec29 v029\n (lam29 (lam29 (lam29 (suc29 (app29 v129 v029)))))\n (lam29 v029))\n\nmul29 : ∀{Γ} → Tm29 Γ (arr29 nat29 (arr29 nat29 nat29)); mul29\n = lam29 (rec29 v029\n (lam29 (lam29 (lam29 (app29 (app29 add29 (app29 v129 v029)) v029))))\n (lam29 zero29))\n\nfact29 : ∀{Γ} → Tm29 Γ (arr29 nat29 nat29); fact29\n = lam29 (rec29 v029 (lam29 (lam29 (app29 (app29 mul29 (suc29 v129)) v029)))\n (suc29 zero29))\n{-# OPTIONS --type-in-type #-}\n\nTy30 : Set\nTy30 =\n (Ty30 : Set)\n (nat top bot : Ty30)\n (arr prod sum : Ty30 → Ty30 → Ty30)\n → Ty30\n\nnat30 : Ty30; nat30 = λ _ nat30 _ _ _ _ _ → nat30\ntop30 : Ty30; top30 = λ _ _ top30 _ _ _ _ → top30\nbot30 : Ty30; bot30 = λ _ _ _ bot30 _ _ _ → bot30\n\narr30 : Ty30 → Ty30 → Ty30; arr30\n = λ A B Ty30 nat30 top30 bot30 arr30 prod sum →\n arr30 (A Ty30 nat30 top30 bot30 arr30 prod sum) (B Ty30 nat30 top30 bot30 arr30 prod sum)\n\nprod30 : Ty30 → Ty30 → Ty30; prod30\n = λ A B Ty30 nat30 top30 bot30 arr30 prod30 sum →\n prod30 (A Ty30 nat30 top30 bot30 arr30 prod30 sum) (B Ty30 nat30 top30 bot30 arr30 prod30 sum)\n\nsum30 : Ty30 → Ty30 → Ty30; sum30\n = λ A B Ty30 nat30 top30 bot30 arr30 prod30 sum30 →\n sum30 (A Ty30 nat30 top30 bot30 arr30 prod30 sum30) (B Ty30 nat30 top30 bot30 arr30 prod30 sum30)\n\nCon30 : Set; Con30\n = (Con30 : Set)\n (nil : Con30)\n (snoc : Con30 → Ty30 → Con30)\n → Con30\n\nnil30 : Con30; nil30\n = λ Con30 nil30 snoc → nil30\n\nsnoc30 : Con30 → Ty30 → Con30; snoc30\n = λ Γ A Con30 nil30 snoc30 → snoc30 (Γ Con30 nil30 snoc30) A\n\nVar30 : Con30 → Ty30 → Set; Var30\n = λ Γ A →\n (Var30 : Con30 → Ty30 → Set)\n (vz : ∀ Γ A → Var30 (snoc30 Γ A) A)\n (vs : ∀ Γ B A → Var30 Γ A → Var30 (snoc30 Γ B) A)\n → Var30 Γ A\n\nvz30 : ∀{Γ A} → Var30 (snoc30 Γ A) A; vz30\n = λ Var30 vz30 vs → vz30 _ _\n\nvs30 : ∀{Γ B A} → Var30 Γ A → Var30 (snoc30 Γ B) A; vs30\n = λ x Var30 vz30 vs30 → vs30 _ _ _ (x Var30 vz30 vs30)\n\nTm30 : Con30 → Ty30 → Set; Tm30\n = λ Γ A →\n (Tm30 : Con30 → Ty30 → Set)\n (var : ∀ Γ A → Var30 Γ A → Tm30 Γ A)\n (lam : ∀ Γ A B → Tm30 (snoc30 Γ A) B → Tm30 Γ (arr30 A B))\n (app : ∀ Γ A B → Tm30 Γ (arr30 A B) → Tm30 Γ A → Tm30 Γ B)\n (tt : ∀ Γ → Tm30 Γ top30)\n (pair : ∀ Γ A B → Tm30 Γ A → Tm30 Γ B → Tm30 Γ (prod30 A B))\n (fst : ∀ Γ A B → Tm30 Γ (prod30 A B) → Tm30 Γ A)\n (snd : ∀ Γ A B → Tm30 Γ (prod30 A B) → Tm30 Γ B)\n (left : ∀ Γ A B → Tm30 Γ A → Tm30 Γ (sum30 A B))\n (right : ∀ Γ A B → Tm30 Γ B → Tm30 Γ (sum30 A B))\n (case : ∀ Γ A B C → Tm30 Γ (sum30 A B) → Tm30 Γ (arr30 A C) → Tm30 Γ (arr30 B C) → Tm30 Γ C)\n (zero : ∀ Γ → Tm30 Γ nat30)\n (suc : ∀ Γ → Tm30 Γ nat30 → Tm30 Γ nat30)\n (rec : ∀ Γ A → Tm30 Γ nat30 → Tm30 Γ (arr30 nat30 (arr30 A A)) → Tm30 Γ A → Tm30 Γ A)\n → Tm30 Γ A\n\nvar30 : ∀{Γ A} → Var30 Γ A → Tm30 Γ A; var30\n = λ x Tm30 var30 lam app tt pair fst snd left right case zero suc rec →\n var30 _ _ x\n\nlam30 : ∀{Γ A B} → Tm30 (snoc30 Γ A) B → Tm30 Γ (arr30 A B); lam30\n = λ t Tm30 var30 lam30 app tt pair fst snd left right case zero suc rec →\n lam30 _ _ _ (t Tm30 var30 lam30 app tt pair fst snd left right case zero suc rec)\n\napp30 : ∀{Γ A B} → Tm30 Γ (arr30 A B) → Tm30 Γ A → Tm30 Γ B; app30\n = λ t u Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec →\n app30 _ _ _ (t Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec)\n (u Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec)\n\ntt30 : ∀{Γ} → Tm30 Γ top30; tt30\n = λ Tm30 var30 lam30 app30 tt30 pair fst snd left right case zero suc rec → tt30 _\n\npair30 : ∀{Γ A B} → Tm30 Γ A → Tm30 Γ B → Tm30 Γ (prod30 A B); pair30\n = λ t u Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec →\n pair30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec)\n (u Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec)\n\nfst30 : ∀{Γ A B} → Tm30 Γ (prod30 A B) → Tm30 Γ A; fst30\n = λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd left right case zero suc rec →\n fst30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd left right case zero suc rec)\n\nsnd30 : ∀{Γ A B} → Tm30 Γ (prod30 A B) → Tm30 Γ B; snd30\n = λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left right case zero suc rec →\n snd30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left right case zero suc rec)\n\nleft30 : ∀{Γ A B} → Tm30 Γ A → Tm30 Γ (sum30 A B); left30\n = λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right case zero suc rec →\n left30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right case zero suc rec)\n\nright30 : ∀{Γ A B} → Tm30 Γ B → Tm30 Γ (sum30 A B); right30\n = λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case zero suc rec →\n right30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case zero suc rec)\n\ncase30 : ∀{Γ A B C} → Tm30 Γ (sum30 A B) → Tm30 Γ (arr30 A C) → Tm30 Γ (arr30 B C) → Tm30 Γ C; case30\n = λ t u v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec →\n case30 _ _ _ _\n (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)\n (u Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)\n (v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)\n\nzero30 : ∀{Γ} → Tm30 Γ nat30; zero30\n = λ Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc rec → zero30 _\n\nsuc30 : ∀{Γ} → Tm30 Γ nat30 → Tm30 Γ nat30; suc30\n = λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec →\n suc30 _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec)\n\nrec30 : ∀{Γ A} → Tm30 Γ nat30 → Tm30 Γ (arr30 nat30 (arr30 A A)) → Tm30 Γ A → Tm30 Γ A; rec30\n = λ t u v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30 →\n rec30 _ _\n (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)\n (u Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)\n (v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)\n\nv030 : ∀{Γ A} → Tm30 (snoc30 Γ A) A; v030\n = var30 vz30\n\nv130 : ∀{Γ A B} → Tm30 (snoc30 (snoc30 Γ A) B) A; v130\n = var30 (vs30 vz30)\n\nv230 : ∀{Γ A B C} → Tm30 (snoc30 (snoc30 (snoc30 Γ A) B) C) A; v230\n = var30 (vs30 (vs30 vz30))\n\nv330 : ∀{Γ A B C D} → Tm30 (snoc30 (snoc30 (snoc30 (snoc30 Γ A) B) C) D) A; v330\n = var30 (vs30 (vs30 (vs30 vz30)))\n\ntbool30 : Ty30; tbool30\n = sum30 top30 top30\n\ntrue30 : ∀{Γ} → Tm30 Γ tbool30; true30\n = left30 tt30\n\ntfalse30 : ∀{Γ} → Tm30 Γ tbool30; tfalse30\n = right30 tt30\n\nifthenelse30 : ∀{Γ A} → Tm30 Γ (arr30 tbool30 (arr30 A (arr30 A A))); ifthenelse30\n = lam30 (lam30 (lam30 (case30 v230 (lam30 v230) (lam30 v130))))\n\ntimes430 : ∀{Γ A} → Tm30 Γ (arr30 (arr30 A A) (arr30 A A)); times430\n = lam30 (lam30 (app30 v130 (app30 v130 (app30 v130 (app30 v130 v030)))))\n\nadd30 : ∀{Γ} → Tm30 Γ (arr30 nat30 (arr30 nat30 nat30)); add30\n = lam30 (rec30 v030\n (lam30 (lam30 (lam30 (suc30 (app30 v130 v030)))))\n (lam30 v030))\n\nmul30 : ∀{Γ} → Tm30 Γ (arr30 nat30 (arr30 nat30 nat30)); mul30\n = lam30 (rec30 v030\n (lam30 (lam30 (lam30 (app30 (app30 add30 (app30 v130 v030)) v030))))\n (lam30 zero30))\n\nfact30 : ∀{Γ} → Tm30 Γ (arr30 nat30 nat30); fact30\n = lam30 (rec30 v030 (lam30 (lam30 (app30 (app30 mul30 (suc30 v130)) v030)))\n (suc30 zero30))\n{-# OPTIONS --type-in-type #-}\n\nTy31 : Set\nTy31 =\n (Ty31 : Set)\n (nat top bot : Ty31)\n (arr prod sum : Ty31 → Ty31 → Ty31)\n → Ty31\n\nnat31 : Ty31; nat31 = λ _ nat31 _ _ _ _ _ → nat31\ntop31 : Ty31; top31 = λ _ _ top31 _ _ _ _ → top31\nbot31 : Ty31; bot31 = λ _ _ _ bot31 _ _ _ → bot31\n\narr31 : Ty31 → Ty31 → Ty31; arr31\n = λ A B Ty31 nat31 top31 bot31 arr31 prod sum →\n arr31 (A Ty31 nat31 top31 bot31 arr31 prod sum) (B Ty31 nat31 top31 bot31 arr31 prod sum)\n\nprod31 : Ty31 → Ty31 → Ty31; prod31\n = λ A B Ty31 nat31 top31 bot31 arr31 prod31 sum →\n prod31 (A Ty31 nat31 top31 bot31 arr31 prod31 sum) (B Ty31 nat31 top31 bot31 arr31 prod31 sum)\n\nsum31 : Ty31 → Ty31 → Ty31; sum31\n = λ A B Ty31 nat31 top31 bot31 arr31 prod31 sum31 →\n sum31 (A Ty31 nat31 top31 bot31 arr31 prod31 sum31) (B Ty31 nat31 top31 bot31 arr31 prod31 sum31)\n\nCon31 : Set; Con31\n = (Con31 : Set)\n (nil : Con31)\n (snoc : Con31 → Ty31 → Con31)\n → Con31\n\nnil31 : Con31; nil31\n = λ Con31 nil31 snoc → nil31\n\nsnoc31 : Con31 → Ty31 → Con31; snoc31\n = λ Γ A Con31 nil31 snoc31 → snoc31 (Γ Con31 nil31 snoc31) A\n\nVar31 : Con31 → Ty31 → Set; Var31\n = λ Γ A →\n (Var31 : Con31 → Ty31 → Set)\n (vz : ∀ Γ A → Var31 (snoc31 Γ A) A)\n (vs : ∀ Γ B A → Var31 Γ A → Var31 (snoc31 Γ B) A)\n → Var31 Γ A\n\nvz31 : ∀{Γ A} → Var31 (snoc31 Γ A) A; vz31\n = λ Var31 vz31 vs → vz31 _ _\n\nvs31 : ∀{Γ B A} → Var31 Γ A → Var31 (snoc31 Γ B) A; vs31\n = λ x Var31 vz31 vs31 → vs31 _ _ _ (x Var31 vz31 vs31)\n\nTm31 : Con31 → Ty31 → Set; Tm31\n = λ Γ A →\n (Tm31 : Con31 → Ty31 → Set)\n (var : ∀ Γ A → Var31 Γ A → Tm31 Γ A)\n (lam : ∀ Γ A B → Tm31 (snoc31 Γ A) B → Tm31 Γ (arr31 A B))\n (app : ∀ Γ A B → Tm31 Γ (arr31 A B) → Tm31 Γ A → Tm31 Γ B)\n (tt : ∀ Γ → Tm31 Γ top31)\n (pair : ∀ Γ A B → Tm31 Γ A → Tm31 Γ B → Tm31 Γ (prod31 A B))\n (fst : ∀ Γ A B → Tm31 Γ (prod31 A B) → Tm31 Γ A)\n (snd : ∀ Γ A B → Tm31 Γ (prod31 A B) → Tm31 Γ B)\n (left : ∀ Γ A B → Tm31 Γ A → Tm31 Γ (sum31 A B))\n (right : ∀ Γ A B → Tm31 Γ B → Tm31 Γ (sum31 A B))\n (case : ∀ Γ A B C → Tm31 Γ (sum31 A B) → Tm31 Γ (arr31 A C) → Tm31 Γ (arr31 B C) → Tm31 Γ C)\n (zero : ∀ Γ → Tm31 Γ nat31)\n (suc : ∀ Γ → Tm31 Γ nat31 → Tm31 Γ nat31)\n (rec : ∀ Γ A → Tm31 Γ nat31 → Tm31 Γ (arr31 nat31 (arr31 A A)) → Tm31 Γ A → Tm31 Γ A)\n → Tm31 Γ A\n\nvar31 : ∀{Γ A} → Var31 Γ A → Tm31 Γ A; var31\n = λ x Tm31 var31 lam app tt pair fst snd left right case zero suc rec →\n var31 _ _ x\n\nlam31 : ∀{Γ A B} → Tm31 (snoc31 Γ A) B → Tm31 Γ (arr31 A B); lam31\n = λ t Tm31 var31 lam31 app tt pair fst snd left right case zero suc rec →\n lam31 _ _ _ (t Tm31 var31 lam31 app tt pair fst snd left right case zero suc rec)\n\napp31 : ∀{Γ A B} → Tm31 Γ (arr31 A B) → Tm31 Γ A → Tm31 Γ B; app31\n = λ t u Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec →\n app31 _ _ _ (t Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec)\n (u Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec)\n\ntt31 : ∀{Γ} → Tm31 Γ top31; tt31\n = λ Tm31 var31 lam31 app31 tt31 pair fst snd left right case zero suc rec → tt31 _\n\npair31 : ∀{Γ A B} → Tm31 Γ A → Tm31 Γ B → Tm31 Γ (prod31 A B); pair31\n = λ t u Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec →\n pair31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec)\n (u Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec)\n\nfst31 : ∀{Γ A B} → Tm31 Γ (prod31 A B) → Tm31 Γ A; fst31\n = λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd left right case zero suc rec →\n fst31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd left right case zero suc rec)\n\nsnd31 : ∀{Γ A B} → Tm31 Γ (prod31 A B) → Tm31 Γ B; snd31\n = λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left right case zero suc rec →\n snd31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left right case zero suc rec)\n\nleft31 : ∀{Γ A B} → Tm31 Γ A → Tm31 Γ (sum31 A B); left31\n = λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right case zero suc rec →\n left31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right case zero suc rec)\n\nright31 : ∀{Γ A B} → Tm31 Γ B → Tm31 Γ (sum31 A B); right31\n = λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case zero suc rec →\n right31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case zero suc rec)\n\ncase31 : ∀{Γ A B C} → Tm31 Γ (sum31 A B) → Tm31 Γ (arr31 A C) → Tm31 Γ (arr31 B C) → Tm31 Γ C; case31\n = λ t u v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec →\n case31 _ _ _ _\n (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)\n (u Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)\n (v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)\n\nzero31 : ∀{Γ} → Tm31 Γ nat31; zero31\n = λ Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc rec → zero31 _\n\nsuc31 : ∀{Γ} → Tm31 Γ nat31 → Tm31 Γ nat31; suc31\n = λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec →\n suc31 _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec)\n\nrec31 : ∀{Γ A} → Tm31 Γ nat31 → Tm31 Γ (arr31 nat31 (arr31 A A)) → Tm31 Γ A → Tm31 Γ A; rec31\n = λ t u v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31 →\n rec31 _ _\n (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)\n (u Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)\n (v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)\n\nv031 : ∀{Γ A} → Tm31 (snoc31 Γ A) A; v031\n = var31 vz31\n\nv131 : ∀{Γ A B} → Tm31 (snoc31 (snoc31 Γ A) B) A; v131\n = var31 (vs31 vz31)\n\nv231 : ∀{Γ A B C} → Tm31 (snoc31 (snoc31 (snoc31 Γ A) B) C) A; v231\n = var31 (vs31 (vs31 vz31))\n\nv331 : ∀{Γ A B C D} → Tm31 (snoc31 (snoc31 (snoc31 (snoc31 Γ A) B) C) D) A; v331\n = var31 (vs31 (vs31 (vs31 vz31)))\n\ntbool31 : Ty31; tbool31\n = sum31 top31 top31\n\ntrue31 : ∀{Γ} → Tm31 Γ tbool31; true31\n = left31 tt31\n\ntfalse31 : ∀{Γ} → Tm31 Γ tbool31; tfalse31\n = right31 tt31\n\nifthenelse31 : ∀{Γ A} → Tm31 Γ (arr31 tbool31 (arr31 A (arr31 A A))); ifthenelse31\n = lam31 (lam31 (lam31 (case31 v231 (lam31 v231) (lam31 v131))))\n\ntimes431 : ∀{Γ A} → Tm31 Γ (arr31 (arr31 A A) (arr31 A A)); times431\n = lam31 (lam31 (app31 v131 (app31 v131 (app31 v131 (app31 v131 v031)))))\n\nadd31 : ∀{Γ} → Tm31 Γ (arr31 nat31 (arr31 nat31 nat31)); add31\n = lam31 (rec31 v031\n (lam31 (lam31 (lam31 (suc31 (app31 v131 v031)))))\n (lam31 v031))\n\nmul31 : ∀{Γ} → Tm31 Γ (arr31 nat31 (arr31 nat31 nat31)); mul31\n = lam31 (rec31 v031\n (lam31 (lam31 (lam31 (app31 (app31 add31 (app31 v131 v031)) v031))))\n (lam31 zero31))\n\nfact31 : ∀{Γ} → Tm31 Γ (arr31 nat31 nat31); fact31\n = lam31 (rec31 v031 (lam31 (lam31 (app31 (app31 mul31 (suc31 v131)) v031)))\n (suc31 zero31))\n{-# OPTIONS --type-in-type #-}\n\nTy32 : Set\nTy32 =\n (Ty32 : Set)\n (nat top bot : Ty32)\n (arr prod sum : Ty32 → Ty32 → Ty32)\n → Ty32\n\nnat32 : Ty32; nat32 = λ _ nat32 _ _ _ _ _ → nat32\ntop32 : Ty32; top32 = λ _ _ top32 _ _ _ _ → top32\nbot32 : Ty32; bot32 = λ _ _ _ bot32 _ _ _ → bot32\n\narr32 : Ty32 → Ty32 → Ty32; arr32\n = λ A B Ty32 nat32 top32 bot32 arr32 prod sum →\n arr32 (A Ty32 nat32 top32 bot32 arr32 prod sum) (B Ty32 nat32 top32 bot32 arr32 prod sum)\n\nprod32 : Ty32 → Ty32 → Ty32; prod32\n = λ A B Ty32 nat32 top32 bot32 arr32 prod32 sum →\n prod32 (A Ty32 nat32 top32 bot32 arr32 prod32 sum) (B Ty32 nat32 top32 bot32 arr32 prod32 sum)\n\nsum32 : Ty32 → Ty32 → Ty32; sum32\n = λ A B Ty32 nat32 top32 bot32 arr32 prod32 sum32 →\n sum32 (A Ty32 nat32 top32 bot32 arr32 prod32 sum32) (B Ty32 nat32 top32 bot32 arr32 prod32 sum32)\n\nCon32 : Set; Con32\n = (Con32 : Set)\n (nil : Con32)\n (snoc : Con32 → Ty32 → Con32)\n → Con32\n\nnil32 : Con32; nil32\n = λ Con32 nil32 snoc → nil32\n\nsnoc32 : Con32 → Ty32 → Con32; snoc32\n = λ Γ A Con32 nil32 snoc32 → snoc32 (Γ Con32 nil32 snoc32) A\n\nVar32 : Con32 → Ty32 → Set; Var32\n = λ Γ A →\n (Var32 : Con32 → Ty32 → Set)\n (vz : ∀ Γ A → Var32 (snoc32 Γ A) A)\n (vs : ∀ Γ B A → Var32 Γ A → Var32 (snoc32 Γ B) A)\n → Var32 Γ A\n\nvz32 : ∀{Γ A} → Var32 (snoc32 Γ A) A; vz32\n = λ Var32 vz32 vs → vz32 _ _\n\nvs32 : ∀{Γ B A} → Var32 Γ A → Var32 (snoc32 Γ B) A; vs32\n = λ x Var32 vz32 vs32 → vs32 _ _ _ (x Var32 vz32 vs32)\n\nTm32 : Con32 → Ty32 → Set; Tm32\n = λ Γ A →\n (Tm32 : Con32 → Ty32 → Set)\n (var : ∀ Γ A → Var32 Γ A → Tm32 Γ A)\n (lam : ∀ Γ A B → Tm32 (snoc32 Γ A) B → Tm32 Γ (arr32 A B))\n (app : ∀ Γ A B → Tm32 Γ (arr32 A B) → Tm32 Γ A → Tm32 Γ B)\n (tt : ∀ Γ → Tm32 Γ top32)\n (pair : ∀ Γ A B → Tm32 Γ A → Tm32 Γ B → Tm32 Γ (prod32 A B))\n (fst : ∀ Γ A B → Tm32 Γ (prod32 A B) → Tm32 Γ A)\n (snd : ∀ Γ A B → Tm32 Γ (prod32 A B) → Tm32 Γ B)\n (left : ∀ Γ A B → Tm32 Γ A → Tm32 Γ (sum32 A B))\n (right : ∀ Γ A B → Tm32 Γ B → Tm32 Γ (sum32 A B))\n (case : ∀ Γ A B C → Tm32 Γ (sum32 A B) → Tm32 Γ (arr32 A C) → Tm32 Γ (arr32 B C) → Tm32 Γ C)\n (zero : ∀ Γ → Tm32 Γ nat32)\n (suc : ∀ Γ → Tm32 Γ nat32 → Tm32 Γ nat32)\n (rec : ∀ Γ A → Tm32 Γ nat32 → Tm32 Γ (arr32 nat32 (arr32 A A)) → Tm32 Γ A → Tm32 Γ A)\n → Tm32 Γ A\n\nvar32 : ∀{Γ A} → Var32 Γ A → Tm32 Γ A; var32\n = λ x Tm32 var32 lam app tt pair fst snd left right case zero suc rec →\n var32 _ _ x\n\nlam32 : ∀{Γ A B} → Tm32 (snoc32 Γ A) B → Tm32 Γ (arr32 A B); lam32\n = λ t Tm32 var32 lam32 app tt pair fst snd left right case zero suc rec →\n lam32 _ _ _ (t Tm32 var32 lam32 app tt pair fst snd left right case zero suc rec)\n\napp32 : ∀{Γ A B} → Tm32 Γ (arr32 A B) → Tm32 Γ A → Tm32 Γ B; app32\n = λ t u Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec →\n app32 _ _ _ (t Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec)\n (u Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec)\n\ntt32 : ∀{Γ} → Tm32 Γ top32; tt32\n = λ Tm32 var32 lam32 app32 tt32 pair fst snd left right case zero suc rec → tt32 _\n\npair32 : ∀{Γ A B} → Tm32 Γ A → Tm32 Γ B → Tm32 Γ (prod32 A B); pair32\n = λ t u Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec →\n pair32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec)\n (u Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec)\n\nfst32 : ∀{Γ A B} → Tm32 Γ (prod32 A B) → Tm32 Γ A; fst32\n = λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd left right case zero suc rec →\n fst32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd left right case zero suc rec)\n\nsnd32 : ∀{Γ A B} → Tm32 Γ (prod32 A B) → Tm32 Γ B; snd32\n = λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left right case zero suc rec →\n snd32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left right case zero suc rec)\n\nleft32 : ∀{Γ A B} → Tm32 Γ A → Tm32 Γ (sum32 A B); left32\n = λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right case zero suc rec →\n left32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right case zero suc rec)\n\nright32 : ∀{Γ A B} → Tm32 Γ B → Tm32 Γ (sum32 A B); right32\n = λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case zero suc rec →\n right32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case zero suc rec)\n\ncase32 : ∀{Γ A B C} → Tm32 Γ (sum32 A B) → Tm32 Γ (arr32 A C) → Tm32 Γ (arr32 B C) → Tm32 Γ C; case32\n = λ t u v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec →\n case32 _ _ _ _\n (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)\n (u Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)\n (v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)\n\nzero32 : ∀{Γ} → Tm32 Γ nat32; zero32\n = λ Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc rec → zero32 _\n\nsuc32 : ∀{Γ} → Tm32 Γ nat32 → Tm32 Γ nat32; suc32\n = λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec →\n suc32 _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec)\n\nrec32 : ∀{Γ A} → Tm32 Γ nat32 → Tm32 Γ (arr32 nat32 (arr32 A A)) → Tm32 Γ A → Tm32 Γ A; rec32\n = λ t u v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32 →\n rec32 _ _\n (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)\n (u Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)\n (v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)\n\nv032 : ∀{Γ A} → Tm32 (snoc32 Γ A) A; v032\n = var32 vz32\n\nv132 : ∀{Γ A B} → Tm32 (snoc32 (snoc32 Γ A) B) A; v132\n = var32 (vs32 vz32)\n\nv232 : ∀{Γ A B C} → Tm32 (snoc32 (snoc32 (snoc32 Γ A) B) C) A; v232\n = var32 (vs32 (vs32 vz32))\n\nv332 : ∀{Γ A B C D} → Tm32 (snoc32 (snoc32 (snoc32 (snoc32 Γ A) B) C) D) A; v332\n = var32 (vs32 (vs32 (vs32 vz32)))\n\ntbool32 : Ty32; tbool32\n = sum32 top32 top32\n\ntrue32 : ∀{Γ} → Tm32 Γ tbool32; true32\n = left32 tt32\n\ntfalse32 : ∀{Γ} → Tm32 Γ tbool32; tfalse32\n = right32 tt32\n\nifthenelse32 : ∀{Γ A} → Tm32 Γ (arr32 tbool32 (arr32 A (arr32 A A))); ifthenelse32\n = lam32 (lam32 (lam32 (case32 v232 (lam32 v232) (lam32 v132))))\n\ntimes432 : ∀{Γ A} → Tm32 Γ (arr32 (arr32 A A) (arr32 A A)); times432\n = lam32 (lam32 (app32 v132 (app32 v132 (app32 v132 (app32 v132 v032)))))\n\nadd32 : ∀{Γ} → Tm32 Γ (arr32 nat32 (arr32 nat32 nat32)); add32\n = lam32 (rec32 v032\n (lam32 (lam32 (lam32 (suc32 (app32 v132 v032)))))\n (lam32 v032))\n\nmul32 : ∀{Γ} → Tm32 Γ (arr32 nat32 (arr32 nat32 nat32)); mul32\n = lam32 (rec32 v032\n (lam32 (lam32 (lam32 (app32 (app32 add32 (app32 v132 v032)) v032))))\n (lam32 zero32))\n\nfact32 : ∀{Γ} → Tm32 Γ (arr32 nat32 nat32); fact32\n = lam32 (rec32 v032 (lam32 (lam32 (app32 (app32 mul32 (suc32 v132)) v032)))\n (suc32 zero32))\n{-# OPTIONS --type-in-type #-}\n\nTy33 : Set\nTy33 =\n (Ty33 : Set)\n (nat top bot : Ty33)\n (arr prod sum : Ty33 → Ty33 → Ty33)\n → Ty33\n\nnat33 : Ty33; nat33 = λ _ nat33 _ _ _ _ _ → nat33\ntop33 : Ty33; top33 = λ _ _ top33 _ _ _ _ → top33\nbot33 : Ty33; bot33 = λ _ _ _ bot33 _ _ _ → bot33\n\narr33 : Ty33 → Ty33 → Ty33; arr33\n = λ A B Ty33 nat33 top33 bot33 arr33 prod sum →\n arr33 (A Ty33 nat33 top33 bot33 arr33 prod sum) (B Ty33 nat33 top33 bot33 arr33 prod sum)\n\nprod33 : Ty33 → Ty33 → Ty33; prod33\n = λ A B Ty33 nat33 top33 bot33 arr33 prod33 sum →\n prod33 (A Ty33 nat33 top33 bot33 arr33 prod33 sum) (B Ty33 nat33 top33 bot33 arr33 prod33 sum)\n\nsum33 : Ty33 → Ty33 → Ty33; sum33\n = λ A B Ty33 nat33 top33 bot33 arr33 prod33 sum33 →\n sum33 (A Ty33 nat33 top33 bot33 arr33 prod33 sum33) (B Ty33 nat33 top33 bot33 arr33 prod33 sum33)\n\nCon33 : Set; Con33\n = (Con33 : Set)\n (nil : Con33)\n (snoc : Con33 → Ty33 → Con33)\n → Con33\n\nnil33 : Con33; nil33\n = λ Con33 nil33 snoc → nil33\n\nsnoc33 : Con33 → Ty33 → Con33; snoc33\n = λ Γ A Con33 nil33 snoc33 → snoc33 (Γ Con33 nil33 snoc33) A\n\nVar33 : Con33 → Ty33 → Set; Var33\n = λ Γ A →\n (Var33 : Con33 → Ty33 → Set)\n (vz : ∀ Γ A → Var33 (snoc33 Γ A) A)\n (vs : ∀ Γ B A → Var33 Γ A → Var33 (snoc33 Γ B) A)\n → Var33 Γ A\n\nvz33 : ∀{Γ A} → Var33 (snoc33 Γ A) A; vz33\n = λ Var33 vz33 vs → vz33 _ _\n\nvs33 : ∀{Γ B A} → Var33 Γ A → Var33 (snoc33 Γ B) A; vs33\n = λ x Var33 vz33 vs33 → vs33 _ _ _ (x Var33 vz33 vs33)\n\nTm33 : Con33 → Ty33 → Set; Tm33\n = λ Γ A →\n (Tm33 : Con33 → Ty33 → Set)\n (var : ∀ Γ A → Var33 Γ A → Tm33 Γ A)\n (lam : ∀ Γ A B → Tm33 (snoc33 Γ A) B → Tm33 Γ (arr33 A B))\n (app : ∀ Γ A B → Tm33 Γ (arr33 A B) → Tm33 Γ A → Tm33 Γ B)\n (tt : ∀ Γ → Tm33 Γ top33)\n (pair : ∀ Γ A B → Tm33 Γ A → Tm33 Γ B → Tm33 Γ (prod33 A B))\n (fst : ∀ Γ A B → Tm33 Γ (prod33 A B) → Tm33 Γ A)\n (snd : ∀ Γ A B → Tm33 Γ (prod33 A B) → Tm33 Γ B)\n (left : ∀ Γ A B → Tm33 Γ A → Tm33 Γ (sum33 A B))\n (right : ∀ Γ A B → Tm33 Γ B → Tm33 Γ (sum33 A B))\n (case : ∀ Γ A B C → Tm33 Γ (sum33 A B) → Tm33 Γ (arr33 A C) → Tm33 Γ (arr33 B C) → Tm33 Γ C)\n (zero : ∀ Γ → Tm33 Γ nat33)\n (suc : ∀ Γ → Tm33 Γ nat33 → Tm33 Γ nat33)\n (rec : ∀ Γ A → Tm33 Γ nat33 → Tm33 Γ (arr33 nat33 (arr33 A A)) → Tm33 Γ A → Tm33 Γ A)\n → Tm33 Γ A\n\nvar33 : ∀{Γ A} → Var33 Γ A → Tm33 Γ A; var33\n = λ x Tm33 var33 lam app tt pair fst snd left right case zero suc rec →\n var33 _ _ x\n\nlam33 : ∀{Γ A B} → Tm33 (snoc33 Γ A) B → Tm33 Γ (arr33 A B); lam33\n = λ t Tm33 var33 lam33 app tt pair fst snd left right case zero suc rec →\n lam33 _ _ _ (t Tm33 var33 lam33 app tt pair fst snd left right case zero suc rec)\n\napp33 : ∀{Γ A B} → Tm33 Γ (arr33 A B) → Tm33 Γ A → Tm33 Γ B; app33\n = λ t u Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec →\n app33 _ _ _ (t Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec)\n (u Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec)\n\ntt33 : ∀{Γ} → Tm33 Γ top33; tt33\n = λ Tm33 var33 lam33 app33 tt33 pair fst snd left right case zero suc rec → tt33 _\n\npair33 : ∀{Γ A B} → Tm33 Γ A → Tm33 Γ B → Tm33 Γ (prod33 A B); pair33\n = λ t u Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec →\n pair33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec)\n (u Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec)\n\nfst33 : ∀{Γ A B} → Tm33 Γ (prod33 A B) → Tm33 Γ A; fst33\n = λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd left right case zero suc rec →\n fst33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd left right case zero suc rec)\n\nsnd33 : ∀{Γ A B} → Tm33 Γ (prod33 A B) → Tm33 Γ B; snd33\n = λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left right case zero suc rec →\n snd33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left right case zero suc rec)\n\nleft33 : ∀{Γ A B} → Tm33 Γ A → Tm33 Γ (sum33 A B); left33\n = λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right case zero suc rec →\n left33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right case zero suc rec)\n\nright33 : ∀{Γ A B} → Tm33 Γ B → Tm33 Γ (sum33 A B); right33\n = λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case zero suc rec →\n right33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case zero suc rec)\n\ncase33 : ∀{Γ A B C} → Tm33 Γ (sum33 A B) → Tm33 Γ (arr33 A C) → Tm33 Γ (arr33 B C) → Tm33 Γ C; case33\n = λ t u v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec →\n case33 _ _ _ _\n (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)\n (u Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)\n (v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)\n\nzero33 : ∀{Γ} → Tm33 Γ nat33; zero33\n = λ Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc rec → zero33 _\n\nsuc33 : ∀{Γ} → Tm33 Γ nat33 → Tm33 Γ nat33; suc33\n = λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec →\n suc33 _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec)\n\nrec33 : ∀{Γ A} → Tm33 Γ nat33 → Tm33 Γ (arr33 nat33 (arr33 A A)) → Tm33 Γ A → Tm33 Γ A; rec33\n = λ t u v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33 →\n rec33 _ _\n (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)\n (u Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)\n (v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)\n\nv033 : ∀{Γ A} → Tm33 (snoc33 Γ A) A; v033\n = var33 vz33\n\nv133 : ∀{Γ A B} → Tm33 (snoc33 (snoc33 Γ A) B) A; v133\n = var33 (vs33 vz33)\n\nv233 : ∀{Γ A B C} → Tm33 (snoc33 (snoc33 (snoc33 Γ A) B) C) A; v233\n = var33 (vs33 (vs33 vz33))\n\nv333 : ∀{Γ A B C D} → Tm33 (snoc33 (snoc33 (snoc33 (snoc33 Γ A) B) C) D) A; v333\n = var33 (vs33 (vs33 (vs33 vz33)))\n\ntbool33 : Ty33; tbool33\n = sum33 top33 top33\n\ntrue33 : ∀{Γ} → Tm33 Γ tbool33; true33\n = left33 tt33\n\ntfalse33 : ∀{Γ} → Tm33 Γ tbool33; tfalse33\n = right33 tt33\n\nifthenelse33 : ∀{Γ A} → Tm33 Γ (arr33 tbool33 (arr33 A (arr33 A A))); ifthenelse33\n = lam33 (lam33 (lam33 (case33 v233 (lam33 v233) (lam33 v133))))\n\ntimes433 : ∀{Γ A} → Tm33 Γ (arr33 (arr33 A A) (arr33 A A)); times433\n = lam33 (lam33 (app33 v133 (app33 v133 (app33 v133 (app33 v133 v033)))))\n\nadd33 : ∀{Γ} → Tm33 Γ (arr33 nat33 (arr33 nat33 nat33)); add33\n = lam33 (rec33 v033\n (lam33 (lam33 (lam33 (suc33 (app33 v133 v033)))))\n (lam33 v033))\n\nmul33 : ∀{Γ} → Tm33 Γ (arr33 nat33 (arr33 nat33 nat33)); mul33\n = lam33 (rec33 v033\n (lam33 (lam33 (lam33 (app33 (app33 add33 (app33 v133 v033)) v033))))\n (lam33 zero33))\n\nfact33 : ∀{Γ} → Tm33 Γ (arr33 nat33 nat33); fact33\n = lam33 (rec33 v033 (lam33 (lam33 (app33 (app33 mul33 (suc33 v133)) v033)))\n (suc33 zero33))\n{-# OPTIONS --type-in-type #-}\n\nTy34 : Set\nTy34 =\n (Ty34 : Set)\n (nat top bot : Ty34)\n (arr prod sum : Ty34 → Ty34 → Ty34)\n → Ty34\n\nnat34 : Ty34; nat34 = λ _ nat34 _ _ _ _ _ → nat34\ntop34 : Ty34; top34 = λ _ _ top34 _ _ _ _ → top34\nbot34 : Ty34; bot34 = λ _ _ _ bot34 _ _ _ → bot34\n\narr34 : Ty34 → Ty34 → Ty34; arr34\n = λ A B Ty34 nat34 top34 bot34 arr34 prod sum →\n arr34 (A Ty34 nat34 top34 bot34 arr34 prod sum) (B Ty34 nat34 top34 bot34 arr34 prod sum)\n\nprod34 : Ty34 → Ty34 → Ty34; prod34\n = λ A B Ty34 nat34 top34 bot34 arr34 prod34 sum →\n prod34 (A Ty34 nat34 top34 bot34 arr34 prod34 sum) (B Ty34 nat34 top34 bot34 arr34 prod34 sum)\n\nsum34 : Ty34 → Ty34 → Ty34; sum34\n = λ A B Ty34 nat34 top34 bot34 arr34 prod34 sum34 →\n sum34 (A Ty34 nat34 top34 bot34 arr34 prod34 sum34) (B Ty34 nat34 top34 bot34 arr34 prod34 sum34)\n\nCon34 : Set; Con34\n = (Con34 : Set)\n (nil : Con34)\n (snoc : Con34 → Ty34 → Con34)\n → Con34\n\nnil34 : Con34; nil34\n = λ Con34 nil34 snoc → nil34\n\nsnoc34 : Con34 → Ty34 → Con34; snoc34\n = λ Γ A Con34 nil34 snoc34 → snoc34 (Γ Con34 nil34 snoc34) A\n\nVar34 : Con34 → Ty34 → Set; Var34\n = λ Γ A →\n (Var34 : Con34 → Ty34 → Set)\n (vz : ∀ Γ A → Var34 (snoc34 Γ A) A)\n (vs : ∀ Γ B A → Var34 Γ A → Var34 (snoc34 Γ B) A)\n → Var34 Γ A\n\nvz34 : ∀{Γ A} → Var34 (snoc34 Γ A) A; vz34\n = λ Var34 vz34 vs → vz34 _ _\n\nvs34 : ∀{Γ B A} → Var34 Γ A → Var34 (snoc34 Γ B) A; vs34\n = λ x Var34 vz34 vs34 → vs34 _ _ _ (x Var34 vz34 vs34)\n\nTm34 : Con34 → Ty34 → Set; Tm34\n = λ Γ A →\n (Tm34 : Con34 → Ty34 → Set)\n (var : ∀ Γ A → Var34 Γ A → Tm34 Γ A)\n (lam : ∀ Γ A B → Tm34 (snoc34 Γ A) B → Tm34 Γ (arr34 A B))\n (app : ∀ Γ A B → Tm34 Γ (arr34 A B) → Tm34 Γ A → Tm34 Γ B)\n (tt : ∀ Γ → Tm34 Γ top34)\n (pair : ∀ Γ A B → Tm34 Γ A → Tm34 Γ B → Tm34 Γ (prod34 A B))\n (fst : ∀ Γ A B → Tm34 Γ (prod34 A B) → Tm34 Γ A)\n (snd : ∀ Γ A B → Tm34 Γ (prod34 A B) → Tm34 Γ B)\n (left : ∀ Γ A B → Tm34 Γ A → Tm34 Γ (sum34 A B))\n (right : ∀ Γ A B → Tm34 Γ B → Tm34 Γ (sum34 A B))\n (case : ∀ Γ A B C → Tm34 Γ (sum34 A B) → Tm34 Γ (arr34 A C) → Tm34 Γ (arr34 B C) → Tm34 Γ C)\n (zero : ∀ Γ → Tm34 Γ nat34)\n (suc : ∀ Γ → Tm34 Γ nat34 → Tm34 Γ nat34)\n (rec : ∀ Γ A → Tm34 Γ nat34 → Tm34 Γ (arr34 nat34 (arr34 A A)) → Tm34 Γ A → Tm34 Γ A)\n → Tm34 Γ A\n\nvar34 : ∀{Γ A} → Var34 Γ A → Tm34 Γ A; var34\n = λ x Tm34 var34 lam app tt pair fst snd left right case zero suc rec →\n var34 _ _ x\n\nlam34 : ∀{Γ A B} → Tm34 (snoc34 Γ A) B → Tm34 Γ (arr34 A B); lam34\n = λ t Tm34 var34 lam34 app tt pair fst snd left right case zero suc rec →\n lam34 _ _ _ (t Tm34 var34 lam34 app tt pair fst snd left right case zero suc rec)\n\napp34 : ∀{Γ A B} → Tm34 Γ (arr34 A B) → Tm34 Γ A → Tm34 Γ B; app34\n = λ t u Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec →\n app34 _ _ _ (t Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec)\n (u Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec)\n\ntt34 : ∀{Γ} → Tm34 Γ top34; tt34\n = λ Tm34 var34 lam34 app34 tt34 pair fst snd left right case zero suc rec → tt34 _\n\npair34 : ∀{Γ A B} → Tm34 Γ A → Tm34 Γ B → Tm34 Γ (prod34 A B); pair34\n = λ t u Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec →\n pair34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec)\n (u Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec)\n\nfst34 : ∀{Γ A B} → Tm34 Γ (prod34 A B) → Tm34 Γ A; fst34\n = λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd left right case zero suc rec →\n fst34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd left right case zero suc rec)\n\nsnd34 : ∀{Γ A B} → Tm34 Γ (prod34 A B) → Tm34 Γ B; snd34\n = λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left right case zero suc rec →\n snd34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left right case zero suc rec)\n\nleft34 : ∀{Γ A B} → Tm34 Γ A → Tm34 Γ (sum34 A B); left34\n = λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right case zero suc rec →\n left34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right case zero suc rec)\n\nright34 : ∀{Γ A B} → Tm34 Γ B → Tm34 Γ (sum34 A B); right34\n = λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case zero suc rec →\n right34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case zero suc rec)\n\ncase34 : ∀{Γ A B C} → Tm34 Γ (sum34 A B) → Tm34 Γ (arr34 A C) → Tm34 Γ (arr34 B C) → Tm34 Γ C; case34\n = λ t u v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec →\n case34 _ _ _ _\n (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)\n (u Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)\n (v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)\n\nzero34 : ∀{Γ} → Tm34 Γ nat34; zero34\n = λ Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc rec → zero34 _\n\nsuc34 : ∀{Γ} → Tm34 Γ nat34 → Tm34 Γ nat34; suc34\n = λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec →\n suc34 _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec)\n\nrec34 : ∀{Γ A} → Tm34 Γ nat34 → Tm34 Γ (arr34 nat34 (arr34 A A)) → Tm34 Γ A → Tm34 Γ A; rec34\n = λ t u v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34 →\n rec34 _ _\n (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)\n (u Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)\n (v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)\n\nv034 : ∀{Γ A} → Tm34 (snoc34 Γ A) A; v034\n = var34 vz34\n\nv134 : ∀{Γ A B} → Tm34 (snoc34 (snoc34 Γ A) B) A; v134\n = var34 (vs34 vz34)\n\nv234 : ∀{Γ A B C} → Tm34 (snoc34 (snoc34 (snoc34 Γ A) B) C) A; v234\n = var34 (vs34 (vs34 vz34))\n\nv334 : ∀{Γ A B C D} → Tm34 (snoc34 (snoc34 (snoc34 (snoc34 Γ A) B) C) D) A; v334\n = var34 (vs34 (vs34 (vs34 vz34)))\n\ntbool34 : Ty34; tbool34\n = sum34 top34 top34\n\ntrue34 : ∀{Γ} → Tm34 Γ tbool34; true34\n = left34 tt34\n\ntfalse34 : ∀{Γ} → Tm34 Γ tbool34; tfalse34\n = right34 tt34\n\nifthenelse34 : ∀{Γ A} → Tm34 Γ (arr34 tbool34 (arr34 A (arr34 A A))); ifthenelse34\n = lam34 (lam34 (lam34 (case34 v234 (lam34 v234) (lam34 v134))))\n\ntimes434 : ∀{Γ A} → Tm34 Γ (arr34 (arr34 A A) (arr34 A A)); times434\n = lam34 (lam34 (app34 v134 (app34 v134 (app34 v134 (app34 v134 v034)))))\n\nadd34 : ∀{Γ} → Tm34 Γ (arr34 nat34 (arr34 nat34 nat34)); add34\n = lam34 (rec34 v034\n (lam34 (lam34 (lam34 (suc34 (app34 v134 v034)))))\n (lam34 v034))\n\nmul34 : ∀{Γ} → Tm34 Γ (arr34 nat34 (arr34 nat34 nat34)); mul34\n = lam34 (rec34 v034\n (lam34 (lam34 (lam34 (app34 (app34 add34 (app34 v134 v034)) v034))))\n (lam34 zero34))\n\nfact34 : ∀{Γ} → Tm34 Γ (arr34 nat34 nat34); fact34\n = lam34 (rec34 v034 (lam34 (lam34 (app34 (app34 mul34 (suc34 v134)) v034)))\n (suc34 zero34))\n{-# OPTIONS --type-in-type #-}\n\nTy35 : Set\nTy35 =\n (Ty35 : Set)\n (nat top bot : Ty35)\n (arr prod sum : Ty35 → Ty35 → Ty35)\n → Ty35\n\nnat35 : Ty35; nat35 = λ _ nat35 _ _ _ _ _ → nat35\ntop35 : Ty35; top35 = λ _ _ top35 _ _ _ _ → top35\nbot35 : Ty35; bot35 = λ _ _ _ bot35 _ _ _ → bot35\n\narr35 : Ty35 → Ty35 → Ty35; arr35\n = λ A B Ty35 nat35 top35 bot35 arr35 prod sum →\n arr35 (A Ty35 nat35 top35 bot35 arr35 prod sum) (B Ty35 nat35 top35 bot35 arr35 prod sum)\n\nprod35 : Ty35 → Ty35 → Ty35; prod35\n = λ A B Ty35 nat35 top35 bot35 arr35 prod35 sum →\n prod35 (A Ty35 nat35 top35 bot35 arr35 prod35 sum) (B Ty35 nat35 top35 bot35 arr35 prod35 sum)\n\nsum35 : Ty35 → Ty35 → Ty35; sum35\n = λ A B Ty35 nat35 top35 bot35 arr35 prod35 sum35 →\n sum35 (A Ty35 nat35 top35 bot35 arr35 prod35 sum35) (B Ty35 nat35 top35 bot35 arr35 prod35 sum35)\n\nCon35 : Set; Con35\n = (Con35 : Set)\n (nil : Con35)\n (snoc : Con35 → Ty35 → Con35)\n → Con35\n\nnil35 : Con35; nil35\n = λ Con35 nil35 snoc → nil35\n\nsnoc35 : Con35 → Ty35 → Con35; snoc35\n = λ Γ A Con35 nil35 snoc35 → snoc35 (Γ Con35 nil35 snoc35) A\n\nVar35 : Con35 → Ty35 → Set; Var35\n = λ Γ A →\n (Var35 : Con35 → Ty35 → Set)\n (vz : ∀ Γ A → Var35 (snoc35 Γ A) A)\n (vs : ∀ Γ B A → Var35 Γ A → Var35 (snoc35 Γ B) A)\n → Var35 Γ A\n\nvz35 : ∀{Γ A} → Var35 (snoc35 Γ A) A; vz35\n = λ Var35 vz35 vs → vz35 _ _\n\nvs35 : ∀{Γ B A} → Var35 Γ A → Var35 (snoc35 Γ B) A; vs35\n = λ x Var35 vz35 vs35 → vs35 _ _ _ (x Var35 vz35 vs35)\n\nTm35 : Con35 → Ty35 → Set; Tm35\n = λ Γ A →\n (Tm35 : Con35 → Ty35 → Set)\n (var : ∀ Γ A → Var35 Γ A → Tm35 Γ A)\n (lam : ∀ Γ A B → Tm35 (snoc35 Γ A) B → Tm35 Γ (arr35 A B))\n (app : ∀ Γ A B → Tm35 Γ (arr35 A B) → Tm35 Γ A → Tm35 Γ B)\n (tt : ∀ Γ → Tm35 Γ top35)\n (pair : ∀ Γ A B → Tm35 Γ A → Tm35 Γ B → Tm35 Γ (prod35 A B))\n (fst : ∀ Γ A B → Tm35 Γ (prod35 A B) → Tm35 Γ A)\n (snd : ∀ Γ A B → Tm35 Γ (prod35 A B) → Tm35 Γ B)\n (left : ∀ Γ A B → Tm35 Γ A → Tm35 Γ (sum35 A B))\n (right : ∀ Γ A B → Tm35 Γ B → Tm35 Γ (sum35 A B))\n (case : ∀ Γ A B C → Tm35 Γ (sum35 A B) → Tm35 Γ (arr35 A C) → Tm35 Γ (arr35 B C) → Tm35 Γ C)\n (zero : ∀ Γ → Tm35 Γ nat35)\n (suc : ∀ Γ → Tm35 Γ nat35 → Tm35 Γ nat35)\n (rec : ∀ Γ A → Tm35 Γ nat35 → Tm35 Γ (arr35 nat35 (arr35 A A)) → Tm35 Γ A → Tm35 Γ A)\n → Tm35 Γ A\n\nvar35 : ∀{Γ A} → Var35 Γ A → Tm35 Γ A; var35\n = λ x Tm35 var35 lam app tt pair fst snd left right case zero suc rec →\n var35 _ _ x\n\nlam35 : ∀{Γ A B} → Tm35 (snoc35 Γ A) B → Tm35 Γ (arr35 A B); lam35\n = λ t Tm35 var35 lam35 app tt pair fst snd left right case zero suc rec →\n lam35 _ _ _ (t Tm35 var35 lam35 app tt pair fst snd left right case zero suc rec)\n\napp35 : ∀{Γ A B} → Tm35 Γ (arr35 A B) → Tm35 Γ A → Tm35 Γ B; app35\n = λ t u Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec →\n app35 _ _ _ (t Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec)\n (u Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec)\n\ntt35 : ∀{Γ} → Tm35 Γ top35; tt35\n = λ Tm35 var35 lam35 app35 tt35 pair fst snd left right case zero suc rec → tt35 _\n\npair35 : ∀{Γ A B} → Tm35 Γ A → Tm35 Γ B → Tm35 Γ (prod35 A B); pair35\n = λ t u Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec →\n pair35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec)\n (u Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec)\n\nfst35 : ∀{Γ A B} → Tm35 Γ (prod35 A B) → Tm35 Γ A; fst35\n = λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd left right case zero suc rec →\n fst35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd left right case zero suc rec)\n\nsnd35 : ∀{Γ A B} → Tm35 Γ (prod35 A B) → Tm35 Γ B; snd35\n = λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left right case zero suc rec →\n snd35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left right case zero suc rec)\n\nleft35 : ∀{Γ A B} → Tm35 Γ A → Tm35 Γ (sum35 A B); left35\n = λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right case zero suc rec →\n left35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right case zero suc rec)\n\nright35 : ∀{Γ A B} → Tm35 Γ B → Tm35 Γ (sum35 A B); right35\n = λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case zero suc rec →\n right35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case zero suc rec)\n\ncase35 : ∀{Γ A B C} → Tm35 Γ (sum35 A B) → Tm35 Γ (arr35 A C) → Tm35 Γ (arr35 B C) → Tm35 Γ C; case35\n = λ t u v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec →\n case35 _ _ _ _\n (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)\n (u Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)\n (v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)\n\nzero35 : ∀{Γ} → Tm35 Γ nat35; zero35\n = λ Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc rec → zero35 _\n\nsuc35 : ∀{Γ} → Tm35 Γ nat35 → Tm35 Γ nat35; suc35\n = λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec →\n suc35 _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec)\n\nrec35 : ∀{Γ A} → Tm35 Γ nat35 → Tm35 Γ (arr35 nat35 (arr35 A A)) → Tm35 Γ A → Tm35 Γ A; rec35\n = λ t u v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35 →\n rec35 _ _\n (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)\n (u Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)\n (v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)\n\nv035 : ∀{Γ A} → Tm35 (snoc35 Γ A) A; v035\n = var35 vz35\n\nv135 : ∀{Γ A B} → Tm35 (snoc35 (snoc35 Γ A) B) A; v135\n = var35 (vs35 vz35)\n\nv235 : ∀{Γ A B C} → Tm35 (snoc35 (snoc35 (snoc35 Γ A) B) C) A; v235\n = var35 (vs35 (vs35 vz35))\n\nv335 : ∀{Γ A B C D} → Tm35 (snoc35 (snoc35 (snoc35 (snoc35 Γ A) B) C) D) A; v335\n = var35 (vs35 (vs35 (vs35 vz35)))\n\ntbool35 : Ty35; tbool35\n = sum35 top35 top35\n\ntrue35 : ∀{Γ} → Tm35 Γ tbool35; true35\n = left35 tt35\n\ntfalse35 : ∀{Γ} → Tm35 Γ tbool35; tfalse35\n = right35 tt35\n\nifthenelse35 : ∀{Γ A} → Tm35 Γ (arr35 tbool35 (arr35 A (arr35 A A))); ifthenelse35\n = lam35 (lam35 (lam35 (case35 v235 (lam35 v235) (lam35 v135))))\n\ntimes435 : ∀{Γ A} → Tm35 Γ (arr35 (arr35 A A) (arr35 A A)); times435\n = lam35 (lam35 (app35 v135 (app35 v135 (app35 v135 (app35 v135 v035)))))\n\nadd35 : ∀{Γ} → Tm35 Γ (arr35 nat35 (arr35 nat35 nat35)); add35\n = lam35 (rec35 v035\n (lam35 (lam35 (lam35 (suc35 (app35 v135 v035)))))\n (lam35 v035))\n\nmul35 : ∀{Γ} → Tm35 Γ (arr35 nat35 (arr35 nat35 nat35)); mul35\n = lam35 (rec35 v035\n (lam35 (lam35 (lam35 (app35 (app35 add35 (app35 v135 v035)) v035))))\n (lam35 zero35))\n\nfact35 : ∀{Γ} → Tm35 Γ (arr35 nat35 nat35); fact35\n = lam35 (rec35 v035 (lam35 (lam35 (app35 (app35 mul35 (suc35 v135)) v035)))\n (suc35 zero35))\n{-# OPTIONS --type-in-type #-}\n\nTy36 : Set\nTy36 =\n (Ty36 : Set)\n (nat top bot : Ty36)\n (arr prod sum : Ty36 → Ty36 → Ty36)\n → Ty36\n\nnat36 : Ty36; nat36 = λ _ nat36 _ _ _ _ _ → nat36\ntop36 : Ty36; top36 = λ _ _ top36 _ _ _ _ → top36\nbot36 : Ty36; bot36 = λ _ _ _ bot36 _ _ _ → bot36\n\narr36 : Ty36 → Ty36 → Ty36; arr36\n = λ A B Ty36 nat36 top36 bot36 arr36 prod sum →\n arr36 (A Ty36 nat36 top36 bot36 arr36 prod sum) (B Ty36 nat36 top36 bot36 arr36 prod sum)\n\nprod36 : Ty36 → Ty36 → Ty36; prod36\n = λ A B Ty36 nat36 top36 bot36 arr36 prod36 sum →\n prod36 (A Ty36 nat36 top36 bot36 arr36 prod36 sum) (B Ty36 nat36 top36 bot36 arr36 prod36 sum)\n\nsum36 : Ty36 → Ty36 → Ty36; sum36\n = λ A B Ty36 nat36 top36 bot36 arr36 prod36 sum36 →\n sum36 (A Ty36 nat36 top36 bot36 arr36 prod36 sum36) (B Ty36 nat36 top36 bot36 arr36 prod36 sum36)\n\nCon36 : Set; Con36\n = (Con36 : Set)\n (nil : Con36)\n (snoc : Con36 → Ty36 → Con36)\n → Con36\n\nnil36 : Con36; nil36\n = λ Con36 nil36 snoc → nil36\n\nsnoc36 : Con36 → Ty36 → Con36; snoc36\n = λ Γ A Con36 nil36 snoc36 → snoc36 (Γ Con36 nil36 snoc36) A\n\nVar36 : Con36 → Ty36 → Set; Var36\n = λ Γ A →\n (Var36 : Con36 → Ty36 → Set)\n (vz : ∀ Γ A → Var36 (snoc36 Γ A) A)\n (vs : ∀ Γ B A → Var36 Γ A → Var36 (snoc36 Γ B) A)\n → Var36 Γ A\n\nvz36 : ∀{Γ A} → Var36 (snoc36 Γ A) A; vz36\n = λ Var36 vz36 vs → vz36 _ _\n\nvs36 : ∀{Γ B A} → Var36 Γ A → Var36 (snoc36 Γ B) A; vs36\n = λ x Var36 vz36 vs36 → vs36 _ _ _ (x Var36 vz36 vs36)\n\nTm36 : Con36 → Ty36 → Set; Tm36\n = λ Γ A →\n (Tm36 : Con36 → Ty36 → Set)\n (var : ∀ Γ A → Var36 Γ A → Tm36 Γ A)\n (lam : ∀ Γ A B → Tm36 (snoc36 Γ A) B → Tm36 Γ (arr36 A B))\n (app : ∀ Γ A B → Tm36 Γ (arr36 A B) → Tm36 Γ A → Tm36 Γ B)\n (tt : ∀ Γ → Tm36 Γ top36)\n (pair : ∀ Γ A B → Tm36 Γ A → Tm36 Γ B → Tm36 Γ (prod36 A B))\n (fst : ∀ Γ A B → Tm36 Γ (prod36 A B) → Tm36 Γ A)\n (snd : ∀ Γ A B → Tm36 Γ (prod36 A B) → Tm36 Γ B)\n (left : ∀ Γ A B → Tm36 Γ A → Tm36 Γ (sum36 A B))\n (right : ∀ Γ A B → Tm36 Γ B → Tm36 Γ (sum36 A B))\n (case : ∀ Γ A B C → Tm36 Γ (sum36 A B) → Tm36 Γ (arr36 A C) → Tm36 Γ (arr36 B C) → Tm36 Γ C)\n (zero : ∀ Γ → Tm36 Γ nat36)\n (suc : ∀ Γ → Tm36 Γ nat36 → Tm36 Γ nat36)\n (rec : ∀ Γ A → Tm36 Γ nat36 → Tm36 Γ (arr36 nat36 (arr36 A A)) → Tm36 Γ A → Tm36 Γ A)\n → Tm36 Γ A\n\nvar36 : ∀{Γ A} → Var36 Γ A → Tm36 Γ A; var36\n = λ x Tm36 var36 lam app tt pair fst snd left right case zero suc rec →\n var36 _ _ x\n\nlam36 : ∀{Γ A B} → Tm36 (snoc36 Γ A) B → Tm36 Γ (arr36 A B); lam36\n = λ t Tm36 var36 lam36 app tt pair fst snd left right case zero suc rec →\n lam36 _ _ _ (t Tm36 var36 lam36 app tt pair fst snd left right case zero suc rec)\n\napp36 : ∀{Γ A B} → Tm36 Γ (arr36 A B) → Tm36 Γ A → Tm36 Γ B; app36\n = λ t u Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec →\n app36 _ _ _ (t Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec)\n (u Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec)\n\ntt36 : ∀{Γ} → Tm36 Γ top36; tt36\n = λ Tm36 var36 lam36 app36 tt36 pair fst snd left right case zero suc rec → tt36 _\n\npair36 : ∀{Γ A B} → Tm36 Γ A → Tm36 Γ B → Tm36 Γ (prod36 A B); pair36\n = λ t u Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec →\n pair36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec)\n (u Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec)\n\nfst36 : ∀{Γ A B} → Tm36 Γ (prod36 A B) → Tm36 Γ A; fst36\n = λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd left right case zero suc rec →\n fst36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd left right case zero suc rec)\n\nsnd36 : ∀{Γ A B} → Tm36 Γ (prod36 A B) → Tm36 Γ B; snd36\n = λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left right case zero suc rec →\n snd36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left right case zero suc rec)\n\nleft36 : ∀{Γ A B} → Tm36 Γ A → Tm36 Γ (sum36 A B); left36\n = λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right case zero suc rec →\n left36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right case zero suc rec)\n\nright36 : ∀{Γ A B} → Tm36 Γ B → Tm36 Γ (sum36 A B); right36\n = λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case zero suc rec →\n right36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case zero suc rec)\n\ncase36 : ∀{Γ A B C} → Tm36 Γ (sum36 A B) → Tm36 Γ (arr36 A C) → Tm36 Γ (arr36 B C) → Tm36 Γ C; case36\n = λ t u v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec →\n case36 _ _ _ _\n (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)\n (u Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)\n (v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)\n\nzero36 : ∀{Γ} → Tm36 Γ nat36; zero36\n = λ Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc rec → zero36 _\n\nsuc36 : ∀{Γ} → Tm36 Γ nat36 → Tm36 Γ nat36; suc36\n = λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec →\n suc36 _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec)\n\nrec36 : ∀{Γ A} → Tm36 Γ nat36 → Tm36 Γ (arr36 nat36 (arr36 A A)) → Tm36 Γ A → Tm36 Γ A; rec36\n = λ t u v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36 →\n rec36 _ _\n (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)\n (u Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)\n (v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)\n\nv036 : ∀{Γ A} → Tm36 (snoc36 Γ A) A; v036\n = var36 vz36\n\nv136 : ∀{Γ A B} → Tm36 (snoc36 (snoc36 Γ A) B) A; v136\n = var36 (vs36 vz36)\n\nv236 : ∀{Γ A B C} → Tm36 (snoc36 (snoc36 (snoc36 Γ A) B) C) A; v236\n = var36 (vs36 (vs36 vz36))\n\nv336 : ∀{Γ A B C D} → Tm36 (snoc36 (snoc36 (snoc36 (snoc36 Γ A) B) C) D) A; v336\n = var36 (vs36 (vs36 (vs36 vz36)))\n\ntbool36 : Ty36; tbool36\n = sum36 top36 top36\n\ntrue36 : ∀{Γ} → Tm36 Γ tbool36; true36\n = left36 tt36\n\ntfalse36 : ∀{Γ} → Tm36 Γ tbool36; tfalse36\n = right36 tt36\n\nifthenelse36 : ∀{Γ A} → Tm36 Γ (arr36 tbool36 (arr36 A (arr36 A A))); ifthenelse36\n = lam36 (lam36 (lam36 (case36 v236 (lam36 v236) (lam36 v136))))\n\ntimes436 : ∀{Γ A} → Tm36 Γ (arr36 (arr36 A A) (arr36 A A)); times436\n = lam36 (lam36 (app36 v136 (app36 v136 (app36 v136 (app36 v136 v036)))))\n\nadd36 : ∀{Γ} → Tm36 Γ (arr36 nat36 (arr36 nat36 nat36)); add36\n = lam36 (rec36 v036\n (lam36 (lam36 (lam36 (suc36 (app36 v136 v036)))))\n (lam36 v036))\n\nmul36 : ∀{Γ} → Tm36 Γ (arr36 nat36 (arr36 nat36 nat36)); mul36\n = lam36 (rec36 v036\n (lam36 (lam36 (lam36 (app36 (app36 add36 (app36 v136 v036)) v036))))\n (lam36 zero36))\n\nfact36 : ∀{Γ} → Tm36 Γ (arr36 nat36 nat36); fact36\n = lam36 (rec36 v036 (lam36 (lam36 (app36 (app36 mul36 (suc36 v136)) v036)))\n (suc36 zero36))\n{-# OPTIONS --type-in-type #-}\n\nTy37 : Set\nTy37 =\n (Ty37 : Set)\n (nat top bot : Ty37)\n (arr prod sum : Ty37 → Ty37 → Ty37)\n → Ty37\n\nnat37 : Ty37; nat37 = λ _ nat37 _ _ _ _ _ → nat37\ntop37 : Ty37; top37 = λ _ _ top37 _ _ _ _ → top37\nbot37 : Ty37; bot37 = λ _ _ _ bot37 _ _ _ → bot37\n\narr37 : Ty37 → Ty37 → Ty37; arr37\n = λ A B Ty37 nat37 top37 bot37 arr37 prod sum →\n arr37 (A Ty37 nat37 top37 bot37 arr37 prod sum) (B Ty37 nat37 top37 bot37 arr37 prod sum)\n\nprod37 : Ty37 → Ty37 → Ty37; prod37\n = λ A B Ty37 nat37 top37 bot37 arr37 prod37 sum →\n prod37 (A Ty37 nat37 top37 bot37 arr37 prod37 sum) (B Ty37 nat37 top37 bot37 arr37 prod37 sum)\n\nsum37 : Ty37 → Ty37 → Ty37; sum37\n = λ A B Ty37 nat37 top37 bot37 arr37 prod37 sum37 →\n sum37 (A Ty37 nat37 top37 bot37 arr37 prod37 sum37) (B Ty37 nat37 top37 bot37 arr37 prod37 sum37)\n\nCon37 : Set; Con37\n = (Con37 : Set)\n (nil : Con37)\n (snoc : Con37 → Ty37 → Con37)\n → Con37\n\nnil37 : Con37; nil37\n = λ Con37 nil37 snoc → nil37\n\nsnoc37 : Con37 → Ty37 → Con37; snoc37\n = λ Γ A Con37 nil37 snoc37 → snoc37 (Γ Con37 nil37 snoc37) A\n\nVar37 : Con37 → Ty37 → Set; Var37\n = λ Γ A →\n (Var37 : Con37 → Ty37 → Set)\n (vz : ∀ Γ A → Var37 (snoc37 Γ A) A)\n (vs : ∀ Γ B A → Var37 Γ A → Var37 (snoc37 Γ B) A)\n → Var37 Γ A\n\nvz37 : ∀{Γ A} → Var37 (snoc37 Γ A) A; vz37\n = λ Var37 vz37 vs → vz37 _ _\n\nvs37 : ∀{Γ B A} → Var37 Γ A → Var37 (snoc37 Γ B) A; vs37\n = λ x Var37 vz37 vs37 → vs37 _ _ _ (x Var37 vz37 vs37)\n\nTm37 : Con37 → Ty37 → Set; Tm37\n = λ Γ A →\n (Tm37 : Con37 → Ty37 → Set)\n (var : ∀ Γ A → Var37 Γ A → Tm37 Γ A)\n (lam : ∀ Γ A B → Tm37 (snoc37 Γ A) B → Tm37 Γ (arr37 A B))\n (app : ∀ Γ A B → Tm37 Γ (arr37 A B) → Tm37 Γ A → Tm37 Γ B)\n (tt : ∀ Γ → Tm37 Γ top37)\n (pair : ∀ Γ A B → Tm37 Γ A → Tm37 Γ B → Tm37 Γ (prod37 A B))\n (fst : ∀ Γ A B → Tm37 Γ (prod37 A B) → Tm37 Γ A)\n (snd : ∀ Γ A B → Tm37 Γ (prod37 A B) → Tm37 Γ B)\n (left : ∀ Γ A B → Tm37 Γ A → Tm37 Γ (sum37 A B))\n (right : ∀ Γ A B → Tm37 Γ B → Tm37 Γ (sum37 A B))\n (case : ∀ Γ A B C → Tm37 Γ (sum37 A B) → Tm37 Γ (arr37 A C) → Tm37 Γ (arr37 B C) → Tm37 Γ C)\n (zero : ∀ Γ → Tm37 Γ nat37)\n (suc : ∀ Γ → Tm37 Γ nat37 → Tm37 Γ nat37)\n (rec : ∀ Γ A → Tm37 Γ nat37 → Tm37 Γ (arr37 nat37 (arr37 A A)) → Tm37 Γ A → Tm37 Γ A)\n → Tm37 Γ A\n\nvar37 : ∀{Γ A} → Var37 Γ A → Tm37 Γ A; var37\n = λ x Tm37 var37 lam app tt pair fst snd left right case zero suc rec →\n var37 _ _ x\n\nlam37 : ∀{Γ A B} → Tm37 (snoc37 Γ A) B → Tm37 Γ (arr37 A B); lam37\n = λ t Tm37 var37 lam37 app tt pair fst snd left right case zero suc rec →\n lam37 _ _ _ (t Tm37 var37 lam37 app tt pair fst snd left right case zero suc rec)\n\napp37 : ∀{Γ A B} → Tm37 Γ (arr37 A B) → Tm37 Γ A → Tm37 Γ B; app37\n = λ t u Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec →\n app37 _ _ _ (t Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec)\n (u Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec)\n\ntt37 : ∀{Γ} → Tm37 Γ top37; tt37\n = λ Tm37 var37 lam37 app37 tt37 pair fst snd left right case zero suc rec → tt37 _\n\npair37 : ∀{Γ A B} → Tm37 Γ A → Tm37 Γ B → Tm37 Γ (prod37 A B); pair37\n = λ t u Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec →\n pair37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec)\n (u Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec)\n\nfst37 : ∀{Γ A B} → Tm37 Γ (prod37 A B) → Tm37 Γ A; fst37\n = λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd left right case zero suc rec →\n fst37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd left right case zero suc rec)\n\nsnd37 : ∀{Γ A B} → Tm37 Γ (prod37 A B) → Tm37 Γ B; snd37\n = λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left right case zero suc rec →\n snd37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left right case zero suc rec)\n\nleft37 : ∀{Γ A B} → Tm37 Γ A → Tm37 Γ (sum37 A B); left37\n = λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right case zero suc rec →\n left37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right case zero suc rec)\n\nright37 : ∀{Γ A B} → Tm37 Γ B → Tm37 Γ (sum37 A B); right37\n = λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case zero suc rec →\n right37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case zero suc rec)\n\ncase37 : ∀{Γ A B C} → Tm37 Γ (sum37 A B) → Tm37 Γ (arr37 A C) → Tm37 Γ (arr37 B C) → Tm37 Γ C; case37\n = λ t u v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec →\n case37 _ _ _ _\n (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)\n (u Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)\n (v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)\n\nzero37 : ∀{Γ} → Tm37 Γ nat37; zero37\n = λ Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc rec → zero37 _\n\nsuc37 : ∀{Γ} → Tm37 Γ nat37 → Tm37 Γ nat37; suc37\n = λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec →\n suc37 _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec)\n\nrec37 : ∀{Γ A} → Tm37 Γ nat37 → Tm37 Γ (arr37 nat37 (arr37 A A)) → Tm37 Γ A → Tm37 Γ A; rec37\n = λ t u v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37 →\n rec37 _ _\n (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)\n (u Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)\n (v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)\n\nv037 : ∀{Γ A} → Tm37 (snoc37 Γ A) A; v037\n = var37 vz37\n\nv137 : ∀{Γ A B} → Tm37 (snoc37 (snoc37 Γ A) B) A; v137\n = var37 (vs37 vz37)\n\nv237 : ∀{Γ A B C} → Tm37 (snoc37 (snoc37 (snoc37 Γ A) B) C) A; v237\n = var37 (vs37 (vs37 vz37))\n\nv337 : ∀{Γ A B C D} → Tm37 (snoc37 (snoc37 (snoc37 (snoc37 Γ A) B) C) D) A; v337\n = var37 (vs37 (vs37 (vs37 vz37)))\n\ntbool37 : Ty37; tbool37\n = sum37 top37 top37\n\ntrue37 : ∀{Γ} → Tm37 Γ tbool37; true37\n = left37 tt37\n\ntfalse37 : ∀{Γ} → Tm37 Γ tbool37; tfalse37\n = right37 tt37\n\nifthenelse37 : ∀{Γ A} → Tm37 Γ (arr37 tbool37 (arr37 A (arr37 A A))); ifthenelse37\n = lam37 (lam37 (lam37 (case37 v237 (lam37 v237) (lam37 v137))))\n\ntimes437 : ∀{Γ A} → Tm37 Γ (arr37 (arr37 A A) (arr37 A A)); times437\n = lam37 (lam37 (app37 v137 (app37 v137 (app37 v137 (app37 v137 v037)))))\n\nadd37 : ∀{Γ} → Tm37 Γ (arr37 nat37 (arr37 nat37 nat37)); add37\n = lam37 (rec37 v037\n (lam37 (lam37 (lam37 (suc37 (app37 v137 v037)))))\n (lam37 v037))\n\nmul37 : ∀{Γ} → Tm37 Γ (arr37 nat37 (arr37 nat37 nat37)); mul37\n = lam37 (rec37 v037\n (lam37 (lam37 (lam37 (app37 (app37 add37 (app37 v137 v037)) v037))))\n (lam37 zero37))\n\nfact37 : ∀{Γ} → Tm37 Γ (arr37 nat37 nat37); fact37\n = lam37 (rec37 v037 (lam37 (lam37 (app37 (app37 mul37 (suc37 v137)) v037)))\n (suc37 zero37))\n{-# OPTIONS --type-in-type #-}\n\nTy38 : Set\nTy38 =\n (Ty38 : Set)\n (nat top bot : Ty38)\n (arr prod sum : Ty38 → Ty38 → Ty38)\n → Ty38\n\nnat38 : Ty38; nat38 = λ _ nat38 _ _ _ _ _ → nat38\ntop38 : Ty38; top38 = λ _ _ top38 _ _ _ _ → top38\nbot38 : Ty38; bot38 = λ _ _ _ bot38 _ _ _ → bot38\n\narr38 : Ty38 → Ty38 → Ty38; arr38\n = λ A B Ty38 nat38 top38 bot38 arr38 prod sum →\n arr38 (A Ty38 nat38 top38 bot38 arr38 prod sum) (B Ty38 nat38 top38 bot38 arr38 prod sum)\n\nprod38 : Ty38 → Ty38 → Ty38; prod38\n = λ A B Ty38 nat38 top38 bot38 arr38 prod38 sum →\n prod38 (A Ty38 nat38 top38 bot38 arr38 prod38 sum) (B Ty38 nat38 top38 bot38 arr38 prod38 sum)\n\nsum38 : Ty38 → Ty38 → Ty38; sum38\n = λ A B Ty38 nat38 top38 bot38 arr38 prod38 sum38 →\n sum38 (A Ty38 nat38 top38 bot38 arr38 prod38 sum38) (B Ty38 nat38 top38 bot38 arr38 prod38 sum38)\n\nCon38 : Set; Con38\n = (Con38 : Set)\n (nil : Con38)\n (snoc : Con38 → Ty38 → Con38)\n → Con38\n\nnil38 : Con38; nil38\n = λ Con38 nil38 snoc → nil38\n\nsnoc38 : Con38 → Ty38 → Con38; snoc38\n = λ Γ A Con38 nil38 snoc38 → snoc38 (Γ Con38 nil38 snoc38) A\n\nVar38 : Con38 → Ty38 → Set; Var38\n = λ Γ A →\n (Var38 : Con38 → Ty38 → Set)\n (vz : ∀ Γ A → Var38 (snoc38 Γ A) A)\n (vs : ∀ Γ B A → Var38 Γ A → Var38 (snoc38 Γ B) A)\n → Var38 Γ A\n\nvz38 : ∀{Γ A} → Var38 (snoc38 Γ A) A; vz38\n = λ Var38 vz38 vs → vz38 _ _\n\nvs38 : ∀{Γ B A} → Var38 Γ A → Var38 (snoc38 Γ B) A; vs38\n = λ x Var38 vz38 vs38 → vs38 _ _ _ (x Var38 vz38 vs38)\n\nTm38 : Con38 → Ty38 → Set; Tm38\n = λ Γ A →\n (Tm38 : Con38 → Ty38 → Set)\n (var : ∀ Γ A → Var38 Γ A → Tm38 Γ A)\n (lam : ∀ Γ A B → Tm38 (snoc38 Γ A) B → Tm38 Γ (arr38 A B))\n (app : ∀ Γ A B → Tm38 Γ (arr38 A B) → Tm38 Γ A → Tm38 Γ B)\n (tt : ∀ Γ → Tm38 Γ top38)\n (pair : ∀ Γ A B → Tm38 Γ A → Tm38 Γ B → Tm38 Γ (prod38 A B))\n (fst : ∀ Γ A B → Tm38 Γ (prod38 A B) → Tm38 Γ A)\n (snd : ∀ Γ A B → Tm38 Γ (prod38 A B) → Tm38 Γ B)\n (left : ∀ Γ A B → Tm38 Γ A → Tm38 Γ (sum38 A B))\n (right : ∀ Γ A B → Tm38 Γ B → Tm38 Γ (sum38 A B))\n (case : ∀ Γ A B C → Tm38 Γ (sum38 A B) → Tm38 Γ (arr38 A C) → Tm38 Γ (arr38 B C) → Tm38 Γ C)\n (zero : ∀ Γ → Tm38 Γ nat38)\n (suc : ∀ Γ → Tm38 Γ nat38 → Tm38 Γ nat38)\n (rec : ∀ Γ A → Tm38 Γ nat38 → Tm38 Γ (arr38 nat38 (arr38 A A)) → Tm38 Γ A → Tm38 Γ A)\n → Tm38 Γ A\n\nvar38 : ∀{Γ A} → Var38 Γ A → Tm38 Γ A; var38\n = λ x Tm38 var38 lam app tt pair fst snd left right case zero suc rec →\n var38 _ _ x\n\nlam38 : ∀{Γ A B} → Tm38 (snoc38 Γ A) B → Tm38 Γ (arr38 A B); lam38\n = λ t Tm38 var38 lam38 app tt pair fst snd left right case zero suc rec →\n lam38 _ _ _ (t Tm38 var38 lam38 app tt pair fst snd left right case zero suc rec)\n\napp38 : ∀{Γ A B} → Tm38 Γ (arr38 A B) → Tm38 Γ A → Tm38 Γ B; app38\n = λ t u Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec →\n app38 _ _ _ (t Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec)\n (u Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec)\n\ntt38 : ∀{Γ} → Tm38 Γ top38; tt38\n = λ Tm38 var38 lam38 app38 tt38 pair fst snd left right case zero suc rec → tt38 _\n\npair38 : ∀{Γ A B} → Tm38 Γ A → Tm38 Γ B → Tm38 Γ (prod38 A B); pair38\n = λ t u Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec →\n pair38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec)\n (u Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec)\n\nfst38 : ∀{Γ A B} → Tm38 Γ (prod38 A B) → Tm38 Γ A; fst38\n = λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd left right case zero suc rec →\n fst38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd left right case zero suc rec)\n\nsnd38 : ∀{Γ A B} → Tm38 Γ (prod38 A B) → Tm38 Γ B; snd38\n = λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left right case zero suc rec →\n snd38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left right case zero suc rec)\n\nleft38 : ∀{Γ A B} → Tm38 Γ A → Tm38 Γ (sum38 A B); left38\n = λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right case zero suc rec →\n left38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right case zero suc rec)\n\nright38 : ∀{Γ A B} → Tm38 Γ B → Tm38 Γ (sum38 A B); right38\n = λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case zero suc rec →\n right38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case zero suc rec)\n\ncase38 : ∀{Γ A B C} → Tm38 Γ (sum38 A B) → Tm38 Γ (arr38 A C) → Tm38 Γ (arr38 B C) → Tm38 Γ C; case38\n = λ t u v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec →\n case38 _ _ _ _\n (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)\n (u Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)\n (v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)\n\nzero38 : ∀{Γ} → Tm38 Γ nat38; zero38\n = λ Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc rec → zero38 _\n\nsuc38 : ∀{Γ} → Tm38 Γ nat38 → Tm38 Γ nat38; suc38\n = λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec →\n suc38 _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec)\n\nrec38 : ∀{Γ A} → Tm38 Γ nat38 → Tm38 Γ (arr38 nat38 (arr38 A A)) → Tm38 Γ A → Tm38 Γ A; rec38\n = λ t u v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38 →\n rec38 _ _\n (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)\n (u Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)\n (v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)\n\nv038 : ∀{Γ A} → Tm38 (snoc38 Γ A) A; v038\n = var38 vz38\n\nv138 : ∀{Γ A B} → Tm38 (snoc38 (snoc38 Γ A) B) A; v138\n = var38 (vs38 vz38)\n\nv238 : ∀{Γ A B C} → Tm38 (snoc38 (snoc38 (snoc38 Γ A) B) C) A; v238\n = var38 (vs38 (vs38 vz38))\n\nv338 : ∀{Γ A B C D} → Tm38 (snoc38 (snoc38 (snoc38 (snoc38 Γ A) B) C) D) A; v338\n = var38 (vs38 (vs38 (vs38 vz38)))\n\ntbool38 : Ty38; tbool38\n = sum38 top38 top38\n\ntrue38 : ∀{Γ} → Tm38 Γ tbool38; true38\n = left38 tt38\n\ntfalse38 : ∀{Γ} → Tm38 Γ tbool38; tfalse38\n = right38 tt38\n\nifthenelse38 : ∀{Γ A} → Tm38 Γ (arr38 tbool38 (arr38 A (arr38 A A))); ifthenelse38\n = lam38 (lam38 (lam38 (case38 v238 (lam38 v238) (lam38 v138))))\n\ntimes438 : ∀{Γ A} → Tm38 Γ (arr38 (arr38 A A) (arr38 A A)); times438\n = lam38 (lam38 (app38 v138 (app38 v138 (app38 v138 (app38 v138 v038)))))\n\nadd38 : ∀{Γ} → Tm38 Γ (arr38 nat38 (arr38 nat38 nat38)); add38\n = lam38 (rec38 v038\n (lam38 (lam38 (lam38 (suc38 (app38 v138 v038)))))\n (lam38 v038))\n\nmul38 : ∀{Γ} → Tm38 Γ (arr38 nat38 (arr38 nat38 nat38)); mul38\n = lam38 (rec38 v038\n (lam38 (lam38 (lam38 (app38 (app38 add38 (app38 v138 v038)) v038))))\n (lam38 zero38))\n\nfact38 : ∀{Γ} → Tm38 Γ (arr38 nat38 nat38); fact38\n = lam38 (rec38 v038 (lam38 (lam38 (app38 (app38 mul38 (suc38 v138)) v038)))\n (suc38 zero38))\n{-# OPTIONS --type-in-type #-}\n\nTy39 : Set\nTy39 =\n (Ty39 : Set)\n (nat top bot : Ty39)\n (arr prod sum : Ty39 → Ty39 → Ty39)\n → Ty39\n\nnat39 : Ty39; nat39 = λ _ nat39 _ _ _ _ _ → nat39\ntop39 : Ty39; top39 = λ _ _ top39 _ _ _ _ → top39\nbot39 : Ty39; bot39 = λ _ _ _ bot39 _ _ _ → bot39\n\narr39 : Ty39 → Ty39 → Ty39; arr39\n = λ A B Ty39 nat39 top39 bot39 arr39 prod sum →\n arr39 (A Ty39 nat39 top39 bot39 arr39 prod sum) (B Ty39 nat39 top39 bot39 arr39 prod sum)\n\nprod39 : Ty39 → Ty39 → Ty39; prod39\n = λ A B Ty39 nat39 top39 bot39 arr39 prod39 sum →\n prod39 (A Ty39 nat39 top39 bot39 arr39 prod39 sum) (B Ty39 nat39 top39 bot39 arr39 prod39 sum)\n\nsum39 : Ty39 → Ty39 → Ty39; sum39\n = λ A B Ty39 nat39 top39 bot39 arr39 prod39 sum39 →\n sum39 (A Ty39 nat39 top39 bot39 arr39 prod39 sum39) (B Ty39 nat39 top39 bot39 arr39 prod39 sum39)\n\nCon39 : Set; Con39\n = (Con39 : Set)\n (nil : Con39)\n (snoc : Con39 → Ty39 → Con39)\n → Con39\n\nnil39 : Con39; nil39\n = λ Con39 nil39 snoc → nil39\n\nsnoc39 : Con39 → Ty39 → Con39; snoc39\n = λ Γ A Con39 nil39 snoc39 → snoc39 (Γ Con39 nil39 snoc39) A\n\nVar39 : Con39 → Ty39 → Set; Var39\n = λ Γ A →\n (Var39 : Con39 → Ty39 → Set)\n (vz : ∀ Γ A → Var39 (snoc39 Γ A) A)\n (vs : ∀ Γ B A → Var39 Γ A → Var39 (snoc39 Γ B) A)\n → Var39 Γ A\n\nvz39 : ∀{Γ A} → Var39 (snoc39 Γ A) A; vz39\n = λ Var39 vz39 vs → vz39 _ _\n\nvs39 : ∀{Γ B A} → Var39 Γ A → Var39 (snoc39 Γ B) A; vs39\n = λ x Var39 vz39 vs39 → vs39 _ _ _ (x Var39 vz39 vs39)\n\nTm39 : Con39 → Ty39 → Set; Tm39\n = λ Γ A →\n (Tm39 : Con39 → Ty39 → Set)\n (var : ∀ Γ A → Var39 Γ A → Tm39 Γ A)\n (lam : ∀ Γ A B → Tm39 (snoc39 Γ A) B → Tm39 Γ (arr39 A B))\n (app : ∀ Γ A B → Tm39 Γ (arr39 A B) → Tm39 Γ A → Tm39 Γ B)\n (tt : ∀ Γ → Tm39 Γ top39)\n (pair : ∀ Γ A B → Tm39 Γ A → Tm39 Γ B → Tm39 Γ (prod39 A B))\n (fst : ∀ Γ A B → Tm39 Γ (prod39 A B) → Tm39 Γ A)\n (snd : ∀ Γ A B → Tm39 Γ (prod39 A B) → Tm39 Γ B)\n (left : ∀ Γ A B → Tm39 Γ A → Tm39 Γ (sum39 A B))\n (right : ∀ Γ A B → Tm39 Γ B → Tm39 Γ (sum39 A B))\n (case : ∀ Γ A B C → Tm39 Γ (sum39 A B) → Tm39 Γ (arr39 A C) → Tm39 Γ (arr39 B C) → Tm39 Γ C)\n (zero : ∀ Γ → Tm39 Γ nat39)\n (suc : ∀ Γ → Tm39 Γ nat39 → Tm39 Γ nat39)\n (rec : ∀ Γ A → Tm39 Γ nat39 → Tm39 Γ (arr39 nat39 (arr39 A A)) → Tm39 Γ A → Tm39 Γ A)\n → Tm39 Γ A\n\nvar39 : ∀{Γ A} → Var39 Γ A → Tm39 Γ A; var39\n = λ x Tm39 var39 lam app tt pair fst snd left right case zero suc rec →\n var39 _ _ x\n\nlam39 : ∀{Γ A B} → Tm39 (snoc39 Γ A) B → Tm39 Γ (arr39 A B); lam39\n = λ t Tm39 var39 lam39 app tt pair fst snd left right case zero suc rec →\n lam39 _ _ _ (t Tm39 var39 lam39 app tt pair fst snd left right case zero suc rec)\n\napp39 : ∀{Γ A B} → Tm39 Γ (arr39 A B) → Tm39 Γ A → Tm39 Γ B; app39\n = λ t u Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec →\n app39 _ _ _ (t Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec)\n (u Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec)\n\ntt39 : ∀{Γ} → Tm39 Γ top39; tt39\n = λ Tm39 var39 lam39 app39 tt39 pair fst snd left right case zero suc rec → tt39 _\n\npair39 : ∀{Γ A B} → Tm39 Γ A → Tm39 Γ B → Tm39 Γ (prod39 A B); pair39\n = λ t u Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec →\n pair39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec)\n (u Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec)\n\nfst39 : ∀{Γ A B} → Tm39 Γ (prod39 A B) → Tm39 Γ A; fst39\n = λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd left right case zero suc rec →\n fst39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd left right case zero suc rec)\n\nsnd39 : ∀{Γ A B} → Tm39 Γ (prod39 A B) → Tm39 Γ B; snd39\n = λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left right case zero suc rec →\n snd39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left right case zero suc rec)\n\nleft39 : ∀{Γ A B} → Tm39 Γ A → Tm39 Γ (sum39 A B); left39\n = λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right case zero suc rec →\n left39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right case zero suc rec)\n\nright39 : ∀{Γ A B} → Tm39 Γ B → Tm39 Γ (sum39 A B); right39\n = λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case zero suc rec →\n right39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case zero suc rec)\n\ncase39 : ∀{Γ A B C} → Tm39 Γ (sum39 A B) → Tm39 Γ (arr39 A C) → Tm39 Γ (arr39 B C) → Tm39 Γ C; case39\n = λ t u v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec →\n case39 _ _ _ _\n (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)\n (u Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)\n (v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)\n\nzero39 : ∀{Γ} → Tm39 Γ nat39; zero39\n = λ Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc rec → zero39 _\n\nsuc39 : ∀{Γ} → Tm39 Γ nat39 → Tm39 Γ nat39; suc39\n = λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec →\n suc39 _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec)\n\nrec39 : ∀{Γ A} → Tm39 Γ nat39 → Tm39 Γ (arr39 nat39 (arr39 A A)) → Tm39 Γ A → Tm39 Γ A; rec39\n = λ t u v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39 →\n rec39 _ _\n (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)\n (u Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)\n (v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)\n\nv039 : ∀{Γ A} → Tm39 (snoc39 Γ A) A; v039\n = var39 vz39\n\nv139 : ∀{Γ A B} → Tm39 (snoc39 (snoc39 Γ A) B) A; v139\n = var39 (vs39 vz39)\n\nv239 : ∀{Γ A B C} → Tm39 (snoc39 (snoc39 (snoc39 Γ A) B) C) A; v239\n = var39 (vs39 (vs39 vz39))\n\nv339 : ∀{Γ A B C D} → Tm39 (snoc39 (snoc39 (snoc39 (snoc39 Γ A) B) C) D) A; v339\n = var39 (vs39 (vs39 (vs39 vz39)))\n\ntbool39 : Ty39; tbool39\n = sum39 top39 top39\n\ntrue39 : ∀{Γ} → Tm39 Γ tbool39; true39\n = left39 tt39\n\ntfalse39 : ∀{Γ} → Tm39 Γ tbool39; tfalse39\n = right39 tt39\n\nifthenelse39 : ∀{Γ A} → Tm39 Γ (arr39 tbool39 (arr39 A (arr39 A A))); ifthenelse39\n = lam39 (lam39 (lam39 (case39 v239 (lam39 v239) (lam39 v139))))\n\ntimes439 : ∀{Γ A} → Tm39 Γ (arr39 (arr39 A A) (arr39 A A)); times439\n = lam39 (lam39 (app39 v139 (app39 v139 (app39 v139 (app39 v139 v039)))))\n\nadd39 : ∀{Γ} → Tm39 Γ (arr39 nat39 (arr39 nat39 nat39)); add39\n = lam39 (rec39 v039\n (lam39 (lam39 (lam39 (suc39 (app39 v139 v039)))))\n (lam39 v039))\n\nmul39 : ∀{Γ} → Tm39 Γ (arr39 nat39 (arr39 nat39 nat39)); mul39\n = lam39 (rec39 v039\n (lam39 (lam39 (lam39 (app39 (app39 add39 (app39 v139 v039)) v039))))\n (lam39 zero39))\n\nfact39 : ∀{Γ} → Tm39 Γ (arr39 nat39 nat39); fact39\n = lam39 (rec39 v039 (lam39 (lam39 (app39 (app39 mul39 (suc39 v139)) v039)))\n (suc39 zero39))\n", "meta": {"hexsha": "9fbf0b4edc1ecb165716e07195513f0aac130634", "size": 280024, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "bench/stlc_lessimpl5k.agda", "max_stars_repo_name": "int-index/smalltt", "max_stars_repo_head_hexsha": "6a87f295148bd753d2519d50c2e1011b64c859ff", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 377, "max_stars_repo_stars_event_min_datetime": "2017-11-26T16:57:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-19T21:31:01.000Z", "max_issues_repo_path": "bench/stlc_lessimpl5k.agda", "max_issues_repo_name": "int-index/smalltt", "max_issues_repo_head_hexsha": "6a87f295148bd753d2519d50c2e1011b64c859ff", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2020-03-16T09:14:57.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-28T21:51:10.000Z", "max_forks_repo_path": "bench/stlc_lessimpl5k.agda", "max_forks_repo_name": "int-index/smalltt", "max_forks_repo_head_hexsha": "6a87f295148bd753d2519d50c2e1011b64c859ff", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 19, "max_forks_repo_forks_event_min_datetime": "2018-12-05T21:11:34.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-03T19:46:54.000Z", "avg_line_length": 42.1659388646, "max_line_length": 102, "alphanum_fraction": 0.6261748993, "num_tokens": 116293, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.34620226997429693}} {"text": "{-# OPTIONS --without-K --rewriting #-}\n\nopen import HoTT\nopen import cohomology.Theory\n\n{- Ordinary cohomology groups of the n-torus Tⁿ = (S¹)ⁿ.\n - We have Cᵏ(Tⁿ) == C⁰(S⁰)^(n choose' k) where _choose'_ defined as below.\n - This argument could give Cᵏ((Sᵐ)ⁿ) with a little more work. -}\n\nmodule cohomology.Torus {i} (OT : OrdinaryTheory i) where\n\nopen OrdinaryTheory OT\nopen import cohomology.Sphere OT\nopen import cohomology.SphereProduct cohomology-theory\n\n\n{- Almost n choose k, but with n choose' O = 0 for any n. -}\n_choose'_ : ℕ → ℤ → ℕ\nn choose' negsucc _ = 0\nn choose' pos O = 0\nn choose' pos (S O) = n\nO choose' pos (S (S k)) = 0\nS n choose' pos (S (S k)) = (n choose' (pos (S (S k)))) + (n choose' (pos (S k)))\n\n\n_-⊙Torus : ℕ → Ptd₀\nO -⊙Torus = ⊙Unit\n(S n) -⊙Torus = ⊙S¹ ⊙× (n -⊙Torus)\n\nC-nTorus : (k : ℤ) (n : ℕ)\n → C k (⊙Lift (n -⊙Torus)) ≃ᴳ C 0 (⊙Lift ⊙S⁰) ^ᴳ (n choose' k)\n\nC-nTorus (negsucc k) O = lift-iso ∘eᴳ trivial-iso-0ᴳ (C-Unit (negsucc k))\n\nC-nTorus (negsucc k) (S n) =\n C (negsucc k) (⊙Lift (S n -⊙Torus))\n ≃ᴳ⟨ C-emap (negsucc k) (⊙lift-equiv ⊙∘e ⊙×-emap (⊙ide _) ⊙lower-equiv) ⟩\n C (negsucc k) (⊙S¹ ⊙× ⊙Lift (n -⊙Torus))\n ≃ᴳ⟨ C-Sphere× (negsucc k) 1 (⊙Lift (n -⊙Torus)) ⟩\n C (negsucc k) (⊙Lift ⊙S¹) ×ᴳ (C (negsucc k) (⊙Lift (n -⊙Torus)) ×ᴳ C (negsucc k) (⊙Susp (⊙Lift (n -⊙Torus))))\n ≃ᴳ⟨ ×ᴳ-emap (trivial-iso-0ᴳ (C-Sphere-≠-is-trivial (negsucc k) 1 (ℤ-negsucc≠pos _ _))) (idiso _) ⟩\n 0ᴳ ×ᴳ (C (negsucc k) (⊙Lift (n -⊙Torus)) ×ᴳ C (negsucc k) (⊙Susp (⊙Lift (n -⊙Torus))))\n ≃ᴳ⟨ ×ᴳ-unit-l _ ⟩\n C (negsucc k) (⊙Lift (n -⊙Torus)) ×ᴳ C (negsucc k) (⊙Susp (⊙Lift (n -⊙Torus)))\n ≃ᴳ⟨ ×ᴳ-emap\n (lower-iso ∘eᴳ C-nTorus (negsucc k) n)\n (C-nTorus (negsucc (S k)) n ∘eᴳ C-Susp (negsucc (S k)) (⊙Lift (n -⊙Torus))) ⟩\n 0ᴳ ×ᴳ Lift-group 0ᴳ\n ≃ᴳ⟨ ×ᴳ-unit-l (Lift-group 0ᴳ) ⟩\n Lift-group 0ᴳ\n ≃ᴳ∎\n\n\n\nC-nTorus (pos O) O = lift-iso ∘eᴳ trivial-iso-0ᴳ (C-Unit 0)\n\nC-nTorus (pos O) (S n) =\n C 0 (⊙Lift (S n -⊙Torus))\n ≃ᴳ⟨ C-emap 0 (⊙lift-equiv ⊙∘e ⊙×-emap (⊙ide _) ⊙lower-equiv) ⟩\n C 0 (⊙S¹ ⊙× ⊙Lift (n -⊙Torus))\n ≃ᴳ⟨ C-Sphere× 0 1 (⊙Lift (n -⊙Torus)) ⟩\n C 0 (⊙Lift ⊙S¹) ×ᴳ (C 0 (⊙Lift (n -⊙Torus)) ×ᴳ C 0 (⊙Susp (⊙Lift (n -⊙Torus))))\n ≃ᴳ⟨ ×ᴳ-emap (trivial-iso-0ᴳ (C-Sphere-≠-is-trivial 0 1 (pos-≠ (ℕ-O≠S _)))) (idiso _) ⟩\n 0ᴳ ×ᴳ (C 0 (⊙Lift (n -⊙Torus)) ×ᴳ C 0 (⊙Susp (⊙Lift (n -⊙Torus))))\n ≃ᴳ⟨ ×ᴳ-unit-l _ ⟩\n C 0 (⊙Lift (n -⊙Torus)) ×ᴳ C 0 (⊙Susp (⊙Lift (n -⊙Torus)))\n ≃ᴳ⟨ ×ᴳ-emap\n (lower-iso ∘eᴳ C-nTorus 0 n)\n (C-nTorus -1 n ∘eᴳ C-Susp -1 (⊙Lift (n -⊙Torus))) ⟩\n 0ᴳ ×ᴳ Lift-group 0ᴳ\n ≃ᴳ⟨ ×ᴳ-unit-l _ ⟩\n Lift-group 0ᴳ\n ≃ᴳ∎\n\nC-nTorus (pos (S O)) O = lift-iso ∘eᴳ trivial-iso-0ᴳ (C-Unit 1)\n\nC-nTorus (pos (S O)) (S n) =\n C 1 (⊙Lift (S n -⊙Torus))\n ≃ᴳ⟨ C-emap 1 (⊙lift-equiv ⊙∘e ⊙×-emap (⊙ide _) ⊙lower-equiv) ⟩\n C 1 (⊙S¹ ⊙× ⊙Lift (n -⊙Torus))\n ≃ᴳ⟨ C-Sphere× 1 1 (⊙Lift (n -⊙Torus)) ⟩\n C 1 (⊙Lift ⊙S¹) ×ᴳ (C 1 (⊙Lift (n -⊙Torus)) ×ᴳ C 1 (⊙Susp (⊙Lift (n -⊙Torus))))\n ≃ᴳ⟨ ×ᴳ-emap (C-Sphere-diag 1)\n ( ×ᴳ-unit-r _\n ∘eᴳ ×ᴳ-emap\n (C-nTorus 1 n)\n (lower-iso\n ∘eᴳ C-nTorus 0 n\n ∘eᴳ C-Susp 0 (⊙Lift (n -⊙Torus)))) ⟩\n C 0 (⊙Lift ⊙S⁰) ×ᴳ (C 0 (⊙Lift ⊙S⁰) ^ᴳ n)\n ≃ᴳ∎\n\nC-nTorus (pos (S (S k))) O = lift-iso ∘eᴳ trivial-iso-0ᴳ (C-Unit (pos (S (S k))))\n\nC-nTorus (pos (S (S k))) (S n) =\n C (pos (S (S k))) (⊙Lift (S n -⊙Torus))\n ≃ᴳ⟨ C-emap (pos (S (S k))) (⊙lift-equiv ⊙∘e ⊙×-emap (⊙ide _) ⊙lower-equiv) ⟩\n C (pos (S (S k))) (⊙S¹ ⊙× ⊙Lift (n -⊙Torus))\n ≃ᴳ⟨ C-Sphere× (pos (S (S k))) 1 (⊙Lift (n -⊙Torus)) ⟩\n C (pos (S (S k))) (⊙Lift ⊙S¹) ×ᴳ (C (pos (S (S k))) (⊙Lift (n -⊙Torus)) ×ᴳ C (pos (S (S k))) (⊙Susp (⊙Lift (n -⊙Torus))))\n ≃ᴳ⟨ ×ᴳ-emap (trivial-iso-0ᴳ (C-Sphere-≠-is-trivial (pos (S (S k))) 1 (pos-≠ (ℕ-S-≠ (ℕ-S≠O k))))) (idiso _) ⟩\n 0ᴳ ×ᴳ (C (pos (S (S k))) (⊙Lift (n -⊙Torus)) ×ᴳ C (pos (S (S k))) (⊙Susp (⊙Lift (n -⊙Torus))))\n ≃ᴳ⟨ ×ᴳ-unit-l _ ⟩\n C (pos (S (S k))) (⊙Lift (n -⊙Torus)) ×ᴳ C (pos (S (S k))) (⊙Susp (⊙Lift (n -⊙Torus)))\n ≃ᴳ⟨ ×ᴳ-emap\n (C-nTorus (pos (S (S k))) n)\n (C-nTorus (pos (S k)) n ∘eᴳ C-Susp (pos (S k)) (⊙Lift (n -⊙Torus))) ⟩\n (C 0 (⊙Lift ⊙S⁰) ^ᴳ (n choose' pos (S (S k)))) ×ᴳ (C 0 (⊙Lift ⊙S⁰) ^ᴳ (n choose' pos (S k)))\n ≃ᴳ⟨ ^ᴳ-+ (C 0 (⊙Lift ⊙S⁰)) (n choose' pos (S (S k))) (n choose' pos (S k)) ⁻¹ᴳ ⟩\n C 0 (⊙Lift ⊙S⁰) ^ᴳ (S n choose' pos (S (S k)))\n ≃ᴳ∎\n", "meta": {"hexsha": "8b7def68136f17fd31fe3f5045f376c3d4e44cf3", "size": 4396, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "theorems/cohomology/Torus.agda", "max_stars_repo_name": "timjb/HoTT-Agda", "max_stars_repo_head_hexsha": "66f800adef943afdf08c17b8ecfba67340fead5e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "theorems/cohomology/Torus.agda", "max_issues_repo_name": "timjb/HoTT-Agda", "max_issues_repo_head_hexsha": 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YES\n2. NO", "lm_q1_score": 0.7634837743174788, "lm_q2_score": 0.45326184801538616, "lm_q1q2_score": 0.34605806647690246}} {"text": "\nmodule UnifyMguPair where\n\nopen import UnifyTerm\nopen import UnifyMgu\nopen import UnifyMguCorrect\n\nopen import Data.Fin using (Fin; suc; zero)\nopen import Data.Nat hiding (_≤_)\nopen import Relation.Binary.PropositionalEquality renaming ([_] to [[_]])\nopen import Function using (_∘_; id; case_of_; _$_)\nopen import Relation.Nullary\nopen import Data.Product\nopen import Data.Empty\nopen import Data.Maybe\nopen import Data.Sum\n\nl1 : ∀ m → Data.Fin.toℕ (Data.Fin.fromℕ m) ≡ m\nl1 zero = refl\nl1 (suc m) = cong suc (l1 m)\n\nl2 : ∀ m → m ≡ m + 0\nl2 zero = refl\nl2 (suc m) = cong suc (l2 m)\n\nfixup : ∀ m → Fin (Data.Fin.toℕ (Data.Fin.fromℕ m) + m) → Fin (m + (m + 0))\nfixup m x rewrite l1 m | sym (l2 m) = x\n\nrevise-down : ∀ {m} → Fin m → Fin (2 * m)\nrevise-down {m} x = Data.Fin.inject+ (m + 0) x\n\nrevise-up : ∀ {m} → Fin m → Fin (2 * m)\nrevise-up {m} x = fixup m (Data.Fin.fromℕ m Data.Fin.+ x)\n\nwrite-variable-down : ∀ {m} → Term m → Term (2 * m)\nwrite-variable-down {m} (i l) = i $ revise-down l\nwrite-variable-down {m} leaf = leaf\nwrite-variable-down {m} (s fork t) = write-variable-down s fork write-variable-down t\n\nwrite-variable-up : ∀ {m} → Term m → Term (2 * m)\nwrite-variable-up {m} (i r) = i (revise-up r)\nwrite-variable-up {m} leaf = leaf\nwrite-variable-up {m} (s fork t) = write-variable-up s fork write-variable-up t\n\nwrite-variables-apart : ∀ {m} (s t : Term m) → Term (2 * m) × Term (2 * m)\nwrite-variables-apart s t = write-variable-down s , write-variable-up t\n\nseparate-substitutions-down : ∀ {m n} → (Fin (2 * m) → Term n) → Fin m → Term n\nseparate-substitutions-down {m} f x = f $ revise-down x\n\nseparate-substitutions-up : ∀ {m n} → (Fin (2 * m) → Term n) → Fin m → Term n\nseparate-substitutions-up {m} f x = f $ revise-up x\n\nseparate-substitutions : ∀ {m n} → (Fin (2 * m) → Term n) → (Fin m → Term n) × (Fin m → Term n)\nseparate-substitutions {m} x = separate-substitutions-down {m} x , separate-substitutions-up {m} x\n\nwrite≡separate : ∀ {m n} (σ : AList (2 * m) n) (t : Term m) → (sub σ ◃_) (write-variable-down t) ≡ ((separate-substitutions-down $ sub σ) ◃_) t\nwrite≡separate {zero} {.0} anil (i x) = refl\nwrite≡separate {suc m} {.(suc (m + suc (m + 0)))} anil (i x) = refl\nwrite≡separate {suc m} {n} (σ asnoc t' / x) (i x₁) = refl\nwrite≡separate σ leaf = refl\nwrite≡separate σ (t₁ fork t₂) = cong₂ _fork_ (write≡separate σ t₁) (write≡separate σ t₂)\n\nwrite≡separate' : ∀ {m n} (σ : AList (2 * m) n) (t : Term m) → (sub σ ◃_) (write-variable-down t) ≡ ((separate-substitutions-down $ sub σ) ◃_) t\nwrite≡separate' {zero} {.0} anil (i x) = refl\nwrite≡separate' {suc m} {.(suc (m + suc (m + 0)))} anil (i x) = refl\nwrite≡separate' {suc m} {n} (σ asnoc t' / x) (i x₁) = refl\nwrite≡separate' σ leaf = refl\nwrite≡separate' σ (t₁ fork t₂) = cong₂ _fork_ (write≡separate σ t₁) (write≡separate σ t₂)\n\nProperty'2 : (m : ℕ) -> Set1\nProperty'2 m = ∀ {n} -> (Fin (2 * m) -> Term n) -> Set\n\nNothing'2 : ∀{m} -> (P : Property'2 m) -> Set\nNothing'2 P = ∀{n} f -> P {n} f -> ⊥\n\nUnifies'2 : ∀ {m} (s t : Term m) -> Property'2 m\nUnifies'2 s t f =\n let --s' , t' = write-variables-apart s t\n f₁ , f₂ = separate-substitutions f\n in f₁ ◃ s ≡ f₂ ◃ t\n\npair-mgu' : ∀ {m} -> (s t : Term m) -> Maybe (∃ (AList (2 * m)))\npair-mgu' {m} s t =\n let s' , t' = write-variables-apart s t\n mgu' = mgu s' t'\n in\n mgu'\n\nup-equality : ∀ {m n} {f : (2 * m) ~> n} (t : Term m) → (f ∘ revise-up) ◃ t ≡ f ◃ write-variable-up t\nup-equality (i x) = refl\nup-equality leaf = refl\nup-equality (t₁ fork t₂) = cong₂ _fork_ (up-equality t₁) (up-equality t₂)\n\ndown-equality : ∀ {m n} {f : (2 * m) ~> n} (t : Term m) → (f ∘ revise-down) ◃ t ≡ f ◃ write-variable-down t\ndown-equality (i x) = refl\ndown-equality leaf = refl\ndown-equality (t₁ fork t₂) = cong₂ _fork_ (down-equality t₁) (down-equality t₂)\n\nrevise-to-write : ∀ {m n} {f : (2 * m) ~> n} (s t : Term m) → (f ∘ revise-down) ◃ s ≡ (f ∘ revise-up) ◃ t → f ◃ write-variable-down s ≡ f ◃ write-variable-up t\nrevise-to-write (i x) (i x₁) x₂ = x₂\nrevise-to-write (i x) leaf x₁ = x₁\nrevise-to-write (i x) (t fork t₁) x₁ = trans x₁ (up-equality (t fork t₁))\nrevise-to-write leaf (i x) x₁ = x₁\nrevise-to-write leaf leaf x = refl\nrevise-to-write leaf (t₁ fork t₂) x = trans x (up-equality (t₁ fork t₂))\nrevise-to-write (s₁ fork s₂) (i x) x₁ = trans (sym (down-equality (s₁ fork s₂))) x₁\nrevise-to-write (s₁ fork s₂) leaf x = trans (sym (down-equality (s₁ fork s₂))) x\nrevise-to-write (s₁ fork s₂) (t₁ fork t₂) x = trans (trans (sym (down-equality (s₁ fork s₂))) x) (up-equality (t₁ fork t₂))\n\nwrite-to-revise : ∀ {m n} {f : (2 * m) ~> n} (s t : Term m) → f ◃ write-variable-down s ≡ f ◃ write-variable-up t → (f ∘ revise-down) ◃ s ≡ (f ∘ revise-up) ◃ t\nwrite-to-revise (i x) (i x₁) x₂ = x₂\nwrite-to-revise (i x) leaf x₁ = x₁\nwrite-to-revise (i x) (t fork t₁) x₁ = trans x₁ (sym (up-equality (t fork t₁)))\nwrite-to-revise leaf (i x) x₁ = x₁\nwrite-to-revise leaf leaf x = refl\nwrite-to-revise leaf (t fork t₁) x = trans x (sym (up-equality (t fork t₁)))\nwrite-to-revise (s₁ fork s₂) (i x) x₁ = trans ((down-equality (s₁ fork s₂))) x₁\nwrite-to-revise (s₁ fork s₂) leaf x = trans ((down-equality (s₁ fork s₂))) x\nwrite-to-revise (s₁ fork s₂) (t fork t₁) x = trans (trans ((down-equality (s₁ fork s₂))) x) (sym (up-equality (t fork t₁)))\n\npair-mgu-c' : ∀ {m} (s t : Term m) ->\n (∃ λ n → ∃ λ σ → (Max⋆ (Unifies'2 s t)) (sub σ) × pair-mgu' s t ≡ just (n , σ))\n ⊎ (Nothing⋆ (Unifies'2 s t) × pair-mgu' s t ≡ nothing)\npair-mgu-c' {m} s t with write-variable-down s | write-variable-up t | inspect write-variable-down s | inspect write-variable-up t\n… | s' | t' | [[ refl ]] | [[ refl ]] with mgu-c s' t'\n… | (inj₁ (n , σ , (σ◃s'=σ◃t' , max-σ) , amgu=just)) =\n inj₁ $\n n ,\n σ ,\n ( write-to-revise s t σ◃s'=σ◃t' ,\n 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YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.346005816693504}} {"text": "{-# OPTIONS --without-K --safe #-}\n\nmodule Definition.Typed where\n\nopen import Definition.Untyped hiding (_∷_)\n\nopen import Tools.Fin\nopen import Tools.Nat\nopen import Tools.Product\n\ninfixl 30 _∙_\ninfix 30 Πⱼ_▹_\ninfix 30 Σⱼ_▹_\ninfix 30 ⟦_⟧ⱼ_▹_\n\n\nprivate\n variable\n n m : Nat\n Γ : Con Term n\n A B F : Term n\n G : Term (1+ n)\n x : Fin n\n\n\n-- Well-typed variables\ndata _∷_∈_ : (x : Fin n) (A : Term n) (Γ : Con Term n) → Set where\n here : x0 ∷ wk1 A ∈ (Γ ∙ A)\n there : (h : x ∷ A ∈ Γ) → (x +1) ∷ wk1 A ∈ (Γ ∙ B)\n\nmutual\n -- Well-formed context\n data ⊢_ : Con Term n → Set where\n ε : ⊢ ε\n _∙_ : ⊢ Γ\n → Γ ⊢ A\n → ⊢ Γ ∙ A\n\n -- Well-formed type\n data _⊢_ (Γ : Con Term n) : Term n → Set where\n Uⱼ : ⊢ Γ → Γ ⊢ U\n ℕⱼ : ⊢ Γ → Γ ⊢ ℕ\n Emptyⱼ : ⊢ Γ → Γ ⊢ Empty\n Unitⱼ : ⊢ Γ → Γ ⊢ Unit\n Πⱼ_▹_ : Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ Π F ▹ G\n Σⱼ_▹_ : Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ Σ F ▹ G\n univ : Γ ⊢ A ∷ U\n → Γ ⊢ A\n\n -- Well-formed term of a type\n data _⊢_∷_ (Γ : Con Term n) : Term n → Term n → Set where\n Πⱼ_▹_ : ∀ {F G}\n → Γ ⊢ F ∷ U\n → Γ ∙ F ⊢ G ∷ U\n → Γ ⊢ Π F ▹ G ∷ U\n Σⱼ_▹_ : ∀ {F G}\n → Γ ⊢ F ∷ U\n → Γ ∙ F ⊢ G ∷ U\n → Γ ⊢ Σ F ▹ G ∷ U\n ℕⱼ : ⊢ Γ → Γ ⊢ ℕ ∷ U\n Emptyⱼ : ⊢ Γ → Γ ⊢ Empty ∷ U\n Unitⱼ : ⊢ Γ → Γ ⊢ Unit ∷ U\n\n var : ∀ {A x}\n → ⊢ Γ\n → x ∷ A ∈ Γ\n → Γ ⊢ var x ∷ A\n\n lamⱼ : ∀ {F G t}\n → Γ ⊢ F\n → Γ ∙ F ⊢ t ∷ G\n → Γ ⊢ lam t ∷ Π F ▹ G\n _∘ⱼ_ : ∀ {g a F G}\n → Γ ⊢ g ∷ Π F ▹ G\n → Γ ⊢ a ∷ F\n → Γ ⊢ g ∘ a ∷ G [ a ]\n\n prodⱼ : ∀ {F G t u}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ∷ F\n → Γ ⊢ u ∷ G [ t ]\n → Γ ⊢ prod t u ∷ Σ F ▹ G\n fstⱼ : ∀ {F G t}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ∷ Σ F ▹ G\n → Γ ⊢ fst t ∷ F\n sndⱼ : ∀ {F G t}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ∷ Σ F ▹ G\n → Γ ⊢ snd t ∷ G [ fst t ]\n\n zeroⱼ : ⊢ Γ\n → Γ ⊢ zero ∷ ℕ\n sucⱼ : ∀ {n}\n → Γ ⊢ n ∷ ℕ\n → Γ ⊢ suc n ∷ ℕ\n natrecⱼ : ∀ {G s z n}\n → Γ ∙ ℕ ⊢ G\n → Γ ⊢ z ∷ G [ zero ]\n → Γ ⊢ s ∷ Π ℕ ▹ (G ▹▹ G [ suc (var x0) ]↑)\n → Γ ⊢ n ∷ ℕ\n → Γ ⊢ natrec G z s n ∷ G [ n ]\n\n Emptyrecⱼ : ∀ {A e}\n → Γ ⊢ A → Γ ⊢ e ∷ Empty → Γ ⊢ Emptyrec A e ∷ A\n\n starⱼ : ⊢ Γ → Γ ⊢ star ∷ Unit\n\n conv : ∀ {t A B}\n → Γ ⊢ t ∷ A\n → Γ ⊢ A ≡ B\n → Γ ⊢ t ∷ B\n\n -- Type equality\n data _⊢_≡_ (Γ : Con Term n) : Term n → Term n → Set where\n univ : ∀ {A B}\n → Γ ⊢ A ≡ B ∷ U\n → Γ ⊢ A ≡ B\n refl : ∀ {A}\n → Γ ⊢ A\n → Γ ⊢ A ≡ A\n sym : ∀ {A B}\n → Γ ⊢ A ≡ B\n → Γ ⊢ B ≡ A\n trans : ∀ {A B C}\n → Γ ⊢ A ≡ B\n → Γ ⊢ B ≡ C\n → Γ ⊢ A ≡ C\n Π-cong : ∀ {F H G E}\n → Γ ⊢ F\n → Γ ⊢ F ≡ H\n → Γ ∙ F ⊢ G ≡ E\n → Γ ⊢ Π F ▹ G ≡ Π H ▹ E\n Σ-cong : ∀ {F H G E}\n → Γ ⊢ F\n → Γ ⊢ F ≡ H\n → Γ ∙ F ⊢ G ≡ E\n → Γ ⊢ Σ F ▹ G ≡ Σ H ▹ E\n\n -- Term equality\n data _⊢_≡_∷_ (Γ : Con Term n) : Term n → Term n → Term n → Set where\n refl : ∀ {t A}\n → Γ ⊢ t ∷ A\n → Γ ⊢ t ≡ t ∷ A\n sym : ∀ {t u A}\n → Γ ⊢ t ≡ u ∷ A\n → Γ ⊢ u ≡ t ∷ A\n trans : ∀ {t u r A}\n → Γ ⊢ t ≡ u ∷ A\n → Γ ⊢ u ≡ r ∷ A\n → Γ ⊢ t ≡ r ∷ A\n conv : ∀ {A B t u}\n → Γ ⊢ t ≡ u ∷ A\n → Γ ⊢ A ≡ B\n → Γ ⊢ t ≡ u ∷ B\n Π-cong : ∀ {E F G H}\n → Γ ⊢ F\n → Γ ⊢ F ≡ H ∷ U\n → Γ ∙ F ⊢ G ≡ E ∷ U\n → Γ ⊢ Π F ▹ G ≡ Π H ▹ E ∷ U\n Σ-cong : ∀ {E F G H}\n → Γ ⊢ F\n → Γ ⊢ F ≡ H ∷ U\n → Γ ∙ F ⊢ G ≡ E ∷ U\n → Γ ⊢ Σ F ▹ G ≡ Σ H ▹ E ∷ U\n app-cong : ∀ {a b f g F G}\n → Γ ⊢ f ≡ g ∷ Π F ▹ G\n → Γ ⊢ a ≡ b ∷ F\n → Γ ⊢ f ∘ a ≡ g ∘ b ∷ G [ a ]\n β-red : ∀ {a t F G}\n → Γ ⊢ F\n → Γ ∙ F ⊢ t ∷ G\n → Γ ⊢ a ∷ F\n → Γ ⊢ (lam t) ∘ a ≡ t [ a ] ∷ G [ a ]\n η-eq : ∀ {f g F G}\n → Γ ⊢ F\n → Γ ⊢ f ∷ Π F ▹ G\n → Γ ⊢ g ∷ Π F ▹ G\n → Γ ∙ F ⊢ wk1 f ∘ var x0 ≡ wk1 g ∘ var x0 ∷ G\n → Γ ⊢ f ≡ g ∷ Π F ▹ G\n fst-cong : ∀ {t t' F G}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ≡ t' ∷ Σ F ▹ G\n → Γ ⊢ fst t ≡ fst t' ∷ F\n snd-cong : ∀ {t t' F G}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ≡ t' ∷ Σ F ▹ G\n → Γ ⊢ snd t ≡ snd t' ∷ G [ fst t ]\n Σ-β₁ : ∀ {F G t u}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ∷ F\n → Γ ⊢ u ∷ G [ t ]\n → Γ ⊢ fst (prod t u) ≡ t ∷ F\n Σ-β₂ : ∀ {F G t u}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ∷ F\n → Γ ⊢ u ∷ G [ t ]\n → Γ ⊢ snd (prod t u) ≡ u ∷ G [ fst (prod t u) ]\n Σ-η : ∀ {p r F G}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ p ∷ Σ F ▹ G\n → Γ ⊢ r ∷ Σ F ▹ G\n → Γ ⊢ fst p ≡ fst r ∷ F\n → Γ ⊢ snd p ≡ snd r ∷ G [ fst p ]\n → Γ ⊢ p ≡ r ∷ Σ F ▹ G\n suc-cong : ∀ {m n}\n → Γ ⊢ m ≡ n ∷ ℕ\n → Γ ⊢ suc m ≡ suc n ∷ ℕ\n natrec-cong : ∀ {z z′ s s′ n n′ F F′}\n → Γ ∙ ℕ ⊢ F ≡ F′\n → Γ ⊢ z ≡ z′ ∷ F [ zero ]\n → Γ ⊢ s ≡ s′ ∷ Π ℕ ▹ (F ▹▹ F [ suc (var x0) ]↑)\n → Γ ⊢ n ≡ n′ ∷ ℕ\n → Γ ⊢ natrec F z s n ≡ natrec F′ z′ s′ n′ ∷ F [ n ]\n natrec-zero : ∀ {z s F}\n → Γ ∙ ℕ ⊢ F\n → Γ ⊢ z ∷ F [ zero ]\n → Γ ⊢ s ∷ Π ℕ ▹ (F ▹▹ F [ suc (var x0) ]↑)\n → Γ ⊢ natrec F z s zero ≡ z ∷ F [ zero ]\n natrec-suc : ∀ {n z s F}\n → Γ ⊢ n ∷ ℕ\n → Γ ∙ ℕ ⊢ F\n → Γ ⊢ z ∷ F [ zero ]\n → Γ ⊢ s ∷ Π ℕ ▹ (F ▹▹ F [ suc (var x0) ]↑)\n → Γ ⊢ natrec F z s (suc n) ≡ (s ∘ n) ∘ (natrec F z s n)\n ∷ F [ suc n ]\n Emptyrec-cong : ∀ {A A' e e'}\n → Γ ⊢ A ≡ A'\n → Γ ⊢ e ≡ e' ∷ Empty\n → Γ ⊢ Emptyrec A e ≡ Emptyrec A' e' ∷ A\n η-unit : ∀ {e e'}\n → Γ ⊢ e ∷ Unit\n → Γ ⊢ e' ∷ Unit\n → Γ ⊢ e ≡ e' ∷ Unit\n\n-- Term reduction\ndata _⊢_⇒_∷_ (Γ : Con Term n) : Term n → Term n → Term n → Set where\n conv : ∀ {A B t u}\n → Γ ⊢ t ⇒ u ∷ A\n → Γ ⊢ A ≡ B\n → Γ ⊢ t ⇒ u ∷ B\n app-subst : ∀ {A B t u a}\n → Γ ⊢ t ⇒ u ∷ Π A ▹ B\n → Γ ⊢ a ∷ A\n → Γ ⊢ t ∘ a ⇒ u ∘ a ∷ B [ a ]\n β-red : ∀ {A B a t}\n → Γ ⊢ A\n → Γ ∙ A ⊢ t ∷ B\n → Γ ⊢ a ∷ A\n → Γ ⊢ (lam t) ∘ a ⇒ t [ a ] ∷ B [ a ]\n fst-subst : ∀ {t t' F G}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ⇒ t' ∷ Σ F ▹ G\n → Γ ⊢ fst t ⇒ fst t' ∷ F\n snd-subst : ∀ {t t' F G}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ⇒ t' ∷ Σ F ▹ G\n → Γ ⊢ snd t ⇒ snd t' ∷ G [ fst t ]\n Σ-β₁ : ∀ {F G t u}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ∷ F\n → Γ ⊢ u ∷ G [ t ]\n → Γ ⊢ fst (prod t u) ⇒ t ∷ F\n Σ-β₂ : ∀ {F G t u}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ t ∷ F\n → Γ ⊢ u ∷ G [ t ]\n -- TODO(WN): Prove that 𝔍 ∷ G [ t ] is admissible\n → Γ ⊢ snd (prod t u) ⇒ u ∷ G [ fst (prod t u) ]\n natrec-subst : ∀ {z s n n′ F}\n → Γ ∙ ℕ ⊢ F\n → Γ ⊢ z ∷ F [ zero ]\n → Γ ⊢ s ∷ Π ℕ ▹ (F ▹▹ F [ suc (var x0) ]↑)\n → Γ ⊢ n ⇒ n′ ∷ ℕ\n → Γ ⊢ natrec F z s n ⇒ natrec F z s n′ ∷ F [ n ]\n natrec-zero : ∀ {z s F}\n → Γ ∙ ℕ ⊢ F\n → Γ ⊢ z ∷ F [ zero ]\n → Γ ⊢ s ∷ Π ℕ ▹ (F ▹▹ F [ suc (var x0) ]↑)\n → Γ ⊢ natrec F z s zero ⇒ z ∷ F [ zero ]\n natrec-suc : ∀ {n z s F}\n → Γ ⊢ n ∷ ℕ\n → Γ ∙ ℕ ⊢ F\n → Γ ⊢ z ∷ F [ zero ]\n → Γ ⊢ s ∷ Π ℕ ▹ (F ▹▹ F [ suc (var x0) ]↑)\n → Γ ⊢ natrec F z s (suc n) ⇒ (s ∘ n) ∘ (natrec F z s n) ∷ F [ suc n ]\n Emptyrec-subst : ∀ {n n′ A}\n → Γ ⊢ A\n → Γ ⊢ n ⇒ n′ ∷ Empty\n → Γ ⊢ Emptyrec A n ⇒ Emptyrec A n′ ∷ A\n\n-- Type reduction\ndata _⊢_⇒_ (Γ : Con Term n) : Term n → Term n → Set where\n univ : ∀ {A B}\n → Γ ⊢ A ⇒ B ∷ U\n → Γ ⊢ A ⇒ B\n\n-- Term reduction closure\ndata _⊢_⇒*_∷_ (Γ : Con Term n) : Term n → Term n → Term n → Set where\n id : ∀ {A t}\n → Γ ⊢ t ∷ A\n → Γ ⊢ t ⇒* t ∷ A\n _⇨_ : ∀ {A t t′ u}\n → Γ ⊢ t ⇒ t′ ∷ A\n → Γ ⊢ t′ ⇒* u ∷ A\n → Γ ⊢ t ⇒* u ∷ A\n\n-- Type reduction closure\ndata _⊢_⇒*_ (Γ : Con Term n) : Term n → Term n → Set where\n id : ∀ {A}\n → Γ ⊢ A\n → Γ ⊢ A ⇒* A\n _⇨_ : ∀ {A A′ B}\n → Γ ⊢ A ⇒ A′\n → Γ ⊢ A′ ⇒* B\n → Γ ⊢ A ⇒* B\n\n-- Type reduction to whnf\n_⊢_↘_ : (Γ : Con Term n) → Term n → Term n → Set\nΓ ⊢ A ↘ B = Γ ⊢ A ⇒* B × Whnf B\n\n-- Term reduction to whnf\n_⊢_↘_∷_ : (Γ : Con Term n) → Term n → Term n → Term n → Set\nΓ ⊢ t ↘ u ∷ A = Γ ⊢ t ⇒* u ∷ A × Whnf u\n\n-- Type eqaulity with well-formed types\n_⊢_:≡:_ : (Γ : Con Term n) → Term n → Term n → Set\nΓ ⊢ A :≡: B = Γ ⊢ A × Γ ⊢ B × (Γ ⊢ A ≡ B)\n\n-- Term equality with well-formed terms\n_⊢_:≡:_∷_ : (Γ : Con Term n) → Term n → Term n → Term n → Set\nΓ ⊢ t :≡: u ∷ A = (Γ ⊢ t ∷ A) × (Γ ⊢ u ∷ A) × (Γ ⊢ t ≡ u ∷ A)\n\n-- Type reduction closure with well-formed types\nrecord _⊢_:⇒*:_ (Γ : Con Term n) (A B : Term n) : Set where\n constructor [_,_,_]\n field\n ⊢A : Γ ⊢ A\n ⊢B : Γ ⊢ B\n D : Γ ⊢ A ⇒* B\n\nopen _⊢_:⇒*:_ using () renaming (D to red; ⊢A to ⊢A-red; ⊢B to ⊢B-red) public\n\n-- Term reduction closure with well-formed terms\nrecord _⊢_:⇒*:_∷_ (Γ : Con Term n) (t u A : Term n) : Set where\n constructor [_,_,_]\n field\n ⊢t : Γ ⊢ t ∷ A\n ⊢u : Γ ⊢ u ∷ A\n d : Γ ⊢ t ⇒* u ∷ A\n\nopen _⊢_:⇒*:_∷_ using () renaming (d to redₜ; ⊢t to ⊢t-redₜ; ⊢u to ⊢u-redₜ) public\n\n-- Well-formed substitutions.\ndata _⊢ˢ_∷_ (Δ : Con Term m) : (σ : Subst m n) (Γ : Con Term n) → Set where\n id : ∀ {σ} → Δ ⊢ˢ σ ∷ ε\n _,_ : ∀ {A σ}\n → Δ ⊢ˢ tail σ ∷ Γ\n → Δ ⊢ head σ ∷ subst (tail σ) A\n → Δ ⊢ˢ σ ∷ Γ ∙ A\n\n-- Conversion of well-formed substitutions.\ndata _⊢ˢ_≡_∷_ (Δ : Con Term m) : (σ σ′ : Subst m n) (Γ : Con Term n) → Set where\n id : ∀ {σ σ′} → Δ ⊢ˢ σ ≡ σ′ ∷ ε\n _,_ : ∀ {A σ σ′}\n → Δ ⊢ˢ tail σ ≡ tail σ′ ∷ Γ\n → Δ ⊢ head σ ≡ head σ′ ∷ subst (tail σ) A\n → Δ ⊢ˢ σ ≡ σ′ ∷ Γ ∙ A\n\n-- Note that we cannot use the well-formed substitutions.\n-- For that, we need to prove the fundamental theorem for substitutions.\n\n⟦_⟧ⱼ_▹_ : (W : BindingType) → ∀ {F G}\n → Γ ⊢ F\n → Γ ∙ F ⊢ G\n → Γ ⊢ ⟦ W ⟧ F ▹ G\n⟦ BΠ ⟧ⱼ ⊢F ▹ ⊢G = Πⱼ ⊢F ▹ ⊢G\n⟦ BΣ ⟧ⱼ ⊢F ▹ ⊢G = Σⱼ ⊢F ▹ ⊢G\n\n⟦_⟧ⱼᵤ_▹_ : (W : BindingType) → ∀ {F G}\n → Γ ⊢ F ∷ U\n → Γ ∙ F ⊢ G ∷ U\n → Γ ⊢ ⟦ W ⟧ F ▹ G ∷ U\n⟦ BΠ ⟧ⱼᵤ ⊢F ▹ ⊢G = Πⱼ ⊢F ▹ ⊢G\n⟦ BΣ ⟧ⱼᵤ ⊢F ▹ ⊢G = Σⱼ ⊢F ▹ ⊢G\n", "meta": {"hexsha": "4b7a71fadcf7db3e1a1bd90d432683f7d08559bb", "size": 12449, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Definition/Typed.agda", "max_stars_repo_name": "fhlkfy/logrel-mltt", "max_stars_repo_head_hexsha": "ea83fc4f618d1527d64ecac82d7d17e2f18ac391", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Definition/Typed.agda", "max_issues_repo_name": "fhlkfy/logrel-mltt", "max_issues_repo_head_hexsha": "ea83fc4f618d1527d64ecac82d7d17e2f18ac391", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Definition/Typed.agda", 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YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3458206285736102}} {"text": "module STLC1.Kovacs.Convertibility where\n\nopen import STLC1.Kovacs.Substitution public\n\n\n--------------------------------------------------------------------------------\n\n\n-- Convertibility (_~_ ; ~refl ; _~⁻¹ ; lam ; app ; β ; η)\ninfix 3 _∼_\ndata _∼_ : ∀ {Γ A} → Γ ⊢ A → Γ ⊢ A → Set\n where\n refl∼ : ∀ {Γ A} → {M : Γ ⊢ A}\n → M ∼ M\n\n _⁻¹∼ : ∀ {Γ A} → {M₁ M₂ : Γ ⊢ A}\n → (p : M₁ ∼ M₂)\n → M₂ ∼ M₁\n\n _⦙∼_ : ∀ {Γ A} → {M₁ M₂ M₃ : Γ ⊢ A}\n → (p : M₁ ∼ M₂) (q : M₂ ∼ M₃)\n → M₁ ∼ M₃\n\n ƛ∼ : ∀ {Γ A B} → {M₁ M₂ : Γ , A ⊢ B}\n → (p : M₁ ∼ M₂)\n → ƛ M₁ ∼ ƛ M₂\n\n _∙∼_ : ∀ {Γ A B} → {M₁ M₂ : Γ ⊢ A ⇒ B} {N₁ N₂ : Γ ⊢ A}\n → (p : M₁ ∼ M₂) (q : N₁ ∼ N₂)\n → M₁ ∙ N₁ ∼ M₂ ∙ N₂\n\n _,∼_ : ∀ {Γ A B} → {M₁ M₂ : Γ ⊢ A} {N₁ N₂ : Γ ⊢ B}\n → (p : M₁ ∼ M₂) (q : N₁ ∼ N₂)\n → M₁ , N₁ ∼ M₂ , N₂\n\n π₁∼ : ∀ {Γ A B} → {M₁ M₂ : Γ ⊢ A ⩕ B}\n → (p : M₁ ∼ M₂)\n → π₁ M₁ ∼ π₁ M₂\n\n π₂∼ : ∀ {Γ A B} → {M₁ M₂ : Γ ⊢ A ⩕ B}\n → (p : M₁ ∼ M₂)\n → π₂ M₁ ∼ π₂ M₂\n\n\n red⇒ : ∀ {Γ A B} → (M : Γ , A ⊢ B) (N : Γ ⊢ A)\n → (ƛ M) ∙ N ∼ cut N M\n\n red⩕₁ : ∀ {Γ A B} → (M : Γ ⊢ A) (N : Γ ⊢ B)\n → π₁ (M , N) ∼ M\n\n red⩕₂ : ∀ {Γ A B} → (M : Γ ⊢ A) (N : Γ ⊢ B)\n → π₂ (M , N) ∼ N\n\n\n exp⇒ : ∀ {Γ A B} → (M : Γ ⊢ A ⇒ B)\n → M ∼ ƛ (wk M ∙ 0)\n\n exp⩕ : ∀ {Γ A B} → (M : Γ ⊢ A ⩕ B)\n → M ∼ π₁ M , π₂ M\n\n exp⫪ : ∀ {Γ} → (M : Γ ⊢ ⫪)\n → M ∼ τ\n\n\n≡→∼ : ∀ {Γ A} → {M₁ M₂ : Γ ⊢ A}\n → M₁ ≡ M₂\n → M₁ ∼ M₂\n≡→∼ refl = refl∼\n\ninstance\n per∼ : ∀ {Γ A} → PER (Γ ⊢ A) _∼_\n per∼ =\n record\n { _⁻¹ = _⁻¹∼\n ; _⦙_ = _⦙∼_\n }\n\n\n--------------------------------------------------------------------------------\n\n\nrenwk : ∀ {Γ Γ′ A B} → (η : Γ′ ⊇ Γ) (M : Γ ⊢ A)\n → (wk {B} ∘ ren η) M ≡\n (ren (liftₑ η) ∘ wk) M\nrenwk η M = ren○ (wkₑ idₑ) η M ⁻¹\n ⦙ (λ η′ → ren (wkₑ η′) M) & ( rid○ η\n ⦙ lid○ η ⁻¹\n )\n ⦙ ren○ (liftₑ η) (wkₑ idₑ) M\n\nrencut : ∀ {Γ Γ′ A B} → (η : Γ′ ⊇ Γ) (M : Γ ⊢ A) (N : Γ , A ⊢ B)\n → (cut (ren η M) ∘ ren (liftₑ η)) N ≡\n (ren η ∘ cut M) N\nrencut η M N = sub◑ (idₛ , ren η M) (liftₑ η) N ⁻¹\n ⦙ (λ σ → sub (σ , ren η M) N) & ( rid◑ η\n ⦙ lid◐ η ⁻¹\n )\n ⦙ sub◐ η (idₛ , M) N\n\n\n-- (~ₑ)\nren∼ : ∀ {Γ Γ′ A} → {M₁ M₂ : Γ ⊢ A}\n → (η : Γ′ ⊇ Γ) → M₁ ∼ M₂\n → ren η M₁ ∼ ren η M₂\nren∼ η refl∼ = refl∼\nren∼ η (p ⁻¹∼) = ren∼ η p ⁻¹\nren∼ η (p ⦙∼ q) = ren∼ η p ⦙ ren∼ η q\nren∼ η (ƛ∼ p) = ƛ∼ (ren∼ (liftₑ η) p)\nren∼ η (p ∙∼ q) = ren∼ η p ∙∼ ren∼ η q\nren∼ η (p ,∼ q) = ren∼ η p ,∼ ren∼ η q\nren∼ η (π₁∼ p) = π₁∼ (ren∼ η p)\nren∼ η (π₂∼ p) = π₂∼ (ren∼ η p)\nren∼ η (red⇒ M N) = coe (((ƛ (ren (liftₑ η) M) ∙ ren η N) ∼_)\n & rencut η N M)\n (red⇒ (ren (liftₑ η) M) (ren η N))\nren∼ η (red⩕₁ M N) = red⩕₁ (ren η M) (ren η N)\nren∼ η (red⩕₂ M N) = red⩕₂ (ren η M) (ren η N)\nren∼ η (exp⇒ M) = coe ((λ M′ → ren η M ∼ ƛ (M′ ∙ 0))\n & renwk η M)\n (exp⇒ (ren η M))\nren∼ η (exp⩕ M) = exp⩕ (ren η M)\nren∼ η (exp⫪ M) = exp⫪ (ren η M)\n\n\n--------------------------------------------------------------------------------\n", "meta": {"hexsha": "a929e767bb2ffca97db693d4425678f7e30a54ef", "size": 3781, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/STLC1/Kovacs/Convertibility.agda", "max_stars_repo_name": "mietek/coquand-kovacs", "max_stars_repo_head_hexsha": "bd626509948fbf8503ec2e31c1852e1ac6edcc79", "max_stars_repo_licenses": ["X11"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/STLC1/Kovacs/Convertibility.agda", "max_issues_repo_name": "mietek/coquand-kovacs", "max_issues_repo_head_hexsha": "bd626509948fbf8503ec2e31c1852e1ac6edcc79", "max_issues_repo_licenses": ["X11"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/STLC1/Kovacs/Convertibility.agda", "max_forks_repo_name": "mietek/coquand-kovacs", "max_forks_repo_head_hexsha": "bd626509948fbf8503ec2e31c1852e1ac6edcc79", "max_forks_repo_licenses": ["X11"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.0079365079, "max_line_length": 80, "alphanum_fraction": 0.266067178, "num_tokens": 1759, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.34569497312102565}} {"text": "\nopen import Oscar.Prelude\nopen import Oscar.Class\nopen import Oscar.Class.Surjection\nopen import Oscar.Data.Proposequality\n\nmodule Oscar.Class.Smap where\n\nopen import Oscar.Class.Hmap public\n\nmodule Smap\n {𝔵₁ 𝔵₁'} {𝔛₁ : Ø 𝔵₁} {𝔛₁' : Ø 𝔵₁'}\n {𝔵₂ 𝔵₂'} {𝔛₂ : Ø 𝔵₂} {𝔛₂' : Ø 𝔵₂'}\n {𝔯₁₂} {𝔯₁₂'}\n (ℜ₁₂ : 𝔛₁ → 𝔛₂ → Ø 𝔯₁₂)\n (ℜ₁₂' : 𝔛₁' → 𝔛₂' → Ø 𝔯₁₂')\n (p₁ : 𝔛₁ → 𝔛₁')\n (p₂ : 𝔛₂ → 𝔛₂')\n where\n class = Hmap.class ℜ₁₂ (λ x y → ℜ₁₂' (p₁ x) (p₂ y))\n type = ∀ {x y} → ℜ₁₂ x y → ℜ₁₂' (p₁ x) (p₂ y)\n method : ⦃ _ : class ⦄ → type\n method = Hmap.method ℜ₁₂ (λ x y → ℜ₁₂' (p₁ x) (p₂ y)) _ _\n\nmodule _\n {𝔵₁ 𝔯₁ 𝔵₂ 𝔯₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}\n {_∼₁_ : 𝔛₁ → 𝔛₁ → Ø 𝔯₁}\n (_∼₂_ : 𝔛₂ → 𝔛₂ → Ø 𝔯₂)\n (μ₁ μ₂ : Surjection.type 𝔛₁ 𝔛₂)\n where\n open Smap _∼₁_ _∼₂_ μ₁ μ₂\n smap⟦_/_/_⟧ : ⦃ _ : class ⦄ → type\n smap⟦_/_/_⟧ = smap\n\nmodule ≡-Smap\n {𝔵₁ 𝔯₁ 𝔵₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}\n (∼₁ : 𝔛₁ → 𝔛₁ → Ø 𝔯₁)\n (μ₁ μ₂ : Surjection.type 𝔛₁ 𝔛₂)\n = Smap ∼₁ _≡_ μ₁ μ₂\n\nmodule _\n {𝔵₁ 𝔯₁ 𝔵₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}\n {_∼₁_ : 𝔛₁ → 𝔛₁ → Ø 𝔯₁}\n (μ₁ μ₂ : Surjection.type 𝔛₁ 𝔛₂)\n where\n open Smap _∼₁_ _≡_ μ₁ μ₂\n ≡-smap⟦_⟧ : ⦃ _ : class ⦄ → type\n ≡-smap⟦_⟧ = smap\n\nmodule Smap!\n {𝔵₁ 𝔯₁ 𝔵₂ 𝔯₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}\n (∼₁ : 𝔛₁ → 𝔛₁ → Ø 𝔯₁)\n (∼₂ : 𝔛₂ → 𝔛₂ → Ø 𝔯₂)\n ⦃ _ : Surjection.class 𝔛₁ 𝔛₂ ⦄\n = Smap ∼₁ ∼₂ surjection surjection\n\nmodule Smaparrow\n {𝔵₁ 𝔵₂ 𝔯 𝔭₁ 𝔭₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}\n (ℜ : 𝔛₁ → 𝔛₁ → Ø 𝔯)\n (𝔓₁ : 𝔛₂ → Ø 𝔭₁)\n (𝔓₂ : 𝔛₂ → Ø 𝔭₂)\n (surjection₁ surjection₂ : Surjection.type 𝔛₁ 𝔛₂)\n = Smap ℜ (Arrow 𝔓₁ 𝔓₂) surjection₁ surjection₂\n\nmodule _\n {𝔵₁ 𝔯 𝔭₁ 𝔭₂} {𝔛₁ : Ø 𝔵₁}\n {ℜ : 𝔛₁ → 𝔛₁ → Ø 𝔯}\n {𝔓₁ : 𝔛₁ → Ø 𝔭₁}\n {𝔓₂ : 𝔛₁ → Ø 𝔭₂}\n where\n smaparrow = Smaparrow.method ℜ 𝔓₁ 𝔓₂ ¡ ¡\n infixr 10 _◃_\n _◃_ = smaparrow\n smaparrow[]syntax = _◃_\n syntax smaparrow[]syntax 𝔛₂ x∼y fx = x∼y ◃[ 𝔛₂ ] fx\n\nmodule Smaparrow!\n {𝔵₁ 𝔵₂ 𝔯 𝔭₁ 𝔭₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}\n (ℜ : 𝔛₁ → 𝔛₁ → Ø 𝔯)\n (𝔓₁ : 𝔛₂ → Ø 𝔭₁)\n (𝔓₂ : 𝔛₂ → Ø 𝔭₂)\n ⦃ _ : Surjection.class 𝔛₁ 𝔛₂ ⦄\n = Smaparrow ℜ 𝔓₁ 𝔓₂ surjection surjection\n\nmodule Smaphomarrow\n {𝔵₁ 𝔯₁ 𝔵₂ 𝔯₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}\n (ℜ : 𝔛₁ → 𝔛₁ → Ø 𝔯₁)\n (𝔓 : 𝔛₂ → Ø 𝔯₂)\n (surjection : Surjection.type 𝔛₁ 𝔛₂)\n = Smaparrow ℜ 𝔓 𝔓 surjection surjection\n\nmodule Smaphomarrow!\n {𝔵₁ 𝔯₁ 𝔵₂ 𝔯₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}\n (ℜ : 𝔛₁ → 𝔛₁ → Ø 𝔯₁)\n (𝔓 : 𝔛₂ → Ø 𝔯₂)\n ⦃ _ : Surjection.class 𝔛₁ 𝔛₂ ⦄\n = Smaphomarrow ℜ 𝔓 surjection\n", "meta": {"hexsha": "ed0e70a86b63d01c8de4f8ce3b65817f3e33d24e", "size": 2350, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "archive/agda-3/src/Oscar/Class/Smap.agda", "max_stars_repo_name": "m0davis/oscar", "max_stars_repo_head_hexsha": "52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb", "max_stars_repo_licenses": ["RSA-MD"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "archive/agda-3/src/Oscar/Class/Smap.agda", "max_issues_repo_name": "m0davis/oscar", "max_issues_repo_head_hexsha": "52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb", "max_issues_repo_licenses": ["RSA-MD"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-04-29T00:35:04.000Z", "max_issues_repo_issues_event_max_datetime": "2019-05-11T23:33:04.000Z", "max_forks_repo_path": "archive/agda-3/src/Oscar/Class/Smap.agda", "max_forks_repo_name": "m0davis/oscar", "max_forks_repo_head_hexsha": "52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb", "max_forks_repo_licenses": ["RSA-MD"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.9795918367, "max_line_length": 59, "alphanum_fraction": 0.5557446809, "num_tokens": 1583, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.685949442167993, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3456541565794323}} {"text": "{-# OPTIONS --cubical --safe #-}\n\nmodule Harmony where\n\nopen import Data.Bool using (Bool; true; false; if_then_else_; _∨_; not; _∧_)\nopen import Data.Fin using (#_; toℕ) renaming (zero to fz; suc to fs)\nopen import Data.List using (List; map; []; _∷_; concatMap; foldr; head; zip; null)\nopen import Data.Maybe using (fromMaybe; is-nothing; Maybe; just; nothing)\nopen import Data.Nat using (ℕ; suc; _∸_; _<ᵇ_)\nopen import Data.Nat.DivMod using (_mod_; _div_)\nopen import Data.Product using (_×_; _,_; proj₁; proj₂; uncurry)\nopen import Data.Vec using (Vec; toList; []; _∷_)\nopen import Function using (_∘_)\n\nopen import BitVec using (BitVec; empty; insert; elements; _∩_; _∈_)\nopen import Counterpoint\nopen import Diatonic using (DiatonicDegree; thirdUp; _≡ᵈ_; degree→PC; major; pitch→DegreeCMajor)\nopen import Interval\nopen import Music\nopen import Note\nopen import Pitch hiding (I)\nopen import Util using (filter; concatMaybe)\n\n-- either 0 or 1 pitch class\npointToPC : Point → List PC\npointToPC (tone p) = pitchToClass p ∷ []\npointToPC (hold p) = pitchToClass p ∷ []\npointToPC rest = []\n\nchordToPCes : {n : ℕ} → Chord n → List PC\nchordToPCes (chord ps) = concatMap pointToPC (toList ps) \n\npitchClassListToSet : List PC → PCSet\npitchClassListToSet = foldr insert empty\n\npitchClassSetToList : PCSet → List PC\npitchClassSetToList ps = fn 0 (toList ps)\n where fn : ℕ → List Bool → List PC\n fn i [] = []\n fn i (false ∷ bs) = fn (suc i) bs\n fn i (true ∷ bs) = i mod s12 ∷ fn (suc i) bs\n\n-- Primary chords, assuming the tonic is pitch class 0.\n\nI-maj I-min IV-maj V-maj : PCSet\n\nI-maj = pitchClassListToSet (# 0 ∷ # 4 ∷ # 7 ∷ [])\nI-min = pitchClassListToSet (# 0 ∷ # 3 ∷ # 7 ∷ [])\nIV-maj = pitchClassListToSet (# 0 ∷ # 5 ∷ # 9 ∷ [])\nV-maj = pitchClassListToSet (# 2 ∷ # 7 ∷ # 11 ∷ [])\n\n-- Triads, without quality\ndata Triad : Set where\n I : Triad\n II : Triad\n III : Triad\n IV : Triad\n V : Triad\n VI : Triad\n VII : Triad\n\n--data Triad : Set where\n-- I : Triad; II : Triad; III : Triad; IV : Triad; V : Triad; VI : Triad; VII : Triad\n\nallTriads : List Triad\nallTriads = I ∷ II ∷ III ∷ IV ∷ V ∷ VI ∷ VII ∷ []\n\ntriadRoot : Triad → DiatonicDegree\ntriadRoot I = (# 0)\ntriadRoot II = (# 1)\ntriadRoot III = (# 2)\ntriadRoot IV = (# 3)\ntriadRoot V = (# 4)\ntriadRoot VI = (# 5)\ntriadRoot VII = (# 6)\n\nrootTriad : DiatonicDegree → Triad\nrootTriad fz = I\nrootTriad ((fs fz)) = II\nrootTriad ((fs (fs fz))) = III\nrootTriad ((fs (fs (fs fz)))) = IV\nrootTriad ((fs (fs (fs (fs fz))))) = V\nrootTriad ((fs (fs (fs (fs (fs fz)))))) = VI\nrootTriad ((fs (fs (fs (fs (fs (fs fz))))))) = VII\n\ntriadDegrees : Triad → Vec DiatonicDegree 3\ntriadDegrees t =\n let root = triadRoot t\n third = thirdUp root\n fifth = thirdUp third\n in root ∷ third ∷ fifth ∷ []\n\ninfix 4 _≡ᵗ_\n_≡ᵗ_ : Triad → Triad → Bool\nt ≡ᵗ u = triadRoot t ≡ᵈ triadRoot u\n\nTriadSet : Set\nTriadSet = BitVec s7\n\ntriadListToSet : List Triad → TriadSet\ntriadListToSet [] = empty\ntriadListToSet (t ∷ ts) = insert (triadRoot t) (triadListToSet ts)\n\ntriadSetToList : TriadSet → List Triad\ntriadSetToList ts = map rootTriad (elements ts)\n\ncontainingTriads : DiatonicDegree → List Triad\ncontainingTriads fz = I ∷ IV ∷ VI ∷ []\ncontainingTriads ((fs fz)) = II ∷ V ∷ VII ∷ []\ncontainingTriads ((fs (fs fz))) = III ∷ VI ∷ I ∷ []\ncontainingTriads ((fs (fs (fs fz)))) = IV ∷ VII ∷ II ∷ []\ncontainingTriads ((fs (fs (fs (fs fz))))) = V ∷ I ∷ III ∷ []\ncontainingTriads ((fs (fs (fs (fs (fs fz)))))) = VI ∷ II ∷ IV ∷ []\ncontainingTriads ((fs (fs (fs (fs (fs (fs fz))))))) = VII ∷ III ∷ V ∷ []\n\n-- from Table of Usual Root Progressions (Major Mode), Harmony (Piston 5e), page 23\nrecord NextTriad : Set where\n constructor nextTriad\n field\n usual : List Triad\n sometimes : List Triad\n rare : List Triad\nopen NextTriad\n\nrootProgression : Triad → NextTriad\nrootProgression I = nextTriad (IV ∷ V ∷ []) (VI ∷ []) (II ∷ III ∷ [])\nrootProgression II = nextTriad (V ∷ []) (IV ∷ VI ∷ []) (I ∷ III ∷ [])\nrootProgression III = nextTriad (VI ∷ []) (IV ∷ []) (I ∷ II ∷ V ∷ [])\nrootProgression IV = nextTriad (V ∷ []) (I ∷ II ∷ []) (III ∷ VI ∷ [])\nrootProgression V = nextTriad (I ∷ []) (IV ∷ VI ∷ []) (II ∷ III ∷ [])\nrootProgression VI = nextTriad (II ∷ V ∷ []) (III ∷ IV ∷ []) (I ∷ [])\nrootProgression VII = nextTriad (I ∷ III ∷ []) (VI ∷ []) (II ∷ IV ∷ V ∷ [])\n\npreviousTriads : Triad → List Triad\npreviousTriads I = V ∷ IV ∷ VII ∷ []\npreviousTriads II = VI ∷ IV ∷ []\npreviousTriads III = VI ∷ VII ∷ []\npreviousTriads IV = I ∷ V ∷ II ∷ III ∷ []\npreviousTriads V = I ∷ IV ∷ II ∷ VI ∷ []\npreviousTriads VI = IV ∷ I ∷ II ∷ V ∷ VII ∷ []\npreviousTriads VII = []\n\nharmonizations : {n : ℕ} → Vec DiatonicDegree n → List (Vec Triad n)\nharmonizations [] = []\nharmonizations (d ∷ []) = map (_∷ []) (containingTriads d)\nharmonizations (d ∷ d' ∷ ds) =\n let tss = harmonizations (d' ∷ ds)\n dTriads = containingTriads d\n in concatMap (λ t → concatMaybe (map (prependTriad t) tss)) dTriads\n where\n prevOk : Triad → Triad → Bool\n prevOk t x = (triadRoot t) ∈ triadListToSet (previousTriads x)\n prependTriad : {n : ℕ} → Triad → Vec Triad (suc n) → Maybe (Vec Triad (suc (suc n)))\n prependTriad t ts = if prevOk t (Data.Vec.head ts) then just (t ∷ ts) else nothing\n\nhalfCadence : {n : ℕ} → Vec Triad n → Bool\nhalfCadence [] = false\nhalfCadence (t ∷ []) = t ≡ᵗ V\nhalfCadence (_ ∷ t ∷ ts) = halfCadence (t ∷ ts)\n\n-- Given a pitch p and a diatontic degree d, return a pitch that\n-- has degree d and is 1-2 octaves lower than p.\npitchLower : Pitch → DiatonicDegree → Pitch\npitchLower p d =\n let (c , o) = absoluteToRelative p\n c' = degree→PC major d\n in relativeToAbsolute (c' , o ∸ 2)\n\n-- Given a soporano voice s a pitch and the other voices\n-- as diatonic degrees of a major scale, voice the\n-- accompaniment in close position.\nvoiceChord : Pitch → Vec DiatonicDegree 3 → Vec Pitch 3\nvoiceChord s (a ∷ t ∷ b ∷ []) =\n let (s' , o) = absoluteToRelative s\n a' = degree→PC major a\n t' = degree→PC major t\n b' = degree→PC major b\n ao = downOctave a' s' o\n to = downOctave t' a' ao\n bo = downOctave b' t' to\n in relativeToAbsolute (a' , ao) ∷\n relativeToAbsolute (t' , to) ∷\n relativeToAbsolute (b' , bo) ∷ []\n where downOctave : PC → PC → Octave → Octave\n downOctave pc₁ pc₂ o =\n if toℕ pc₁ <ᵇ toℕ pc₂ then o\n else (o ∸ 1)\n\n-- Given a soprano pitch p and a triad harmonization t,\n-- generate a list of possible bass notes.\n-- Assumes p is in t. Only the root of the triad is\n-- allowed to be doubled.\n-- Each bass note is pitched 1-2 octaves below p.\nbassNotes : Pitch → Triad → List Pitch\nbassNotes p t =\n let sop = pitch→DegreeCMajor p\n root = triadRoot t\n ds = toList (triadDegrees t)\n ds' = filter (λ d → (sop ≡ᵈ root) ∨ not (sop ≡ᵈ d)) ds\n in map (pitchLower p) ds'\n\n-- Given a soprano pitch p and a triad harmonization t,\n-- generate a harmonizing chord in root position.\n-- Assumes p is in t. Only the root of the triad is\n-- allowed to be doubled.\n-- Each bass note is pitched 1-2 octaves below p.\n-- Alto and Tenor fit inside.\n-- Currently root or third is preferred for alto.\nharmonizingChord : Pitch → Triad → Vec Pitch 3\nharmonizingChord p t =\n let sop = pitch→DegreeCMajor p\n root = triadRoot t\n third = thirdUp root\n fifth = thirdUp third\n alto = if sop ≡ᵈ root then third else root\n tenor = if sop ≡ᵈ fifth then third else fifth\n in voiceChord p (alto ∷ tenor ∷ root ∷ [])\n where\n remove : DiatonicDegree → Vec DiatonicDegree 3 → Vec DiatonicDegree 2\n remove sop (d ∷ d₁ ∷ d₂ ∷ []) =\n if d ≡ᵈ sop then d₁ ∷ d₂ ∷ []\n else (if d₁ ≡ᵈ sop then d ∷ d₂ ∷ [] else d ∷ d₁ ∷ [])\n\n-- Create 4 part harmonizations ending in V for a melody in C major.\nvoicedHarmonizations : {n : ℕ} → Vec Pitch n → List (Vec (Vec Pitch 4) n)\nvoicedHarmonizations {n} ps =\n let ds = Data.Vec.map pitch→DegreeCMajor ps\n hs : List (Vec Triad n)\n hs = filter halfCadence (harmonizations ds)\n in map (λ ts → Data.Vec.map (λ pt → proj₁ pt ∷ harmonizingChord (proj₁ pt) (proj₂ pt))\n (Data.Vec.zip ps ts)) hs\n\n-- Check interval between each pair of voices.\nintervalsOkFilter : Vec Pitch 4 → Bool\nintervalsOkFilter (s ∷ a ∷ t ∷ b ∷ []) =\n null (concatMaybe (map (intervalCheck ∘ toPitchInterval)\n-- ((s , a) ∷ (s , t) ∷ (s , b) ∷ (a , t) ∷ (a , b) ∷ (t , b) ∷ [])))\n ((s , a) ∷ (s , b) ∷ (s , t) ∷ [])))\n\nfilterIntervalsOk : {n : ℕ} → List (Vec (Vec Pitch 4) n) → List (Vec (Vec Pitch 4) n)\nfilterIntervalsOk xs =\n let f : List (Vec Pitch 4) → Bool\n f xs = foldr _∧_ true (map intervalsOkFilter xs)\n in filter (f ∘ toList) xs\n\nmotionErrors : {n : ℕ} → Vec (Vec Pitch 4) n → List MotionError\nmotionErrors xs =\n let ss = Data.Vec.map (Data.Vec.head) xs\n as = Data.Vec.map (Data.Vec.head ∘ Data.Vec.tail) xs\n ts = Data.Vec.map (Data.Vec.head ∘ Data.Vec.tail ∘ Data.Vec.tail) xs\n bs = Data.Vec.map (Data.Vec.head ∘ Data.Vec.tail ∘ Data.Vec.tail ∘ Data.Vec.tail) xs\n\n sas = map toPitchInterval (toList (Data.Vec.zip as ss))\n sts = map toPitchInterval (toList (Data.Vec.zip ts ss))\n sbs = map toPitchInterval (toList (Data.Vec.zip bs ss))\n ats = map toPitchInterval (toList (Data.Vec.zip ts as))\n abs = map toPitchInterval (toList (Data.Vec.zip bs as))\n tbs = map toPitchInterval (toList (Data.Vec.zip bs ts))\n in concatMap checkMotion (sas ∷ sts ∷ sbs ∷ ats ∷ abs ∷ tbs ∷ [])\n\n--filterSBMotionOk : {n : ℕ} → List (Vec (Vec Pitch 4) n) → List (Vec (Vec Pitch 4) n)\n--filterSBMotionOk = filter motionOkFilter\n\n-- Given a soprano line with harmonization, generate\n-- a list of possible bass lines 1-1 with soprano notes.\n-- The SB counterpoint must satisfy 1st species\n-- interval and motion rules.\nbassLines : List (Pitch × Triad) → List (List Pitch)\n-- We need to make the base case a singleton list of an empty list for\n-- the general case to work. Possibly look into modifying the general\n-- case to handle a base case of an empty list.\nbassLines [] = [] ∷ []\nbassLines ((sop , triad) ∷ pts) =\n let pss = bassLines pts\n basses = bassNotes sop triad\n intervalOkSBs : List PitchInterval -- list of bass notes with interval (to sop) that pass intervalCheck\n intervalOkSBs = filter (is-nothing ∘ intervalCheck) (map (toPitchInterval ∘ (_, sop)) basses)\n intervalOkBs = map proj₁ intervalOkSBs\n intervalOkBassLines = concatMap (λ ps → (map (_∷ ps) intervalOkBs)) pss\n in filter (mCheck sop (Data.Maybe.map proj₁ (head pts))) intervalOkBassLines\n where\n -- Given a soprano pitch, possibly a second soprano pitch and a list of\n -- bass pitches, if the second soprano pitch exists and there are at\n -- least two bass pitches, check that the motion from the first SB pair to\n -- the second is allowed. If there aren't two SB pairs, return true.\n mCheck : Pitch → Maybe Pitch → List Pitch → Bool\n mCheck _ nothing _ = true\n mCheck _ (just _) [] = true\n mCheck _ (just _) (_ ∷ []) = true\n mCheck s₁ (just s₂) (b₁ ∷ b₂ ∷ _) =\n let sb₁ = toPitchInterval (b₁ , s₁)\n sb₂ = toPitchInterval (b₂ , s₂)\n in (is-nothing ∘ uncurry motionCheck) (sb₁ , sb₂)\n\n-- Given a soprano line with harmonization, generate\n-- a list of possible triads 1-1 with soprano notes.\n-- All pairwise counterpoint must satisfy 1st species\n-- interval and motion rules.\nchordProg : List (Pitch × Triad) → List (List Pitch)\n-- We need to make the base case a singleton list of an empty list for\n-- the general case to work. Possibly look into modifying the general\n-- case to handle a base case of an empty list.\nchordProg [] = [] ∷ []\nchordProg ((sop , triad) ∷ pts) =\n let pss = chordProg pts\n basses = bassNotes sop triad\n intervalOkSBs : List PitchInterval -- list of bass notes with interval (to sop) that pass intervalCheck\n intervalOkSBs = filter (is-nothing ∘ intervalCheck) (map (toPitchInterval ∘ (_, sop)) basses)\n intervalOkBs = map proj₁ intervalOkSBs\n intervalOkBassLines = concatMap (λ ps → (map (_∷ ps) intervalOkBs)) pss\n in filter (mCheck sop (Data.Maybe.map proj₁ (head pts))) intervalOkBassLines\n where\n -- Given a soprano pitch, possibly a second soprano pitch and a list of\n -- bass pitches, if the second soprano pitch exists and there are at\n -- least two bass pitches, check that the motion from the first SB pair to\n -- the second is allowed. If there aren't two SB pairs, return true.\n mCheck : Pitch → Maybe Pitch → List Pitch → Bool\n mCheck _ nothing _ = true\n mCheck _ (just _) [] = true\n mCheck _ (just _) (_ ∷ []) = true\n mCheck s₁ (just s₂) (b₁ ∷ b₂ ∷ _) =\n let sb₁ = toPitchInterval (b₁ , s₁)\n sb₂ = toPitchInterval (b₂ , s₂)\n in (is-nothing ∘ uncurry motionCheck) (sb₁ , sb₂)\n", "meta": {"hexsha": "a91bbd7a3e261c2dd8a1acfbcd4350ee9d7fd891", "size": 13427, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "agda/Harmony.agda", "max_stars_repo_name": "halfaya/MusicTools", "max_stars_repo_head_hexsha": "04896c61b603d46011b7d718fcb47dd756e66021", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 28, "max_stars_repo_stars_event_min_datetime": "2017-04-21T09:08:52.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-04T18:04:07.000Z", "max_issues_repo_path": "agda/Harmony.agda", "max_issues_repo_name": "halfaya/MusicTools", "max_issues_repo_head_hexsha": "04896c61b603d46011b7d718fcb47dd756e66021", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-11-13T01:26:20.000Z", "max_issues_repo_issues_event_max_datetime": "2020-11-17T00:58:55.000Z", "max_forks_repo_path": "agda/Harmony.agda", "max_forks_repo_name": "halfaya/MusicTools", "max_forks_repo_head_hexsha": "04896c61b603d46011b7d718fcb47dd756e66021", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-01-12T17:02:36.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-10T04:04:40.000Z", "avg_line_length": 41.3138461538, "max_line_length": 109, "alphanum_fraction": 0.6145825575, "num_tokens": 4449, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.34545842328678156}} {"text": "module Issue1760g where\n\ndata ⊥ : Set where\n\n{-# NO_POSITIVITY_CHECK #-}\n{-# NON_TERMINATING #-}\nmutual\n record U : Set where\n constructor roll\n field ap : U → U\n\n lemma : U → ⊥\n lemma (roll u) = lemma (u (roll u))\n\n bottom : ⊥\n bottom = lemma (roll λ x → x)\n", "meta": {"hexsha": "ef04ee4da2b2cf472d7d6110e0b7e8bce9e3d754", "size": 274, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/Succeed/Issue1760g.agda", "max_stars_repo_name": "hborum/agda", "max_stars_repo_head_hexsha": "aac88412199dd4cbcb041aab499d8a6b7e3f4a2e", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2015-03-28T14:51:03.000Z", "max_stars_repo_stars_event_max_datetime": "2015-12-07T20:14:00.000Z", "max_issues_repo_path": "test/Succeed/Issue1760g.agda", "max_issues_repo_name": "hborum/agda", "max_issues_repo_head_hexsha": "aac88412199dd4cbcb041aab499d8a6b7e3f4a2e", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2018-11-14T15:31:44.000Z", "max_issues_repo_issues_event_max_datetime": "2019-04-01T19:39:26.000Z", "max_forks_repo_path": "test/Succeed/Issue1760g.agda", "max_forks_repo_name": "Agda-zh/agda", "max_forks_repo_head_hexsha": "231d6ad8e77b67ff8c4b1cb35a6c31ccd988c3e9", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-03-05T20:02:38.000Z", "max_forks_repo_forks_event_max_datetime": "2019-03-05T20:02:38.000Z", "avg_line_length": 16.1176470588, "max_line_length": 37, "alphanum_fraction": 0.5948905109, "num_tokens": 88, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.72487026428967, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.3454584176216055}} {"text": "module Thesis.SIRelBigStep.IlcSILR where\n\nopen import Data.Unit.Base hiding (_≤_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.Lang public\nopen import Thesis.SIRelBigStep.DLang public\n\nopen import Thesis.SIRelBigStep.ArithExtra public\n\nrrelT3-skeleton :\n ∀ {τ Γ} →\n ((v1 : Val τ) → (dv : DVal τ) → (v2 : Val τ) → (k : ℕ) → Set) →\n (t1 : Term Γ τ) (dt : DTerm Γ τ) (t2 : Term Γ τ)\n (ρ1 : ⟦ Γ ⟧Context) (dρ : ChΔ Γ) (ρ2 : ⟦ Γ ⟧Context) →\n ℕ → Set\nrrelT3-skeleton {τ} ternary-rel t1 dt t2 ρ1 dρ ρ2 k =\n (v1 v2 : Val τ) →\n ∀ j (j>=_ _>>=ˡ_ _>>=ʳ_\n_>>=_ : ∀ {P Q S T} → <:-Close P (S ⇒ T) → (P ⊂: Q) → <:-Close Q (S ⇒ T)\n(defn p p₁ p₂) >>= P⊂Q with P⊂Q p\n(defn p p₁ p₂) >>= P⊂Q | defn q q₁ q₂ = defn q (<:-trans p₁ q₁) (<:-trans q₂ p₂)\n\n_>>=ˡ_ : ∀ {P R S T} → <:-Close P (S ⇒ T) → (R <: S) → <:-Close P (R ⇒ T)\n(defn p p₁ p₂) >>=ˡ q = defn p (<:-trans q p₁) p₂\n\n_>>=ʳ_ : ∀ {P S T U} → <:-Close P (S ⇒ T) → (T <: U) → <:-Close P (S ⇒ U)\n(defn p p₁ p₂) >>=ʳ q = defn p p₁ (<:-trans p₂ q)\n\n-- Properties of ⊂:\n⊂:-refl : ∀ {P} → P ⊂: P\n⊂:-refl p = just p\n\n_[∪]_ : ∀ {P Q R S T U} → <:-Close P (R ⇒ S) → <:-Close Q (T ⇒ U) → <:-Close (∪-Lift P Q) ((R ∪ T) ⇒ (S ∪ U))\n(defn p p₁ p₂) [∪] (defn q q₁ q₂) = defn (union p q) (<:-union p₁ q₁) (<:-union p₂ q₂)\n\n_[∩]_ : ∀ {P Q R S T U} → <:-Close P (R ⇒ S) → <:-Close Q (T ⇒ U) → <:-Close (∩-Lift P Q) ((R ∩ T) ⇒ (S ∩ U))\n(defn p p₁ p₂) [∩] (defn q q₁ q₂) = defn (intersect p q) (<:-intersect p₁ q₁) (<:-intersect p₂ q₂)\n\n⊂:-∩-saturate-inj : ∀ {P} → P ⊂: ∩-Saturate P\n⊂:-∩-saturate-inj p = defn (base p) <:-refl <:-refl\n\n⊂:-∪-saturate-inj : ∀ {P} → P ⊂: ∪-Saturate P\n⊂:-∪-saturate-inj p = just (base p)\n\n⊂:-∩-lift-saturate : ∀ {P} → ∩-Lift (∩-Saturate P) (∩-Saturate P) ⊂: ∩-Saturate P\n⊂:-∩-lift-saturate (intersect p q) = just (intersect p q)\n\n⊂:-∪-lift-saturate : ∀ {P} → ∪-Lift (∪-Saturate P) (∪-Saturate P) ⊂: ∪-Saturate P\n⊂:-∪-lift-saturate (union p q) = just (union p q)\n\n⊂:-∩-lift : ∀ {P Q R S} → (P ⊂: Q) → (R ⊂: S) → (∩-Lift P R ⊂: ∩-Lift Q S)\n⊂:-∩-lift P⊂Q R⊂S (intersect n o) = P⊂Q n [∩] R⊂S o\n\n⊂:-∪-lift : ∀ {P Q R S} → (P ⊂: Q) → (R ⊂: S) → (∪-Lift P R ⊂: ∪-Lift Q S)\n⊂:-∪-lift P⊂Q R⊂S (union n o) = P⊂Q n [∪] R⊂S o\n\n⊂:-∩-saturate : ∀ {P Q} → (P ⊂: Q) → (∩-Saturate P ⊂: ∩-Saturate Q)\n⊂:-∩-saturate P⊂Q (base p) = P⊂Q p >>= ⊂:-∩-saturate-inj\n⊂:-∩-saturate P⊂Q (intersect p q) = (⊂:-∩-saturate P⊂Q p [∩] ⊂:-∩-saturate P⊂Q q) >>= ⊂:-∩-lift-saturate\n\n⊂:-∪-saturate : ∀ {P Q} → (P ⊂: Q) → (∪-Saturate P ⊂: ∪-Saturate Q)\n⊂:-∪-saturate P⊂Q (base p) = P⊂Q p >>= ⊂:-∪-saturate-inj\n⊂:-∪-saturate P⊂Q (union p q) = (⊂:-∪-saturate P⊂Q p [∪] ⊂:-∪-saturate P⊂Q q) >>= ⊂:-∪-lift-saturate\n\n⊂:-∩-saturate-indn : ∀ {P Q} → (P ⊂: Q) → (∩-Lift Q Q ⊂: Q) → (∩-Saturate P ⊂: Q)\n⊂:-∩-saturate-indn P⊂Q QQ⊂Q (base p) = P⊂Q p\n⊂:-∩-saturate-indn P⊂Q QQ⊂Q (intersect p q) = (⊂:-∩-saturate-indn P⊂Q QQ⊂Q p [∩] ⊂:-∩-saturate-indn P⊂Q QQ⊂Q q) >>= QQ⊂Q\n\n⊂:-∪-saturate-indn : ∀ {P Q} → (P ⊂: Q) → (∪-Lift Q Q ⊂: Q) → (∪-Saturate P ⊂: Q)\n⊂:-∪-saturate-indn P⊂Q QQ⊂Q (base p) = P⊂Q p\n⊂:-∪-saturate-indn P⊂Q QQ⊂Q (union p q) = (⊂:-∪-saturate-indn P⊂Q QQ⊂Q p [∪] ⊂:-∪-saturate-indn P⊂Q QQ⊂Q q) >>= QQ⊂Q\n\n∪-saturate-resp-∩-saturation : ∀ {P} → (∩-Lift P P ⊂: P) → (∩-Lift (∪-Saturate P) (∪-Saturate P) ⊂: ∪-Saturate P)\n∪-saturate-resp-∩-saturation ∩P⊂P (intersect (base p) (base q)) = ∩P⊂P (intersect p q) >>= ⊂:-∪-saturate-inj\n∪-saturate-resp-∩-saturation ∩P⊂P (intersect p (union q q₁)) = (∪-saturate-resp-∩-saturation ∩P⊂P (intersect p q) [∪] ∪-saturate-resp-∩-saturation ∩P⊂P (intersect p q₁)) >>= ⊂:-∪-lift-saturate >>=ˡ <:-∩-distl-∪ >>=ʳ ∩-distl-∪-<:\n∪-saturate-resp-∩-saturation ∩P⊂P (intersect (union p p₁) q) = (∪-saturate-resp-∩-saturation ∩P⊂P (intersect p q) [∪] ∪-saturate-resp-∩-saturation ∩P⊂P (intersect p₁ q)) >>= ⊂:-∪-lift-saturate >>=ˡ <:-∩-distr-∪ >>=ʳ ∩-distr-∪-<:\n\nov-language : ∀ {F t} → FunType F → (∀ {S T} → Overloads F (S ⇒ T) → Language (S ⇒ T) t) → Language F t\nov-language (S ⇒ T) p = p here\nov-language (F ∩ G) p = (ov-language F (p ∘ left) , ov-language G (p ∘ right))\n\nov-<: : ∀ {F R S T U} → FunType F → Overloads F (R ⇒ S) → ((R ⇒ S) <: (T ⇒ U)) → F <: (T ⇒ U)\nov-<: F here p = p\nov-<: (F ∩ G) (left o) p = <:-trans <:-∩-left (ov-<: F o p)\nov-<: (F ∩ G) (right o) p = <:-trans <:-∩-right (ov-<: G o p)\n\n<:ᵒ-impl-<: : ∀ {F G} → FunType F → FunType G → (F <:ᵒ G) → (F <: G)\n<:ᵒ-impl-<: F (T ⇒ U) F>= ⊂:-overloads-left\n⊂:-overloads-⋒ (R ⇒ S) (G ∩ H) (intersect here (right o)) = ⊂:-overloads-⋒ (R ⇒ S) H (intersect here o) >>= ⊂:-overloads-right\n⊂:-overloads-⋒ (E ∩ F) G (intersect (left n) o) = ⊂:-overloads-⋒ E G (intersect n o) >>= ⊂:-overloads-left\n⊂:-overloads-⋒ (E ∩ F) G (intersect (right n) o) = ⊂:-overloads-⋒ F G (intersect n o) >>= ⊂:-overloads-right\n\n⊂:-⋒-overloads : ∀ {F G} → FunType F → FunType G → Overloads (F ⋒ G) ⊂: ∩-Lift (Overloads F) (Overloads G)\n⊂:-⋒-overloads (R ⇒ S) (T ⇒ U) here = defn (intersect here here) (∩ⁿ-<:-∩ R T) (∩-<:-∩ⁿ S U)\n⊂:-⋒-overloads (R ⇒ S) (G ∩ H) (left o) = ⊂:-⋒-overloads (R ⇒ S) G o >>= ⊂:-∩-lift ⊂:-refl ⊂:-overloads-left \n⊂:-⋒-overloads (R ⇒ S) (G ∩ H) (right o) = ⊂:-⋒-overloads (R ⇒ S) H o >>= ⊂:-∩-lift ⊂:-refl ⊂:-overloads-right\n⊂:-⋒-overloads (E ∩ F) G (left o) = ⊂:-⋒-overloads E G o >>= ⊂:-∩-lift ⊂:-overloads-left ⊂:-refl\n⊂:-⋒-overloads (E ∩ F) G (right o) = ⊂:-⋒-overloads F G o >>= ⊂:-∩-lift ⊂:-overloads-right ⊂:-refl\n\n⊂:-overloads-⋓ : ∀ {F G} → FunType F → FunType G → ∪-Lift (Overloads F) (Overloads G) ⊂: Overloads (F ⋓ G)\n⊂:-overloads-⋓ (R ⇒ S) (T ⇒ U) (union here here) = defn here (∪-<:-∪ⁿ R T) (∪ⁿ-<:-∪ S U)\n⊂:-overloads-⋓ (R ⇒ S) (G ∩ H) (union here (left o)) = ⊂:-overloads-⋓ (R ⇒ S) G (union here o) >>= ⊂:-overloads-left\n⊂:-overloads-⋓ (R ⇒ S) (G ∩ H) (union here (right o)) = ⊂:-overloads-⋓ (R ⇒ S) H (union here o) >>= ⊂:-overloads-right\n⊂:-overloads-⋓ (E ∩ F) G (union (left n) o) = ⊂:-overloads-⋓ E G (union n o) >>= ⊂:-overloads-left\n⊂:-overloads-⋓ (E ∩ F) G (union (right n) o) = ⊂:-overloads-⋓ F G (union n o) >>= ⊂:-overloads-right\n\n⊂:-⋓-overloads : ∀ {F G} → FunType F → FunType G → Overloads (F ⋓ G) ⊂: ∪-Lift (Overloads F) (Overloads G)\n⊂:-⋓-overloads (R ⇒ S) (T ⇒ U) here = defn (union here here) (∪ⁿ-<:-∪ R T) (∪-<:-∪ⁿ S U)\n⊂:-⋓-overloads (R ⇒ S) (G ∩ H) (left o) = ⊂:-⋓-overloads (R ⇒ S) G o >>= ⊂:-∪-lift ⊂:-refl ⊂:-overloads-left\n⊂:-⋓-overloads (R ⇒ S) (G ∩ H) (right o) = ⊂:-⋓-overloads (R ⇒ S) H o >>= ⊂:-∪-lift ⊂:-refl ⊂:-overloads-right\n⊂:-⋓-overloads (E ∩ F) G (left o) = ⊂:-⋓-overloads E G o >>= ⊂:-∪-lift ⊂:-overloads-left ⊂:-refl\n⊂:-⋓-overloads (E ∩ F) G (right o) = ⊂:-⋓-overloads F G o >>= ⊂:-∪-lift ⊂:-overloads-right ⊂:-refl\n\n∪-saturate-overloads : ∀ {F} → FunType F → Overloads (∪-saturate F) ⊂: ∪-Saturate (Overloads F)\n∪-saturate-overloads (S ⇒ T) here = just (base here)\n∪-saturate-overloads (F ∩ G) (left (left o)) = ∪-saturate-overloads F o >>= ⊂:-∪-saturate ⊂:-overloads-left\n∪-saturate-overloads (F ∩ G) (left (right o)) = ∪-saturate-overloads G o >>= ⊂:-∪-saturate ⊂:-overloads-right\n∪-saturate-overloads (F ∩ G) (right o) =\n ⊂:-⋓-overloads (normal-∪-saturate F) (normal-∪-saturate G) o >>=\n ⊂:-∪-lift (∪-saturate-overloads F) (∪-saturate-overloads G) >>=\n ⊂:-∪-lift (⊂:-∪-saturate ⊂:-overloads-left) (⊂:-∪-saturate ⊂:-overloads-right) >>=\n ⊂:-∪-lift-saturate\n\noverloads-∪-saturate : ∀ {F} → FunType F → ∪-Saturate (Overloads F) ⊂: Overloads (∪-saturate F)\noverloads-∪-saturate F = ⊂:-∪-saturate-indn (inj F) (step F) where\n\n inj : ∀ {F} → FunType F → Overloads F ⊂: Overloads (∪-saturate F)\n inj (S ⇒ T) here = just here\n inj (F ∩ G) (left p) = inj F p >>= ⊂:-overloads-left >>= ⊂:-overloads-left\n inj (F ∩ G) (right p) = inj G p >>= ⊂:-overloads-right >>= ⊂:-overloads-left\n\n step : ∀ {F} → FunType F → ∪-Lift (Overloads (∪-saturate F)) (Overloads (∪-saturate F)) ⊂: Overloads (∪-saturate F)\n step (S ⇒ T) (union here here) = defn here (<:-∪-lub <:-refl <:-refl) <:-∪-left\n step (F ∩ G) (union (left (left p)) (left (left q))) = step F (union p q) >>= ⊂:-overloads-left >>= ⊂:-overloads-left\n step (F ∩ G) (union (left (left p)) (left (right q))) = ⊂:-overloads-⋓ (normal-∪-saturate F) (normal-∪-saturate G) (union p q) >>= ⊂:-overloads-right\n step (F ∩ G) (union (left (right p)) (left (left q))) = ⊂:-overloads-⋓ (normal-∪-saturate F) (normal-∪-saturate G) (union q p) >>= ⊂:-overloads-right >>=ˡ <:-∪-symm >>=ʳ <:-∪-symm\n step (F ∩ G) (union (left (right p)) (left (right q))) = step G (union p q) >>= ⊂:-overloads-right >>= ⊂:-overloads-left\n step (F ∩ G) (union p (right q)) with ⊂:-⋓-overloads (normal-∪-saturate F) (normal-∪-saturate G) q\n step (F ∩ G) (union (left (left p)) (right q)) | defn (union q₁ q₂) q₃ q₄ =\n (step F (union p q₁) [∪] just q₂) >>=\n ⊂:-overloads-⋓ (normal-∪-saturate F) (normal-∪-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-union <:-refl q₃) <:-∪-assocl >>=ʳ\n <:-trans <:-∪-assocr (<:-union <:-refl q₄)\n step (F ∩ G) (union (left (right p)) (right q)) | defn (union q₁ q₂) q₃ q₄ =\n (just q₁ [∪] step G (union p q₂)) >>=\n ⊂:-overloads-⋓ (normal-∪-saturate F) (normal-∪-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-union <:-refl q₃) (<:-∪-lub (<:-trans <:-∪-left <:-∪-right) (<:-∪-lub <:-∪-left (<:-trans <:-∪-right <:-∪-right))) >>=ʳ\n <:-trans (<:-∪-lub (<:-trans <:-∪-left <:-∪-right) (<:-∪-lub <:-∪-left (<:-trans <:-∪-right <:-∪-right))) (<:-union <:-refl q₄)\n step (F ∩ G) (union (right p) (right q)) | defn (union q₁ q₂) q₃ q₄ with ⊂:-⋓-overloads (normal-∪-saturate F) (normal-∪-saturate G) p\n step (F ∩ G) (union (right p) (right q)) | defn (union q₁ q₂) q₃ q₄ | defn (union p₁ p₂) p₃ p₄ =\n (step F (union p₁ q₁) [∪] step G (union p₂ q₂)) >>=\n ⊂:-overloads-⋓ (normal-∪-saturate F) (normal-∪-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-union p₃ q₃) (<:-∪-lub (<:-union <:-∪-left <:-∪-left) (<:-union <:-∪-right <:-∪-right)) >>=ʳ\n <:-trans (<:-∪-lub (<:-union <:-∪-left <:-∪-left) (<:-union <:-∪-right <:-∪-right)) (<:-union p₄ q₄)\n step (F ∩ G) (union (right p) q) with ⊂:-⋓-overloads (normal-∪-saturate F) (normal-∪-saturate G) p\n step (F ∩ G) (union (right p) (left (left q))) | defn (union p₁ p₂) p₃ p₄ =\n (step F (union p₁ q) [∪] just p₂) >>=\n ⊂:-overloads-⋓ (normal-∪-saturate F) (normal-∪-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-union p₃ <:-refl) (<:-∪-lub (<:-union <:-∪-left <:-refl) (<:-trans <:-∪-right <:-∪-left)) >>=ʳ\n <:-trans (<:-∪-lub (<:-union <:-∪-left <:-refl) (<:-trans <:-∪-right <:-∪-left)) (<:-union p₄ <:-refl)\n step (F ∩ G) (union (right p) (left (right q))) | defn (union p₁ p₂) p₃ p₄ =\n (just p₁ [∪] step G (union p₂ q)) >>=\n ⊂:-overloads-⋓ (normal-∪-saturate F) (normal-∪-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-union p₃ <:-refl) <:-∪-assocr >>=ʳ\n <:-trans <:-∪-assocl (<:-union p₄ <:-refl)\n step (F ∩ G) (union (right p) (right q)) | defn (union p₁ p₂) p₃ p₄ with ⊂:-⋓-overloads (normal-∪-saturate F) (normal-∪-saturate G) q\n step (F ∩ G) (union (right p) (right q)) | defn (union p₁ p₂) p₃ p₄ | defn (union q₁ q₂) q₃ q₄ =\n (step F (union p₁ q₁) [∪] step G (union p₂ q₂)) >>=\n ⊂:-overloads-⋓ (normal-∪-saturate F) (normal-∪-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-union p₃ q₃) (<:-∪-lub (<:-union <:-∪-left <:-∪-left) (<:-union <:-∪-right <:-∪-right)) >>=ʳ\n <:-trans (<:-∪-lub (<:-union <:-∪-left <:-∪-left) (<:-union <:-∪-right <:-∪-right)) (<:-union p₄ q₄)\n\n∪-saturated : ∀ {F} → FunType F → ∪-Lift (Overloads (∪-saturate F)) (Overloads (∪-saturate F)) ⊂: Overloads (∪-saturate F)\n∪-saturated F o =\n ⊂:-∪-lift (∪-saturate-overloads F) (∪-saturate-overloads F) o >>=\n ⊂:-∪-lift-saturate >>=\n overloads-∪-saturate F\n\n∩-saturate-overloads : ∀ {F} → FunType F → Overloads (∩-saturate F) ⊂: ∩-Saturate (Overloads F)\n∩-saturate-overloads (S ⇒ T) here = just (base here)\n∩-saturate-overloads (F ∩ G) (left (left o)) = ∩-saturate-overloads F o >>= ⊂:-∩-saturate ⊂:-overloads-left\n∩-saturate-overloads (F ∩ G) (left (right o)) = ∩-saturate-overloads G o >>= ⊂:-∩-saturate ⊂:-overloads-right\n∩-saturate-overloads (F ∩ G) (right o) =\n ⊂:-⋒-overloads (normal-∩-saturate F) (normal-∩-saturate G) o >>=\n ⊂:-∩-lift (∩-saturate-overloads F) (∩-saturate-overloads G) >>=\n ⊂:-∩-lift (⊂:-∩-saturate ⊂:-overloads-left) (⊂:-∩-saturate ⊂:-overloads-right) >>=\n ⊂:-∩-lift-saturate\n\noverloads-∩-saturate : ∀ {F} → FunType F → ∩-Saturate (Overloads F) ⊂: Overloads (∩-saturate F)\noverloads-∩-saturate F = ⊂:-∩-saturate-indn (inj F) (step F) where\n \n inj : ∀ {F} → FunType F → Overloads F ⊂: Overloads (∩-saturate F)\n inj (S ⇒ T) here = just here\n inj (F ∩ G) (left p) = inj F p >>= ⊂:-overloads-left >>= ⊂:-overloads-left\n inj (F ∩ G) (right p) = inj G p >>= ⊂:-overloads-right >>= ⊂:-overloads-left\n\n step : ∀ {F} → FunType F → ∩-Lift (Overloads (∩-saturate F)) (Overloads (∩-saturate F)) ⊂: Overloads (∩-saturate F)\n step (S ⇒ T) (intersect here here) = defn here <:-∩-left (<:-∩-glb <:-refl <:-refl) \n step (F ∩ G) (intersect (left (left p)) (left (left q))) = step F (intersect p q) >>= ⊂:-overloads-left >>= ⊂:-overloads-left\n step (F ∩ G) (intersect (left (left p)) (left (right q))) = ⊂:-overloads-⋒ (normal-∩-saturate F) (normal-∩-saturate G) (intersect p q) >>= ⊂:-overloads-right\n step (F ∩ G) (intersect (left (right p)) (left (left q))) = ⊂:-overloads-⋒ (normal-∩-saturate F) (normal-∩-saturate G) (intersect q p) >>= ⊂:-overloads-right >>=ˡ <:-∩-symm >>=ʳ <:-∩-symm\n step (F ∩ G) (intersect (left (right p)) (left (right q))) = step G (intersect p q) >>= ⊂:-overloads-right >>= ⊂:-overloads-left\n step (F ∩ G) (intersect (right p) q) with ⊂:-⋒-overloads (normal-∩-saturate F) (normal-∩-saturate G) p\n step (F ∩ G) (intersect (right p) (left (left q))) | defn (intersect p₁ p₂) p₃ p₄ =\n (step F (intersect p₁ q) [∩] just p₂) >>=\n ⊂:-overloads-⋒ (normal-∩-saturate F) (normal-∩-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-intersect p₃ <:-refl) (<:-∩-glb (<:-intersect <:-∩-left <:-refl) (<:-trans <:-∩-left <:-∩-right)) >>=ʳ\n <:-trans (<:-∩-glb (<:-intersect <:-∩-left <:-refl) (<:-trans <:-∩-left <:-∩-right)) (<:-intersect p₄ <:-refl)\n step (F ∩ G) (intersect (right p) (left (right q))) | defn (intersect p₁ p₂) p₃ p₄ =\n (just p₁ [∩] step G (intersect p₂ q)) >>=\n ⊂:-overloads-⋒ (normal-∩-saturate F) (normal-∩-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-intersect p₃ <:-refl) <:-∩-assocr >>=ʳ\n <:-trans <:-∩-assocl (<:-intersect p₄ <:-refl)\n step (F ∩ G) (intersect (right p) (right q)) | defn (intersect p₁ p₂) p₃ p₄ with ⊂:-⋒-overloads (normal-∩-saturate F) (normal-∩-saturate G) q\n step (F ∩ G) (intersect (right p) (right q)) | defn (intersect p₁ p₂) p₃ p₄ | defn (intersect q₁ q₂) q₃ q₄ =\n (step F (intersect p₁ q₁) [∩] step G (intersect p₂ q₂)) >>=\n ⊂:-overloads-⋒ (normal-∩-saturate F) (normal-∩-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-intersect p₃ q₃) (<:-∩-glb (<:-intersect <:-∩-left <:-∩-left) (<:-intersect <:-∩-right <:-∩-right)) >>=ʳ\n <:-trans (<:-∩-glb (<:-intersect <:-∩-left <:-∩-left) (<:-intersect <:-∩-right <:-∩-right)) (<:-intersect p₄ q₄)\n step (F ∩ G) (intersect p (right q)) with ⊂:-⋒-overloads (normal-∩-saturate F) (normal-∩-saturate G) q\n step (F ∩ G) (intersect (left (left p)) (right q)) | defn (intersect q₁ q₂) q₃ q₄ =\n (step F (intersect p q₁) [∩] just q₂) >>=\n ⊂:-overloads-⋒ (normal-∩-saturate F) (normal-∩-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-intersect <:-refl q₃) <:-∩-assocl >>=ʳ\n <:-trans <:-∩-assocr (<:-intersect <:-refl q₄)\n step (F ∩ G) (intersect (left (right p)) (right q)) | defn (intersect q₁ q₂) q₃ q₄ =\n (just q₁ [∩] step G (intersect p q₂) ) >>=\n ⊂:-overloads-⋒ (normal-∩-saturate F) (normal-∩-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-intersect <:-refl q₃) (<:-∩-glb (<:-trans <:-∩-right <:-∩-left) (<:-∩-glb <:-∩-left (<:-trans <:-∩-right <:-∩-right))) >>=ʳ\n <:-∩-glb (<:-trans <:-∩-right <:-∩-left) (<:-trans (<:-∩-glb <:-∩-left (<:-trans <:-∩-right <:-∩-right)) q₄)\n step (F ∩ G) (intersect (right p) (right q)) | defn (intersect q₁ q₂) q₃ q₄ with ⊂:-⋒-overloads (normal-∩-saturate F) (normal-∩-saturate G) p\n step (F ∩ G) (intersect (right p) (right q)) | defn (intersect q₁ q₂) q₃ q₄ | defn (intersect p₁ p₂) p₃ p₄ =\n (step F (intersect p₁ q₁) [∩] step G (intersect p₂ q₂)) >>=\n ⊂:-overloads-⋒ (normal-∩-saturate F) (normal-∩-saturate G) >>=\n ⊂:-overloads-right >>=ˡ\n <:-trans (<:-intersect p₃ q₃) (<:-∩-glb (<:-intersect <:-∩-left <:-∩-left) (<:-intersect <:-∩-right <:-∩-right)) >>=ʳ\n <:-trans (<:-∩-glb (<:-intersect <:-∩-left <:-∩-left) (<:-intersect <:-∩-right <:-∩-right)) (<:-intersect p₄ q₄)\n\nsaturate-overloads : ∀ {F} → FunType F → Overloads (saturate F) ⊂: ∪-Saturate (∩-Saturate (Overloads F))\nsaturate-overloads F o = ∪-saturate-overloads (normal-∩-saturate F) o >>= (⊂:-∪-saturate (∩-saturate-overloads F))\n\noverloads-saturate : ∀ {F} → FunType F → ∪-Saturate (∩-Saturate (Overloads F)) ⊂: Overloads (saturate F)\noverloads-saturate F o = ⊂:-∪-saturate (overloads-∩-saturate F) o >>= overloads-∪-saturate (normal-∩-saturate F)\n\n-- Saturated F whenever\n-- * if F has overloads (R ⇒ S) and (T ⇒ U) then F has an overload which is a subtype of ((R ∩ T) ⇒ (S ∩ U))\n-- * ditto union\ndata Saturated (F : Type) : Set where\n\n defn : \n\n (∀ {R S T U} → Overloads F (R ⇒ S) → Overloads F (T ⇒ U) → <:-Close (Overloads F) ((R ∩ T) ⇒ (S ∩ U))) →\n (∀ {R S T U} → Overloads F (R ⇒ S) → Overloads F (T ⇒ U) → <:-Close (Overloads F) ((R ∪ T) ⇒ (S ∪ U))) →\n -----------\n Saturated F\n\n-- saturated F is saturated!\nsaturated : ∀ {F} → FunType F → Saturated (saturate F)\nsaturated F = defn\n (λ n o → (saturate-overloads F n [∩] saturate-overloads F o) >>= ∪-saturate-resp-∩-saturation ⊂:-∩-lift-saturate >>= overloads-saturate F)\n (λ n o → ∪-saturated (normal-∩-saturate F) (union n o))\n", "meta": {"hexsha": "13f7d171ffe571ab31261d45297a5b85c08369d1", "size": 25260, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "prototyping/Properties/TypeSaturation.agda", "max_stars_repo_name": "Libertus-Lab/luau", "max_stars_repo_head_hexsha": "f1b46f4b967f11fabe666da1de0e71b225368260", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-11-06T08:03:00.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-06T08:03:00.000Z", "max_issues_repo_path": "prototyping/Properties/TypeSaturation.agda", "max_issues_repo_name": "sthagen/Roblox-luau", "max_issues_repo_head_hexsha": "39fbd2146a379fb0878369b48764cd7e8772c0fb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, 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YES\n2. NO\n\n", "lm_q1_score": 0.7745833737577158, "lm_q2_score": 0.4455295350395727, "lm_q1q2_score": 0.34509977035965866}} {"text": "{-# OPTIONS --without-K #-}\nmodule well-typed-syntax where\n\ninfixl 2 _▻_\ninfixl 3 _‘’_\ninfixl 3 _‘’₁_\ninfixl 3 _‘’₂_\ninfixl 3 _‘’₃_\ninfixl 3 _‘’ₐ_\ninfixr 1 _‘→’_\ninfixl 3 _‘‘’’_\ninfixl 3 _w‘‘’’_\ninfixr 1 _‘‘→'’’_\ninfixr 1 _w‘‘→'’’_\n\nmutual\n data Context : Set where\n ε : Context\n _▻_ : (Γ : Context) → Typ Γ → Context\n\n data Typ : Context → Set where\n _‘’_ : ∀ {Γ A} → Typ (Γ ▻ A) → Term {Γ} A → Typ Γ\n _‘’₁_ : ∀ {Γ A B} → (C : Typ (Γ ▻ A ▻ B)) → (a : Term {Γ} A) → Typ (Γ ▻ B ‘’ a)\n _‘’₂_ : ∀ {Γ A B C} → (D : Typ (Γ ▻ A ▻ B ▻ C)) → (a : Term {Γ} A) → Typ (Γ ▻ B ‘’ a ▻ C ‘’₁ a)\n _‘’₃_ : ∀ {Γ A B C D} → (E : Typ (Γ ▻ A ▻ B ▻ C ▻ D)) → (a : Term {Γ} A) → Typ (Γ ▻ B ‘’ a ▻ C ‘’₁ a ▻ D ‘’₂ a)\n W : ∀ {Γ A} → Typ Γ → Typ (Γ ▻ A)\n W1 : ∀ {Γ A B} → Typ (Γ ▻ B) → Typ (Γ ▻ A ▻ (W {Γ = Γ} {A = A} B))\n W2 : ∀ {Γ A B C} → Typ (Γ ▻ B ▻ C) → Typ (Γ ▻ A ▻ W B ▻ W1 C)\n _‘→’_ : ∀ {Γ} (A : Typ Γ) → Typ (Γ ▻ A) → Typ Γ\n ‘Σ’ : ∀ {Γ} (T : Typ Γ) → Typ (Γ ▻ T) → Typ Γ\n ‘Context’ : ∀ {Γ} → Typ Γ\n ‘Typ’ : ∀ {Γ} → Typ (Γ ▻ ‘Context’)\n ‘Term’ : ∀ {Γ} → Typ (Γ ▻ ‘Context’ ▻ ‘Typ’)\n\n\n data Term : ∀ {Γ} → Typ Γ → Set where\n w : ∀ {Γ A B} → Term {Γ} B → Term {Γ = Γ ▻ A} (W {Γ = Γ} {A = A} B)\n ‘λ∙’ : ∀ {Γ A B} → Term {Γ = (Γ ▻ A)} B → Term {Γ} (A ‘→’ B)\n _‘’ₐ_ : ∀ {Γ A B} → (f : Term {Γ} (A ‘→’ B)) → (x : Term {Γ} A) → Term {Γ} (B ‘’ x)\n ‘VAR₀’ : ∀ {Γ T} → Term {Γ = Γ ▻ T} (W T)\n ⌜_⌝c : ∀ {Γ} → Context → Term {Γ} ‘Context’\n ⌜_⌝T : ∀ {Γ Γ'} → Typ Γ' → Term {Γ} (‘Typ’ ‘’ ⌜ Γ' ⌝c)\n ⌜_⌝t : ∀ {Γ Γ'} {T : Typ Γ'} → Term T → Term {Γ} (‘Term’ ‘’₁ ⌜ Γ' ⌝c ‘’ ⌜ T ⌝T)\n ‘quote-term’ : ∀ {Γ Γ'} {A : Typ Γ'} → Term {Γ} (‘Term’ ‘’₁ ⌜ Γ' ⌝c ‘’ ⌜ A ⌝T ‘→’ W (‘Term’ ‘’₁ ⌜ Γ ⌝c ‘’ ⌜ ‘Term’ ‘’₁ ⌜ Γ' ⌝c ‘’ ⌜ A ⌝T ⌝T))\n ‘quote-sigma’ : ∀ {Γ Γ'} → Term {Γ} (‘Σ’ ‘Context’ ‘Typ’ ‘→’ W (‘Term’ ‘’₁ ⌜ Γ' ⌝c ‘’ ⌜ ‘Σ’ ‘Context’ ‘Typ’ ⌝T))\n {-‘substTyp’ : ∀ {Γ' Γ} {A : Typ Γ}\n → Term {Γ'} (‘Typ’ ‘’ ⌜ Γ ▻ A ⌝c\n ‘→’ W (‘Term’ ‘’₁ ⌜ Γ ⌝c ‘’ ⌜ A ⌝T\n ‘→’ W (‘Typ’ ‘’ ⌜ Γ ⌝c)))-}\n ‘cast’ : Term {ε} (‘Σ’ ‘Context’ ‘Typ’ ‘→’ W (‘Typ’ ‘’ ⌜ ε ▻ ‘Σ’ ‘Context’ ‘Typ’ ⌝c))\n SW : ∀ {Γ A B} {a : Term {Γ} A} → Term {Γ} (W B ‘’ a) → Term {Γ} B\n weakenTyp-substTyp-tProd : ∀ {Γ T T' A B} {a : Term {Γ} T} → Term {Γ = Γ ▻ T'} (W ((A ‘→’ B) ‘’ a)) → Term {Γ ▻ T'} (W ((A ‘’ a) ‘→’ (B ‘’₁ a)))\n substTyp-weakenTyp1-VAR₀ : ∀ {Γ A T} → Term {Γ ▻ A} (W1 T ‘’ ‘VAR₀’) → Term {Γ ▻ A} T\n weakenTyp-tProd : ∀ {Γ A B C} → Term {Γ = Γ ▻ C} (W (A ‘→’ B)) → Term {Γ = Γ ▻ C} (W A ‘→’ W1 B)\n weakenTyp-tProd-inv : ∀ {Γ A B C} → Term {Γ = Γ ▻ C} (W A ‘→’ W1 B) → Term {Γ = Γ ▻ C} (W (A ‘→’ B))\n weakenTyp-weakenTyp-tProd : ∀ {Γ A B C D} → Term {Γ ▻ C ▻ D} (W (W (A ‘→’ B))) → Term {Γ ▻ C ▻ D} (W (W A ‘→’ W1 B))\n substTyp1-tProd : ∀ {Γ T T' A B} {a : Term {Γ} T} → Term {Γ ▻ T' ‘’ a} ((A ‘→’ B) ‘’₁ a) → Term {Γ ▻ T' ‘’ a} ((A ‘’₁ a) ‘→’ (B ‘’₂ a))\n weakenTyp1-tProd : ∀ {Γ C D A B} → Term {Γ ▻ C ▻ W D} (W1 (A ‘→’ B)) → Term {Γ ▻ C ▻ W D} (W1 A ‘→’ W2 B)\n substTyp2-tProd : ∀ {Γ T T' T'' A B} {a : Term {Γ} T} → Term {Γ ▻ T' ‘’ a ▻ T'' ‘’₁ a} ((A ‘→’ B) ‘’₂ a) → Term {Γ ▻ T' ‘’ a ▻ T'' ‘’₁ a} ((A ‘’₂ a) ‘→’ (B ‘’₃ a))\n substTyp1-substTyp-weakenTyp-inv : ∀ {Γ C T A} {a : Term {Γ} C} {b : Term {Γ} (T ‘’ a)} → Term {Γ} (A ‘’ a) → Term {Γ} (W A ‘’₁ a ‘’ b)\n substTyp1-substTyp-weakenTyp : ∀ {Γ C T A} {a : Term {Γ} C} {b : Term {Γ} (T ‘’ a)} → Term {Γ} (W A ‘’₁ a ‘’ b) → Term {Γ} (A ‘’ a)\n weakenTyp-weakenTyp-substTyp1-substTyp-weakenTyp : ∀ {Γ C T A D E} {a : Term {Γ} C} {b : Term {Γ} (T ‘’ a)} → Term {Γ ▻ D ▻ E} (W (W (W A ‘’₁ a ‘’ b))) → Term {Γ ▻ D ▻ E} (W (W (A ‘’ a)))\n weakenTyp-substTyp2-substTyp1-substTyp-weakenTyp-inv : ∀ {Γ A B C T T'} {a : Term {Γ} A} {b : Term {Γ} (B ‘’ a)} {c : Term {Γ} (C ‘’₁ a ‘’ b)}\n → Term {Γ ▻ T'} (W (T ‘’₁ a ‘’ b))\n → Term {Γ ▻ T'} (W (W T ‘’₂ a ‘’₁ b ‘’ c))\n substTyp2-substTyp1-substTyp-weakenTyp : ∀ {Γ A B C T} {a : Term {Γ} A} {b : Term {Γ} (B ‘’ a)} {c : Term {Γ} (C ‘’₁ a ‘’ b)}\n → Term {Γ} (W T ‘’₂ a ‘’₁ b ‘’ c)\n → Term {Γ} (T ‘’₁ a ‘’ b)\n weakenTyp-substTyp2-substTyp1-substTyp-tProd : ∀ {Γ T T' T'' T''' A B} {a : Term {Γ} T} {b : Term {Γ} (T' ‘’ a)} {c : Term {Γ} (T'' ‘’₁ a ‘’ b)}\n → Term {Γ ▻ T'''} (W ((A ‘→’ B) ‘’₂ a ‘’₁ b ‘’ c))\n → Term {Γ ▻ T'''} ((W (A ‘’₂ a ‘’₁ b ‘’ c)) ‘→’ (W1 (B ‘’₃ a ‘’₂ b ‘’₁ c)))\n weakenTyp2-weakenTyp1 : ∀ {Γ A B C D} → Term {Γ ▻ A ▻ W B ▻ W1 C} (W2 (W D)) → Term {Γ ▻ A ▻ W B ▻ W1 C} (W (W1 D))\n weakenTyp1-weakenTyp : ∀ {Γ A B C} → Term {Γ ▻ A ▻ W B} (W1 (W C)) → Term {Γ ▻ A ▻ W B} (W (W C))\n weakenTyp1-weakenTyp-inv : ∀ {Γ A B C} → Term {Γ ▻ A ▻ W B} (W (W C)) → Term {Γ ▻ A ▻ W B} (W1 (W C))\n weakenTyp1-weakenTyp1-weakenTyp : ∀ {Γ A B C T} → Term {Γ ▻ A ▻ B ▻ W (W C)} (W1 (W1 (W T))) → Term {Γ ▻ A ▻ B ▻ W (W C)} (W1 (W (W T)))\n substTyp1-weakenTyp1 : ∀ {Γ A B C} {a : Term {Γ} A} → Term {Γ ▻ W B ‘’ a} (W1 C ‘’₁ a) → Term {Γ ▻ B} C\n weakenTyp1-substTyp-weakenTyp1-inv : ∀ {Γ A T'' T' T} {a : Term {Γ} A}\n → Term {Γ ▻ T'' ▻ W (T' ‘’ a)} (W1 (W (T ‘’ a)))\n → Term {Γ ▻ T'' ▻ W (T' ‘’ a)} (W1 (W T ‘’₁ a))\n weakenTyp1-substTyp-weakenTyp1 : ∀ {Γ A T'' T' T} {a : Term {Γ} A}\n → Term {Γ ▻ T'' ▻ W (T' ‘’ a)} (W1 (W T ‘’₁ a))\n → Term {Γ ▻ T'' ▻ W (T' ‘’ a)} (W1 (W (T ‘’ a)))\n weakenTyp-substTyp-substTyp-weakenTyp1 : ∀ {Γ T' B A} {b : Term {Γ} B} {a : Term {Γ ▻ B} (W A)} {T : Typ (Γ ▻ A)}\n → Term {Γ ▻ T'} (W (W1 T ‘’ a ‘’ b))\n → Term {Γ ▻ T'} (W (T ‘’ (SW ((‘λ∙’ a) ‘’ₐ b))))\n weakenTyp-substTyp-substTyp-weakenTyp1-inv : ∀ {Γ T' B A} {b : Term {Γ} B} {a : Term {Γ ▻ B} (W A)} {T : Typ (Γ ▻ A)}\n → Term {Γ ▻ T'} (W (T ‘’ (SW ((‘λ∙’ a) ‘’ₐ b))))\n → Term {Γ ▻ T'} (W (W1 T ‘’ a ‘’ b))\n substTyp-weakenTyp1-weakenTyp : ∀ {Γ T} {A : Typ Γ} {B : Typ Γ}\n → {a : Term {Γ = Γ ▻ T} (W {Γ = Γ} {A = T} B)}\n → Term {Γ = Γ ▻ T} (W1 (W A) ‘’ a)\n → Term {Γ = Γ ▻ T} (W A)\n substTyp3-substTyp2-substTyp1-substTyp-weakenTyp : ∀ {Γ A B C D T T'} {a : Term {Γ} A} {b : Term {Γ} (B ‘’ a)} {c : Term {Γ} (C ‘’₁ a ‘’ b)}\n {d : Term {Γ = (Γ ▻ T')} (W (D ‘’₂ a ‘’₁ b ‘’ c))}\n → Term {Γ = (Γ ▻ T')} (W1 (W T ‘’₃ a ‘’₂ b ‘’₁ c) ‘’ d)\n → Term {Γ = (Γ ▻ T')} (W (T ‘’₂ a ‘’₁ b ‘’ c))\n weakenTyp-substTyp2-substTyp1-substTyp-weakenTyp1 : ∀ {Γ A B C T T'} {a : Term {Γ} A} {b : Term (B ‘’ a)} {c : Term (C ‘’ a)}\n → Term {Γ = (Γ ▻ T')} (W (W1 T ‘’₂ a ‘’₁ b ‘’ substTyp1-substTyp-weakenTyp-inv c))\n → Term {Γ = (Γ ▻ T')} (W (T ‘’₁ a ‘’ c))\n substTyp1-substTyp-tProd : ∀ {Γ T T' A B a b} → Term ((_‘→’_ {Γ = Γ ▻ T ▻ T'} A B) ‘’₁ a ‘’ b) → Term (_‘→’_ {Γ = Γ} (A ‘’₁ a ‘’ b) (B ‘’₂ a ‘’₁ b))\n substTyp2-substTyp-substTyp-weakenTyp1-weakenTyp-weakenTyp : ∀ {Γ A} {T : Typ (Γ ▻ A)} {T' C B} {a : Term {Γ} A} {b : Term {Γ = (Γ ▻ C ‘’ a)} (B ‘’₁ a)}\n {c : Term {Γ = (Γ ▻ T')} (W (C ‘’ a))}\n → Term {Γ = (Γ ▻ T')} (W1 (W (W T) ‘’₂ a ‘’ b) ‘’ c)\n → Term {Γ = (Γ ▻ T')} (W (T ‘’ a))\n substTyp1-substTyp-weakenTyp2-weakenTyp : ∀ {Γ T' A B T} {a : Term {Γ ▻ T'} (W A)} {b : Term {Γ ▻ T'} (W1 B ‘’ a)}\n → Term {Γ ▻ T'} (W2 (W T) ‘’₁ a ‘’ b)\n → Term {Γ ▻ T'} (W1 T ‘’ a)\n weakenTyp-weakenTyp1-weakenTyp : ∀ {Γ A B C D} → Term {Γ ▻ A ▻ W B ▻ W1 C} (W (W1 (W D))) → Term {Γ ▻ A ▻ W B ▻ W1 C} (W (W (W D)))\n beta-under-subst : ∀ {Γ A B B'} {g : Term {Γ} (A ‘→’ W B)} {x : Term {Γ} A}\n → Term (B' ‘’ SW (‘λ∙’ (SW (‘λ∙’ (weakenTyp1-weakenTyp (substTyp-weakenTyp1-VAR₀ (weakenTyp-tProd (w (weakenTyp-tProd (w g))) ‘’ₐ ‘VAR₀’))) ‘’ₐ ‘VAR₀’)) ‘’ₐ x))\n → Term (B' ‘’ SW (g ‘’ₐ x))\n ‘proj₁'’ : ∀ {Γ} {T : Typ Γ} {P : Typ (Γ ▻ T)} → Term (‘Σ’ T P ‘→’ W T)\n ‘proj₂'’ : ∀ {Γ} {T : Typ Γ} {P : Typ (Γ ▻ T)} → Term {Γ ▻ ‘Σ’ T P} (W1 P ‘’ SW (‘λ∙’ (weakenTyp1-weakenTyp (substTyp-weakenTyp1-VAR₀ (weakenTyp-tProd (w (weakenTyp-tProd (w ‘proj₁'’))) ‘’ₐ ‘VAR₀’))) ‘’ₐ ‘VAR₀’))\n ‘existT’ : ∀ {Γ T P} (x : Term {Γ} T) (p : Term (P ‘’ x)) → Term (‘Σ’ T P)\n {- these are redundant, but useful for not having to normalize the subsequent ones -}\n _‘‘’’_ : ∀ {Γ} {A : Typ Γ}\n → Term {ε} (‘Typ’ ‘’ ⌜ Γ ▻ A ⌝c)\n → Term {ε} (‘Term’ ‘’₁ ⌜ Γ ⌝c ‘’ ⌜ A ⌝T)\n → Term {ε} (‘Typ’ ‘’ ⌜ Γ ⌝c)\n _w‘‘’’_ : ∀ {X Γ} {A : Typ Γ}\n → Term {ε ▻ X} (W (‘Typ’ ‘’ ⌜ Γ ▻ A ⌝c))\n → Term {ε ▻ X} (W (‘Term’ ‘’₁ ⌜ Γ ⌝c ‘’ ⌜ A ⌝T))\n → Term {ε ▻ X} (W (‘Typ’ ‘’ ⌜ Γ ⌝c))\n _‘‘→'’’_ : ∀ {Γ}\n → Term {ε} (‘Typ’ ‘’ Γ)\n → Term {ε} (‘Typ’ ‘’ Γ)\n → Term {ε} (‘Typ’ ‘’ Γ)\n _w‘‘→'’’_ : ∀ {X Γ}\n → Term {ε ▻ X} (W (‘Typ’ ‘’ Γ))\n → Term {ε ▻ X} (W (‘Typ’ ‘’ Γ))\n → Term {ε ▻ X} (W (‘Typ’ ‘’ Γ))\n w→ : ∀ {Γ A B C} → Term (A ‘→’ W B) → Term {Γ = Γ ▻ C} (W A ‘→’ W (W B))\n {- things that were postulates, but are no longer -}\n ‘‘→'’’→w‘‘→'’’ : ∀ {T'}\n {b : Term {ε} (‘Typ’ ‘’ ⌜ ε ⌝c)}\n {c : Term {ε ▻ T'} (W (‘Typ’ ‘’ ⌜ ε ⌝c))}\n {e : Term {ε} T'}\n → Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ (SW (‘λ∙’ (c w‘‘→'’’ w b) ‘’ₐ e))\n ‘→’ W (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ (SW (‘λ∙’ c ‘’ₐ e) ‘‘→'’’ b)))\n w‘‘→'’’→‘‘→'’’ : ∀ {T'}\n {b : Term {ε} (‘Typ’ ‘’ ⌜ ε ⌝c)}\n {c : Term {ε ▻ T'} (W (‘Typ’ ‘’ ⌜ ε ⌝c))}\n {e : Term {ε} T'}\n → Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ (SW (‘λ∙’ c ‘’ₐ e) ‘‘→'’’ b)\n ‘→’ W (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ (SW (‘λ∙’ (c w‘‘→'’’ w b) ‘’ₐ e))))\n ‘tApp-nd’ : ∀ {Γ} {A : Term {ε} (‘Typ’ ‘’ Γ)} {B : Term {ε} (‘Typ’ ‘’ Γ)} →\n Term {ε} (‘Term’ ‘’₁ Γ ‘’ (A ‘‘→'’’ B)\n ‘→’ W (‘Term’ ‘’₁ Γ ‘’ A\n ‘→’ W (‘Term’ ‘’₁ Γ ‘’ B)))\n ⌜←'⌝ : ∀ {H X} →\n Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ (⌜ H ⌝T ‘‘→'’’ ⌜ X ⌝T)\n ‘→’ W (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ ⌜ H ‘→’ W X ⌝T))\n ⌜→'⌝ : ∀ {H X} →\n Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ ⌜ H ‘→’ W X ⌝T\n ‘→’ W (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ (⌜ H ⌝T ‘‘→'’’ ⌜ X ⌝T)))\n ‘‘fcomp-nd’’ : ∀ {A B C} →\n Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ (A ‘‘→'’’ C)\n ‘→’ W (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ (C ‘‘→'’’ B)\n ‘→’ W (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ (A ‘‘→'’’ B))))\n ⌜‘’⌝ : ∀ {B A} {b : Term {ε} B} →\n Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’\n (⌜ A ‘’ b ⌝T ‘‘→'’’ ⌜ A ⌝T ‘‘’’ ⌜ b ⌝t))\n ⌜‘’⌝' : ∀ {B A} {b : Term {ε} B} →\n Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’\n (⌜ A ⌝T ‘‘’’ ⌜ b ⌝t ‘‘→'’’ ⌜ A ‘’ b ⌝T))\n ‘cast-refl’ : ∀ {T : Typ (ε ▻ ‘Σ’ ‘Context’ ‘Typ’)} →\n Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’\n ((⌜ T ‘’ ‘existT’ ⌜ ε ▻ ‘Σ’ ‘Context’ ‘Typ’ ⌝c ⌜ T ⌝T ⌝T)\n ‘‘→'’’\n (SW (‘cast’ ‘’ₐ ‘existT’ ⌜ ε ▻ ‘Σ’ ‘Context’ ‘Typ’ ⌝c ⌜ T ⌝T)\n ‘‘’’ SW (‘quote-sigma’ ‘’ₐ ‘existT’ ⌜ ε ▻ ‘Σ’ ‘Context’ ‘Typ’ ⌝c ⌜ T ⌝T))))\n ‘cast-refl'’ : ∀ {T : Typ (ε ▻ ‘Σ’ ‘Context’ ‘Typ’)} →\n Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’\n ((SW (‘cast’ ‘’ₐ ‘existT’ ⌜ ε ▻ ‘Σ’ ‘Context’ ‘Typ’ ⌝c ⌜ T ⌝T)\n ‘‘’’ SW (‘quote-sigma’ ‘’ₐ ‘existT’ ⌜ ε ▻ ‘Σ’ ‘Context’ ‘Typ’ ⌝c ⌜ T ⌝T))\n ‘‘→'’’\n (⌜ T ‘’ ‘existT’ ⌜ ε ▻ ‘Σ’ ‘Context’ ‘Typ’ ⌝c ⌜ T ⌝T ⌝T)))\n ‘s→→’ : ∀ {T B}\n {b : Term {ε} (T ‘→’ W (‘Typ’ ‘’ ⌜ ε ▻ B ⌝c))}\n {c : Term {ε} (T ‘→’ W (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ ⌜ B ⌝T))}\n {v : Term {ε} T} →\n (Term {ε} (‘Term’ ‘’₁ ⌜ ε ⌝c\n ‘’ ((SW (((‘λ∙’ (SW (w→ b ‘’ₐ ‘VAR₀’) w‘‘’’ SW (w→ c ‘’ₐ ‘VAR₀’)) ‘’ₐ v))))\n ‘‘→'’’ (SW (b ‘’ₐ v) ‘‘’’ SW (c ‘’ₐ v)))))\n ‘s←←’ : ∀ {T B}\n {b : Term {ε} (T ‘→’ W (‘Typ’ ‘’ ⌜ ε ▻ B ⌝c))}\n {c : Term {ε} (T ‘→’ W (‘Term’ ‘’₁ ⌜ ε ⌝c ‘’ ⌜ B ⌝T))}\n {v : Term 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"max_issues_repo_issues_event_max_datetime": "2015-07-17T20:20:43.000Z", "max_forks_repo_path": "internal/well-typed-syntax.agda", "max_forks_repo_name": "JasonGross/lob", "max_forks_repo_head_hexsha": "716129208eaf4fe3b5f629f95dde4254805942b3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2015-07-17T18:53:37.000Z", "max_forks_repo_forks_event_max_datetime": "2015-07-17T18:53:37.000Z", "avg_line_length": 61.3020833333, "max_line_length": 216, "alphanum_fraction": 0.3701784197, "num_tokens": 6522, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.45713671682749485, "lm_q1q2_score": 0.345099358580539}} {"text": "module Type.NbE where\n\nopen import Context\nopen import Type.Core\n\nopen import Function\nopen import Data.Empty\nopen import Data.Sum.Base\n\ninfix 3 _⊢ᵗⁿᵉ_ _⊢ᵗⁿᶠ_ _⊨ᵗ_\ninfixl 9 _[_]ᵗ\n\nmutual\n Starⁿᵉ : Conᵏ -> Set\n Starⁿᵉ Θ = Θ ⊢ᵗⁿᵉ ⋆\n\n Starⁿᶠ : Conᵏ -> Set\n Starⁿᶠ Θ = Θ ⊢ᵗⁿᶠ ⋆\n\n data _⊢ᵗⁿᵉ_ Θ : Kind -> Set where\n Varⁿᵉ : ∀ {σ} -> σ ∈ Θ -> Θ ⊢ᵗⁿᵉ σ\n _∙ⁿᵉ_ : ∀ {σ τ} -> Θ ⊢ᵗⁿᵉ σ ⇒ᵏ τ -> Θ ⊢ᵗⁿᶠ σ -> Θ ⊢ᵗⁿᵉ τ\n _⇒ⁿᵉ_ : Starⁿᵉ Θ -> Starⁿᵉ Θ -> Starⁿᵉ Θ\n πⁿᵉ : ∀ σ -> Starⁿᵉ (Θ ▻ σ) -> Starⁿᵉ Θ\n μⁿᵉ : ∀ {κ} -> Θ ⊢ᵗⁿᶠ (κ ⇒ᵏ ⋆) ⇒ᵏ κ ⇒ᵏ ⋆ -> Θ ⊢ᵗⁿᶠ κ -> Starⁿᵉ Θ\n\n data _⊢ᵗⁿᶠ_ Θ : Kind -> Set where\n Neⁿᶠ : ∀ {σ} -> Θ ⊢ᵗⁿᵉ σ -> Θ ⊢ᵗⁿᶠ σ\n Lamⁿᶠ_ : ∀ {σ τ} -> Θ ▻ σ ⊢ᵗⁿᶠ τ -> Θ ⊢ᵗⁿᶠ σ ⇒ᵏ τ\n\nmutual\n embᵗⁿᵉ : ∀ {Θ σ} -> Θ ⊢ᵗⁿᵉ σ -> Θ ⊢ᵗ σ\n embᵗⁿᵉ (Varⁿᵉ v) = Var v\n embᵗⁿᵉ (φ ∙ⁿᵉ α) = embᵗⁿᵉ φ ∙ embᵗⁿᶠ α\n embᵗⁿᵉ (α ⇒ⁿᵉ β) = embᵗⁿᵉ α ⇒ embᵗⁿᵉ β\n embᵗⁿᵉ (πⁿᵉ σ α) = π σ (embᵗⁿᵉ α)\n embᵗⁿᵉ (μⁿᵉ ψ α) = μ (embᵗⁿᶠ ψ) (embᵗⁿᶠ α)\n\n embᵗⁿᶠ : ∀ {Θ σ} -> Θ ⊢ᵗⁿᶠ σ -> Θ ⊢ᵗ σ\n embᵗⁿᶠ (Neⁿᶠ α) = embᵗⁿᵉ α\n embᵗⁿᶠ (Lamⁿᶠ β) = Lam (embᵗⁿᶠ β)\n\nmutual\n renᵗⁿᵉ : ∀ {Θ Ξ σ} -> Θ ⊆ Ξ -> Θ ⊢ᵗⁿᵉ σ -> Ξ ⊢ᵗⁿᵉ σ\n renᵗⁿᵉ ι (Varⁿᵉ v) = Varⁿᵉ (renᵛ ι v)\n renᵗⁿᵉ ι (φ ∙ⁿᵉ α) = renᵗⁿᵉ ι φ ∙ⁿᵉ renᵗⁿᶠ ι α\n renᵗⁿᵉ ι (α ⇒ⁿᵉ β) = renᵗⁿᵉ ι α ⇒ⁿᵉ renᵗⁿᵉ ι β\n renᵗⁿᵉ ι (πⁿᵉ σ α) = πⁿᵉ σ (renᵗⁿᵉ (keep ι) α)\n renᵗⁿᵉ ι (μⁿᵉ ψ α) = μⁿᵉ (renᵗⁿᶠ ι ψ) (renᵗⁿᶠ ι α)\n\n renᵗⁿᶠ : ∀ {Θ Ξ σ} -> Θ ⊆ Ξ -> Θ ⊢ᵗⁿᶠ σ -> Ξ ⊢ᵗⁿᶠ σ\n renᵗⁿᶠ ι (Neⁿᶠ α) = Neⁿᶠ (renᵗⁿᵉ ι α)\n renᵗⁿᶠ ι (Lamⁿᶠ β) = Lamⁿᶠ (renᵗⁿᶠ (keep ι) β)\n\nmutual\n _⊨ᵗ_ : Conᵏ -> Kind -> Set\n Θ ⊨ᵗ σ = Θ ⊢ᵗⁿᵉ σ ⊎ Kripke Θ σ\n\n Kripke : Conᵏ -> Kind -> Set\n Kripke Θ ⋆ = ⊥\n Kripke Θ (σ ⇒ᵏ τ) = ∀ {Ξ} -> Θ ⊆ Ξ -> Ξ ⊨ᵗ σ -> Ξ ⊨ᵗ τ\n\nNeˢ : ∀ {σ Θ} -> Θ ⊢ᵗⁿᵉ σ -> Θ ⊨ᵗ σ\nNeˢ = inj₁\n\nVarˢ : ∀ {σ Θ} -> σ ∈ Θ -> Θ ⊨ᵗ σ\nVarˢ = Neˢ ∘ Varⁿᵉ\n\nrenᵗˢ : ∀ {σ Θ Δ} -> Θ ⊆ Δ -> Θ ⊨ᵗ σ -> Δ ⊨ᵗ σ\nrenᵗˢ ι (inj₁ α) = inj₁ (renᵗⁿᵉ ι α)\nrenᵗˢ {⋆} ι (inj₂ ())\nrenᵗˢ {σ ⇒ᵏ τ} ι (inj₂ k) = inj₂ λ κ -> k (κ ∘ ι)\n\nreadbackᵗ : ∀ {σ Θ} -> Θ ⊨ᵗ σ -> Θ ⊢ᵗⁿᶠ σ\nreadbackᵗ (inj₁ α) = Neⁿᶠ α\nreadbackᵗ {⋆} (inj₂ ())\nreadbackᵗ {σ ⇒ᵏ τ} (inj₂ k) = Lamⁿᶠ (readbackᵗ (k topᵒ (Varˢ vz)))\n\n_∙ˢ_ : ∀ {Θ σ τ} -> Θ ⊨ᵗ σ ⇒ᵏ τ -> Θ ⊨ᵗ σ -> Θ ⊨ᵗ τ\ninj₁ f ∙ˢ x = Neˢ (f ∙ⁿᵉ readbackᵗ x)\ninj₂ k ∙ˢ x = k idᵒ x\n\ngroundⁿᵉ : ∀ {Θ} -> Θ ⊨ᵗ ⋆ -> Θ ⊢ᵗⁿᵉ ⋆\ngroundⁿᵉ (inj₁ α) = α\ngroundⁿᵉ (inj₂ ())\n\nenvironmentᵗˢ : Environment _⊨ᵗ_\nenvironmentᵗˢ = record\n { varᵈ = Varˢ\n ; renᵈ = renᵗˢ\n }\n\nmodule _ where\n open Environment environmentᵗˢ\n\n mutual\n evalᵗ : ∀ {Θ Ξ σ} -> Ξ ⊢ᵉ Θ -> Θ ⊢ᵗ σ -> Ξ ⊨ᵗ σ\n evalᵗ ρ (Var v) = lookupᵉ v ρ\n evalᵗ ρ (Lam β) = inj₂ λ ι α -> evalᵗ (renᵉ ι ρ ▷ α) β\n evalᵗ ρ (φ ∙ α) = evalᵗ ρ φ ∙ˢ evalᵗ ρ α\n evalᵗ ρ (α ⇒ β) = inj₁ $ groundⁿᵉ (evalᵗ ρ α) ⇒ⁿᵉ groundⁿᵉ (evalᵗ ρ β)\n evalᵗ ρ (π σ α) = inj₁ $ πⁿᵉ σ ∘ groundⁿᵉ $ evalᵗ (keepᵉ ρ) α\n evalᵗ ρ (μ ψ α) = inj₁ $ μⁿᵉ (nfᵗ ρ ψ) (nfᵗ ρ α)\n\n nfᵗ : ∀ {Θ Ξ σ} -> Ξ ⊢ᵉ Θ -> Θ ⊢ᵗ σ -> Ξ ⊢ᵗⁿᶠ σ\n nfᵗ ρ = readbackᵗ ∘ evalᵗ ρ\n\n normalize : ∀ {Θ σ} -> Θ ⊢ᵗ σ -> Θ ⊢ᵗ σ\n normalize = embᵗⁿᶠ ∘ nfᵗ idᵉ\n\n_[_]ᵗ : ∀ {Θ σ τ} -> Θ ▻ σ ⊢ᵗ τ -> Θ ⊢ᵗ σ -> Θ ⊢ᵗ τ\nβ [ α ]ᵗ = normalize $ Lam β ∙ α\n{-# DISPLAY normalize (Lam β ∙ α) = β [ α ]ᵗ #-}\n", "meta": {"hexsha": "258b7073d15fec2918fe8c674574bacee511a414", "size": 3168, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "papers/unraveling-recursion/code/src/Type/NbE.agda", "max_stars_repo_name": "AriFordsham/plutus", "max_stars_repo_head_hexsha": "f7d34336cd3d65f62b0da084a16f741dc9156413", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1299, "max_stars_repo_stars_event_min_datetime": "2018-10-02T13:41:39.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T01:10:02.000Z", "max_issues_repo_path": "papers/unraveling-recursion/code/src/Type/NbE.agda", "max_issues_repo_name": "AriFordsham/plutus", "max_issues_repo_head_hexsha": "f7d34336cd3d65f62b0da084a16f741dc9156413", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 2493, "max_issues_repo_issues_event_min_datetime": "2018-09-28T19:28:17.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T15:31:31.000Z", "max_forks_repo_path": "papers/unraveling-recursion/code/src/Type/NbE.agda", "max_forks_repo_name": "AriFordsham/plutus", "max_forks_repo_head_hexsha": "f7d34336cd3d65f62b0da084a16f741dc9156413", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 399, "max_forks_repo_forks_event_min_datetime": "2018-10-05T09:36:10.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T11:18:25.000Z", "avg_line_length": 27.7894736842, "max_line_length": 74, "alphanum_fraction": 0.5132575758, "num_tokens": 2511, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.4843800842769844, "lm_q1q2_score": 0.34499107950905167}} {"text": "{-# OPTIONS --without-K #-}\n\nopen import BaseOver\n\nmodule Spaces.Flattening {i j k}\n (A : Set i) (B : Set j) (f g : B → A)\n (C : A → Set k) (D : (b : B) → C (f b) ≃ C (g b)) where\n\nopen import Spaces.FlatteningTypes A B f g C D\n\n-- The family of paths used in the definition of [flatten]\npaths-flatten : (b : B) → (cct (f b) == cct (g b) [ (λ w → (P w → Wt)) ↓ pp b ])\npaths-flatten b =\n ↓-app→cst-in (λ q → ppt b _ ∘' ap (cct (g b)) (↓-pp-out q))\n\nflatten-curried : (w : W) → (P w → Wt)\nflatten-curried = W-rec _ cct paths-flatten\n\nflatten : Σ W P → Wt\nflatten (w , x) = flatten-curried w x\n\nunflatten : Wt → Σ W P\nunflatten = Wt-rec-nondep _ (λ a c → (cc a , c))\n (λ b d → Σ-eq (pp b) (↓-pp-in refl))\n\n--\n\nflatten-unflatten : (w : Wt) → flatten (unflatten w) ≡ w\nflatten-unflatten =\n Wt-rec _\n (λ _ _ → refl)\n (λ b d → ↓-◯=id-in unflatten flatten\n (ap flatten (ap unflatten (ppt b d))\n ≡⟨ Wt-rec-nondep-β _ (λ a c → (cc a , c)) (λ b d → Σ-eq (pp b) (↓-pp-in refl)) b d |in-ctx ap flatten ⟩\n ap flatten (Σ-eq (pp b) (↓-pp-in refl))\n ≡⟨ split-ap2 flatten (pp b) (↓-pp-in refl) ⟩\n ↓-app→cst-out (apd flatten-curried (pp b)) (↓-pp-in refl)\n ≡⟨ W-rec-β _ cct paths-flatten b |in-ctx (λ u → ↓-app→cst-out u (↓-pp-in refl)) ⟩\n ↓-app→cst-out (paths-flatten b) (↓-pp-in refl)\n ≡⟨ refl ⟩\n ↓-app→cst-out (↓-app→cst-in\n (λ q → ppt b _ ∘' ap (cct (g b)) (↓-pp-out q))) (↓-pp-in refl)\n ≡⟨ ↓-app→cst-β (λ q → ppt b _ ∘' ap (cct (g b)) (↓-pp-out q)) (↓-pp-in refl) ⟩\n ppt b d ∘' ap (cct (g b)) (↓-pp-out (↓-pp-in refl))\n ≡⟨ ↓-pp-β refl |in-ctx (λ u → ppt b d ∘' ap (cct (g b)) u) ⟩\n ppt b d ∎))\n\nunflatten-flatten-curried : (w : W) (x : P w)\n → unflatten (flatten-curried w x) ≡ (w , x)\nunflatten-flatten-curried =\n W-rec _ (λ a x → refl)\n (λ b → ↓-Π-in\n (λ q → ↓-◯=id-in flatten unflatten\n (ap unflatten (ap flatten (Σ-eq (pp b) q))\n ≡⟨ split-ap2 flatten (pp b) q |in-ctx ap unflatten ⟩\n ap unflatten (↓-app→cst-out (apd flatten-curried (pp b)) q)\n ≡⟨ W-rec-β _ cct paths-flatten b |in-ctx (λ u → ap unflatten (↓-app→cst-out u q)) ⟩\n ap unflatten (↓-app→cst-out (paths-flatten b) q)\n ≡⟨ refl ⟩\n ap unflatten (↓-app→cst-out (↓-app→cst-in (λ qq → ppt b _ ∘' ap (cct (g b)) (↓-pp-out qq))) q)\n ≡⟨ ↓-app→cst-β (λ qq → ppt b _ ∘' ap (cct (g b)) (↓-pp-out qq)) q |in-ctx ap unflatten ⟩\n ap unflatten (ppt b _ ∘' ap (cct (g b)) (↓-pp-out q))\n ≡⟨ ap-∘' unflatten (ppt b _) (ap (cct (g b)) (↓-pp-out q)) ⟩\n ap unflatten (ppt b _) ∘' ap unflatten (ap (cct (g b)) (↓-pp-out q))\n ≡⟨ Wt-rec-nondep-β _ (λ a c → (cc a , c)) (λ b d → Σ-eq (pp b) (↓-pp-in refl)) b _ |in-ctx (λ u → u ∘' ap unflatten (ap (cct (g b)) (↓-pp-out q))) ⟩\n Σ-eq (pp b) (↓-pp-in refl) ∘' ap unflatten (ap (cct (g b)) (↓-pp-out q))\n ≡⟨ compose-ap unflatten (cct (g b)) (↓-pp-out q) |in-ctx (λ u → (Σ-eq (pp b) (↓-pp-in refl) ∘' u)) ⟩\n Σ-eq (pp b) (↓-pp-in refl) ∘' ap (unflatten ◯ cct (g b)) (↓-pp-out q)\n ≡⟨ refl ⟩\n Σ-eq (pp b) (↓-pp-in refl) ∘' ap (λ x → (cc (g b), x)) (↓-pp-out q)\n ≡⟨ ap-cst,id P (↓-pp-out q) |in-ctx (λ u → Σ-eq (pp b) (↓-pp-in refl) ∘' u) ⟩\n Σ-eq (pp b) (↓-pp-in refl) ∘' Σ-eq refl (↓-pp-out q)\n ≡⟨ Σ-∘' (↓-pp-in refl) (↓-pp-out q) ⟩\n Σ-eq (pp b) (↓-pp-in refl ∘'dep ↓-pp-out q)\n ≡⟨ to-transp-weird q (pp-path _ _) |in-ctx Σ-eq (pp b) ⟩\n Σ-eq (pp b) q ∎)))\n\nunflatten-flatten : (wx : Σ W P) → unflatten (flatten wx) ≡ wx\nunflatten-flatten (w , x) = unflatten-flatten-curried w x\n\neqv : Σ W P ≃ Wt\neqv = (flatten , iso-is-eq flatten unflatten flatten-unflatten unflatten-flatten)\n", "meta": {"hexsha": "fdf8c689b57f4d8f0022bfe03508b7a220fcaae4", "size": 3886, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "old/Spaces/Flattening.agda", "max_stars_repo_name": "UlrikBuchholtz/HoTT-Agda", "max_stars_repo_head_hexsha": "f8fa68bf753d64d7f45556ca09d0da7976709afa", "max_stars_repo_licenses": ["MIT"], 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"alphanum_fraction": 0.4886773031, "num_tokens": 1634, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.34489499481174735}} {"text": "{-# OPTIONS --safe #-}\nmodule Cubical.Algebra.DirectSum.DirectSumHIT.PseudoNormalForm where\n\nopen import Cubical.Foundations.Prelude\nopen import Cubical.Foundations.HLevels\n\nopen import Cubical.Data.Nat renaming (_+_ to _+n_ ; _·_ to _·n_)\nopen import Cubical.Data.Sigma\nopen import Cubical.Data.List\nopen import Cubical.Data.Vec.DepVec\n\nopen import Cubical.HITs.PropositionalTruncation as PT\n\nopen import Cubical.Algebra.AbGroup\nopen import Cubical.Algebra.AbGroup.Instances.DirectSumHIT\nopen import Cubical.Algebra.DirectSum.DirectSumHIT.Base\n\nprivate variable\n ℓ : Level\n\nopen AbGroupStr\nopen AbGroupTheory\n\n\n-----------------------------------------------------------------------------\n-- Notation\n\nmodule DefPNF\n (G : (n : ℕ) → Type ℓ)\n (Gstr : (n : ℕ) → AbGroupStr (G n))\n where\n\n open AbGroupStr (snd (⊕HIT-AbGr ℕ G Gstr)) using ()\n renaming\n ( 0g to 0⊕HIT\n ; _+_ to _+⊕HIT_\n ; -_ to -⊕HIT_\n ; +Assoc to +⊕HIT-Assoc\n ; +IdR to +⊕HIT-IdR\n ; +IdL to +⊕HIT-IdL\n ; +InvR to +⊕HIT-InvR\n ; +InvL to +⊕HIT-InvL\n ; +Comm to +⊕HIT-Comm\n ; is-set to isSet⊕HIT)\n\n\n-----------------------------------------------------------------------------\n-- Lemma\n\n -- def pseudo normal form\n sumHIT : {n : ℕ} → depVec G n → ⊕HIT ℕ G Gstr\n sumHIT {0} ⋆ = 0⊕HIT\n sumHIT {suc n} (a □ dv) = (base n a) +⊕HIT (sumHIT dv)\n\n -- 0 and sum\n replicate0g : (n : ℕ) → depVec G n\n replicate0g (zero) = ⋆\n replicate0g (suc n) = (0g (Gstr n)) □ (replicate0g n)\n\n sumHIT0g : (n : ℕ) → sumHIT (replicate0g n) ≡ 0⊕HIT\n sumHIT0g (zero) = refl\n sumHIT0g (suc n) = cong₂ _+⊕HIT_ (base-neutral n) (sumHIT0g n)\n ∙ +⊕HIT-IdL _\n\n -- extension and sum\n extendDVL : (k l : ℕ) → (dv : depVec G l) → depVec G (k +n l)\n extendDVL zero l dv = dv\n extendDVL (suc k) l dv = (0g (Gstr (k +n l))) □ (extendDVL k l dv)\n\n extendDVLeq : (k l : ℕ) → (dv : depVec G l) → sumHIT (extendDVL k l dv) ≡ sumHIT dv\n extendDVLeq (zero) l dv = refl\n extendDVLeq (suc k) l dv = cong (λ X → X +⊕HIT sumHIT (extendDVL k l dv)) (base-neutral (k +n l))\n ∙ +⊕HIT-IdL _\n ∙ extendDVLeq k l dv\n\n extendDVR : (k l : ℕ) → (dv : depVec G k) → depVec G (k +n l)\n extendDVR k l dv = subst (λ X → depVec G X) (+-comm l k) (extendDVL l k dv)\n\n extendDVReq : (k l : ℕ) → (dv : depVec G k) → sumHIT (extendDVR k l dv) ≡ sumHIT dv\n extendDVReq k l dv = J (λ m p → sumHIT (subst (λ X → depVec G X) p (extendDVL l k dv)) ≡ sumHIT dv)\n (sumHIT (subst (λ X → depVec G X) refl (extendDVL l k dv))\n ≡⟨ cong sumHIT (transportRefl (extendDVL l k dv)) ⟩\n sumHIT (extendDVL l k dv)\n ≡⟨ extendDVLeq l k dv ⟩\n sumHIT dv ∎)\n (+-comm l k)\n\n -- pointwise add\n _pt+DV_ : {n : ℕ} → (dva dvb : depVec G n) → depVec G n\n _pt+DV_ {0} ⋆ ⋆ = ⋆\n _pt+DV_ {suc n} (a □ dva) (b □ dvb) = Gstr n ._+_ a b □ (dva pt+DV dvb)\n\n sumHIT+ : {n : ℕ} → (dva dvb : depVec G n) → sumHIT (dva pt+DV dvb) ≡ sumHIT dva +⊕HIT sumHIT dvb\n sumHIT+ {0} ⋆ ⋆ = sym (+⊕HIT-IdR _)\n sumHIT+ {suc n} (a □ dva) (b □ dvb) = cong₂ _+⊕HIT_ (sym (base-add _ _ _)) (sumHIT+ dva dvb)\n ∙ comm-4 (⊕HIT-AbGr ℕ G Gstr) _ _ _ _\n\n\n-----------------------------------------------------------------------------\n-- Case Traduction\n\n {- WARNING :\n The pseudo normal form is not unique.\n It is actually not unique enough so that it is not possible to raise one from ⊕HIT.\n Hence we actually need to make it a prop to be able to eliminate.\n -}\n\n untruncatedPNF : (x : ⊕HIT ℕ G Gstr) → Type ℓ\n untruncatedPNF x = Σ[ m ∈ ℕ ] Σ[ dv ∈ depVec G m ] x ≡ sumHIT dv\n\n PNF : (x : ⊕HIT ℕ G Gstr) → Type ℓ\n PNF x = ∥ untruncatedPNF x ∥₁\n\n untruncatedPNF2 : (x y : ⊕HIT ℕ G Gstr) → Type ℓ\n untruncatedPNF2 x y = Σ[ m ∈ ℕ ] Σ[ a ∈ depVec G m ] Σ[ b ∈ depVec G m ] (x ≡ sumHIT a) × (y ≡ sumHIT b)\n\n PNF2 : (x y : ⊕HIT ℕ G Gstr) → Type ℓ\n PNF2 x y = ∥ untruncatedPNF2 x y ∥₁\n\n-----------------------------------------------------------------------------\n-- Translation\n\n ⊕HIT→PNF : (x : ⊕HIT ℕ G Gstr) → ∥ Σ[ m ∈ ℕ ] Σ[ a ∈ depVec G m ] x ≡ sumHIT a ∥₁\n ⊕HIT→PNF = DS-Ind-Prop.f _ _ _ _\n (λ _ → squash₁)\n ∣ (0 , (⋆ , refl)) ∣₁\n base→PNF\n add→PNF\n where\n base→PNF : (n : ℕ) → (a : G n) → PNF (base n a)\n base→PNF n a = ∣ (suc n) , ((a □ replicate0g n) , sym (cong (λ X → base n a +⊕HIT X) (sumHIT0g n)\n ∙ +⊕HIT-IdR _)) ∣₁\n\n add→PNF : {U V : ⊕HIT ℕ G Gstr} → (ind-U : PNF U) → (ind-V : PNF V) → PNF (U +⊕HIT V)\n add→PNF {U} {V} = elim2 (λ _ _ → squash₁)\n (λ { (k , dva , p) →\n λ { (l , dvb , q) →\n ∣ ((k +n l)\n , (((extendDVR k l dva) pt+DV (extendDVL k l dvb))\n , cong₂ _+⊕HIT_ p q\n ∙ cong₂ _+⊕HIT_ (sym (extendDVReq k l dva)) (sym (extendDVLeq k l dvb))\n ∙ sym (sumHIT+ (extendDVR k l dva) (extendDVL k l dvb)) )) ∣₁}})\n\n\n\n ⊕HIT→PNF2 : (x y : ⊕HIT ℕ G Gstr) → ∥ Σ[ m ∈ ℕ ] Σ[ a ∈ depVec G m ] Σ[ b ∈ depVec G m ] (x ≡ sumHIT a) × (y ≡ sumHIT b) ∥₁\n ⊕HIT→PNF2 x y = helper (⊕HIT→PNF x) (⊕HIT→PNF y)\n where\n helper : PNF x → PNF y →\n ∥ Σ[ m ∈ ℕ ] Σ[ a ∈ depVec G m ] Σ[ b ∈ depVec G m ] (x ≡ sumHIT a) × (y ≡ sumHIT b) ∥₁\n helper = elim2 (λ _ _ → squash₁)\n (λ { (k , dva , p) →\n λ { (l , dvb , q) →\n ∣ ((k +n l)\n , ((extendDVR k l dva)\n , (extendDVL k l dvb\n , p ∙ sym (extendDVReq k l dva)\n , q ∙ sym (extendDVLeq k l dvb)))) ∣₁}})\n\n-----------------------------------------------------------------------------\n-- Some idea\n\n{-\n This file should be generalizable to a general decidable index by adding a second vector\n-}\n\n\n{-\n It maybe possible to give a normal for without need the prop truncation.\n The issue with the current one is that we rely on a underline data type depVec\n which forces us to give an explict length. That's what forces the ∥_∥₁.\n Hence by getting rid of it and be rewrittinf the term it might be possible\n to get a normal form without the PT.\n\n Indeed this basically about pemuting and summing them by G n\n ∑ base (σ i) a (σ i) -> ∑[i ∈ ℕ] ∑[j ∈ I] base i (b i j) -> ∑ base i (c i)\n where a b c are informal \"sequences\"\n\n Then prove that if we extract the integer, we get an inceasing list\n with no coefficient being present twice.\n-}\n", "meta": {"hexsha": "8aef7f3b7ad5495ce014d419b7e3fca9f46d4c0a", "size": 6709, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Cubical/Algebra/DirectSum/DirectSumHIT/PseudoNormalForm.agda", "max_stars_repo_name": "thomas-lamiaux/cubical", "max_stars_repo_head_hexsha": "58c0b83bb0fed0dc683f3d29b1709effe51c1689", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Cubical/Algebra/DirectSum/DirectSumHIT/PseudoNormalForm.agda", "max_issues_repo_name": "thomas-lamiaux/cubical", "max_issues_repo_head_hexsha": "58c0b83bb0fed0dc683f3d29b1709effe51c1689", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Cubical/Algebra/DirectSum/DirectSumHIT/PseudoNormalForm.agda", "max_forks_repo_name": "thomas-lamiaux/cubical", "max_forks_repo_head_hexsha": "58c0b83bb0fed0dc683f3d29b1709effe51c1689", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.8626373626, "max_line_length": 125, "alphanum_fraction": 0.5029065434, "num_tokens": 2548, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.34474859140422703}} {"text": "{-\n\nIndex a structure S by the type variable: X ↦ X → S X\n\n-}\n{-# OPTIONS --cubical --no-import-sorts --safe #-}\nmodule Cubical.Structures.Relational.UnaryOp where\n\nopen import Cubical.Foundations.Prelude\nopen import Cubical.Foundations.Isomorphism\nopen import Cubical.Foundations.Equiv\nopen import Cubical.Foundations.Function\nopen import Cubical.Foundations.HLevels\nopen import Cubical.Foundations.Structure\nopen import Cubical.Foundations.RelationalStructure\nopen import Cubical.Foundations.Univalence\nopen import Cubical.Functions.FunExtEquiv\nopen import Cubical.Data.Sigma\nopen import Cubical.Relation.Binary.Base\nopen import Cubical.Relation.ZigZag.Base\nopen import Cubical.HITs.SetQuotients\nopen import Cubical.HITs.PropositionalTruncation as Trunc\n\nopen import Cubical.Structures.NAryOp\n\nprivate\n variable\n ℓ ℓ₁ ℓ₁' : Level\n\n-- Structured relations\n\npreservesSetsUnaryFun : {S : Type ℓ → Type ℓ₁}\n → preservesSets S → preservesSets (NAryFunStructure 1 S)\npreservesSetsUnaryFun p setX = isSetΠ λ _ → p setX\n\nUnaryFunRelStr : {S : Type ℓ → Type ℓ₁} {ℓ₁' : Level}\n → StrRel S ℓ₁' → StrRel (NAryFunStructure 1 S) (ℓ-max ℓ ℓ₁')\nUnaryFunRelStr ρ R f g =\n ∀ {x y} → R x y → ρ R (f x) (g y)\n\nopen BinaryRelation\nopen isEquivRel\nopen isQuasiEquivRel\nopen SuitableStrRel\n\nprivate\n composeWith[_] : {A : Type ℓ} (R : EquivPropRel A ℓ)\n → compPropRel (R .fst) (quotientPropRel (R .fst .fst)) .fst ≡ graphRel [_]\n composeWith[_] R =\n funExt₂ λ a t →\n hPropExt squash (squash/ _ _)\n (Trunc.rec (squash/ _ _) (λ {(b , r , p) → eq/ a b r ∙ p }))\n (λ p → ∣ a , R .snd .reflexive a , p ∣)\n\nunaryFunSuitableRel : {S : Type ℓ → Type ℓ₁} (p : preservesSets S) {ρ : StrRel S ℓ₁'}\n → SuitableStrRel S ρ\n → SuitableStrRel (NAryFunStructure 1 S) (UnaryFunRelStr ρ)\nunaryFunSuitableRel pres {ρ} θ .quo (X , f) R h .fst =\n f₀ ,\n λ {x} → J (λ y p → ρ (graphRel [_]) (f x) (f₀ y)) (θ .quo (X , f x) R (href x) .fst .snd)\n where\n href = h ∘ R .snd .reflexive\n\n f₀ : _\n f₀ [ x ] = θ .quo (X , f x) R (href x) .fst .fst\n f₀ (eq/ x₀ x₁ r i) = path i\n where\n path : θ .quo (X , f x₀) R (href x₀) .fst .fst ≡ θ .quo (X , f x₁) R (href x₁) .fst .fst\n path =\n cong fst\n (θ .quo (X , f x₀) R (href x₀) .snd\n ( θ .quo (X , f x₁) R (href x₁) .fst .fst\n , subst\n (λ T → ρ T (f x₀) (θ .quo (X , f x₁) R (href x₁) .fst .fst))\n (composeWith[_] R)\n (θ .transitive (R .fst) (quotientPropRel (R .fst .fst))\n (h r)\n (θ .quo (X , f x₁) R (href x₁) .fst .snd))\n ))\n f₀ (squash/ _ _ p q j i) =\n pres squash/ _ _ (cong f₀ p) (cong f₀ q) j i\nunaryFunSuitableRel pres {ρ} θ .quo (X , f) R h .snd (f' , c) =\n Σ≡Prop\n (λ _ → isPropImplicitΠ λ _ → isPropImplicitΠ λ _ → isPropΠ λ _ →\n θ .prop (λ _ _ → squash/ _ _) _ _)\n (funExt\n (elimProp (λ _ → pres squash/ _ _)\n (λ x → cong fst (θ .quo (X , f x) R (href x) .snd (f' [ x ] , c refl)))))\n where\n href = h ∘ R .snd .reflexive\nunaryFunSuitableRel pres {ρ} θ .symmetric R h {x} {y} r = θ .symmetric R (h r)\nunaryFunSuitableRel pres {ρ} θ .transitive R R' h h' {x} {z} =\n Trunc.rec\n (θ .prop (λ _ _ → squash) _ _)\n (λ {(y , r , r') → θ .transitive R R' (h r) (h' r')})\nunaryFunSuitableRel pres {ρ} θ .prop propR f g =\n isPropImplicitΠ λ x →\n isPropImplicitΠ λ y →\n isPropΠ λ _ → θ .prop propR (f x) (g y)\n\nunaryFunRelMatchesEquiv : {S : Type ℓ → Type ℓ₁}\n (ρ : StrRel S ℓ₁') {ι : StrEquiv S ℓ₁'}\n → StrRelMatchesEquiv ρ ι\n → StrRelMatchesEquiv (UnaryFunRelStr ρ) (UnaryFunEquivStr ι)\nunaryFunRelMatchesEquiv ρ μ (X , f) (Y , g) e =\n compEquiv (isoToEquiv isom) (equivPi λ _ → μ _ _ e)\n where\n open Iso\n\n isom : Iso _ _\n isom .fun h x = h refl\n isom .inv k {x} = J (λ y _ → ρ (graphRel (e .fst)) (f x) (g y)) (k x)\n isom .rightInv k i x = JRefl (λ y _ → ρ (graphRel (e .fst)) (f x) (g y)) (k x) i\n isom .leftInv h =\n implicitFunExt λ {x} →\n implicitFunExt λ {y} →\n funExt λ p →\n J (λ y p → isom .inv (isom .fun h) p ≡ h p)\n (funExt⁻ (isom .rightInv (isom .fun h)) x)\n p\n", "meta": {"hexsha": "1f6e424848353bd91b65d15742217d9bf77637da", "size": 4086, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Cubical/Structures/Relational/UnaryOp.agda", "max_stars_repo_name": "RobertHarper/cubical", "max_stars_repo_head_hexsha": "d13941587a58895b65f714f1ccc9c1f5986b109c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Cubical/Structures/Relational/UnaryOp.agda", "max_issues_repo_name": "RobertHarper/cubical", "max_issues_repo_head_hexsha": "d13941587a58895b65f714f1ccc9c1f5986b109c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Cubical/Structures/Relational/UnaryOp.agda", "max_forks_repo_name": "RobertHarper/cubical", "max_forks_repo_head_hexsha": "d13941587a58895b65f714f1ccc9c1f5986b109c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-11-22T02:02:01.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-22T02:02:01.000Z", "avg_line_length": 33.7685950413, "max_line_length": 92, "alphanum_fraction": 0.6130690162, "num_tokens": 1622, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3447238421626679}} {"text": "{-# OPTIONS --safe --warning=error --without-K #-}\n\nopen import Agda.Primitive\n\nmodule Basic where\n\ndata False : Set where\n\nrecord True : Set where\n\ndata _||_ {a b : _} (A : Set a) (B : Set b) : Set (a ⊔ b) where\n inl : A → A || B\n inr : B → A || B\ninfix 1 _||_\n", "meta": {"hexsha": "c08a95f2521f1c4f21fe02995126202479f51389", "size": 264, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Basic.agda", "max_stars_repo_name": "Smaug123/CubicalTutorial", "max_stars_repo_head_hexsha": "3d56e649152d2cd906281943860ca19da1d1f344", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-04-26T17:02:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-16T23:14:13.000Z", "max_issues_repo_path": "Basic.agda", "max_issues_repo_name": "Smaug123/CubicalTutorial", "max_issues_repo_head_hexsha": "3d56e649152d2cd906281943860ca19da1d1f344", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Basic.agda", "max_forks_repo_name": "Smaug123/CubicalTutorial", "max_forks_repo_head_hexsha": "3d56e649152d2cd906281943860ca19da1d1f344", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 17.6, "max_line_length": 63, "alphanum_fraction": 0.5833333333, "num_tokens": 94, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.34462911788977846}} {"text": "{-# OPTIONS --cubical #-}\nmodule _ where\nopen import Agda.Primitive.Cubical renaming (primINeg to ~_; primIMax to _∨_; primIMin to _∧_)\nopen import Agda.Builtin.Cubical.Path\nopen import Agda.Builtin.Cubical.Sub\nopen import Agda.Builtin.Cubical.Sub using () renaming (Sub to _[_↦_]; primSubOut to ouc)\nopen import Agda.Primitive renaming (_⊔_ to ℓ-max)\nopen import Agda.Builtin.Sigma\n\ntranspFill : ∀ {ℓ} {A' : Set ℓ} (φ : I)\n (A : (i : I) → Set ℓ [ φ ↦ (\\ _ → A') ]) →\n (u0 : ouc (A i0)) →\n PathP (λ i → ouc (A i)) u0 (primTransp (λ i → ouc (A i)) φ u0)\ntranspFill φ A u0 i = primTransp (\\ j → ouc (A (i ∧ j))) (~ i ∨ φ) u0\n\n-- private\n-- internalFiber : ∀ {ℓ ℓ'} {A : Set ℓ} {B : Set ℓ'} (f : A → B) (y : B) → Set (ℓ-max ℓ ℓ')\n-- internalFiber {A = A} f y = Σ A \\ x → y ≡ f x\n\n-- infix 4 _≃_\n\n-- postulate\n-- _≃_ : ∀ {ℓ ℓ'} (A : Set ℓ) (B : Set ℓ') → Set (ℓ-max ℓ ℓ')\n-- equivFun : ∀ {ℓ ℓ'} {A : Set ℓ} {B : Set ℓ'} → A ≃ B → A → B\n-- equivProof : ∀ {la lt} (T : Set la) (A : Set lt) → (w : T ≃ A) → (a : A)\n-- → ∀ ψ → (Partial ψ (internalFiber (equivFun w) a)) → internalFiber (equivFun w) a\n\n-- {-# BUILTIN EQUIV _≃_ #-}\n-- {-# BUILTIN EQUIVFUN equivFun #-}\n-- {-# BUILTIN EQUIVPROOF equivProof #-}\n\n-- -- This is a module so we can easily rename the primitives.\n-- module GluePrims where\n-- primitive\n-- primGlue : ∀ {ℓ ℓ'} (A : Set ℓ) {φ : I}\n-- → (T : Partial φ (Set ℓ')) → (e : PartialP φ (λ o → T o ≃ A))\n-- → Set ℓ'\n-- prim^glue : ∀ {ℓ ℓ'} {A : Set ℓ} {φ : I}\n-- → {T : Partial φ (Set ℓ')} → {e : PartialP φ (λ o → T o ≃ A)}\n-- → PartialP φ T → A → primGlue A T e\n-- prim^unglue : ∀ {ℓ ℓ'} {A : Set ℓ} {φ : I}\n-- → {T : Partial φ (Set ℓ')} → {e : PartialP φ (λ o → T o ≃ A)}\n-- → primGlue A T e → A\n\n-- -- Needed for transp in Glue.\n-- primFaceForall : (I → I) → I\n\nopen import Agda.Builtin.Cubical.Glue public\n renaming ( prim^glue to glue\n ; prim^unglue to unglue)\n\n-- We uncurry Glue to make it a bit more pleasant to use\nGlue : ∀ {ℓ ℓ'} (A : Set ℓ) {φ : I}\n → (Te : Partial φ (Σ (Set ℓ') \\ T → T ≃ A))\n → Set ℓ'\nGlue A Te = primGlue A (λ x → Te x .fst) (λ x → Te x .snd)\n\nmodule TestHComp {ℓ ℓ'} (A : Set ℓ) {φ : I} (Te : Partial φ (Σ (Set ℓ') \\ T → T ≃ A))\n (ψ : I) (u : I → Partial ψ (Glue A Te)) (u0 : Sub (Glue A Te) ψ (u i0) ) where\n result : Glue A Te\n result = glue {φ = φ} (\\ { (φ = i1) → primHComp {A = Te itIsOne .fst} u (primSubOut u0) })\n (primHComp {A = A} (\\ i → \\ { (ψ = i1) → unglue {φ = φ} (u i itIsOne)\n ; (φ = i1) → equivFun (Te itIsOne .snd)\n (primHComp (\\ j → \\ { (ψ = i1) → u (i ∧ j) itIsOne\n ; (i = i0) → primSubOut u0 })\n (primSubOut u0)) })\n (unglue {φ = φ} (primSubOut u0)))\n\n test : primHComp {A = Glue A Te} {ψ} u (primSubOut u0) ≡ result\n test i = primHComp {A = Glue A Te} {ψ} u (primSubOut u0)\n\n\nmodule TestTransp {ℓ ℓ'} (A : Set ℓ) {φ : I} (Te : Partial φ (Σ (Set ℓ') \\ T → T ≃ A))\n (u0 : (Glue A Te)) where\n ψ = i0\n\n a0 = unglue {φ = φ} u0\n a1 = primComp (\\ _ → A)\n φ\n (\\ { i (φ = i1) → equivFun (Te itIsOne .snd) (transpFill {A' = Te itIsOne .fst} ψ (\\ i → inc (Te itIsOne .fst)) u0 i) })\n a0\n\n pair : PartialP φ λ o → Helpers.fiber (Te o .snd .fst) a1\n pair o = equivProof (Te o .fst) A (Te o .snd) a1 φ \\ { (φ = i1) → _ , Helpers.refl }\n\n result : Glue A Te\n result = glue {φ = φ} (λ o → pair o .fst) (primHComp (\\ { j (φ = i1) → pair itIsOne .snd j}) a1)\n\n test : primTransp (\\ _ → Glue A Te) ψ u0 ≡ result\n test = Helpers.refl\n", "meta": {"hexsha": "9e265f7df66723521e48e4b48991e55e9983298e", "size": 3953, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/Succeed/Issue3399.agda", "max_stars_repo_name": "caryoscelus/agda", "max_stars_repo_head_hexsha": "98d6f195fe672e54ef0389b4deb62e04e3e98327", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "test/Succeed/Issue3399.agda", "max_issues_repo_name": "caryoscelus/agda", "max_issues_repo_head_hexsha": "98d6f195fe672e54ef0389b4deb62e04e3e98327", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "test/Succeed/Issue3399.agda", "max_forks_repo_name": "caryoscelus/agda", "max_forks_repo_head_hexsha": "98d6f195fe672e54ef0389b4deb62e04e3e98327", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.4395604396, "max_line_length": 129, "alphanum_fraction": 0.4687579054, "num_tokens": 1520, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.34462911788977846}} {"text": "{-# OPTIONS --without-K --safe #-}\n\nopen import Definition.Typed.EqualityRelation\n\nmodule Definition.LogicalRelation.Fundamental.Reducibility {{eqrel : EqRelSet}} where\nopen EqRelSet {{...}}\n\nopen import Definition.Untyped\nopen import Definition.Typed\nopen import Definition.LogicalRelation\nopen import Definition.LogicalRelation.Substitution\nopen import Definition.LogicalRelation.Substitution.Reducibility\nopen import Definition.LogicalRelation.Fundamental\n\nopen import Tools.Product\n\n\n-- Well-formed types are reducible.\nreducible : ∀ {A Γ} → Γ ⊢ A → Γ ⊩⟨ ¹ ⟩ A\nreducible A = let [Γ] , [A] = fundamental A\n in reducibleᵛ [Γ] [A]\n\n-- Well-formed equality is reducible.\nreducibleEq : ∀ {A B Γ} → Γ ⊢ A ≡ B\n → ∃₂ λ [A] ([B] : Γ ⊩⟨ ¹ ⟩ B) → Γ ⊩⟨ ¹ ⟩ A ≡ B / [A]\nreducibleEq {A} {B} A≡B =\n let [Γ] , [A] , [B] , [A≡B] = fundamentalEq A≡B\n in reducibleᵛ [Γ] [A]\n , reducibleᵛ [Γ] [B]\n , reducibleEqᵛ {A} {B} [Γ] [A] [A≡B]\n\n-- Well-formed terms are reducible.\nreducibleTerm : ∀ {t A Γ} → Γ ⊢ t ∷ A → ∃ λ [A] → Γ ⊩⟨ ¹ ⟩ t ∷ A / [A]\nreducibleTerm {t} {A} ⊢t =\n let [Γ] , [A] , [t] = fundamentalTerm ⊢t\n in reducibleᵛ [Γ] [A] , reducibleTermᵛ {t} {A} [Γ] [A] [t]\n\n-- Well-formed term equality is reducible.\nreducibleEqTerm : ∀ {t u A Γ} → Γ ⊢ t ≡ u ∷ A → ∃ λ [A] → Γ ⊩⟨ ¹ ⟩ t ≡ u ∷ A / [A]\nreducibleEqTerm {t} {u} {A} t≡u =\n let [Γ] , modelsTermEq [A] [t] [u] [t≡u] = fundamentalTermEq t≡u\n in reducibleᵛ [Γ] [A] , reducibleEqTermᵛ {t} {u} {A} [Γ] [A] [t≡u]\n", "meta": {"hexsha": "b8439af7902035abe497c83ced92f4f230f3968e", "size": 1491, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Definition/LogicalRelation/Fundamental/Reducibility.agda", "max_stars_repo_name": "Vtec234/logrel-mltt", "max_stars_repo_head_hexsha": "4746894adb5b8edbddc8463904ee45c2e9b29b69", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Definition/LogicalRelation/Fundamental/Reducibility.agda", "max_issues_repo_name": "Vtec234/logrel-mltt", "max_issues_repo_head_hexsha": "4746894adb5b8edbddc8463904ee45c2e9b29b69", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Definition/LogicalRelation/Fundamental/Reducibility.agda", "max_forks_repo_name": "Vtec234/logrel-mltt", "max_forks_repo_head_hexsha": "4746894adb5b8edbddc8463904ee45c2e9b29b69", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.6744186047, "max_line_length": 85, "alphanum_fraction": 0.6130114017, "num_tokens": 626, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417487156366, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3446291108731628}} {"text": "------------------------------------------------------------------------\n-- A variant of weak bisimilarity that can be used to relate the\n-- number of steps in two computations\n------------------------------------------------------------------------\n\n{-# OPTIONS --sized-types #-}\n\nopen import Prelude hiding (_+_; _*_)\n\nmodule Delay-monad.Quantitative-weak-bisimilarity {a} {A : Type a} where\n\nopen import Equality.Propositional\nopen import Logical-equivalence using (_⇔_)\nopen import Prelude.Size\n\nopen import Conat equality-with-J as Conat\n using (Conat; zero; suc; force; ⌜_⌝; _+_; _*_;\n [_]_≤_; step-≤; step-∼≤; _∎≤; step-∼; _∎∼)\nopen import Function-universe equality-with-J as F hiding (id; _∘_)\n\nopen import Delay-monad\nopen import Delay-monad.Bisimilarity as B\n using (now; later; laterˡ; laterʳ; force)\n\n------------------------------------------------------------------------\n-- The relation\n\nmutual\n\n -- Quantitative weak bisimilarity. [ ∞ ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y\n -- is a variant of x B.≈ y for which the number of later\n -- constructors in x is bounded by nˡ plus 1 + mˡ times the number\n -- of later constructors in y, and the number of later constructors\n -- in y is bounded by nʳ plus 1 + mʳ times the number of later\n -- constructors in x (see ≈⇔≈×steps≤steps² below).\n\n infix 4 [_∣_∣_∣_∣_]_≈_ [_∣_∣_∣_∣_]_≈′_\n\n data [_∣_∣_∣_∣_]_≈_\n (i : Size) (mˡ mʳ : Conat ∞) :\n Conat ∞ → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a where\n now : ∀ {x nˡ nʳ} → [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] now x ≈ now x\n later : ∀ {x y nˡ nʳ} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ + mˡ ∣ nʳ + mʳ ] x .force ≈′ y .force →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] later x ≈ later y\n laterˡ : ∀ {x y nˡ nʳ} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ .force ∣ nʳ ] x .force ≈ y →\n [ i ∣ mˡ ∣ mʳ ∣ suc nˡ ∣ nʳ ] later x ≈ y\n laterʳ : ∀ {x y nˡ nʳ} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ .force ] x ≈ y .force →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ suc nʳ ] x ≈ later y\n\n record [_∣_∣_∣_∣_]_≈′_\n (i : Size) (mˡ mʳ nˡ nʳ : Conat ∞)\n (x y : Delay A ∞) : Type a where\n coinductive\n field\n force : {j : Size< i} → [ j ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y\n\nopen [_∣_∣_∣_∣_]_≈′_ public\n\n-- Specialised variants of [_∣_∣_∣_∣_]_≈_ and [_∣_∣_∣_∣_]_≈′_.\n\ninfix 4 [_∣_∣_]_≈_ [_∣_∣_]_≈′_\n\n[_∣_∣_]_≈_ : Size → Conat ∞ → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ mˡ ∣ mʳ ] x ≈ y = [ i ∣ mˡ ∣ mʳ ∣ zero ∣ zero ] x ≈ y\n\n[_∣_∣_]_≈′_ : Size → Conat ∞ → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ mˡ ∣ mʳ ] x ≈′ y = [ i ∣ mˡ ∣ mʳ ∣ zero ∣ zero ] x ≈′ y\n\n-- Quantitative expansion.\n\ninfix 4 [_∣_∣_]_≳_ [_∣_∣_]_≳′_ [_∣_]_≳_ [_∣_]_≳′_\n\n[_∣_∣_]_≳_ : Size → Conat ∞ → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ m ∣ n ] x ≳ y = [ i ∣ m ∣ zero ∣ n ∣ zero ] x ≈ y\n\n[_∣_∣_]_≳′_ : Size → Conat ∞ → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ m ∣ n ] x ≳′ y = [ i ∣ m ∣ zero ∣ n ∣ zero ] x ≈′ y\n\n[_∣_]_≳_ : Size → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ m ] x ≳ y = [ i ∣ m ∣ zero ] x ≳ y\n\n[_∣_]_≳′_ : Size → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ m ] x ≳′ y = [ i ∣ m ∣ zero ] x ≳′ y\n\n-- The converse of quantitative expansion.\n\ninfix 4 [_∣_∣_]_≲_ [_∣_∣_]_≲′_ [_∣_]_≲_ [_∣_]_≲′_\n\n[_∣_∣_]_≲_ : Size → Conat ∞ → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ m ∣ n ] x ≲ y = [ i ∣ m ∣ n ] y ≳ x\n\n[_∣_∣_]_≲′_ : Size → Conat ∞ → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ m ∣ n ] x ≲′ y = [ i ∣ m ∣ n ] y ≳′ x\n\n[_∣_]_≲_ : Size → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ m ] x ≲ y = [ i ∣ m ] y ≳ x\n\n[_∣_]_≲′_ : Size → Conat ∞ → Delay A ∞ → Delay A ∞ → Type a\n[ i ∣ m ] x ≲′ y = [ i ∣ m ] y ≳′ x\n\n------------------------------------------------------------------------\n-- Conversions\n\n-- Weakening.\n\nweaken :\n ∀ {i mˡ mˡ′ mʳ mʳ′ nˡ nˡ′ nʳ nʳ′ x y} →\n [ ∞ ] mˡ ≤ mˡ′ → [ ∞ ] mʳ ≤ mʳ′ →\n [ ∞ ] nˡ ≤ nˡ′ → [ ∞ ] nʳ ≤ nʳ′ →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n [ i ∣ mˡ′ ∣ mʳ′ ∣ nˡ′ ∣ nʳ′ ] x ≈ y\nweaken pˡ pʳ = λ where\n qˡ qʳ now → now\n (suc qˡ) qʳ (laterˡ r) → laterˡ (weaken pˡ pʳ (qˡ .force) qʳ r)\n qˡ (suc qʳ) (laterʳ r) → laterʳ (weaken pˡ pʳ qˡ (qʳ .force) r)\n qˡ qʳ (later r) → later λ { .force →\n weaken pˡ pʳ\n (qˡ Conat.+-mono pˡ)\n (qʳ Conat.+-mono pʳ)\n (r .force) }\n\nweakenˡʳ :\n ∀ {i mˡ mʳ nˡ nˡ′ nʳ nʳ′ x y} →\n [ ∞ ] nˡ ≤ nˡ′ → [ ∞ ] nʳ ≤ nʳ′ →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ′ ∣ nʳ′ ] x ≈ y\nweakenˡʳ = weaken (_ ∎≤) (_ ∎≤)\n\nweakenˡ :\n ∀ {i mˡ mʳ nˡ nˡ′ nʳ x y} →\n [ ∞ ] nˡ ≤ nˡ′ →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ′ ∣ nʳ ] x ≈ y\nweakenˡ p = weakenˡʳ p (_ ∎≤)\n\nweakenʳ :\n ∀ {i mˡ mʳ nˡ nʳ nʳ′ x y} →\n [ ∞ ] nʳ ≤ nʳ′ →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ′ ] x ≈ y\nweakenʳ p = weakenˡʳ (_ ∎≤) p\n\n-- Strong bisimilarity is contained in quantitative weak bisimilarity.\n\n∼→≈ : ∀ {i mˡ mʳ nˡ nʳ x y} →\n B.[ i ] x ∼ y → [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y\n∼→≈ now = now\n∼→≈ (later p) = later λ { .force → ∼→≈ (p .force) }\n\n-- Quantitative expansion is contained in expansion.\n\n≳→≳ : ∀ {i m n x y} →\n [ i ∣ m ∣ n ] x ≳ y → B.[ i ] x ≳ y\n≳→≳ now = now\n≳→≳ (later p) = later λ { .force → ≳→≳ (p .force) }\n≳→≳ (laterˡ p) = laterˡ (≳→≳ p)\n\n-- Quantitative weak bisimilarity is contained in weak bisimilarity.\n\n≈→≈ : ∀ {i mˡ mʳ nˡ nʳ x y} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y → B.[ i ] x ≈ y\n≈→≈ now = now\n≈→≈ (later p) = later λ { .force → ≈→≈ (p .force) }\n≈→≈ (laterˡ p) = laterˡ (≈→≈ p)\n≈→≈ (laterʳ p) = laterʳ (≈→≈ p)\n\n-- In some cases expansion is contained in quantitative expansion.\n\n≳→≳-steps :\n ∀ {m x y i} → B.[ i ] x ≳ y → [ i ∣ m ∣ steps x ] x ≳ y\n≳→≳-steps now = now\n≳→≳-steps (laterˡ p) = laterˡ (≳→≳-steps p)\n≳→≳-steps {m} (later {x = x} p) = later λ { .force →\n weakenˡ lemma (≳→≳-steps (p .force)) }\n where\n lemma =\n steps (x .force) ≤⟨ Conat.≤suc ⟩\n steps (later x) ≤⟨ Conat.m≤m+n ⟩\n steps (later x) + m ∎≤\n\n-- In some cases weak bisimilarity is contained in quantitative weak\n-- bisimilarity.\n\n≈→≈-steps :\n ∀ {mˡ mʳ x y i} →\n B.[ i ] x ≈ y → [ i ∣ mˡ ∣ mʳ ∣ steps x ∣ steps y ] x ≈ y\n≈→≈-steps now = now\n≈→≈-steps (laterˡ p) = laterˡ (≈→≈-steps p)\n≈→≈-steps (laterʳ p) = laterʳ (≈→≈-steps p)\n≈→≈-steps {mˡ} {mʳ} (later {x = x} {y = y} p) = later λ { .force →\n weakenˡʳ x-lemma y-lemma (≈→≈-steps (p .force)) }\n where\n x-lemma =\n steps (x .force) ≤⟨ Conat.≤suc ⟩\n steps (later x) ≤⟨ Conat.m≤m+n ⟩\n steps (later x) + mˡ ∎≤\n\n y-lemma =\n steps (y .force) ≤⟨ Conat.≤suc ⟩\n steps (later y) ≤⟨ Conat.m≤m+n ⟩\n steps (later y) + mʳ ∎≤\n\n-- In some cases quantitative weak bisimilarity is contained in strong\n-- bisimilarity.\n\nnever≈→∼ :\n ∀ {i mˡ mʳ nˡ nʳ x} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] never ≈ x → B.[ i ] never ∼ x\nnever≈→∼ (later p) = later λ { .force → never≈→∼ (p .force) }\nnever≈→∼ (laterˡ p) = never≈→∼ p\nnever≈→∼ (laterʳ p) = later λ { .force → never≈→∼ p }\n\n≈never→∼ :\n ∀ {i mˡ mʳ nˡ nʳ x} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ never → B.[ i ] x ∼ never\n≈never→∼ (later p) = later λ { .force → ≈never→∼ (p .force) }\n≈never→∼ (laterˡ p) = later λ { .force → ≈never→∼ p }\n≈never→∼ (laterʳ p) = ≈never→∼ p\n\n≈→∼ : ∀ {i x y} → [ i ∣ zero ∣ zero ] x ≈ y → B.[ i ] x ∼ y\n≈→∼ now = now\n≈→∼ (later p) = later λ { .force → ≈→∼ (p .force) }\n\n------------------------------------------------------------------------\n-- Reflexivity, symmetry/antisymmetry, transitivity\n\n-- Quantitative weak bisimilarity is reflexive.\n\nreflexive-≈ : ∀ {i mˡ mʳ nˡ nʳ} x → [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ x\nreflexive-≈ (now _) = now\nreflexive-≈ (later x) = later λ { .force → reflexive-≈ (x .force) }\n\n-- Quantitative weak bisimilarity is symmetric (in a certain sense).\n\nsymmetric-≈ :\n ∀ {i mˡ mʳ nˡ nʳ x y} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n [ i ∣ mʳ ∣ mˡ ∣ nʳ ∣ nˡ ] y ≈ x\nsymmetric-≈ now = now\nsymmetric-≈ (later p) = later λ { .force → symmetric-≈ (p .force) }\nsymmetric-≈ (laterˡ p) = laterʳ (symmetric-≈ p)\nsymmetric-≈ (laterʳ p) = laterˡ (symmetric-≈ p)\n\n-- Four variants of transitivity.\n\ntransitive-≳∼ :\n ∀ {i m n x y z} →\n [ i ∣ m ∣ n ] x ≳ y → B.[ i ] y ∼ z → [ i ∣ m ∣ n ] x ≳ z\ntransitive-≳∼ = λ where\n now now → now\n (laterˡ p) q → laterˡ (transitive-≳∼ p q)\n (later p) (later q) → later λ { .force →\n transitive-≳∼ (p .force) (q .force) }\n\ntransitive-∼≲ :\n ∀ {i m n x y z} →\n B.[ i ] x ∼ y → [ i ∣ m ∣ n ] y ≲ z → [ i ∣ m ∣ n ] x ≲ z\ntransitive-∼≲ p q = transitive-≳∼ q (B.symmetric p)\n\ntransitive-≈∼ :\n ∀ {i mˡ mʳ nˡ nʳ x y z} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y → y B.∼ z →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ z\ntransitive-≈∼ = λ where\n now now → now\n (later p) (later q) → later λ { .force →\n transitive-≈∼ (p .force) (q .force) }\n (laterˡ p) q → laterˡ (transitive-≈∼ p q)\n (laterʳ p) (later q) → laterʳ (transitive-≈∼ p (q .force))\n\ntransitive-∼≈ :\n ∀ {i mˡ mʳ nˡ nʳ x y z} →\n x B.∼ y → [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] y ≈ z →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ z\ntransitive-∼≈ = λ where\n now now → now\n (later p) (later q) → later λ { .force →\n transitive-∼≈ (p .force) (q .force) }\n (later p) (laterˡ q) → laterˡ (transitive-∼≈ (p .force) q)\n p (laterʳ q) → laterʳ (transitive-∼≈ p q)\n\n-- Equational reasoning combinators.\n\ninfix -1 _∎ˢ\ninfixr -2 step-∼≈ˢ step-≳∼ˢ step-≈∼ˢ _≳⟨⟩ˢ_ step-≡≈ˢ _∼⟨⟩ˢ_\n\n_∎ˢ : ∀ {i mˡ mʳ nˡ nʳ} x → [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ x\n_∎ˢ = reflexive-≈\n\nstep-∼≈ˢ : ∀ {i mˡ mʳ nˡ nʳ} x {y z} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] y ≈ z → x B.∼ y →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ z\nstep-∼≈ˢ _ y≈z x∼y = transitive-∼≈ x∼y y≈z\n\nsyntax step-∼≈ˢ x y≈z x∼y = x ∼⟨ x∼y ⟩ˢ y≈z\n\nstep-≳∼ˢ : ∀ {i m n} x {y z} →\n B.[ i ] y ∼ z → [ i ∣ m ∣ n ] x ≳ y →\n [ i ∣ m ∣ n ] x ≳ z\nstep-≳∼ˢ _ y∼z x≳y = transitive-≳∼ x≳y y∼z\n\nsyntax step-≳∼ˢ x y∼z x≳y = x ≳⟨ x≳y ⟩ˢ y∼z\n\nstep-≈∼ˢ : ∀ {i mˡ mʳ nˡ nʳ} x {y z} →\n y B.∼ z → [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ z\nstep-≈∼ˢ _ y∼z x≈y = transitive-≈∼ x≈y y∼z\n\nsyntax step-≈∼ˢ x y∼z x≈y = x ≈⟨ x≈y ⟩ˢ y∼z\n\n_≳⟨⟩ˢ_ : ∀ {i mˡ mʳ nˡ nʳ} x {y} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] drop-later x ≈ y →\n [ i ∣ mˡ ∣ mʳ ∣ ⌜ 1 ⌝ + nˡ ∣ nʳ ] x ≈ y\nnow _ ≳⟨⟩ˢ p = weakenˡ Conat.≤suc p\nlater _ ≳⟨⟩ˢ p = laterˡ p\n\nstep-≡≈ˢ : ∀ {i mˡ mʳ nˡ nʳ} x {y z} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] y ≈ z → x ≡ y →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ z\nstep-≡≈ˢ _ y≈z refl = y≈z\n\nsyntax step-≡≈ˢ x y≈z x≡y = x ≡⟨ x≡y ⟩ˢ y≈z\n\n_∼⟨⟩ˢ_ : ∀ {i mˡ mʳ nˡ nʳ} x {y} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y\n_ ∼⟨⟩ˢ x≈y = x≈y\n\n------------------------------------------------------------------------\n-- Some results related to the steps function\n\n-- If y is a quantitative expansion of x, then it contains at least as\n-- many later constructors as x.\n\nsteps-mono :\n ∀ {i m n x y} → [ i ∣ m ∣ n ] x ≲ y → [ i ] steps x ≤ steps y\nsteps-mono = B.steps-mono ∘ ≳→≳\n\n-- If [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y holds, then the number of later\n-- constructors in x is bounded by nˡ plus 1 + mˡ times the number of\n-- later constructors in y.\n\nsteps-+-*ʳ :\n ∀ {mˡ mʳ nˡ nʳ i x y} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n [ i ] steps x ≤ nˡ + (⌜ 1 ⌝ + mˡ) * steps y\nsteps-+-*ʳ {mˡ} {mʳ} {nˡ} {nʳ} = λ where\n now → zero\n\n (later {x = x} {y = y} p) →\n steps (later x) ∼⟨ suc (λ { .force → _ ∎∼ }) ⟩≤\n suc (λ { .force → steps (x .force) }) ≤⟨ suc (λ { .force → steps-+-*ʳ (p .force) }) ⟩\n suc (λ { .force → nˡ + mˡ + (⌜ 1 ⌝ + mˡ) * steps (y .force) }) ∼⟨ suc (λ { .force → Conat.symmetric-∼ (Conat.+-assoc nˡ) }) ⟩≤\n ⌜ 1 ⌝ + nˡ + (mˡ + (⌜ 1 ⌝ + mˡ) * steps (y .force)) ∼⟨ Conat.suc+∼+suc ⟩≤\n nˡ + ((⌜ 1 ⌝ + mˡ) + (⌜ 1 ⌝ + mˡ) * steps (y .force)) ∼⟨ (nˡ ∎∼) Conat.+-cong Conat.symmetric-∼ Conat.*suc∼+* ⟩≤\n nˡ + (⌜ 1 ⌝ + mˡ) * steps (later y) ∎≤\n\n (laterˡ {x = x} {y = y} {nˡ = nˡ} p) →\n steps (later x) ∼⟨ suc (λ { .force → _ ∎∼ }) ⟩≤\n ⌜ 1 ⌝ + steps (x .force) ≤⟨ (_ Conat.∎≤) Conat.+-mono steps-+-*ʳ p ⟩\n ⌜ 1 ⌝ + (nˡ .force + (⌜ 1 ⌝ + mˡ) * steps y) ∼⟨ suc (λ { .force → _ ∎∼ }) ⟩≤\n suc nˡ + (⌜ 1 ⌝ + mˡ) * steps y ∎≤\n\n (laterʳ {x = x} {y = y} {nʳ = nʳ} p) →\n steps x ≤⟨ steps-+-*ʳ p ⟩\n nˡ + (⌜ 1 ⌝ + mˡ) * steps (y .force) ≤⟨ (nˡ ∎≤) Conat.+-mono Conat.m≤n+m ⟩\n nˡ + ((⌜ 1 ⌝ + mˡ) + (⌜ 1 ⌝ + mˡ) * steps (y .force)) ∼⟨ (nˡ ∎∼) Conat.+-cong Conat.symmetric-∼ Conat.*suc∼+* ⟩≤\n nˡ + (⌜ 1 ⌝ + mˡ) * steps (later y) ∎≤\n\n-- If [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y holds, then the number of later\n-- constructors in y is bounded by nʳ plus 1 + mʳ times the number of\n-- later constructors in x.\n\nsteps-+-*ˡ :\n ∀ {mˡ mʳ nˡ nʳ i x y} →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n [ i ] steps y ≤ nʳ + (⌜ 1 ⌝ + mʳ) * steps x\nsteps-+-*ˡ = steps-+-*ʳ ∘ symmetric-≈\n\n-- [ ∞ ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y holds iff x and y are weakly\n-- bisimilar and the number of later constructors in x and y are\n-- related in a certain way.\n\n≈⇔≈×steps≤steps² :\n ∀ {mˡ mʳ nˡ nʳ x y} →\n [ ∞ ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y ⇔\n x B.≈ y ×\n [ ∞ ] steps x ≤ nˡ + (⌜ 1 ⌝ + mˡ) * steps y ×\n [ ∞ ] steps y ≤ nʳ + (⌜ 1 ⌝ + mʳ) * steps x\n≈⇔≈×steps≤steps² {mˡ} {mʳ} {x = x} {y} = record\n { to = λ p → ≈→≈ p , steps-+-*ʳ p , steps-+-*ˡ p\n ; from = λ { (p , q , r) → from p q r }\n }\n where\n from-lemma :\n ∀ {m n} {x y : Delay′ A ∞} {i} {j : Size< i} →\n [ i ] steps (later x) ≤ n + (⌜ 1 ⌝ + m) * steps (later y) →\n [ j ] steps (x .force) ≤ n + m + (⌜ 1 ⌝ + m) * steps (y .force)\n from-lemma {m} {n} {x} {y} hyp = Conat.cancel-suc-≤ lemma .force\n where\n lemma =\n ⌜ 1 ⌝ + steps (x .force) ∼⟨ suc (λ { .force → _ ∎∼ }) ⟩≤\n steps (later x) ≤⟨ hyp ⟩\n n + (⌜ 1 ⌝ + m) * steps (later y) ∼⟨ (n ∎∼) Conat.+-cong Conat.*suc∼+* ⟩≤\n n + ((⌜ 1 ⌝ + m) + (⌜ 1 ⌝ + m) * steps (y .force)) ∼⟨ Conat.symmetric-∼ Conat.suc+∼+suc ⟩≤\n ⌜ 1 ⌝ + n + (m + (⌜ 1 ⌝ + m) * steps (y .force)) ∼⟨ suc (λ { .force → _ ∎∼ }) ⟩≤\n ⌜ 1 ⌝ + (n + (m + (⌜ 1 ⌝ + m) * steps (y .force))) ∼⟨ suc (λ { .force → Conat.+-assoc n }) ⟩≤\n ⌜ 1 ⌝ + (n + m + (⌜ 1 ⌝ + m) * steps (y .force)) ∎≤\n\n from :\n ∀ {nˡ nʳ i x y} →\n B.[ i ] x ≈ y →\n [ ∞ ] steps x ≤ nˡ + (⌜ 1 ⌝ + mˡ) * steps y →\n [ ∞ ] steps y ≤ nʳ + (⌜ 1 ⌝ + mʳ) * steps x →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y\n from now _ _ = now\n\n from (later p) q r = later λ { .force →\n from (p .force) (from-lemma q) (from-lemma r) }\n\n from (laterˡ {y = later _} p) q r = later λ { .force →\n from (B.laterʳ⁻¹ p) (from-lemma q) (from-lemma r) }\n\n from (laterʳ {x = later _} p) q r = later λ { .force →\n from (B.laterˡ⁻¹ p) (from-lemma q) (from-lemma r) }\n\n from {nˡ = zero} (laterˡ {y = now _} p) ()\n from {nˡ = suc _} (laterˡ {y = now _} p) (suc q) _ =\n laterˡ (from p (q .force) zero)\n\n from {nʳ = zero} (laterʳ {x = now _} p) _ ()\n from {nʳ = suc _} (laterʳ {x = now _} p) _ (suc r) =\n laterʳ (from p zero (r .force))\n\n-- [ ∞ ∣ m ∣ n ] x ≳ y holds iff x is an expansion of y and the number\n-- of later constructors in x and y are related in a certain way.\n\n≳⇔≳×steps≤steps :\n ∀ {m n x y} →\n [ ∞ ∣ m ∣ n ] x ≳ y ⇔\n x B.≳ y × [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y\n≳⇔≳×steps≤steps {m} {n} {x} {y} =\n [ ∞ ∣ m ∣ n ] x ≳ y ↝⟨ ≈⇔≈×steps≤steps² ⟩\n\n x B.≈ y ×\n [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y ×\n [ ∞ ] steps y ≤ (⌜ 1 ⌝ + ⌜ 0 ⌝) * steps x ↝⟨ F.id ×-cong F.id ×-cong (_ ∎∼) Conat.≤-cong-∼ lemma ⟩\n\n x B.≈ y ×\n [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y ×\n [ ∞ ] steps y ≤ steps x ↝⟨ record { to = B.symmetric; from = B.symmetric } ×-cong from-isomorphism ×-comm ⟩\n\n y B.≈ x ×\n [ ∞ ] steps y ≤ steps x ×\n [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y ↔⟨ ×-assoc ⟩\n\n (y B.≈ x ×\n [ ∞ ] steps y ≤ steps x) ×\n [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y ↝⟨ inverse B.≲⇔≈×steps≤steps ×-cong F.id ⟩□\n\n x B.≳ y × [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y □\n where\n lemma =\n (⌜ 1 ⌝ + ⌜ 0 ⌝) * steps x ∼⟨ suc (λ { .force → _ ∎∼ }) Conat.*-cong (_ ∎∼) ⟩\n ⌜ 1 ⌝ * steps x ∼⟨ Conat.*-left-identity _ ⟩\n steps x ∎∼\n\n-- [ ∞ ∣ m ∣ n ] x ≳ y holds iff x is weakly bisimilar to y and the\n-- number of later constructors in x and y are related in a certain\n-- way.\n\n≳⇔≈×steps≤steps² :\n ∀ {m n x y} →\n [ ∞ ∣ m ∣ n ] x ≳ y ⇔\n x B.≈ y ×\n [ ∞ ] steps y ≤ steps x ×\n [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y\n≳⇔≈×steps≤steps² {m} {n} {x} {y} =\n [ ∞ ∣ m ∣ n ] x ≳ y ↝⟨ ≳⇔≳×steps≤steps ⟩\n\n x B.≳ y ×\n [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y ↝⟨ B.≲⇔≈×steps≤steps ×-cong F.id ⟩\n\n (y B.≈ x ×\n [ ∞ ] steps y ≤ steps x) ×\n [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y ↔⟨ inverse ×-assoc ⟩\n\n y B.≈ x ×\n [ ∞ ] steps y ≤ steps x ×\n [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y ↝⟨ record { to = B.symmetric; from = B.symmetric } ×-cong F.id ⟩□\n\n x B.≈ y ×\n [ ∞ ] steps y ≤ steps x ×\n [ ∞ ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y □\n\n-- The left-to-right direction of ≳⇔≳×steps≤steps can be made\n-- size-preserving.\n\n≳→≳×steps≤steps :\n ∀ {i m n x y} →\n [ i ∣ m ∣ n ] x ≳ y →\n B.[ i ] x ≳ y × [ i ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y\n≳→≳×steps≤steps x≳y = ≳→≳ x≳y , steps-+-*ʳ x≳y\n\n-- The right-to-left direction of ≳⇔≳×steps≤steps can be made\n-- size-preserving iff A is uninhabited.\n\n≳×steps≤steps→≳⇔uninhabited :\n (∀ {i m n x y} →\n B.[ i ] x ≳ y ×\n [ i ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y →\n [ i ∣ m ∣ n ] x ≳ y)\n ⇔\n ¬ A\n≳×steps≤steps→≳⇔uninhabited = record\n { to = flip to\n ; from =\n ¬ A ↝⟨ (λ ¬A {_ _ _ _ _} → ∼→≈ (B.uninhabited→trivial ¬A _ _)) ⟩\n\n (∀ {i m n x y} → [ i ∣ m ∣ n ] x ≳ y) ↝⟨ (λ hyp {_ _ _ _ _} _ → hyp {_}) ⟩□\n\n (∀ {i m n x y} →\n B.[ i ] x ≳ y ×\n [ i ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y →\n [ i ∣ m ∣ n ] x ≳ y) □\n }\n where\n strengthen-≳now :\n ∀ {i m n x y} →\n [ i ∣ m ∣ n ] x ≳ now y →\n [ ∞ ∣ m ∣ n ] x ≳ now y\n strengthen-≳now now = now\n strengthen-≳now (laterˡ p) = laterˡ (strengthen-≳now p)\n\n to :\n A →\n ¬ (∀ {i m n x y} →\n B.[ i ] x ≳ y ×\n [ i ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y →\n [ i ∣ m ∣ n ] x ≳ y)\n to x =\n (∀ {i m n x y} →\n B.[ i ] x ≳ y ×\n [ i ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y →\n [ i ∣ m ∣ n ] x ≳ y) ↝⟨ (λ hyp → curry hyp) ⟩\n\n (∀ {i m n} →\n B.[ i ] f m ≳ now x →\n [ i ] steps (f m) ≤ n + zero →\n [ i ∣ zero ∣ n ] f m ≳ now x) ↝⟨ (λ hyp {_ _ _} p → hyp (≳now _) (complicate p)) ⟩\n\n (∀ {i m n} → [ i ] ⌜ m ⌝ ≤ n → [ i ∣ zero ∣ n ] f m ≳ now x) ↝⟨ strengthen-≳now ∘_ ⟩\n\n (∀ {i m n} → [ i ] ⌜ m ⌝ ≤ n → [ ∞ ∣ zero ∣ n ] f m ≳ now x) ↝⟨ (λ hyp {_ _ _} p → steps-+-*ʳ (hyp p)) ⟩\n\n (∀ {i m n} → [ i ] ⌜ m ⌝ ≤ n → [ ∞ ] steps (f m) ≤ n + zero) ↝⟨ (λ hyp {_ _ _} p → simplify (hyp p)) ⟩\n\n (∀ {i m n} → [ i ] ⌜ m ⌝ ≤ n → [ ∞ ] ⌜ m ⌝ ≤ n) ↝⟨ (λ hyp → hyp) ⟩\n\n (∀ {i} → [ i ] ⌜ 2 ⌝ ≤ ⌜ 1 ⌝ → [ ∞ ] ⌜ 2 ⌝ ≤ ⌜ 1 ⌝) ↝⟨ Conat.no-strengthening-≤-21 ⟩□\n\n ⊥ □\n where\n f : ∀ {i} → ℕ → Delay A i\n f zero = now x\n f (suc n) = later λ { .force → f n }\n\n ≳now : ∀ {i} n → B.[ i ] f n ≳ now x\n ≳now zero = now\n ≳now (suc n) = laterˡ (≳now n)\n\n ∼steps : ∀ {i} n → Conat.[ i ] ⌜ n ⌝ ∼ steps (f n)\n ∼steps zero = zero\n ∼steps (suc n) = suc λ { .force → ∼steps n }\n\n complicate :\n ∀ {m n i} → [ i ] ⌜ m ⌝ ≤ n → [ i ] steps (f m) ≤ n + zero\n complicate {m} {n} p =\n steps (f m) ∼⟨ Conat.symmetric-∼ (∼steps m) ⟩≤\n ⌜ m ⌝ ≤⟨ p ⟩\n n ∼⟨ Conat.symmetric-∼ (Conat.+-right-identity _) ⟩≤\n n + zero ∎≤\n\n simplify :\n ∀ {m n i} → [ i ] steps (f m) ≤ n + zero → [ i ] ⌜ m ⌝ ≤ n\n simplify {m} {n} p =\n ⌜ m ⌝ ∼⟨ ∼steps m ⟩≤\n steps (f m) ≤⟨ p ⟩\n n + zero ∼⟨ Conat.+-right-identity _ ⟩≤\n n ∎≤\n\n-- The left-to-right direction of ≈⇔≈×steps≤steps² can be made\n-- size-preserving.\n\n≈→≈×steps≤steps² :\n ∀ {i mˡ mʳ nˡ nʳ x y} →\n [ ∞ ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y →\n B.[ i ] x ≈ y ×\n [ i ] steps x ≤ nˡ + (⌜ 1 ⌝ + mˡ) * steps y ×\n [ i ] steps y ≤ nʳ + (⌜ 1 ⌝ + mʳ) * steps x\n≈→≈×steps≤steps² x≈y = ≈→≈ x≈y , steps-+-*ʳ x≈y , steps-+-*ˡ x≈y\n\n-- The right-to-left direction of ≈⇔≈×steps≤steps² can be made\n-- size-preserving iff A is uninhabited.\n\n≈×steps≤steps²→≈⇔uninhabited :\n (∀ {i mˡ mʳ nˡ nʳ x y} →\n B.[ i ] x ≈ y ×\n [ i ] steps x ≤ nˡ + (⌜ 1 ⌝ + mˡ) * steps y ×\n [ i ] steps y ≤ nʳ + (⌜ 1 ⌝ + mʳ) * steps x →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y)\n ⇔\n ¬ A\n≈×steps≤steps²→≈⇔uninhabited = record\n { to =\n (∀ {i mˡ mʳ nˡ nʳ x y} →\n B.[ i ] x ≈ y ×\n [ i ] steps x ≤ nˡ + (⌜ 1 ⌝ + mˡ) * steps y ×\n [ i ] steps y ≤ nʳ + (⌜ 1 ⌝ + mʳ) * steps x →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y) ↝⟨ (λ { hyp (p , q) → hyp (B.≳→ p , q , lemma (B.steps-mono p)) }) ⟩\n\n (∀ {i m n x y} →\n B.[ i ] x ≳ y ×\n [ i ] steps x ≤ n + (⌜ 1 ⌝ + m) * steps y →\n [ i ∣ m ∣ n ] x ≳ y) ↝⟨ _⇔_.to ≳×steps≤steps→≳⇔uninhabited ⟩□\n\n ¬ A □\n ; from =\n ¬ A ↝⟨ (λ ¬A {_ _ _ _ _} → ∼→≈ (B.uninhabited→trivial ¬A _ _)) ⟩\n\n (∀ {i mˡ mʳ nˡ nʳ x y} → [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y) ↝⟨ (λ hyp {_ _ _ _ _ _ _} _ → hyp {_}) ⟩□\n\n (∀ {i mˡ mʳ nˡ nʳ x y} →\n B.[ i ] x ≈ y ×\n [ i ] steps x ≤ nˡ + (⌜ 1 ⌝ + mˡ) * steps y ×\n [ i ] steps y ≤ nʳ + (⌜ 1 ⌝ + mʳ) * steps x →\n [ i ∣ mˡ ∣ mʳ ∣ nˡ ∣ nʳ ] x ≈ y) □\n }\n where\n lemma :\n ∀ {m n i} →\n [ i ] m ≤ n →\n [ i ] m ≤ (⌜ 1 ⌝ + ⌜ 0 ⌝) * n\n lemma {m} {n} p =\n m ≤⟨ p ⟩\n n ∼⟨ Conat.symmetric-∼ (Conat.*-left-identity _) ⟩≤\n ⌜ 1 ⌝ * n ∼⟨ Conat.symmetric-∼ (Conat.+-right-identity _) Conat.*-cong (_ ∎∼) ⟩≤\n (⌜ 1 ⌝ + ⌜ 0 ⌝) * n ∎≤\n", "meta": {"hexsha": "bcf8ff43241d79983f8c1d4dc62013c3272e2112", "size": 22824, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Delay-monad/Quantitative-weak-bisimilarity.agda", "max_stars_repo_name": "nad/delay-monad", "max_stars_repo_head_hexsha": "495f9996673d0f1f34ce202902daaa6c39f8925e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/Delay-monad/Quantitative-weak-bisimilarity.agda", "max_issues_repo_name": "nad/delay-monad", "max_issues_repo_head_hexsha": "495f9996673d0f1f34ce202902daaa6c39f8925e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Delay-monad/Quantitative-weak-bisimilarity.agda", "max_forks_repo_name": "nad/delay-monad", "max_forks_repo_head_hexsha": "495f9996673d0f1f34ce202902daaa6c39f8925e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.0061349693, "max_line_length": 138, "alphanum_fraction": 0.4198212408, "num_tokens": 11812, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.344623109741221}} {"text": "------------------------------------------------------------------------\n-- Total recognisers which can handle left recursion\n------------------------------------------------------------------------\n\n-- The recognisers are parametrised on the alphabet.\n\nmodule TotalRecognisers.LeftRecursion (Tok : Set) where\n\nopen import Algebra\nopen import Codata.Musical.Notation\nopen import Data.Bool as Bool hiding (_∧_; _≤_)\nimport Data.Bool.Properties as Bool\nprivate\n module BoolCS = CommutativeSemiring Bool.∧-∨-commutativeSemiring\nopen import Function.Base\nopen import Function.Equality using (_⟨$⟩_)\nopen import Function.Equivalence as Eq\n using (_⇔_; equivalence; module Equivalence)\n renaming (_∘_ to _⟨∘⟩_)\nopen import Data.List using (List; []; _∷_; _++_; [_])\nimport Data.List.Properties\nprivate\n module ListMonoid {A : Set} =\n Monoid (Data.List.Properties.++-monoid A)\nopen import Data.Product as Prod\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable as Decidable\n\n------------------------------------------------------------------------\n-- A \"right-strict\" variant of _∧_\n\n-- If the left-strict variant of _∧_ were used to type _·_ below, then\n-- the inferred definition of D-nullable would not be total; it would\n-- contain expressions of the form \"D-nullable t (♭ p₁) ∧ false\". With\n-- the right-strict definition of _∧_ such expressions reduce to\n-- \"false\".\n\ninfixr 6 _∧_\n\n_∧_ : Bool → Bool → Bool\nb ∧ true = b\nb ∧ false = false\n\n-- A lemma.\n\nleft-zero : ∀ b → false ∧ b ≡ false\nleft-zero true = refl\nleft-zero false = refl\n\n------------------------------------------------------------------------\n-- Recogniser combinators\n\ninfixl 10 _·_\ninfixl 5 _∣_\n\nmutual\n\n -- The index is true if the corresponding language contains the empty\n -- string (is nullable).\n\n data P : Bool → Set where\n fail : P false\n empty : P true\n sat : (Tok → Bool) → P false\n _∣_ : ∀ {n₁ n₂} → P n₁ → P n₂ → P (n₁ ∨ n₂)\n _·_ : ∀ {n₁ n₂} → ∞⟨ n₂ ⟩P n₁ → ∞⟨ n₁ ⟩P n₂ → P (n₁ ∧ n₂)\n nonempty : ∀ {n} → P n → P false\n cast : ∀ {n₁ n₂} → n₁ ≡ n₂ → P n₁ → P n₂\n\n -- Delayed if the index is /false/.\n\n ∞⟨_⟩P : Bool → Bool → Set\n ∞⟨ false ⟩P n = ∞ (P n)\n ∞⟨ true ⟩P n = P n\n\n-- Note that fail, nonempty and cast could be defined as derived\n-- combinators. (For cast this is obvious, fail could be defined\n-- either using sat or the combinator leftRight below, and nonempty is\n-- defined in the module AlternativeNonempty. Note also that the proof\n-- in TotalRecognisers.LeftRecursion.ExpressiveStrength does not rely\n-- on these constructors.) However, Agda uses /guarded/ corecursion,\n-- so the fact that nonempty and cast are constructors can be very\n-- convenient when constructing other recognisers.\n\n-- For an example of the use of nonempty, see the Kleene star example\n-- in TotalRecognisers.LeftRecursion.Lib. For examples of the use of\n-- cast, see TotalRecognisers.LeftRecursion.ExpressiveStrength and\n-- TotalRecognisers.LeftRecursion.NotOnlyContextFree.\n\n------------------------------------------------------------------------\n-- Helpers\n\n♭? : ∀ {b n} → ∞⟨ b ⟩P n → P n\n♭? {b = false} x = ♭ x\n♭? {b = true} x = x\n\n♯? : ∀ {b n} → P n → ∞⟨ b ⟩P n\n♯? {b = false} x = ♯ x\n♯? {b = true} x = x\n\nforced? : ∀ {b n} → ∞⟨ b ⟩P n → Bool\nforced? {b = b} _ = b\n\n-- A lemma.\n\n♭?♯? : ∀ b {n} {p : P n} → ♭? {b} (♯? p) ≡ p\n♭?♯? false = refl\n♭?♯? true = refl\n\n------------------------------------------------------------------------\n-- Semantics\n\n-- The semantics is defined inductively: s ∈ p iff the string s is\n-- contained in the language defined by p.\n\ninfix 4 _∈_\n\ndata _∈_ : ∀ {n} → List Tok → P n → Set where\n empty : [] ∈ empty\n sat : ∀ {f t} → T (f t) → [ t ] ∈ sat f\n ∣-left : ∀ {s n₁ n₂} {p₁ : P n₁} {p₂ : P n₂} →\n s ∈ p₁ → s ∈ p₁ ∣ p₂\n ∣-right : ∀ {s n₁ n₂} {p₁ : P n₁} {p₂ : P n₂} →\n s ∈ p₂ → s ∈ p₁ ∣ p₂\n _·_ : ∀ {s₁ s₂ n₁ n₂}\n {p₁ : ∞⟨ n₂ ⟩P n₁} {p₂ : ∞⟨ n₁ ⟩P n₂} →\n s₁ ∈ ♭? p₁ → s₂ ∈ ♭? p₂ → s₁ ++ s₂ ∈ p₁ · p₂\n nonempty : ∀ {n t s} {p : P n} →\n t ∷ s ∈ p → t ∷ s ∈ nonempty p\n cast : ∀ {n₁ n₂ s} {p : P n₁} {eq : n₁ ≡ n₂} →\n s ∈ p → s ∈ cast eq p\n\ninfix 4 _≤_ _≈_\n\n-- p₁ ≤ p₂ iff the language (defined by) p₂ contains all the strings\n-- in the language p₁.\n\n_≤_ : ∀ {n₁ n₂} → P n₁ → P n₂ → Set\np₁ ≤ p₂ = ∀ {s} → s ∈ p₁ → s ∈ p₂\n\n-- p₁ ≈ p₂ iff the languages p₁ and p₂ contain the same strings.\n\n_≈_ : ∀ {n₁ n₂} → P n₁ → P n₂ → Set\np₁ ≈ p₂ = ∀ {s} → s ∈ p₁ ⇔ s ∈ p₂\n\n-- p₁ ≈ p₂ iff both p₁ ≤ p₂ and p₂ ≤ p₁.\n\n≈⇔≤≥ : ∀ {n₁ n₂} {p₁ : P n₁} {p₂ : P n₂} →\n p₁ ≈ p₂ ⇔ (p₁ ≤ p₂ × p₂ ≤ p₁)\n≈⇔≤≥ = equivalence\n (λ p₁≈p₂ → ((λ {s} → _⟨$⟩_ (Equivalence.to (p₁≈p₂ {s = s})))\n , λ {s} → _⟨$⟩_ (Equivalence.from (p₁≈p₂ {s = s}))))\n (λ p₁≤≥p₂ {s} → equivalence (proj₁ p₁≤≥p₂ {s = s})\n (proj₂ p₁≤≥p₂ {s = s}))\n\n-- Some lemmas.\n\ncast∈ : ∀ {n} {p p′ : P n} {s s′} → s ≡ s′ → p ≡ p′ → s ∈ p → s′ ∈ p′\ncast∈ refl refl s∈ = s∈\n\ndrop-♭♯ : ∀ n {n′} {p : P n′} → ♭? (♯? {n} p) ≤ p\ndrop-♭♯ n = cast∈ refl (♭?♯? n)\n\nadd-♭♯ : ∀ n {n′} {p : P n′} → p ≤ ♭? (♯? {n} p)\nadd-♭♯ n = cast∈ refl (sym $ ♭?♯? n)\n\n------------------------------------------------------------------------\n-- Example: A definition which is left and right recursive\n\nleftRight : P false\nleftRight = ♯ leftRight · ♯ leftRight\n\n-- Note that leftRight is equivalent to fail, so fail does not need to\n-- be a primitive combinator.\n\nleftRight≈fail : leftRight ≈ fail\nleftRight≈fail = equivalence ≤fail (λ ())\n where\n ≤fail : ∀ {s A} → s ∈ leftRight → A\n ≤fail (∈₁ · ∈₂) = ≤fail ∈₁\n\n-- For more examples, see TotalRecognisers.LeftRecursion.Lib.\n\n------------------------------------------------------------------------\n-- Nullability\n\n-- The nullability index is correct.\n\n⇒ : ∀ {n} {p : P n} → [] ∈ p → n ≡ true\n⇒ pr = ⇒′ pr refl\n where\n ⇒′ : ∀ {n s} {p : P n} → s ∈ p → s ≡ [] → n ≡ true\n ⇒′ empty refl = refl\n ⇒′ (sat _) ()\n ⇒′ (∣-left pr₁) refl with ⇒ pr₁\n ⇒′ (∣-left pr₁) refl | refl = refl\n ⇒′ (∣-right pr₂) refl with ⇒ pr₂\n ⇒′ (∣-right {n₁ = n₁} pr₂) refl | refl = proj₂ BoolCS.zero n₁\n ⇒′ (nonempty p) ()\n ⇒′ (cast {eq = refl} p) refl = ⇒′ p refl\n ⇒′ (_·_ {[]} pr₁ pr₂) refl = cong₂ _∧_ (⇒ pr₁) (⇒ pr₂)\n ⇒′ (_·_ {_ ∷ _} pr₁ pr₂) ()\n\n⇐ : ∀ {n} (p : P n) → n ≡ true → [] ∈ p\n⇐ fail ()\n⇐ empty refl = empty\n⇐ (sat f) ()\n⇐ (_∣_ {true} p₁ p₂) refl = ∣-left (⇐ p₁ refl)\n⇐ (_∣_ {false} {true} p₁ p₂) refl = ∣-right {p₁ = p₁} (⇐ p₂ refl)\n⇐ (_∣_ {false} {false} p₁ p₂) ()\n⇐ (nonempty p) ()\n⇐ (cast refl p) refl = cast (⇐ p refl)\n⇐ (_·_ {.true} {true} p₁ p₂) refl = ⇐ p₁ refl · ⇐ p₂ refl\n⇐ (_·_ {_} {false} p₁ p₂) ()\n\nindex-correct : ∀ {n} {p : P n} → [] ∈ p ⇔ n ≡ true\nindex-correct = equivalence ⇒ (⇐ _)\n\n-- We can decide if the empty string belongs to a given language.\n\nnullable? : ∀ {n} (p : P n) → Dec ([] ∈ p)\nnullable? {n} p = Decidable.map (Eq.sym index-correct) (Bool._≟_ n true)\n\n------------------------------------------------------------------------\n-- Derivative\n\n-- The index of the derivative.\n\nD-nullable : ∀ {n} → Tok → P n → Bool\nD-nullable t fail = false\nD-nullable t empty = false\nD-nullable t (sat f) = f t\nD-nullable t (p₁ ∣ p₂) = D-nullable t p₁ ∨ D-nullable t p₂\nD-nullable t (nonempty p) = D-nullable t p\nD-nullable t (cast _ p) = D-nullable t p\nD-nullable t (p₁ · p₂) with forced? p₁ | forced? p₂\n... | true | false = D-nullable t p₁\n... | false | false = false\n... | true | true = D-nullable t p₁ ∨ D-nullable t p₂\n... | false | true = D-nullable t p₂\n\n-- D t p is the \"derivative\" of p with respect to t. It is specified\n-- by the equivalence s ∈ D t p ⇔ t ∷ s ∈ p (proved below).\n\nD : ∀ {n} (t : Tok) (p : P n) → P (D-nullable t p)\nD t fail = fail\nD t empty = fail\nD t (sat f) with f t\n... | true = empty\n... | false = fail\nD t (p₁ ∣ p₂) = D t p₁ ∣ D t p₂\nD t (nonempty p) = D t p\nD t (cast _ p) = D t p\nD t (p₁ · p₂) with forced? p₁ | forced? p₂\n... | true | false = D t p₁ · ♯? (♭ p₂)\n... | false | false = ♯ D t (♭ p₁) · ♯? (♭ p₂)\n... | true | true = D t p₁ · ♯? p₂ ∣ D t p₂\n... | false | true = ♯ D t (♭ p₁) · ♯? p₂ ∣ D t p₂\n\n-- D is correct.\n\nD-sound : ∀ {n s t} {p : P n} → s ∈ D t p → t ∷ s ∈ p\nD-sound s∈ = D-sound′ _ _ s∈\n where\n sat-lemma : ∀ {s} f t → s ∈ D t (sat f) → T (f t) × s ≡ []\n sat-lemma f t ∈ with f t\n sat-lemma f t empty | true = (_ , refl)\n sat-lemma f t () | false\n\n D-sound′ : ∀ {s n} (p : P n) t → s ∈ D t p → t ∷ s ∈ p\n D-sound′ fail t ()\n D-sound′ empty t ()\n D-sound′ (sat f) t s∈ with sat-lemma f t s∈\n ... | (ok , refl) = sat ok\n D-sound′ (p₁ ∣ p₂) t (∣-left ∈₁) = ∣-left (D-sound′ p₁ t ∈₁)\n D-sound′ (p₁ ∣ p₂) t (∣-right ∈₂) = ∣-right {p₁ = p₁} (D-sound′ p₂ t ∈₂)\n D-sound′ (nonempty p) t ∈ = nonempty (D-sound ∈)\n D-sound′ (cast _ p) t ∈ = cast (D-sound ∈)\n D-sound′ (p₁ · p₂) t s∈ with forced? p₁ | forced? p₂\n D-sound′ (p₁ · p₂) t (∣-left (∈₁ · ∈₂)) | true | true = D-sound ∈₁ · drop-♭♯ (D-nullable t p₁) ∈₂\n D-sound′ (p₁ · p₂) t (∣-right ∈₂) | true | true = ⇐ p₁ refl · D-sound′ p₂ t ∈₂\n D-sound′ (p₁ · p₂) t (∣-left (∈₁ · ∈₂)) | false | true = D-sound ∈₁ · drop-♭♯ (D-nullable t (♭ p₁)) ∈₂\n D-sound′ (p₁ · p₂) t (∣-right ∈₂) | false | true = ⇐ (♭ p₁) refl · D-sound′ p₂ t ∈₂\n D-sound′ (p₁ · p₂) t (∈₁ · ∈₂) | true | false = D-sound ∈₁ · drop-♭♯ (D-nullable t p₁ ) ∈₂\n D-sound′ (p₁ · p₂) t (∈₁ · ∈₂) | false | false = D-sound ∈₁ · drop-♭♯ (D-nullable t (♭ p₁)) ∈₂\n\nD-complete : ∀ {n s t} {p : P n} → t ∷ s ∈ p → s ∈ D t p\nD-complete {t = t} t∷s∈ = D-complete′ _ t∷s∈ refl\n where\n D-complete′ : ∀ {s s′ n} (p : P n) → s′ ∈ p → s′ ≡ t ∷ s → s ∈ D t p\n D-complete′ fail () refl\n D-complete′ empty () refl\n D-complete′ (sat f) (sat ok) refl with f t\n D-complete′ (sat f) (sat ok) refl | true = empty\n D-complete′ (sat f) (sat ()) refl | false\n D-complete′ (p₁ ∣ p₂) (∣-left ∈₁) refl = ∣-left (D-complete ∈₁)\n D-complete′ (p₁ ∣ p₂) (∣-right ∈₂) refl = ∣-right {p₁ = D t p₁} (D-complete ∈₂)\n D-complete′ (nonempty p) (nonempty ∈) refl = D-complete ∈\n D-complete′ (cast _ p) (cast ∈) refl = D-complete ∈\n D-complete′ (p₁ · p₂) _ _ with forced? p₁ | forced? p₂\n D-complete′ (p₁ · p₂) (_·_ {[]} ∈₁ ∈₂) refl | true | true = ∣-right {p₁ = D t p₁ · _} (D-complete ∈₂)\n D-complete′ (p₁ · p₂) (_·_ {._ ∷ _} ∈₁ ∈₂) refl | true | true = ∣-left (D-complete ∈₁ · add-♭♯ (D-nullable t p₁) ∈₂)\n D-complete′ (p₁ · p₂) (_·_ {[]} ∈₁ ∈₂) refl | true | false with ⇒ ∈₁\n D-complete′ (p₁ · p₂) (_·_ {[]} ∈₁ ∈₂) refl | true | false | ()\n D-complete′ (p₁ · p₂) (_·_ {._ ∷ _} ∈₁ ∈₂) refl | true | false = D-complete ∈₁ · add-♭♯ (D-nullable t p₁) ∈₂\n D-complete′ (p₁ · p₂) (_·_ {[]} ∈₁ ∈₂) refl | false | true = ∣-right {p₁ = _·_ {n₂ = false} _ _} (D-complete ∈₂)\n D-complete′ (p₁ · p₂) (_·_ {._ ∷ _} ∈₁ ∈₂) refl | false | true = ∣-left (D-complete ∈₁ · add-♭♯ (D-nullable t (♭ p₁)) ∈₂)\n D-complete′ (p₁ · p₂) (_·_ {[]} ∈₁ ∈₂) refl | false | false with ⇒ ∈₁\n D-complete′ (p₁ · p₂) (_·_ {[]} ∈₁ ∈₂) refl | false | false | ()\n D-complete′ (p₁ · p₂) (_·_ {._ ∷ _} ∈₁ ∈₂) refl | false | false = D-complete ∈₁ · add-♭♯ (D-nullable t (♭ p₁)) ∈₂\n\nD-correct : ∀ {n s t} {p : P n} → s ∈ D t p ⇔ t ∷ s ∈ p\nD-correct = equivalence D-sound D-complete\n\n------------------------------------------------------------------------\n-- _∈_ is decidable\n\n-- _∈?_ runs a recogniser. Note that the result is yes or no plus a\n-- /proof/ verifying that the answer is correct.\n\ninfix 4 _∈?_\n\n_∈?_ : ∀ {n} (s : List Tok) (p : P n) → Dec (s ∈ p)\n[] ∈? p = nullable? p\nt ∷ s ∈? p with s ∈? D t p\nt ∷ s ∈? p | yes s∈Dtp = yes (D-sound s∈Dtp)\nt ∷ s ∈? p | no s∉Dtp = no (s∉Dtp ∘ D-complete)\n\n-- The last three lines could be replaced by the following one:\n--\n-- t ∷ s ∈? p = Decidable.map D-correct (s ∈? D t p)\n\n------------------------------------------------------------------------\n-- Alternative characterisation of equality\n\ninfix 5 _∷_\ninfix 4 _≈′_\n\n-- Two recognisers/languages are equal if their nullability indices\n-- are equal and all their derivatives are equal (coinductively). Note\n-- that the elements of this type are bisimulations.\n\ndata _≈′_ {n₁ n₂} (p₁ : P n₁) (p₂ : P n₂) : Set where\n _∷_ : n₁ ≡ n₂ → (∀ t → ∞ (D t p₁ ≈′ D t p₂)) → p₁ ≈′ p₂\n\n-- This definition is equivalent to the one above.\n\n≈′-sound : ∀ {n₁ n₂} {p₁ : P n₁} {p₂ : P n₂} → p₁ ≈′ p₂ → p₁ ≈ p₂\n≈′-sound (refl ∷ rest) {[]} = Eq.sym index-correct ⟨∘⟩ index-correct\n≈′-sound (refl ∷ rest) {t ∷ s} =\n D-correct ⟨∘⟩ ≈′-sound (♭ (rest t)) ⟨∘⟩ Eq.sym D-correct\n\nsame-nullability : ∀ {n₁ n₂} {p₁ : P n₁} {p₂ : P n₂} →\n p₁ ≈ p₂ → n₁ ≡ n₂\nsame-nullability p₁≈p₂ =\n Bool.⇔→≡ (index-correct ⟨∘⟩ p₁≈p₂ ⟨∘⟩ Eq.sym index-correct)\n\nD-cong : ∀ {n₁ n₂} {p₁ : P n₁} {p₂ : P n₂} {t} →\n p₁ ≈ p₂ → D t p₁ ≈ D t p₂\nD-cong p₁≈p₂ = Eq.sym D-correct ⟨∘⟩ p₁≈p₂ ⟨∘⟩ D-correct\n\n≈′-complete : ∀ {n₁ n₂} {p₁ : P n₁} {p₂ : P n₂} → p₁ ≈ p₂ → p₁ ≈′ p₂\n≈′-complete p₁≈p₂ =\n same-nullability p₁≈p₂ ∷ λ _ → ♯ ≈′-complete (D-cong p₁≈p₂)\n\n≈′-correct : ∀ {n₁ n₂} {p₁ : P n₁} {p₂ : P n₂} → p₁ ≈′ p₂ ⇔ p₁ ≈ p₂\n≈′-correct = equivalence ≈′-sound ≈′-complete\n\n------------------------------------------------------------------------\n-- The combinator nonempty does not need to be primitive\n\n-- The variant of nonempty which is defined below (nonempty′) makes\n-- many recognisers larger, though.\n\nmodule AlternativeNonempty where\n\n nonempty′ : ∀ {n} → P n → P false\n nonempty′ fail = fail\n nonempty′ empty = fail\n nonempty′ (sat f) = sat f\n nonempty′ (p₁ ∣ p₂) = nonempty′ p₁ ∣ nonempty′ p₂\n nonempty′ (nonempty p) = nonempty′ p\n nonempty′ (cast eq p) = nonempty′ p\n nonempty′ (p₁ · p₂) with forced? p₁ | forced? p₂\n ... | false | _ = p₁ · p₂\n ... | true | false = p₁ · p₂\n ... | true | true = nonempty′ p₁ ∣ nonempty′ p₂\n ∣ ♯ nonempty′ p₁ · ♯ nonempty′ p₂\n\n sound : ∀ {n} {p : P n} → nonempty′ p ≤ nonempty p\n sound {s = []} pr with ⇒ pr\n ... | ()\n sound {s = _ ∷ _} pr = nonempty (sound′ _ pr refl)\n where\n sound′ : ∀ {n t s s′} (p : P n) →\n s′ ∈ nonempty′ p → s′ ≡ t ∷ s → t ∷ s ∈ p\n sound′ fail () refl\n sound′ empty () refl\n sound′ (sat f) (sat ok) refl = sat ok\n sound′ (p₁ ∣ p₂) (∣-left pr) refl = ∣-left (sound′ p₁ pr refl)\n sound′ (p₁ ∣ p₂) (∣-right pr) refl = ∣-right {p₁ = p₁} (sound′ p₂ pr refl)\n sound′ (nonempty p) pr refl = nonempty (sound′ p pr refl)\n sound′ (cast _ p) pr refl = cast (sound′ p pr refl)\n sound′ (p₁ · p₂) pr _ with forced? p₁ | forced? p₂\n sound′ (p₁ · p₂) pr refl | false | _ = pr\n sound′ (p₁ · p₂) pr refl | true | false = pr\n sound′ (p₁ · p₂) (∣-left (∣-left pr)) refl | true | true = cast∈ (proj₂ ListMonoid.identity _) refl $\n sound′ p₁ pr refl · ⇐ p₂ refl\n sound′ (p₁ · p₂) (∣-left (∣-right pr)) refl | true | true = ⇐ p₁ refl · sound′ p₂ pr refl\n sound′ (p₁ · p₂) (∣-right (_·_ {[]} pr₁ pr₂)) refl | true | true with ⇒ pr₁\n ... | ()\n sound′ (p₁ · p₂) (∣-right (_·_ {_ ∷ _} pr₁ pr₂)) refl | true | true with sound {p = p₂} pr₂\n ... | nonempty pr₂′ = sound′ p₁ pr₁ refl · pr₂′\n\n complete : ∀ {n} {p : P n} → nonempty p ≤ nonempty′ p\n complete (nonempty pr) = complete′ _ pr refl\n where\n complete′ : ∀ {n t s s′} (p : P n) →\n s ∈ p → s ≡ t ∷ s′ → t ∷ s′ ∈ nonempty′ p\n complete′ fail () refl\n complete′ empty () refl\n complete′ (sat f) (sat ok) refl = sat ok\n complete′ (p₁ ∣ p₂) (∣-left pr) refl = ∣-left (complete′ p₁ pr refl)\n complete′ (p₁ ∣ p₂) (∣-right pr) refl = ∣-right {n₁ = false} (complete′ p₂ pr refl)\n complete′ (nonempty p) (nonempty pr) refl = complete′ p pr refl\n complete′ (cast _ p) (cast pr) refl = complete′ p pr refl\n complete′ (p₁ · p₂) pr _ with forced? p₁ | forced? p₂\n complete′ (p₁ · p₂) pr refl | false | _ = pr\n complete′ (p₁ · p₂) pr refl | true | false = pr\n complete′ (p₁ · p₂) (_·_ {[]} pr₁ pr₂) refl | true | true = ∣-left (∣-right {n₁ = false} (complete′ p₂ pr₂ refl))\n complete′ (p₁ · p₂) (_·_ {_ ∷ _} {[]} pr₁ pr₂) refl | true | true = cast∈ (sym $ proj₂ ListMonoid.identity _) refl $\n ∣-left (∣-left {n₂ = false} (complete′ p₁ pr₁ refl))\n complete′ (p₁ · p₂) (_·_ {_ ∷ _} {_ ∷ _} pr₁ pr₂) refl | true | true = ∣-right {n₁ = false} (complete′ p₁ pr₁ refl ·\n complete′ p₂ pr₂ refl)\n\n correct : ∀ {n} {p : P n} → nonempty′ p ≈ nonempty p\n correct = equivalence sound complete\n", "meta": {"hexsha": "9e54962440762ae3df52cb8ff8e7ada4b332c403", "size": 18180, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "TotalRecognisers/LeftRecursion.agda", "max_stars_repo_name": "nad/parser-combinators", "max_stars_repo_head_hexsha": "76774f54f466cfe943debf2da731074fe0c33644", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-07-03T08:56:13.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-03T08:56:13.000Z", "max_issues_repo_path": "TotalRecognisers/LeftRecursion.agda", "max_issues_repo_name": "nad/parser-combinators", "max_issues_repo_head_hexsha": "76774f54f466cfe943debf2da731074fe0c33644", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "TotalRecognisers/LeftRecursion.agda", "max_forks_repo_name": "nad/parser-combinators", "max_forks_repo_head_hexsha": "76774f54f466cfe943debf2da731074fe0c33644", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.4123006834, "max_line_length": 134, "alphanum_fraction": 0.4728822882, "num_tokens": 6790, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3445189262749576}} {"text": "{-# OPTIONS --safe #-}\n\n-- This is a file for dealing with Monuses: these are monoids that are like the\n-- positive half of a group. Much of my info on them comes from these papers:\n--\n-- * Wehrung, Friedrich. ‘Injective Positively Ordered Monoids I’. Journal of\n-- Pure and Applied Algebra 83, no. 1 (11 November 1992): 43–82.\n-- https://doi.org/10.1016/0022-4049(92)90104-N.\n-- * Wehrung, Friedrich. ‘Embedding Simple Commutative Monoids into Simple\n-- Refinement Monoids’. Semigroup Forum 56, no. 1 (January 1998): 104–29.\n-- https://doi.org/10.1007/s00233-002-7008-0.\n-- * Amer, K. ‘Equationally Complete Classes of Commutative Monoids with Monus’.\n-- Algebra Universalis 18, no. 1 (1 February 1984): 129–31.\n-- https://doi.org/10.1007/BF01182254.\n-- * Wehrung, Friedrich. ‘Metric Properties of Positively Ordered Monoids’.\n-- Forum Mathematicum 5, no. 5 (1993).\n-- https://doi.org/10.1515/form.1993.5.183.\n-- * Wehrung, Friedrich. ‘Restricted Injectivity, Transfer Property and\n-- Decompositions of Separative Positively Ordered Monoids.’ Communications in\n-- Algebra 22, no. 5 (1 January 1994): 1747–81.\n-- https://doi.org/10.1080/00927879408824934.\n--\n-- These monoids have a preorder defined on them, the algebraic preorder:\n-- \n-- x ≤ y = ∃ z × (y ≡ x ∙ z)\n--\n-- The _∸_ operator extracts the z from above, if it exists.\n\nmodule Algebra.Monus where\n\nopen import Prelude\nopen import Algebra\nopen import Relation.Binary\nopen import Path.Reasoning\nopen import Function.Reasoning\n\n-- Positively ordered monoids.\n--\n-- These are monoids which have a preorder that respects the monoid operation\n-- in a straightforward way.\nrecord POM ℓ₁ ℓ₂ : Type (ℓsuc (ℓ₁ ℓ⊔ ℓ₂)) where\n field commutativeMonoid : CommutativeMonoid ℓ₁\n open CommutativeMonoid commutativeMonoid public\n field preorder : Preorder 𝑆 ℓ₂\n open Preorder preorder public renaming (refl to ≤-refl)\n field\n positive : ∀ x → ε ≤ x\n ≤-cong : ∀ x {y z} → y ≤ z → x ∙ y ≤ x ∙ z\n\n x≤x∙y : ∀ {x y} → x ≤ x ∙ y\n x≤x∙y {x} {y} = subst (_≤ x ∙ y) (∙ε x) (≤-cong x (positive y))\n\n ≤-congʳ : ∀ x {y z} → y ≤ z → y ∙ x ≤ z ∙ x\n ≤-congʳ x {y} {z} p = subst₂ _≤_ (comm x y) (comm x z) (≤-cong x p)\n\n alg-≤-trans : ∀ {x y z k₁ k₂} → y ≡ x ∙ k₁ → z ≡ y ∙ k₂ → z ≡ x ∙ (k₁ ∙ k₂)\n alg-≤-trans {x} {y} {z} {k₁} {k₂} y≡x∙k₁ z≡y∙k₂ =\n z ≡⟨ z≡y∙k₂ ⟩\n y ∙ k₂ ≡⟨ cong (_∙ k₂) y≡x∙k₁ ⟩\n (x ∙ k₁) ∙ k₂ ≡⟨ assoc x k₁ k₂ ⟩\n x ∙ (k₁ ∙ k₂) ∎\n\n infix 4 _≺_\n _≺_ : 𝑆 → 𝑆 → Type _\n x ≺ y = ∃ k × (y ≡ x ∙ k) × (k ≢ ε)\n\n-- Total Antisymmetric POM\nrecord TAPOM ℓ₁ ℓ₂ : Type (ℓsuc (ℓ₁ ℓ⊔ ℓ₂)) where\n field pom : POM ℓ₁ ℓ₂\n open POM pom public using (preorder; _≤_; ≤-cong; ≤-congʳ; x≤x∙y; commutativeMonoid; positive)\n open CommutativeMonoid commutativeMonoid public\n field\n _≤|≥_ : Total _≤_\n antisym : Antisymmetric _≤_\n\n totalOrder : TotalOrder 𝑆 ℓ₂ ℓ₂\n totalOrder = fromPartialOrder (record { preorder = preorder ; antisym = antisym }) _≤|≥_\n open TotalOrder totalOrder public hiding (_≤|≥_; antisym) renaming (refl to ≤-refl)\n\n-- Every commutative monoid generates a positively ordered monoid\n-- called the \"algebraic\" or \"minimal\" pom\nmodule AlgebraicPOM {ℓ} (mon : CommutativeMonoid ℓ) where\n commutativeMonoid = mon\n open CommutativeMonoid mon\n\n infix 4 _≤_\n _≤_ : 𝑆 → 𝑆 → Type _\n x ≤ y = ∃ z × (y ≡ x ∙ z)\n\n -- The snd here is the same proof as alg-≤-trans, so could be refactored out.\n ≤-trans : Transitive _≤_\n ≤-trans (k₁ , _) (k₂ , _) .fst = k₁ ∙ k₂\n ≤-trans {x} {y} {z} (k₁ , y≡x∙k₁) (k₂ , z≡y∙k₂) .snd =\n z ≡⟨ z≡y∙k₂ ⟩\n y ∙ k₂ ≡⟨ cong (_∙ k₂) y≡x∙k₁ ⟩\n (x ∙ k₁) ∙ k₂ ≡⟨ assoc x k₁ k₂ ⟩\n x ∙ (k₁ ∙ k₂) ∎\n\n preorder : Preorder 𝑆 ℓ\n Preorder._≤_ preorder = _≤_\n Preorder.refl preorder = ε , sym (∙ε _)\n Preorder.trans preorder = ≤-trans\n\n positive : ∀ x → ε ≤ x\n positive x = x , sym (ε∙ x)\n\n ≤-cong : ∀ x {y z} → y ≤ z → x ∙ y ≤ x ∙ z\n ≤-cong x (k , z≡y∙k) = k , cong (x ∙_) z≡y∙k ; sym (assoc x _ k)\n\nalgebraic-pom : ∀ {ℓ} → CommutativeMonoid ℓ → POM ℓ ℓ\nalgebraic-pom mon = record { AlgebraicPOM mon }\n\n-- Total Minimal POM\nrecord TMPOM ℓ : Type (ℓsuc ℓ) where\n field commutativeMonoid : CommutativeMonoid ℓ\n\n pom : POM _ _\n pom = algebraic-pom commutativeMonoid\n\n open POM pom public\n\n infix 4 _≤|≥_\n field _≤|≥_ : Total _≤_\n\n <⇒≺ : ∀ x y → y ≰ x → x ≺ y\n <⇒≺ x y x-is-inj (pre⊙∘-equiv {Z = ⊙Trunc 1 (⊙Ω (⊙Susp (de⊙ X)))} ⊙eq) (⊙<– ⊙eq) ⊙decodeN $\n (⊙<– ⊙eq) ⊙∘ ⊙encodeN\n =⟨ ⊙<–-inv-l ⊙eq ⟩\n ⊙idf _\n =⟨ ! ⊙decodeN-⊙encodeN ⟩\n ⊙decodeN ⊙∘ ⊙encodeN =∎\n\n ⊙encodeN-⊙decodeN : ⊙encodeN ⊙∘ ⊙decodeN == ⊙idf _\n ⊙encodeN-⊙decodeN =\n ⊙encodeN ⊙∘ ⊙decodeN\n =⟨ ap (⊙encodeN ⊙∘_) (! ⊙<–-⊙eq) ⟩\n ⊙encodeN ⊙∘ ⊙<– ⊙eq\n =⟨ ⊙<–-inv-r ⊙eq ⟩\n ⊙idf _ =∎\n\n ⊙eq⁻¹ : ⊙Trunc 1 X ⊙≃ ⊙Trunc 1 (⊙Ω (⊙Susp (de⊙ X)))\n ⊙eq⁻¹ = ⊙decodeN , snd (eq ⁻¹)\n\n iso : Ω^S-group 0 (⊙Trunc 1 (⊙Ω (⊙Susp (de⊙ X))))\n ≃ᴳ Ω^S-group 0 (⊙Trunc 1 X)\n iso = Ω^S-group-emap 0 ⊙eq\n\n abstract\n π₂-Susp : πS 1 (⊙Susp (de⊙ X)) ≃ᴳ πS 0 X\n π₂-Susp =\n πS 1 (⊙Susp (de⊙ X))\n ≃ᴳ⟨ πS-Ω-split-iso 0 (⊙Susp (de⊙ X)) ⟩\n πS 0 (⊙Ω (⊙Susp (de⊙ X)))\n ≃ᴳ⟨ Ω^S-group-Trunc-fuse-diag-iso 0 (⊙Ω (⊙Susp (de⊙ X))) ⁻¹ᴳ ⟩\n Ω^S-group 0 (⊙Trunc 1 (⊙Ω (⊙Susp (de⊙ X))))\n ≃ᴳ⟨ iso ⟩\n Ω^S-group 0 (⊙Trunc 1 X)\n ≃ᴳ⟨ Ω^S-group-Trunc-fuse-diag-iso 0 X ⟩\n πS 0 X ≃ᴳ∎\n\nmodule Pi2HSuspNaturality {i} {X Y : Ptd i}\n (f : X ⊙→ Y)\n {{_ : has-level 1 (de⊙ X)}} {{_ : has-level 1 (de⊙ Y)}}\n {{_ : is-connected 0 (de⊙ X)}} {{_ : is-connected 0 (de⊙ Y)}}\n (H-X : HSS X) (H-Y : HSS Y) where\n\n import homotopy.SuspAdjointLoop as SAL\n private\n module Π₂X = Pi2HSusp H-X\n module Π₂Y = Pi2HSusp H-Y\n\n ⊙decodeN-natural :\n Π₂Y.⊙decodeN ◃⊙∘\n ⊙Trunc-fmap f ◃⊙idf\n =⊙∘\n ⊙Trunc-fmap (⊙Ω-fmap (⊙Susp-fmap (fst f))) ◃⊙∘\n Π₂X.⊙decodeN ◃⊙idf\n ⊙decodeN-natural = =⊙∘-in $\n Π₂Y.⊙decodeN ⊙∘ ⊙Trunc-fmap f\n =⟨ ⊙λ= (⊙Trunc-fmap-⊙∘ (SAL.η Y) f) ⟩\n ⊙Trunc-fmap (SAL.η Y ⊙∘ f)\n =⟨ ap ⊙Trunc-fmap (SAL.η-natural f) ⟩\n ⊙Trunc-fmap (⊙Ω-fmap (⊙Susp-fmap (fst f)) ⊙∘ SAL.η X)\n =⟨ ! (⊙λ= (⊙Trunc-fmap-⊙∘ (⊙Ω-fmap (⊙Susp-fmap (fst f))) (SAL.η X))) ⟩\n ⊙Trunc-fmap (⊙Ω-fmap (⊙Susp-fmap (fst f))) ⊙∘ Π₂X.⊙decodeN =∎\n\n ⊙encodeN-natural :\n Π₂Y.⊙encodeN ◃⊙∘\n ⊙Trunc-fmap (⊙Ω-fmap (⊙Susp-fmap (fst f))) ◃⊙idf\n =⊙∘\n ⊙Trunc-fmap f ◃⊙∘\n Π₂X.⊙encodeN ◃⊙idf\n ⊙encodeN-natural =\n Π₂Y.⊙encodeN ◃⊙∘\n ⊙Trunc-fmap (⊙Ω-fmap (⊙Susp-fmap (fst f))) ◃⊙idf\n =⊙∘⟨ 2 & 0 & =⊙∘-in {gs = Π₂X.⊙decodeN ◃⊙∘ Π₂X.⊙encodeN ◃⊙idf} $\n ! Π₂X.⊙decodeN-⊙encodeN ⟩\n Π₂Y.⊙encodeN ◃⊙∘\n ⊙Trunc-fmap (⊙Ω-fmap (⊙Susp-fmap (fst f))) ◃⊙∘\n Π₂X.⊙decodeN ◃⊙∘\n Π₂X.⊙encodeN ◃⊙idf\n =⊙∘⟨ 1 & 2 & !⊙∘ ⊙decodeN-natural ⟩\n Π₂Y.⊙encodeN ◃⊙∘\n Π₂Y.⊙decodeN ◃⊙∘\n ⊙Trunc-fmap f ◃⊙∘\n Π₂X.⊙encodeN ◃⊙idf\n =⊙∘⟨ 0 & 2 & =⊙∘-in {gs = ⊙idf-seq} Π₂Y.⊙encodeN-⊙decodeN ⟩\n ⊙Trunc-fmap f ◃⊙∘\n Π₂X.⊙encodeN ◃⊙idf ∎⊙∘\n", "meta": {"hexsha": "d86e4e7396f144839331da3f3f2dff69a829f69c", "size": 9398, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "theorems/homotopy/Pi2HSusp.agda", "max_stars_repo_name": "AntoineAllioux/HoTT-Agda", "max_stars_repo_head_hexsha": "1037d82edcf29b620677a311dcfd4fc2ade2faa6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 294, "max_stars_repo_stars_event_min_datetime": "2015-01-09T16:23:23.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T13:54:45.000Z", "max_issues_repo_path": "theorems/homotopy/Pi2HSusp.agda", "max_issues_repo_name": "AntoineAllioux/HoTT-Agda", "max_issues_repo_head_hexsha": "1037d82edcf29b620677a311dcfd4fc2ade2faa6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 31, "max_issues_repo_issues_event_min_datetime": "2015-03-05T20:09:00.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-03T19:15:25.000Z", "max_forks_repo_path": "theorems/homotopy/Pi2HSusp.agda", "max_forks_repo_name": "AntoineAllioux/HoTT-Agda", "max_forks_repo_head_hexsha": "1037d82edcf29b620677a311dcfd4fc2ade2faa6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 50, "max_forks_repo_forks_event_min_datetime": "2015-01-10T01:48:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-14T03:03:25.000Z", "avg_line_length": 31.9659863946, "max_line_length": 90, "alphanum_fraction": 0.4922323899, "num_tokens": 4517, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3439531166114479}} {"text": "\nmodule UniDB.Morph.WeakenPrime where\n\nopen import UniDB.Spec\nopen import UniDB.Morph.ShiftsPrime\nopen import Function\n\n--------------------------------------------------------------------------------\n\ndata Weaken` : MOR where\n baseW : {γ : Dom} → Weaken` γ γ\n stepW : {γ₁ γ₂ : Dom} → Weaken` γ₁ γ₂ → Weaken` γ₁ (suc γ₂)\n\nlkWeakenIx` : {γ₁ γ₂ : Dom} → Weaken` γ₁ γ₂ → Ix γ₁ → Ix γ₂\nlkWeakenIx` baseW i = i\nlkWeakenIx` (stepW ξ) i = suc (lkWeakenIx` ξ i)\n\ninstance\n iLkWeaken` : {T : STX} {{vrT : Vr T}} → Lk T Weaken`\n lk {{iLkWeaken`}} ξ i = vr (lkWeakenIx` ξ i)\n\n iWkmWeaken` : Wkm Weaken`\n Wkm.wkm iWkmWeaken` zero = baseW\n Wkm.wkm iWkmWeaken` (suc δ) = stepW (wkm {Weaken`} δ)\n\n iIdmWeaken` : Idm Weaken`\n idm {{iIdmWeaken`}} γ = baseW\n\n iWkWeaken` : {γ₁ : Dom} → Wk (Weaken` γ₁)\n wk₁ {{iWkWeaken`}} ξ = stepW ξ\n wk {{iWkWeaken`}} zero x = x\n wk {{iWkWeaken`}} (suc δ) x = wk₁ (wk δ x)\n wk-zero {{iWkWeaken`}} x = refl\n wk-suc {{iWkWeaken`}} δ x = refl\n\nlkWeakenIx`-wkm : {γ : Dom} (δ : Dom) (i : Ix γ) →\n lkWeakenIx` (wkm {Weaken`} δ) i ≡ wk δ i\nlkWeakenIx`-wkm zero i = refl\nlkWeakenIx`-wkm (suc δ) i = cong suc (lkWeakenIx`-wkm δ i)\n\ninstance\n iLkWkmWeaken` : {T : STX} {{vrT : Vr T}} → LkWkm T Weaken`\n lk-wkm {{iLkWkmWeaken` {T}}} δ i = cong (vr {T}) (lkWeakenIx`-wkm δ i)\n\n iCompWeaken` : Comp Weaken`\n _⊙_ {{iCompWeaken`}} ξ baseW = ξ\n _⊙_ {{iCompWeaken`}} ξ (stepW ξ₂) = stepW (ξ ⊙ ξ₂)\n\nwk-comp` : {γ₁ γ₂ γ₃ : Dom} (ξ₁ : Weaken` γ₁ γ₂) (ξ₂ : Weaken` γ₂ γ₃)\n (δ : Dom) → wk δ (ξ₁ ⊙ ξ₂) ≡ ξ₁ ⊙ wk δ ξ₂\nwk-comp` ξ₁ ξ₂ zero = refl\nwk-comp` ξ₁ ξ₂ (suc δ) = cong stepW (wk-comp` ξ₁ ξ₂ δ)\n\nlkIx-wk-weaken` : {γ₁ γ₂ : Dom} (δ : Dom) (ξ : Weaken` γ₁ γ₂) (i : Ix γ₁) →\n lk {Ix} {Weaken`} (wk δ ξ) i ≡ wk δ (lk {Ix} {Weaken`} ξ i)\nlkIx-wk-weaken` zero ξ i = refl\nlkIx-wk-weaken` (suc δ) ξ i = cong suc (lkIx-wk-weaken` δ ξ i)\n\nmodule _ (T : STX) {{vrT : Vr T}} {{wkT : Wk T}} {{wkVrT : WkVr T}} where\n\n lk-wk₁-weaken` :\n {γ₁ γ₂ : Dom} (ξ : Weaken` γ₁ γ₂) (i : Ix γ₁) →\n lk {T} {Weaken`} (wk₁ ξ) i ≡ wk₁ (lk {T} {Weaken`} ξ i)\n lk-wk₁-weaken` ξ i = sym (wk₁-vr {T} (lk {Ix} {Weaken`} ξ i) )\n\n lk-wk-weaken` :\n {γ₁ γ₂ : Dom} (δ : Dom) (ξ : Weaken` γ₁ γ₂) (i : Ix γ₁) →\n lk {T} {Weaken`} (wk δ ξ) i ≡ wk δ (lk {T} {Weaken`} ξ i)\n lk-wk-weaken` δ ξ i = trans\n (cong (vr {T}) (lkIx-wk-weaken` δ ξ i))\n (sym (wk-vr {T} δ (lk {Ix} {Weaken`} ξ i)))\n\nlkWeakenIx`-comp :\n {γ₁ γ₂ γ₃ : Dom} (ξ₁ : Weaken` γ₁ γ₂) (ξ₂ : Weaken` γ₂ γ₃)\n (i : Ix γ₁) →\n lkWeakenIx` ((iCompWeaken` Comp.⊙ ξ₁) ξ₂) i ≡\n lkWeakenIx` ξ₂ (lkWeakenIx` ξ₁ i)\nlkWeakenIx`-comp ξ₁ baseW i = refl\nlkWeakenIx`-comp ξ₁ (stepW ξ₂) i = cong suc (lkWeakenIx`-comp ξ₁ ξ₂ i)\n\ninstance\n iLkRenWeaken` : {T : STX} {{vrT : Vr T}} → LkRen T Weaken`\n lk-ren {{iLkRenWeaken`}} ξ i = refl\n\n iLkRenCompWeaken` : {T : STX} {{vrT : Vr T}} → LkRenComp T Weaken`\n lk-ren-comp {{iLkRenCompWeaken` {T}}} ξ₁ ξ₂ i =\n cong (vr {T}) (lkWeakenIx`-comp ξ₁ ξ₂ i)\n\n iCompIdmWeaken` : CompIdm Weaken`\n ⊙-idm {{iCompIdmWeaken`}} ξ = refl\n idm-⊙ {{iCompIdmWeaken`}} baseW = refl\n idm-⊙ {{iCompIdmWeaken`}} (stepW ξ) = cong stepW (idm-⊙ {Weaken`} ξ)\n\n iCompAssocWeaken` : CompAssoc Weaken`\n ⊙-assoc {{iCompAssocWeaken`}} ξ₁ ξ₂ baseW = refl\n ⊙-assoc {{iCompAssocWeaken`}} ξ₁ ξ₂ (stepW ξ₃) =\n cong stepW (⊙-assoc {Weaken`} ξ₁ ξ₂ ξ₃)\n\n⊙-wkm-weaken` : {γ₁ γ₂ : Dom} (ξ : Weaken` γ₁ γ₂) (δ : Dom) →\n ξ ⊙ wkm {Weaken`} δ ≡ wk δ ξ\n⊙-wkm-weaken` ξ zero = refl\n⊙-wkm-weaken` ξ (suc δ) = cong stepW (⊙-wkm-weaken` ξ δ)\n\nextensionality-weaken` : {γ₁ γ₂ : Dom} (ξ₁ ξ₂ : Weaken` γ₁ γ₂)\n (hyp : (i : Ix γ₁) → lk {Ix} ξ₁ i ≡ lk {Ix} ξ₂ i) →\n ξ₁ ≡ ξ₂\nextensionality-weaken` baseW baseW hyp = refl\nextensionality-weaken` baseW (stepW ξ₂) hyp = case hyp zero of λ ()\nextensionality-weaken` (stepW ξ₁) baseW hyp = case hyp zero of λ ()\nextensionality-weaken` (stepW ξ₁) (stepW ξ₂) hyp =\n cong stepW (extensionality-weaken` ξ₁ ξ₂ (suc-inj ∘ hyp))\n\n--------------------------------------------------------------------------------\n", "meta": {"hexsha": "87137066ca572551e6f9aa5500ad5dad56fbb231", "size": 4056, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "UniDB/Morph/WeakenPrime.agda", "max_stars_repo_name": "skeuchel/unidb-agda", "max_stars_repo_head_hexsha": "7ae52205db44ad4f463882ba7e5082120fb76349", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "UniDB/Morph/WeakenPrime.agda", "max_issues_repo_name": "skeuchel/unidb-agda", "max_issues_repo_head_hexsha": "7ae52205db44ad4f463882ba7e5082120fb76349", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "UniDB/Morph/WeakenPrime.agda", "max_forks_repo_name": "skeuchel/unidb-agda", "max_forks_repo_head_hexsha": "7ae52205db44ad4f463882ba7e5082120fb76349", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.5789473684, "max_line_length": 80, "alphanum_fraction": 0.5643491124, "num_tokens": 1939, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.34388304749192006}} {"text": "------------------------------------------------------------------------\n-- An alternative definition of the partiality monad: a variant of the\n-- delay monad quotiented by a notion of weak bisimilarity\n------------------------------------------------------------------------\n\n{-# OPTIONS --erased-cubical --sized-types #-}\n\nmodule Partiality-monad.Coinductive.Alternative where\n\nopen import Equality.Propositional.Cubical\nopen import Logical-equivalence using (_⇔_)\nopen import Prelude hiding (⊥)\n\nopen import Bijection equality-with-J using (_↔_)\nopen import Function-universe equality-with-J hiding (⊥↔⊥)\nopen import H-level equality-with-J\nopen import H-level.Truncation.Propositional equality-with-paths\nopen import Quotient equality-with-paths\n\nimport Delay-monad.Alternative as A\nimport Delay-monad.Alternative.Equivalence as A\nimport Delay-monad.Alternative.Weak-bisimilarity as A\nimport Delay-monad.Bisimilarity as B\nimport Partiality-monad.Coinductive as C\n\n-- The partiality monad, defined as the alternative definition of the\n-- delay monad quotiented by weak bisimilarity.\n\n_⊥ : ∀ {a} → Type a → Type a\nA ⊥ = A.Delay A / A._≈_\n\n-- The partiality monad is a set.\n\n⊥-is-set : ∀ {a} {A : Type a} → Is-set (A ⊥)\n⊥-is-set = /-is-set\n\n-- This definition of the partiality monad is isomorphic to the one in\n-- Partiality-monad.Coinductive, for sets, assuming extensionality.\n\n⊥↔⊥ : ∀ {a} {A : Type a} →\n Is-set A →\n B.Extensionality a →\n A ⊥ ↔ A C.⊥\n⊥↔⊥ {A = A} A-set delay-ext = D↔D /-cong lemma\n where\n D↔D = A.Delay↔Delay delay-ext\n\n lemma : (x y : A.Delay A) →\n x A.≈ y ⇔ ∥ _↔_.to D↔D x B.≈ _↔_.to D↔D y ∥\n lemma x y =\n x A.≈ y ↔⟨ inverse $ ∥∥↔ (A.≈-propositional x y) ⟩\n ∥ x A.≈ y ∥ ↝⟨ ∥∥-cong-⇔ (A.≈⇔≈ A-set x y) ⟩□\n ∥ _↔_.to D↔D x B.≈ _↔_.to D↔D y ∥ □\n", "meta": {"hexsha": "15cc22a92483117e9f1055a3d083e6070a7dd6a0", "size": 1853, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Partiality-monad/Coinductive/Alternative.agda", "max_stars_repo_name": "nad/partiality-monad", "max_stars_repo_head_hexsha": "f69749280969f9093e5e13884c6feb0ad2506eae", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-05-21T22:59:18.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-03T08:56:08.000Z", "max_issues_repo_path": "src/Partiality-monad/Coinductive/Alternative.agda", "max_issues_repo_name": "nad/partiality-monad", "max_issues_repo_head_hexsha": "f69749280969f9093e5e13884c6feb0ad2506eae", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Partiality-monad/Coinductive/Alternative.agda", "max_forks_repo_name": "nad/partiality-monad", "max_forks_repo_head_hexsha": "f69749280969f9093e5e13884c6feb0ad2506eae", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.3148148148, "max_line_length": 81, "alphanum_fraction": 0.6049649217, "num_tokens": 613, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.34388304749192006}} {"text": "-- MIT License\n\n-- Copyright (c) 2021 Luca Ciccone and Luca Padovani\n\n-- Permission is hereby granted, free of charge, to any person\n-- obtaining a copy of this software and associated documentation\n-- files (the \"Software\"), to deal in the Software without\n-- restriction, including without limitation the rights to use,\n-- copy, modify, merge, publish, distribute, sublicense, and/or sell\n-- copies of the Software, and to permit persons to whom the\n-- Software is furnished to do so, subject to the following\n-- conditions:\n\n-- The above copyright notice and this permission notice shall be\n-- included in all copies or substantial portions of the Software.\n\n-- THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND,\n-- EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES\n-- OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND\n-- NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT\n-- HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,\n-- WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n-- FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR\n-- OTHER DEALINGS IN THE SOFTWARE.\n\n{-# OPTIONS --guardedness --sized-types #-}\n\nopen import Data.Product\nopen import Data.Empty\nopen import Data.Sum\nopen import Data.Vec\nopen import Data.List as List\nopen import Data.Unit\nopen import Data.Fin\nopen import Data.Bool renaming (Bool to 𝔹)\nopen import Relation.Unary using (_∈_; _⊆_;_∉_)\nopen import Relation.Binary.Construct.Closure.ReflexiveTransitive\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Nullary\nopen import Size\nopen import Codata.Thunk\n\nopen import is-lib.InfSys\nopen import Common using (Message)\n\nmodule FairSubtyping-IS {𝕋 : Set} (message : Message 𝕋) where\n\n open Message message\n open import SessionType message\n open import Session message\n open import Transitions message\n open import Convergence message\n open import Divergence message\n open import Discriminator message\n open import Action message using (Action)\n open import Subtyping message\n open import FairSubtyping message as FS\n open import HasTrace message\n open import Compliance message\n open import FairCompliance message\n open import Trace message\n open import FairCompliance-IS message\n\n private\n U : Set\n U = SessionType × SessionType\n\n data FSubIS-RN : Set where\n nil-any end-def : FSubIS-RN\n ii oo : FSubIS-RN\n\n data FSubCOIS-RN : Set where\n co-conv : FSubCOIS-RN\n\n nil-any-r : FinMetaRule U\n nil-any-r .Ctx = SessionType\n nil-any-r .comp T =\n [] ,\n ------------------\n (nil , T)\n\n end-def-r : FinMetaRule U\n end-def-r .Ctx = Σ[ (T , S) ∈ SessionType × SessionType ] End T × Defined S\n end-def-r .comp ((T , S) , _) =\n [] ,\n ------------------\n (T , S)\n\n ii-r : MetaRule U\n ii-r .Ctx = Σ[ (f , g) ∈ Continuation × Continuation ] dom f ⊆ dom g\n ii-r .Pos ((f , _) , _) = Σ[ t ∈ 𝕋 ] t ∈ dom f\n ii-r .prems ((f , g) , _) (t , _) = f t .force , g t .force\n ii-r .conclu ((f , g) , _) = inp f , inp g\n\n oo-r : MetaRule U\n oo-r .Ctx = Σ[ (f , g) ∈ Continuation × Continuation ] dom g ⊆ dom f × Witness g\n oo-r .Pos ((_ , g) , _) = Σ[ t ∈ 𝕋 ] t ∈ dom g\n oo-r .prems ((f , g) , _) (t , _) = f t .force , g t .force\n oo-r .conclu ((f , g) , _) = out f , out g\n\n co-conv-r : FinMetaRule U\n co-conv-r .Ctx = Σ[ (T , S) ∈ SessionType × SessionType ] T ↓ S\n co-conv-r .comp ((T , S) , _) =\n [] ,\n ------------------\n (T , S)\n\n FSubIS : IS U\n FSubIS .Names = FSubIS-RN\n FSubIS .rules nil-any = from nil-any-r\n FSubIS .rules end-def = from end-def-r\n FSubIS .rules ii = ii-r\n FSubIS .rules oo = oo-r\n\n FSubCOIS : IS U\n FSubCOIS .Names = FSubCOIS-RN\n FSubCOIS .rules co-conv = from co-conv-r\n\n _≤F_ : SessionType → SessionType → Set\n T ≤F S = FCoInd⟦ FSubIS , FSubCOIS ⟧ (T , S)\n\n _≤Fᵢ_ : SessionType → SessionType → Set\n T ≤Fᵢ S = Ind⟦ FSubIS ∪ FSubCOIS ⟧ (T , S)\n\n _≤Fc_ : SessionType → SessionType → Set\n T ≤Fc S = CoInd⟦ FSubIS ⟧ (T , S)\n\n\n {- Specification using _⊢_ is correct wrt FairSubtypingS -}\n\n FSSpec-⊢ : U → Set\n FSSpec-⊢ (T , S) = ∀{R} → R ⊢ T → R ⊢ S\n \n spec-sound : ∀{T S} → FairSubtypingS T S → FSSpec-⊢ (T , S)\n spec-sound fs fc = fc-complete (fs (fc-sound fc))\n\n spec-complete : ∀{T S} → FSSpec-⊢ (T , S) → FairSubtypingS T S\n spec-complete fs fc = fc-sound (fs (fc-complete fc))\n\n ------------------------------------------------------\n\n {- Soundness -}\n  -- Using bounded coinduction wrt SpecAux\n\n ≤Fᵢ->↓ : ∀{S T} → S ≤Fᵢ T → S ↓ T\n ≤Fᵢ->↓ (fold (inj₁ nil-any , _ , refl , _)) = nil-converges\n ≤Fᵢ->↓ (fold (inj₁ end-def , (_ , (end , def)) , refl , _)) = end-converges end def\n ≤Fᵢ->↓ (fold (inj₁ ii , _ , refl , pr)) = converge (pre-conv-inp-back λ x → ↓->preconv (≤Fᵢ->↓ (pr (_ , x))))\n ≤Fᵢ->↓ (fold (inj₁ oo , (_ , (incl , (t , ok-t))) , refl , pr)) = \n converge \n λ _ _ → [] , t , none , (_ , incl ok-t , step (out (incl ok-t)) refl) , (_ , ok-t , step (out ok-t) refl) , ≤Fᵢ->↓ (pr (t , ok-t))\n ≤Fᵢ->↓ (fold (inj₂ co-conv , (_ , conv) , refl , _)) = conv\n\n SpecAux : U → Set\n SpecAux (R , T) = Σ[ S ∈ SessionType ] S ≤F T × R ⊢ S \n\n ≤Fᵢ->defined : ∀{S T} → Defined S → S ≤Fᵢ T → Defined T\n ≤Fᵢ->defined def fs = conv->defined def (≤Fᵢ->↓ fs)\n\n spec-bounded-rec : ∀{R S} T → T ≤Fᵢ S → R ⊢ T → R ⊢ᵢ S\n spec-bounded-rec _ fs fc = \n let _ , reds , succ = con-sound (≤Fᵢ->↓ fs) (fc-sound fc) in\n maysucceed->⊢ᵢ reds succ\n\n spec-bounded : SpecAux ⊆ λ (R , S) → R ⊢ᵢ S\n spec-bounded (T , fs , fc) = spec-bounded-rec T (fcoind-to-ind fs) fc\n\n spec-cons : SpecAux ⊆ ISF[ FCompIS ] SpecAux\n spec-cons {(R , T)} (S , fs , fc) with fc .CoInd⟦_⟧.unfold\n spec-cons {(R , T)} (S , fs , fc) | client-end , ((_ , (win , def)) , _) , refl , _ = \n client-end , ((R , _) , (win , ≤Fᵢ->defined def (fcoind-to-ind fs))) , refl , λ ()\n spec-cons {(out r , _)} ((inp f) , fs , fc) | oi , (((.r , .f) , wit-r) , _) , refl , pr with fs .CoInd⟦_⟧.unfold\n ... | end-def , (((.(inp f) , _) , (inp e , _)) , _) , refl , _ = ⊥-elim (e _ (proj₂ (fc->defined (pr wit-r))))\n ... | ii , (((.f , g) , _) , _) , refl , pr' = oi , (_ , wit-r) , refl , λ wit → _ , pr' (_ , proj₂ (fc->defined (pr wit))) , pr wit\n spec-cons {(inp r , T)} (out f , fs , fc) | io , (((.r , .f) , wit-f) , _) , refl , pr with fs .CoInd⟦_⟧.unfold\n ... | end-def , (((.(out f) , _) , (out e , _)) , _) , refl , _ = ⊥-elim (e _ (proj₂ wit-f))\n ... | oo , (((.f , g) , (incl , wit-g)) , _) , refl , pr' = io , (_ , wit-g) , refl , λ wit → _ , pr' wit , pr (_ , incl (proj₂ wit))\n\n spec-aux-sound : SpecAux ⊆ λ (R , S) → R ⊢ S\n spec-aux-sound = bounded-coind[ FCompIS , FCompCOIS ] SpecAux spec-bounded spec-cons\n\n fs-sound : ∀{T S} → T ≤F S → FSSpec-⊢ (T , S)\n fs-sound {T} fs fc = spec-aux-sound (T , fs , fc)\n\n {- Soundness & Completeness of Sub wrt ≤Fc -}\n\n ≤Fc->sub : ∀{S T} → S ≤Fc T → ∀ {i} → Sub S T i\n ≤Fc->sub fs with fs .CoInd⟦_⟧.unfold\n ... | nil-any , _ , refl , _ = nil<:any\n ... | end-def , (_ , (end , def)) , refl , _ = end<:def end def\n ... | ii , ((f , g) , incl) , refl , pr = inp<:inp incl λ x → λ where .force → if-def x\n where \n if-def : (t : 𝕋) → ∀{i} → Sub (f t .force) (g t .force) i\n if-def t with t ∈? f\n ... | yes ok-t = ≤Fc->sub (pr (_ , ok-t))\n ... | no no-t = subst (λ x → Sub x (g t .force) _) (sym (not-def->nil no-t)) nil<:any\n ... | oo , (_ , (incl , wit)) , refl , pr = out<:out wit incl λ ok-x → λ where .force → ≤Fc->sub (pr (_ , ok-x))\n\n sub->≤Fc : ∀{S T} → (∀{i} → Sub S T i) → S ≤Fc T\n CoInd⟦_⟧.unfold (sub->≤Fc fs) with fs\n ... | nil<:any = nil-any , _ , refl , λ ()\n ... | end<:def end def = end-def , (_ , (end , def)) , refl , λ ()\n ... | inp<:inp incl pr = ii , (_ , incl) , refl , λ (p , _) → sub->≤Fc (pr p .force)\n ... | out<:out wit incl pr = oo , (_ , (incl , wit)) , refl , λ (_ , ok) → sub->≤Fc (pr ok .force)\n\n {- Auxiliary -}\n\n -- Only premise for rules using sample-cont in ⊢\n sample-cont-prem : ∀{f : Continuation}{t R} → R ⊢ f t .force \n → (pos : Σ[ p ∈ 𝕋 ] p ∈ dom (sample-cont t R nil)) → (sample-cont t R nil) (proj₁ pos) .force ⊢ f (proj₁ pos) .force\n sample-cont-prem {f} {t} pr (p , ok-p) with p ?= t\n ... | yes refl = pr\n sample-cont-prem {f} {t} pr (p , ()) | no ¬eq\n\n -- Premises using sample-cont-dual in ⊢\n sample-cont-prems : ∀{f : Continuation}{t} → t ∉ dom f \n → (pos : Σ[ p ∈ 𝕋 ] p ∈ dom f) → (sample-cont t nil win) (proj₁ pos) .force ⊢ f (proj₁ pos) .force\n sample-cont-prems {f} {t} no-t (p , ok-p) with p ?= t\n ... | yes refl = ⊥-elim (no-t ok-p)\n ... | no ¬eq = win⊢def ok-p\n\n -- Premises using sample-cont-dual in ⊢\n sample-cont-prems' : ∀{f : Continuation}{t R} → R ⊢ f t .force \n → (pos : Σ[ p ∈ 𝕋 ] p ∈ dom f) → (sample-cont t R win) (proj₁ pos) .force ⊢ f (proj₁ pos) .force\n sample-cont-prems' {f} {t} pr (p , ok-p) with p ?= t\n ... | yes refl = pr\n ... | no ¬eq = win⊢def ok-p\n\n spec-inp->incl : ∀{f g} → FSSpec-⊢ (inp f , inp g) → dom f ⊆ dom g\n spec-inp->incl {f} {g} fs {t} ok-t with fs (apply-fcoind oi ((sample-cont t win nil , f) , wit-cont out) (sample-cont-prem {f} {t} (win⊢def ok-t))) .CoInd⟦_⟧.unfold\n ... | client-end , ((_ , (out e , _)) , _) , refl , _ = ⊥-elim (e t (proj₂ (wit-cont out)))\n ... | oi , _ , refl , pr = proj₂ (fc->defined (pr (t , proj₂ (wit-cont out))))\n\n spec-out->incl : ∀{f g} → FSSpec-⊢ (out f , out g) → Witness f → dom g ⊆ dom f\n spec-out->incl {f} {g} fs wit {t} ok-t with t ∈? f\n ... | yes ok = ok\n ... | no no-t with (fs (apply-fcoind io ((sample-cont t nil win , f) , wit) (sample-cont-prems {f} {t} no-t))) .CoInd⟦_⟧.unfold\n ... | client-end , ((_ , (() , _)) , _) , refl , _\n ... | io , _ , refl , pr = ⊥-elim (cont-not-def (proj₁ (fc->defined (pr (t , ok-t)))))\n\n spec-out->wit : ∀{f g} → Witness f → FSSpec-⊢ (out f , out g) → Witness g\n spec-out->wit {f} {g} wit-f fs with Empty? g\n ... | inj₂ wit = wit\n ... | inj₁ e with (fs (apply-fcoind io ((full-cont win , f) , wit-f) λ (_ , ok) → win⊢def ok)) .CoInd⟦_⟧.unfold \n ... | client-end , ((_ , (() , _)) , _) , refl , _\n ... | io , ((_ , wit-g) , _) , refl , _ = ⊥-elim (e _ (proj₂ wit-g))\n \n {- Boundedness & Consistency -}\n \n fsspec-cons : FSSpec-⊢ ⊆ ISF[ FSubIS ] FSSpec-⊢\n fsspec-cons {nil , T} fs = nil-any , _ , refl , λ ()\n fsspec-cons {inp f , nil} fs with (fs (apply-fcoind client-end ((win , _) , (Win-win , inp)) λ ())) .CoInd⟦_⟧.unfold\n ... | client-end , ((_ , (_ , ())) , _) , refl , _\n fsspec-cons {inp f , inp g} fs = ii , ((f , g) , spec-inp->incl fs) , refl , \n λ (p , _) {R} fc-r-f → \n let wit = wit-cont (proj₁ (fc->defined fc-r-f)) in\n let fc-Or-Ig = fs (apply-fcoind oi ((sample-cont p R nil , f) , wit) (sample-cont-prem {f} {p} fc-r-f)) in\n let fc-r-g = ⊢-after-out {sample-cont p R nil} {g} {p} (proj₂ wit) fc-Or-Ig in\n subst (λ x → x ⊢ g p .force) (sym force-eq) fc-r-g\n fsspec-cons {inp f , out g} fs with Empty? f\n ... | inj₁ e = end-def , (_ , (inp e , out)) , refl , λ ()\n ... | inj₂ (t , ok-t) with (fs (apply-fcoind oi ((sample-cont t win nil , f) , wit-cont out) (sample-cont-prem {f} {t} (win⊢def ok-t)))) .CoInd⟦_⟧.unfold\n ... | client-end , ((_ , (out e , out)) , _) , refl , _ = ⊥-elim (e t (proj₂ (wit-cont out)))\n fsspec-cons {out f , nil} fs with (fs (apply-fcoind client-end ((win , _) , (Win-win , out)) λ ())) .CoInd⟦_⟧.unfold\n ... | client-end , ((_ , (_ , ())) , _) , refl , _\n fsspec-cons {out f , inp g} fs with Empty? f\n ... | inj₁ e = end-def , (_ , (out e , inp)) , refl , λ ()\n ... | inj₂ (t , ok-t) with (fs (apply-fcoind io ((full-cont win , f) , (t , ok-t)) λ (_ , ok-p) → win⊢def ok-p)) .CoInd⟦_⟧.unfold\n ... | client-end , ((_ , (() , inp)) , _) , refl , _\n fsspec-cons {out f , out g} fs with Empty? f\n ... | inj₁ e = end-def , (_ , (out e , out)) , refl , λ ()\n ... | inj₂ (t , ok-t) = \n let wit-g = spec-out->wit (t , ok-t) fs in\n let incl = spec-out->incl fs (t , ok-t) in \n oo , ((f , g) , (incl , wit-g)) , refl , λ (p , ok-p) {R} fc-r-f → \n let fc-Ir-Og = fs (apply-fcoind io ((sample-cont p R win , f) , (t , ok-t)) (sample-cont-prems' {f} {p} fc-r-f)) in\n let fc-r-g = ⊢-after-in {sample-cont p R win} {g} {p} ok-p fc-Ir-Og in\n subst (λ x → x ⊢ g p .force) (sym force-eq) fc-r-g\n\n fsspec->sub : ∀{S T} → FSSpec-⊢ (S , T) → S ≤Fc T\n fsspec->sub = coind[ FSubIS ] FSSpec-⊢ fsspec-cons\n\n postulate\n not-conv-div : ∀{T S} → ¬ T ↓ S → T ↑ S\n\n fs-convergence : ∀{T S} → FairSubtypingS T S → T ↓ S\n fs-convergence {T} {S} fs with Common.excluded-middle {T ↓ S}\n fs-convergence {T} {S} fs | yes p = p\n fs-convergence {T} {S} fs | no p =\n let div = not-conv-div p in\n let sub = ≤Fc->sub (fsspec->sub (spec-sound fs)) in\n let d-comp = discriminator-compliant sub div in\n let ¬d-comp = discriminator-not-compliant sub div in\n ⊥-elim (¬d-comp (fs d-comp))\n \n fsspec-bounded : ∀{S T} → FSSpec-⊢ (S , T) → S ≤Fᵢ T\n fsspec-bounded fs = apply-ind (inj₂ co-conv) (_ , (fs-convergence (spec-complete fs))) λ ()\n\n {- Completeness -}\n\n fs-complete : ∀{S T} → FSSpec-⊢ (S , T) → S ≤F T\n fs-complete = bounded-coind[ FSubIS , FSubCOIS ] FSSpec-⊢ fsspec-bounded fsspec-cons", "meta": {"hexsha": "4077504f226f0806b3bef3a51e6e79ec040fc91c", "size": 13148, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/FairSubtyping-IS.agda", "max_stars_repo_name": "boystrange/FairSubtypingAgda", "max_stars_repo_head_hexsha": "c4b78e70c3caf68d509f4360b9171d9f80ecb825", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2021-07-29T14:32:30.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-24T14:38:47.000Z", "max_issues_repo_path": "src/FairSubtyping-IS.agda", "max_issues_repo_name": "boystrange/FairSubtypingAgda", "max_issues_repo_head_hexsha": "c4b78e70c3caf68d509f4360b9171d9f80ecb825", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, 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YES\n2. NO\n\n", "lm_q1_score": 0.6926419704455588, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.34361540757062065}} {"text": "open import Prelude hiding (subst)\n\nmodule Implicits.Substitutions.MetaType where\n\nopen import Implicits.Syntax.Type\nopen import Implicits.Syntax.MetaType\n\nopen import Data.Fin.Substitution\nopen import Data.Star as Star hiding (map)\nopen import Data.Star.Properties\nopen import Data.Nat.Properties.Simple\nopen import Data.Vec hiding ([_])\n\nmodule MetaTypeTypeSubst where\n\n MetaSub : (ℕ → ℕ → Set) → ℕ → ℕ → ℕ → Set\n MetaSub T m ν μ = Sub (T m) ν μ\n\n record MetaLift (T : ℕ → ℕ → Set) : Set where\n field\n simple : ∀ {m} → Simple (T m)\n lift : ∀ {m ν} → T m ν → MetaType m ν\n\n module MetaTypeApp {T} (l : MetaLift T) where\n open MetaLift l\n\n infixl 8 _/_\n\n mutual\n _s/_ : ∀ {m μ ν} → MetaSimpleType m ν → MetaSub T m ν μ → MetaType m μ\n tvar x s/ σ = lift (lookup x σ)\n mvar x s/ σ = simpl (mvar x)\n (a →' b) s/ σ = simpl ((a / σ) →' (b / σ))\n tc c s/ σ = simpl (tc c)\n\n _/_ : ∀ {m μ ν} → MetaType m ν → MetaSub T m ν μ → MetaType m μ\n simpl x / σ = (x s/ σ)\n (a ⇒ b) / σ = (a / σ) ⇒ (b / σ)\n (∀' a) / σ = ∀' (a / σ ↑)\n where\n open Simple simple\n\n module _ {m : ℕ} where\n open Application (record { _/_ = _/_ {m = m} }) public using (_/✶_)\n open Simple (simple {m})\n\n →'-/✶-↑✶ : ∀ k {n n' a b} (ρs : Subs (T m) n n') →\n (simpl (a →' b)) /✶ ρs ↑✶ k ≡ simpl ((a /✶ ρs ↑✶ k) →' (b /✶ ρs ↑✶ k))\n →'-/✶-↑✶ k ε = refl\n →'-/✶-↑✶ k (r ◅ ρs) = cong₂ _/_ (→'-/✶-↑✶ k ρs) refl\n\n ⇒-/✶-↑✶ : ∀ k {n n' a b} (ρs : Subs (T m) n n') →\n (a ⇒ b) /✶ ρs ↑✶ k ≡ (a /✶ ρs ↑✶ k) ⇒ (b /✶ ρs ↑✶ k)\n ⇒-/✶-↑✶ k ε = refl\n ⇒-/✶-↑✶ k (r ◅ ρs) = cong₂ _/_ (⇒-/✶-↑✶ k ρs) refl\n\n tc-/✶-↑✶ : ∀ k {c n n'} (ρs : Subs (T m) n n') →\n (simpl (tc c)) /✶ ρs ↑✶ k ≡ simpl (tc c)\n tc-/✶-↑✶ k ε = refl\n tc-/✶-↑✶ k (r ◅ ρs) = cong₂ _/_ (tc-/✶-↑✶ k ρs) refl \n\n mvar-/✶-↑✶ : ∀ k {n n' c} (ρs : Subs (T m) n n') →\n (simpl (mvar c)) /✶ ρs ↑✶ k ≡ simpl (mvar c)\n mvar-/✶-↑✶ k ε = refl\n mvar-/✶-↑✶ k (r ◅ ρs) = cong₂ _/_ (mvar-/✶-↑✶ k ρs) refl \n\n ∀'-/✶-↑✶ : ∀ k {n n' a} (ρs : Subs (T m) n n') →\n (∀' a) /✶ ρs ↑✶ k ≡ ∀' (a /✶ ρs ↑✶ (suc k))\n ∀'-/✶-↑✶ k ε = refl\n ∀'-/✶-↑✶ k (x ◅ ρs) = cong₂ _/_ (∀'-/✶-↑✶ k ρs) refl\n\n Fin′ : ℕ → ℕ → Set\n Fin′ m ν = Fin ν\n\n module Lifted {m} {T} (lift : MetaLift T) where\n application : Application (MetaType m) (T m)\n application = record { _/_ = MetaTypeApp._/_ lift }\n\n open MetaLift lift public\n open Application application public\n open Simple (simple {m}) public\n\n varLift : MetaLift Fin′\n varLift = record {\n simple = record { var = Prelude.id ; weaken = suc }\n ; lift = (λ n → simpl (tvar n))\n }\n\n infixl 8 _/Var_\n\n _/Var_ : ∀ {m n k} → MetaType m n → Sub Fin n k → MetaType m k\n _/Var_ = MetaTypeApp._/_ varLift\n\n simple : ∀ {m} → Simple (MetaType m)\n simple = record { var = λ x → simpl (tvar x); weaken = λ x → x /Var VarSubst.wk }\n\n termLift : MetaLift MetaType\n termLift = record { simple = simple; lift = Prelude.id }\n\n private\n module ExpandSubst {n : ℕ} where\n app : Application (MetaType n) (MetaType n)\n app = record { _/_ = MetaTypeApp._/_ termLift }\n\n subst : Subst (MetaType n)\n subst = record\n { simple = simple\n ; application = app\n }\n\n open Subst subst public\n\n open ExpandSubst public hiding (var; simple)\n\nmodule MetaTypeMetaSubst where\n\n MetaSub : (ℕ → ℕ → Set) → ℕ → ℕ → ℕ → Set\n MetaSub T ν m m' = Sub (flip T ν) m m'\n\n record ExpandSimple (T : ℕ → ℕ → Set) : Set where\n field\n simple : ∀ {ν} → Simple (flip T ν)\n\n module _ {ν : ℕ} where\n open Simple (simple {ν}) public\n\n record ExpandApp (T : ℕ → ℕ → Set) : Set where\n field\n app : ∀ {ν} → Application (flip MetaType ν) (flip T ν)\n\n module _ {ν : ℕ} where\n open Application (app {ν}) public\n\n record MetaLift (T : ℕ → ℕ → Set) : Set where\n field\n exp-simple : ExpandSimple T\n lift : ∀ {m ν} → T m ν → MetaType m ν\n tpweaken : ∀ {m ν} → T m ν → T m (suc ν)\n\n open ExpandSimple exp-simple using (simple; _↑; weaken) renaming (var to evar)\n \n _↑tp : ∀ {m m' ν} → MetaSub T ν m m' → MetaSub T (suc ν) m m'\n _↑tp s = map tpweaken s\n\n field\n comm-weaken-tpweaken : ∀ {m ν} (a : T m ν) → weaken (tpweaken a) ≡ tpweaken (weaken a)\n tpweaken-var : ∀ {ν m} (n : Fin m) → (tpweaken {ν = ν} (evar n)) ≡ evar n\n\n _↑tp⋆_ : ∀ {m m' ν} → MetaSub T ν m m' → (k : ℕ) → MetaSub T (k + ν) m m'\n s ↑tp⋆ zero = s\n s ↑tp⋆ suc k = (s ↑tp⋆ k) ↑tp\n\n module MetaTypeApp {T} (l : MetaLift T) where\n open MetaLift l\n\n infixl 8 _/_\n\n mutual\n _s/_ : ∀ {m n ν} → MetaSimpleType m ν → MetaSub T ν m n → MetaType n ν\n tvar x s/ σ = simpl (tvar x)\n mvar x s/ σ = lift (lookup x σ)\n (a →' b) s/ σ = simpl ((a / σ) →' (b / σ))\n tc c s/ σ = simpl (tc c)\n\n _/_ : ∀ {m n ν} → MetaType m ν → MetaSub T ν m n → MetaType n ν\n simpl x / σ = (x s/ σ)\n (a ⇒ b) / σ = (a / σ) ⇒ (b / σ)\n (∀' a) / σ = ∀' (a / σ ↑tp)\n\n open ExpandApp (record { app = record { _/_ = _/_ } }) hiding (_/_)\n open ExpandSimple exp-simple\n\n Fin′ : ℕ → ℕ → Set\n Fin′ m ν = Fin m\n\n module Lifted {T} (lift : MetaLift T) where\n open ExpandApp (record { app = record { _/_ = MetaTypeApp._/_ lift }}) public\n open MetaLift lift public\n open ExpandSimple exp-simple public\n\n module _ where\n private\n exp-simple : ExpandSimple Fin′\n exp-simple = record { simple = record { var = Prelude.id ; weaken = suc }}\n\n open ExpandSimple exp-simple\n\n varLift : MetaLift Fin′\n varLift = record {\n tpweaken = Prelude.id\n ; exp-simple = exp-simple\n ; lift = (λ n → simpl (mvar n))\n ; comm-weaken-tpweaken = λ s → refl\n ; tpweaken-var = λ n → refl }\n\n infixl 8 _/Var_\n\n _/Var_ : ∀ {m m' ν} → MetaType m ν → Sub Fin m m' → MetaType m' ν\n _/Var_ = MetaTypeApp._/_ varLift\n\n simple : ∀ {ν} → Simple (flip MetaType ν)\n simple = record { var = λ x → simpl (mvar x); weaken = λ x → x /Var VarSubst.wk }\n\n module _ where\n exp-simple : ExpandSimple MetaType\n exp-simple = record { simple = simple }\n\n open ExpandSimple exp-simple\n module MTTS = MetaTypeTypeSubst\n\n _↑⋆tp_ : ∀ {m m' ν} → MetaSub MetaType ν m m' → ∀ k → MetaSub MetaType (k + ν) m m'\n x ↑⋆tp zero = x\n x ↑⋆tp (suc k) = map MTTS.weaken (x ↑⋆tp k)\n\n _↑tp : ∀ {m m' ν} → MetaSub MetaType ν m m' → MetaSub MetaType (suc ν) m m'\n x ↑tp = x ↑⋆tp 1\n\n tp-weaken = MTTS.weaken\n\n comm-tp/var-/var : ∀ {ν ν' m m'} (a : MetaType m ν) {s : Sub Fin ν ν'} {s' : Sub Fin m m'} →\n (a MTTS./Var s) /Var s' ≡ (a /Var s') MTTS./Var s\n comm-tp/var-/var (a ⇒ b) = cong₂ _⇒_ (comm-tp/var-/var a) (comm-tp/var-/var b)\n comm-tp/var-/var (∀' a) = cong ∀' (comm-tp/var-/var a)\n comm-tp/var-/var (simpl (tvar x)) = refl\n comm-tp/var-/var (simpl (mvar x)) = refl\n comm-tp/var-/var (simpl (a →' b)) = cong₂ (λ u v → simpl (u →' v)) (comm-tp/var-/var a)\n (comm-tp/var-/var b)\n comm-tp/var-/var (simpl (tc c)) = refl\n\n comm-weaken-tpweaken : ∀ {m ν} (a : MetaType m ν) → weaken (tp-weaken a) ≡ tp-weaken (weaken a)\n comm-weaken-tpweaken (a ⇒ b) = cong₂ _⇒_ (comm-weaken-tpweaken a) (comm-weaken-tpweaken b)\n comm-weaken-tpweaken (∀' a) = cong ∀' (comm-tp/var-/var a)\n comm-weaken-tpweaken (simpl (tvar x)) = refl\n comm-weaken-tpweaken (simpl (mvar x)) = refl\n comm-weaken-tpweaken (simpl (a →' b)) = cong₂ (λ u v → simpl (u →' v)) (comm-weaken-tpweaken a)\n (comm-weaken-tpweaken b)\n comm-weaken-tpweaken (simpl (tc c)) = refl\n\n open import Data.Vec.Properties\n\n termLift : MetaLift MetaType\n termLift = record {\n tpweaken = MetaTypeTypeSubst.weaken\n ; exp-simple = exp-simple\n ; lift = Prelude.id \n ; comm-weaken-tpweaken = comm-weaken-tpweaken\n ; tpweaken-var = λ n → refl}\n\n private\n module ExpandSubst {n : ℕ} where\n app : Application (flip MetaType n) (flip MetaType n)\n app = record { _/_ = MetaTypeApp._/_ termLift }\n\n subst : Subst (flip MetaType n)\n subst = record\n { simple = simple\n ; application = app\n }\n\n open Subst subst public\n\n open ExpandSubst public hiding (var; simple)\n\n open-meta-k : ∀ {m ν} k → (a : MetaType m (k + suc ν)) → MetaType (suc m) (k + ν)\n open-meta-k {m} {ν} k a = (weaken a) MetaTypeTypeSubst./ \n (MetaTypeTypeSubst.sub (simpl (mvar zero)) MetaTypeTypeSubst.↑⋆ k)\n\n open-meta : ∀ {m ν} → (a : MetaType m (suc ν)) → MetaType (suc m) ν\n open-meta a = open-meta-k zero a\n\n _◁m₁ : ∀ {ν m} (r : MetaType m ν) → ℕ\n _◁m₁ (a ⇒ b) = b ◁m₁ \n _◁m₁ (∀' r) = 1 + r ◁m₁ \n _◁m₁ (simpl x) = zero\n\n -- heads of metatypes\n _◁m : ∀ {ν m} (r : MetaType m ν) → (MetaType ((r ◁m₁) + m) ν)\n (a ⇒ b) ◁m = b ◁m\n ∀' r ◁m = open-meta (r ◁m)\n simpl x ◁m = simpl x\n\n smeta-weaken : ∀ {m ν} → MetaSimpleType m ν → MetaSimpleType (suc m) ν\n smeta-weaken (tc x) = tc x\n smeta-weaken (tvar n) = tvar n\n smeta-weaken (mvar m) = mvar (suc m)\n smeta-weaken (a →' b) = weaken a →' weaken b\n", "meta": {"hexsha": "f4d11d9aca983392d4c35069845acd568b4b7b43", "size": 9281, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Implicits/Substitutions/MetaType.agda", "max_stars_repo_name": "metaborg/ts.agda", "max_stars_repo_head_hexsha": "7fe638b87de26df47b6437f5ab0a8b955384958d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2019-04-05T17:57:11.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-07T04:08:41.000Z", "max_issues_repo_path": "src/Implicits/Substitutions/MetaType.agda", "max_issues_repo_name": "metaborg/ts.agda", "max_issues_repo_head_hexsha": "7fe638b87de26df47b6437f5ab0a8b955384958d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Implicits/Substitutions/MetaType.agda", "max_forks_repo_name": "metaborg/ts.agda", "max_forks_repo_head_hexsha": "7fe638b87de26df47b6437f5ab0a8b955384958d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.3379790941, "max_line_length": 99, "alphanum_fraction": 0.5276371081, "num_tokens": 3604, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6926419704455588, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.34361540757062065}} {"text": "open import Data.Product using ( proj₁ ; proj₂ )\nopen import Data.Sum using ( _⊎_ ; inj₁ ; inj₂ )\nopen import Relation.Binary.PropositionalEquality using ( refl )\nopen import Web.Semantic.DL.ABox.Interp using ( ⌊_⌋ ; ind ; _*_ )\nopen import Web.Semantic.DL.ABox.Interp.Morphism using ( _,_ )\nopen import Web.Semantic.DL.ABox.Model using\n ( _⊨a_ ; on-bnode ; bnodes ; _,_ ; ⊨a-resp-≲ )\nopen import Web.Semantic.DL.Category.Composition using ( _∙_ )\nopen import Web.Semantic.DL.Category.Properties.Composition.Lemmas using\n ( compose-left ; compose-right ; compose-resp-⊨a )\nopen import Web.Semantic.DL.Category.Properties.Tensor.Lemmas using\n ( tensor-up ; tensor-down ; tensor-resp-⊨a )\nopen import Web.Semantic.DL.Category.Object using ( Object ; IN ; fin )\nopen import Web.Semantic.DL.Category.Morphism using\n ( _⇒_ ; BN ; impl ; _⊑_ ; _≣_ ; _,_ )\nopen import Web.Semantic.DL.Category.Tensor using ( _⊗_ ; _⟨⊗⟩_ )\nopen import Web.Semantic.DL.Category.Unit using ( I )\nopen import Web.Semantic.DL.Category.Wiring using\n ( wires-≈ ; wires-≈⁻¹ ; symm )\nopen import Web.Semantic.DL.Signature using ( Signature )\nopen import Web.Semantic.DL.TBox using ( TBox )\nopen import Web.Semantic.DL.TBox.Interp using ( Δ ; _⊨_≈_ ; ≈-refl ; ≈-sym )\nopen import Web.Semantic.DL.TBox.Interp.Morphism using ( ≲-refl )\nopen import Web.Semantic.Util using\n ( _∘_ ; False ; ⊎-swap\n ; _⊕_⊕_ ; inode ; bnode ; enode ; left ; right ; up ; down )\n\nmodule Web.Semantic.DL.Category.Properties.Tensor.SymmNatural\n {Σ : Signature} {S T : TBox Σ} where\n\nsymm-natural : ∀ {A₁ A₂ B₁ B₂ : Object S T} (F₁ : A₁ ⇒ B₁) (F₂ : A₂ ⇒ B₂) →\n ((F₁ ⟨⊗⟩ F₂) ∙ symm B₁ B₂ ≣ symm A₁ A₂ ∙ (F₂ ⟨⊗⟩ F₁))\nsymm-natural {A₁} {A₂} {B₁} {B₂} F₁ F₂ = (LHS⊑RHS , RHS⊑LHS) where\n\n LHS⊑RHS : (F₁ ⟨⊗⟩ F₂) ∙ symm B₁ B₂ ⊑ symm A₁ A₂ ∙ (F₂ ⟨⊗⟩ F₁)\n LHS⊑RHS J J⊨STA J⊨LHS = (f , J⊨RHS) where\n\n f : False ⊕ (IN A₂ ⊎ IN A₁) ⊕ (BN F₂ ⊎ BN F₁) → Δ ⌊ J ⌋\n f (inode ())\n f (bnode (inj₁ x)) = ind J (inode (inj₂ x))\n f (bnode (inj₂ x)) = ind J (inode (inj₁ x))\n f (enode (inj₁ v)) = ind J (bnode (inode (inj₂ v)))\n f (enode (inj₂ v)) = ind J (bnode (inode (inj₁ v)))\n\n lemma₀ : ∀ x → \n ⌊ J ⌋ ⊨ ind J (inode (⊎-swap x)) ≈ f (bnode x)\n lemma₀ (inj₁ x) = ≈-refl ⌊ J ⌋\n lemma₀ (inj₂ y) = ≈-refl ⌊ J ⌋\n\n lemma₁ : ∀ x → \n ⌊ J ⌋ ⊨ ind J (left (up x)) ≈ on-bnode f (ind J) (right (down x))\n lemma₁ (inode x) = ≈-refl ⌊ J ⌋\n lemma₁ (bnode v) = ≈-refl ⌊ J ⌋\n lemma₁ (enode y) = wires-≈ ⊎-swap (proj₂ (fin (B₂ ⊗ B₁)) (inj₂ y)) \n (compose-right (F₁ ⟨⊗⟩ F₂) (symm B₁ B₂) J J⊨LHS)\n\n lemma₂ : ∀ x → \n ⌊ J ⌋ ⊨ ind J (left (down x)) ≈ on-bnode f (ind J) (right (up x))\n lemma₂ (inode x) = ≈-refl ⌊ J ⌋\n lemma₂ (bnode v) = ≈-refl ⌊ J ⌋\n lemma₂ (enode y) = wires-≈ ⊎-swap (proj₂ (fin (B₂ ⊗ B₁)) (inj₁ y))\n (compose-right (F₁ ⟨⊗⟩ F₂) (symm B₁ B₂) J J⊨LHS)\n\n J⊨RHS : bnodes J f ⊨a impl (symm A₁ A₂ ∙ (F₂ ⟨⊗⟩ F₁))\n J⊨RHS = compose-resp-⊨a (symm A₁ A₂) (F₂ ⟨⊗⟩ F₁) (bnodes J f) \n (wires-≈⁻¹ ⊎-swap lemma₀ (proj₁ (fin (A₂ ⊗ A₁)))) \n (tensor-resp-⊨a F₂ F₁ (right * bnodes J f) \n (⊨a-resp-≲ (≲-refl ⌊ J ⌋ , lemma₂) (impl F₂) \n (tensor-down F₁ F₂ (left * J) \n (compose-left (F₁ ⟨⊗⟩ F₂) (symm B₁ B₂) J J⊨LHS)))\n (⊨a-resp-≲ (≲-refl ⌊ J ⌋ , lemma₁) (impl F₁)\n (tensor-up F₁ F₂ (left * J)\n (compose-left (F₁ ⟨⊗⟩ F₂) (symm B₁ B₂) J J⊨LHS))))\n\n RHS⊑LHS : symm A₁ A₂ ∙ (F₂ ⟨⊗⟩ F₁) ⊑ (F₁ ⟨⊗⟩ F₂) ∙ symm B₁ B₂\n RHS⊑LHS J J⊨STA J⊨RHS = (f , J⊨LHS) where\n\n f : ((BN F₁ ⊎ BN F₂) ⊕ (IN B₁ ⊎ IN B₂) ⊕ False) → Δ ⌊ J ⌋\n f (inode (inj₁ v)) = ind J (bnode (enode (inj₂ v)))\n f (inode (inj₂ v)) = ind J (bnode (enode (inj₁ v)))\n f (bnode (inj₁ y)) = ind J (enode (inj₂ y))\n f (bnode (inj₂ y)) = ind J (enode (inj₁ y))\n f (enode ())\n\n lemma₀ : ∀ x → ⌊ J ⌋ ⊨ f (bnode (⊎-swap x)) ≈ ind J (enode x)\n lemma₀ (inj₁ y) = ≈-refl ⌊ J ⌋\n lemma₀ (inj₂ y) = ≈-refl ⌊ J ⌋\n\n lemma₁ : ∀ x →\n ⌊ J ⌋ ⊨ ind J (right (down x)) ≈ on-bnode f (ind J) (left (up x))\n lemma₁ (inode x) = ≈-sym ⌊ J ⌋\n (wires-≈ ⊎-swap (proj₂ (fin (A₂ ⊗ A₁)) (inj₂ x))\n (compose-left (symm A₁ A₂) (F₂ ⟨⊗⟩ F₁) J J⊨RHS))\n lemma₁ (bnode v) = ≈-refl ⌊ J ⌋\n lemma₁ (enode y) = ≈-refl ⌊ J ⌋\n\n lemma₂ : ∀ x → \n ⌊ J ⌋ ⊨ ind J (right (up x)) ≈ on-bnode f (ind J) (left (down x))\n lemma₂ (inode x) = ≈-sym ⌊ J ⌋\n (wires-≈ ⊎-swap (proj₂ (fin (A₂ ⊗ A₁)) (inj₁ x))\n (compose-left (symm A₁ A₂) (F₂ ⟨⊗⟩ F₁) J J⊨RHS))\n lemma₂ (bnode v) = ≈-refl ⌊ J ⌋\n lemma₂ (enode y) = ≈-refl ⌊ J ⌋\n\n J⊨LHS : bnodes J f ⊨a impl ((F₁ ⟨⊗⟩ F₂) ∙ symm B₁ B₂)\n J⊨LHS = compose-resp-⊨a (F₁ ⟨⊗⟩ F₂) (symm B₁ B₂) (bnodes J f)\n (tensor-resp-⊨a F₁ F₂ (left * bnodes J f)\n (⊨a-resp-≲ (≲-refl ⌊ J ⌋ , lemma₁) (impl F₁)\n (tensor-down F₂ F₁ (right * J)\n (compose-right (symm A₁ A₂) (F₂ ⟨⊗⟩ F₁) J J⊨RHS)))\n (⊨a-resp-≲ (≲-refl ⌊ J ⌋ , lemma₂) (impl F₂)\n (tensor-up F₂ F₁ (right * J)\n (compose-right (symm A₁ A₂) (F₂ ⟨⊗⟩ F₁) J J⊨RHS))))\n (wires-≈⁻¹ ⊎-swap lemma₀ (proj₁ (fin (B₂ ⊗ B₁))))", "meta": {"hexsha": "fc2722e1c16622617635bf186fc33d7f23716c0c", "size": 5096, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Web/Semantic/DL/Category/Properties/Tensor/SymmNatural.agda", "max_stars_repo_name": "agda/agda-web-semantic", "max_stars_repo_head_hexsha": "8ddbe83965a616bff6fc7a237191fa261fa78bab", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2015-09-13T17:46:41.000Z", "max_stars_repo_stars_event_max_datetime": "2020-03-14T14:21:08.000Z", "max_issues_repo_path": "src/Web/Semantic/DL/Category/Properties/Tensor/SymmNatural.agda", "max_issues_repo_name": "agda/agda-web-semantic", "max_issues_repo_head_hexsha": "8ddbe83965a616bff6fc7a237191fa261fa78bab", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2018-11-14T02:32:28.000Z", "max_issues_repo_issues_event_max_datetime": "2021-01-04T20:57:19.000Z", "max_forks_repo_path": "src/Web/Semantic/DL/Category/Properties/Tensor/SymmNatural.agda", "max_forks_repo_name": "agda/agda-web-semantic", "max_forks_repo_head_hexsha": "8ddbe83965a616bff6fc7a237191fa261fa78bab", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2017-12-03T14:52:09.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-12T11:40:03.000Z", "avg_line_length": 44.701754386, "max_line_length": 76, "alphanum_fraction": 0.5494505495, "num_tokens": 2456, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7577943712746407, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.34347927713960125}} {"text": "-- {-# OPTIONS -v tc.conv.level:60 #-}\n-- {-# OPTIONS -v tc.conv:30 #-}\n\n{- Agda development version: Wed Oct 30 16:30:06 GMT 2013\n\n The last line of code triggers the following error,\n but replacing '_' with 'a' typechecks just fine.\n\n Bug.agda:32,8-11\n tt != a of type ⊤\n when checking that the expression s _ has type P tt → P a\n\n Changing 'Set (q a)' to 'Set' in line 26 suppresses the error.\n-}\n\n-- Andreas, 2013-10-31 Fixed by retrying sort comparison after\n-- successful type comparison (which might have solve the missing level metas).\n\nmodule Issue930 where\n\nopen import Common.Level\n\ndata ⊤ : Set where\n tt : ⊤\n\npostulate\n q : ⊤ → Level\n P : (a : ⊤) → Set (q a)\n s : (a : ⊤) → P tt → P a\n a : ⊤\n g : (P tt → P a) → ⊤\n\nv : ⊤\nv = g (s _)\n\n{-\ncoerce term v = s ?1\n from type t1 = P tt → P ?1\n to type t2 = P tt → P a\nequalSort\n Set (q tt ⊔ q ?1) == Set (q a ⊔ q tt)\ncompareAtom q tt == q a : Level\ncompareTerm tt == a : ⊤\nsort comparison failed -- THIS ERROR IS CAUGHT, BUT RETHROWN AT THE END\ncompareTerm P tt → P ?1 =< P tt → P a : Set (q tt ⊔ q ?1)\ncompare function types\n t1 = P tt → P ?1\n t2 = P tt → P a\nequalSort\n Set (q ?1) == Set (q a)\ncompareTerm ?1 == a : ⊤\nattempting shortcut ?1 := a\nsolving _13 := a\n\n-}\n", "meta": {"hexsha": "f1ff8f5b2fda1c842a2af126b2ece0d275044909", "size": 1269, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/Succeed/Issue930.agda", "max_stars_repo_name": "shlevy/agda", "max_stars_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1989, "max_stars_repo_stars_event_min_datetime": "2015-01-09T23:51:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T18:20:48.000Z", "max_issues_repo_path": "test/Succeed/Issue930.agda", "max_issues_repo_name": "shlevy/agda", "max_issues_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 4066, "max_issues_repo_issues_event_min_datetime": "2015-01-10T11:24:51.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T21:14:49.000Z", "max_forks_repo_path": "test/Succeed/Issue930.agda", "max_forks_repo_name": "Agda-zh/agda", "max_forks_repo_head_hexsha": "231d6ad8e77b67ff8c4b1cb35a6c31ccd988c3e9", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 371, "max_forks_repo_forks_event_min_datetime": "2015-01-03T14:04:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-30T19:00:30.000Z", "avg_line_length": 22.6607142857, "max_line_length": 79, "alphanum_fraction": 0.5957446809, "num_tokens": 453, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34342015362247647}} {"text": "open import Relation.Binary.PropositionalEquality using (_≡_; refl; subst)\nopen import Data.Sum\n\nimport SingleSorted.AlgebraicTheory as SS\n\nmodule SingleSorted.Combinators where\n\n\nmodule Sum {𝓈} (Σ₁ Σ₂ : SS.Signature) (T₁ : SS.Theory 𝓈 Σ₁) (T₂ : SS.Theory 𝓈 Σ₂) where\n\n -- disjoint sum of signatures\n S : SS.Signature\n S = record { oper = SS.Signature.oper Σ₁ ⊎ SS.Signature.oper Σ₂\n ; oper-arity = [ SS.Signature.oper-arity Σ₁ , SS.Signature.oper-arity Σ₂ ]\n }\n\n inj-term-l : ∀ {Γ : SS.Context} → SS.Signature.Term Σ₁ Γ → SS.Signature.Term S Γ\n inj-term-l {Γ} (SS.Signature.tm-var x) = SS.Signature.tm-var x\n inj-term-l {Γ} (SS.Signature.tm-oper f ts) = SS.Signature.tm-oper (inj₁ f) λ{ i → inj-term-l (ts i)}\n\n inj-term-r : ∀ {Γ : SS.Context} → SS.Signature.Term Σ₂ Γ → SS.Signature.Term S Γ\n inj-term-r {Γ} (SS.Signature.tm-var x) = SS.Signature.tm-var x\n inj-term-r {Γ} (SS.Signature.tm-oper f ts) = SS.Signature.tm-oper (inj₂ f) λ{ i → inj-term-r (ts i)}\n\n coerce₁ : SS.Signature.Equation Σ₁ → SS.Signature.Equation S\n coerce₁ eq = record { eq-ctx = SS.Signature.Equation.eq-ctx eq\n ; eq-lhs = inj-term-l (SS.Signature.Equation.eq-lhs eq)\n ; eq-rhs = inj-term-l (SS.Signature.Equation.eq-rhs eq)\n }\n\n coerce₂ : SS.Signature.Equation Σ₂ → SS.Signature.Equation S\n coerce₂ eq = record { eq-ctx = SS.Signature.Equation.eq-ctx eq\n ; eq-lhs = inj-term-r (SS.Signature.Equation.eq-lhs eq)\n ; eq-rhs = inj-term-r (SS.Signature.Equation.eq-rhs eq)\n }\n\n -- define a theory with the set of axioms a union of the axioms of both theories\n T : SS.Theory 𝓈 S\n T = record { ax = SS.Theory.ax T₁ ⊎ SS.Theory.ax T₂\n ; ax-eq = [ (λ a → coerce₁ (SS.Theory.ax-eq T₁ a)) , (λ a → coerce₂ (SS.Theory.ax-eq T₂ a)) ]\n }\n", "meta": {"hexsha": "3c53b1d074db999c6dd5a0098696b17100128487", "size": 1895, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/SingleSorted/Combinators.agda", "max_stars_repo_name": "cilinder/formaltt", "max_stars_repo_head_hexsha": "0a9d25e6e3965913d9b49a47c88cdfb94b55ffeb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 21, "max_stars_repo_stars_event_min_datetime": "2021-02-16T14:07:06.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-19T15:50:08.000Z", "max_issues_repo_path": "src/SingleSorted/Combinators.agda", "max_issues_repo_name": "andrejbauer/formaltt", "max_issues_repo_head_hexsha": "2aaf850bb1a262681c5a232cdefae312f921b9d4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-04-30T14:18:25.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-14T16:15:17.000Z", "max_forks_repo_path": "src/SingleSorted/Combinators.agda", "max_forks_repo_name": "andrejbauer/formaltt", "max_forks_repo_head_hexsha": "2aaf850bb1a262681c5a232cdefae312f921b9d4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2021-02-16T13:43:07.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-24T02:51:43.000Z", "avg_line_length": 45.119047619, "max_line_length": 106, "alphanum_fraction": 0.6121372032, "num_tokens": 624, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160664, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.34342015362247635}} {"text": "----------------------------------------------------------------------\n-- Functional small-step semantics\n----------------------------------------------------------------------\n\nmodule SystemF.Reduction where\n\nopen import Codata.Musical.Notation\nopen import Category.Monad\nopen import Category.Monad.Partiality.All\nopen import Data.Maybe as Maybe using (Maybe; just; nothing; map)\nopen import Data.Maybe.Relation.Unary.All as MaybeAll using (nothing; just)\nopen import Data.Maybe.Relation.Unary.Any as MaybeAny using (just)\nopen import Data.Nat using (ℕ; _+_)\nopen import Data.Unit using (tt)\nopen import Data.Vec using ([])\nopen import Relation.Binary.PropositionalEquality as P using (_≡_; _≢_)\nopen import Relation.Nullary\n\nopen import PartialityAndFailure as PF\nopen PF.Equality hiding (fail)\nopen PF.Equivalence\nprivate\n open module M {f} = RawMonad (PF.monad {f}) using (_>>=_; return)\n\nopen import SystemF.Type\nopen import SystemF.Term\nopen import SystemF.WtTerm\nopen import SystemF.Eval hiding (type-soundness)\n\nopen TypeSubst using () renaming (_[/_] to _[/tp_])\nopen TermTypeSubst using () renaming (_[/_] to _[/tmTp_])\nopen TermTermSubst using () renaming (_[/_] to _[/tmTm_])\nopen WtTermTypeSubst using () renaming (_[/_]′ to _[/⊢tmTp_])\nopen WtTermTermSubst using () renaming (_[/_] to _[/⊢tmTm_])\n\n----------------------------------------------------------------------\n-- Functional call-by-value small-step semantics in the partiality\n-- monad\n\n-- The functional presentation of the small-step semantics below is\n-- heavily inspired by Danielsson's ICFP'12 paper \"Operational\n-- Semantics Using the Partiality Monad\". Whereas the paper\n-- illustrates the technique to give a functional abstract machine\n-- semantics (i.e. the semantics of a \"VM\"), we skip the compilation\n-- step and directly reduce terms, which results in a (functional)\n-- structural operational semantics (SOS). We adopt many of the\n-- conventions used in the accompanying code, which can be found at\n--\n-- http://www.cse.chalmers.se/~nad/publications/danielsson-semantics-partiality-monad.tgz\n--\n-- As pointed out in Danielsson's paper, the functional presentation\n-- of the semantics feels rather natural in that it follows the form\n-- of an interpreter, and it has the added advantage of proving that\n-- the semantics are deterministic and computable \"for free\".\n--\n-- For more information about Danielson's paper see\n--\n-- http://www.cse.chalmers.se/~nad/publications/danielsson-semantics-partiality-monad.html\n\n\n----------------------------------------------------------------------\n-- Small-step call-by-value reduction\n\n-- Results of reduction steps\ndata Result (m n : ℕ) : Set where\n continue : (t : Term m n) → Result m n -- further reducible term\n done : (v : Val m n) → Result m n -- irreducible value\n\n-- Take a single reduction step (if possible).\nstep : ∀ {m n} → Term m n → Maybe (Result m n)\nstep (var x) = nothing\nstep (Λ t) = just (done (Λ t))\nstep (λ' a t) = just (done (λ' a t))\nstep (μ a t) = just (continue (t [/tmTm μ a t ]))\nstep (t [ a ]) with step t\n... | just (continue t′) = just (continue (t′ [ a ]))\n... | just (done (Λ t′)) = just (continue (t′ [/tmTp a ]))\n... | just (done _) = nothing\n... | nothing = nothing\nstep {m} {n} (s · t) with step s\n... | just (continue s′) = just (continue (s′ · t))\n... | just (done v) = nested v\n where -- Call-by-value: only reduce (s · t) if s is a value.\n nested : Val m n → Maybe (Result m n)\n nested v with step t\n nested v | (just (continue t′)) = just (continue (⌜ v ⌝ · t′))\n nested (λ' _ s′) | (just (done v)) = just (continue (s′ [/tmTm ⌜ v ⌝ ]))\n nested _ | (just (done _)) = nothing\n nested _ | nothing = nothing\n... | nothing = nothing\nstep (fold a t) with step t\n... | just (continue t′) = just (continue (fold a t′))\n... | just (done v) = just (done (fold a v))\n... | nothing = nothing\nstep (unfold a t) with step t\n... | just (continue t′) = just (continue (unfold a t′))\n... | just (done (fold _ v)) = just (done v)\n... | just (done _) = nothing\n... | nothing = nothing\n\n\n----------------------------------------------------------------------\n-- Type soundness\n\ninfix 4 _⊢res_∈_ _⊢res?_∈_\n\n-- Well-typedness lifted to results of reduction steps.\ndata _⊢res_∈_ {m n} (Γ : Ctx m n) : Result m n → Type n → Set where\n ⊢continue : ∀ {t a} → Γ ⊢ t ∈ a → Γ ⊢res continue t ∈ a\n ⊢done : ∀ {v a} → Γ ⊢val v ∈ a → Γ ⊢res done v ∈ a\n\n-- Well-typedness lifted to possibly undefined reduction steps.\n_⊢res?_∈_ : ∀ {m n} → Ctx m n → Maybe (Result m n) → Type n → Set\nΓ ⊢res? r? ∈ a = MaybeAll.All (λ r → Γ ⊢res r ∈ a) r?\n\n-- Preservation of well-typedness: a well-typed term reduces in one\n-- step to a result of the same type or fails to reduce.\n⊢step : ∀ {m n} {Γ : Ctx m n} {t a} → Γ ⊢ t ∈ a → Γ ⊢res? step t ∈ a\n⊢step (var x) = nothing\n⊢step (Λ ⊢t) = just (⊢done (Λ ⊢t))\n⊢step (λ' a ⊢t) = just (⊢done (λ' a ⊢t))\n⊢step (μ a ⊢t) = just (⊢continue (⊢t [/⊢tmTm μ a ⊢t ]))\n⊢step {t = t [ a ]} (⊢t [ .a ]) with step t | ⊢step ⊢t\n... | just ._ | just (⊢continue ⊢t′) = just (⊢continue (⊢t′ [ a ]))\n... | just ._ | just (⊢done (Λ ⊢t′)) = just (⊢continue (⊢t′ [/⊢tmTp a ]))\n... | nothing | nothing = nothing\n⊢step {m} {n} {Γ} {s · t} {b} (⊢s · ⊢t) with step s | ⊢step ⊢s\n... | just ._ | just (⊢continue ⊢s′) = just (⊢continue (⊢s′ · ⊢t))\n... | just (done (λ' a s′)) | just (⊢done (λ' .a ⊢s′)) = nested\n where\n nested : Γ ⊢res? step ((λ' a s′) · t) ∈ b\n nested with step t | ⊢step ⊢t\n ... | just ._ | just (⊢continue ⊢t′) = just (⊢continue ((λ' a ⊢s′) · ⊢t′))\n ... | just ._ | just (⊢done v) = just (⊢continue (⊢s′ [/⊢tmTm ⊢⌜ v ⌝ ]))\n ... | nothing | nothing = nothing\n... | nothing | nothing = nothing\n⊢step {t = fold a t} (fold .a ⊢t) with step t | ⊢step ⊢t\n... | just ._ | just (⊢continue ⊢t′) = just (⊢continue (fold a ⊢t′))\n... | just ._ | just (⊢done ⊢v) = just (⊢done (fold a ⊢v))\n... | nothing | nothing = nothing\n⊢step {t = unfold a t} (unfold .a ⊢t) with step t | ⊢step ⊢t\n... | just ._ | just (⊢continue ⊢t′) = just (⊢continue (unfold a ⊢t′))\n... | just ._ | just (⊢done (fold .a ⊢v)) = just (⊢done ⊢v)\n... | nothing | nothing = nothing\n\n-- Progress: reduction of well-typed closed terms does not fail.\nprogress : ∀ {t} {a : Type 0} → [] ⊢ t ∈ a → Maybe.Is-just (step t)\nprogress (var ())\nprogress (Λ t) = just tt\nprogress (λ' a t) = just tt\nprogress (μ a t) = just tt\nprogress {t [ a ]} (⊢t [ .a ]) with step t | ⊢step ⊢t | progress ⊢t\n... | just ._ | just (⊢continue _) | just tt = just tt\n... | just ._ | just (⊢done (Λ _)) | just tt = just tt\nprogress {s · t} (⊢s · ⊢t) with step s | ⊢step ⊢s | progress ⊢s\n... | just ._ | just (⊢continue _) | just tt = just tt\n... | just (done (λ' a s′)) | just (⊢done (λ' .a _)) | just tt = nested\n where\n nested : Maybe.Is-just (step ((λ' a s′) · t))\n nested with step t | ⊢step ⊢t | progress ⊢t\n ... | just ._ | just (⊢continue _) | just tt = just tt\n ... | just ._ | just (⊢done _) | just tt = just tt\nprogress {fold a t} (fold .a ⊢t) with step t | ⊢step ⊢t | progress ⊢t\n... | just ._ | just (⊢continue _) | just tt = just tt\n... | just ._ | just (⊢done _) | just tt = just tt\nprogress {unfold a t} (unfold .a ⊢t) with step t | ⊢step ⊢t | progress ⊢t\n... | just ._ | just (⊢continue _) | just tt = just tt\n... | just ._ | just (⊢done (fold .a _)) | just tt = just tt\n\ninfix 7 _↓ ⊢_↓\n\n-- Evaluation of untyped (open) terms in the partiality monad via\n-- repeated reduction.\n_↓ : ∀ {m n} → Term m n → Comp m n\nt ↓ with step t\n... | just (continue t′) = later (♯ (t′ ↓))\n... | just (done v) = return v\n... | nothing = fail\n\n-- Evaluation of closed terms preserves well-typedness.\n⊢_↓ : ∀ {t a} → [] ⊢ t ∈ a → [] ⊢comp t ↓ ∈ a\n⊢_↓ {t} ⊢t with step t | ⊢step ⊢t | progress ⊢t\n... | just (continue t′) | just (⊢continue ⊢t′) | just tt = later (♯ ⊢ ⊢t′ ↓)\n... | just (done v) | just (⊢done ⊢v) | just tt = now (just ⊢v)\n... | nothing | _ | ()\n\n-- Type soundness: evaluation of well-typed terms does not fail.\ntype-soundness : ∀ {t a} → [] ⊢ t ∈ a → ¬ t ↓ ≈ fail\ntype-soundness ⊢t = does-not-fail ⊢ ⊢t ↓\n\n\n----------------------------------------------------------------------\n-- Strong bisimilarity of big-step and small-step semantics\n\nopen PF.AlternativeEquality\n renaming (return to returnP; fail to failP; _>>=_ to _>>=P_)\n\n-- Lemma: values don't reduce.\nstep-val : ∀ {m n} (v : Val m n) → step ⌜ v ⌝ ≡ just (done v)\nstep-val (Λ a) = P.refl\nstep-val (λ' a t) = P.refl\nstep-val (fold a v) with step ⌜ v ⌝ | step-val v\n... | just (continue t) | ()\n... | just (done w) | w≡v = P.cong (map (lower a)) w≡v\n where lower : ∀ {m n} → Type (1 + n) → Result m n → Result m n\n lower a (done v) = done (fold a v)\n lower a (continue t) = continue t\n... | nothing | ()\n\n-- Lemma: _↓ \"preserves\" values.\n↓-val : ∀ {m n} (v : Val m n) → ⌜ v ⌝ ↓ ≡ return v\n↓-val v with step ⌜ v ⌝ | step-val v\n↓-val v | just (continue t) | ()\n↓-val v | just (done w) | w≡v = P.cong toComp w≡v\n where toComp : ∀ {m n} → Maybe (Result m n) → Comp m n\n toComp (just (continue t′)) = later (♯ (t′ ↓))\n toComp (just (done v)) = return v\n toComp nothing = fail\n↓-val v | nothing | ()\n\nmutual\n infix 7 _[_]⇓≅↓′ _·⇓≅↓′_\n\n -- A helper lemma relating reduction and evaluation of type\n -- application.\n _[_]⇓≅↓′ : ∀ {m n} (t : Term m n) (a : Type n) →\n (t ↓) [ a ]⇓ ≅P (t [ a ]) ↓\n t [ a ]⇓≅↓′ with step t\n ... | just (continue t′) = later (♯ (t′ [ a ]⇓≅↓′))\n ... | just (done (Λ t′)) = later (♯ ⇓≅↓′ (t′ [/tmTp a ]) )\n ... | just (done (λ' _ _)) = failP\n ... | just (done (fold _ _)) = failP\n ... | nothing = failP\n\n -- A helper lemma relating reduction and evaluation of term\n -- application.\n _·⇓≅↓′_ : ∀ {m n} (s : Term m n) (t : Term m n) →\n (s ↓) ·⇓ (t ↓) ≅P (s · t) ↓\n s ·⇓≅↓′ t with step s\n s ·⇓≅↓′ t | just (continue s′) = later (♯ (s′ ·⇓≅↓′ t))\n s ·⇓≅↓′ t | just (done v) with step t\n s ·⇓≅↓′ t | just (done v) | just (continue t′) = later (♯ subst v)\n where\n subst : ∀ v → (return v) ·⇓ (t′ ↓) ≅P (⌜ v ⌝ · t′) ↓\n subst v with ⌜ v ⌝ ↓ | ↓-val v | ⌜ v ⌝ ·⇓≅↓′ t′\n subst v | now (just .v) | P.refl | v≅t′ = v≅t′\n subst _ | now nothing | () | _\n subst _ | later x₁ | () | _\n s ·⇓≅↓′ t | just (done (Λ s′)) | just (done w) = failP\n s ·⇓≅↓′ t | just (done (λ' a s′)) | just (done w) =\n later (♯ ⇓≅↓′ (s′ [/tmTm ⌜ w ⌝ ]))\n s ·⇓≅↓′ t | just (done (fold x v)) | just (done w) = failP\n s ·⇓≅↓′ t | just (done v) | nothing = failP\n s ·⇓≅↓′ t | nothing = failP\n\n -- A helper lemma relating reduction and evaluation of recursive\n -- type folding.\n fold⇓≅↓′ : ∀ {m n} (a : Type (1 + n)) (t : Term m n) →\n fold⇓ a (t ↓) ≅P (fold a t) ↓\n fold⇓≅↓′ a t with step t\n ... | just (continue t′) = later (♯ fold⇓≅↓′ a t′)\n ... | just (done v) = returnP P.refl\n ... | nothing = failP\n\n -- A helper lemma relating reduction and evaluation of recursive\n -- type unfolding.\n unfold⇓≅↓′ : ∀ {m n} (a : Type (1 + n)) (t : Term m n) →\n unfold⇓ a (t ↓) ≅P (unfold a t) ↓\n unfold⇓≅↓′ a t with step t\n ... | just (continue t′) = later (♯ unfold⇓≅↓′ a t′)\n ... | just (done (Λ _)) = failP\n ... | just (done (λ' _ _)) = failP\n ... | just (done (fold _ v)) = returnP P.refl\n ... | nothing = failP\n\n -- Big-step evaluation and small-step reduction are strongly bisimliar.\n ⇓≅↓′ : ∀ {m n} (t : Term m n) → t ⇓ ≅P t ↓\n ⇓≅↓′ (var x) = failP\n ⇓≅↓′ (Λ t) = returnP P.refl\n ⇓≅↓′ (λ' a t) = returnP P.refl\n ⇓≅↓′ (μ a t) = later (♯ ⇓≅↓′ (t [/tmTm μ a t ]))\n ⇓≅↓′ (t [ a ]) =\n (t [ a ]) ⇓ ≅⟨ complete ([]-comp t a) ⟩\n (t ⇓) [ a ]⇓ ≅⟨ (⇓≅↓′ t) >>=P (λ v → (_ ∎)) ⟩\n (t ↓) [ a ]⇓ ≅⟨ t [ a ]⇓≅↓′ ⟩\n (t [ a ]) ↓ ∎\n ⇓≅↓′ (s · t) =\n (s · t) ⇓ ≅⟨ complete (·-comp s t) ⟩\n (s ⇓) ·⇓ (t ⇓) ≅⟨ (⇓≅↓′ s >>=P λ v → ⇓≅↓′ t >>=P λ w → (_ ∎)) ⟩\n (s ↓) ·⇓ (t ↓) ≅⟨ s ·⇓≅↓′ t ⟩\n (s · t) ↓ ∎\n ⇓≅↓′ (fold a t) =\n (fold a t) ⇓ ≅⟨ complete (fold-comp a t) ⟩\n fold⇓ a (t ⇓) ≅⟨ (⇓≅↓′ t) >>=P (λ v → (_ ∎)) ⟩\n fold⇓ a (t ↓) ≅⟨ fold⇓≅↓′ a t ⟩\n (fold a t) ↓ ∎\n ⇓≅↓′ (unfold a t) =\n (unfold a t) ⇓ ≅⟨ complete (unfold-comp a t) ⟩\n unfold⇓ a (t ⇓) ≅⟨ (⇓≅↓′ t) >>=P (λ v → (_ ∎)) ⟩\n unfold⇓ a (t ↓) ≅⟨ unfold⇓≅↓′ a t ⟩\n (unfold a t) ↓ ∎\n\n-- Big-step evaluation and small-step reduction are strongly bisimliar.\n⇓≅↓ : ∀ {m n} (t : Term m n) → t ⇓ ≅ t ↓\n⇓≅↓ t = sound (⇓≅↓′ t)\n", "meta": {"hexsha": "eea0a4932fa88df0f45e6ee26a74b02042405df1", "size": 13081, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/SystemF/Reduction.agda", "max_stars_repo_name": "sstucki/system-f-agda", "max_stars_repo_head_hexsha": "ea262cf7714cdb762643f10275c568596f57cd1d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 68, "max_stars_repo_stars_event_min_datetime": "2015-05-26T13:12:56.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-01T01:25:16.000Z", "max_issues_repo_path": "src/SystemF/Reduction.agda", 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YES\n2. YES\n\n", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3431612818573753}} {"text": "{-# OPTIONS --without-K --safe #-}\n\nmodule Dodo.Binary.Transitive where\n\n-- Stdlib imports\nimport Relation.Binary.PropositionalEquality as Eq\nopen Eq using (_≡_; _≢_; refl)\nopen import Level using (Level; _⊔_)\nopen import Function using (flip; _∘_)\nopen import Data.Sum using (_⊎_; inj₁; inj₂)\nopen import Data.Product using (_,_; _×_; proj₁; proj₂; ∃-syntax)\nopen import Relation.Unary using (Pred; _∈_)\nopen import Relation.Binary using (Rel; REL)\nopen import Relation.Binary using (Transitive)\nopen import Relation.Binary.Construct.Closure.Transitive using (TransClosure; [_]; _∷_; _∷ʳ_; _++_)\n-- Local imports\nopen import Dodo.Unary.Equality\nopen import Dodo.Binary.Equality\nopen import Dodo.Binary.Domain\n\n\n⁺-flip : {a ℓ : Level} {A : Set a} → {R : Rel A ℓ}\n → {x y : A}\n → TransClosure R x y\n → TransClosure (flip R) y x\n⁺-flip [ x∼y ] = [ x∼y ]\n⁺-flip (x∼z ∷ z∼⁺y) = ⁺-flip z∼⁺y ∷ʳ x∼z\n\n⁺-map : {a b ℓ₁ ℓ₂ : Level} {A : Set a} {B : Set b}\n → {R : Rel A ℓ₁} {Q : Rel B ℓ₂}\n → (f : A → B)\n → (map : ∀ {x y : A} → R x y → Q (f x) (f y))\n → {x y : A}\n → TransClosure R x y\n → TransClosure Q (f x) (f y)\n⁺-map _ map [ Rxy ] = [ map Rxy ]\n⁺-map _ map ( Rxz ∷ R⁺zy ) = map Rxz ∷ ⁺-map _ map R⁺zy\n\n⁺-dom : ∀ {a ℓ : Level} {A : Set a} {R : Rel A ℓ}\n → {x y : A}\n → TransClosure R x y\n ------------------\n → x ∈ dom R\n⁺-dom {R = R} [ Rxy ] = take-dom R Rxy\n⁺-dom {R = R} ( Rxz ∷ R⁺zy ) = take-dom R Rxz\n\n⁺-codom : ∀ {a ℓ : Level} {A : Set a} {R : Rel A ℓ}\n → {x y : A}\n → TransClosure R x y\n ------------------\n → y ∈ codom R\n⁺-codom {R = R} [ Rxy ] = take-codom R Rxy\n⁺-codom {R = R} ( Rxz ∷ R⁺zy ) = ⁺-codom R⁺zy\n\n⁺-udrˡ : ∀ {a ℓ : Level} {A : Set a} {R : Rel A ℓ}\n → {x y : A}\n → TransClosure R x y\n ------------------\n → x ∈ udr R\n⁺-udrˡ = inj₁ ∘ ⁺-dom\n\n⁺-udrʳ : ∀ {a ℓ : Level} {A : Set a} {R : Rel A ℓ}\n → {x y : A}\n → TransClosure R x y\n ------------------\n → y ∈ udr R\n⁺-udrʳ = inj₂ ∘ ⁺-codom\n\n⁺-predʳ : {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {R : Rel A ℓ₂}\n → (f : ∀ {x y : A} → P x → R x y → P y)\n → {x y : A}\n → TransClosure R x y\n → P x\n → P y\n⁺-predʳ f [ Rxy ] Px = f Px Rxy\n⁺-predʳ f ( Rxz ∷ R⁺zy ) Px = ⁺-predʳ f R⁺zy (f Px Rxz)\n\n⁺-predˡ : {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {R : Rel A ℓ₂}\n → (f : ∀ {x y : A} → P y → R x y → P x)\n → {x y : A}\n → TransClosure R x y\n → P y\n → P x\n⁺-predˡ f [ Rxy ] Px = f Px Rxy\n⁺-predˡ f ( Rxz ∷ R⁺zy ) Px = f (⁺-predˡ f R⁺zy Px) Rxz\n\n⁺-lift-predʳ : {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {R : Rel A ℓ₂}\n → (f : ∀ {x y : A} → R x y → P y)\n → {x y : A}\n → TransClosure R x y\n → P y\n⁺-lift-predʳ f [ Rxy ] = f Rxy\n⁺-lift-predʳ f ( Rxz ∷ R⁺zy ) = ⁺-lift-predʳ f R⁺zy\n\n⁺-lift-predˡ : {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {R : Rel A ℓ₂}\n → (f : ∀ {x y : A} → R x y → P x)\n → {x y : A}\n → TransClosure R x y\n → P x\n⁺-lift-predˡ f [ Rxy ] = f Rxy\n⁺-lift-predˡ f ( Rxz ∷ R⁺zy ) = f Rxz\n\n\nmodule _ {a ℓ : Level} {A : Set a} {R : Rel A ℓ} where\n\n ⁺-idem : TransClosure R ⇔₂ TransClosure (TransClosure R)\n ⁺-idem = ⇔: ⊆-proof ⊇-proof\n where\n ⊆-proof : TransClosure R ⊆₂' TransClosure (TransClosure R)\n ⊆-proof _ _ R⁺xy = [ R⁺xy ]\n \n ⊇-proof : TransClosure (TransClosure R) ⊆₂' TransClosure R\n ⊇-proof _ _ [ R⁺xy ] = R⁺xy\n ⊇-proof _ y ( _∷_ {_} {z} R⁺xz R⁺⁺zy ) = R⁺xz ++ ⊇-proof z y R⁺⁺zy\n\n ⁺-join : ∀ {x y : A} → TransClosure (TransClosure R) x y → TransClosure R x y\n ⁺-join = ⇔₂-apply-⊇₂ ⁺-idem\n\n\nmodule _ {a ℓ : Level} {A : Set a} {R : Rel A ℓ} where\n\n ⁺-preserves-dom : dom R ⇔₁ dom (TransClosure R)\n ⁺-preserves-dom = ⇔: ⊆-proof ⊇-proof\n where\n ⊆-proof : dom R ⊆₁' dom (TransClosure R)\n ⊆-proof x (y , Rxy) = (y , [ Rxy ])\n\n ⊇-proof : dom (TransClosure R) ⊆₁' dom R\n ⊇-proof x (y , R⁺xy) = ⁺-dom R⁺xy\n\n ⁺-preserves-codom : codom R ⇔₁ codom (TransClosure R)\n ⁺-preserves-codom = ⇔: ⊆-proof ⊇-proof\n where\n ⊆-proof : codom R ⊆₁' codom (TransClosure R)\n ⊆-proof y (x , Rxy) = (x , [ Rxy ])\n\n ⊇-proof : codom (TransClosure R) ⊆₁' codom R\n ⊇-proof y (x , R⁺xy) = ⁺-codom R⁺xy\n\n ⁺-preserves-udr : udr R ⇔₁ udr (TransClosure R)\n ⁺-preserves-udr = ⇔: ⊆-proof ⊇-proof\n where\n ⊆-proof : udr R ⊆₁' udr (TransClosure R)\n ⊆-proof _ (inj₁ x∈dom) = inj₁ (⇔₁-apply-⊆₁ ⁺-preserves-dom x∈dom)\n ⊆-proof _ (inj₂ x∈codom) = inj₂ (⇔₁-apply-⊆₁ ⁺-preserves-codom x∈codom)\n \n ⊇-proof : udr (TransClosure R) ⊆₁' udr R\n ⊇-proof _ (inj₁ x∈dom) = inj₁ (⇔₁-apply-⊇₁ ⁺-preserves-dom x∈dom)\n ⊇-proof _ (inj₂ x∈codom) = inj₂ (⇔₁-apply-⊇₁ ⁺-preserves-codom x∈codom)\n\n\nmodule _ {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Rel A ℓ₁} {Q : Rel A ℓ₂} where\n\n ⁺-preserves-⊆₂ : P ⊆₂ Q → TransClosure P ⊆₂ TransClosure Q\n ⁺-preserves-⊆₂ P⊆Q = ⊆: lemma\n where\n lemma : TransClosure P ⊆₂' TransClosure Q\n lemma _ _ [ Pxy ] = [ ⊆₂-apply P⊆Q Pxy ]\n lemma _ y ( _∷_ {_} {z} Pxz P⁺zy ) = ⊆₂-apply P⊆Q Pxz ∷ lemma z y P⁺zy\n\n\nmodule _ {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Rel A ℓ₁} {Q : Rel A ℓ₂} where\n\n ⁺-preserves-⇔₂ : P ⇔₂ Q → TransClosure P ⇔₂ TransClosure Q\n ⁺-preserves-⇔₂ = ⇔₂-compose ⁺-preserves-⊆₂ ⁺-preserves-⊆₂\n\n\nmodule _ {a ℓ : Level} {A : Set a} {R : Rel A ℓ} where\n\n ⁺-trans-⇔₂ : Transitive R → R ⇔₂ TransClosure R\n ⁺-trans-⇔₂ transR = ⇔: ⊆-proof ⊇-proof\n where\n ⊆-proof : R ⊆₂' TransClosure R\n ⊆-proof _ _ = [_]\n \n ⊇-proof : TransClosure R ⊆₂' R\n ⊇-proof _ _ [ Rxy ] = Rxy\n ⊇-proof _ y (_∷_ {_} {w} Rxw R⁺wy) = transR Rxw (⊇-proof w y R⁺wy)\n\n ⁺-trans-⊆₂ : Transitive R → TransClosure R ⊆₂ R\n ⁺-trans-⊆₂ transR = ⇔₂-to-⊇₂ (⁺-trans-⇔₂ transR)\n\n ⁺-join-trans : Transitive R → {x y : A} → TransClosure R x y → R x y\n ⁺-join-trans = ⊆₂-apply ∘ ⁺-trans-⊆₂\n\n ⁺-unconsʳ :\n {x y : A}\n → TransClosure R x y\n → R x y ⊎ ∃[ z ] (TransClosure R x z × R z y)\n ⁺-unconsʳ [ Rxy ] = inj₁ Rxy\n ⁺-unconsʳ ( Rxz ∷ R⁺zy ) with ⁺-unconsʳ R⁺zy\n ... | inj₁ Rzy = inj₂ (_ , [ Rxz ] , Rzy)\n ... | inj₂ (v , R⁺zv , Rvy) = inj₂ (v , Rxz ∷ R⁺zv , Rvy)\n", "meta": {"hexsha": "8d900d640ac4d7e13600136824cebe4f51dc4bca", "size": 5970, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Dodo/Binary/Transitive.agda", "max_stars_repo_name": "sourcedennis/agda-dodo", "max_stars_repo_head_hexsha": "376f0ccee1e1aa31470890e494bcb534324f598a", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/Dodo/Binary/Transitive.agda", "max_issues_repo_name": "sourcedennis/agda-dodo", "max_issues_repo_head_hexsha": "376f0ccee1e1aa31470890e494bcb534324f598a", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Dodo/Binary/Transitive.agda", "max_forks_repo_name": "sourcedennis/agda-dodo", "max_forks_repo_head_hexsha": "376f0ccee1e1aa31470890e494bcb534324f598a", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.7731958763, "max_line_length": 99, "alphanum_fraction": 0.5308207705, "num_tokens": 2879, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34309547480636743}} {"text": "{-# OPTIONS --without-K --safe #-}\nmodule PiFracDynDef where\nopen import Data.Bool\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Sum\nopen import Data.Product\nopen import Data.Maybe\nopen import Function\nopen import Relation.Binary.PropositionalEquality\n renaming ([_] to R[_])\nopen import Relation.Binary.Core\nopen import Relation.Nullary\n\ninfix 70 _×ᵤ_\ninfix 60 _+ᵤ_\ninfix 40 _↔_\ninfixr 50 _⊚_\n\ndata ◯ : Set where\n ○ : ◯\n\n-- Pi\n\nmutual\n data 𝕌 : Set where\n 𝟘 : 𝕌\n 𝟙 : 𝕌\n _+ᵤ_ : 𝕌 → 𝕌 → 𝕌\n _×ᵤ_ : 𝕌 → 𝕌 → 𝕌\n 𝟙/_ : 𝕌 → 𝕌\n\n ⟦_⟧ : 𝕌 → Set\n ⟦ 𝟘 ⟧ = ⊥\n ⟦ 𝟙 ⟧ = ⊤\n ⟦ t₁ +ᵤ t₂ ⟧ = ⟦ t₁ ⟧ ⊎ ⟦ t₂ ⟧\n ⟦ t₁ ×ᵤ t₂ ⟧ = ⟦ t₁ ⟧ × ⟦ t₂ ⟧\n ⟦ 𝟙/ t ⟧ = ◯\n\n data _↔_ : 𝕌 → 𝕌 → Set where\n unite₊l : {t : 𝕌} → 𝟘 +ᵤ t ↔ t\n uniti₊l : {t : 𝕌} → t ↔ 𝟘 +ᵤ t\n unite₊r : {t : 𝕌} → t +ᵤ 𝟘 ↔ t\n uniti₊r : {t : 𝕌} → t ↔ t +ᵤ 𝟘\n swap₊ : {t₁ t₂ : 𝕌} → t₁ +ᵤ t₂ ↔ t₂ +ᵤ t₁\n assocl₊ : {t₁ t₂ t₃ : 𝕌} → t₁ +ᵤ (t₂ +ᵤ t₃) ↔ (t₁ +ᵤ t₂) +ᵤ t₃\n assocr₊ : {t₁ t₂ t₃ : 𝕌} → (t₁ +ᵤ t₂) +ᵤ t₃ ↔ t₁ +ᵤ (t₂ +ᵤ t₃)\n unite⋆l : {t : 𝕌} → 𝟙 ×ᵤ t ↔ t\n uniti⋆l : {t : 𝕌} → t ↔ 𝟙 ×ᵤ t\n unite⋆r : {t : 𝕌} → t ×ᵤ 𝟙 ↔ t\n uniti⋆r : {t : 𝕌} → t ↔ t ×ᵤ 𝟙\n swap⋆ : {t₁ t₂ : 𝕌} → t₁ ×ᵤ t₂ ↔ t₂ ×ᵤ t₁\n assocl⋆ : {t₁ t₂ t₃ : 𝕌} → t₁ ×ᵤ (t₂ ×ᵤ t₃) ↔ (t₁ ×ᵤ t₂) ×ᵤ t₃\n assocr⋆ : {t₁ t₂ t₃ : 𝕌} → (t₁ ×ᵤ t₂) ×ᵤ t₃ ↔ t₁ ×ᵤ (t₂ ×ᵤ t₃)\n absorbr : {t : 𝕌} → 𝟘 ×ᵤ t ↔ 𝟘\n absorbl : {t : 𝕌} → t ×ᵤ 𝟘 ↔ 𝟘\n factorzr : {t : 𝕌} → 𝟘 ↔ t ×ᵤ 𝟘\n factorzl : {t : 𝕌} → 𝟘 ↔ 𝟘 ×ᵤ t\n dist : {t₁ t₂ t₃ : 𝕌} → (t₁ +ᵤ t₂) ×ᵤ t₃ ↔ (t₁ ×ᵤ t₃) +ᵤ (t₂ ×ᵤ t₃)\n factor : {t₁ t₂ t₃ : 𝕌} → (t₁ ×ᵤ t₃) +ᵤ (t₂ ×ᵤ t₃) ↔ (t₁ +ᵤ t₂) ×ᵤ t₃\n distl : {t₁ t₂ t₃ : 𝕌} → t₁ ×ᵤ (t₂ +ᵤ t₃) ↔ (t₁ ×ᵤ t₂) +ᵤ (t₁ ×ᵤ t₃)\n factorl : {t₁ t₂ t₃ : 𝕌 } → (t₁ ×ᵤ t₂) +ᵤ (t₁ ×ᵤ t₃) ↔ t₁ ×ᵤ (t₂ +ᵤ t₃)\n id↔ : {t : 𝕌} → t ↔ t\n _⊚_ : {t₁ t₂ t₃ : 𝕌} → (t₁ ↔ t₂) → (t₂ ↔ t₃) → (t₁ ↔ t₃)\n _⊕_ : {t₁ t₂ t₃ t₄ : 𝕌} → (t₁ ↔ t₃) → (t₂ ↔ t₄) → (t₁ +ᵤ t₂ ↔ t₃ +ᵤ t₄)\n _⊗_ : {t₁ t₂ t₃ t₄ : 𝕌} → (t₁ ↔ t₃) → (t₂ ↔ t₄) → (t₁ ×ᵤ t₂ ↔ t₃ ×ᵤ t₄)\n η : {t : 𝕌} {t≠0 : ¬ card t ≡ 0} → 𝟙 ↔ t ×ᵤ (𝟙/ t)\n ε : {t : 𝕌} {t≠0 : ¬ card t ≡ 0} → t ×ᵤ (𝟙/ t) ↔ 𝟙\n\n-- Number of points in type\n card : (t : 𝕌) → ℕ\n card 𝟘 = 0\n card 𝟙 = 1\n card (t₁ +ᵤ t₂) = card t₁ + card t₂\n card (t₁ ×ᵤ t₂) = card t₁ * card t₂\n card (𝟙/ t) = 1\n\n-- If number of points is zero then it is impossible to find a value\n-- of the type\n0empty : {t : 𝕌} → card t ≡ 0 → (v : ⟦ t ⟧) → ⊥\n0empty {𝟘} _ ()\n0empty {𝟙} () tt\n0empty {t₁ +ᵤ t₂} s (inj₁ v₁)\n with card t₁ | card t₂ | inspect card t₁\n0empty {t₁ +ᵤ t₂} refl (inj₁ v₁) | 0 | 0 | R[ s₁ ] =\n 0empty {t₁} s₁ v₁\n0empty {t₁ +ᵤ t₂} s (inj₂ v₂)\n with card t₁ | card t₂ | inspect card t₂\n0empty {t₁ +ᵤ t₂} refl (inj₂ v₂) | ℕ.zero | ℕ.zero | R[ s₂ ] =\n 0empty {t₂} s₂ v₂\n0empty {t₁ ×ᵤ t₂} s (v₁ , v₂)\n with card t₁ | card t₂ | inspect card t₁ | inspect card t₂\n0empty {t₁ ×ᵤ t₂} refl (v₁ , v₂) | ℕ.zero | _ | R[ s₁ ] | _ =\n 0empty {t₁} s₁ v₁\n0empty {t₁ ×ᵤ t₂} s (v₁ , v₂) | ℕ.suc n₁ | ℕ.zero | R[ s₁ ] | R[ s₂ ] =\n 0empty {t₂} s₂ v₂\n0empty {𝟙/ t} () f\n\ndefault : (t : 𝕌) → {t≠0 : ¬ card t ≡ 0} → ⟦ t ⟧\ndefault 𝟘 {t≠0} = ⊥-elim (t≠0 refl) \ndefault 𝟙 = tt\ndefault (t₁ +ᵤ t₂) {p≠0} with card t₁ | card t₂ | inspect card t₁ | inspect card t₂\n... | 0 | 0 | R[ s₁ ] | R[ s₂ ] = ⊥-elim (p≠0 refl)\n... | 0 | suc n | R[ s₁ ] | R[ s₂ ] =\n inj₂ (default t₂ {λ t2≡0 → ⊥-elim (p≠0 (trans (sym s₂) t2≡0))})\n... | suc m | 0 | R[ s₁ ] | R[ s₂ ] =\n inj₁ (default t₁ {λ t1≡0 →\n ⊥-elim (p≠0 ((trans (sym (trans s₁ (sym (+-identityʳ (suc m))))) t1≡0)))})\n... | suc m | suc n | R[ s₁ ] | R[ s₂ ] =\n inj₁ (default t₁ {λ t1≡0 → ⊥-elim (1+n≢0 (trans (sym s₁) t1≡0))})\ndefault (t₁ ×ᵤ t₂) {p≠0} with card t₁ | card t₂ | inspect card t₁ | inspect card t₂\n... | 0 | 0 | R[ s₁ ] | R[ s₂ ] = ⊥-elim (p≠0 refl)\n... | 0 | suc n | R[ s₁ ] | R[ s₂ ] = ⊥-elim (p≠0 refl)\n... | suc m | 0 | R[ s₁ ] | R[ s₂ ] = ⊥-elim (p≠0 (*-zeroʳ (suc m)))\n... | suc m | suc n | R[ s₁ ] | R[ s₂ ] =\n default t₁ {λ t1≡0 → ⊥-elim (1+n≢0 (trans (sym s₁) t1≡0))},\n default t₂ {λ t2≡0 → ⊥-elim (1+n≢0 (trans (sym s₂) t2≡0))}\ndefault (𝟙/ t) = ○ \n\n𝕌dec : (t : 𝕌) → Decidable (_≡_ {A = ⟦ t ⟧})\n𝕌dec 𝟘 ()\n𝕌dec 𝟙 tt tt = yes refl\n𝕌dec (t₁ +ᵤ t₂) (inj₁ x) (inj₁ y) with 𝕌dec t₁ x y\n𝕌dec (t₁ +ᵤ t₂) (inj₁ x) (inj₁ .x) | yes refl = yes refl\n𝕌dec (t₁ +ᵤ t₂) (inj₁ x) (inj₁ y) | no ¬p = no (λ {refl → ¬p refl})\n𝕌dec (t₁ +ᵤ t₂) (inj₁ x) (inj₂ y) = no (λ ())\n𝕌dec (t₁ +ᵤ t₂) (inj₂ x) (inj₁ y) = no (λ ())\n𝕌dec (t₁ +ᵤ t₂) (inj₂ x) (inj₂ y) with 𝕌dec t₂ x y\n𝕌dec (t₁ +ᵤ t₂) (inj₂ x) (inj₂ .x) | yes refl = yes refl\n𝕌dec (t₁ +ᵤ t₂) (inj₂ x) (inj₂ y) | no ¬p = no (λ {refl → ¬p refl})\n𝕌dec (t₁ ×ᵤ t₂) (x₁ , y₁) (x₂ , y₂) with 𝕌dec t₁ x₁ x₂ | 𝕌dec t₂ y₁ y₂\n𝕌dec (t₁ ×ᵤ t₂) (x₁ , y₁) (.x₁ , .y₁) | yes refl | yes refl = yes refl\n𝕌dec (t₁ ×ᵤ t₂) (x₁ , y₁) (.x₁ , y₂) | yes refl | no ¬p = no (λ p → ¬p (cong proj₂ p))\n𝕌dec (t₁ ×ᵤ t₂) (x₁ , y₁) (x₂ , .y₁) | no ¬p | yes refl = no (λ p → ¬p (cong proj₁ p))\n𝕌dec (t₁ ×ᵤ t₂) (x₁ , y₁) (x₂ , y₂) | no ¬p | no ¬p₁ = no (λ p → ¬p (cong proj₁ p))\n𝕌dec (𝟙/ t) ○ ○ = yes refl\n\n_≟ᵤ_ : {t : 𝕌} → Decidable (_≡_ {A = ⟦ t ⟧})\n_≟ᵤ_ {t} v w = 𝕌dec t v w\n\ninterp : {t₁ t₂ : 𝕌} → (t₁ ↔ t₂) → ⟦ t₁ ⟧ → Maybe ⟦ t₂ ⟧\ninterp unite₊l (inj₁ ())\ninterp unite₊l (inj₂ v) = just v\ninterp uniti₊l v = just (inj₂ v)\ninterp unite₊r (inj₁ v) = just v\ninterp unite₊r (inj₂ ())\ninterp uniti₊r v = just (inj₁ v)\ninterp swap₊ (inj₁ v) = just (inj₂ v)\ninterp swap₊ (inj₂ v) = just (inj₁ v)\ninterp assocl₊ (inj₁ v) = just (inj₁ (inj₁ v))\ninterp assocl₊ (inj₂ (inj₁ v)) = just (inj₁ (inj₂ v))\ninterp assocl₊ (inj₂ (inj₂ v)) = just (inj₂ v)\ninterp assocr₊ (inj₁ (inj₁ v)) = just (inj₁ v)\ninterp assocr₊ (inj₁ (inj₂ v)) = just (inj₂ (inj₁ v))\ninterp assocr₊ (inj₂ v) = just (inj₂ (inj₂ v))\ninterp unite⋆l v = just (proj₂ v)\ninterp uniti⋆l v = just (tt , v)\ninterp unite⋆r v = just (proj₁ v)\ninterp uniti⋆r v = just (v , tt)\ninterp swap⋆ (v₁ , v₂) = just (v₂ , v₁)\ninterp assocl⋆ (v₁ , v₂ , v₃) = just ((v₁ , v₂) , v₃)\ninterp assocr⋆ ((v₁ , v₂) , v₃) = just (v₁ , v₂ , v₃)\ninterp absorbr (() , v)\ninterp absorbl (v , ())\ninterp factorzr ()\ninterp factorzl ()\ninterp dist (inj₁ v₁ , v₃) = just (inj₁ (v₁ , v₃))\ninterp dist (inj₂ v₂ , v₃) = just (inj₂ (v₂ , v₃))\ninterp factor (inj₁ (v₁ , v₃)) = just (inj₁ v₁ , v₃)\ninterp factor (inj₂ (v₂ , v₃)) = just (inj₂ v₂ , v₃)\ninterp distl (v₁ , inj₁ v₂) = just (inj₁ (v₁ , v₂))\ninterp distl (v₁ , inj₂ v₃) = just (inj₂ (v₁ , v₃))\ninterp factorl (inj₁ (v₁ , v₂)) = just (v₁ , inj₁ v₂)\ninterp factorl (inj₂ (v₁ , v₃)) = just (v₁ , inj₂ v₃)\ninterp id↔ v = just v\ninterp (c₁ ⊚ c₂) v = interp c₁ v >>= interp c₂\ninterp (c₁ ⊕ c₂) (inj₁ v) = interp c₁ v >>= just ∘ inj₁\ninterp (c₁ ⊕ c₂) (inj₂ v) = interp c₂ v >>= just ∘ inj₂\ninterp (c₁ ⊗ c₂) (v₁ , v₂) = interp c₁ v₁ >>= (λ v₁' → interp c₂ v₂ >>= λ v₂' → just (v₁' , v₂'))\ninterp (η {t} {t≠0}) tt = just (default t {t≠0} , ○)\ninterp (ε {t} {t≠0}) (v' , ○) with 𝕌dec t (default t {t≠0}) v'\ninterp (ε {t}) (v' , ○) | yes _ = just tt\ninterp (ε {t}) (v' , ○) | no _ = nothing -- if v ≡ v' then tt else throw Error\n \n--- Examples\n\n𝟚 : 𝕌\n𝟚 = 𝟙 +ᵤ 𝟙\n\n𝔽 𝕋 : ⟦ 𝟚 ⟧\n𝔽 = inj₁ tt\n𝕋 = inj₂ tt\n\nxorr xorl : 𝟚 ×ᵤ 𝟚 ↔ 𝟚 ×ᵤ 𝟚\nxorr = dist ⊚ (id↔ ⊕ (id↔ ⊗ swap₊)) ⊚ factor\nxorl = distl ⊚ (id↔ ⊕ (swap₊ ⊗ id↔)) ⊚ factorl\n\n\n𝟚≠0 : ¬ (card 𝟚 ≡ 0)\n𝟚≠0 ()\n\nη𝟚 : 𝟙 ↔ 𝟚 ×ᵤ (𝟙/ 𝟚)\nη𝟚 = η {t≠0 = 𝟚≠0}\n\nε𝟚 : 𝟚 ×ᵤ (𝟙/ 𝟚) ↔ 𝟙\nε𝟚 = ε {t≠0 = 𝟚≠0}\n\n-- ─────┬────⊕─── ───────\n-- | | ⨉\n-- ┌──⊕────┴─── ───┐\n-- └────────────────┘\nid' : 𝟚 ↔ 𝟚\nid' = uniti⋆r ⊚ (id↔ ⊗ η𝟚) ⊚ assocl⋆ ⊚\n ((xorr ⊚ xorl ⊚ swap⋆) ⊗ id↔) ⊚\n assocr⋆ ⊚ (id↔ ⊗ ε𝟚) ⊚ unite⋆r\n\nex1 : interp id' 𝕋 ≡ just 𝕋\nex1 = refl\n\nex2 : interp id' 𝔽 ≡ just 𝔽\nex2 = refl\n\n-- ┌────── ───────┐\n-- └──────╲╱───────┘\n-- ╱╲\n-- ┌───── ──────┐\n-- └───────────────┘\nswitch : 𝟙 ↔ 𝟙\nswitch = uniti⋆r ⊚ (η𝟚 ⊗ η𝟚) ⊚ assocl⋆ ⊚\n (((swap⋆ ⊗ id↔) ⊚ assocr⋆ ⊚\n (id↔ ⊗ swap⋆) ⊚ assocl⋆ ⊚ (swap⋆ ⊗ id↔)) ⊗ id↔) ⊚ \n assocr⋆ ⊚ (ε𝟚 ⊗ ε𝟚) ⊚ unite⋆r\n\nbad : 𝟚 ↔ 𝟚\nbad = uniti⋆r ⊚ (id↔ ⊗ η𝟚) ⊚ assocl⋆ ⊚\n ((xorr ⊚ swap⋆) ⊗ id↔) ⊚\n assocr⋆ ⊚ (id↔ ⊗ ε𝟚) ⊚ unite⋆r\n\nex3 : interp bad 𝔽 ≡ just 𝔽\nex3 = refl\n\nex4 : interp bad 𝕋 ≡ nothing\nex4 = refl\n\n{--\nshouldn't_type_check : 𝟙 ↔ 𝟙\nshouldn't_type_check = η {v = 𝔽} ⊚ ε {v = 𝕋}\n\nex5 : interp shouldn't_type_check tt ≡ nothing\nex5 = refl\n\nmore : 𝟙 ↔ 𝟙\nmore = η {v = 𝔽} ⊚ (swap₊ ⊗ id↔) ⊚ ε {v = 𝕋}\n\nex6 : interp more tt ≡ just tt\nex6 = refl\n--}\n", "meta": {"hexsha": "d96ef8f34577ab5b16e4bce8a5706cbfcc069d32", "size": 8490, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "fracGC/PiFracDynDef.agda", "max_stars_repo_name": "JacquesCarette/pi-dual", "max_stars_repo_head_hexsha": "003835484facfde0b770bc2b3d781b42b76184c1", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 14, 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0.4970553592, "num_tokens": 4974, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.3430337678044994}} {"text": "------------------------------------------------------------------------\n-- The partiality monads in Partiality-monad.Inductive and\n-- Partiality-monad.Coinductive are pointwise equivalent, for sets,\n-- assuming extensionality and countable choice\n------------------------------------------------------------------------\n\n{-# OPTIONS --cubical --sized-types #-}\n\nopen import Prelude hiding (⊥; ↑)\n\nmodule Partiality-monad.Equivalence {a} {A : Type a} where\n\nopen import Equality.Propositional.Cubical\nopen import Logical-equivalence using (_⇔_)\nopen import Prelude.Size\n\nopen import Bijection equality-with-J using (_↔_)\nopen import Embedding equality-with-J\n using (Is-embedding; Injective≃Is-embedding)\nopen import Equivalence equality-with-J as Eq\n using (_≃_; Is-equivalence)\nopen import Function-universe equality-with-J as F hiding (id; _∘_)\nopen import H-level equality-with-J\nopen import H-level.Closure equality-with-J\nopen import H-level.Truncation.Propositional equality-with-paths\n as Trunc\nopen import Injection equality-with-J using (Injective)\nopen import Quotient equality-with-paths as Quotient hiding ([_])\nopen import Univalence-axiom equality-with-J\n\nimport Delay-monad as D\nopen import Delay-monad.Alternative as A using (_↓_; _↑)\nimport Delay-monad.Alternative.Equivalence as A\nimport Delay-monad.Alternative.Partial-order as A\nimport Delay-monad.Alternative.Termination as A\nimport Delay-monad.Alternative.Weak-bisimilarity as A\nimport Delay-monad.Bisimilarity as B\nimport Partiality-monad.Coinductive as C\nimport Partiality-monad.Coinductive.Alternative as CA\nopen import Partiality-monad.Inductive as I\n hiding (_⊥; _⊑_; Increasing-sequence; _⇓_)\nopen import Partiality-monad.Inductive.Alternative-order\nopen import Partiality-monad.Inductive.Eliminators\n\n------------------------------------------------------------------------\n-- A function from the partiality monad defined in\n-- Partiality-monad.Coinductive.Alternative to the inductive\n-- partiality monad\n\n-- Turns potential values into partial values.\n\nMaybe→⊥ : Maybe A → A I.⊥\nMaybe→⊥ nothing = never\nMaybe→⊥ (just y) = now y\n\n-- Maybe→⊥ is monotone.\n\nMaybe→⊥-mono : ∀ {x y} → A.LE x y → Maybe→⊥ x I.⊑ Maybe→⊥ y\nMaybe→⊥-mono (inj₁ refl) = ⊑-refl _\nMaybe→⊥-mono (inj₂ (refl , y≢n)) = never⊑ _\n\n-- Maybe→⊥ can be used to turn increasing sequences of one kind into\n-- increasing sequences of another kind.\n\nDelay→Inc-seq : A.Delay A → I.Increasing-sequence A\nDelay→Inc-seq (f , inc) = Maybe→⊥ ∘ f , Maybe→⊥-mono ∘ inc\n\n-- Turns increasing sequences of potential values into partial values.\n\nDelay→⊥ : A.Delay A → A I.⊥\nDelay→⊥ = ⨆ ∘ Delay→Inc-seq\n\n-- Delay→⊥ is monotone (if A is a set).\n\nDelay→⊥-mono :\n Is-set A → ∀ x y → x A.∥⊑∥ y → Delay→⊥ x I.⊑ Delay→⊥ y\nDelay→⊥-mono A-set x@(f , _) y@(g , _) x⊑y = ∃⊑→⨆⊑⨆ inc\n where\n inc : ∀ m → ∃ λ n → Maybe→⊥ (f m) I.⊑ Maybe→⊥ (g n)\n inc m with inspect (f m)\n inc m | nothing , fm↑ = 0 , (Maybe→⊥ (f m) ≡⟨ cong Maybe→⊥ fm↑ ⟩⊑\n never ⊑⟨ never⊑ _ ⟩■\n Maybe→⊥ (g 0) ■)\n inc m | just z , fm↓z =\n k , (Maybe→⊥ (f m) ≡⟨ cong Maybe→⊥ fm↓z ⟩⊑\n Maybe→⊥ (just z) ≡⟨ cong Maybe→⊥ (sym $ proj₂ y⇓z) ⟩⊑\n Maybe→⊥ (g k) ■)\n where\n y⇓z = $⟨ ∣ m , fm↓z ∣ ⟩\n x A.∥⇓∥ z ↝⟨ x⊑y z ⟩\n y A.∥⇓∥ z ↝⟨ _⇔_.from (A.⇓⇔∥⇓∥ A-set y) ⟩□\n y A.⇓ z □\n\n k : ℕ\n k = proj₁ y⇓z\n\n-- Delay→⊥ maps weakly bisimilar values to equal values (if A is a\n-- set).\n\nDelay→⊥-≈→≡ : Is-set A → ∀ x y → x A.≈ y → Delay→⊥ x ≡ Delay→⊥ y\nDelay→⊥-≈→≡ A-set x y =\n x A.≈ y ↝⟨ Σ-map (Delay→⊥-mono A-set x y) (Delay→⊥-mono A-set y x) ⟩\n Delay→⊥ x I.⊑ Delay→⊥ y × Delay→⊥ x I.⊒ Delay→⊥ y ↝⟨ uncurry antisymmetry ⟩□\n Delay→⊥ x ≡ Delay→⊥ y □\n\n-- If A is a set, then values in A CA.⊥ can be mapped to values in the\n-- inductive partiality monad.\n\n⊥→⊥ : Is-set A → A CA.⊥ → A I.⊥\n⊥→⊥ A-set = Quotient.rec λ where\n .[]ʳ → Delay→⊥\n .[]-respects-relationʳ {x = x} {y = y} → Delay→⊥-≈→≡ A-set x y\n .is-setʳ → ⊥-is-set\n\n------------------------------------------------------------------------\n-- A lemma\n\n-- I._⇓_ and A._∥⇓∥_ are pointwise logically equivalent (via Delay→⊥),\n-- for sets.\n\n⇓⇔∥⇓∥ :\n Is-set A →\n ∀ x {y} → Delay→⊥ x I.⇓ y ⇔ x A.∥⇓∥ y\n⇓⇔∥⇓∥ A-set x@(f , _) {y} =\n Delay→⊥ x I.⇓ y ↔⟨⟩\n ⨆ (Delay→Inc-seq x) I.⇓ y ↔⟨ ⨆⇓≃∥∃⇓∥ ⟩\n ∥ (∃ λ n → Maybe→⊥ (f n) I.⇓ y) ∥ ↝⟨ ∥∥-cong-⇔ (∃-cong λ _ → record { to = to _; from = cong Maybe→⊥ }) ⟩\n ∥ (∃ λ n → f n ↓ y) ∥ ↝⟨ F.id ⟩□\n x A.∥⇓∥ y □\n where\n to : ∀ n → Maybe→⊥ (f n) I.⇓ y → f n ↓ y\n to n f⇓y with f n\n ... | nothing = ⊥-elim $ now≢never _ (sym f⇓y)\n ... | just y′ =\n just y′ ≡⟨ cong just y′≡y ⟩∎\n just y ∎\n where\n y′≡y = $⟨ f⇓y ⟩\n now y′ ≡ now y ↔⟨ now≡now≃∥≡∥ ⟩\n ∥ y′ ≡ y ∥ ↝⟨ ∥∥↔ A-set ⟩□\n y′ ≡ y □\n\n------------------------------------------------------------------------\n-- ⊥→⊥ is injective\n\n-- Delay→⊥ is (kind of) injective (if A is a set).\n\nDelay→⊥-injective :\n Is-set A →\n ∀ x y → Delay→⊥ x ≡ Delay→⊥ y → x A.≈ y\nDelay→⊥-injective A-set x y x≡y =\n lemma A-set x y (≡→⊑ x≡y)\n , lemma A-set y x (≡→⊑ (sym x≡y))\n where\n ≡→⊑ : ∀ {x y} → x ≡ y → x I.⊑ y\n ≡→⊑ refl = ⊑-refl _\n\n lemma :\n Is-set A →\n ∀ x y → Delay→⊥ x I.⊑ Delay→⊥ y → x A.∥⊑∥ y\n lemma A-set x y x⊑y z =\n x A.∥⇓∥ z ↝⟨ _⇔_.from (⇓⇔∥⇓∥ A-set x) ⟩\n Delay→⊥ x I.⇓ z ↔⟨ ⇓≃now⊑ ⟩\n now z I.⊑ Delay→⊥ x ↝⟨ flip ⊑-trans x⊑y ⟩\n now z I.⊑ Delay→⊥ y ↔⟨ inverse ⇓≃now⊑ ⟩\n Delay→⊥ y I.⇓ z ↝⟨ _⇔_.to (⇓⇔∥⇓∥ A-set y) ⟩□\n y A.∥⇓∥ z □\n\n-- ⊥→⊥ A-set is injective.\n\n⊥→⊥-injective :\n (A-set : Is-set A) →\n Injective (⊥→⊥ A-set)\n⊥→⊥-injective A-set {x} {y} = Quotient.elim-prop\n {P = λ x → ⊥→⊥ A-set x ≡ ⊥→⊥ A-set y → x ≡ y}\n (λ where\n .[]ʳ x → Quotient.elim-prop\n {P = λ y → Delay→⊥ x ≡ ⊥→⊥ A-set y → Quotient.[ x ] ≡ y}\n (λ where\n .[]ʳ y → []-respects-relation ∘\n Delay→⊥-injective A-set x y\n .is-propositionʳ _ →\n Π-closure ext 1 λ _ →\n /-is-set)\n y\n .is-propositionʳ _ →\n Π-closure ext 1 λ _ →\n /-is-set)\n x\n\n------------------------------------------------------------------------\n-- ⊥→⊥ is surjective\n\n-- Delay→⊥ is surjective (if A is a set, assuming countable choice).\n\nDelay→⊥-surjective :\n Is-set A →\n Axiom-of-countable-choice a →\n Surjective Delay→⊥\nDelay→⊥-surjective A-set cc =\n ⊥-rec-⊥ (record\n { pe = constant-sequence nothing\n ; po = constant-sequence ∘ just\n ; pl = λ s →\n (∀ n → ∥ (∃ λ x → Delay→⊥ x ≡ s [ n ]) ∥) ↝⟨ cc ⟩\n ∥ (∀ n → ∃ λ x → Delay→⊥ x ≡ s [ n ]) ∥ ↔⟨ ∥∥-cong ΠΣ-comm ⟩\n ∥ (∃ λ f → ∀ n → Delay→⊥ (f n) ≡ s [ n ]) ∥ ↝⟨ ∥∥-map (uncurry $ flip construct s) ⟩□\n ∥ (∃ λ x → Delay→⊥ x ≡ ⨆ s) ∥ □\n ; pp = λ _ → truncation-is-proposition\n })\n where\n -- The increasing sequences (of type A.Delay A) returned by this\n -- function are constant.\n\n constant-sequence : ∀ x → ∥ (∃ λ y → Delay→⊥ y ≡ Maybe→⊥ x) ∥\n constant-sequence x = ∣ (const x , const (inj₁ refl)) , ⨆-const ∣\n\n -- Given a sequence and an increasing sequence s that are pointwise\n -- equal (via Delay→⊥) one can construct an increasing sequence that\n -- is equal (via Delay→⊥) to ⨆ s.\n\n construct :\n (f : ℕ → A.Delay A)\n (s : I.Increasing-sequence A) →\n (∀ n → Delay→⊥ (f n) ≡ s [ n ]) →\n ∃ λ x → Delay→⊥ x ≡ ⨆ s\n construct f s h = x , x-correct\n where\n -- We use f and an isomorphism between ℕ and ℕ × ℕ to construct a\n -- function from ℕ to Maybe A.\n\n f₂ : ℕ → ℕ → Maybe A\n f₂ m n = proj₁ (f m) n\n\n f₁ : ℕ → Maybe A\n f₁ =\n ℕ ↔⟨ ℕ↔ℕ² ⟩\n ℕ × ℕ ↝⟨ uncurry f₂ ⟩□\n Maybe A □\n\n -- All values that this function can terminate with are equal.\n\n termination-value-unique-f₂ :\n ∀ {m n y m′ n′ y′} →\n f₂ m n ↓ y × f₂ m′ n′ ↓ y′ →\n y ≡ y′\n termination-value-unique-f₂ {m} {n} {y} {m′} {n′} {y′} =\n f₂ m n ↓ y × f₂ m′ n′ ↓ y′ ↝⟨ f₂↓→⨆s⇓ ×-cong f₂↓→⨆s⇓ ⟩\n ⨆ s I.⇓ y × ⨆ s I.⇓ y′ ↝⟨ uncurry termination-value-merely-unique ⟩\n ∥ y ≡ y′ ∥ ↔⟨ ∥∥↔ A-set ⟩□\n y ≡ y′ □\n where\n f₂↓→⨆s⇓ : ∀ {y m n} → f₂ m n ↓ y → ⨆ s I.⇓ y\n f₂↓→⨆s⇓ {y} {m} f₂↓ =\n terminating-element-is-⨆ s\n (s [ m ] ≡⟨ sym (h m) ⟩\n Delay→⊥ (f m) ≡⟨ _⇔_.from (⇓⇔∥⇓∥ A-set (f m)) ∣ _ , f₂↓ ∣ ⟩∎\n now y ∎)\n\n termination-value-unique-f₁ :\n ∀ {y y′} →\n (∃ λ n → f₁ n ↓ y) →\n (∃ λ n → f₁ n ↓ y′) →\n y ≡ y′\n termination-value-unique-f₁ (_ , ↓y) (_ , ↓y′) =\n termination-value-unique-f₂ (↓y , ↓y′)\n\n abstract\n\n -- Thus the function can be completed to an increasing sequence.\n\n completed-f₁ : ∃ λ x → ∀ {y} → x A.⇓ y ⇔ ∃ λ n → f₁ n ↓ y\n completed-f₁ = A.complete-function f₁ termination-value-unique-f₁\n\n -- The increasing sequence that is returned above.\n\n x : A.Delay A\n x = proj₁ completed-f₁\n\n -- Every potential value in the increasing sequence x is smaller\n -- than or equal to (via Maybe→⊥) some value in s.\n\n x⊑s : ∀ m → ∃ λ n → Maybe→⊥ (proj₁ x m) I.⊑ s [ n ]\n x⊑s m with inspect (proj₁ x m)\n x⊑s m | nothing , x↑ =\n zero\n , (Maybe→⊥ (proj₁ x m) ≡⟨ cong Maybe→⊥ x↑ ⟩⊑\n never ⊑⟨ never⊑ _ ⟩■\n s [ 0 ] ■)\n x⊑s m | just y , x↓ =\n n₁\n , (Maybe→⊥ (proj₁ x m) ≡⟨ cong Maybe→⊥ x↓ ⟩⊑\n now y ≡⟨ sym f⇓ ⟩⊑\n Delay→⊥ (f n₁) ≡⟨ h n₁ ⟩⊑\n s [ n₁ ] ■)\n where\n f₁↓ : ∃ λ n → f₁ n ↓ y\n f₁↓ = _⇔_.to (proj₂ completed-f₁) (_ , x↓)\n\n n = proj₁ f₁↓\n n₁ = proj₁ (_↔_.to ℕ↔ℕ² n)\n n₂ = proj₂ (_↔_.to ℕ↔ℕ² n)\n\n f⇓ : Delay→⊥ (f n₁) I.⇓ y\n f⇓ =\n _≃_.from ⇓≃now⊑\n (now y ≡⟨ cong Maybe→⊥ $ sym $ proj₂ f₁↓ ⟩⊑\n Maybe→⊥ (f₁ n) ⊑⟨⟩\n Maybe→⊥ (f₂ n₁ n₂) ⊑⟨⟩\n Maybe→⊥ (proj₁ (f n₁) n₂) ⊑⟨⟩\n Delay→Inc-seq (f n₁) [ n₂ ] ⊑⟨ upper-bound (Delay→Inc-seq (f n₁)) _ ⟩■\n Delay→⊥ (f n₁) ■)\n\n -- Furthermore every potential value in f is smaller than or equal\n -- to x.\n\n f⊑x : ∀ m → f m A.⊑ x\n f⊑x m y =\n f m A.⇓ y ↝⟨ Σ-map (_↔_.from ℕ↔ℕ² ∘ (m ,_)) (λ {n} →\n\n f₂ m n ↓ y ↝⟨ ≡⇒→ (cong ((_↓ y) ∘ uncurry f₂) $ sym $ _↔_.right-inverse-of ℕ↔ℕ² (m , n)) ⟩\n uncurry f₂ (_↔_.to ℕ↔ℕ² (_↔_.from ℕ↔ℕ² (m , n))) ↓ y ↝⟨ F.id ⟩□\n f₁ (_↔_.from ℕ↔ℕ² (m , n)) ↓ y □) ⟩\n\n (∃ λ n → f₁ n ↓ y) ↝⟨ _⇔_.from (proj₂ completed-f₁) ⟩□\n x A.⇓ y □\n\n -- Thus x is correctly defined.\n\n x-correct : Delay→⊥ x ≡ ⨆ s\n x-correct = antisymmetry\n (⊑→⨆⊑⨆ λ n →\n Delay→Inc-seq x [ n ] ⊑⟨ proj₂ (x⊑s n) ⟩■\n s [ proj₁ (x⊑s n) ] ■)\n (least-upper-bound _ _ λ m →\n s [ m ] ≡⟨ sym (h m) ⟩⊑\n Delay→⊥ (f m) ⊑⟨ Delay→⊥-mono A-set (f m) x (∥∥-map ∘ f⊑x m) ⟩■\n Delay→⊥ x ■)\n\n-- ⊥→⊥ A-set is surjective (assuming countable choice).\n\n⊥→⊥-surjective :\n (A-set : Is-set A) →\n Axiom-of-countable-choice a →\n Surjective (⊥→⊥ A-set)\n⊥→⊥-surjective A-set cc x =\n ∥∥-map (λ { (pre , can-pre≡x) → Quotient.[ pre ] , can-pre≡x })\n (Delay→⊥-surjective A-set cc x)\n\n------------------------------------------------------------------------\n-- ⊥→⊥ is an equivalence\n\n-- ⊥→⊥ A-set is an equivalence (assuming countable choice).\n\n⊥→⊥-equiv :\n (A-set : Is-set A) →\n Axiom-of-countable-choice a →\n Is-equivalence (⊥→⊥ A-set)\n⊥→⊥-equiv A-set cc = $⟨ _,_ {B = const _}\n (⊥→⊥-surjective A-set cc)\n (λ {_ _} → ⊥→⊥-injective A-set) ⟩\n Surjective (⊥→⊥ A-set) × Injective (⊥→⊥ A-set) ↝⟨ Σ-map id (_≃_.to (Injective≃Is-embedding ext /-is-set ⊥-is-set _)) ⟩\n Surjective (⊥→⊥ A-set) × Is-embedding (⊥→⊥ A-set) ↝⟨ _≃_.to surjective×embedding≃equivalence ⟩□\n Is-equivalence (⊥→⊥ A-set) □\n\n-- Thus the inductive definition of the partiality monad is equivalent\n-- to the definition in Partiality-monad.Coinductive.Alternative, for\n-- sets, assuming countable choice.\n\n⊥≃⊥′ :\n Is-set A →\n Axiom-of-countable-choice a →\n A CA.⊥ ≃ A I.⊥\n⊥≃⊥′ A-set choice = Eq.⟨ _ , ⊥→⊥-equiv A-set choice ⟩\n\n------------------------------------------------------------------------\n-- The two definitions of the partiality monad are equivalent\n\n-- The inductive and coinductive definitions of the partiality monad\n-- are pointwise equivalent, for sets, assuming extensionality and\n-- countable choice.\n\n⊥≃⊥ :\n Is-set A →\n B.Extensionality a →\n Axiom-of-countable-choice a →\n A I.⊥ ≃ A C.⊥\n⊥≃⊥ A-set delay-ext choice =\n A I.⊥ ↝⟨ inverse (⊥≃⊥′ A-set choice) ⟩\n A CA.⊥ ↔⟨ CA.⊥↔⊥ A-set delay-ext ⟩□\n A C.⊥ □\n\n-- The previous result has a number of preconditions. None of these\n-- preconditions are needed to translate from the delay monad to the\n-- quotient inductive-inductive partiality monad.\n\nDelay→⊥′ : D.Delay A ∞ → A I.⊥\nDelay→⊥′ =\n D.Delay A ∞ ↝⟨ _⇔_.from A.Delay⇔Delay ⟩\n A.Delay A ↝⟨ Delay→⊥ ⟩□\n A I.⊥ □\n\n-- This translation turns weakly bisimilar computations into equal\n-- computations, assuming that the underlying type is a set.\n\nDelay→⊥′-≈→≡ :\n Is-set A →\n ∀ {x y} → x B.≈ y → Delay→⊥′ x ≡ Delay→⊥′ y\nDelay→⊥′-≈→≡ A-set {x} {y} =\n x B.≈ y ↝⟨ _⇔_.to (A.≈⇔≈′ A-set) ⟩\n _⇔_.from A.Delay⇔Delay x A.≈ _⇔_.from A.Delay⇔Delay y ↝⟨ Delay→⊥-≈→≡ A-set (_⇔_.from A.Delay⇔Delay x) (_⇔_.from A.Delay⇔Delay y) ⟩□\n Delay→⊥′ x ≡ Delay→⊥′ y □\n\n-- One can also translate from the coinductive to the inductive\n-- partiality monad, as long as the underlying type is a set.\n\n⊥→⊥′ : Is-set A → A C.⊥ → A I.⊥\n⊥→⊥′ A-set = Quotient.rec λ where\n .[]ʳ → Delay→⊥′\n .[]-respects-relationʳ → Trunc.rec I.⊥-is-set (Delay→⊥′-≈→≡ A-set)\n .is-setʳ → I.⊥-is-set\n", "meta": {"hexsha": "a583edf50e55859f8cc5d905b6d36f39f2733f33", "size": 14579, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Partiality-monad/Equivalence.agda", "max_stars_repo_name": "nad/partiality-monad", "max_stars_repo_head_hexsha": "f69749280969f9093e5e13884c6feb0ad2506eae", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-05-21T22:59:18.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-03T08:56:08.000Z", "max_issues_repo_path": "src/Partiality-monad/Equivalence.agda", "max_issues_repo_name": "nad/partiality-monad", "max_issues_repo_head_hexsha": "f69749280969f9093e5e13884c6feb0ad2506eae", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Partiality-monad/Equivalence.agda", "max_forks_repo_name": "nad/partiality-monad", "max_forks_repo_head_hexsha": "f69749280969f9093e5e13884c6feb0ad2506eae", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.5473933649, "max_line_length": 148, "alphanum_fraction": 0.481583099, "num_tokens": 6075, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3429747339241959}} {"text": "{-# OPTIONS --without-K --safe #-}\n\n-- | Operations that ensure a cycle traverses a particular element\n-- at most once.\nmodule Dodo.Binary.Cycle where\n\n-- Stdlib import\nimport Relation.Binary.PropositionalEquality as Eq\nopen Eq using (_≡_; _≢_; refl)\nopen import Level using (Level; _⊔_)\nopen import Function using (_∘_)\nopen import Data.Product using (_×_; _,_; proj₁; proj₂)\nopen import Data.Sum using (_⊎_; inj₁; inj₂)\nopen import Relation.Nullary using (yes; no)\nopen import Relation.Binary using (Rel)\nopen import Relation.Binary.Construct.Closure.Transitive using (TransClosure; [_]; _∷_; _∷ʳ_; _++_)\n-- Local imports\nopen import Dodo.Unary.Dec\nopen import Dodo.Binary.Transitive\n\n\n-- # Definitions\n\n-- | The given relation, with proofs that neither element equals `z`.\nExcludeRel : {a ℓ : Level} {A : Set a} → (R : Rel A ℓ) → (z : A) → Rel A (a ⊔ ℓ)\nExcludeRel R z x y = R x y × x ≢ z × y ≢ z\n\n-- | A cycle of R which passes through `z` /exactly once/.\ndata PassCycle {a ℓ : Level} {A : Set a} (R : Rel A ℓ) (z : A) : Set (a ⊔ ℓ) where\n cycle₁ : R z z → PassCycle R z\n cycle₂ : {x : A} → x ≢ z → R z x → R x z → PassCycle R z\n cycleₙ : {x y : A} → R z x → TransClosure (ExcludeRel R z) x y → R y z → PassCycle R z\n\n\n-- # Functions\n\n-- | A cycle of a relation either /does not/ pass through an element `y`, or it\n-- can be made to pass through `y` /exactly once/.\n--\n-- Note that if the original cycle passes `y` /more than once/, then only one\n-- cycle of the multi-cycle through `y` may be taken.\ndivert-cycle : {a ℓ : Level} {A : Set a}\n → {R : Rel A ℓ}\n → {x : A}\n → TransClosure R x x\n → {y : A}\n → DecPred (_≡ y)\n → PassCycle R y ⊎ TransClosure (ExcludeRel R y) x x\ndivert-cycle {x = x} [ Rxx ] eq-xm with eq-xm x\n... | yes refl = inj₁ (cycle₁ Rxx)\n... | no x≢y = inj₂ [ ( Rxx , x≢y , x≢y ) ]\ndivert-cycle {A = A} {R = R} {x} ( Rxw ∷ R⁺wx ) {y} eq-dec = lemma Rxw R⁺wx\n where\n -- Chain that starts with `y`.\n --\n -- `Ryx` and `R⁺xz` are acculumators.\n lemma-incl : {x z : A} → R y x → TransClosure (ExcludeRel R y) x z → TransClosure R z y → PassCycle R y\n lemma-incl Ryx R⁺xz [ Rzy ] = cycleₙ Ryx R⁺xz Rzy\n lemma-incl Ryx R⁺xz (_∷_ {_} {w} Rzw R⁺wy) with eq-dec w\n lemma-incl Ryx R⁺xz (_∷_ {_} {w} Rzw R⁺wy) | yes refl = cycleₙ Ryx R⁺xz Rzw\n lemma-incl Ryx R⁺xz (_∷_ {_} {w} Rzw R⁺wy) | no w≢y =\n let z≢y = ⁺-lift-predʳ (proj₂ ∘ proj₂) R⁺xz\n in lemma-incl Ryx (R⁺xz ∷ʳ (Rzw , z≢y , w≢y)) R⁺wy\n\n -- First step of a chain that starts with `y`.\n lemma-incl₀ : {x : A} → R y x → TransClosure R x y → PassCycle R y\n lemma-incl₀ {x} Ryx R⁺xy with eq-dec x\n lemma-incl₀ {x} Ryx R⁺xy | yes refl = cycle₁ Ryx\n lemma-incl₀ {x} Ryx [ Rxy ] | no x≢y = cycle₂ x≢y Ryx Rxy\n lemma-incl₀ {x} Ryx (_∷_ {_} {z} Rxz R⁺zy) | no x≢y with eq-dec z\n lemma-incl₀ {x} Ryx (_∷_ {x} {z} Rxz R⁺zy) | no x≢y | yes refl = cycle₂ x≢y Ryx Rxz\n lemma-incl₀ {x} Ryx (_∷_ {x} {z} Rxz R⁺zy) | no x≢y | no z≢y = lemma-incl Ryx [ Rxz , x≢y , z≢y ] R⁺zy\n\n -- Chain that does /not/ (yet) pass through `y`.\n lemma-excl : {z : A} → TransClosure (ExcludeRel R y) x z → TransClosure R z x → PassCycle R y ⊎ TransClosure (ExcludeRel R y) x x\n lemma-excl R⁺xz [ Rzx ] =\n let z≢y = ⁺-lift-predʳ (proj₂ ∘ proj₂) R⁺xz\n x≢y = ⁺-lift-predˡ (proj₁ ∘ proj₂) R⁺xz\n in inj₂ (R⁺xz ∷ʳ (Rzx , z≢y , x≢y))\n lemma-excl R⁺xz (_∷_ {_} {w} Rzw R⁺wx) with eq-dec w\n lemma-excl R⁺xz (_∷_ {_} {_} Rzw [ Rwx ]) | yes refl = inj₁ (cycleₙ Rwx R⁺xz Rzw)\n lemma-excl R⁺xz (_∷_ {_} {_} Rzw (Rwq ∷ R⁺qx)) | yes refl = inj₁ (lemma-incl₀ Rwq (R⁺qx ++ (⁺-map _ proj₁ R⁺xz) ∷ʳ Rzw))\n lemma-excl R⁺xz (_∷_ {_} {_} Rzw R⁺wx) | no w≢y =\n let z≢y = ⁺-lift-predʳ (proj₂ ∘ proj₂) R⁺xz\n in lemma-excl (R⁺xz ∷ʳ (Rzw , z≢y , w≢y)) R⁺wx\n \n lemma : {w : A} → R x w → TransClosure R w x → PassCycle R y ⊎ TransClosure (ExcludeRel R y) x x\n lemma {_} Rxw R⁺wx with eq-dec x\n lemma {_} Rxw R⁺wx | yes refl = inj₁ (lemma-incl₀ Rxw R⁺wx)\n lemma {w} Rxw R⁺wx | no x≢y with eq-dec w\n lemma {_} Rxw [ Rwx ] | no x≢y | yes refl = inj₁ (cycle₂ x≢y Rwx Rxw)\n lemma {_} Rxw ( Rwz ∷ R⁺zx ) | no x≢y | yes refl = inj₁ (lemma-incl₀ Rwz (R⁺zx ∷ʳ Rxw))\n lemma {_} Rxw R⁺wx | no x≢y | no w≢y = lemma-excl [ Rxw , x≢y , w≢y ] R⁺wx\n", "meta": {"hexsha": "d73ab4df3527f4f64ab1f64faca745fc9bf6c4be", "size": 4301, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Dodo/Binary/Cycle.agda", "max_stars_repo_name": "sourcedennis/agda-dodo", "max_stars_repo_head_hexsha": "376f0ccee1e1aa31470890e494bcb534324f598a", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/Dodo/Binary/Cycle.agda", "max_issues_repo_name": "sourcedennis/agda-dodo", "max_issues_repo_head_hexsha": "376f0ccee1e1aa31470890e494bcb534324f598a", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Dodo/Binary/Cycle.agda", "max_forks_repo_name": "sourcedennis/agda-dodo", "max_forks_repo_head_hexsha": "376f0ccee1e1aa31470890e494bcb534324f598a", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.7553191489, "max_line_length": 131, "alphanum_fraction": 0.5893978145, "num_tokens": 1822, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5350984286266116, "lm_q1q2_score": 0.3428032392466955}} {"text": "------------------------------------------------------------------------\n-- Lemmas related to strong similarity for CCS\n------------------------------------------------------------------------\n\n{-# OPTIONS --sized-types #-}\n\nopen import Prelude\n\nmodule Similarity.CCS {ℓ} {Name : Type ℓ} where\n\nopen import Equality.Propositional\nopen import Prelude.Size\n\nopen import Function-universe equality-with-J hiding (id; _∘_)\n\nimport Bisimilarity.CCS as BL\nimport Bisimilarity.Equational-reasoning-instances\nopen import Equational-reasoning\nopen import Labelled-transition-system.CCS Name\nimport Similarity.Equational-reasoning-instances\n\nopen import Bisimilarity CCS as B using (_∼_; _∼′_)\nopen import Similarity CCS\n\n------------------------------------------------------------------------\n-- Congruence lemmas\n\nprivate\n module CL {i} = BL.Cong-lemmas [ i ]_≤′_ challenge\n\nmutual\n\n -- _∣_ preserves similarity.\n\n infix 6 _∣-cong_ _∣-cong′_\n\n _∣-cong_ : ∀ {i P P′ Q Q′} →\n [ i ] P ≤ P′ → [ i ] Q ≤ Q′ → [ i ] P ∣ Q ≤ P′ ∣ Q′\n P≤P′ ∣-cong Q≤Q′ = ⟨ CL.∣-cong _∣-cong′_ P≤P′ Q≤Q′ ⟩\n\n _∣-cong′_ : ∀ {i P P′ Q Q′} →\n [ i ] P ≤′ P′ → [ i ] Q ≤′ Q′ → [ i ] P ∣ Q ≤′ P′ ∣ Q′\n force (P≤P′ ∣-cong′ Q≤Q′) = force P≤P′ ∣-cong force Q≤Q′\n\n-- _⊕_ preserves similarity.\n\ninfix 8 _⊕-cong_ _⊕-cong′_\n\n_⊕-cong_ : ∀ {i P P′ Q Q′} →\n [ i ] P ≤ P′ → [ i ] Q ≤ Q′ → [ i ] P ⊕ Q ≤ P′ ⊕ Q′\nP≤P′ ⊕-cong Q≤Q′ = ⟨ CL.⊕-cong P≤P′ Q≤Q′ ⟩\n\n_⊕-cong′_ : ∀ {i P P′ Q Q′} →\n [ i ] P ≤′ P′ → [ i ] Q ≤′ Q′ → [ i ] P ⊕ Q ≤′ P′ ⊕ Q′\nforce (P≤P′ ⊕-cong′ Q≤Q′) = force P≤P′ ⊕-cong force Q≤Q′\n\n-- _·_ preserves similarity.\n\ninfix 12 _·-cong_ _·-cong′_\n\n_·-cong_ :\n ∀ {i μ μ′ P P′} →\n μ ≡ μ′ → [ i ] force P ≤′ force P′ → [ i ] μ · P ≤ μ′ · P′\n_·-cong_ {i} refl P≤P′ = ⟨ CL.·-cong {i = i} P≤P′ ⟩\n\n_·-cong′_ :\n ∀ {i μ μ′ P P′} →\n μ ≡ μ′ → [ i ] force P ≤′ force P′ → [ i ] μ · P ≤′ μ′ · P′\nforce (μ≡μ′ ·-cong′ P≤P′) = μ≡μ′ ·-cong P≤P′\n\n-- _∙_ preserves similarity.\n\ninfix 12 _∙-cong_ _∙-cong′_\n\n_∙-cong_ : ∀ {i μ μ′ P P′} →\n μ ≡ μ′ → [ i ] P ≤ P′ → [ i ] μ ∙ P ≤ μ′ ∙ P′\nrefl ∙-cong P≤P′ = refl ·-cong convert {a = ℓ} P≤P′\n\n_∙-cong′_ : ∀ {i μ μ′ P P′} →\n μ ≡ μ′ → [ i ] P ≤′ P′ → [ i ] μ ∙ P ≤′ μ′ ∙ P′\nforce (μ≡μ′ ∙-cong′ P≤P′) = μ≡μ′ ∙-cong force P≤P′\n\n-- _∙ turns equal actions into similar processes.\n\ninfix 12 _∙-cong _∙-cong′\n\n_∙-cong : ∀ {μ μ′} → μ ≡ μ′ → μ ∙ ≤ μ′ ∙\nrefl ∙-cong = reflexive\n\n_∙-cong′ : ∀ {μ μ′} → μ ≡ μ′ → μ ∙ ≤′ μ′ ∙\nrefl ∙-cong′ = reflexive\n\nmutual\n\n -- ⟨ν_⟩ preserves similarity.\n\n ⟨ν_⟩-cong : ∀ {i a a′ P P′} →\n a ≡ a′ → [ i ] P ≤ P′ → [ i ] ⟨ν a ⟩ P ≤ ⟨ν a′ ⟩ P′\n ⟨ν refl ⟩-cong P≤P′ = ⟨ CL.⟨ν⟩-cong ⟨ν refl ⟩-cong′ P≤P′ ⟩\n\n ⟨ν_⟩-cong′ : ∀ {i a a′ P P′} →\n a ≡ a′ → [ i ] P ≤′ P′ → [ i ] ⟨ν a ⟩ P ≤′ ⟨ν a′ ⟩ P′\n force (⟨ν a≡a′ ⟩-cong′ P≤P′) = ⟨ν a≡a′ ⟩-cong (force P≤P′)\n\nmutual\n\n -- !_ preserves similarity.\n\n infix 10 !-cong_ !-cong′_\n\n !-cong_ : ∀ {i P P′} →\n [ i ] P ≤ P′ → [ i ] ! P ≤ ! P′\n !-cong P≤P′ = ⟨ CL.!-cong BL.6-1-3-2 _∣-cong′_ !-cong′_ P≤P′ ⟩\n\n !-cong′_ : ∀ {i P P′} → [ i ] P ≤′ P′ → [ i ] ! P ≤′ ! P′\n force (!-cong′ P≤P′) = !-cong force P≤P′\n\n-- _[_] preserves similarity.\n\nmutual\n\n infix 5 _[_]-cong _[_]-cong′\n\n _[_]-cong :\n ∀ {i n Ps Qs}\n (C : Context ∞ n) → (∀ x → [ i ] Ps x ≤ Qs x) →\n [ i ] C [ Ps ] ≤ C [ Qs ]\n hole x [ Ps≤Qs ]-cong = Ps≤Qs x\n ∅ [ Ps≤Qs ]-cong = reflexive\n C₁ ∣ C₂ [ Ps≤Qs ]-cong = (C₁ [ Ps≤Qs ]-cong) ∣-cong (C₂ [ Ps≤Qs ]-cong)\n C₁ ⊕ C₂ [ Ps≤Qs ]-cong = (C₁ [ Ps≤Qs ]-cong) ⊕-cong (C₂ [ Ps≤Qs ]-cong)\n μ · C [ Ps≤Qs ]-cong = refl ·-cong λ { .force → force C [ Ps≤Qs ]-cong }\n ⟨ν a ⟩ C [ Ps≤Qs ]-cong = ⟨ν refl ⟩-cong (C [ Ps≤Qs ]-cong)\n ! C [ Ps≤Qs ]-cong = !-cong (C [ Ps≤Qs ]-cong)\n\n _[_]-cong′ :\n ∀ {i n Ps Qs}\n (C : Context ∞ n) → (∀ x → [ i ] Ps x ≤′ Qs x) →\n [ i ] C [ Ps ] ≤′ C [ Qs ]\n force (C [ Ps≤Qs ]-cong′) = C [ (λ x → force (Ps≤Qs x)) ]-cong\n\n------------------------------------------------------------------------\n-- Other results\n\n-- P is similar to P ⊕ Q.\n\n≤-⊕-left : ∀ {i P Q} → [ i ] P ≤ P ⊕ Q\n≤-⊕-left = ⟨ (λ P⟶P′ → _ , sum-left P⟶P′ , reflexive) ⟩\n\n-- Q is similar to P ⊕ Q.\n\n≤-⊕-right : ∀ {i P Q} → [ i ] Q ≤ P ⊕ Q\n≤-⊕-right = ⟨ (λ Q⟶Q′ → _ , sum-right Q⟶Q′ , reflexive) ⟩\n\n-- If Name is inhabited, then there are two processes that are similar\n-- in both directions, but not bisimilar.\n--\n-- I took this example from Wikipedia; I suspect that it is due to\n-- Milner.\n\n≤≥≁ : Name → ∃ λ P → ∃ λ Q → P ≤ Q × Q ≤ P × ¬ P ∼ Q\n≤≥≁ x = machine₁ , machine₂\n , machine₁≤machine₂ , machine₂≤machine₁ , machine₁≁machine₂\n where\n\n -- Some names (with kinds), constructed using x.\n\n pay coffee tea : Name-with-kind\n\n pay = x , true\n coffee = pay\n tea = x , false\n\n -- Two vending machines.\n\n machine₁ machine₂ machine₁′ machine₂′ : Proc ∞\n\n machine₁′ = name pay ∙ (coffee ∙ ⊕ tea ∙)\n machine₁ = ! machine₁′\n\n machine₂′ = (name pay ∙ (coffee ∙) ⊕ name pay ∙ (tea ∙))\n ⊕\n machine₁′\n machine₂ = ! machine₂′\n\n -- A lemma.\n\n machine₁⟶ :\n ∀ {P μ} → machine₁ [ μ ]⟶ P →\n μ ≡ name pay × P ∼ machine₁ ∣ (coffee ∙ ⊕ tea ∙)\n machine₁⟶ tr = case BL.6-1-3-2 tr of λ where\n (inj₁ (_ , action , P∼)) → refl , P∼\n (inj₂ (_ , _ , _ , _ , action , tr , _)) →\n ⊥-elim (names-are-not-inverted tr)\n\n -- The first machine is similar to the second one.\n\n machine₁≤machine₂ : ∀ {i} → [ i ] machine₁ ≤ machine₂\n machine₁≤machine₂ {i} =\n StepC.⟨ (λ {P} tr →\n case machine₁⟶ tr of λ where\n (refl , P∼) →\n _\n , (machine₂ [ name pay ]⟶⟨ replication (par-right (sum-right action)) ⟩\n machine₂ ∣ coffee ∙ ⊕ tea ∙)\n , (P ∼⟨ ≤: convert {a = ℓ} P∼ ⟩\n machine₁ ∣ coffee ∙ ⊕ tea ∙ ∼⟨ machine₁≤′machine₂ ∣-cong′ (_ ■) ⟩■\n machine₂ ∣ coffee ∙ ⊕ tea ∙))\n ⟩\n where\n machine₁≤′machine₂ : [ i ] machine₁ ≤′ machine₂\n force machine₁≤′machine₂ = machine₁≤machine₂\n\n -- The second machine is similar to the first one.\n\n machine₂≤machine₁ : ∀ {i} → [ i ] machine₂ ≤ machine₁\n machine₂≤machine₁ {i} = StepC.⟨ helper ∘ BL.6-1-3-2 ⟩\n where\n machine₂≤′machine₁ : [ i ] machine₂ ≤′ machine₁\n force machine₂≤′machine₁ = machine₂≤machine₁\n\n lemma =\n machine₁ [ name pay ]⟶⟨ replication (par-right action) ⟩\n machine₁ ∣ coffee ∙ ⊕ tea ∙\n\n helper :\n ∀ {P μ} →\n (∃ λ P′ → machine₂′ [ μ ]⟶ P′ × P ∼ machine₂ ∣ P′)\n ⊎\n (μ ≡ τ × ∃ λ P′ → ∃ λ P″ → ∃ λ a →\n machine₂′ [ name a ]⟶ P′ × machine₂′ [ name (co a) ]⟶ P″ ×\n P ∼ (machine₂ ∣ P′) ∣ P″) →\n ∃ λ Q → machine₁ [ μ ]⟶ Q × [ i ] P ≤′ Q\n helper {P} (inj₁ (_ , sum-left (sum-left action) , P∼)) =\n _\n , lemma\n , (P ∼⟨ ≤: convert {a = ℓ} P∼ ⟩\n machine₂ ∣ coffee ∙ ∼⟨ machine₂≤′machine₁ ∣-cong′ convert {a = ℓ} ≤-⊕-left ⟩■\n machine₁ ∣ coffee ∙ ⊕ tea ∙)\n\n helper {P} (inj₁ (_ , sum-left (sum-right action) , P∼)) =\n _\n , lemma\n , (P ∼⟨ ≤: convert {a = ℓ} P∼ ⟩\n machine₂ ∣ tea ∙ ∼⟨ machine₂≤′machine₁ ∣-cong′ convert {a = ℓ} ≤-⊕-right ⟩■\n machine₁ ∣ coffee ∙ ⊕ tea ∙)\n\n helper {P} (inj₁ (_ , sum-right action , P∼)) =\n _\n , lemma\n , (P ∼⟨ ≤: convert {a = ℓ} P∼ ⟩\n machine₂ ∣ coffee ∙ ⊕ tea ∙ ∼⟨ machine₂≤′machine₁ ∣-cong′ (_ ■) ⟩■\n machine₁ ∣ coffee ∙ ⊕ tea ∙)\n\n helper (inj₂ (_ , _ , _ , _ , sum-left (sum-left action) , sum-left (sum-left tr) , _)) = ⊥-elim (names-are-not-inverted tr)\n helper (inj₂ (_ , _ , _ , _ , sum-left (sum-left action) , sum-left (sum-right tr) , _)) = ⊥-elim (names-are-not-inverted tr)\n helper (inj₂ (_ , _ , _ , _ , sum-left (sum-left action) , sum-right tr , _)) = ⊥-elim (names-are-not-inverted tr)\n helper (inj₂ (_ , _ , _ , _ , sum-left (sum-right action) , sum-left (sum-left tr) , _)) = ⊥-elim (names-are-not-inverted tr)\n helper (inj₂ (_ , _ , _ , _ , sum-left (sum-right action) , sum-left (sum-right tr) , _)) = ⊥-elim (names-are-not-inverted tr)\n helper (inj₂ (_ , _ , _ , _ , sum-left (sum-right action) , sum-right tr , _)) = ⊥-elim (names-are-not-inverted tr)\n helper (inj₂ (_ , _ , _ , _ , sum-right action , sum-left (sum-left tr) , _)) = ⊥-elim (names-are-not-inverted tr)\n helper (inj₂ (_ , _ , _ , _ , sum-right action , sum-left (sum-right tr) , _)) = ⊥-elim (names-are-not-inverted tr)\n helper (inj₂ (_ , _ , _ , _ , sum-right action , sum-right tr , _)) = ⊥-elim (names-are-not-inverted tr)\n\n -- The two machines are not bisimilar.\n\n machine₁≁machine₂ : ¬ machine₁ ∼ machine₂\n machine₁≁machine₂ =\n machine₁ ∼ machine₂ ↝⟨ (λ hyp → B.right-to-left hyp\n (replication (par-right (sum-left (sum-left action))))) ⟩\n\n (∃ λ P → machine₁ [ name pay ]⟶ P × P ∼′ machine₂ ∣ coffee ∙) ↝⟨ Σ-map id (Σ-map (proj₂ ∘ machine₁⟶) id) ⟩\n\n (∃ λ P → P ∼ machine₁ ∣ (coffee ∙ ⊕ tea ∙) ×\n P ∼′ machine₂ ∣ coffee ∙) ↝⟨ (λ { (_ , P∼ , P∼′) → transitive {a = ℓ} (symmetric P∼) (convert {a = ℓ} P∼′) }) ⟩\n\n machine₁ ∣ (coffee ∙ ⊕ tea ∙) ∼ machine₂ ∣ coffee ∙ ↝⟨ (λ hyp → Σ-map id proj₁ $\n B.left-to-right hyp (par-right (sum-right action))) ⟩\n\n (∃ λ P → machine₂ ∣ coffee ∙ [ name tea ]⟶ P) ↝⟨ helper ∘ proj₂ ⟩\n\n pay ≡ tea ⊎ coffee ≡ tea ↝⟨ [ (λ ()) , (λ ()) ] ⟩□\n\n ⊥ □\n where\n helper : ∀ {P} →\n machine₂ ∣ coffee ∙ [ name tea ]⟶ P →\n pay ≡ tea ⊎ coffee ≡ tea\n helper (par-right tr) = inj₂ $ cancel-name $ ·-only tr\n helper (par-left tr) =\n inj₁ $ cancel-name $\n !-only (⊕-only (⊕-only ·-only ·-only) ·-only) tr\n", "meta": {"hexsha": "af175ea58f1622e204c594deca21f08b7feeedf9", "size": 10284, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Similarity/CCS.agda", "max_stars_repo_name": "nad/up-to", "max_stars_repo_head_hexsha": "b936ff85411baf3401ad85ce85d5ff2e9aa0ca14", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": 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YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5350984286266116, "lm_q1q2_score": 0.3428032392466955}} {"text": "open import Data.Bool using (Bool; true; false; _∧_)\nimport Relation.Binary.PropositionalEquality as Eq\nopen Eq using (_≡_; refl; sym; subst)\nopen Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _≡⟨_⟩_; _∎)\nopen import Data.Unit using (⊤; tt)\nopen import Data.Nat using (ℕ; zero; suc; _≤_; _≥_; _>_)\nopen import Data.Product using (_×_; _,_; proj₁; proj₂; Σ-syntax; ∃; ∃-syntax)\nopen import Data.Sum using (_⊎_; inj₁; inj₂)\nopen import Data.List using (List; []; [_]; _∷_; _∷ʳ_; _++_)\nopen import Data.List.Reverse using (Reverse; reverseView)\nopen import Function using (_$_)\n\nmodule SnapshotConsistency\n (Addr : Set) (_≟_ : Addr → Addr → Bool) (_≤?MAXADDR : Addr → Bool) (_≤?MAXWCNT : ℕ → Bool)\n (Data : Set) (defaultData : Data)\n where\n\ninfixl 20 _•_\ninfixl 20 _⊙_\ninfixl 20 _++RTC_\ninfixl 20 _<≐>_\n\nvariable\n addr : Addr\n dat : Data\n\n_≐_ : {A B : Set} → (A → B) → (A → B) → Set\ns ≐ t = ∀ a → s a ≡ t a\n\nsym-≐ : {A B : Set} {s t : A → B} → s ≐ t → t ≐ s\nsym-≐ eq = λ{x → sym (eq x)}\n\n_<≐>_ : {A B : Set} {s t u : A → B} → s ≐ t → t ≐ u → s ≐ u\n_<≐>_ {A} {B} {s} {t} {u} e q = λ{x → begin s x ≡⟨ e x ⟩ t x ≡⟨ q x ⟩ u x ∎}\n\ndata SnocList (A : Set) : Set where\n [] : SnocList A\n _•_ : (as : SnocList A) → (a : A) → SnocList A\n\n_⊙_ : {A : Set} → SnocList A → SnocList A → SnocList A\nxs ⊙ [] = xs\nxs ⊙ (ys • y) = (xs ⊙ ys) • y\n\ndata All {A : Set} (P : A → Set) : SnocList A → Set where\n [] : All P []\n _∷_ : ∀ {xs : SnocList A} {x : A} → All P xs → P x → All P (xs • x)\n\n_++All_ : {A : Set} {P : A → Set} {xs ys : SnocList A} → All P xs → All P ys → All P (xs ⊙ ys)\nall₁ ++All [] = all₁\nall₁ ++All (all₂ ∷ x) = all₁ ++All all₂ ∷ x\n\nmapAll : {A : Set} {P Q : A → Set} {xs : SnocList A}\n → ({x : A} → P x → Q x) → All P xs → All Q xs\nmapAll pq [] = []\nmapAll pq (all ∷ x) = (mapAll pq all) ∷ (pq x)\n\ndata Action : Set where\n w[_↦_] : (addr : Addr) (dat : Data) → Action\n f : Action\n r : Action\n wᶜ[_↦_] : (addr : Addr) (dat : Data) → Action\n fᶜ : Action\n rᶜ : Action\n cp : Action\n er : Action\n cpᶜ : Action\n erᶜ : Action\n\nvariable\n ac : Action\n\ndata Regular : Action → Set where\n w : Regular w[ addr ↦ dat ]\n cp : Regular cp\n er : Regular er\n\ndata Write : Action → Set where\n w : Write w[ addr ↦ dat ]\n\ndata Snapshot : Action → Set where\n f : Snapshot f\n\ndata RecoveryCrash : Action → Set where\n rᶜ : RecoveryCrash rᶜ\n\ndata RegularSuccess : Action → Set where\n w : RegularSuccess w[ addr ↦ dat ]\n f : RegularSuccess f\n\ndata Regular×Snapshot : Action → Set where\n w : Regular×Snapshot w[ addr ↦ dat ]\n cp : Regular×Snapshot cp\n er : Regular×Snapshot er\n f : Regular×Snapshot f\n\ndata Regular×SnapshotCrash : Action → Set where\n wᶜ : Regular×SnapshotCrash wᶜ[ addr ↦ dat ]\n fᶜ : Regular×SnapshotCrash fᶜ\n cpᶜ : Regular×SnapshotCrash cpᶜ\n erᶜ : Regular×SnapshotCrash erᶜ\n\nTrace = SnocList Action\n\nvariable\n ef : Trace\n ef₁ : Trace\n ef₂ : Trace\n ef₃ : Trace\n frag : Trace\n frag-w : Trace\n frag-rᶜ : Trace\n flist : SnocList Action\n flist-w : SnocList Action\n flist-rᶜ : SnocList Action\n\n--Reflexive Transitive Closure\ndata RTC {A S : Set} (R : S → A → S → Set) : S → SnocList A → S → Set where\n ∅ : ∀ {s : S} → RTC R s [] s\n _•_ : ∀ {s t u : S} {acs : SnocList A} {ac : A}\n → RTC R s acs t → R t ac u → RTC R s (acs • ac) u\n\n_++RTC_ : {A S : Set} {R : S → A → S → Set} {s t u : S} {ef₁ ef₂ : SnocList A}\n → RTC R s ef₁ t → RTC R t ef₂ u → RTC R s (ef₁ ⊙ ef₂) u\ntc-s-t ++RTC ∅ = tc-s-t\ntc-s-t ++RTC (tc-t-u • rr) = (tc-s-t ++RTC tc-t-u) • rr\n\nsplitRTC : {A S : Set} {R : S → A → S → Set} {s s' : S} → (splitOn : SnocList A) → {rest : SnocList A}\n → ( fr : RTC R s (splitOn ⊙ rest) s') → Σ[ s'' ∈ S ] Σ[ fr₁ ∈ RTC R s splitOn s'' ] Σ[ fr₂ ∈ RTC R s'' rest s' ] (fr ≡ (fr₁ ++RTC fr₂))\nsplitRTC ef₁ {rest = []} t = (_ , t , ∅ , refl)\nsplitRTC ef₁ {rest = (ef₂ • ac)} (t • rr) with splitRTC ef₁ t\n... | s'' , t₁ , t₂ , refl = s'' , t₁ , t₂ • rr , refl\n\ndata OneRecovery : Trace → Set where\n wᶜ : {tr₁ tr₂ tr₃ : Trace} → All Regular×Snapshot tr₁ → All Regular tr₂ → All RecoveryCrash tr₃\n → OneRecovery (tr₁ ⊙ ([] • f ⊙ tr₂) ⊙ ([] • wᶜ[ addr ↦ dat ] ⊙ tr₃ • r))\n fᶜ : {tr₁ tr₂ tr₃ : Trace} → All Regular×Snapshot tr₁ → All Regular tr₂ → All RecoveryCrash tr₃\n → OneRecovery (tr₁ ⊙ ([] • f ⊙ tr₂) ⊙ ([] • fᶜ ⊙ tr₃ • r))\n wᶜ-nof : {tr₂ tr₃ : Trace} → All Regular tr₂ → All RecoveryCrash tr₃\n → OneRecovery (tr₂ ⊙ ([] • wᶜ[ addr ↦ dat ] ⊙ tr₃ • r))\n fᶜ-nof : {tr₂ tr₃ : Trace} → All Regular tr₂ → All RecoveryCrash tr₃\n → OneRecovery (tr₂ ⊙ ([] • fᶜ ⊙ tr₃ • r))\n\ndata MultiRecovery : Trace → Set where\n init : {tr : Trace} → All RecoveryCrash tr → MultiRecovery (tr • r)\n one : {tr₁ tr₂ : Trace} → MultiRecovery tr₁ → OneRecovery tr₂ → MultiRecovery (tr₁ ⊙ tr₂)\n\ndata 1RFrags {S : Set} {R : S → Action → S → Set} : {s s' : S} {tr : Trace} → OneRecovery tr → RTC R s tr s' → Set where\n wᶜ : {tr₁ tr₂ tr₃ : Trace} {all₁ : All Regular×Snapshot tr₁} {all₂ : All Regular tr₂} {all₃ : All RecoveryCrash tr₃}\n → {s₁ s₂ s₃ s₄ : S} (fr₁ : RTC R s₁ tr₁ s₂) (fr₂ : RTC R s₂ ([] • f ⊙ tr₂) s₃) (fr₃ : RTC R s₃ ([] • wᶜ[ addr ↦ dat ] ⊙ tr₃ • r) s₄)\n → 1RFrags (wᶜ all₁ all₂ all₃) (fr₁ ++RTC fr₂ ++RTC fr₃)\n fᶜ : {tr₁ tr₂ tr₃ : Trace} {all₁ : All Regular×Snapshot tr₁} {all₂ : All Regular tr₂} {all₃ : All RecoveryCrash tr₃}\n → {s₁ s₂ s₃ s₄ : S} (fr₁ : RTC R s₁ tr₁ s₂) (fr₂ : RTC R s₂ ([] • f ⊙ tr₂) s₃) (fr₃ : RTC R s₃ ([] • fᶜ ⊙ tr₃ • r) s₄)\n → 1RFrags (fᶜ all₁ all₂ all₃) (fr₁ ++RTC fr₂ ++RTC fr₃)\n wᶜ-nof : {tr₂ tr₃ : Trace} {all₂ : All Regular tr₂} {all₃ : All RecoveryCrash tr₃}\n → {s₂ s₃ s₄ : S} (fr₂ : RTC R s₂ tr₂ s₃) (fr₃ : RTC R s₃ ([] • wᶜ[ addr ↦ dat ] ⊙ tr₃ • r) s₄)\n → 1RFrags (wᶜ-nof all₂ all₃) (fr₂ ++RTC fr₃)\n fᶜ-nof : {tr₂ tr₃ : Trace} {all₂ : All Regular tr₂} {all₃ : All RecoveryCrash tr₃}\n → {s₂ s₃ s₄ : S} (fr₂ : RTC R s₂ tr₂ s₃) (fr₃ : RTC R s₃ ([] • fᶜ ⊙ tr₃ • r) s₄)\n → 1RFrags (fᶜ-nof all₂ all₃) (fr₂ ++RTC fr₃)\n\nview1R : {tr : Trace} (1r : OneRecovery tr) {S : Set} {R : S → Action → S → Set} {s s' : S} (fr : RTC R s tr s') → 1RFrags 1r fr\nview1R (wᶜ {tr₁ = tr₁} {tr₂ = tr₂} {tr₃ = tr₃} all₁ all₂ all₃) {s = s₁} {s' = s₄} fr\n with splitRTC (tr₁ ⊙ ([] • f ⊙ tr₂)) {rest = [] • wᶜ[ _ ↦ _ ] ⊙ tr₃ • r} fr\n... | s₃ , fr-l , fr₃ , refl with splitRTC tr₁ {rest = [] • f ⊙ tr₂} fr-l\n... | s₂ , fr₁ , fr₂ , refl = wᶜ fr₁ fr₂ fr₃\nview1R (fᶜ {tr₁ = tr₁} {tr₂ = tr₂} {tr₃ = tr₃} all₁ all₂ all₃) {s = s₁} {s' = s₄} fr\n with splitRTC (tr₁ ⊙ ([] • f ⊙ tr₂)) {rest = [] • fᶜ ⊙ tr₃ • r} fr\n... | s₃ , fr-l , fr₃ , refl with splitRTC tr₁ {rest = [] • f ⊙ tr₂} fr-l\n... | s₂ , fr₁ , fr₂ , refl = fᶜ fr₁ fr₂ fr₃\nview1R (wᶜ-nof {tr₂ = tr₂} {tr₃ = tr₃} all₂ all₃) fr with splitRTC tr₂ fr\n... | _ , fr₁ , fr₂ , refl = wᶜ-nof fr₁ fr₂\nview1R (fᶜ-nof {tr₂ = tr₂} {tr₃ = tr₃} all₂ all₃) fr with splitRTC tr₂ fr\n... | _ , fr₁ , fr₂ , refl = fᶜ-nof fr₁ fr₂\n\ndata MRFrags {S : Set} {R : S → Action → S → Set} : {s s' : S} {tr : Trace} → MultiRecovery tr → RTC R s tr s' → Set where\n init : {tr : Trace} {all : All RecoveryCrash tr} {s s' : S} (fr : RTC R s (tr • r) s') → MRFrags (init all) fr\n one : {tr₁ : Trace} {mr : MultiRecovery tr₁} {s₁ s₂ : S} {fr₁ : RTC R s₁ tr₁ s₂} → MRFrags mr fr₁\n → {tr₂ : Trace} {1r : OneRecovery tr₂} {s₃ : S} (fr₂ : RTC R s₂ tr₂ s₃) → 1RFrags 1r fr₂ → MRFrags (one mr 1r) (fr₁ ++RTC fr₂)\n\nviewMR : {tr : Trace} (mr : MultiRecovery tr) {S : Set} {R : S → Action → S → Set} {s s' : S} (fr : RTC R s tr s') → MRFrags mr fr\nviewMR (init all) fr = init fr\nviewMR (one {tr₁ = tr₁} mr all) fr with splitRTC tr₁ fr\n... | _ , fr-l , fr-r , refl = one (viewMR mr fr-l) fr-r (view1R all fr-r)\n\nlastr : {S : Set} {s s' : S} {R : S → Action → S → Set} {tr : Trace} (mr : MultiRecovery tr)\n → (fr : RTC R s tr s') → (frs : MRFrags mr fr) → Σ[ s'' ∈ S ] (R s'' r s')\nlastr .(init _) .fr (init fr) with fr\n... | _ • x = _ , x\nlastr ._ ._ (one _ ._ (wᶜ _ _ (fr₃ • x))) = _ , x\nlastr ._ ._ (one _ ._ (fᶜ _ _ (fr₃ • x))) = _ , x\nlastr ._ ._ (one _ ._ (wᶜ-nof _ (fr₃ • x))) = _ , x\nlastr ._ ._ (one _ ._ (fᶜ-nof _ (fr₃ • x))) = _ , x\n\nSnapshotConsistency : {S : Set} {s s' : S} {R : S → Action → S → Set} (ER : S → S → Set)\n {tr : Trace} → (1r : OneRecovery tr) → (fr : RTC R s tr s') → 1RFrags 1r fr → Set\nSnapshotConsistency ER ._ ._ (wᶜ {s₂ = s₂} {s₄ = s₄} _ _ _) = ER s₂ s₄\nSnapshotConsistency ER ._ ._ (fᶜ {s₂ = s₂} {s₃} {s₄} _ _ _) = ER s₃ s₄ ⊎ ER s₂ s₄\nSnapshotConsistency ER ._ ._ (wᶜ-nof {s₂ = s₂} {s₄ = s₄} _ _) = ER s₂ s₄\nSnapshotConsistency ER ._ ._ (fᶜ-nof {s₂ = s₂} {s₃} {s₄} _ _) = ER s₃ s₄ ⊎ ER s₂ s₄\n\nmodule Spec where\n\n record State : Set where\n field\n volatile : Addr → Data\n stable : Addr → Data\n w-count : ℕ\n\n Init : State → Set\n Init s = (addr : Addr) → (State.stable s addr ≡ defaultData)\n\n variable\n t : State\n t' : State\n\n update : (Addr → Data) → ℕ → Addr → Data → (Addr → Data)\n update s wcnt addr dat i with (addr ≤?MAXADDR) ∧ (wcnt ≤?MAXWCNT)\n update s wcnt addr dat i | false = s i\n update s wcnt addr dat i | true with addr ≟ i\n update s wcnt addr dat i | true | true = dat\n update s wcnt addr dat i | true | false = s i\n\n data Step (s s' : State) : Action → Set where\n w : update (State.volatile s) (State.w-count s) addr dat ≐ State.volatile s'\n → State.stable s ≐ State.stable s'\n → suc (State.w-count s) ≡ State.w-count s'\n → Step s s' w[ addr ↦ dat ]\n f : State.volatile s ≐ State.volatile s'\n → State.volatile s ≐ State.stable s'\n → State.w-count s' ≡ zero\n → Step s s' f\n r : State.stable s ≐ State.volatile s'\n → State.stable s ≐ State.stable s'\n → State.w-count s' ≡ zero\n → Step s s' r\n wᶜ : State.stable s ≐ State.stable s'\n → Step s s' (wᶜ[ addr ↦ dat ])\n fᶜ : State.volatile s ≐ State.stable s' ⊎ State.stable s ≐ State.stable s'\n → Step s s' fᶜ\n rᶜ : State.stable s ≐ State.stable s'\n → Step s s' rᶜ\n cp : State.volatile s ≐ State.volatile s'\n → State.stable s ≐ State.stable s'\n → State.w-count s ≡ State.w-count s'\n → Step s s' cp\n er : State.volatile s ≐ State.volatile s'\n → State.stable s ≐ State.stable s'\n → State.w-count s ≡ State.w-count s'\n → Step s s' er\n cpᶜ : State.stable s ≐ State.stable s' → Step s s' cpᶜ\n erᶜ : State.stable s ≐ State.stable s' → Step s s' erᶜ\n\n _⟦_⟧▸_ : State → Action → State → Set\n s ⟦ ac ⟧▸ s' = Step s s' ac\n\n _⟦_⟧*▸_ = RTC _⟦_⟧▸_\n\n record StbP (ac : Action) : Set where --Stability Reserving Actions\n field\n preserve : {s s' : State} → s ⟦ ac ⟧▸ s' → (State.stable s ≐ State.stable s')\n\n instance\n stb-r : StbP r\n stb-r = record { preserve = λ{(r _ ss _) → ss} }\n stb-w : StbP w[ addr ↦ dat ]\n stb-w = record { preserve = λ{(w _ ss _ ) → ss} }\n stb-wᶜ : StbP wᶜ[ addr ↦ dat ]\n stb-wᶜ = record { preserve = λ{(wᶜ ss) → ss} }\n stb-rᶜ : StbP rᶜ\n stb-rᶜ = record { preserve = λ{(rᶜ ss) → ss} }\n stb-cp : StbP cp\n stb-cp = record { preserve = λ{(cp _ ss _) → ss}}\n stb-er : StbP er\n stb-er = record { preserve = λ{(er _ ss _) → ss}}\n stb-cpᶜ : StbP cpᶜ\n stb-cpᶜ = record { preserve = λ{(cpᶜ ss) → ss}}\n stb-erᶜ : StbP erᶜ\n stb-erᶜ = record { preserve = λ{(erᶜ ss) → ss}}\n\n idemₛ : {tr : Trace} → All StbP tr\n → ∀ {s s' : State} → s ⟦ tr ⟧*▸ s'\n → State.stable s ≐ State.stable s'\n idemₛ [] ∅ = λ{_ → refl}\n idemₛ (all ∷ x) (s2s'' • s''2s') =\n idemₛ all s2s'' <≐> StbP.preserve x s''2s'\n\n r→rs : Regular ac → Regular×Snapshot ac\n r→rs w = w\n r→rs cp = cp\n r→rs er = er\n\n n→sp : Regular ac → StbP ac\n n→sp w = stb-w\n n→sp cp = stb-cp\n n→sp er = stb-er\n\n rᶜ→sp : RecoveryCrash ac → StbP ac\n rᶜ→sp rᶜ = stb-rᶜ\n\n SpecSC-wᶜ : ∀ {s₂ s₃ s₄ : State} {tr-w tr-rᶜ}\n → {{_ : All Regular tr-w}} {{_ : All RecoveryCrash tr-rᶜ}}\n → s₂ ⟦ [] • f ⊙ tr-w ⟧*▸ s₃ → s₃ ⟦ [] • wᶜ[ addr ↦ dat ] ⊙ tr-rᶜ • r ⟧*▸ s₄\n → State.volatile s₂ ≐ State.volatile s₄\n SpecSC-wᶜ {{ all₂ }} {{ all₃ }} s₂▸s₃ (s₃▹ • r sv _ _)\n with splitRTC ([] • f) s₂▸s₃ | splitRTC ([] • wᶜ[ _ ↦ _ ]) s₃▹\n ... | s₂' , ∅ • (f vv vs _) , s₂'▸s₃ , _ | s₃' , ∅ • (wᶜ ss) , s₃'▸s₄▹ , _ =\n vs <≐> idemₛ (mapAll n→sp all₂) s₂'▸s₃ <≐>\n ss <≐> idemₛ (mapAll rᶜ→sp all₃) s₃'▸s₄▹ <≐> sv\n\n SpecSC-wᶜ-nof : ∀ {s₁ s₂ s₃ s₄ : State} {tr-w tr-rᶜ}\n → {{_ : All Regular tr-w}} {{_ : All RecoveryCrash tr-rᶜ}}\n → s₁ ⟦ r ⟧▸ s₂ → s₂ ⟦ tr-w ⟧*▸ s₃ → s₃ ⟦ [] • wᶜ[ addr ↦ dat ] ⊙ tr-rᶜ • r ⟧*▸ s₄\n → State.volatile s₂ ≐ State.volatile s₄\n SpecSC-wᶜ-nof {{ all₂ }} {{ all₃ }} (r sv' ss' _) s₂▸s₃ (s₃▹ • r sv _ _)\n with splitRTC ([] • wᶜ[ _ ↦ _ ]) s₃▹\n ... | s₃' , ∅ • (wᶜ ss) , s₃'▸s₄▹ , _ =\n sym-≐ sv' <≐> ss' <≐> idemₛ (mapAll n→sp all₂) s₂▸s₃ <≐>\n ss <≐> idemₛ (mapAll rᶜ→sp all₃) s₃'▸s₄▹ <≐> sv\n\n SpecSC-fᶜ : ∀ {s₂ s₃ s : State} {tr-w tr-rᶜ}\n → {{_ : All Regular tr-w}} {{_ : All RecoveryCrash tr-rᶜ}}\n → s₂ ⟦ ([] • f) ⊙ tr-w ⟧*▸ s₃ → s₃ ⟦ ([] • fᶜ) ⊙ tr-rᶜ • r ⟧*▸ s\n → State.volatile s₃ ≐ State.volatile s ⊎ State.volatile s₂ ≐ State.volatile s\n SpecSC-fᶜ {{all₁}} {{all₂}} s₂▸s₃ (s₃▸s • r sv ss _)\n with splitRTC ([] • f) s₂▸s₃ | splitRTC ([] • fᶜ) s₃▸s\n ... | _ , ∅ • f vv vs _ , ▸s₃ , _ | s₃' , ∅ • fᶜ (inj₁ vsᶜ) , s₃'▸s , _ =\n inj₁ $ vsᶜ <≐> idemₛ (mapAll rᶜ→sp all₂) s₃'▸s <≐> sv\n ... | _ , ∅ • f vv vs _ , ▸s₃ , _ | s₃' , ∅ • fᶜ (inj₂ ssᶜ) , s₃'▸s , _ =\n inj₂ $ vs <≐>\n idemₛ (mapAll n→sp all₁) ▸s₃ <≐> ssᶜ <≐>\n idemₛ (mapAll rᶜ→sp all₂) s₃'▸s <≐> sv\n\n SpecSC-fᶜ-nof : ∀ {s₁ s₂ s₃ s₄ : State} {tr-w tr-rᶜ}\n → {{_ : All Regular tr-w}} {{_ : All RecoveryCrash tr-rᶜ}}\n → s₁ ⟦ r ⟧▸ s₂ → s₂ ⟦ tr-w ⟧*▸ s₃ → s₃ ⟦ [] • fᶜ ⊙ tr-rᶜ • r ⟧*▸ s₄\n → State.volatile s₃ ≐ State.volatile s₄ ⊎ State.volatile s₂ ≐ State.volatile s₄\n SpecSC-fᶜ-nof {{ all₂ }} {{ all₃ }} (r sv' ss' _) s₂▸s₃ (s₃▹ • r sv _ _)\n with splitRTC ([] • fᶜ) s₃▹\n ... | s₃' , ∅ • fᶜ (inj₁ vsᶜ) , s₃'▸s , _ = inj₁ $ vsᶜ <≐> idemₛ (mapAll rᶜ→sp all₃) s₃'▸s <≐> sv\n ... | s₃' , ∅ • fᶜ (inj₂ ssᶜ) , s₃'▸s , _ = inj₂ $ sym-≐ sv' <≐> ss' <≐>\n idemₛ (mapAll n→sp all₂) s₂▸s₃ <≐> ssᶜ <≐>\n idemₛ (mapAll rᶜ→sp all₃) s₃'▸s <≐> sv\n\n SC : {t₀ t t' : State} {tr₀ tr : Trace} → (mr : MultiRecovery tr₀) → (1r : OneRecovery tr)\n → Init t₀ → (fr₀ : t₀ ⟦ tr₀ ⟧*▸ t) → MRFrags mr fr₀ → (fr : t ⟦ tr ⟧*▸ t') → (frs : 1RFrags 1r fr)\n → SnapshotConsistency (λ t t' → State.volatile t ≐ State.volatile t') 1r fr frs\n SC mr _ init-t₀ fr₀ frs₀ ._ (wᶜ {all₂ = all₂} {all₃} fr₁ fr₂ fr₃) = SpecSC-wᶜ {{all₂}} {{all₃}} fr₂ fr₃\n SC mr _ init-t₀ fr₀ frs₀ ._ (fᶜ {all₂ = all₂} {all₃} fr₁ fr₂ fr₃) = SpecSC-fᶜ {{all₂}} {{all₃}} fr₂ fr₃\n SC mr _ init-t₀ fr₀ frs₀ ._ (wᶜ-nof {all₂ = all₂} {all₃} fr₂ fr₃) = SpecSC-wᶜ-nof {{all₂}} {{all₃}} (proj₂ (lastr mr fr₀ frs₀)) fr₂ fr₃\n SC mr _ init-t₀ fr₀ frs₀ ._ (fᶜ-nof {all₂ = all₂} {all₃} fr₂ fr₃) = SpecSC-fᶜ-nof {{all₂}} {{all₃}} (proj₂ (lastr mr fr₀ frs₀)) fr₂ fr₃\n\nopen Spec hiding (SC)\n\nmodule Prog\n (runSpec : (t : State) (ac : Action) → ∃[ t' ] (t ⟦ ac ⟧▸ t'))\n (RawStateᴾ : Set) (_⟦_⟧ᴿ▸_ : RawStateᴾ → Action → RawStateᴾ → Set)\n (RI CI : RawStateᴾ → Set)\n (AR CR : RawStateᴾ → State → Set)\n (RIRI : {s s' : RawStateᴾ} {ac : Action} → Regular×Snapshot ac → s ⟦ ac ⟧ᴿ▸ s' → RI s → RI s')\n (ARAR : {s s' : RawStateᴾ} {t t' : State} {ac : Action} → Regular×Snapshot ac\n → s ⟦ ac ⟧ᴿ▸ s' → t ⟦ ac ⟧▸ t' → RI s × AR s t → AR s' t')\n (RICI : {s s' : RawStateᴾ} {ac : Action} → Regular×SnapshotCrash ac → s ⟦ ac ⟧ᴿ▸ s' → RI s → CI s')\n (ARCR : {s s' : RawStateᴾ} {t t' : State} {ac : Action} → Regular×SnapshotCrash ac\n → s ⟦ ac ⟧ᴿ▸ s' → t ⟦ ac ⟧▸ t' → RI s × AR s t → CR s' t')\n (CIRI : {s s' : RawStateᴾ} → s ⟦ r ⟧ᴿ▸ s' → CI s → RI s')\n (CRAR : {s s' : RawStateᴾ} {t t' : State} → s ⟦ r ⟧ᴿ▸ s' → t ⟦ r ⟧▸ t' → CI s × CR s t → AR s' t')\n (CICI : {s s' : RawStateᴾ} → s ⟦ rᶜ ⟧ᴿ▸ s' → CI s → CI s')\n (CRCR : {s s' : RawStateᴾ} {t t' : State} → s ⟦ rᶜ ⟧ᴿ▸ s' → t ⟦ rᶜ ⟧▸ t' → CI s × CR s t → CR s' t')\n (read : RawStateᴾ → Addr → Data)\n (AR⇒ObsEquiv : {s : RawStateᴾ} {t : State} → RI s × AR s t → read s ≐ State.volatile t)\n (Initᴿ : RawStateᴾ → Set)\n (initᴿ-CI : (s : RawStateᴾ) → Initᴿ s → CI s)\n (initᴿ-CR : (s : RawStateᴾ) → Initᴿ s → (t : State) → Init t → CR s t)\n (t-init : Σ[ t ∈ State ] Init t)\n where\n\n variable\n rs : RawStateᴾ\n rs₁ : RawStateᴾ\n rinv : RI rs\n cinv : CI rs\n rs' : RawStateᴾ\n rs'' : RawStateᴾ\n rs''' : RawStateᴾ\n rinv' : RI rs'\n cinv' : CI rs'\n\n _⟦_⟧ᴿ*▸_ = RTC _⟦_⟧ᴿ▸_\n\n data Inv (rs : RawStateᴾ) : Set where\n normal : RI rs → Inv rs\n crash : CI rs → Inv rs\n\n Stateᴾ : Set\n Stateᴾ = Σ[ rs ∈ RawStateᴾ ] Inv rs\n\n data Initᴾ : Stateᴾ → Set where\n init : Initᴿ rs → Initᴾ (rs , crash cinv)\n\n variable\n s : Stateᴾ\n s' : Stateᴾ\n s'' : Stateᴾ\n s''' : Stateᴾ\n\n data _⟦_⟧ᴾ▸_ : Stateᴾ → Action → Stateᴾ → Set where\n w : rs ⟦ w[ addr ↦ dat ] ⟧ᴿ▸ rs' → (rs , normal rinv) ⟦ w[ addr ↦ dat ] ⟧ᴾ▸ (rs' , normal rinv')\n f : rs ⟦ f ⟧ᴿ▸ rs' → (rs , normal rinv) ⟦ f ⟧ᴾ▸ (rs' , normal rinv')\n wᶜ : rs ⟦ wᶜ[ addr ↦ dat ] ⟧ᴿ▸ rs' → (rs , normal rinv) ⟦ wᶜ[ addr ↦ dat ] ⟧ᴾ▸ (rs' , crash cinv')\n fᶜ : rs ⟦ fᶜ ⟧ᴿ▸ rs' → (rs , normal rinv) ⟦ fᶜ ⟧ᴾ▸ (rs' , crash cinv')\n rᶜ : rs ⟦ rᶜ ⟧ᴿ▸ rs' → (rs , crash cinv) ⟦ rᶜ ⟧ᴾ▸ (rs' , crash cinv')\n r : rs ⟦ r ⟧ᴿ▸ rs' → (rs , crash cinv) ⟦ r ⟧ᴾ▸ (rs' , normal rinv')\n cp : rs ⟦ cp ⟧ᴿ▸ rs' → (rs , normal rinv) ⟦ cp ⟧ᴾ▸ (rs' , normal rinv')\n er : rs ⟦ er ⟧ᴿ▸ rs' → (rs , normal rinv) ⟦ er ⟧ᴾ▸ (rs' , normal rinv')\n cpᶜ : rs ⟦ cpᶜ ⟧ᴿ▸ rs' → (rs , normal rinv) ⟦ cpᶜ ⟧ᴾ▸ (rs' , crash cinv')\n erᶜ : rs ⟦ erᶜ ⟧ᴿ▸ rs' → (rs , normal rinv) ⟦ erᶜ ⟧ᴾ▸ (rs' , crash cinv')\n\n _⟦_⟧ᴾ*▸_ = RTC _⟦_⟧ᴾ▸_\n\n lift-n×s : {tr : Trace} {{_ : All Regular×Snapshot tr}} → rs ⟦ tr ⟧ᴿ*▸ rs' →\n ∃[ rinv' ] ((rs , normal rinv) ⟦ tr ⟧ᴾ*▸ (rs' , normal rinv'))\n lift-n×s ∅ = _ , ∅\n lift-n×s {{all ∷ w}} (rs*▸rs'' • rs''▸rs') =\n let (rinv'' , s*▸s'') = lift-n×s {{all}} rs*▸rs''\n in RIRI w rs''▸rs' rinv'' , s*▸s'' • w rs''▸rs'\n lift-n×s {{all ∷ f}} (rs*▸rs'' • rs''▸rs') =\n let (rinv'' , s*▸s'') = lift-n×s {{all}} rs*▸rs''\n in RIRI f rs''▸rs' rinv'' , s*▸s'' • f rs''▸rs'\n lift-n×s {{all ∷ cp}} (rs*▸rs'' • rs''▸rs') =\n let (rinv'' , s*▸s'') = lift-n×s {{all}} rs*▸rs''\n in RIRI cp rs''▸rs' rinv'' , s*▸s'' • cp rs''▸rs'\n lift-n×s {{all ∷ er}} (rs*▸rs'' • rs''▸rs') =\n let (rinv'' , s*▸s'') = lift-n×s {{all}} rs*▸rs''\n in RIRI er rs''▸rs' rinv'' , s*▸s'' • er rs''▸rs'\n\n lift-n : {tr : Trace} {{_ : All Regular tr}} → rs ⟦ tr ⟧ᴿ*▸ rs'\n → ∃[ rinv' ] ((rs , normal rinv) ⟦ tr ⟧ᴾ*▸ (rs' , normal rinv'))\n lift-n {{all}} rs*▸rs' =\n lift-n×s {{(mapAll (λ{w → w; cp → cp; er → er}) all)}} rs*▸rs'\n\n lift-rᶜ : {tr : Trace} {{_ : All RecoveryCrash tr}} → rs ⟦ tr ⟧ᴿ*▸ rs' →\n ∃[ cinv' ] ((rs , crash cinv) ⟦ tr ⟧ᴾ*▸ (rs' , crash cinv'))\n lift-rᶜ ∅ = _ , ∅\n lift-rᶜ {{all ∷ rᶜ}} (rs*▸rs'' • rs''▸rs') =\n let (cinv'' , s*▸s'') = lift-rᶜ {{all}} rs*▸rs''\n in CICI rs''▸rs' cinv'' , s*▸s'' • rᶜ rs''▸rs'\n\n lift-mr : {tr : Trace} (mr : MultiRecovery tr) (fr : rs ⟦ tr ⟧ᴿ*▸ rs') → MRFrags mr fr → Initᴿ rs\n → ∃[ cinv ] ∃[ rinv' ] let s = (rs , crash cinv) in (s ⟦ tr ⟧ᴾ*▸ (rs' , normal rinv')) × Initᴾ s\n lift-mr ._ ._ (init {all = all} fr) init-rs with fr\n ... | fr₀ • rr with lift-rᶜ {cinv = initᴿ-CI _ init-rs} {{all}} fr₀\n ... | cinv₀ , fr₀ᴾ = _ , CIRI rr cinv₀ , fr₀ᴾ • r rr , init init-rs\n lift-mr ._ ._ (one frs₀ fr frs) init-rs with lift-mr _ _ frs₀ init-rs\n lift-mr ._ ._ (one frs₀ ._ (wᶜ {tr₃ = tr₃} {all₁ = all₁} {all₂} {all₃} fr₁ fr₂ fr₃)) init-rs | cinv₀ , rinv₀ , fr₀ᴾ , init-s with splitRTC ([] • (wᶜ[ _ ↦ _ ])) {rest = (tr₃ • r)} fr₃\n ... | rs'' , ∅ • s₃▸s₃' , s₃'▸r • r▸rs' , eq with lift-n×s {rinv = rinv₀} {{all₁}} fr₁\n ... | rinv₁ , frP₁ with lift-n×s {rinv = rinv₁} {{ ([] ∷ f) ++All mapAll r→rs all₂ }} fr₂\n ... | rinv₂ , frP₂ with RICI wᶜ s₃▸s₃' rinv₂\n ... | cinv₂' with lift-rᶜ {cinv = cinv₂'} {{all₃}} s₃'▸r\n ... | cinv₃ , frP₃ with CIRI r▸rs' cinv₃\n ... | rinv₄ = cinv₀ , rinv₄ , fr₀ᴾ ++RTC (frP₁ ++RTC frP₂ ++RTC ((∅ • wᶜ s₃▸s₃') ++RTC (frP₃ • r r▸rs'))), init-s\n lift-mr ._ ._ (one frs₀ ._ (fᶜ {tr₃ = tr₃} {all₁ = all₁} {all₂} {all₃} fr₁ fr₂ fr₃)) init-rs | cinv₀ , rinv₀ , fr₀ᴾ , init-s with splitRTC ([] • fᶜ) {rest = (tr₃ • r)} fr₃\n ... | rs'' , ∅ • s₃▸s₃' , s₃'▸r • r▸rs' , eq with lift-n×s {rinv = rinv₀} {{all₁}} fr₁\n ... | rinv₁ , frP₁ with lift-n×s {rinv = rinv₁} {{ ([] ∷ f) ++All mapAll r→rs all₂ }} fr₂\n ... | rinv₂ , frP₂ with RICI fᶜ s₃▸s₃' rinv₂\n ... | cinv₂' with lift-rᶜ {cinv = cinv₂'} {{all₃}} s₃'▸r\n ... | cinv₃ , frP₃ with CIRI r▸rs' cinv₃\n ... | rinv₄ = cinv₀ , rinv₄ , fr₀ᴾ ++RTC (frP₁ ++RTC frP₂ ++RTC ((∅ • fᶜ s₃▸s₃') ++RTC (frP₃ • r r▸rs'))), init-s\n lift-mr ._ ._ (one frs₀ ._ (wᶜ-nof {tr₃ = tr₃} {all₂ = all₂} {all₃} fr₂ fr₃)) init-rs | cinv₀ , rinv₀ , fr₀ᴾ , init-s with splitRTC ([] • wᶜ[ _ ↦ _ ]) {rest = (tr₃ • r)} fr₃\n ... | rs'' , ∅ • s₃▸s₃' , s₃'▸r • r▸rs' , eq with lift-n×s {rinv = rinv₀} {{ mapAll r→rs all₂ }} fr₂\n ... | rinv₂ , frP₂ with RICI wᶜ s₃▸s₃' rinv₂\n ... | cinv₂' with lift-rᶜ {cinv = cinv₂'} {{all₃}} s₃'▸r\n ... | cinv₃ , frP₃ with CIRI r▸rs' cinv₃\n ... | rinv₄ = cinv₀ , rinv₄ , fr₀ᴾ ++RTC (frP₂ ++RTC ((∅ • wᶜ s₃▸s₃') ++RTC (frP₃ • r r▸rs'))), init-s\n lift-mr ._ ._ (one frs₀ ._ (fᶜ-nof {tr₃ = tr₃} {all₂ = all₂} {all₃} fr₂ fr₃)) init-rs | cinv₀ , rinv₀ , fr₀ᴾ , init-s with splitRTC ([] • fᶜ) {rest = (tr₃ • r)} fr₃\n ... | rs'' , ∅ • s₃▸s₃' , s₃'▸r • r▸rs' , eq with lift-n×s {rinv = rinv₀} {{ mapAll r→rs all₂ }} fr₂\n ... | rinv₂ , frP₂ with RICI fᶜ s₃▸s₃' rinv₂\n ... | cinv₂' with lift-rᶜ {cinv = cinv₂'} {{all₃}} s₃'▸r\n ... | cinv₃ , frP₃ with CIRI r▸rs' cinv₃\n ... | rinv₄ = cinv₀ , rinv₄ , fr₀ᴾ ++RTC (frP₂ ++RTC ((∅ • fᶜ s₃▸s₃') ++RTC (frP₃ • r r▸rs'))), init-s\n\n ObsEquiv : Stateᴾ → State → Set\n ObsEquiv (rs , _) t = read rs ≐ State.volatile t\n\n data SR : Stateᴾ → State → Set where\n ar : AR rs t → SR (rs , normal rinv) t\n cr : CR rs t → SR (rs , crash cinv) t\n\n simSR : SR s t → s ⟦ ac ⟧ᴾ▸ s' → ∃[ t' ] (t ⟦ ac ⟧▸ t' × SR s' t')\n simSR {s , normal rinv} {t} (ar AR-rs-t) (w {addr = addr} {dat = dat} rs▸rs') =\n let (t' , t▸t') = runSpec t w[ addr ↦ dat ]\n in t' , t▸t' , ar (ARAR w rs▸rs' t▸t' (rinv , AR-rs-t))\n simSR {s , normal rinv} {t} (ar AR-rs-t) (f rs▸rs') =\n let (t' , t▸t') = runSpec t f\n in t' , t▸t' , ar (ARAR f rs▸rs' t▸t' (rinv , AR-rs-t))\n simSR {s , normal rinv} {t} (ar AR-rs-t) (wᶜ {addr = addr} {dat = dat} rs▸rs') =\n let (t' , t▸t') = runSpec t wᶜ[ addr ↦ dat ]\n in t' , t▸t' , cr (ARCR wᶜ rs▸rs' t▸t' (rinv , AR-rs-t))\n simSR {s , normal rinv} {t} (ar AR-rs-t) (fᶜ rs▸rs') =\n let (t' , t▸t') = runSpec t fᶜ\n in t' , t▸t' , cr (ARCR fᶜ rs▸rs' t▸t' (rinv , AR-rs-t))\n simSR {s , crash cinv} {t} (cr CR-rs-t) (rᶜ rs▸rs') =\n let (t' , t▸t') = runSpec t rᶜ\n in t' , t▸t' , cr (CRCR rs▸rs' t▸t' (cinv , CR-rs-t))\n simSR {s , crash cinv} {t} (cr CR-rs-t) (r rs▸rs') =\n let (t' , t▸t') = runSpec t r\n in t' , t▸t' , ar (CRAR rs▸rs' t▸t' (cinv , CR-rs-t))\n simSR {s , normal rinv} {t} (ar AR-rs-t) (cp rs▸rs') =\n let (t' , t▸t') = runSpec t cp\n in t' , t▸t' , ar (ARAR cp rs▸rs' t▸t' (rinv , AR-rs-t))\n simSR {s , normal rinv} {t} (ar AR-rs-t) (er rs▸rs') =\n let (t' , t▸t') = runSpec t er\n in t' , t▸t' , ar (ARAR er rs▸rs' t▸t' (rinv , AR-rs-t))\n simSR {s , normal rinv} {t} (ar AR-rs-t) (cpᶜ rs▸rs') =\n let (t' , t▸t') = runSpec t cpᶜ\n in t' , t▸t' , cr (ARCR cpᶜ rs▸rs' t▸t' (rinv , AR-rs-t))\n simSR {s , normal rinv} {t} (ar AR-rs-t) (erᶜ rs▸rs') =\n let (t' , t▸t') = runSpec t erᶜ\n in t' , t▸t' , cr (ARCR erᶜ rs▸rs' t▸t' (rinv , AR-rs-t))\n\n runSimSR : SR s t → s ⟦ ef ⟧ᴾ*▸ s' → ∃[ t' ] (t ⟦ ef ⟧*▸ t' × SR s' t')\n runSimSR SR-s-t ∅ = _ , ∅ , SR-s-t\n runSimSR SR-s-t (s*▸s'' • s''▸s') =\n let (t'' , t*▸t'' , SR-s''-t'') = runSimSR SR-s-t s*▸s''\n (t' , t''▸t' , SR-s'-t' ) = simSR SR-s''-t'' s''▸s'\n in _ , (t*▸t'' • t''▸t') , SR-s'-t'\n\n Conformant-all : {tr : Trace} {s s' : Stateᴾ} → s ⟦ tr ⟧ᴾ*▸ s' → {t t' : State} → t ⟦ tr ⟧*▸ t' → Set\n Conformant-all {s' = s'} ∅ {t' = t'} ∅ = ⊤\n Conformant-all {s' = s'} (frP • _) {t' = t'} (frS • _) = Conformant-all frP frS × ObsEquiv s' t'\n\n Conformant-1R : {tr : Trace} (1r : OneRecovery tr)\n → {s s' : Stateᴾ} (frP : s ⟦ tr ⟧ᴾ*▸ s') → 1RFrags 1r frP\n → {t t' : State } (frS : t ⟦ tr ⟧*▸ t') → 1RFrags 1r frS → Set\n Conformant-1R ._ {s' = s'} ._ (wᶜ frP₁ frP₂ frP₃) {t' = t'} ._ (wᶜ frS₁ frS₂ frS₃) = Conformant-all (frP₁ ++RTC frP₂) (frS₁ ++RTC frS₂) × ObsEquiv s' t'\n Conformant-1R ._ {s' = s'} ._ (fᶜ frP₁ frP₂ frP₃) {t' = t'} ._ (fᶜ frS₁ frS₂ frS₃) = Conformant-all (frP₁ ++RTC frP₂) (frS₁ ++RTC frS₂) × ObsEquiv s' t'\n Conformant-1R ._ {s' = s'} ._ (wᶜ-nof frP₂ frP₃) {t' = t'} ._ (wᶜ-nof frS₂ frS₃) = Conformant-all frP₂ frS₂ × ObsEquiv s' t'\n Conformant-1R ._ {s' = s'} ._ (fᶜ-nof frP₂ frP₃) {t' = t'} ._ (fᶜ-nof frS₂ frS₃) = Conformant-all frP₂ frS₂ × ObsEquiv s' t'\n\n Conformant-all-intermediate : {s₁ s₂ s₃ : Stateᴾ} {t₁ t₂ t₃ : State} {tr tr' : Trace}\n (frP : s₁ ⟦ tr ⟧ᴾ*▸ s₂) (frS : t₁ ⟦ tr ⟧*▸ t₂) (frP' : s₂ ⟦ tr' ⟧ᴾ*▸ s₃) (frS' : t₂ ⟦ tr' ⟧*▸ t₃)\n → Conformant-all (frP ++RTC frP') (frS ++RTC frS') → ObsEquiv s₁ t₁ → ObsEquiv s₂ t₂\n Conformant-all-intermediate ∅ ∅ _ _ conf oe = oe\n Conformant-all-intermediate (frP • sP) (frS • sS) ∅ ∅ conf oe = proj₂ conf\n Conformant-all-intermediate (frP • sP) (frS • sS) (frP' • sP') (frS' • sS') conf oe = Conformant-all-intermediate (frP • sP) (frS • sS) frP' frS' (proj₁ conf) oe\n\n conf-all++ : {s₁ s₂ s₃ : Stateᴾ} {t₁ t₂ t₃ : State} {tr tr' : Trace}\n (frP : s₁ ⟦ tr ⟧ᴾ*▸ s₂) (frS : t₁ ⟦ tr ⟧*▸ t₂) (frP' : s₂ ⟦ tr' ⟧ᴾ*▸ s₃) (frS' : t₂ ⟦ tr' ⟧*▸ t₃)\n → Conformant-all frP frS → Conformant-all frP' frS'\n → Conformant-all (frP ++RTC frP') (frS ++RTC frS')\n conf-all++ frP frS ∅ ∅ conf conf' = conf\n conf-all++ frP frS (frP' • p) (frS' • s) conf (conf' , oe) = conf-all++ frP frS frP' frS' conf conf' , oe\n\n Conformant : {tr : Trace} (mr : MultiRecovery tr)\n {s s' : Stateᴾ} (frP : s ⟦ tr ⟧ᴾ*▸ s') → MRFrags mr frP\n → {t t' : State } (frS : t ⟦ tr ⟧*▸ t') → MRFrags mr frS → Set\n Conformant (init _) {s' = s'} _ _ {t' = t'} _ _ = ObsEquiv s' t'\n Conformant (one mr 1r) .(_ ++RTC frP₂) (one {s₂ = s''} frPs frP₂ frPs₂) .(_ ++RTC frS₂) (one {s₂ = t''} frSs frS₂ frSs₂) =\n Conformant mr _ frPs _ frSs × ObsEquiv s'' t'' × Conformant-1R 1r frP₂ frPs₂ frS₂ frSs₂\n\n BC-all : {tr : Trace} → All Regular×Snapshot tr → {s s' : Stateᴾ} {t : State} →\n SR s t → ObsEquiv s t → (frP : s ⟦ tr ⟧ᴾ*▸ s') →\n Σ[ t' ∈ State ] Σ[ frS ∈ t ⟦ tr ⟧*▸ t' ] SR s' t' × ObsEquiv s' t' × Conformant-all frP frS\n BC-all [] sr oe-s-t ∅ = _ , ∅ , sr , oe-s-t , tt\n BC-all (all ∷ _) sr oe-s-t (frP • _) with BC-all all sr oe-s-t frP\n BC-all (all ∷ _) sr oe-s-t (frP • s''▸s') | t'' , frS'' , sr'' , oe'' , conf'' with simSR sr'' s''▸s'\n BC-all (all ∷ w) sr oe-s-t (frP • w {rinv' = rinv'} rs''▸s') | t'' , frS'' , sr'' , oe'' , conf'' | t' , t''▸t' , ar ar' =\n let oe' = AR⇒ObsEquiv (rinv' , ar') in t' , frS'' • t''▸t' , ar ar' , oe' , conf'' , oe'\n BC-all (all ∷ cp) sr oe-s-t (frP • cp {rinv' = rinv'} rs''▸s') | t'' , frS'' , sr'' , oe'' , conf'' | t' , t''▸t' , ar ar' =\n let oe' = AR⇒ObsEquiv (rinv' , ar') in t' , frS'' • t''▸t' , ar ar' , oe' , conf'' , oe'\n BC-all (all ∷ er) sr oe-s-t (frP • er {rinv' = rinv'} rs''▸s') | t'' , frS'' , sr'' , oe'' , conf'' | t' , t''▸t' , ar ar' =\n let oe' = AR⇒ObsEquiv (rinv' , ar') in t' , frS'' • t''▸t' , ar ar' , oe' , conf'' , oe'\n BC-all (all ∷ f) sr oe-s-t (frP • f {rinv' = rinv'} rs''▸s') | t'' , frS'' , sr'' , oe'' , conf'' | t' , t''▸t' , ar ar' =\n let oe' = AR⇒ObsEquiv (rinv' , ar') in t' , frS'' • t''▸t' , ar ar' , oe' , conf'' , oe'\n\n BC-1R : {tr : Trace} → (1r : OneRecovery tr) → {s s' : Stateᴾ} {t : State} →\n SR s t → ObsEquiv s t → (frP : s ⟦ tr ⟧ᴾ*▸ s') (frPs : 1RFrags 1r frP) →\n Σ[ t' ∈ State ] Σ[ frS ∈ t ⟦ tr ⟧*▸ t' ] Σ[ frSs ∈ 1RFrags 1r frS ] SR s' t' × ObsEquiv s' t' × Conformant-1R 1r frP frPs frS frSs\n BC-1R 1r sr s=t frP (wᶜ {all₁ = all₁} {all₂ = all₂} fr₁ fr₂ fr₃@(fr₃' • r {rinv' = rinv'} _))\n with BC-all all₁ sr s=t fr₁\n ... | t₁ , frS₁ , sr-s₁-t₁ , s₁=t₁ , conf₁\n with BC-all (([] ∷ f) ++All mapAll r→rs all₂) sr-s₁-t₁ s₁=t₁ fr₂\n ... | t₂ , frS₂ , sr-s₂-t₂ , s₂=t₂ , conf₂ with runSimSR sr-s₂-t₂ fr₃\n ... | t' , frS' , ar ar-s'-t' =\n let oe' = AR⇒ObsEquiv (rinv' , ar-s'-t')\n in t' , frS₁ ++RTC frS₂ ++RTC frS' , wᶜ frS₁ frS₂ frS' , ar ar-s'-t' , oe' , conf-all++ fr₁ frS₁ fr₂ frS₂ conf₁ conf₂ , oe'\n BC-1R 1r sr s=t frP (fᶜ {all₁ = all₁} {all₂ = all₂} fr₁ fr₂ fr₃@(fr₃' • r {rinv' = rinv'} _))\n with BC-all all₁ sr s=t fr₁\n ... | t₁ , frS₁ , sr-s₁-t₁ , s₁=t₁ , conf₁\n with BC-all (([] ∷ f) ++All mapAll r→rs all₂) sr-s₁-t₁ s₁=t₁ fr₂\n ... | t₂ , frS₂ , sr-s₂-t₂ , s₂=t₂ , conf₂ with runSimSR sr-s₂-t₂ fr₃\n ... | t' , frS' , ar ar-s'-t' =\n let oe' = AR⇒ObsEquiv (rinv' , ar-s'-t')\n in t' , frS₁ ++RTC frS₂ ++RTC frS' , fᶜ frS₁ frS₂ frS' , ar ar-s'-t' , oe' , conf-all++ fr₁ frS₁ fr₂ frS₂ conf₁ conf₂ , oe'\n BC-1R 1r sr s=t frP (wᶜ-nof {all₂ = all₂} fr₂ fr₃@(fr₃' • r {rinv' = rinv'} _))\n with BC-all (mapAll r→rs all₂) sr s=t fr₂\n ... | _ , frS₂ , sr-s₂-t₂ , s₂=t₂ , conf₂ with runSimSR sr-s₂-t₂ fr₃\n ... | t' , frS' , ar ar-s'-t' =\n let oe' = AR⇒ObsEquiv (rinv' , ar-s'-t')\n in t' , frS₂ ++RTC frS' , wᶜ-nof frS₂ frS' , ar ar-s'-t' , oe' , conf₂ , oe'\n BC-1R 1r sr s=t frP (fᶜ-nof {all₂ = all₂} fr₂ fr₃@(fr₃' • r {rinv' = rinv'} _))\n with BC-all (mapAll r→rs all₂) sr s=t fr₂\n ... | _ , frS₂ , sr-s₂-t₂ , s₂=t₂ , conf₂ with runSimSR sr-s₂-t₂ fr₃\n ... | t' , frS' , ar ar-s'-t' =\n let oe' = AR⇒ObsEquiv (rinv' , ar-s'-t')\n in t' , frS₂ ++RTC frS' , fᶜ-nof frS₂ frS' , ar ar-s'-t' , oe' , conf₂ , oe'\n\n BC-ind : {tr : Trace} (mr : MultiRecovery tr) {s s' : Stateᴾ} → Initᴾ s → (frP : s ⟦ tr ⟧ᴾ*▸ s') (frPs : MRFrags mr frP)\n → Σ[ t ∈ State ] Init t × Σ[ t' ∈ State ] SR s' t' × ObsEquiv s' t' × Σ[ frS ∈ t ⟦ tr ⟧*▸ t' ] Σ[ frSs ∈ MRFrags mr frS ] Conformant mr frP frPs frS frSs\n BC-ind (init all) {s = s₀} (init rs₀) (frP • rP@(r {rinv' = rinv'} rs)) frPs with runSimSR (cr (initᴿ-CR (proj₁ s₀) rs₀ (proj₁ t-init) (proj₂ t-init))) frP\n ... | t'' , frS , sr' with simSR sr' rP\n ... | t' , rS , (ar ar-rs'-t') =\n let eq = AR⇒ObsEquiv (rinv' , ar-rs'-t')\n in proj₁ t-init , proj₂ t-init , t' , ar ar-rs'-t' , eq , frS • rS , init (frS • rS) , eq\n BC-ind (one mr x) init-s _ (one {mr = mr₁} {fr₁ = frP₁} frPs₁ {1r = 1r} frP₂ frPs₂) with BC-ind mr init-s _ frPs₁\n ... | t , init-t , t'' , sr'' , oe'' , frS₁ , frSs₁ , conf₁ with BC-1R 1r sr'' oe'' frP₂ frPs₂\n ... | t' , frS₂ , frSs₂ , sr' , oe' , conf₂ = t , init-t , t' , sr' , oe' , frS₁ ++RTC frS₂ , one frSs₁ frS₂ frSs₂ , conf₁ , oe'' , conf₂\n\n BC-mr : {tr : Trace} (mr : MultiRecovery tr) {s s' : Stateᴾ} → Initᴾ s → (frP : s ⟦ tr ⟧ᴾ*▸ s') → (frPs : MRFrags mr frP)\n → Σ[ t ∈ State ] Init t × Σ[ t' ∈ State ] Σ[ frS ∈ t ⟦ tr ⟧*▸ t' ] Σ[ frSs ∈ MRFrags mr frS ] Conformant mr frP frPs frS frSs\n BC-mr mr init-s frP frPs =\n let (t , init-t , t' , _ , _ , frS , frSs , conf) = BC-ind mr init-s frP frPs\n in (t , init-t , t' , frS , frSs , conf)\n\n BC : {tr : Trace} (mr : MultiRecovery tr) {s s' : Stateᴾ} → Initᴾ s → (frP : s ⟦ tr ⟧ᴾ*▸ s') (frPs : MRFrags mr frP)\n → {tr' : Trace} {s'' : Stateᴾ} → All Regular×Snapshot tr' → (frP' : s' ⟦ tr' ⟧ᴾ*▸ s'')\n → Σ[ t ∈ State ] Init t × Σ[ t' ∈ State ] Σ[ frS ∈ t ⟦ tr ⟧*▸ t' ] Σ[ frSs ∈ MRFrags mr frS ] Σ[ t'' ∈ State ] Σ[ frS' ∈ t' ⟦ tr' ⟧*▸ t'' ]\n Conformant mr frP frPs frS frSs × Conformant-all frP' frS'\n BC mr init-s frP frPs all frP' =\n let (t , init-t , t' , sr' , oe' , frS , frSs , conf) = BC-ind mr init-s frP frPs\n (t'' , frS' , _ , _ , conf') = BC-all all sr' oe' frP'\n in (t , init-t , t' , frS , frSs , t'' , frS' , conf , conf')\n\n SC : {s₀ s s' : Stateᴾ} {tr₀ tr : Trace} → (mr : MultiRecovery tr₀) → (1r : OneRecovery tr)\n → Initᴾ s₀ → (fr₀ : s₀ ⟦ tr₀ ⟧ᴾ*▸ s) → MRFrags mr fr₀ → (frP : s ⟦ tr ⟧ᴾ*▸ s') → (frPs : 1RFrags 1r frP)\n → SnapshotConsistency (λ{(rs , _) (rs' , _) → read rs ≐ read rs'}) 1r frP frPs\n SC mr 1r init-s₀ frP₀ frPs₀ frP frPs with BC-mr (one mr 1r) init-s₀ (frP₀ ++RTC frP) (one frPs₀ frP frPs)\n SC mr ._ init-s₀ frP₀ frPs₀ ._ (wᶜ frP₁ frP₂ frP₃) |\n t₀ , init-t₀ , t' , ._ , one frSs ._ (wᶜ {all₁ = all₁} {all₂ = all₂} {all₃ = all₃} frS₁ frS₂ frS₃) , (_ , oe-s-t , conf , oe-s'-t')\n = Conformant-all-intermediate frP₁ frS₁ frP₂ frS₂ conf oe-s-t <≐> SpecSC-wᶜ ⦃ all₂ ⦄ ⦃ all₃ ⦄ frS₂ frS₃ <≐> sym-≐ oe-s'-t'\n SC mr ._ init-s₀ frP₀ frPs₀ ._ (fᶜ frP₁ frP₂ frP₃) |\n t₀ , init-t₀ , t' , ._ , one frSs ._ (fᶜ {all₁ = all₁} {all₂ = all₂} {all₃ = all₃} frS₁ frS₂ frS₃) , (_ , oe-s-t , conf , oe-s'-t')\n with SpecSC-fᶜ {{all₂}} {{all₃}} frS₂ frS₃\n ... | inj₁ req = inj₁ $ Conformant-all-intermediate (frP₁ ++RTC frP₂) (frS₁ ++RTC frS₂) ∅ ∅ conf oe-s-t <≐> req <≐> sym-≐ oe-s'-t'\n ... | inj₂ req = inj₂ $ Conformant-all-intermediate frP₁ frS₁ frP₂ frS₂ conf oe-s-t <≐> req <≐> sym-≐ oe-s'-t'\n SC mr ._ init-s₀ frP₀ frPs₀ ._ (wᶜ-nof frP₂ frP₃) |\n t₀ , init-t₀ , t' , ._ , one {fr₁ = frS₀} frSs ._ (wᶜ-nof {all₂ = all₂} {all₃ = all₃} frS₂ frS₃) , (_ , oe-s-t , conf , oe-s'-t')\n = oe-s-t <≐> SpecSC-wᶜ-nof ⦃ all₂ ⦄ ⦃ all₃ ⦄ (proj₂ (lastr mr frS₀ frSs)) frS₂ frS₃ <≐> sym-≐ oe-s'-t'\n SC mr ._ init-s₀ frP₀ frPs₀ ._ (fᶜ-nof frP₂ frP₃) |\n t₀ , init-t₀ , t' , ._ , one frSs ._ (fᶜ-nof {all₂ = all₂} {all₃ = all₃} frS₂ frS₃) , (_ , oe-s-t , conf , oe-s'-t')\n with SpecSC-fᶜ-nof {{all₂}} {{all₃}} (proj₂ (lastr mr _ frSs)) frS₂ frS₃\n ... | inj₁ req = inj₁ $ Conformant-all-intermediate frP₂ frS₂ ∅ ∅ conf oe-s-t <≐> req <≐> sym-≐ oe-s'-t'\n ... | inj₂ req = inj₂ $ oe-s-t <≐> req <≐> sym-≐ oe-s'-t'\n", "meta": {"hexsha": "46cb5ae5eb20b27f8da99b0ff7286fa5bb331e23", "size": 35283, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "agda/SnapshotConsistency.agda", "max_stars_repo_name": "yunshengtw/scftl", "max_stars_repo_head_hexsha": "c3ea3347071461241e3799e2cac46859e0bb64da", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-12-06T16:09:29.000Z", 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YES\n2. YES", "lm_q1_score": 0.6548947155710234, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3427852205407398}} {"text": "open import Level.NP\nopen import Type\nopen import Relation.Binary.Logical\nopen import Relation.Binary.PropositionalEquality\n\nmodule Explore.Universe.Logical (X : ★) where\n\nopen import Explore.Universe.Type\nopen import Explore.Universe X\nopen import Explore.Core\n\nmodule From⟦X⟧ (⟦X⟧ : ⟦★₀⟧ X X) where\n\n -- TODO _⟦≃⟧_ : (⟦Rel⟧ ⟦★₀⟧) ₀ _≃_ _≃_\n\n data ⟦U⟧ : ⟦★₁⟧ U U\n ⟦El⟧ : (⟦U⟧ ⟦→⟧ ⟦★₀⟧) El El\n\n data ⟦U⟧ where\n ⟦𝟘ᵁ⟧ : ⟦U⟧ 𝟘ᵁ 𝟘ᵁ\n ⟦𝟙ᵁ⟧ : ⟦U⟧ 𝟙ᵁ 𝟙ᵁ\n ⟦𝟚ᵁ⟧ : ⟦U⟧ 𝟚ᵁ 𝟚ᵁ\n _⟦×ᵁ⟧_ : ⟦Op₂⟧ {_} {_} {₁} ⟦U⟧ _×ᵁ_ _×ᵁ_\n _⟦⊎ᵁ⟧_ : ⟦Op₂⟧ {_} {_} {₁} ⟦U⟧ _⊎ᵁ_ _⊎ᵁ_\n ⟦Σᵁ⟧ : (⟨ u ∶ ⟦U⟧ ⟩⟦→⟧ (⟦El⟧ u ⟦→⟧ ⟦U⟧) ⟦→⟧ ⟦U⟧) Σᵁ Σᵁ\n ⟦Xᵁ⟧ : ⟦U⟧ Xᵁ Xᵁ\n -- ⟦≃ᵁ⟧ : (⟨ u ∶ ⟦U⟧ ⟩⟦→⟧ (⟨ A ∶ ⟦★₀⟧ ⟩⟦→⟧ ⟦El⟧ u ⟦≃⟧ A ⟦→⟧ ⟦U⟧)) ≃ᵁ ≃ᵁ\n\n ⟦El⟧ ⟦𝟘ᵁ⟧ = _≡_\n ⟦El⟧ ⟦𝟙ᵁ⟧ = _≡_\n ⟦El⟧ ⟦𝟚ᵁ⟧ = _≡_\n ⟦El⟧ (u₀ ⟦×ᵁ⟧ u₁) = ⟦El⟧ u₀ ⟦×⟧ ⟦El⟧ u₁\n ⟦El⟧ (u₀ ⟦⊎ᵁ⟧ u₁) = ⟦El⟧ u₀ ⟦⊎⟧ ⟦El⟧ u₁\n ⟦El⟧ (⟦Σᵁ⟧ u f) = ⟦Σ⟧ (⟦El⟧ u) λ x → ⟦El⟧ (f x)\n ⟦El⟧ ⟦Xᵁ⟧ = ⟦X⟧\n -- ⟦El⟧ (⟦≃ᵁ⟧ u A e) = A\n\n module From⟦Xᵉ⟧\n {⟦X⟧ : ⟦★₀⟧ X X}\n {ℓ₀ ℓ₁} ℓᵣ\n {Xᵉ : Explore X}\n (⟦Xᵉ⟧ : ⟦Explore⟧ {ℓ₀} {ℓ₁} ℓᵣ ⟦X⟧ Xᵉ Xᵉ) where\n open From⟦X⟧ ⟦X⟧ public\n\n ⟦explore⟧ : ∀ {u₀ u₁} (u : ⟦U⟧ u₀ u₁) → ⟦Explore⟧ {ℓ₀} {ℓ₁} ℓᵣ (⟦El⟧ u) (explore u₀) (explore u₁)\n ⟦explore⟧ ⟦𝟘ᵁ⟧ = ⟦𝟘ᵉ⟧ {ℓ₀} {ℓ₁} {ℓᵣ}\n ⟦explore⟧ ⟦𝟙ᵁ⟧ = ⟦𝟙ᵉ⟧ {ℓ₀} {ℓ₁} {ℓᵣ} {_≡_} {refl}\n ⟦explore⟧ ⟦𝟚ᵁ⟧ = ⟦𝟚ᵉ⟧ {ℓ₀} {ℓ₁} {ℓᵣ} {_≡_} {refl} {refl}\n ⟦explore⟧ (u₀ ⟦×ᵁ⟧ u₁) = ⟦explore×⟧ {ℓ₀} {ℓ₁} {ℓᵣ} (⟦explore⟧ u₀) (⟦explore⟧ u₁)\n ⟦explore⟧ (u₀ ⟦⊎ᵁ⟧ u₁) = ⟦explore⊎⟧ {ℓ₀} {ℓ₁} {ℓᵣ} (⟦explore⟧ u₀) (⟦explore⟧ u₁)\n ⟦explore⟧ (⟦Σᵁ⟧ u f) = ⟦exploreΣ⟧ {ℓ₀} {ℓ₁} {ℓᵣ} (⟦explore⟧ u) (⟦explore⟧ ∘ f)\n ⟦explore⟧ ⟦Xᵁ⟧ = ⟦Xᵉ⟧\n -- ⟦explore⟧ (⟦≃ᵁ⟧ u A e) = {!⟦explore-iso⟧ e!}\n\n\n{-\n⟦U⟧-sound : ∀ {{_ : FunExt}} {x y} → ⟦U⟧ x y → x ≡ y\n⟦U⟧-refl : ∀ x → ⟦U⟧ x x\n\n{-\n⟦El⟧-refl : ∀ x → {!⟦El⟧ x x!}\n⟦El⟧-refl = {!!}\n-}\n\n⟦U⟧-sound ⟦𝟘ᵁ⟧ = refl\n⟦U⟧-sound ⟦𝟙ᵁ⟧ = refl\n⟦U⟧-sound ⟦𝟚ᵁ⟧ = refl\n⟦U⟧-sound (u ⟦×ᵁ⟧ u₁) = ap₂ _×ᵁ_ (⟦U⟧-sound u) (⟦U⟧-sound u₁)\n⟦U⟧-sound (u ⟦⊎ᵁ⟧ u₁) = ap₂ _⊎ᵁ_ (⟦U⟧-sound u) (⟦U⟧-sound u₁)\n⟦U⟧-sound (⟦Σᵁ⟧ {u₀} {u₁} u {f₀} {f₁} fᵣ) = apd₂ Σᵁ (⟦U⟧-sound u) (tr-→ El (const U) (⟦U⟧-sound u) f₀ ∙ λ= (λ A → ap (λ z → z (f₀ (tr El (! ⟦U⟧-sound u) A))) (tr-const (⟦U⟧-sound u)) ∙ ⟦U⟧-sound (fᵣ {!!}))) -- (λ= (λ y → let foo = xᵣ {{!!}} {y} {!xᵣ!} in {!tr-→ El (const U) (⟦U⟧-sound u)!}))\n\n⟦U⟧-refl 𝟘ᵁ = ⟦𝟘ᵁ⟧\n⟦U⟧-refl 𝟙ᵁ = ⟦𝟙ᵁ⟧\n⟦U⟧-refl 𝟚ᵁ = ⟦𝟚ᵁ⟧\n⟦U⟧-refl (x ×ᵁ x₁) = ⟦U⟧-refl x ⟦×ᵁ⟧ ⟦U⟧-refl x₁\n⟦U⟧-refl (x ⊎ᵁ x₁) = ⟦U⟧-refl x ⟦⊎ᵁ⟧ ⟦U⟧-refl x₁\n⟦U⟧-refl (Σᵁ x f) = ⟦Σᵁ⟧ (⟦U⟧-refl x) (λ y → {!⟦U⟧-refl ?!})\n-}\n", "meta": {"hexsha": "31670ca7b32664780b6dcdc87c801330413c1b95", "size": 2602, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "experiments/Explore/Universe/Logical.agda", "max_stars_repo_name": "crypto-agda/explore", "max_stars_repo_head_hexsha": "16bc8333503ff9c00d47d56f4ec6113b9269a43e", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2016-06-05T09:25:32.000Z", "max_stars_repo_stars_event_max_datetime": "2017-06-28T19:19:29.000Z", "max_issues_repo_path": "experiments/Explore/Universe/Logical.agda", "max_issues_repo_name": "crypto-agda/explore", "max_issues_repo_head_hexsha": "16bc8333503ff9c00d47d56f4ec6113b9269a43e", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-03-16T14:24:04.000Z", "max_issues_repo_issues_event_max_datetime": "2019-03-16T14:24:04.000Z", "max_forks_repo_path": "experiments/Explore/Universe/Logical.agda", "max_forks_repo_name": "crypto-agda/explore", "max_forks_repo_head_hexsha": "16bc8333503ff9c00d47d56f4ec6113b9269a43e", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.9367088608, "max_line_length": 292, "alphanum_fraction": 0.4454265949, "num_tokens": 2063, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.43782349911420193, "lm_q1q2_score": 0.34266802516057054}} {"text": "open import MLib.Algebra.PropertyCode\nopen import MLib.Algebra.PropertyCode.Structures\n\nmodule MLib.Matrix.Tensor {c ℓ} (struct : Struct bimonoidCode c ℓ) where\n\nopen import MLib.Prelude\nopen import MLib.Matrix.Core\nopen import MLib.Matrix.Equality struct\nopen import MLib.Matrix.Mul struct\nopen import MLib.Algebra.Operations struct\n\nopen Table using (head; tail; rearrange; fromList; toList; _≗_; replicate)\nopen Nat using () renaming (_+_ to _+ℕ_; _*_ to _*ℕ_)\n\nopen FunctionProperties\n\nopen import MLib.Fin.Parts.Simple\n\n-- Tensor product\n\n_⊠_ : ∀ {m n p q} → Matrix S m n → Matrix S p q → Matrix S (m *ℕ p) (n *ℕ q)\n(A ⊠ B) i j =\n let i₁ , i₂ = toParts i\n j₁ , j₂ = toParts j\n in A i₁ j₁ *′ B i₂ j₂\n\nprivate\n ≡⇒≡×≡×≡ :\n ∀ {a b c} {A : Set a} {B : Set b} {C : Set c}\n {i i′ : A} {j j′ : B} {k k′ : C} →\n (i , j , k) ≡ (i′ , j′ , k′) →\n i ≡ i′ × j ≡ j′ × k ≡ k′\n ≡⇒≡×≡×≡ = Σ.map id Σ.≡⇒≡×≡ ∘ Σ.≡⇒≡×≡\n\nopen _≃_\n\nmodule _ ⦃ props : Has (associative on * ∷ []) ⦄ {m n p q r s} where\n\n ⊠-associative :\n (A : Matrix S m n) (B : Matrix S p q) (C : Matrix S r s) →\n (A ⊠ B) ⊠ C ≃ A ⊠ (B ⊠ C)\n ⊠-associative A B C .m≡p = Nat.*-assoc m p r\n ⊠-associative A B C .n≡q = Nat.*-assoc n q s\n ⊠-associative A B C .equal {i} {i′} {j} {j′} i≅i′ j≅j′ =\n let i₁ , i₂ , i₃ = toParts³ m p r i\n j₁ , j₂ , j₃ = toParts³ n q s j\n\n i′₁ , i′₂ , i′₃ = toParts³′ m p r i′\n j′₁ , j′₂ , j′₃ = toParts³′ n q s j′\n\n i₁-eq , i₂-eq , i₃-eq = ≡⇒≡×≡×≡ (toParts-assoc m p r i≅i′)\n j₁-eq , j₂-eq , j₃-eq = ≡⇒≡×≡×≡ (toParts-assoc n q s j≅j′)\n\n open EqReasoning S.setoid\n in begin\n ((A ⊠ B) ⊠ C) i j ≡⟨⟩\n A i₁ j₁ *′ B i₂ j₂ *′ C i₃ j₃ ≈⟨ from props (associative on *) _ _ _ ⟩\n A i₁ j₁ *′ (B i₂ j₂ *′ C i₃ j₃) ≡⟨ ≡.cong₂ _*′_ (≡.cong₂ A i₁-eq j₁-eq) (≡.cong₂ _*′_ (≡.cong₂ B i₂-eq j₂-eq) (≡.cong₂ C i₃-eq j₃-eq)) ⟩\n A i′₁ j′₁ *′ (B i′₂ j′₂ *′ C i′₃ j′₃) ≡⟨⟩\n (A ⊠ (B ⊠ C)) i′ j′ ∎\n\n⊠-cong : ∀ {m n m′ n′} {p q p′ q′} {A : Matrix S m n} {A′ : Matrix S m′ n′} {B : Matrix S p q} {B′ : Matrix S p′ q′} → A ≃ A′ → B ≃ B′ → (A ⊠ B) ≃ (A′ ⊠ B′)\n⊠-cong A≃A′ B≃B′ with A≃A′ .m≡p | B≃B′ .m≡p | A≃A′ .n≡q | B≃B′ .n≡q\n⊠-cong {A = A} {A′} {B} {B′} A≃A′ B≃B′ | ≡.refl | ≡.refl | ≡.refl | ≡.refl = lem\n where\n lem : (A ⊠ B) ≃ (A′ ⊠ B′)\n lem .m≡p = ≡.refl\n lem .n≡q = ≡.refl\n lem .equal ≅.refl ≅.refl = cong * (A≃A′ .equal ≅.refl ≅.refl) (B≃B′ .equal ≅.refl ≅.refl)\n\n⊠-identityˡ :\n ⦃ props : Has (1# is leftIdentity for * ∷ []) ⦄ →\n ∀ {m n} (A : Matrix S m n) → 1● {1} ⊠ A ≃ A\n⊠-identityˡ A .m≡p = Nat.*-identityˡ _\n⊠-identityˡ A .n≡q = Nat.*-identityˡ _\n⊠-identityˡ ⦃ props ⦄ A .equal {i} {i′} {j} {j′} i≅i′ j≅j′ =\n let i₁ , i₂ = toParts {1} i\n j₁ , j₂ = toParts {1} j\n -- x : i₁ ≡ zero\n -- x′ : i₂ ≡ i′\n -- y : j₁ ≡ zero\n -- y′ : j₂ ≡ j′\n x , x′ = Σ.≡⇒≡×≡ (toParts-1ˡ i i′ i≅i′)\n y , y′ = Σ.≡⇒≡×≡ (toParts-1ˡ j j′ j≅j′)\n\n open EqReasoning S.setoid\n in begin\n (1● {1} ⊠ A) i j ≡⟨⟩\n 1● {1} i₁ j₁ *′ A i₂ j₂ ≡⟨ ≡.cong₂ _*′_ (≡.cong₂ (1● {1}) x y) (≡.cong₂ A x′ y′) ⟩\n 1● {1} zero zero *′ A i′ j′ ≡⟨⟩\n 1′ *′ A i′ j′ ≈⟨ from props (1# is leftIdentity for *) _ ⟩\n A i′ j′ ∎\n\n⊠-identityʳ :\n ⦃ props : Has (1# is rightIdentity for * ∷ []) ⦄ →\n ∀ {m n} (A : Matrix S m n) → A ⊠ 1● {1} ≃ A\n⊠-identityʳ A .m≡p = Nat.*-identityʳ _\n⊠-identityʳ A .n≡q = Nat.*-identityʳ _\n⊠-identityʳ A .equal {i} {i′} {j} {j′} i≅i′ j≅j′ with Σ.≡⇒≡×≡ (toParts-1ʳ i i′ i≅i′) | Σ.≡⇒≡×≡ (toParts-1ʳ j j′ j≅j′)\n⊠-identityʳ ⦃ props ⦄ A .equal {i} {i′} {j} {j′} i≅i′ j≅j′ | x , x′ | y , y′\n rewrite x | x′ | y | y′ = from props (1# is rightIdentity for *) _\n", "meta": {"hexsha": "cf6b878078a92eb4ff4a32b7ebca5f3be5d4fa12", "size": 3738, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/MLib/Matrix/Tensor.agda", "max_stars_repo_name": "bch29/agda-matrices", "max_stars_repo_head_hexsha": "e26ae2e0aa7721cb89865aae78625a2f3fd2b574", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/MLib/Matrix/Tensor.agda", "max_issues_repo_name": "bch29/agda-matrices", "max_issues_repo_head_hexsha": "e26ae2e0aa7721cb89865aae78625a2f3fd2b574", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/MLib/Matrix/Tensor.agda", "max_forks_repo_name": "bch29/agda-matrices", "max_forks_repo_head_hexsha": "e26ae2e0aa7721cb89865aae78625a2f3fd2b574", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.6470588235, "max_line_length": 156, "alphanum_fraction": 0.4794007491, "num_tokens": 2004, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.3426342022420749}} {"text": "open import Data.Nat using (ℕ; _+_) renaming (_≤?_ to _≤?ₙ_)\nopen import Data.Bool using (Bool; true; false; not; _∧_)\nopen import Data.String using (String)\nopen import Data.Sum using (_⊎_; [_,_]; inj₁; inj₂)\nopen import Relation.Binary using (Decidable)\nopen import Relation.Nullary using (yes; no; ¬_)\nopen import Relation.Nullary.Negation using (contradiction)\nopen import Data.List using (List; []; _∷_)\nopen import Relation.Nullary.Decidable using (⌊_⌋)\nopen import Data.Empty using (⊥-elim)\nopen import Data.Product using (_×_; -,_; _-,-_; ∃; ∃-syntax) renaming (_,_ to _,,_)\nopen import Relation.Binary.PropositionalEquality using (_≡_; refl; sym)\nopen import Level using (Level; suc; _⊔_)\nopen import Agda.Builtin.Sigma hiding (_,_) -- renaming (_,_ to _,,_)\nopen import Data.Maybe using (Maybe; just; nothing; Is-just)\nimport Relation.Binary.PropositionalEquality as Eq\n\nopen import IMP hiding (com; state) ---using (aexp; aval; bexp; bval)\n\npostulate\n _≟_ : Decidable {A = ℕ} _≡_\n\naddr = ℕ\n\nheap = addr → Maybe val\n\nstore = vname → val\n\nstate = store × heap\n\nassn : ∀{l} → Set (suc l)\nassn {a} = store → heap → Set a\n\nemp : assn\nemp _ h = ∀ a → h a ≡ nothing\n\n_⊢>_ : IMP.aexp → IMP.aexp → assn\n_⊢>_ a a′ s h = h (aval a s) ≡ just (aval a′ s)\n × ∀ (a″) → ¬(a″ ≡ aval a s) → h a″ ≡ nothing\n\n_⊆_ : heap → heap → Set\nh₀ ⊆ h = ∀ a → (h a ≡ nothing → h₀ a ≡ nothing)\n × (∀{o} → h₀ a ≡ just o → h₀ a ≡ h a)\n\nheap-union : heap → heap → heap → addr → Set\nheap-union h h₁ h₂ a with h a\n... | just o = (h₁ a ≡ just o × h₂ a ≡ nothing)\n ⊎ (h₁ a ≡ nothing × h₂ a ≡ just o)\n... | nothing = h₁ a ≡ nothing × h₂ a ≡ nothing\n\n_∼_⊥_ : heap → heap → heap → Set\nh ∼ h₁ ⊥ h₂ = ∀ a → heap-union h h₁ h₂ a\n\nunion-subset : ∀ h {h₁ h₂}\n → h ∼ h₁ ⊥ h₂\n → h₁ ⊆ h\nunion-subset h x a with h a | x a\nunion-subset h x a | just x₁ | inj₁ (fst₁ ,, snd₁) = (λ ()) ,, (λ {x} x₁ → fst₁)\nunion-subset h x a | just x₁ | inj₂ (fst₁ ,, snd₁) rewrite fst₁ = (λ _ → refl) ,, (λ {x} ())\nunion-subset h x a | nothing | fst₁ ,, snd₁ = (λ x → fst₁) ,, (λ {x} x₁ → fst₁)\n\n_*_ : ∀{l} → assn {l} → assn {l} → assn {l}\n_*_ P Q s h = ∃[ h₁ ] ∃[ h₂ ] ((h ∼ h₁ ⊥ h₂)\n × P s h₁\n × Q s h₂)\n\n_dom_ : addr → heap → Set\na dom h = ∃[ v ] (h a ≡ just v)\n\n_¬dom_ : addr → heap → Set\na ¬dom h = h a ≡ nothing\n\n_[_::=ₕ_] : heap → addr → val → heap\n(h [ X ::=ₕ n ]) Y with Y ≟ X\n... | yes _ = just n\n... | no _ = h Y\n\n_/[_] : heap → addr → heap\n(h /[ X ]) Y with Y ≟ X\n... | yes _ = nothing\n... | no _ = h Y\n\ndata com : Set where\n SKIP : com\n _::=_ : String → aexp → com\n _::_ : com → com → com \n IF_THEN_ELSE_ : bexp → com → com → com\n WHILE_DO_ : bexp → com → com\n _::=cons_ : String → addr → com\n _::=[_] : String → aexp → com\n [_]::=_ : aexp → aexp → com\n dispose_ : aexp → com\n\ndata ⊢[_]_[_] {l} : assn {l} → com → assn {l} → Set (suc l) where\n Skip : ∀{P}\n → ⊢[ P ] SKIP [ P ]\n Loc : ∀{Q a x}\n → ⊢[ (λ s → Q (s [ x ::= aval a s ])) ] (x ::= a) [ Q ]\n Comp : ∀{P Q R c₁ c₂}\n → ⊢[ P ] c₁ [ Q ]\n → ⊢[ Q ] c₂ [ R ]\n → ⊢[ P ] c₁ :: c₂ [ R ]\n If : ∀{P b c₁ Q c₂}\n → ⊢[ (λ s h → P s h × bval b s ≡ true) ] c₁ [ Q ]\n → ⊢[ (λ s h → P s h × bval b s ≡ false) ] c₂ [ Q ]\n → ⊢[ P ] (IF b THEN c₁ ELSE c₂) [ Q ]\n While : ∀{P b c}\n → ⊢[ (λ s h → P s h × bval b s ≡ true) ] c [ P ]\n → ⊢[ P ] (WHILE b DO c) [ (λ s h → P s h × bval b s ≡ false) ]\n Conseq : ∀{P Q P′ Q′ : assn} {c}\n → (∀ s h → P′ s h → P s h)\n → ⊢[ P ] c [ Q ]\n → (∀ s h → Q s h → Q′ s h)\n → ⊢[ P′ ] c [ Q′ ]\n Frame : ∀{A B R c}\n → ⊢[ A ] c [ B ]\n → ⊢[ A * R ] c [ B * R ]\n\ndata config : Set where\n ⦅_,_,_⦆ : com → store → heap → config\n abort : config\n\ndata _⇒_ : config → config → Set where\n Loc : ∀{x a s h}\n → ⦅ x ::= a , s , h ⦆ ⇒ ⦅ SKIP , (s [ x ::= aval a s ]) , h ⦆\n Comp₁ : ∀{c s h}\n → ⦅ SKIP :: c , s , h ⦆ ⇒ ⦅ c , s , h ⦆\n Comp₂ : ∀{c₁ c₁′ c₂ s s′ h h′}\n → ⦅ c₁ , s , h ⦆ ⇒ ⦅ c₁′ , s′ , h′ ⦆\n → ⦅ c₁ :: c₂ , s , h ⦆ ⇒ ⦅ c₁′ :: c₂ , s′ , h′ ⦆\n CompFail : ∀{c₁ c₂ s h}\n → ⦅ c₁ , s , h ⦆ ⇒ abort\n → ⦅ c₁ :: c₂ , s , h ⦆ ⇒ abort\n IfTrue : ∀{b s c₁ c₂ h}\n → bval b s ≡ true\n → ⦅ IF b THEN c₁ ELSE c₂ , s , h ⦆ ⇒ ⦅ c₁ , s , h ⦆\n IfFalse : ∀{b s c₁ c₂ h}\n → bval b s ≡ false\n → ⦅ IF b THEN c₁ ELSE c₂ , s , h ⦆ ⇒ ⦅ c₂ , s , h ⦆ \n While : ∀{b s c h}\n → ⦅ WHILE b DO c , s , h ⦆ ⇒ ⦅ IF b THEN (c :: (WHILE b DO c)) ELSE SKIP , s , h ⦆\n Cons : ∀{l h s x h′ s′}\n → l ¬dom h\n → h′ ≡ h [ l ::=ₕ 0 ]\n → s′ ≡ s [ x ::= l ] \n → ⦅ x ::=cons l , s , h ⦆ ⇒ ⦅ SKIP , s′ , h′ ⦆\n Lookup : ∀{a s x h v s′}\n → (aval a s) dom h\n → s′ ≡ s [ x ::= v ]\n → ⦅ x ::=[ a ] , s , h ⦆ ⇒ ⦅ SKIP , s′ , h ⦆\n LookupFail : ∀{a s x h}\n → (aval a s) ¬dom h\n → ⦅ x ::=[ a ] , s , h ⦆ ⇒ abort\n Write : ∀{a s a′ h}\n → (aval a s) dom h\n → ⦅ [ a ]::= a′ , s , h ⦆ ⇒ ⦅ SKIP , s , h [ aval a s ::=ₕ aval a′ s ] ⦆\n WriteFail : ∀{a s a′ x h}\n → (aval a s) ¬dom h\n → ⦅ [ x ]::= a′ , s , h ⦆ ⇒ abort\n Dispose : ∀{a s h}\n → (aval a s) dom h\n → ⦅ dispose a , s , h ⦆ ⇒ ⦅ SKIP , s , h /[ aval a s ] ⦆\n DisposeFail : ∀{a s h}\n → (aval a s) ¬dom h\n → ⦅ dispose a , s , h ⦆ ⇒ abort\n\ndata _⇒*_ : config → config → Set where\n _∎ : ∀ c → c ⇒* c\n _→⟨_⟩_ : ∀ c {c′ c″}\n → c ⇒ c′\n → c′ ⇒* c″\n → c ⇒* c″ \n\nSafe : config → Set\nSafe c = ¬ (c ⇒* abort)\n\nlemma1 : ∀{c s h₀ h}\n → h₀ ⊆ h\n → ⦅ c , s , h ⦆ ⇒ abort\n → ⦅ c , s , h₀ ⦆ ⇒ abort\nlemma1 sub (CompFail r) = CompFail (lemma1 sub r)\nlemma1 sub (LookupFail {a}{s} r) = LookupFail (fst (sub (aval a s)) r)\nlemma1 sub (WriteFail {a}{s} r) = WriteFail {a} (fst (sub (aval a s)) r)\nlemma1 sub (DisposeFail {a}{s} r) = DisposeFail (fst (sub (aval a s)) r)\n\nsubset-update : ∀ l {h h₀ v}\n → h₀ ⊆ h\n → (h₀ [ l ::=ₕ v ]) ⊆ (h [ l ::=ₕ v ])\nsubset-update l b Y with Y ≟ l\n... | yes p = (λ x → x) ,, λ {o} _ → refl\n... | no ¬p = b Y\n\nsubset-delete : ∀ v {h h₀}\n → h₀ ⊆ h\n → (h₀ /[ v ]) ⊆ (h /[ v ])\nsubset-delete l b Y with Y ≟ l\n... | yes p = (λ x → refl) ,, (λ {x} x₁ → refl)\n... | no ¬p = b Y\n\nlemma2 : ∀ h₀ {h c c′ s s′ h′}\n → h₀ ⊆ h\n → ⦅ c , s , h ⦆ ⇒ ⦅ c′ , s′ , h′ ⦆\n → ⦅ c , s , h₀ ⦆ ⇒ abort\n ⊎ (∃[ h′₀ ] (h′₀ ⊆ h′\n × ⦅ c , s , h₀ ⦆ ⇒ ⦅ c′ , s′ , h′₀ ⦆))\nlemma2 h₀ x Loc = inj₂ (h₀ ,, x ,, Loc)\nlemma2 h₀ x Comp₁ = inj₂ (h₀ ,, x ,, Comp₁)\nlemma2 h₀ x (Comp₂ x₁) with lemma2 h₀ x x₁\nlemma2 h₀ x (Comp₂ x₁) | inj₁ x₂ = inj₁ (CompFail x₂)\nlemma2 h₀ x (Comp₂ x₁) | inj₂ (h′₀ ,, sub ,, red) = inj₂ (h′₀ ,, sub ,, Comp₂ red)\nlemma2 h₀ x (IfTrue x₁) = inj₂ (h₀ ,, x ,, IfTrue x₁)\nlemma2 h₀ x (IfFalse x₁) = inj₂ (h₀ ,, x ,, IfFalse x₁)\nlemma2 h₀ x While = inj₂ (h₀ ,, x ,, While)\nlemma2 h₀ x (Cons {l} x₁ A B) rewrite A | B = inj₂ ( (h₀ [ l ::=ₕ 0 ])\n ,, subset-update l x\n ,, Cons (fst (x l) x₁) refl refl)\nlemma2 h₀ x (Lookup {a}{s} (_ ,, p) A) rewrite A with h₀ (aval a s) | Eq.inspect h₀ (aval a s)\n... | nothing | Eq.[ eq ] = inj₁ (LookupFail eq)\n... | just o | Eq.[ eq ] = inj₂ ( h₀\n ,, x\n ,, Lookup (-, Eq.trans (snd (x (aval a s)) eq) p) refl)\nlemma2 h₀ x (Write {a}{s}{a′} (_ ,, x₁)) with h₀ (aval a s) | Eq.inspect h₀ (aval a s)\n... | nothing | Eq.[ eq ] = inj₁ (WriteFail {a} eq)\n... | just o | Eq.[ eq ] = inj₂ ( (h₀ [ aval a s ::=ₕ aval a′ s ])\n ,, subset-update (aval a s) x\n ,, Write (-, (Eq.trans (snd (x (aval a s)) eq) x₁)))\nlemma2 h₀ x (Dispose {a}{s} (_ ,, x₁)) with h₀ (aval a s) | Eq.inspect h₀ (aval a s)\n... | nothing | Eq.[ eq ] = inj₁ (DisposeFail eq)\n... | just o | Eq.[ eq ] = inj₂ ( (h₀ /[ aval a s ])\n ,, subset-delete (aval a s) x\n ,, Dispose (-, (Eq.trans (snd (x (aval a s)) eq) x₁)))\n\nframe1sub : ∀{c s h H}\n → h ⊆ H\n → Safe ⦅ c , s , h ⦆\n → Safe ⦅ c , s , H ⦆\nframe1sub {c}{s}{h}{H} x x₁ (_→⟨_⟩_ .(⦅ c , s , H ⦆) {⦅ x₄ , x₅ , x₆ ⦆} x₂ x₃) with lemma2 h x x₂\n... | inj₁ x₇ = x₁ (⦅ c , s , h ⦆ →⟨ x₇ ⟩ (abort ∎))\n... | inj₂ (_ ,, fst₂ ,, snd₁) = frame1sub fst₂ (λ z → x₁ (_ →⟨ snd₁ ⟩ z)) x₃\nframe1sub {c}{s}{h}{H} x x₁ (_→⟨_⟩_ .(⦅ c , s , H ⦆) {abort} x₂ x₃) = x₁ (_ →⟨ lemma1 x x₂ ⟩ (abort ∎))\n\nframe1 : ∀{c s h H z}\n → H ∼ h ⊥ z\n → Safe ⦅ c , s , h ⦆\n → Safe ⦅ c , s , H ⦆\nframe1 {H = H} x x₁ x₂ = frame1sub (union-subset H x) x₁ x₂\n\nheap-union-update : ∀{l h h₀ h₁ v}\n → l ¬dom h₁\n → h ∼ h₀ ⊥ h₁\n → (h [ l ::=ₕ v ]) ∼ (h₀ [ l ::=ₕ v ]) ⊥ h₁\nheap-union-update {l}{h}{v = v} d x a with (h [ l ::=ₕ v ]) a | Eq.inspect (h [ l ::=ₕ v ]) a\nheap-union-update {l}{h} d x a | just x₁ | Eq.[ eq ] with a ≟ l | h a | Eq.inspect h a | x a\nheap-union-update {l}{h} d x a | just x₁ | Eq.[ eq ] | yes p | just x₂ | Eq.[ eq2 ] | inj₁ x₃ = inj₁ (eq ,, snd x₃)\nheap-union-update {l}{h} d x a | just x₁ | Eq.[ eq ] | yes p | just x₂ | Eq.[ eq2 ] | inj₂ y rewrite p = inj₁ (eq ,, d)\nheap-union-update {l}{h} d x a | just x₁ | Eq.[ eq ] | yes p | nothing | Eq.[ eq2 ] | C = inj₁ (eq ,, snd C)\nheap-union-update {l}{h} d x a | just x₁ | Eq.[ eq ] | no ¬p | just x₂ | Eq.[ eq2 ] | inj₁ x₃ rewrite (Eq.trans (sym eq) eq2) = inj₁ x₃\nheap-union-update {l}{h} d x a | just x₁ | Eq.[ eq ] | no ¬p | just x₂ | Eq.[ eq2 ] | inj₂ y rewrite (Eq.trans (sym eq) eq2) = inj₂ y\nheap-union-update {l}{h} d x a | just x₁ | Eq.[ eq ] | no ¬p | nothing | Eq.[ eq2 ] | C rewrite (Eq.trans (sym eq) eq2) = inj₁ C\nheap-union-update {l}{h} d x a | nothing | Eq.[ eq ] with a ≟ l | h a | Eq.inspect h a | x a\nheap-union-update {l}{h} d x a | nothing | Eq.[ eq ] | no ¬p | just x₁ | Eq.[ eq2 ] | inj₁ x₂ = Eq.trans (Eq.trans (fst x₂) (sym eq2)) eq ,, snd x₂\nheap-union-update {l}{h} d x a | nothing | Eq.[ eq ] | no ¬p | just x₁ | Eq.[ eq2 ] | inj₂ y = fst y ,, Eq.trans (snd y) (Eq.trans (sym eq2) eq)\nheap-union-update {l}{h} d x a | nothing | Eq.[ eq ] | no ¬p | nothing | Eq.[ eq2 ] | E = E\n\nheap-union-delete : ∀{h h₀ h₁ v}\n → v ¬dom h₁\n → h ∼ h₀ ⊥ h₁\n → (h /[ v ]) ∼ (h₀ /[ v ]) ⊥ h₁\nheap-union-delete {h}{h₀}{h₁}{v} d x a with (h /[ v ]) a | Eq.inspect (h /[ v ]) a\nheap-union-delete {h} {h₀} {h₁} {v} d x a | just x₁ | Eq.[ eq ] with a ≟ v | h a | Eq.inspect h a | x a\nheap-union-delete {h} {h₀} {h₁} {v} d x a | just x₁ | Eq.[ () ] | yes p | just x₂ | Eq.[ eq2 ] | R\nheap-union-delete {h} {h₀} {h₁} {v} d x a | just x₁ | Eq.[ eq ] | no ¬p | just x₂ | Eq.[ eq2 ] | inj₁ x₃ rewrite (Eq.trans (sym eq) eq2) = inj₁ x₃\nheap-union-delete {h} {h₀} {h₁} {v} d x a | just x₁ | Eq.[ eq ] | no ¬p | just x₂ | Eq.[ eq2 ] | inj₂ y rewrite (Eq.trans (sym eq) eq2) = inj₂ y\nheap-union-delete {h} {h₀} {h₁} {v} d x a | just x₁ | Eq.[ eq ] | no ¬p | nothing | Eq.[ eq2 ] | fst₁ ,, snd₁ = inj₂ (fst₁ ,, Eq.trans snd₁ (Eq.trans (sym eq2) eq))\nheap-union-delete {h} {h₀} {h₁} {v} d x a | nothing | Eq.[ eq ] with a ≟ v | h a | Eq.inspect h a | x a\nheap-union-delete {h} {h₀} {h₁} {v} d x a | nothing | Eq.[ eq ] | yes p | just x₁ | Eq.[ eq2 ] | inj₁ x₂ = refl ,, snd x₂\nheap-union-delete {h} {h₀} {h₁} {v} d x a | nothing | Eq.[ eq ] | yes p | just x₁ | Eq.[ eq2 ] | inj₂ y rewrite p = refl ,, d\nheap-union-delete {h} {h₀} {h₁} {v} d x a | nothing | Eq.[ eq ] | yes p | nothing | Eq.[ eq2 ] | E = refl ,, snd E\nheap-union-delete {h} {h₀} {h₁} {v} d x a | nothing | Eq.[ eq ] | no ¬p | just x₁ | Eq.[ eq2 ] | inj₁ x₂ = Eq.trans (fst x₂) (Eq.trans (sym eq2) eq) ,, snd x₂\nheap-union-delete {h} {h₀} {h₁} {v} d x a | nothing | Eq.[ eq ] | no ¬p | just x₁ | Eq.[ eq2 ] | inj₂ y = fst y ,, Eq.trans (snd y) (Eq.trans (sym eq2) eq)\nheap-union-delete {h} {h₀} {h₁} {v} d x a | nothing | Eq.[ eq ] | no ¬p | nothing | Eq.[ eq2 ] | E = E\n\nunion-exclusionᵣ : ∀{l h h₀ h₁}\n → h ∼ h₀ ⊥ h₁\n → l ¬dom h\n → l ¬dom h₁\nunion-exclusionᵣ {l}{h} A B with h l | A l\n... | nothing | fst₁ ,, snd₁ = snd₁\n\n\nunion-presenceᵣ : ∀{l h h₀ h₁}\n → h ∼ h₀ ⊥ h₁\n → l dom h\n → l dom h₀\n → l ¬dom h₁\nunion-presenceᵣ {l}{h} A B C with h l | A l\nunion-presenceᵣ {l} {h} A B C | just x | inj₁ (fst₁ ,, snd₁) = snd₁\nunion-presenceᵣ {l} {h} A B (fst₁ ,, snd₁) | just x | inj₂ (fst₂ ,, snd₂) with Eq.trans (sym fst₂) snd₁\n... | ()\n\n\nunion-reduction : ∀{a b c s h′ h h₀ h′₀ h₁}\n → ⦅ c , s , h₀ ⦆ ⇒ ⦅ a , b , h′₀ ⦆\n → ⦅ c , s , h ⦆ ⇒ ⦅ a , b , h′ ⦆\n → h ∼ h₀ ⊥ h₁\n → h′ ∼ h′₀ ⊥ h₁\nunion-reduction Loc Loc C = C\nunion-reduction Comp₁ Comp₁ C = C\nunion-reduction (Comp₂ A) (Comp₂ B) C = union-reduction A B C\nunion-reduction (IfTrue x) (IfTrue x₁) C = C\nunion-reduction (IfTrue x) (IfFalse x₁) C = C\nunion-reduction (IfFalse x) (IfTrue x₁) C = C\nunion-reduction (IfFalse x) (IfFalse x₁) C = C\nunion-reduction While While C = C\nunion-reduction (Cons d A B) (Cons d2 A′ B′) C a rewrite A | B | A′ | B′ = heap-union-update (union-exclusionᵣ C d2) C a\nunion-reduction (Lookup e A) (Lookup e2 A′) C rewrite A | A′ = C\nunion-reduction (Write x) (Write x₁) C a = heap-union-update (union-presenceᵣ C x₁ x) C a\nunion-reduction (Dispose x) (Dispose x₁) C a = heap-union-delete (union-presenceᵣ C x₁ x) C a\n\n\nframe2 : ∀{c s h h₀ h₁ h′ s′}\n → Safe ⦅ c , s , h₀ ⦆\n → h ∼ h₀ ⊥ h₁\n → ⦅ c , s , h ⦆ ⇒* ⦅ SKIP , s′ , h′ ⦆\n → ∃[ h′₀ ] ( ⦅ c , s , h₀ ⦆ ⇒* ⦅ SKIP , s′ , h′₀ ⦆\n × h′ ∼ h′₀ ⊥ h₁ )\nframe2 s t (.(⦅ SKIP , _ , _ ⦆) ∎) = _ ,, (_ ∎) ,, t\nframe2 {h₀ = h₀} s t (_→⟨_⟩_ .(⦅ _ , _ , _ ⦆) {⦅ _ , _ , _ ⦆} x r) with lemma2 h₀ (union-subset _ t) x\n... | inj₁ x₄ = ⊥-elim (s (_ →⟨ x₄ ⟩ (abort ∎)))\n... | inj₂ (fst₁ ,, fst₂ ,, snd₁) with frame2 (λ z → s (_ →⟨ snd₁ ⟩ z)) (union-reduction snd₁ x t) r\n... | fst₃ ,, fst₄ ,, snd₂ = fst₃ ,, (_ →⟨ snd₁ ⟩ fst₄) ,, snd₂\nframe2 s t (_→⟨_⟩_ .(⦅ _ , _ , _ ⦆) {abort} x (.abort →⟨ () ⟩ r))\n\n⊨[_]_[_] : assn → com → assn → Set\n⊨[ A ] c [ B ] = ∀{s h}\n → A s h\n → Safe ⦅ c , s , h ⦆\n × (∀{s′ h′} → ⦅ c , s , h ⦆ ⇒* ⦅ SKIP , s′ , h′ ⦆ → B s′ h′)\n\nNotInfluenced : assn → com → Set\nNotInfluenced R c = ∀{s s′ z h₀ h′₀ hᵣ}\n → z ∼ h′₀ ⊥ hᵣ\n → ⦅ c , s , h₀ ⦆ ⇒* ⦅ SKIP , s′ , h′₀ ⦆ \n → R s hᵣ → R s′ hᵣ\n\nframe-soundness : ∀{A B R : assn} {c}\n → NotInfluenced R c\n → ⊨[ A ] c [ B ]\n → ⊨[ A * R ] c [ B * R ]\nframe-soundness {A}{B}{R}{c} Inf H {s}{h} 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YES\n2. NO\n\n", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3425424305375825}} {"text": "module _ where\n\nmodule M where\n\n data D : Set where\n d : D\n\n private\n instance\n x : D\n x = d\n\n! : ⦃ _ : M.D ⦄ → M.D\n! ⦃ x ⦄ = x\n\ny : M.D\ny = !\n", "meta": {"hexsha": "ff56a7f89465e71cb620a2722f06a8795bf2697b", "size": 161, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/Fail/Issue1913.agda", "max_stars_repo_name": "shlevy/agda", "max_stars_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1989, "max_stars_repo_stars_event_min_datetime": "2015-01-09T23:51:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T18:20:48.000Z", "max_issues_repo_path": "test/Fail/Issue1913.agda", "max_issues_repo_name": "shlevy/agda", "max_issues_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 4066, "max_issues_repo_issues_event_min_datetime": "2015-01-10T11:24:51.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T21:14:49.000Z", "max_forks_repo_path": "test/Fail/Issue1913.agda", "max_forks_repo_name": "Agda-zh/agda", "max_forks_repo_head_hexsha": "231d6ad8e77b67ff8c4b1cb35a6c31ccd988c3e9", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 371, "max_forks_repo_forks_event_min_datetime": "2015-01-03T14:04:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-30T19:00:30.000Z", "avg_line_length": 8.9444444444, "max_line_length": 21, "alphanum_fraction": 0.4285714286, "num_tokens": 71, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6959583124210896, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3425424243299271}} {"text": "{-# OPTIONS --rewriting #-}\n{-# OPTIONS --allow-unsolved-metas #-}\n\nopen import Common.Equality\n\n{-# BUILTIN REWRITE _≡_ #-}\n\npostulate\n A : Set\n f : A → A\n a b c : A\n fa-to-b : f a ≡ b\n fx-to-c : ∀ x → f x ≡ c\n\n{-# REWRITE fa-to-b #-}\n{-# REWRITE fx-to-c #-}\n\ntest₁ : f a ≡ b\ntest₁ = refl\n\nx : A\n\ntest₂ : f x ≡ c\ntest₂ = refl\n\nx = a\n", "meta": {"hexsha": "c3b5f1ed04ffe9f53a89c99e71d25fa0063f450d", "size": 339, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/Fail/RewriteNondeterministic.agda", "max_stars_repo_name": "shlevy/agda", "max_stars_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1989, "max_stars_repo_stars_event_min_datetime": "2015-01-09T23:51:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T18:20:48.000Z", "max_issues_repo_path": "test/Fail/RewriteNondeterministic.agda", "max_issues_repo_name": "shlevy/agda", "max_issues_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 4066, "max_issues_repo_issues_event_min_datetime": "2015-01-10T11:24:51.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T21:14:49.000Z", "max_forks_repo_path": "test/Fail/RewriteNondeterministic.agda", "max_forks_repo_name": "Agda-zh/agda", "max_forks_repo_head_hexsha": "231d6ad8e77b67ff8c4b1cb35a6c31ccd988c3e9", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 371, "max_forks_repo_forks_event_min_datetime": "2015-01-03T14:04:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-30T19:00:30.000Z", "avg_line_length": 12.5555555556, "max_line_length": 38, "alphanum_fraction": 0.5309734513, "num_tokens": 129, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7185943805178139, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.34246745912349813}} {"text": "{-# OPTIONS --cubical --no-import-sorts --allow-unsolved-metas #-}\n\nmodule Test.Number where\n\nopen import Agda.Primitive renaming (_⊔_ to ℓ-max; lsuc to ℓ-suc; lzero to ℓ-zero)\n\nprivate\n variable\n ℓ ℓ' ℓ'' : Level\n\nopen import Cubical.Foundations.Everything renaming (_⁻¹ to _⁻¹ᵖ; assoc to ∙-assoc)\nopen import Cubical.Relation.Nullary.Base -- ¬_\nopen import Cubical.Relation.Binary.Base -- Rel\nopen import Cubical.Data.Unit.Base -- Unit\nopen import Cubical.Data.Empty -- ⊥\nopen import Cubical.Data.Sum.Base renaming (_⊎_ to infixr 4 _⊎_)\nopen import Cubical.Data.Sigma.Base renaming (_×_ to infixr 4 _×_)\nopen import Cubical.Data.Empty renaming (elim to ⊥-elim) -- `⊥` and `elim`\nopen import Function.Base using (it; _∋_; _$_)\n\n-- open import Data.Nat.Base using (ℕ) renaming (_≤_ to _≤ₙ_)\n-- open import Cubical.Data.Nat using (ℕ; zero; suc) renaming (_+_ to _+ₙ_)\n-- open import Cubical.Data.Nat.Order renaming (zero-≤ to z≤n; suc-≤-suc to s≤s; _≤_ to _≤ₙ_; _<_ to _<ₙ_)\n-- open import Cubical.Data.Fin.Base\n-- import Cubical.Data.Fin.Properties\n-- open import Cubical.Data.Nat using (ℕ; zero; suc) renaming (_+_ to _+ₙ_)\n-- open import Cubical.Data.Nat.Properties using (+-suc; injSuc; snotz; +-comm; +-assoc; +-zero; inj-m+)\n-- open import Cubical.Data.Nat.Order renaming (zero-≤ to z≤n; suc-≤-suc to s≤s; _≤_ to _≤ₙ_; _<_ to _<ₙ_; _≟_ to _≟ₙ_)\n-- open import Data.Nat.Base using (ℕ; z≤n; s≤s; zero; suc) renaming (_≤_ to _≤ₙ_; _<_ to _<ₙ_; _+_ to _+ₙ_)\n-- open import Agda.Builtin.Bool renaming (true to TT; false to FF)\n-- import Cubical.Data.Fin.Properties\n-- open import Data.Nat.Properties using (+-mono-<)\n\n-- open import Bundles\n\nopen import Number.Postulates\nopen import Number.Structures\nopen import Number.Bundles\nopen import Number.Inclusions\nopen import Number.Base\nopen import Number.Coercions\nopen import Number.Operations\n\nopen ℕⁿ\nopen ℤᶻ\nopen ℚᶠ\nopen ℝʳ\nopen ℂᶜ\n\nimport Data.Nat.Properties\n\n\n-- NOTE: well, for 15 allowed coercions, we might just enumerate them\n-- unfortunately with overlapping patterns a style as in `Cl` is not possible\n-- we need to explicitly write out all the 5×5 combinations\n-- or, we implement a min operator which might work even with overlapping patterns\n\n-- num {isNat ,, p} (x ,, q) = x\n-- num {isInt ,, p} (x ,, q) = x\n-- num {isRat ,, p} (x ,, q) = x\n-- num {isReal ,, p} (x ,, q) = x\n-- num {isComplex ,, p} (x ,, q) = x\n\n\n-- TODO: name this \"inject\" instead of \"coerce\"\n-- TODO: make the module ℤ and the Carrier ℤ.ℤ\n-- TODO: for a binary relation `a # b` it would be nice to have a way to compose ≡-pathes to the left and the right\n-- similar to how ∙ can be used for pathes\n-- this reasoning might extend to transitive relations\n-- `cong₂ _#_ refl x` and `cong₂ _#_ x refl` to this (together with `transport`)\n-- NOTE: maybe ℕ↪ℤ should be a postfix operation\n\n-- module _ where\n-- module ℕ' = ROrderedCommSemiring ℕ.Bundle\n-- module ℤ' = ROrderedCommRing ℤ.Bundle\n-- module ℚ' = ROrderedField ℚ.Bundle\n-- module ℝ' = ROrderedField ℝ.Bundle\n-- module ℂ' = RField ℂ.Bundle--\n\n\n\n-- coerce-OCSR : ∀{l p} {ll : NumberKind} {𝕏OCSR 𝕐OCSR : ROrderedCommSemiring {ℝℓ} {ℝℓ'}}\n-- → (x : Number (l ,, p))\n-- → {f : Il l → Il ll}\n-- → IsROrderedCommSemiringInclusion 𝕏OCSR 𝕐OCSR f\n-- → Ip ll p (f (num x))\n-- coerce-OCSR {l} {ll} {p} {𝕏OCSR} {𝕐OCSR} {f} (x ,, q) = ?\n\n{-\nprivate\n instance\n z≤n' : ∀ {n} → zero ≤ₙ n\n z≤n' {n} = z≤n\n s≤s' : ∀ {m n} {{m≤n : m ≤ₙ n}} → suc m ≤ₙ suc n\n s≤s' {m} {n} {{m≤n}} = s≤s m≤n\n-}\n\n{-\n-- TODO: why does `it` not work here?\n⁻¹-Levels : (a : NumberKind) → Σ[ b ∈ NumberKind ] a ≤ₙₗ b\n⁻¹-Levels isNat = isRat , z≤n -- it\n⁻¹-Levels isInt = isRat , s≤s z≤n -- s≤s' {{z≤n'}}\n⁻¹-Levels isRat = isRat , s≤s (s≤s z≤n)\n⁻¹-Levels isReal = isReal , s≤s (s≤s (s≤s z≤n)) -- it\n⁻¹-Levels isComplex = isComplex , s≤s (s≤s (s≤s (s≤s z≤n)))\n\n⁻¹-Levels' : (a : NumberKind) → NumberKind\n⁻¹-Levels' x = maxₙₗ x isRat\n-}\n\nopen PatternsType\n\n{-\nprivate\n pattern X = anyPositivity\n pattern X⁺⁻ = isNonzero\n pattern X₀⁺ = isNonnegative\n pattern X⁺ = isPositive\n pattern X⁻ = isNegative\n pattern X₀⁻ = isNonpositive\n-}\n\n{-\n⁻¹-Types : NumberProp → Maybe NumberProp\n⁻¹-Types (level ,, X ) = nothing\n⁻¹-Types (level ,, X₀⁺) = nothing\n⁻¹-Types (level ,, X₀⁻) = nothing\n⁻¹-Types (level ,, p ) = just (fst (⁻¹-Levels level) ,, p)\n-}\n\n-- ∀{{ q : Unit }} → Number (level ,, X⁺⁻)\n-- ∀{{ q : Unit }} → Number (level ,, X⁺ )\n-- ∀{{ q : Unit }} → Number (level ,, X⁻ )\n\n-- pattern [ℝ₀⁺] = (isReal , X₀⁺)\n-- [ℝ₀⁺] = Number (isReal , isNonnegativeᵒʳ)\n-- [ℝ⁺] = Number (isReal , isPositiveᵒʳ)\n-- [ℕ⁺] = Number (isNat , isPositiveᵒʳ)\n-- [ℝ] = Number (isReal , anyPositivityᵒʳ)\n\nopen import Number.Prettyprint\n\n-- {-# DISPLAY maxₙₗ' isReal isReal = isReal #-}\n-- {-# DISPLAY Number (isReal , isNonnegative) = [ℝ₀⁺] #-}\n-- {-# DISPLAY Number (isReal , isPositive) = [ℝ⁺] #-}\n\n\n[1ʳ] : [ℝ⁺]\n[1ʳ] = 1ʳ ,, ℝ.0<1\n\n[1]-Type : (l : NumberKind) → Type (NumberLevel l)\n[1]-Type isNat = [ℕ⁺]\n[1]-Type isInt = [ℤ⁺]\n[1]-Type isRat = [ℚ⁺]\n[1]-Type isReal = [ℝ⁺]\n[1]-Type isComplex = [ℂ⁺⁻]\n\n-- NOTE: this is ambiguous with generic operations such as _+_\n[1] : ∀{l} → [1]-Type l\n[1] {isNat} = 1ⁿ ,, ℕ.0<1\n[1] {isInt} = 1ᶻ ,, ℤ.0<1\n[1] {isRat} = 1ᶠ ,, ℚ.0<1\n[1] {isReal} = 1ʳ ,, ℝ.0<1\n[1] {isComplex} = 1ᶜ ,, ℂ.1#0\n\n\n-- test101 : Number (isNat , isPositiveᵒʳ) → Number (isReal , isNonnegativeᵒʳ) → {!!}\n\nopen import Function.Base using (typeOf)\n\n\ntest201 : [ℕ⁺] → [ℝ₀⁺] → [ℝ]\n-- As-patterns (or @-patterns) go well with resolving things in our approach\ntest201 n@(nn ,, np) r@(rn ,, rp) = let\n-- generic operations are provided\n-- q : [ℕ⁺]\n-- z : [ℝ₀⁺]\n q = n + n\n z = r + r\n\n-- we can project-out the underlying number of a `Number` with `num`\n-- zʳ : ℝ\n zʳ = num z\n\n-- and we can project-out the property of a `Number` with `prp`\n-- zp : 0ʳ ≤ʳ (rn +ʳ rn)\n zp = prp z\n\n-- since the generic `_+_` makes use of `_+ʳ_` on ℝ, we get definitional equality\n _ : zʳ ≡ rn +ʳ rn\n _ = refl\n\n-- we can turn a generic number into a Σ pair with `pop`\n-- qʳ : ℕ₀\n-- qʳ = nn +ⁿ nn\n-- qp : 0ⁿ <ⁿ (nn +ⁿ nn)\n-- qp = +-<-<-implies-<ʳ nn nn np np\n (qʳ , qp) = pop q\n\n-- and we can create a number with `_,,_`\n-- this needs some type annotation for help\n q' : typeOf q\n q' = qʳ ,, qp\n\n-- if the two parts of q and q' are in scope, then we get definitional equality\n _ : q ≡ q'\n _ = refl\n\n-- r is nonnegative from [ℝ₀⁺], [1ʳ] is positive from [ℝ⁺]\n-- and _+_ makes use of the fact that \"positive + nonnegative = positive\"\n-- y : [ℝ⁺]\n-- y = (rn +ʳ 1ʳ) ,, +-≤-<-implies-<ʳ rn 1ʳ rp 0<1\n y = r + [1ʳ]\n\n-- _+_ automatically coerces n from ℕ⁺ to ℝ⁺\n-- and uses the fact that \"positive + nonnegative = positive\"\n-- n+r : [ℝ⁺]\n-- n+r = (ℕ↪ℝ nn +ʳ rn) ,, +-<-≤-implies-<ʳ (ℕ↪ℝ nn) rn (coerce-ℕ↪ℝ (nn ,, np)) rp\n n+r = n + r\n\n-- generic relations like _<_ also make use of their underlying relations\n-- and therefore we also get definitional equality, no matter how the relation is stated\n pp : [1ʳ] < (r + [1ʳ])\n pp = {!!}\n pp' : 1ʳ <ʳ num (r + [1ʳ])\n pp' = {!!}\n pp'' : 1ʳ <ʳ (rn +ʳ 1ʳ )\n pp'' = {!!}\n _ : (pp ≡ pp') × (pp ≡ pp'')\n _ = refl , refl\n in {! - [1ʳ]!}\n\n\n_ = {! ℕ!}\n\n\n{-\n\ndistance : ∀(x y : [ℝ]) → [ℝ]\ndistance x y = max (x + (- y)) (- (x + (- y)))\n\nIsCauchy : (x : ℕ → ℝ) → Type (ℓ-max ℓ' ℚℓ)\nIsCauchy x = ∀(ε : [ℚ⁺]) → ∃[ N ∈ ℕ ] ∀(m n : ℕ) → N ≤ⁿ m → N ≤ⁿ n → distance (x m) (x n) < ε\n\n-}\n\ntest : [ℕ⁺] → [ℝ₀⁺] → [ℝ]\ntest n@(nn ,, np) r@(rn ,, rp) = let\n q : [ℕ⁺]\n q = n + n\n z : [ℝ₀⁺]\n z = r + r\n k : [ℝ⁺]\n k = n + r\n in {!!}\n", "meta": {"hexsha": "461a9ffca51e2d2b722fc0ed21ccc04cd53aac49", "size": 7749, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "agda/Test/Number.agda", "max_stars_repo_name": "mchristianl/synthetic-reals", "max_stars_repo_head_hexsha": "10206b5c3eaef99ece5d18bf703c9e8b2371bde4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-07-31T18:15:26.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-19T12:15:21.000Z", "max_issues_repo_path": "agda/Test/Number.agda", "max_issues_repo_name": "mchristianl/synthetic-reals", "max_issues_repo_head_hexsha": "10206b5c3eaef99ece5d18bf703c9e8b2371bde4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "agda/Test/Number.agda", "max_forks_repo_name": "mchristianl/synthetic-reals", "max_forks_repo_head_hexsha": "10206b5c3eaef99ece5d18bf703c9e8b2371bde4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.26953125, "max_line_length": 119, "alphanum_fraction": 0.5869144406, "num_tokens": 3141, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3423259679420455}} {"text": "------------------------------------------------------------------------\n-- Laws related to _>>=_\n------------------------------------------------------------------------\n\nmodule TotalParserCombinators.Laws.Monad where\n\nopen import Algebra\nopen import Category.Monad\nopen import Codata.Musical.Notation\nopen import Data.List\nopen import Data.List.Categorical\n using () renaming (module MonadProperties to ListMonad)\nimport Data.List.Relation.Binary.BagAndSetEquality as BSEq\nopen import Function\nopen import Level\n\nprivate\n module BagMonoid {k} {A : Set} =\n CommutativeMonoid (BSEq.commutativeMonoid k A)\n open RawMonad {f = zero} Data.List.Categorical.monad\n using () renaming (_>>=_ to _>>=′_)\n\nopen import TotalParserCombinators.Derivative\nopen import TotalParserCombinators.Congruence as Eq\n hiding (return; fail) renaming (_∣_ to _∣′_)\nimport TotalParserCombinators.Laws.AdditiveMonoid as AdditiveMonoid\nopen import TotalParserCombinators.Laws.Derivative as Derivative\nopen import TotalParserCombinators.Laws.ReturnStar as Return⋆\nopen import TotalParserCombinators.Lib\nopen import TotalParserCombinators.Parser\n\n------------------------------------------------------------------------\n-- _>>=_, return, _∣_ and fail form a monad with a zero and a plus\n\n-- If the laws below are combined with the additive monoid laws this\n-- means that we have something which resembles an idempotent semiring\n-- (if we restrict ourselves to language equivalence).\n\n-- The zero laws are proved elsewhere.\n\nopen Derivative public\n using () renaming (left-zero->>= to left-zero;\n right-zero->>= to right-zero)\n\nleft-distributive : ∀ {Tok R₁ R₂ xs} {f g : R₁ → List R₂}\n (p₁ : Parser Tok R₁ xs)\n (p₂ : (x : R₁) → Parser Tok R₂ (f x))\n (p₃ : (x : R₁) → Parser Tok R₂ (g x)) →\n p₁ >>= (λ x → p₂ x ∣ p₃ x) ≅P p₁ >>= p₂ ∣ p₁ >>= p₃\nleft-distributive {xs = xs} {f} {g} p₁ p₂ p₃ =\n BSEq.>>=-left-distributive xs {f = f} ∷ λ t → ♯ (\n D t (p₁ >>= (λ x → p₂ x ∣ p₃ x)) ≅⟨ D->>= p₁ (λ x → p₂ x ∣ p₃ x) ⟩\n\n D t p₁ >>= (λ x → p₂ x ∣ p₃ x) ∣\n return⋆ xs >>= (λ x → D t (p₂ x) ∣ D t (p₃ x)) ≅⟨ left-distributive (D t p₁) p₂ p₃ ∣′\n left-distributive (return⋆ xs)\n (λ x → D t (p₂ x)) (λ x → D t (p₃ x)) ⟩\n (D t p₁ >>= p₂ ∣ D t p₁ >>= p₃) ∣\n (return⋆ xs >>= (λ x → D t (p₂ x)) ∣\n return⋆ xs >>= (λ x → D t (p₃ x))) ≅⟨ AdditiveMonoid.swap\n (D t p₁ >>= p₂) (D t p₁ >>= p₃)\n (return⋆ xs >>= (λ x → D t (p₂ x)))\n (return⋆ xs >>= (λ x → D t (p₃ x))) ⟩\n (D t p₁ >>= p₂ ∣ return⋆ xs >>= (λ x → D t (p₂ x))) ∣\n (D t p₁ >>= p₃ ∣ return⋆ xs >>= (λ x → D t (p₃ x))) ≅⟨ sym (D->>= p₁ p₂ ∣′ D->>= p₁ p₃) ⟩\n\n D t (p₁ >>= p₂) ∣ D t (p₁ >>= p₃) ∎)\n\nright-distributive : ∀ {Tok R₁ R₂ xs₁ xs₂} {f : R₁ → List R₂}\n (p₁ : Parser Tok R₁ xs₁)\n (p₂ : Parser Tok R₁ xs₂)\n (p₃ : (x : R₁) → Parser Tok R₂ (f x)) →\n (p₁ ∣ p₂) >>= p₃ ≅P p₁ >>= p₃ ∣ p₂ >>= p₃\nright-distributive {xs₁ = xs₁} {xs₂} {f} p₁ p₂ p₃ =\n BagMonoid.reflexive (ListMonad.right-distributive xs₁ xs₂ f) ∷ λ t → ♯ (\n D t ((p₁ ∣ p₂) >>= p₃) ≅⟨ D->>= (p₁ ∣ p₂) p₃ ⟩\n\n (D t p₁ ∣ D t p₂) >>= p₃ ∣\n return⋆ (xs₁ ++ xs₂) >>= (λ x → D t (p₃ x)) ≅⟨ ((D t p₁ ∣ D t p₂) >>= p₃ ∎) ∣′\n [ ○ - ○ - ○ - ○ ] Return⋆.distrib-∣ xs₁ xs₂ >>=\n (λ x → D t (p₃ x) ∎) ⟩\n (D t p₁ ∣ D t p₂) >>= p₃ ∣\n (return⋆ xs₁ ∣ return⋆ xs₂) >>= (λ x → D t (p₃ x)) ≅⟨ right-distributive (D t p₁) (D t p₂) p₃ ∣′\n right-distributive (return⋆ xs₁) (return⋆ xs₂)\n (λ x → D t (p₃ x)) ⟩\n ((D t p₁ >>= p₃) ∣ (D t p₂ >>= p₃)) ∣\n (return⋆ xs₁ >>= (λ x → D t (p₃ x)) ∣\n return⋆ xs₂ >>= (λ x → D t (p₃ x))) ≅⟨ AdditiveMonoid.swap\n (D t p₁ >>= p₃) (D t p₂ >>= p₃)\n (return⋆ xs₁ >>= (λ x → D t (p₃ x)))\n (return⋆ xs₂ >>= (λ x → D t (p₃ x))) ⟩\n (D t p₁ >>= p₃ ∣ return⋆ xs₁ >>= (λ x → D t (p₃ x))) ∣\n (D t p₂ >>= p₃ ∣ return⋆ xs₂ >>= (λ x → D t (p₃ x))) ≅⟨ sym (D->>= p₁ p₃ ∣′ D->>= p₂ p₃) ⟩\n\n D t (p₁ >>= p₃) ∣ D t (p₂ >>= p₃) ∎)\n\nleft-identity : ∀ {Tok R₁ R₂} {f : R₁ → List R₂}\n (x : R₁) (p : (x : R₁) → Parser Tok R₂ (f x)) →\n return x >>= p ≅P p x\nleft-identity {f = f} x p =\n BagMonoid.reflexive (ListMonad.left-identity x f) ∷ λ t → ♯ (\n D t (return x >>= p) ≅⟨ D->>= (return x) p ⟩\n fail >>= p ∣ return⋆ [ x ] >>= (λ x → D t (p x)) ≅⟨ left-zero p ∣′\n [ ○ - ○ - ○ - ○ ] AdditiveMonoid.right-identity (return x) >>=\n (λ x → D t (p x) ∎) ⟩\n fail ∣ return x >>= (λ x → D t (p x)) ≅⟨ AdditiveMonoid.left-identity (return x >>= (λ x → D t (p x))) ⟩\n return x >>= (λ x → D t (p x)) ≅⟨ left-identity x (λ x → D t (p x)) ⟩\n D t (p x) ∎)\n\nright-identity : ∀ {Tok R xs}\n (p : Parser Tok R xs) → p >>= return ≅P p\nright-identity {xs = xs} p =\n BagMonoid.reflexive (ListMonad.right-identity xs) ∷ λ t → ♯ (\n D t (p >>= return) ≅⟨ D->>= p return ⟩\n D t p >>= return ∣ return⋆ xs >>= (λ _ → fail) ≅⟨ right-identity (D t p) ∣′ right-zero (return⋆ xs) ⟩\n D t p ∣ fail ≅⟨ AdditiveMonoid.right-identity (D t p) ⟩\n D t p ∎)\n\nassociative : ∀ {Tok R₁ R₂ R₃ xs}\n {f : R₁ → List R₂} {g : R₂ → List R₃}\n (p₁ : Parser Tok R₁ xs)\n (p₂ : (x : R₁) → Parser Tok R₂ (f x))\n (p₃ : (x : R₂) → Parser Tok R₃ (g x)) →\n p₁ >>= (λ x → p₂ x >>= p₃) ≅P p₁ >>= p₂ >>= p₃\nassociative {xs = xs} {f} {g} p₁ p₂ p₃ =\n BagMonoid.reflexive (ListMonad.associative xs f g) ∷ λ t → ♯ (\n D t (p₁ >>= λ x → p₂ x >>= p₃) ≅⟨ D->>= p₁ (λ x → p₂ x >>= p₃) ⟩\n\n D t p₁ >>= (λ x → p₂ x >>= p₃) ∣\n return⋆ xs >>= (λ x → D t (p₂ x >>= p₃)) ≅⟨ associative (D t p₁) p₂ p₃ ∣′\n [ ○ - ○ - ○ - ○ ] return⋆ xs ∎ >>= (λ x → D->>= (p₂ x) p₃) ⟩\n D t p₁ >>= p₂ >>= p₃ ∣\n return⋆ xs >>=\n (λ x → D t (p₂ x) >>= p₃ ∣\n return⋆ (f x) >>= λ x → D t (p₃ x)) ≅⟨ (D t p₁ >>= p₂ >>= p₃ ∎) ∣′\n left-distributive (return⋆ xs)\n (λ x → D t (p₂ x) >>= p₃)\n (λ x → return⋆ (f x) >>= (λ x → D t (p₃ x))) ⟩\n D t p₁ >>= p₂ >>= p₃ ∣\n (return⋆ xs >>= (λ x → D t (p₂ x) >>= p₃) ∣\n return⋆ xs >>= (λ x → return⋆ (f x) >>= λ x → D t (p₃ x))) ≅⟨ (D t p₁ >>= p₂ >>= p₃ ∎) ∣′\n (associative (return⋆ xs) (λ x → D t (p₂ x)) p₃ ∣′\n associative (return⋆ xs) (return⋆ ∘ f) (λ x → D t (p₃ x))) ⟩\n D t p₁ >>= p₂ >>= p₃ ∣\n (return⋆ xs >>= (λ x → D t (p₂ x)) >>= p₃ ∣\n return⋆ xs >>= (return⋆ ∘ f) >>= λ x → D t (p₃ x)) ≅⟨ sym $ AdditiveMonoid.associative\n (D t p₁ >>= p₂ >>= p₃)\n (return⋆ xs >>= (λ x → D t (p₂ x)) >>= p₃)\n (return⋆ xs >>= (return⋆ ∘ f) >>= (λ x → D t (p₃ x))) ⟩\n D t p₁ >>= p₂ >>= p₃ ∣\n return⋆ xs >>= (λ x → D t (p₂ x)) >>= p₃ ∣\n return⋆ xs >>= (return⋆ ∘ f) >>= (λ x → D t (p₃ x)) ≅⟨ sym (right-distributive\n (D t p₁ >>= p₂)\n (return⋆ xs >>= (λ x → D t (p₂ x)))\n p₃) ∣′\n [ ○ - ○ - ○ - ○ ] sym (Return⋆.distrib->>= xs f) >>=\n (λ x → D t (p₃ x) ∎) ⟩\n (D t p₁ >>= p₂ ∣\n return⋆ xs >>= (λ x → D t (p₂ x))) >>= p₃ ∣\n return⋆ (xs >>=′ f) >>= (λ x → D t (p₃ x)) ≅⟨ [ ○ - ○ - ○ - ○ ] sym (D->>= p₁ p₂) >>= (λ x → p₃ x ∎) ∣′\n (return⋆ (xs >>=′ f) >>= (λ x → D t (p₃ x)) ∎) ⟩\n D t (p₁ >>= p₂) >>= p₃ ∣\n return⋆ (xs >>=′ f) >>= (λ x → D t (p₃ x)) ≅⟨ sym $ D->>= (p₁ >>= p₂) p₃ ⟩\n\n D t (p₁ >>= p₂ >>= p₃) ∎)\n", "meta": {"hexsha": "4c918f7bef70e728c552a93f130f86c487035e7f", "size": 9805, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "TotalParserCombinators/Laws/Monad.agda", "max_stars_repo_name": "nad/parser-combinators", "max_stars_repo_head_hexsha": "76774f54f466cfe943debf2da731074fe0c33644", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-07-03T08:56:13.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-03T08:56:13.000Z", "max_issues_repo_path": "TotalParserCombinators/Laws/Monad.agda", "max_issues_repo_name": "nad/parser-combinators", "max_issues_repo_head_hexsha": "76774f54f466cfe943debf2da731074fe0c33644", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "TotalParserCombinators/Laws/Monad.agda", "max_forks_repo_name": "nad/parser-combinators", "max_forks_repo_head_hexsha": "76774f54f466cfe943debf2da731074fe0c33644", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 59.0662650602, "max_line_length": 132, "alphanum_fraction": 0.3491075982, "num_tokens": 3270, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175005616829, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3420619302881984}} {"text": "------------------------------------------------------------------------\n-- The rec construction can be encoded using λ-terms\n------------------------------------------------------------------------\n\nmodule Recursion-without-rec where\n\nopen import Equality.Propositional.Cubical\nopen import Prelude hiding (id; swap)\n\nimport Finite-subset.Listed equality-with-paths as S\n\n-- To simplify the development, let's work with actual natural numbers\n-- as variables and constants (see\n-- Atom.one-can-restrict-attention-to-χ-ℕ-atoms).\n\nopen import Atom\n\nopen import Alpha-equivalence χ-ℕ-atoms\nopen import Chi χ-ℕ-atoms\nopen import Compatibility χ-ℕ-atoms\nopen import Constants χ-ℕ-atoms\nopen import Free-variables χ-ℕ-atoms\nopen import Reasoning χ-ℕ-atoms\nopen import Substitution χ-ℕ-atoms\nopen import Values χ-ℕ-atoms\n\nopen χ-atoms χ-ℕ-atoms\n\nopen import Combinators using (id; id-closed)\n\nprivate\n variable\n A : Type\n R : A → A → Type\n x y z z₁ z₂ : Var\n e e′ v : Exp\n\n------------------------------------------------------------------------\n-- \"Plain\" lambda terms\n\n-- A predicate that holds for plain lambda terms, i.e. terms that are\n-- built up using only var, lambda and apply.\n\nPlain : Exp → Type\nPlain (var _) = ⊤\nPlain (lambda _ e) = Plain e\nPlain (apply e₁ e₂) = Plain e₁ × Plain e₂\nPlain (case _ _) = ⊥\nPlain (rec _ _) = ⊥\nPlain (const _ _) = ⊥\n\n-- Plain is preserved by substitutions.\n\nplain-subst : ∀ e → Plain e → Plain e′ → Plain (e [ x ← e′ ])\nplain-subst (apply e₁ e₂) (e₁-ok , e₂-ok) e′-ok =\n plain-subst e₁ e₁-ok e′-ok , plain-subst e₂ e₂-ok e′-ok\nplain-subst {x = x} (lambda y e) e-ok e′-ok with x V.≟ y\n… | yes _ = e-ok\n… | no _ = plain-subst e e-ok e′-ok\nplain-subst {x = x} (var y) _ e′-ok with x V.≟ y\n… | yes _ = e′-ok\n… | no _ = _\n\n------------------------------------------------------------------------\n-- A variant of a fixpoint combinator\n\n-- A variant of the call-by-value fixpoint combinator Θᵥ that (at the\n-- time of writing) is presented on the Wikipedia page about\n-- fixed-point combinators\n-- (https://en.wikipedia.org/wiki/Fixed-point_combinator). There Θᵥ is\n-- defined to be the application of λxy.y(λz.xxyz) to itself. I have\n-- dropped the final z, and made the choice of variable name for z\n-- customisable.\n\nmutual\n\n F : Var → Exp\n F z = apply (f z) (f z)\n\n f : Var → Exp\n f z =\n lambda v-x (lambda v-y (\n apply (var v-y) (\n lambda z (\n apply (apply (var v-x) (var v-x)) (var v-y)))))\n\n-- The expressions f z, F z and id are closed.\n\nf-closed : Closed (f z)\nf-closed =\n Closed′-closed-under-lambda $\n Closed′-closed-under-lambda $\n Closed′-closed-under-apply\n (from-⊎ (closed′? (var v-y) (v-y ∷ v-x ∷ [])))\n (Closed′-closed-under-lambda $\n Closed′-closed-under-apply\n (Closed′-closed-under-apply\n (Closed′-closed-under-var (inj₂ (inj₂ (inj₁ refl))))\n (Closed′-closed-under-var (inj₂ (inj₂ (inj₁ refl)))))\n (Closed′-closed-under-var (inj₂ (inj₁ refl))))\n\nF-closed : Closed (F z)\nF-closed = Closed′-closed-under-apply f-closed f-closed\n\n-- F z₁ and f z₁ are α-equivalent to F z₂ and f z₂, respectively,\n-- assuming that z₁ and z₂ are distinct from v-x and v-y.\n\nf≈αf :\n z₁ ≢ v-x → z₁ ≢ v-y → z₂ ≢ v-x → z₂ ≢ v-y →\n Alpha R (f z₁) (f z₂)\nf≈αf {z₁ = z₁} {z₂ = z₂} {R = R} z₁≢x z₁≢y z₂≢x z₂≢y =\n lambda (\n lambda (\n apply (var y∼y₁) (\n lambda (\n apply (\n apply (var x∼x) (var x∼x)) (\n var y∼y₂)))))\n where\n x∼x : (R [ v-x ∼ v-x ] [ v-y ∼ v-y ] [ z₁ ∼ z₂ ]) v-x v-x\n x∼x = inj₂ (z₁≢x , z₂≢x , inj₂ ((λ ()) , (λ ()) , inj₁ (refl , refl)))\n\n y∼y₁ : (R [ v-x ∼ v-x ] [ v-y ∼ v-y ]) v-y v-y\n y∼y₁ = inj₁ (refl , refl)\n\n y∼y₂ : (R [ v-x ∼ v-x ] [ v-y ∼ v-y ] [ z₁ ∼ z₂ ]) v-y v-y\n y∼y₂ = inj₂ (z₁≢y , z₂≢y , y∼y₁)\n\nF≈αF :\n z₁ ≢ v-x → z₁ ≢ v-y → z₂ ≢ v-x → z₂ ≢ v-y →\n Alpha R (F z₁) (F z₂)\nF≈αF z₁≢x z₁≢y z₂≢x z₂≢y =\n apply (f≈αf z₁≢x z₁≢y z₂≢x z₂≢y) (f≈αf z₁≢x z₁≢y z₂≢x z₂≢y)\n\n------------------------------------------------------------------------\n-- A plain alternative to rec\n\n-- An expression former that has the same semantics as rec, but that\n-- takes plain expressions to plain expressions.\n--\n-- Note that substitution does not necessarily behave in the same way\n-- for plain-rec as for rec (see below).\n\nplain-rec : Var → Exp → Exp\nplain-rec x e =\n let z , _ = fresh′ (S.from-List (v-x ∷ v-y ∷ [])) e in\n apply (lambda z (apply (F z) (lambda x (e [ x ← apply (var x) id ]))))\n id\n\n-- If e is a plain lambda term, then plain-rec x e is a plain\n-- lambda term.\n\nplain-rec-plain : ∀ e → Plain e → Plain (plain-rec x e)\nplain-rec-plain e ok = (_ , plain-subst e ok _) , _\n\n-- The semantic rule given for rec is admissible for plain-rec.\n\nplain-rec-⇓ :\n ∀ e → e [ x ← plain-rec x e ] ⇓ v → plain-rec x e ⇓ v\nplain-rec-⇓ {x = x} {v = v} e ⇓v =\n plain-rec x e ⟶⟨⟩\n\n apply (lambda z′\n (apply (F z′) (lambda x (e [ x ← apply (var x) id ]))))\n id ⟶⟨ apply lambda lambda ⟩\n\n apply (F z′) (lambda x (e [ x ← apply (var x) id ])) [ z′ ← id ] ≡⟨ subst-∉ z′ _ z∉apply ⟩⟶\n\n apply (F z′) (lambda x (e [ x ← apply (var x) id ])) ⟶⟨ []⇓ (apply← ∙) F⇓ ⟩\n\n apply (lambda v-y\n (apply (var v-y) (lambda z′ (apply (F z′) (var v-y)))))\n (lambda x (e [ x ← apply (var x) id ])) ⟶⟨ apply lambda lambda ⟩\n\n apply (var v-y) (lambda z′ (apply (F z′) (var v-y)))\n [ v-y ← lambda x (e [ x ← apply (var x) id ]) ] ≡⟨ cong (apply _) (lambda-step-≢ y≢z) ⟩⟶\n\n apply (lambda x (e [ x ← apply (var x) id ]))\n (lambda z′ (apply (F z′) (lambda x (e [ x ← apply (var x) id ])))) ⟶⟨ apply lambda lambda ⟩\n\n e [ x ← apply (var x) id ]\n [ x ← lambda z′\n (apply (F z′) (lambda x (e [ x ← apply (var x) id ]))) ] ≡⟨ fusion e ⟩⟶\n\n e [ x ← apply (var x) id\n [ x ← lambda z′\n (apply (F z′)\n (lambda x (e [ x ← apply (var x) id ]))) ] ] ≡⟨ cong₂ (λ e₁ e₂ → e [ x ← apply e₁ e₂ ])\n (var-step-≡ (refl {x = x}))\n (subst-closed x _ id-closed) ⟩⟶\n\n e [ x ← plain-rec x e ] ⇓⟨ ⇓v ⟩■\n\n v\n where\n z,f = fresh′ (S.from-List (v-x ∷ v-y ∷ [])) e\n\n z′ : Var\n z′ = proj₁ z,f\n\n z∉e : ¬ z′ ∈FV e\n z∉e = proj₁ (proj₂ z,f)\n\n x≢z : v-x ≢ z′\n x≢z x≡z = proj₂ (proj₂ z,f) (S.≡→∈∷ (sym x≡z))\n\n y≢z : v-y ≢ z′\n y≢z y≡z = proj₂ (proj₂ z,f) (S.∈→∈∷ (S.≡→∈∷ (sym y≡z)))\n\n F⇓ :\n F z′ ⇓\n lambda v-y (apply (var v-y) (lambda z′ (apply (F z′) (var v-y))))\n F⇓ =\n apply lambda lambda\n (lambda v-y\n (apply (var v-y)\n (lambda z′ (apply (apply (var v-x) (var v-x)) (var v-y))))\n [ v-x ← f z′ ] ⟶⟨⟩\n\n lambda v-y\n (apply (var v-y)\n (lambda z′ (apply (apply (var v-x) (var v-x)) (var v-y))\n [ v-x ← f z′ ])) ≡⟨ cong (lambda _) $ cong (apply _) $\n lambda-step-≢ x≢z ⟩⟶\n\n lambda v-y (apply (var v-y) (lambda z′ (apply (F z′) (var v-y)))) ■⟨ lambda _ _ ⟩)\n\n z∉apply :\n ¬ z′ ∈FV apply (F z′) (lambda x (e [ x ← apply (var x) id ]))\n z∉apply (apply-left z∈Fz) =\n F-closed z′ (λ ()) z∈Fz\n z∉apply (apply-right (lambda z≢x z∈)) with subst-∈FV x e z∈\n … | inj₁ (z∈e , _) = z∉e z∈e\n … | inj₂ (apply-left (var z≡x)) = z≢x z≡x\n … | inj₂ (apply-right z∈id) = id-closed z′ (λ ()) z∈id\n\n-- Substitution of closed expressions is not in general defined in the\n-- same way for plain-rec as for rec.\n\n¬-plain-rec-subst :\n ¬ (∀ y e x e′ →\n Closed e′ →\n plain-rec y e [ x ← e′ ] ≡\n plain-rec y (if x V.≟ y then e else e [ x ← e′ ]))\n¬-plain-rec-subst plain-rec-subst =\n not-equal (plain-rec-subst y′ e₁ x′ e₂ id-closed)\n where\n y′ = v-y\n x′ = v-z\n e₂ = id\n e₁ = var x′\n\n not-equal :\n plain-rec y′ e₁ [ x′ ← e₂ ] ≢\n plain-rec y′ (if x′ V.≟ y′ then e₁ else e₁ [ x′ ← e₂ ])\n not-equal ()\n\n-- However, it is defined in the same way /up to α-equivalence/.\n\nplain-rec-subst :\n ∀ x →\n Closed e′ →\n plain-rec y e [ x ← e′ ] ≈α\n plain-rec y (if x V.≟ y then e else e [ x ← e′ ])\nplain-rec-subst {e′ = e′} {y = y} {e = e} x cl-e′ =\n apply (lambda z¹\n (apply (F z¹) (lambda y (e [ y ← apply (var y) id ]))))\n id [ x ← e′ ] ≡⟨ cong (apply _) $\n subst-closed x _ id-closed ⟩α\n apply (lambda z¹\n (apply (F z¹) (lambda y (e [ y ← apply (var y) id ])))\n [ x ← e′ ])\n id ≡⟨ cong (λ e → apply (lambda z¹ e) id) $\n lemma₁ (x V.≟ z¹) (x V.≟ y) ⟩α\n apply (lambda z¹\n (apply (F z¹)\n (lambda y ((if x V.≟ y then e else e [ x ← e′ ])\n [ y ← apply (var y) id ]))))\n id ≈⟨ apply (lambda (apply (F≈αF z¹≢x z¹≢y z²≢x z²≢y)\n (refl-Alpha _ lemma₂)))\n refl-α ⟩α∎\n apply (lambda z²\n (apply (F z²)\n (lambda y ((if x V.≟ y then e else e [ x ← e′ ])\n [ y ← apply (var y) id ]))))\n id ∎\n where\n z¹,f₁ = fresh′ (S.from-List (v-x ∷ v-y ∷ [])) e\n\n z¹ : Var\n z¹ = proj₁ z¹,f₁\n\n z¹∉e : ¬ z¹ ∈FV e\n z¹∉e = proj₁ (proj₂ z¹,f₁)\n\n z¹≢x : z¹ ≢ v-x\n z¹≢x z¹≡x = proj₂ (proj₂ z¹,f₁) (S.≡→∈∷ z¹≡x)\n\n z¹≢y : z¹ ≢ v-y\n z¹≢y z¹≡y = proj₂ (proj₂ z¹,f₁) (S.∈→∈∷ (S.≡→∈∷ z¹≡y))\n\n z²,f₂ =\n fresh′ (S.from-List (v-x ∷ v-y ∷ []))\n (if x V.≟ y then e else e [ x ← e′ ])\n\n z² : Var\n z² = proj₁ z²,f₂\n\n z²∉ : ¬ z² ∈FV if x V.≟ y then e else e [ x ← e′ ]\n z²∉ = proj₁ (proj₂ z²,f₂)\n\n z²≢x : z² ≢ v-x\n z²≢x z²≡x = proj₂ (proj₂ z²,f₂) (S.≡→∈∷ z²≡x)\n\n z²≢y : z² ≢ v-y\n z²≢y z²≡y = proj₂ (proj₂ z²,f₂) (S.∈→∈∷ (S.≡→∈∷ z²≡y))\n\n x∉y-id : x ≢ y → ¬ x ∈FV apply (var y) id\n x∉y-id x≢y (apply-left (var x≡y)) = x≢y x≡y\n x∉y-id _ (apply-right x∈id) = id-closed x (λ ()) x∈id\n\n lemma₁ :\n (x≟z¹ : Dec (x ≡ z¹)) (x≟y : Dec (x ≡ y)) →\n if x≟z¹\n then apply (F z¹) (lambda y (e [ y ← apply (var y) id ]))\n else apply (F z¹) (lambda y (e [ y ← apply (var y) id ]))\n [ x ← e′ ] ≡\n apply (F z¹)\n (lambda y ((if x≟y then e else e [ x ← e′ ])\n [ y ← apply (var y) id ]))\n lemma₁ (yes _) (yes _) = refl\n\n lemma₁ (yes x≡z¹) (no _) =\n apply (F z¹) (lambda y (e [ y ← apply (var y) id ])) ≡⟨ cong (apply _) $ cong (lambda _) $ cong (_[ _ ← _ ]) $ sym $\n subst-∉ x e (z¹∉e ∘ subst (_∈FV _) x≡z¹) ⟩∎\n apply (F z¹) (lambda y (e [ x ← e′ ] [ y ← apply (var y) id ])) ∎\n\n lemma₁ (no _) (yes x≡y) =\n apply (F z¹ [ x ← e′ ])\n (lambda y (e [ y ← apply (var y) id ]) [ x ← e′ ]) ≡⟨ cong (λ e″ → apply e″ (lambda y (e [ y ← apply (var y) id ])\n [ x ← e′ ])) $\n subst-closed x e′ F-closed ⟩\n\n apply (F z¹) (lambda y (e [ y ← apply (var y) id ]) [ x ← e′ ]) ≡⟨ cong (apply (F z¹)) $\n lambda-step-≡ x≡y ⟩∎\n apply (F z¹) (lambda y (e [ y ← apply (var y) id ])) ∎\n\n lemma₁ (no _) (no x≢y) =\n apply (F z¹ [ x ← e′ ])\n (lambda y (e [ y ← apply (var y) id ]) [ x ← e′ ]) ≡⟨ cong (λ e″ → apply e″ (lambda y (e [ y ← apply (var y) id ])\n [ x ← e′ ])) $\n subst-closed x e′ F-closed ⟩\n\n apply (F z¹) (lambda y (e [ y ← apply (var y) id ]) [ x ← e′ ]) ≡⟨ cong (apply (F z¹)) $\n lambda-step-≢ x≢y ⟩\n\n apply (F z¹) (lambda y (e [ y ← apply (var y) id ] [ x ← e′ ])) ≡⟨ cong (apply _) $ cong (lambda _) $ sym $\n swap x≢y (x∉y-id x≢y) (cl-e′ _ (λ ())) e ⟩∎\n apply (F z¹) (lambda y (e [ x ← e′ ] [ y ← apply (var y) id ])) ∎\n\n lemma₂ :\n ∀ z →\n z ∈FV lambda y ((if x V.≟ y then e else e [ x ← e′ ])\n [ y ← apply (var y) id ]) →\n (_≡_ [ z¹ ∼ z² ]) z z\n lemma₂ z (lambda z≢y z∈) =\n case subst-∈FV y _ z∈ of λ where\n (inj₁ (z∈ , _)) →\n inj₂ ( z¹≢ (x V.≟ y) z∈\n , z²∉ ∘ flip (subst (_∈FV _)) z∈ ∘ sym\n , refl\n )\n\n (inj₂ (apply-left (var z≡y))) →\n ⊥-elim $ z≢y z≡y\n\n (inj₂ (apply-right ∈id)) →\n ⊥-elim $ id-closed _ (λ ()) ∈id\n where\n z¹≢ :\n ∀ x≟y →\n z ∈FV if x≟y then e else e [ x ← e′ ] →\n z¹ ≢ z\n z¹≢ (yes _) z∈ = z¹∉e ∘ flip (subst (_∈FV _)) z∈ ∘ sym\n z¹≢ (no _) z∈ = case subst-∈FV x e z∈ of λ where\n (inj₁ (z∈ , _)) →\n z¹∉e ∘ flip (subst (_∈FV _)) z∈ ∘ sym\n (inj₂ z∈e′) →\n ⊥-elim $ cl-e′ _ (λ ()) z∈e′\n", "meta": {"hexsha": "dd7baa0a367ca4c363e0842fd44f7a4ea1c32ec5", "size": 13794, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/Recursion-without-rec.agda", "max_stars_repo_name": "nad/chi", "max_stars_repo_head_hexsha": "30966769b8cbd46aa490b6964a4aa0e67a7f9ab1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-05-21T22:58:07.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-20T16:27:00.000Z", "max_issues_repo_path": "src/Recursion-without-rec.agda", "max_issues_repo_name": "nad/chi", "max_issues_repo_head_hexsha": "30966769b8cbd46aa490b6964a4aa0e67a7f9ab1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-05-21T23:29:54.000Z", "max_issues_repo_issues_event_max_datetime": "2020-06-08T11:08:25.000Z", "max_forks_repo_path": "src/Recursion-without-rec.agda", "max_forks_repo_name": "nad/chi", "max_forks_repo_head_hexsha": "30966769b8cbd46aa490b6964a4aa0e67a7f9ab1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.3692307692, "max_line_length": 132, "alphanum_fraction": 0.431491953, "num_tokens": 5029, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.34201891910339244}} {"text": "-- Andreas, 2017-08-25, issue #1611.\n-- Fixed by Jesper Cockx as #2621.\n\nopen import Common.Prelude\n\ndata D : Bool → Set where\n dt : D true\n df : D false\n\nWorks : ∀{b} → D b → Set\nWorks dt = Bool\nWorks df = Bool\n\nFails : ∀{b : Bool} → D _ → Set\nFails dt = Bool\nFails df = Bool\n\n-- WAS:\n-- false != true of type Bool\n-- when checking that the pattern df has type D true\n\n-- SHOULD BE:\n-- Don't know whether to split on dt\n\n-- NOW:\n-- I'm not sure if there should be a case for the constructor dt,\n-- because I get stuck when trying to solve the following unification\n-- problems (inferred index ≟ expected index):\n-- true ≟ _9\n-- when checking that the pattern dt has type D _9\n", "meta": {"hexsha": "5807f1e612c335f5d19f20564143c6bd15729b9a", "size": 681, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/Fail/Issue1611.agda", "max_stars_repo_name": "cruhland/agda", "max_stars_repo_head_hexsha": "7f58030124fa99dfbf8db376659416f3ad8384de", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1989, "max_stars_repo_stars_event_min_datetime": "2015-01-09T23:51:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T18:20:48.000Z", "max_issues_repo_path": "test/Fail/Issue1611.agda", "max_issues_repo_name": "cruhland/agda", "max_issues_repo_head_hexsha": "7f58030124fa99dfbf8db376659416f3ad8384de", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4066, "max_issues_repo_issues_event_min_datetime": "2015-01-10T11:24:51.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T21:14:49.000Z", "max_forks_repo_path": "test/Fail/Issue1611.agda", "max_forks_repo_name": "cruhland/agda", "max_forks_repo_head_hexsha": "7f58030124fa99dfbf8db376659416f3ad8384de", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 371, "max_forks_repo_forks_event_min_datetime": "2015-01-03T14:04:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-30T19:00:30.000Z", "avg_line_length": 21.9677419355, "max_line_length": 69, "alphanum_fraction": 0.6666666667, "num_tokens": 205, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.34183661188386244}} {"text": "{-# OPTIONS --without-K --rewriting #-}\n\nopen import HoTT\n\nmodule homotopy.PushoutSplit where\n\n-- g h\n-- D --> B --> C K = A ⊔^D B / (f,g) d₁ = A <- D -> B\n-- f| | | L = K ⊔^B C / (right,h) d₂ = K <- B -> C\n-- v v v d = A <- D -> C\n-- A --> K --> L\n--\nmodule PushoutRSplit {i j k l} {A : Type i} {B : Type j} {C : Type k}\n {D : Type l} (f : D → A) (g : D → B) (h : B → C) where\n\n private\n d₁ : Span\n d₁ = span A B D f g\n\n d₂ : Span\n d₂ = span (Pushout d₁) C B right h\n\n d : Span\n d = span A C D f (h ∘ g)\n\n split-span-map : SpanMap d d₂\n split-span-map = span-map left (idf C) g (comm-sqr λ a → glue a) (comm-sqr λ _ → idp)\n\n module Split = PushoutFmap split-span-map\n\n split : Pushout d → Pushout d₂\n split = Split.f\n\n inner-span-map : SpanMap d₁ d\n inner-span-map = span-map (idf A) h (idf D) (comm-sqr λ _ → idp) (comm-sqr λ _ → idp)\n\n module Inner = PushoutFmap inner-span-map\n\n inner : Pushout d₁ → Pushout d\n inner = Inner.f\n\n module Merge = PushoutRec {d = d₂} {D = Pushout d}\n inner right (λ _ → idp)\n\n merge : Pushout d₂ → Pushout d\n merge = Merge.f\n\n private\n square-extend-tr : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ b : A}\n {p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}\n {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁} (q : a₁₀ == b)\n → Square p₀₋ p₋₀ p₋₁ p₁₋\n → Square p₀₋ (p₋₀ ∙ q) p₋₁ (! q ∙' p₁₋)\n square-extend-tr idp ids = ids\n\n split-inner : (k : Pushout d₁) → split (inner k) == left k\n split-inner = Pushout-elim\n (λ a → idp)\n (λ b → ! (glue b))\n (λ d → ↓-='-from-square $\n (ap-∘ split inner (glue d)\n ∙ ap (ap split) (Inner.glue-β d) ∙ Split.glue-β d)\n ∙v⊡ square-extend-tr (glue (g d)) vid-square)\n\n abstract\n split-merge : (l : Pushout d₂) → split (merge l) == l\n split-merge = Pushout-elim\n split-inner\n (λ c → idp)\n (λ b → ↓-∘=idf-from-square split merge $\n ap (ap split) (Merge.glue-β b) ∙v⊡ bl-square (glue b))\n\n\n merge-split : (l : Pushout d) → merge (split l) == l\n merge-split = Pushout-elim\n (λ a → idp)\n (λ c → idp)\n (λ d → ↓-∘=idf-in' merge split $\n ap merge (ap split (glue d))\n =⟨ ap (ap merge) (Split.glue-β d) ⟩\n ap merge (ap left (glue d) ∙ glue (g d))\n =⟨ ap-∙ merge (ap left (glue d)) (glue (g d)) ⟩\n ap merge (ap left (glue d)) ∙ ap merge (glue (g d))\n =⟨ ap2 _∙_ (∘-ap merge left (glue d)) (Merge.glue-β (g d)) ⟩\n ap inner (glue d) ∙ idp\n =⟨ ∙-unit-r _ ⟩\n ap inner (glue d)\n =⟨ Inner.glue-β d ⟩\n glue d ∎)\n\n\n split-equiv : Pushout d ≃ Pushout d₂\n split-equiv = equiv split merge split-merge merge-split\n\n{-\n two-pushouts-left : lift ∘ left == left ∘ left\n [ (λ E → (A → E)) ↓ two-pushouts ]\n two-pushouts-left = codomain-over-equiv _ _\n\n two-pushouts-right : lift ∘ right == right [ (λ E → (D → E)) ↓ two-pushouts ]\n two-pushouts-right = codomain-over-equiv _ _\n\n two-pushouts-inner : lift ∘ inner == left\n [ (λ E → (Pushout d₁ → E)) ↓ two-pushouts ]\n two-pushouts-inner = codomain-over-equiv _ _ ▹ λ= split-inner\n-}\n\nrsplit-equiv = PushoutRSplit.split-equiv\n\n-- h\n-- D --> C K = A ⊔^D C / (f,h) d₁ = A <- D -> C\n-- f| | L = B ⊔^A K / (g,left) d₂ = B <- A -> K\n-- v v d = B <- D -> C\n-- A --> K\n-- g| |\n-- v v\n-- B --> L\n\nmodule PushoutLSplit {i j k l} {A : Type i} {B : Type j} {C : Type k}\n {D : Type l} (f : D → A) (g : A → B) (h : D → C) where\n\n private\n d₁ : Span\n d₁ = span A C D f h\n\n d₂ : Span\n d₂ = span B (Pushout d₁) A g left\n\n d : Span\n d = span B C D (g ∘ f) h\n\n split-span-map : SpanMap d d₂\n split-span-map = span-map (idf B) right f (comm-sqr λ _ → idp) (comm-sqr λ d → ! (glue d))\n\n module Split = PushoutFmap split-span-map\n\n split : Pushout d → Pushout d₂\n split = Split.f\n\n inner-span-map : SpanMap d₁ d\n inner-span-map = span-map g (idf C) (idf D) (comm-sqr λ _ → idp) (comm-sqr λ _ → idp)\n\n module Inner = PushoutFmap inner-span-map\n\n inner : Pushout d₁ → Pushout d\n inner = Inner.f\n\n module Merge = PushoutRec {d = d₂} {D = Pushout d}\n left inner (λ _ → idp)\n\n merge : Pushout d₂ → Pushout d\n merge = Merge.f\n\n private\n square-extend-tl : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ b : A}\n {p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}\n {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁} (q : b == a₀₀)\n → Square p₀₋ p₋₀ p₋₁ p₁₋\n → Square (q ∙' p₀₋) (q ∙' p₋₀) p₋₁ p₁₋\n square-extend-tl idp ids = ids\n\n split-inner : (k : Pushout d₁) → split (inner k) == right k\n split-inner = Pushout-elim\n (λ a → glue a)\n (λ c → idp)\n (λ d → ↓-='-from-square $\n (ap-∘ split inner (glue d)\n ∙ ap (ap split) (Inner.glue-β d) ∙ Split.glue-β d\n ∙ ap (λ p → glue (f d) ∙' ap right p) (!-! (glue d)))\n ∙v⊡ square-extend-tl (glue (f d)) vid-square)\n\n abstract\n split-merge : (l : Pushout d₂) → split (merge l) == l\n split-merge = Pushout-elim\n (λ b → idp)\n split-inner\n (λ a → ↓-∘=idf-from-square split merge $\n ap (ap split) (Merge.glue-β a) ∙v⊡ br-square (glue a))\n\n\n merge-split : (l : Pushout d) → merge (split l) == l\n merge-split = Pushout-elim\n (λ b → idp)\n (λ c → idp)\n (λ d → ↓-∘=idf-in' merge split $\n ap merge (ap split (glue d))\n =⟨ ap (ap merge) (Split.glue-β d) ⟩\n ap merge (glue (f d) ∙' ap right (! (! (glue d))))\n =⟨ ap-∙' merge (glue (f d)) (ap right (! (! (glue d)))) ⟩\n ap merge (glue (f d)) ∙' ap merge (ap right (! (! (glue d))))\n =⟨ ap2 _∙'_ (Merge.glue-β (f d)) (∘-ap merge right (! (! (glue d)))) ⟩\n idp ∙' ap inner (! (! (glue d)))\n =⟨ ∙'-unit-l _ ⟩\n ap inner (! (! (glue d)))\n =⟨ ap (ap inner) (!-! (glue d)) ⟩\n ap inner (glue d)\n =⟨ Inner.glue-β d ⟩\n glue d ∎)\n\n\n split-equiv : Pushout d ≃ Pushout d₂\n split-equiv = equiv split merge split-merge merge-split\n\nlsplit-equiv = PushoutLSplit.split-equiv\n\n\n{- TODO Update this part\n\n-- g h\n-- Y --> Z --> W K = X ⊔^Y Y / (f,g) ps₁ = X <- Y -> Z\n-- f| | | L = Z ⊔^Z W / (left,h) ps₂ = K <- Z -> W\n-- v v v ps = X <- Y -> W\n-- X --> K --> L\n--\nmodule TwoPushoutsPtd {i j k l} {X : Ptd i} {Y : Ptd j} {Z : Ptd k} {W : Ptd l}\n (f : Y ⊙→ X) (g : Y ⊙→ Z) (h : Z ⊙→ W) where\n\n private\n ps₁ = ⊙span X Z Y f g\n ps₂ = ⊙span (⊙Pushout ps₁) W Z (⊙right ps₁) h\n ps = ⊙span X W Y f (h ⊙∘ g)\n\n open TwoPushoutsEquiv (fst f) (fst g) (fst h)\n\n two-pushouts-ptd :\n ⊙Lift {j = lmax l (lmax k (lmax j i))} (⊙Pushout ps)\n == ⊙Pushout ps₂\n two-pushouts-ptd = ⊙ua (two-pushouts-equiv ∘e lift-equiv) idp\n\n two-pushouts-⊙left :\n ⊙lift ⊙∘ ⊙left ps == ⊙left ps₂ ⊙∘ ⊙left ps₁\n [ (λ V → X ⊙→ V) ↓ two-pushouts-ptd ]\n two-pushouts-⊙left = codomain-over-⊙equiv _ _ _\n\n two-pushouts-⊙right :\n ⊙lift ⊙∘ ⊙right ps == ⊙right ps₂\n [ (λ V → W ⊙→ V) ↓ two-pushouts-ptd ]\n two-pushouts-⊙right =\n codomain-over-⊙equiv _ _ _ ▹ pair= idp (lemma f g h)\n where\n lemma : {X : Ptd i} {Y : Ptd j} {Z : Ptd k} {W : Ptd l}\n (f : Y ⊙→ X) (g : Y ⊙→ Z) (h : Z ⊙→ W)\n → ap (TwoPushoutsEquiv.split (fst f) (fst g) (fst h)\n ∘ lower {j = lmax l (lmax k (lmax j i))})\n (snd (⊙lift ⊙∘ ⊙right (⊙span X W Y f (h ⊙∘ g))))\n ∙ idp\n == ap right (! (snd h)) ∙ ! (glue (pt Z))\n ∙' ap left (snd (⊙right (⊙span X Z Y f g)))\n lemma {Y = Y} (f , idp) (g , idp) (h , idp) =\n ap (2P.split ∘ lower) (ap lift (! (glue (pt Y))) ∙ idp) ∙ idp\n =⟨ ∙-unit-r _ ⟩\n ap (2P.split ∘ lower) (ap lift (! (glue (pt Y))) ∙ idp)\n =⟨ ∙-unit-r _ |in-ctx (ap (2P.split ∘ lower)) ⟩\n ap (2P.split ∘ lower) (ap lift (! (glue (pt Y))))\n =⟨ ∘-ap (2P.split ∘ lower) lift _ ⟩\n ap 2P.split (! (glue (pt Y)))\n =⟨ ap-! 2P.split (glue (pt Y)) ⟩\n ! (ap 2P.split (glue (pt Y)))\n =⟨ 2P.Split.glue-β (pt Y) |in-ctx ! ⟩\n ! (ap left (glue (pt Y)) ∙ glue (g (pt Y)))\n =⟨ !-∙ (ap left (glue (pt Y))) (glue (g (pt Y))) ⟩\n ! (glue (g (pt Y))) ∙ ! (ap left (glue (pt Y)))\n =⟨ !-ap left (glue (pt Y)) |in-ctx (λ w → ! (glue (g (pt Y))) ∙ w) ⟩\n ! (glue (g (pt Y))) ∙ ap left (! (glue (pt Y)))\n =⟨ ∙=∙' (! (glue (g (pt Y)))) (ap left (! (glue (pt Y)))) ⟩\n ! (glue (g (pt Y))) ∙' ap left (! (glue (pt Y))) ∎\n where\n module 2P = TwoPushoutsEquiv f g h\n\n two-pushouts-⊙inner : ⊙lift ⊙∘ (inner , idp) == ⊙left ps₂\n [ (λ V → ⊙Pushout ps₁ ⊙→ V) ↓ two-pushouts-ptd ]\n two-pushouts-⊙inner =\n codomain-over-⊙equiv _ _ _ ▹ ⊙λ= split-inner idp\n-}\n\n{-\nopen TwoPushoutsEquiv using () renaming (split-equiv to vsplit-pushouts-equiv)\n -- two-pushouts; two-pushouts-left; two-pushouts-right; two-pushouts-inner\n\nopen TwoPushoutsPtd\n using (two-pushouts-ptd; two-pushouts-⊙left; two-pushouts-⊙right;\n two-pushouts-⊙inner)\n-}\n", "meta": {"hexsha": "22328209efeebad14a270a06b810de5cb386d949", "size": 9098, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "theorems/homotopy/PushoutSplit.agda", "max_stars_repo_name": "mikeshulman/HoTT-Agda", "max_stars_repo_head_hexsha": "e7d663b63d89f380ab772ecb8d51c38c26952dbb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "theorems/homotopy/PushoutSplit.agda", "max_issues_repo_name": "mikeshulman/HoTT-Agda", "max_issues_repo_head_hexsha": "e7d663b63d89f380ab772ecb8d51c38c26952dbb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "theorems/homotopy/PushoutSplit.agda", "max_forks_repo_name": "mikeshulman/HoTT-Agda", "max_forks_repo_head_hexsha": "e7d663b63d89f380ab772ecb8d51c38c26952dbb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-12-26T21:31:57.000Z", "max_forks_repo_forks_event_max_datetime": "2018-12-26T21:31:57.000Z", "avg_line_length": 32.0352112676, "max_line_length": 92, "alphanum_fraction": 0.4919762585, "num_tokens": 3693, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3418366118838624}} {"text": "\n-- Internal hom in families\nmodule SOAS.Abstract.Hom {T : Set} where\n\nopen import SOAS.Common\nopen import SOAS.Construction.Structure\nopen import SOAS.Context\nopen import SOAS.Variable\nopen import SOAS.Families.Core {T}\nopen import SOAS.Families.Isomorphism\nopen import SOAS.Families.BCCC\n\nopen import SOAS.Construction.Skew.SkewClosed\n\nopen import Categories.Functor.Bifunctor\nopen import Categories.NaturalTransformation.Dinatural using (dtHelper)\n\n\n-- Heterogeneous action of a sorted family on a family\n⟨_,_⟩ : Familyₛ → Family → Family\n⟨ 𝒳 , Y ⟩ Γ = (Γ ~[ 𝒳 ]↝_) ⇾ Y\n\n⟨-,-⟩F : Bifunctor 𝔽amₛ.op 𝔽amilies 𝔽amilies\n⟨-,-⟩F = record\n { F₀ = λ{ (𝒳 , Y) → ⟨ 𝒳 , Y ⟩ }\n ; F₁ = λ{ (f , g) o {Δ} σ → g (o (f ∘ σ)) }\n ; identity = refl\n ; homomorphism = refl\n ; F-resp-≈ = λ{ {f = f , g} (p , p′) {Γ} {o} → dext′ (trans (cong (g ∘ o) (dext′ p)) p′) }\n }\n\n-- Arrow mapping\n⟨_,_⟩₁ : {𝒳 𝒳′ : Familyₛ} {Y Y′ : Family} → 𝒳′ ⇾̣ 𝒳 → Y ⇾ Y′ → (⟨ 𝒳 , Y ⟩ ⇾ ⟨ 𝒳′ , Y′ ⟩)\n⟨ f , g ⟩₁ = Functor.₁ ⟨-,-⟩F (f , g)\n\n\n-- Internal hom of sorted families\n〖_,_〗 : Familyₛ → Familyₛ → Familyₛ\n〖 X , Y 〗 τ = ⟨ X , Y τ ⟩\n\n〖-,-〗F : Bifunctor 𝔽amₛ.op 𝔽amiliesₛ 𝔽amiliesₛ\n〖-,-〗F = record\n { F₀ = λ{ (X , Y) → 〖 X , Y 〗 }\n ; F₁ = λ{ (f , g) o {Δ} σ → g (o (f ∘ σ)) }\n ; identity = refl\n ; homomorphism = refl\n ; F-resp-≈ = λ{ {f = f , g} (p , p′) {x = h} → dext′ (trans (cong (g ∘ h) (dext′ p)) p′) }\n }\n\n-- Arrow mapping\n〖_,_〗₁ : {𝒳 𝒳′ 𝒴 𝒴′ : Familyₛ} → 𝒳′ ⇾̣ 𝒳 → 𝒴 ⇾̣ 𝒴′ → (〖 𝒳 , 𝒴 〗 ⇾̣ 〖 𝒳′ , 𝒴′ 〗)\n〖 f , g 〗₁ h σ = g (h (f ∘ σ))\n\n〖_,_〗ₗ : {𝒳 𝒳′ : Familyₛ} → 𝒳′ ⇾̣ 𝒳 → (𝒴 : Familyₛ) → (〖 𝒳 , 𝒴 〗 ⇾̣ 〖 𝒳′ , 𝒴 〗)\n〖 f , Z 〗ₗ h σ = h (f ∘ σ)\n\n〖_,_〗ᵣ : {𝒴 𝒴′ : Familyₛ} → (𝒳 : Familyₛ) → 𝒴 ⇾̣ 𝒴′ → (〖 𝒳 , 𝒴 〗 ⇾̣ 〖 𝒳 , 𝒴′ 〗)\n〖 X , g 〗ᵣ h σ = g (h σ)\n\n\n-- | Structure morphisms\n\ni : (𝒳 : Familyₛ) → 〖 ℐ , 𝒳 〗 ⇾̣ 𝒳\ni 𝒳 o = o id\n\ni′ : (X : Family) → ⟨ ℐ , X ⟩ ⇾ X\ni′ X o = o id\n\nj : (𝒳 : Familyₛ) → ℐ ⇾̣ 〖 𝒳 , 𝒳 〗\nj 𝒳 v σ = σ v\n\nL : (𝒳 𝒴 𝒵 : Familyₛ) → 〖 𝒴 , 𝒵 〗 ⇾̣ 〖 〖 𝒳 , 𝒴 〗 , 〖 𝒳 , 𝒵 〗 〗\nL 𝒳 Y Z o ς σ = o (λ v → ς v σ)\n\nL′ : (𝒳 𝒴 : Familyₛ)(Z : Family) → ⟨ 𝒴 , Z ⟩ ⇾ ⟨ 〖 𝒳 , 𝒴 〗 , ⟨ 𝒳 , Z ⟩ ⟩\nL′ 𝒳 𝒴 Z o ς σ = o (λ v → ς v σ)\n\n\n-- Category of sorted families is skew-closed under the internal hom\n𝔽amₛ:SkewClosed : SkewClosed 𝔽amiliesₛ\n𝔽amₛ:SkewClosed = record\n { [-,-] = 〖-,-〗F\n ; unit = ℐ\n ; identity = ntHelper record { η = i ; commute = λ f → refl }\n ; diagonal = dtHelper record { α = j ; commute = λ f → refl }\n ; L = L\n ; L-commute = refl\n ; Lj≈j = refl\n ; ijL≈id = refl\n ; iL≈i = refl\n ; ij≈id = refl\n ; pentagon = refl\n }\n\n\nprivate\n variable\n 𝒳 𝒴 𝒵 : Familyₛ\n Y Z : Family\n\n-- ⟨X,-⟩ distributes over and factors out of products\n⟨𝒳,Y×Z⟩≅⟨𝒳,Y⟩×⟨𝒳,Z⟩ : ⟨ 𝒳 , (Y ×ₘ Z) ⟩ ≅ₘ ⟨ 𝒳 , Y ⟩ ×ₘ ⟨ 𝒳 , Z ⟩\n⟨𝒳,Y×Z⟩≅⟨𝒳,Y⟩×⟨𝒳,Z⟩ = record\n { from = λ h → (λ ρ → proj₁ (h ρ)) , λ ϱ → proj₂ (h ϱ)\n ; to = λ{ (bx , by) ρ → bx ρ , by ρ}\n ; iso = record { isoˡ = refl ; isoʳ = refl }\n }\n\n-- ⟨X,-⟩ factors out of coproducts\n⟨𝒳,Y⟩+⟨𝒳,Z⟩⇾⟨𝒳,Y+Z⟩ : ⟨ 𝒳 , Y ⟩ +ₘ ⟨ 𝒳 , Z ⟩ ⇾ ⟨ 𝒳 , (Y +ₘ Z) ⟩\n⟨𝒳,Y⟩+⟨𝒳,Z⟩⇾⟨𝒳,Y+Z⟩ (inj₁ ox) σ = inj₁ (ox σ)\n⟨𝒳,Y⟩+⟨𝒳,Z⟩⇾⟨𝒳,Y+Z⟩ (inj₂ oy) ς = inj₂ (oy ς)\n\n-- Same properties for the hom\n〖𝒳,𝒴×̣𝒵〗≅̣〖𝒳,𝒴〗×̣〖𝒳,𝒵〗 : 〖 𝒳 , 𝒴 ×̣ₘ 𝒵 〗 ≅̣ₘ 〖 𝒳 , 𝒴 〗 ×̣ₘ 〖 𝒳 , 𝒵 〗\n〖𝒳,𝒴×̣𝒵〗≅̣〖𝒳,𝒴〗×̣〖𝒳,𝒵〗 = ≅ₘ→≅̣ₘ ⟨𝒳,Y×Z⟩≅⟨𝒳,Y⟩×⟨𝒳,Z⟩\n\n〖𝒳,𝒴〗+̣〖𝒳,𝒵〗⇾̣〖𝒳,𝒴+̣𝒵〗 : 〖 𝒳 , 𝒴 〗 +̣ₘ 〖 𝒳 , 𝒵 〗 ⇾̣ 〖 𝒳 , (𝒴 +̣ₘ 𝒵) 〗\n〖𝒳,𝒴〗+̣〖𝒳,𝒵〗⇾̣〖𝒳,𝒴+̣𝒵〗 = ⟨𝒳,Y⟩+⟨𝒳,Z⟩⇾⟨𝒳,Y+Z⟩\n", "meta": {"hexsha": "2bc9f2675dc71821adc483355ac4a31744b78017", "size": 3343, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "SOAS/Abstract/Hom.agda", "max_stars_repo_name": "JoeyEremondi/agda-soas", "max_stars_repo_head_hexsha": "ff1a985a6be9b780d3ba2beff68e902394f0a9d8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 39, "max_stars_repo_stars_event_min_datetime": "2021-11-09T20:39:55.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-19T17:33:12.000Z", "max_issues_repo_path": "SOAS/Abstract/Hom.agda", "max_issues_repo_name": "JoeyEremondi/agda-soas", "max_issues_repo_head_hexsha": "ff1a985a6be9b780d3ba2beff68e902394f0a9d8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-11-21T12:19:32.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-21T12:19:32.000Z", "max_forks_repo_path": "SOAS/Abstract/Hom.agda", "max_forks_repo_name": "JoeyEremondi/agda-soas", "max_forks_repo_head_hexsha": "ff1a985a6be9b780d3ba2beff68e902394f0a9d8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2021-11-09T20:39:59.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-24T12:49:17.000Z", "avg_line_length": 27.8583333333, "max_line_length": 92, "alphanum_fraction": 0.4851929405, "num_tokens": 2147, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7606506526772884, "lm_q2_score": 0.44939263446475963, "lm_q1q2_score": 0.34183080071398547}} {"text": "{-# OPTIONS --cubical --no-import-sorts --safe #-}\nmodule Cubical.Data.Empty.Base where\n\nopen import Cubical.Core.Everything\n\ndata ⊥ : Type₀ where\n\nrec : ∀ {ℓ} {A : Type ℓ} → ⊥ → A\nrec ()\n\nelim : ∀ {ℓ} {A : ⊥ → Type ℓ} → (x : ⊥) → A x\nelim ()\n", "meta": {"hexsha": "43cef061357e8a84fde5deb41f9c263bef3f7bc8", "size": 243, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Cubical/Data/Empty/Base.agda", "max_stars_repo_name": "Schippmunk/cubical", "max_stars_repo_head_hexsha": "c345dc0c49d3950dc57f53ca5f7099bb53a4dc3a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Cubical/Data/Empty/Base.agda", "max_issues_repo_name": "Schippmunk/cubical", "max_issues_repo_head_hexsha": "c345dc0c49d3950dc57f53ca5f7099bb53a4dc3a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2022-01-27T02:07:48.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-27T02:07:48.000Z", "max_forks_repo_path": "Cubical/Data/Empty/Base.agda", "max_forks_repo_name": "Schippmunk/cubical", "max_forks_repo_head_hexsha": "c345dc0c49d3950dc57f53ca5f7099bb53a4dc3a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-11-22T02:02:01.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-22T02:02:01.000Z", "avg_line_length": 18.6923076923, "max_line_length": 50, "alphanum_fraction": 0.5720164609, "num_tokens": 91, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3415946969709353}} {"text": "open import Categories\nopen import Monads\n\nmodule Monads.CatofAdj.TermAdjObj {a b}{C : Cat {a}{b}}(M : Monad C) where\n\nopen import Library\nopen import Functors\nopen import Naturals\nopen import Adjunctions\nopen import Monads.CatofAdj M\nopen import Categories.Terminal\nopen import Monads.EM M\nopen import Monads.EM.Adjunction M\nopen import Adjunctions.Adj2Mon\n\nopen Cat\nopen Fun\nopen Monad M\nopen NatT\nopen Adj\n\nlemX : R EMAdj ○ L EMAdj ≅ TFun M\nlemX = FunctorEq _ _ refl refl\n\nEMObj : Obj CatofAdj \nEMObj = record { \n D = EM;\n adj = EMAdj;\n law = lemX;\n ηlaw = idl C;\n bindlaw = λ{X Y f} →\n cong bind \n (stripsubst (Hom C X) f (fcong Y (cong OMap (sym lemX))))}\n\nopen ObjAdj\nopen Adj\n\nalaw1lem : ∀{c d}{D : Cat {c}{d}}\n (T : Fun C C)(L : Fun C D)(R : Fun D C)\n (p : R ○ L ≅ T)\n (η : ∀ {X} → Hom C X (OMap T X)) → \n (right : ∀ {X Y} → Hom C X (OMap R Y) → Hom D (OMap L X) Y) → \n (left : ∀ {X Y} → Hom D (OMap L X) Y → Hom C X (OMap R Y)) → \n (ηlaw : ∀ {X} → left (iden D {OMap L X}) ≅ η {X}) → \n ∀ {X}{Z}{f : Hom C Z (OMap R X)} → \n (nat : comp C (HMap R (right f)) (comp C (left (iden D)) (iden C))\n ≅\n left (comp D (right f) (comp D (iden D) (HMap L (iden C))))) →\n (lawb : left (right f) ≅ f) → \n f \n ≅\n comp C (subst (λ Z₁ → Hom C Z₁ (OMap R X)) \n (fcong Z (cong OMap p))\n (HMap R (right f))) \n η\nalaw1lem {D = D} .(R ○ L) L R refl η right left ηlaw {X}{Z}{f} nat lawb =\n trans (trans (trans (sym lawb) \n (cong left \n (trans (sym (idr D)) \n (cong (comp D (right f)) \n (trans (sym (fid L))\n (sym (idl D)))))))\n (trans (sym nat) \n (cong (comp C (HMap R (right f))) \n (idr C)))) \n (cong (comp C (HMap R (right f))) ηlaw)\n\nalaw2lem : ∀{c d}{D : Cat {c}{d}}\n (T : Fun C C)(L : Fun C D)(R : Fun D C)\n (p : R ○ L ≅ T) → \n (right : ∀ {X Y} → Hom C X (OMap R Y) → Hom D (OMap L X) Y) → \n (bind : ∀ {X Y} → Hom C X (OMap T Y) → Hom C (OMap T X) (OMap T Y)) → \n (natright : {X₁ X' : Obj C} {Y Y' : Obj D} (f₁ : Hom C X' X₁)\n (g : Hom D Y Y') (h : Hom C X₁ (OMap R Y)) →\n right (comp C (HMap R g) (comp C h f₁)) \n ≅\n comp D g (comp D (right h) (HMap L f₁))) → \n ∀ {X}{Z} {W} {k : Hom C Z (OMap T W)}{f : Hom C W (OMap R X)} → \n (bindlaw : HMap R (right (subst (Hom C Z) (fcong W (cong OMap (sym p))) k)) \n ≅ bind k) → \n subst (λ Z → Hom C Z (OMap R X))\n (fcong Z (cong OMap p))\n (HMap R\n (right\n (comp C\n (subst (λ Z → Hom C Z (OMap R X))\n (fcong W (cong OMap p)) (HMap R (right f)))\n k)))\n ≅\n comp C\n (subst (λ Z → Hom C Z (OMap R X))\n (fcong W (cong OMap p)) (HMap R (right f)))\n (bind k)\nalaw2lem {D = D} .(R ○ L) L R refl right bind natright {X}{Z}{W}{k}{f} bindlaw =\n trans (trans (cong (HMap R) \n (trans (cong (λ k₁ → right (comp C (HMap R (right f)) k₁))\n (sym (idr C))) \n (trans (natright (iden C) (right f) k) \n (trans (cong (λ h → comp D \n (right f) \n (comp D \n (right k) \n h))\n (fid L))\n (trans (sym (ass D)) \n (idr D))))))\n (fcomp R)) \n (cong (comp C (HMap R (right f))) bindlaw)\n\n\nahomlem : ∀{c d}{D : Cat {c}{d}}\n (T : Fun C C)(L : Fun C D)(R : Fun D C)(p : R ○ L ≅ T) → \n (right : ∀ {X Y} → Hom C X (OMap R Y) → Hom D (OMap L X) Y) →\n (natright : {X₁ X' : Obj C} {Y Y' : Obj D} (f₁ : Hom C X' X₁)\n (g : Hom D Y Y') (h : Hom C X₁ (OMap R Y)) →\n right (comp C (HMap R g) (comp C h f₁)) \n ≅\n comp D g (comp D (right h) (HMap L f₁))) → \n {X : Obj D}{Y : Obj D}{f : Hom D X Y} →\n {Z : Obj C} {f₁ : Hom C Z (OMap R X)} →\n comp C (HMap R f)\n (subst (λ Z₁ → Hom C Z₁ (OMap R X))\n (fcong Z (cong OMap p))\n (HMap R (right f₁))) \n ≅\n subst (λ Z₁ → Hom C Z₁ (OMap R Y))\n (fcong Z (cong OMap p))\n (HMap R (right (comp C (HMap R f) f₁)))\nahomlem {D = D} .(R ○ L) L R refl right natright {X}{Y}{f}{Z}{g} = \n trans (sym (fcomp R))\n (cong (HMap R)\n (sym\n (trans (cong (λ g₁ → right (comp C (HMap R f) g₁)) (sym (idr C)))\n (trans (natright (iden C) f g)\n (trans (cong (λ h → comp D f (comp D (right g) h)) (fid L))\n (trans (sym (ass D)) (idr D)))))))\n\n\nLlawlem : ∀{c d}{D : Cat {c}{d}}\n (T : Fun C C)(L : Fun C D)(R : Fun D C)(p : R ○ L ≅ T) → \n (right : ∀ {X Y} → Hom C X (OMap R Y) → Hom D (OMap L X) Y) → \n (bind : ∀ {X Y} → Hom C X (OMap T Y) → Hom C (OMap T X) (OMap T Y)) → \n (bindlaw : {X Y : Obj C} {f : Hom C X (OMap T Y)} →\n HMap R\n (right\n (subst (Hom C X) (fcong Y (cong OMap (sym p))) f))\n ≅ bind f) → \n ∀{X Z} → \n {f : Hom C Z (OMap R (OMap L X))}\n {f' : Hom C Z (OMap T X)} → (q : f ≅ f') → \n subst (λ Z₁ → Hom C Z₁ (OMap R (OMap L X)))\n (fcong Z (cong OMap p)) (HMap R (right f))\n ≅ bind f'\nLlawlem .(R ○ L) L R refl right bind bindlaw {X}{Z}{f}{.f} refl = bindlaw\n\n", "meta": {"hexsha": "fe1c90c4353df997cdefc5e1d7733a6d7ad77d20", "size": 5480, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Monads/CatofAdj/TermAdjObj.agda", "max_stars_repo_name": "jmchapman/Relative-Monads", 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"max_forks_repo_forks_event_max_datetime": "2019-11-04T21:33:13.000Z", "avg_line_length": 35.1282051282, "max_line_length": 80, "alphanum_fraction": 0.4359489051, "num_tokens": 2096, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.42632159254749036, "lm_q1q2_score": 0.34135228947358576}} {"text": "{-# OPTIONS --rewriting --prop #-}\n\nopen import common\nopen import syntx\n\n{- The sort corresponding to judgments -}\n\ndata JudgmentSort : Set where\n Ty : JudgmentSort\n Tm : JudgmentSort\n Ty= : JudgmentSort\n Tm= : JudgmentSort\n\nJudgmentArityArgs = ArityArgs JudgmentSort\nJudgmentArity = Arity JudgmentSort\n\n{-\nJudgments are indexed by the signature, their ambient context, the length of their local context,\nand their sort.\n\nWe can see judgments as consisting of two contexts, one normal context (the ambient context) and\nthen one dependent context (the local context). The reason is that all typing rules occur in an\nambient context which never changes, and sometimes add new assumptions (to the local context).\nTherefore we will never have to check that the ambient contexts are equal, it will be forced by the\ntyping.\n\nIndexing judgments by sorts is very good to get rid of absurd cases, when giving typing rules and\nthat some judgments are supposed to have certain sorts.\n-}\n\ndata Judgment (Σ : Signature) {m : ℕ} (Γ : Ctx Σ m) (n : ℕ) : JudgmentSort → Set where\n _⊢_ : (Δ : DepCtx Σ m n) → TyExpr Σ (m + n) → Judgment Σ Γ n Ty\n _⊢_:>_ : (Δ : DepCtx Σ m n) → TmExpr Σ (m + n) → TyExpr Σ (m + n) → Judgment Σ Γ n Tm\n _⊢_==_ : (Δ : DepCtx Σ m n) → TyExpr Σ (m + n) → TyExpr Σ (m + n) → Judgment Σ Γ n Ty=\n _⊢_==_:>_ : (Δ : DepCtx Σ m n) → TmExpr Σ (m + n) → TmExpr Σ (m + n) → TyExpr Σ (m + n)\n → Judgment Σ Γ n Tm=\n\n\n\n{-\nA derivation rule consists of a partial function taking a tuple of judgments (of the correct\narities) and returning another judgment. Moreover, a derivation rule is extendable to any other\nsignature the original signature maps to.\n\nThe type [DerivationRulePremises Σ Γ args] represents tuples of judgments of arities [args] (and in\nsignature [Σ] and with ambient context [Γ])\n\nThe type [DerivationRule Σ ar n] represents derivation rules in signature [Σ], of arity [ar] and in\nscope [n]. It lives in [Set₁] because it quantifies over arbitrary signatures that [Σ] maps into.\n-}\n\ndata DerivationRulePremises (Σ : Signature) {n : ℕ} (Γ : Ctx Σ n) : JudgmentArityArgs → Set where\n [] : DerivationRulePremises Σ Γ []\n _,_ : {m : ℕ} {k : JudgmentSort} {args : JudgmentArityArgs}\n → DerivationRulePremises Σ Γ args\n → Judgment Σ Γ m k\n → DerivationRulePremises Σ Γ (args , (m , k))\n\nrecord DerivationRule (Σ : Signature) (ar : JudgmentArity) : Set₁ where\n field\n rule : {Σ' : Signature} {n : ℕ} → (Σ →Sig Σ') n → (Γ : Ctx Σ' n)\n → DerivationRulePremises Σ' Γ (args ar) → Partial (Judgment Σ' Γ 0 (sort ar))\nopen DerivationRule public\n\n{- A derivability structure consists of a bunch of derivation rules, indexed by their arities -}\n\ndata Tag : Set where\n S T C Eq : Tag\n\nrecord DerivabilityStructure (Σ : Signature) : Set₁ where\n field\n Rules : Tag → JudgmentArity → Set\n derivationRule : {t : Tag} {ar : JudgmentArity} (r : Rules t ar) → DerivationRule Σ ar\nopen DerivabilityStructure public\n\n\n{- We can move the local context to the end of the ambient context -}\n\nmodule _ {Σ : Signature} {m : ℕ} {Γ : Ctx Σ m} where\n\n Γ+ : {l : ℕ} (Δ : DepCtx Σ m l) → Ctx Σ (m + l)\n Γ+ ◇ = Γ\n Γ+ (Δ , A) = (Γ+ Δ , A)\n\n exchangeCtx : {n : ℕ} {k : JudgmentSort} → Judgment Σ Γ n k → Ctx Σ (m + n)\n exchangeCtx (Δ ⊢ A) = Γ+ Δ\n exchangeCtx (Δ ⊢ u :> A) = Γ+ Δ\n exchangeCtx (Δ ⊢ A == B) = Γ+ Δ\n exchangeCtx (Δ ⊢ u == v :> A) = Γ+ Δ\n\n exchange : {n : ℕ} {k : JudgmentSort} → (j : Judgment Σ Γ n k) → Judgment Σ (exchangeCtx j) 0 k\n exchange (Δ ⊢ A) = ◇ ⊢ A\n exchange (Δ ⊢ u :> A) = ◇ ⊢ u :> A\n exchange (Δ ⊢ A == B) = ◇ ⊢ A == B\n exchange (Δ ⊢ u == v :> A) = ◇ ⊢ u == v :> A\n\n\n{-\nA judgment can be derivable in one different way:\n- if it has a trivial local context, then it should be obtained by applying a rule [r] from the\n derivability structure to a list of judgments [js] which are all derivable [js-der] and for which\n the rule is defined [def].\n\nThe type [DerivableArgs E js] represents the fact that all of the judgments in [js] are derivables.\nThe type [Derivable E j] represents the fact that the judgment [j] is derivable.\n-}\n\ndata Derivable {Σ : Signature} (E : DerivabilityStructure Σ)\n : {m : ℕ} {Γ : Ctx Σ m} {k : JudgmentSort} → Judgment Σ Γ 0 k → Prop\n\ndata DerivableArgs {Σ : Signature} (E : DerivabilityStructure Σ) {m : ℕ} {Γ : Ctx Σ m}\n : {ar : JudgmentArityArgs} → DerivationRulePremises Σ Γ ar → Prop where\n [] : DerivableArgs E []\n _,_ : {n : ℕ} {k : JudgmentSort} {j : Judgment Σ Γ n k}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (exchange j)\n → DerivableArgs E (js , j)\n\ndata Derivable {Σ} E where\n apr : (t : Tag) {ar : JudgmentArity} (r : Rules E t ar) {m : ℕ} {Γ : Ctx Σ m}\n {js : DerivationRulePremises Σ Γ (args ar)}\n (js-der : DerivableArgs E js) {{def : isDefined (rule (derivationRule E r) idSig Γ js)}}\n → Derivable E (rule (derivationRule E r) idSig Γ js $ def)\n\n\n{- Special cases of [_,_], used to make Agda not blow up -}\n\n_,0Ty_ : ∀ {Σ} {E} {m} {Γ : Ctx Σ m} {A : TyExpr Σ m}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (◇ ⊢ A)\n → DerivableArgs E (js , ◇ ⊢ A)\ndjs ,0Ty dj = djs , dj\n\n_,0Ty=_ : ∀ {Σ} {E} {m} {Γ : Ctx Σ m} {A B : _}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (◇ ⊢ A == B)\n → DerivableArgs E (js , ◇ ⊢ A == B)\ndjs ,0Ty= dj = djs , dj\n\n_,0Tm_ : ∀ {Σ} {E} {m} {Γ : Ctx Σ m} {u : _} {A : _}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (◇ ⊢ u :> A)\n → DerivableArgs E (js , ◇ ⊢ u :> A)\ndjs ,0Tm dj = djs , dj\n\n_,0Tm=_ : ∀ {Σ} {E} {m} {Γ : Ctx Σ m} {u v : _} {A : _}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (◇ ⊢ u == v :> A)\n → DerivableArgs E (js , ◇ ⊢ u == v :> A)\ndjs ,0Tm= dj = djs , dj\n\n_,1Ty_ : ∀ {Σ} {E} {m} {Γ : Ctx Σ m} {A} {B}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (exchange ((◇ , A) ⊢ B))\n → DerivableArgs E (js , (◇ , A) ⊢ B)\ndjs ,1Ty dj = djs , dj\n\n_,1Ty=_ : ∀ {Σ} {E} {m} {Γ : Ctx Σ m} {A} {B C}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (exchange ((◇ , A) ⊢ B == C))\n → DerivableArgs E (js , (◇ , A) ⊢ B == C)\ndjs ,1Ty= dj = djs , dj\n\n_,1Tm_ : ∀ {Σ} {E} {m} {Γ : Ctx Σ m} {u : _} {A : _} {B : _}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (exchange ((◇ , B) ⊢ u :> A))\n → DerivableArgs E (js , (◇ , B) ⊢ u :> A)\ndjs ,1Tm dj = djs , dj\n\n_,1Tm=_ : ∀ {Σ} {E} {m} {Γ : Ctx Σ m} {u v : _} {A : _} {B : _}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (exchange ((◇ , B) ⊢ u == v :> A))\n → DerivableArgs E (js , (◇ , B) ⊢ u == v :> A)\ndjs ,1Tm= dj = djs , dj\n\n\n_,2Tm_ : ∀ {Σ} {E} {m} {Γ : Ctx Σ m} {u : _} {A : _} {B : _} {C : _}\n {ar : JudgmentArityArgs} {js : DerivationRulePremises Σ Γ ar}\n → DerivableArgs E js\n → Derivable E (exchange ((◇ , B , C) ⊢ u :> A))\n → DerivableArgs E (js , (◇ , B , C) ⊢ u :> A)\ndjs ,2Tm dj = djs , dj\n\ninfixl 4 _,0Ty_ _,0Ty=_ _,0Tm_ _,0Tm=_\n _,1Ty_ _,1Ty=_ _,1Tm_ _,1Tm=_\n _,2Tm_\n", "meta": {"hexsha": "3e0c63e785baaf2836658e4087760b60f6c49827", "size": 7657, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "derivability.agda", "max_stars_repo_name": "guillaumebrunerie/general-type-theories", "max_stars_repo_head_hexsha": "f9bfefd0a70ae5bdc3906829ee1165c731882bca", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "derivability.agda", "max_issues_repo_name": 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YES\n2. YES", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.34135105954015343}} {"text": "{-# OPTIONS --without-K --rewriting #-}\n\nopen import lib.Basics\nopen import lib.two-semi-categories.Functor\nopen import lib.two-semi-categories.FundamentalCategory\nopen import lib.two-semi-categories.FunctorInverse\nopen import lib.types.Pi using ()\n\nmodule lib.two-semi-categories.FunextFunctors where\n\nmodule FunextFunctors {i j} (A : Type i) (B : Type j) {{B-level : has-level 2 B}} where\n\n open import lib.two-semi-categories.FunCategory\n\n private\n\n app=-pres-comp : ∀ {f g h : A → B} (α : f == g) (β : g == h) → app= (α ∙ β) == (λ a → app= α a ∙ app= β a)\n app=-pres-comp α β = λ= (λ a → ap-∙ (λ f → f a) α β)\n\n abstract\n app=-pres-comp-coh : ∀ {f g h i : A → B} (α : f == g) (β : g == h) (γ : h == i)\n → app=-pres-comp (α ∙ β) γ ◃∙\n ap (λ s a → s a ∙ app= γ a) (app=-pres-comp α β) ◃∙\n λ= (λ a → ∙-assoc (app= α a) (app= β a) (app= γ a)) ◃∎\n =ₛ\n ap app= (∙-assoc α β γ) ◃∙\n app=-pres-comp α (β ∙ γ) ◃∙\n ap (λ s a → app= α a ∙ s a) (app=-pres-comp β γ) ◃∎\n app=-pres-comp-coh {f} idp idp γ =\n app=-pres-comp idp γ ◃∙\n ap (λ s a → s a ∙ app= γ a) (app=-pres-comp idp idp) ◃∙\n λ= (λ a → idp) ◃∎\n =ₛ⟨ 2 & 1 & =ₛ-in {t = []} (! (λ=-η idp)) ⟩\n app=-pres-comp idp γ ◃∙\n ap (λ s a → s a ∙ app= γ a) (λ= (λ a → idp {a = idp {a = f a}})) ◃∎\n =ₛ₁⟨ 1 & 1 & ap (ap (λ s a → s a ∙ app= γ a)) (! (λ=-η idp)) ⟩\n app=-pres-comp idp γ ◃∙\n idp ◃∎\n =ₛ₁⟨ 1 & 1 & ap (ap (λ s → s)) (λ=-η idp) ⟩\n app=-pres-comp idp γ ◃∙\n ap (λ s → s) (λ= (λ a → idp {a = app= γ a})) ◃∎\n =ₛ⟨ 0 & 0 & contract ⟩\n idp ◃∙\n app=-pres-comp idp γ ◃∙\n ap (λ s → s) (λ= (λ a → idp {a = app= γ a})) ◃∎ ∎ₛ\n\n app=-functor : TwoSemiFunctor (2-type-fundamental-cat (A → B))\n (fun-cat A (2-type-fundamental-cat B))\n app=-functor =\n record\n { F₀ = idf (A → B)\n ; F₁ = app=\n ; pres-comp = app=-pres-comp\n ; pres-comp-coh = app=-pres-comp-coh\n }\n\n private\n module app=-functor =\n TwoSemiFunctor app=-functor\n module app=-inverse =\n FunctorInverse\n app=-functor\n (idf-is-equiv _)\n (λ f g → snd app=-equiv)\n\n λ=-functor : TwoSemiFunctor (fun-cat A (2-type-fundamental-cat B))\n (2-type-fundamental-cat (A → B))\n λ=-functor = app=-inverse.functor\n\n module λ=-functor = TwoSemiFunctor λ=-functor\n\n abstract\n λ=-functor-pres-comp=λ=-∙ : ∀ {f g h : A → B} (α : f ∼ g) (β : g ∼ h)\n → λ=-functor.pres-comp α β == =ₛ-out (λ=-∙ α β)\n λ=-functor-pres-comp=λ=-∙ α β = =ₛ-out {t = =ₛ-out (λ=-∙ α β) ◃∎} $\n λ=-functor.pres-comp α β ◃∎\n =ₛ⟨ app=-inverse.pres-comp-β α β ⟩\n idp ◃∙\n ap2 (λ s t → λ= (λ a → s a ∙ t a)) (! (λ= (app=-β α))) (! (λ= (app=-β β))) ◃∙\n ap λ= (! (λ= (λ a → ap-∙ (λ f → f a) (λ= α) (λ= β)))) ◃∙\n ! (λ=-η (λ= α ∙ λ= β)) ◃∎\n =ₛ⟨ 0 & 1 & expand [] ⟩\n ap2 (λ s t → λ= (λ a → s a ∙ t a)) (! (λ= (app=-β α))) (! (λ= (app=-β β))) ◃∙\n ap λ= (! (λ= (λ a → ap-∙ (λ f → f a) (λ= α) (λ= β)))) ◃∙\n ! (λ=-η (λ= α ∙ λ= β)) ◃∎\n =ₛ₁⟨ 0 & 1 & step₈ ⟩\n ap λ= (! (λ= (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a')))) ◃∙\n ap λ= (! (λ= (λ a' → ap-∙ (λ γ → γ a') (λ= α) (λ= β)))) ◃∙\n ! (λ=-η (λ= α ∙ λ= β)) ◃∎\n =ₛ⟨ 0 & 2 &\n ap-seq-=ₛ λ= $ ∙-!-seq $\n λ= (λ a' → ap-∙ (λ γ → γ a') (λ= α) (λ= β)) ◃∙\n λ= (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a')) ◃∎ ⟩\n ap λ= (! (λ= (λ a' → ap-∙ (λ γ → γ a') (λ= α) (λ= β)) ∙\n λ= (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a')))) ◃∙\n ! (λ=-η (λ= α ∙ λ= β)) ◃∎\n =ₛ₁⟨ 0 & 1 & ap-! λ= (λ= (λ a' → ap-∙ (λ γ → γ a') (λ= α) (λ= β)) ∙\n λ= (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a'))) ⟩\n ! (ap λ= (λ= (λ a' → ap-∙ (λ γ → γ a') (λ= α) (λ= β)) ∙\n λ= (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a')))) ◃∙\n ! (λ=-η (λ= α ∙ λ= β)) ◃∎\n =ₛ₁⟨ 0 & 1 &\n ap (! ∘ ap λ=) $ =ₛ-out $\n ∙-λ= (λ a' → ap-∙ (λ γ → γ a') (λ= α) (λ= β))\n (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a')) ⟩\n ! (ap λ= (λ= (λ a' → ap-∙ (λ γ → γ a') (λ= α) (λ= β) ∙\n ap2 _∙_ (app=-β α a') (app=-β β a')))) ◃∙\n ! (λ=-η (λ= α ∙ λ= β)) ◃∎\n =ₛ⟨ =ₛ-in $\n ∙-! (ap λ= (λ= (λ a' → ap-∙ (λ γ → γ a') (λ= α) (λ= β) ∙\n ap2 _∙_ (app=-β α a') (app=-β β a'))))\n (λ=-η (λ= α ∙ λ= β)) ⟩\n ! (λ=-η (λ= α ∙ λ= β) ∙\n ap λ= (λ= (λ a' → ap-∙ (λ γ → γ a') (λ= α) (λ= β) ∙\n ap2 _∙_ (app=-β α a') (app=-β β a')))) ◃∎ ∎ₛ\n where\n step₈' : ap2 (λ s t a → s a ∙ t a) (λ= (app=-β α)) (λ= (app=-β β)) ==\n λ= (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a'))\n step₈' =\n –>-is-inj app=-equiv _ _ $ λ= $ λ a →\n app= (ap2 (λ s t a' → s a' ∙ t a') (λ= (app=-β α)) (λ= (app=-β β))) a\n =⟨ ap-ap2 (λ f → f a) (λ s t a' → s a' ∙ t a') (λ= (app=-β α)) (λ= (app=-β β)) ⟩\n ap2 (λ s t → s a ∙ t a) (λ= (app=-β α)) (λ= (app=-β β))\n =⟨ ! (ap2-ap-lr _∙_ (λ f → f a) (λ f → f a) (λ= (app=-β α)) (λ= (app=-β β))) ⟩\n ap2 _∙_ (app= (λ= (app=-β α)) a) (app= (λ= (app=-β β)) a)\n =⟨ ap2 (ap2 _∙_) (app=-β (app=-β α) a) (app=-β (app=-β β) a) ⟩\n ap2 _∙_ (app=-β α a) (app=-β β a)\n =⟨ ! (app=-β (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a')) a) ⟩\n app= (λ= (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a'))) a =∎\n step₈ : ap2 (λ s t → λ= (λ a → s a ∙ t a)) (! (λ= (app=-β α))) (! (λ= (app=-β β))) ==\n ap λ= (! (λ= (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a'))))\n step₈ =\n ap2 (λ s t → λ= (λ a → s a ∙ t a)) (! (λ= (app=-β α))) (! (λ= (app=-β β)))\n =⟨ ! (ap-ap2 λ= (λ s t a → s a ∙ t a) (! (λ= (app=-β α))) (! (λ= (app=-β β)))) ⟩\n ap λ= (ap2 (λ s t a → s a ∙ t a) (! (λ= (app=-β α))) (! (λ= (app=-β β))))\n =⟨ ap (ap λ=) (ap2-! (λ s t a → s a ∙ t a) (λ= (app=-β α)) (λ= (app=-β β))) ⟩\n ap λ= (! (ap2 (λ s t a → s a ∙ t a) (λ= (app=-β α)) (λ= (app=-β β))))\n =⟨ ap (ap λ= ∘ !) step₈' ⟩\n ap λ= (! (λ= (λ a' → ap2 _∙_ (app=-β α a') (app=-β β a')))) =∎\n", "meta": {"hexsha": "fae3251e530f2c5dd7d0a2acbe3d3c997713f2fa", "size": 6291, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "core/lib/two-semi-categories/FunextFunctors.agda", "max_stars_repo_name": "AntoineAllioux/HoTT-Agda", "max_stars_repo_head_hexsha": "1037d82edcf29b620677a311dcfd4fc2ade2faa6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 294, "max_stars_repo_stars_event_min_datetime": "2015-01-09T16:23:23.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T13:54:45.000Z", "max_issues_repo_path": "core/lib/two-semi-categories/FunextFunctors.agda", "max_issues_repo_name": "AntoineAllioux/HoTT-Agda", "max_issues_repo_head_hexsha": "1037d82edcf29b620677a311dcfd4fc2ade2faa6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 31, "max_issues_repo_issues_event_min_datetime": "2015-03-05T20:09:00.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-03T19:15:25.000Z", "max_forks_repo_path": "core/lib/two-semi-categories/FunextFunctors.agda", "max_forks_repo_name": "AntoineAllioux/HoTT-Agda", "max_forks_repo_head_hexsha": "1037d82edcf29b620677a311dcfd4fc2ade2faa6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 50, "max_forks_repo_forks_event_min_datetime": "2015-01-10T01:48:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-14T03:03:25.000Z", "avg_line_length": 44.9357142857, "max_line_length": 110, "alphanum_fraction": 0.3751390876, "num_tokens": 2983, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.34126806686115424}} {"text": "open import Function using (case_of_; _∘_)\nopen import Data.List using (List; _++_; map) renaming (_∷_ to _,_; _∷ʳ_ to _,′_; [] to ∅)\nopen import Data.List.Properties using (map-++-commute)\nopen import Data.Product using () renaming (_×_ to _x'_)\nopen import Relation.Binary.PropositionalEquality as PropEq using (_≡_; refl; sym; cong)\n\nmodule LinearLogic (U : Set) (R : U) (⟦_⟧ᵁ : U → Set) where\n\ninfixr 40 ¬_\ninfix 30 _⊗_\ninfixr 20 _⊸_\ninfix 4 _⊢_\n\ndata Type : Set where\n el : (A : U) → Type\n ⊥ : Type\n _⊗_ : Type → Type → Type\n _⊸_ : Type → Type → Type\n\ndata _⊢_ : ∀ (X : List Type) (A : Type) → Set where\n var : ∀ {A} → A , ∅ ⊢ A\n abs : ∀ {X A B} → A , X ⊢ B → X ⊢ A ⊸ B\n app : ∀ {X Y A B} → X ⊢ A ⊸ B → Y ⊢ A → X ++ Y ⊢ B\n pair : ∀ {X Y A B} → X ⊢ A → Y ⊢ B → X ++ Y ⊢ A ⊗ B\n case : ∀ {X Y A B C } → X ⊢ A ⊗ B → A , B , Y ⊢ C → X ++ Y ⊢ C\n exch : ∀ {X Y Z W A} → (X ++ Z) ++ (Y ++ W) ⊢ A\n → (X ++ Y) ++ (Z ++ W) ⊢ A\n\n¬_ : Type → Type\n¬ A = A ⊸ ⊥\n\nexch₀ : ∀ {A B C X} → A , B , X ⊢ C → B , A , X ⊢ C\nexch₀ {A} {B} {X = X} t = exch {∅} {B , ∅} {A , ∅} {X} t\n\nswap : ∀ {A B} → ∅ ⊢ A ⊗ B ⊸ B ⊗ A\nswap {A} {B} = abs (case var (exch₀ (pair var var)))\n\nraise : ∀ {A B X} → X ⊢ A → X ⊢ (A ⊸ B) ⊸ B\nraise t = abs (app var t)\n\n++-assoc : ∀ {a} {A : Set a} (X Y Z : List A) → X ++ (Y ++ Z) ≡ (X ++ Y) ++ Z\n++-assoc ∅ Y Z = refl\n++-assoc (x , X) Y Z = cong (_,_ x) (++-assoc X Y Z)\n\nxs++[]=xs : ∀ {a} {A : Set a} (xs : List A) → xs ++ ∅ ≡ xs\nxs++[]=xs ∅ = refl\nxs++[]=xs (x , xs) = cong (_,_ x) (xs++[]=xs xs)\n\nto-front : ∀ {X A B} → A , X ⊢ B → X ,′ A ⊢ B\nto-front {X} {A} {B} t = lem1 lem2\n where\n lem1 : A , (X ++ ∅) ⊢ B → X ,′ A ⊢ B\n lem1 = exch {∅} {X} {A , ∅} {∅}\n lem2 : A , (X ++ ∅) ⊢ B\n lem2 rewrite xs++[]=xs X = t\n\nto-back : ∀ {X A B} → X ,′ A ⊢ B → A , X ⊢ B\nto-back {X} {A} {B} t = lem2\n where\n lem1 : A , X ++ ∅ ⊢ B\n lem1 = exch {∅} {A , ∅} {X} {∅} t\n lem2 : A , X ⊢ B\n lem2 rewrite sym (xs++[]=xs (A , X)) = lem1\n\nYX↝XY : ∀ {A} X Y → Y ++ X ⊢ A → X ++ Y ⊢ A\nYX↝XY {A} X Y t = lem₃\n where\n lem₁ : Y ++ X ++ ∅ ⊢ A\n lem₁ rewrite xs++[]=xs X = t\n lem₂ : X ++ Y ++ ∅ ⊢ A\n lem₂ = exch {∅} {X} {Y} {∅} lem₁\n lem₃ : X ++ Y ⊢ A\n lem₃ = PropEq.subst (λ Y → X ++ Y ⊢ A) (xs++[]=xs Y) lem₂\n\nY[XZ]↝X[YZ] : ∀ {A} X Y Z → Y ++ (X ++ Z) ⊢ A → X ++ (Y ++ Z) ⊢ A\nY[XZ]↝X[YZ] {A} X Y Z t = exch {∅} {X} {Y} {Z} t\n\n[YX]Z↝[XY]Z : ∀ {A} X Y Z → (Y ++ X) ++ Z ⊢ A → (X ++ Y) ++ Z ⊢ A\n[YX]Z↝[XY]Z {A} X Y Z t = lem₃\n where\n lem₁ : Y ++ (X ++ Z) ⊢ A\n lem₁ rewrite ++-assoc Y X Z = t\n lem₂ : X ++ (Y ++ Z) ⊢ A\n lem₂ = Y[XZ]↝X[YZ] X Y Z lem₁\n lem₃ : (X ++ Y) ++ Z ⊢ A\n lem₃ rewrite sym (++-assoc X Y Z) = lem₂\n\n[XZ]Y↝[XY]Z : ∀ {A} X Y Z → (X ++ Z) ++ Y ⊢ A → (X ++ Y) ++ Z ⊢ A\n[XZ]Y↝[XY]Z {A} X Y Z t = lem₃\n where\n lem₁ : (X ++ Z) ++ Y ++ ∅ ⊢ A\n lem₁ rewrite xs++[]=xs Y = t\n lem₂ : (X ++ Y) ++ Z ++ ∅ ⊢ A\n lem₂ = exch {X} {Y} {Z} {∅} lem₁\n lem₃ : (X ++ Y) ++ Z ⊢ A\n lem₃ = PropEq.subst (λ Z → (X ++ Y) ++ Z ⊢ A) (xs++[]=xs Z) lem₂\n\nX[ZY]↝X[YZ] : ∀ {A} X Y Z → X ++ (Z ++ Y) ⊢ A → X ++ (Y ++ Z) ⊢ A\nX[ZY]↝X[YZ] {A} X Y Z t = lem₃\n where\n lem₁ : (X ++ Z) ++ Y ⊢ A\n lem₁ rewrite sym (++-assoc X Z Y) = t\n lem₂ : (X ++ Y) ++ Z ⊢ A\n lem₂ = [XZ]Y↝[XY]Z X Y Z lem₁\n lem₃ : X ++ Y ++ Z ⊢ A\n lem₃ rewrite ++-assoc X Y Z = lem₂\n\nXYZW↝XWZY : ∀ {A} X Y Z W → (X ++ Y) ++ (Z ++ W) ⊢ A → (X ++ W) ++ (Z ++ Y) ⊢ A\nXYZW↝XWZY {A} X Y Z W t = lem₃\n where\n lem₁ : (X ++ Y) ++ (W ++ Z) ⊢ A\n lem₁ = X[ZY]↝X[YZ] (X ++ Y) W Z t\n lem₂ : (X ++ W) ++ (Y ++ Z) ⊢ A\n lem₂ = exch {X} {W} {Y} {Z} lem₁\n lem₃ : (X ++ W) ++ (Z ++ Y) ⊢ A\n lem₃ = X[ZY]↝X[YZ] (X ++ W) Z Y lem₂\n\nXYZW↝YWXZ : ∀ {A} X Y Z W → (X ++ Y) ++ (Z ++ W) ⊢ A → (Y ++ W) ++ (X ++ Z) ⊢ A\nXYZW↝YWXZ {A} X Y Z W t = lem₃\n where\n lem₁ : (Y ++ X) ++ (Z ++ W) ⊢ A\n lem₁ = [YX]Z↝[XY]Z Y X (Z ++ W) t\n lem₂ : (Y ++ X) ++ (W ++ Z) ⊢ A\n lem₂ = X[ZY]↝X[YZ] (Y ++ X) W Z lem₁\n lem₃ : (Y ++ W) ++ (X ++ Z) ⊢ A\n lem₃ = exch {Y} {W} {X} {Z} lem₂\n\nXYZW↝ZXWY : ∀ {A} X Y Z W → (X ++ Y) ++ (Z ++ W) ⊢ A → (Z ++ X) ++ (W ++ Y) ⊢ A\nXYZW↝ZXWY {A} X Y Z W t = lem₃\n where\n lem₁ : (X ++ Z) ++ (Y ++ W) ⊢ A\n lem₁ = exch {X} {Z} {Y} {W} t\n lem₂ : (Z ++ X) ++ (Y ++ W) ⊢ A\n lem₂ = [YX]Z↝[XY]Z Z X (Y ++ W) lem₁\n lem₃ : (Z ++ X) ++ (W ++ Y) ⊢ A\n lem₃ = X[ZY]↝X[YZ] (Z ++ X) W Y lem₂\n\nXYZW↝ZYXW : ∀ {A} X Y Z W → (X ++ Y) ++ (Z ++ W) ⊢ A → (Z ++ Y) ++ (X ++ W) ⊢ A\nXYZW↝ZYXW {A} X Y Z W t = lem₃\n where\n lem₁ : (Y ++ X) ++ (Z ++ W) ⊢ A\n lem₁ = [YX]Z↝[XY]Z Y X (Z ++ W) t\n lem₂ : (Y ++ Z) ++ (X ++ W) ⊢ A\n lem₂ = exch {Y} {Z} {X} {W} lem₁\n lem₃ : (Z ++ Y) ++ (X ++ W) ⊢ A\n lem₃ = [YX]Z↝[XY]Z Z Y (X ++ W) lem₂\n\npair-left : ∀ {X A B C} → A , B , X ⊢ C → A ⊗ B , X ⊢ C\npair-left t = case var t\n\npair-left′ : ∀ {X A B C} → X ++ (A , B , ∅) ⊢ C → X ,′ A ⊗ B ⊢ C\npair-left′ {X} {A} {B} {C} = lem₃\n where\n lem₁ : X ,′ A ,′ B ⊢ C → X ,′ A ⊗ B ⊢ C\n lem₁ t = to-front (pair-left (to-back {B , X} {A} (to-back {X ,′ A} {B} t)))\n lem₂ : ∀ {a} {A : Set a} xs (y z : A) → xs ,′ y ,′ z ≡ xs ++ (y , z , ∅)\n lem₂ ∅ y z = refl\n lem₂ (x , xs) y z = cong (_,_ x) (lem₂ xs y z)\n lem₃ : X ++ (A , B , ∅) ⊢ C → X ,′ A ⊗ B ⊢ C\n lem₃ rewrite sym (lem₂ X A B) = lem₁\n\nopen import IntuitionisticLogic U ⟦_⟧ᵁ as IL renaming (Type to TypeIL; _⊗_ to _×_)\nopen IL.Explicit\n hiding (swap; swap′)\n renaming (_⊢_ to _⊢IL_; ReifyType to ReifyTypeIL; ReifyCtxt to ReiftCtxtIL; [_] to reifyIL)\n\ninstance\n ReifyType : Reify Type TypeIL\n ReifyType = record { ⟦_⟧ = ⟦_⟧ }\n where\n\n ⟦_⟧ : Type → TypeIL\n ⟦ ⊥ ⟧ = el R\n ⟦ el A ⟧ = el A\n ⟦ A ⊗ B ⟧ = ⟦ A ⟧ × ⟦ B ⟧\n ⟦ A ⊸ B ⟧ = ⟦ A ⟧ ⇒ ⟦ B ⟧\n\nopen Reify {{...}} using (⟦_⟧)\n\ninstance\n ReifyCtxt : Reify (List Type) (List TypeIL)\n ReifyCtxt = record { ⟦_⟧ = map ⟦_⟧ }\n\n⟦X++Y⟧=⟦X⟧++⟦Y⟧ : (X Y : List Type) → ⟦ X ++ Y ⟧ ≡ ⟦ X ⟧ ++ ⟦ Y ⟧\n⟦X++Y⟧=⟦X⟧++⟦Y⟧ X Y = map-++-commute ⟦_⟧ X Y\n\ntoIL : ∀ {X A} → X ⊢ A → ⟦ X ⟧ ⊢IL ⟦ A ⟧\ntoIL var = var\ntoIL (abs t) = abs (toIL t)\ntoIL (app {X} {Y} s t) rewrite ⟦X++Y⟧=⟦X⟧++⟦Y⟧ X Y = app (toIL s) (toIL t)\ntoIL (pair {X} {Y} s t) rewrite ⟦X++Y⟧=⟦X⟧++⟦Y⟧ X Y = pair (toIL s) (toIL t)\ntoIL (case {X} {Y} s t) rewrite ⟦X++Y⟧=⟦X⟧++⟦Y⟧ X Y = case (toIL s) (toIL t)\ntoIL (exch {X} {Y} {Z} {W} {A} t) = lem4\n where\n lem1 : ⟦ (X ++ Z) ++ (Y ++ W) ⟧ ⊢IL ⟦ A ⟧\n lem1 = toIL t\n lem2 : (⟦ X ⟧ ++ ⟦ Z ⟧) ++ (⟦ Y ⟧ ++ ⟦ W ⟧) ⊢IL ⟦ A ⟧\n lem2 rewrite sym (⟦X++Y⟧=⟦X⟧++⟦Y⟧ X Z)\n | sym (⟦X++Y⟧=⟦X⟧++⟦Y⟧ Y W)\n | sym (⟦X++Y⟧=⟦X⟧++⟦Y⟧ (X ++ Z) (Y ++ W)) = lem1\n lem3 : (⟦ X ⟧ ++ ⟦ Y ⟧) ++ (⟦ Z ⟧ ++ ⟦ W ⟧) ⊢IL ⟦ A ⟧\n lem3 = exch {⟦ X ⟧} {⟦ Y ⟧} {⟦ Z ⟧} {⟦ W ⟧} lem2\n lem4 : ⟦ (X ++ Y) ++ (Z ++ W) ⟧ ⊢IL ⟦ A ⟧\n lem4 rewrite ⟦X++Y⟧=⟦X⟧++⟦Y⟧ (X ++ Y) (Z ++ W)\n | ⟦X++Y⟧=⟦X⟧++⟦Y⟧ X Y\n | ⟦X++Y⟧=⟦X⟧++⟦Y⟧ Z W = lem3\n\n[_] : {A : Type} {X : List Type} → X ⊢ A → (Ctxt ⟦ ⟦ X ⟧ ⟧ → ⟦ ⟦ A ⟧ ⟧)\n[_] = reifyIL ∘ toIL\n\nswap′ : {A B : Type} → ⟦ ⟦ A ⟧ ⟧ x' ⟦ ⟦ B ⟧ ⟧ → ⟦ ⟦ B ⟧ ⟧ x' ⟦ ⟦ A ⟧ ⟧\nswap′ {A} {B} = [ swap {A} {B} ] ∅\n\n", "meta": {"hexsha": "0e3e8265200a5b4def54f58a7667f0a8c7078c3d", "size": 7106, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/LinearLogic.agda", "max_stars_repo_name": "wenkokke/msla2014", "max_stars_repo_head_hexsha": "b880cf25ed8e81b9a965ea9aad18377008d68a9f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-11-17T23:04:39.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-29T09:07:45.000Z", "max_issues_repo_path": "src/LinearLogic.agda", "max_issues_repo_name": "wenkokke/msla2014", "max_issues_repo_head_hexsha": "b880cf25ed8e81b9a965ea9aad18377008d68a9f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/LinearLogic.agda", "max_forks_repo_name": "wenkokke/msla2014", "max_forks_repo_head_hexsha": "b880cf25ed8e81b9a965ea9aad18377008d68a9f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.2056074766, "max_line_length": 93, "alphanum_fraction": 0.4012102449, "num_tokens": 3834, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.34126806686115424}} {"text": "\nopen import Oscar.Prelude\nopen import Oscar.Class.Successor₀\nopen import Oscar.Class.Successor₁\nopen import Oscar.Class.Injectivity\nopen import Oscar.Class.Pure\n\nmodule Oscar.Class.Thickandthin where\n\nmodule _\n {x} {X : Ø x} {a} (A : X → Ø a) {b} (B : X → Ø b)\n where\n record [𝓣hin] : Ø₀ where\n no-eta-equality\n constructor ∁\n module _\n ⦃ _ : 𝓢uccessor₀ X ⦄\n where\n 𝔱hin : ∀ (m : X) → A (⇑₀ m) → B m → Ø b\n 𝔱hin m = λ _ _ → B (⇑₀ m)\n 𝓽hin = ∀ {m : X} → A (⇑₀ m) → B m → B (⇑₀ m)\n record 𝓣hin ⦃ _ : [𝓣hin] ⦄ : Ø x ∙̂ a ∙̂ b where\n field\n thin : 𝓽hin\n instance `𝓘njection₂ : ∀ {m} → 𝓘njection₂ (𝔱hin m)\n `𝓘njection₂ = ∁ thin\nopen 𝓣hin ⦃ … ⦄ public\n\nmodule _\n {x} {X : Ø x} {a} (A : X → Ø a) {b} (B : X → Ø b)\n where\n record [𝓣hick] : Ø₀ where\n no-eta-equality\n constructor ∁\n module _\n ⦃ _ : 𝓢uccessor₀ X ⦄\n where\n 𝓽hick = ∀ {m} → A m → B (⇑₀ m) → B m\n record 𝓣hick ⦃ _ : [𝓣hick] ⦄ : Ø x ∙̂ a ∙̂ b where field thick : 𝓽hick\nopen 𝓣hick ⦃ … ⦄ public\n\nmodule _\n {x} {X : Ø x}\n {a} (A : X → Ø a)\n {b} (B : X → Ø b)\n {ℓ} (_≈_ : ∀ {x} → B x → B x → Ø ℓ)\n where\n record [𝓣hick/thin=1] : Ø₀ where\n no-eta-equality\n constructor ∁\n module _\n ⦃ _ : 𝓢uccessor₀ X ⦄\n ⦃ _ : [𝓢uccessor₁] A ⦄\n ⦃ _ : 𝓢uccessor₁ A ⦄\n ⦃ _ : [𝓣hin] A B ⦄\n ⦃ _ : 𝓣hin A B ⦄\n ⦃ _ : [𝓣hick] A B ⦄\n ⦃ _ : 𝓣hick A B ⦄\n where\n 𝓽hick/thin=1 = ∀ {m} (x : A m) (y : B m) → thick x (thin (⇑₁ x) y) ≈ y\n record 𝓣hick/thin=1 : Ø x ∙̂ a ∙̂ b ∙̂ ℓ where field thick/thin=1 : 𝓽hick/thin=1\nopen 𝓣hick/thin=1 ⦃ … ⦄ public\n\nmodule _\n {x} {X : Ø x}\n {a} (A : X → Ø a)\n {b} (B : X → Ø b)\n {c} (C : Ø b → Ø c)\n where\n record [𝓒heck] : Ø₀ where\n no-eta-equality\n constructor ∁\n module _\n ⦃ _ : 𝓢uccessor₀ X ⦄\n where\n 𝓬heck = ∀ {m} → A (⇑₀ m) → B (⇑₀ m) → C (B m)\n record 𝓒heck ⦃ _ : [𝓒heck] ⦄ : Ø x ∙̂ a ∙̂ b ∙̂ c where field check : 𝓬heck\nopen 𝓒heck ⦃ … ⦄ public\n\ncheck[_] : ∀\n {x} {X : Ø x}\n {a} {A : X → Ø a}\n {b} {B : X → Ø b}\n {c} (C : Ø b → Ø c)\n ⦃ _ : [𝓒heck] A B C ⦄\n ⦃ _ : 𝓢uccessor₀ X ⦄\n ⦃ _ : 𝓒heck A B C ⦄\n → 𝓬heck A B C\ncheck[ _ ] = check\n\nmodule _\n {x} {X : Ø x}\n {a} (A : X → Ø a)\n {b} (B : X → Ø b)\n {c} (C : Ø b → Ø c)\n {ℓ} (_≈_ : ∀ {x} → C (B x) → C (B x) → Ø ℓ)\n where\n record [𝓒heck/thin=1] : Ø₀ where\n no-eta-equality\n constructor ∁\n module _\n ⦃ _ : 𝓢uccessor₀ X ⦄\n ⦃ _ : [𝓣hin] A B ⦄\n ⦃ _ : 𝓣hin A B ⦄\n ⦃ _ : [𝓒heck] A B C ⦄\n ⦃ _ : 𝓒heck A B C ⦄\n ⦃ _ : 𝓟ure C ⦄\n where\n 𝓬heck/thin=1 = ∀ {n} (x : A (⇑₀ n)) (y : B n) → check x (thin x y) ≈ pure y\n record 𝓒heck/thin=1 ⦃ _ : [𝓒heck/thin=1] ⦄ : Ø x ∙̂ a ∙̂ b ∙̂ c ∙̂ ℓ where field check/thin=1 : 𝓬heck/thin=1\nopen 𝓒heck/thin=1 ⦃ … ⦄ public\n\ncheck/thin=1[_] : ∀\n {x} {X : Ø x}\n {a} {A : X → Ø a}\n {b} {B : X → Ø b}\n {c} {C : Ø b → Ø c}\n {ℓ} (_≈_ : ∀ {x} → C (B x) → C (B x) → Ø ℓ)\n ⦃ _ : 𝓢uccessor₀ X ⦄\n ⦃ _ : [𝓣hin] A B ⦄\n ⦃ _ : 𝓣hin A B ⦄\n ⦃ _ : [𝓒heck] A B C ⦄\n ⦃ _ : 𝓒heck A B C ⦄\n ⦃ _ : 𝓟ure C ⦄\n ⦃ _ : [𝓒heck/thin=1] A B C _≈_ ⦄\n ⦃ _ : 𝓒heck/thin=1 A B C _≈_ ⦄\n → 𝓬heck/thin=1 A B C _≈_\ncheck/thin=1[ _ ] = check/thin=1\n\nrecord IsThickandthin\n {x a b c ℓb ℓc}\n {X : Ø x}\n (A : X → Ø a)\n (B : X → Ø b)\n (_≈B_ : ∀ {x} → B x → B x → Ø ℓb)\n (C : Ø b → Ø c)\n (_≈C_ : ∀ {x} → C (B x) → C (B x) → Ø ℓc)\n : Ø x ∙̂ a ∙̂ ↑̂ b ∙̂ ℓb ∙̂ c ∙̂ ℓc where\n constructor ∁\n field\n overlap ⦃ `𝓢uccessor₀ ⦄ : 𝓢uccessor₀ X\n overlap ⦃ `[𝓢uccessor₁] ⦄ : [𝓢uccessor₁] A\n overlap ⦃ `𝓢uccessor₁ ⦄ : 𝓢uccessor₁ A\n overlap ⦃ `[𝓣hick] ⦄ : [𝓣hick] A B\n overlap ⦃ `𝓣hick ⦄ : 𝓣hick A B\n overlap ⦃ `[𝓣hin] ⦄ : [𝓣hin] A B\n overlap ⦃ `𝓣hin ⦄ : 𝓣hin A B\n overlap ⦃ `[𝓘njectivity₂,₁] ⦄ : ∀ {m} → [𝓘njectivity₂,₁] (𝔱hin A B m) _≈B_ _≈B_\n overlap ⦃ `𝓘njectivity₂,₁ ⦄ : ∀ {m} → 𝓘njectivity₂,₁ (𝔱hin A B m) _≈B_ _≈B_\n overlap ⦃ `[𝓒heck] ⦄ : [𝓒heck] A B C\n overlap ⦃ `𝓒heck ⦄ : 𝓒heck A B C\n overlap ⦃ `[𝓣hick/thin=1] ⦄ : [𝓣hick/thin=1] A B _≈B_\n overlap ⦃ `𝓣hick/thin=1 ⦄ : 𝓣hick/thin=1 A B _≈B_\n overlap ⦃ `[𝓒heck/thin=1] ⦄ : [𝓒heck/thin=1] A B C _≈C_\n overlap ⦃ `𝓟ure ⦄ : 𝓟ure C\n overlap ⦃ `𝓒heck/thin=1 ⦄ : 𝓒heck/thin=1 A B C _≈C_\n\nrecord Thickandthin x a b ℓb c ℓc : Ø ↑̂ (x ∙̂ a ∙̂ b ∙̂ ℓb ∙̂ c ∙̂ ℓc) where\n constructor ∁\n field\n {X} : Ø x\n A : X → Ø a\n B : X → Ø b\n _≈B_ : ∀ {x} → B x → B x → Ø ℓb\n C : Ø b → Ø c\n _≈C_ : ∀ {x} → C (B x) → C (B x) → Ø ℓc\n ⦃ `IsThickandthin ⦄ : IsThickandthin A B _≈B_ C _≈C_\n", "meta": {"hexsha": "06281ff4a3b24df6b720b8c6673f9ed66ec9906f", "size": 4473, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "archive/agda-3/src/Oscar/Class/Thickandthin.agda", "max_stars_repo_name": "m0davis/oscar", "max_stars_repo_head_hexsha": "52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb", "max_stars_repo_licenses": ["RSA-MD"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": 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YES\n2. NO\n\n", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.46101677931231594, "lm_q1q2_score": 0.3412328005167754}} {"text": "open import Mockingbird.Forest using (Forest)\n\n-- Aristocratic Birds\nmodule Mockingbird.Problems.Chapter19 {ℓb ℓ} (forest : Forest {ℓb} {ℓ}) where\n\nopen import Data.Product using (_×_; _,_; ∃-syntax)\nopen import Function using (_$_)\nopen import Level using (_⊔_)\nopen import Data.Vec using ([]; _∷_)\nopen import Relation.Unary using (Pred; _∈_; _⊆_; _⊇_)\n\nopen Forest forest\nopen import Mockingbird.Forest.Birds forest\nopen import Mockingbird.Forest.Combination.Vec forest\nopen import Mockingbird.Forest.Combination.Vec.Properties forest\nopen import Mockingbird.Forest.Extensionality forest\nimport Mockingbird.Problems.Chapter11 forest as Chapter₁₁\nimport Mockingbird.Problems.Chapter12 forest as Chapter₁₂\n\nproblem₁″ : ⦃ _ : HasEagle ⦄ ⦃ _ : HasCardinalOnceRemoved ⦄ ⦃ _ : HasCardinalTwiceRemoved ⦄ ⦃ _ : HasWarbler ⦄ → HasJaybird\nproblem₁″ = record\n { J = C** ∙ (W ∙ (C* ∙ E))\n ; isJaybird = λ x y z w → begin\n C** ∙ (W ∙ (C* ∙ E)) ∙ x ∙ y ∙ z ∙ w ≈⟨ isCardinalTwiceRemoved (W ∙ (C* ∙ E)) x y z w ⟩\n W ∙ (C* ∙ E) ∙ x ∙ y ∙ w ∙ z ≈⟨ congʳ $ congʳ $ congʳ $ isWarbler (C* ∙ E) x ⟩\n C* ∙ E ∙ x ∙ x ∙ y ∙ w ∙ z ≈⟨ congʳ $ congʳ $ isCardinalOnceRemoved E x x y ⟩\n E ∙ x ∙ y ∙ x ∙ w ∙ z ≈⟨ isEagle x y x w z ⟩\n x ∙ y ∙ (x ∙ w ∙ z) ∎\n }\n\nproblem₁′ : ⦃ _ : HasBluebird ⦄ ⦃ _ : HasCardinal ⦄ ⦃ _ : HasWarbler ⦄ → HasJaybird\nproblem₁′ = record\n { J = B ∙ (B ∙ C) ∙ (W ∙ (B ∙ C ∙ (B ∙ (B ∙ B ∙ B))))\n ; isJaybird = isJaybird ⦃ problem₁″ ⦄\n } where\n instance\n hasEagle = Chapter₁₁.problem₇\n hasCardinalOnceRemoved = Chapter₁₁.problem₃₁\n hasCardinalTwiceRemoved = Chapter₁₁.problem₃₅-C**\n\nproblem₁ : ⦃ _ : HasBluebird ⦄ ⦃ _ : HasThrush ⦄ ⦃ _ : HasMockingbird ⦄ → HasJaybird\nproblem₁ = problem₁′\n where\n instance\n hasCardinal = Chapter₁₁.problem₂₁′\n hasWarbler = Chapter₁₂.problem₇\n\nproblem₂ : ⦃ _ : HasJaybird ⦄ ⦃ _ : HasIdentity ⦄ → HasQuixoticBird\nproblem₂ = record\n { Q₁ = J ∙ I\n ; isQuixoticBird = λ x y z → begin\n J ∙ I ∙ x ∙ y ∙ z ≈⟨ isJaybird I x y z ⟩\n I ∙ x ∙ (I ∙ z ∙ y) ≈⟨ congʳ $ isIdentity x ⟩\n x ∙ (I ∙ z ∙ y) ≈⟨ congˡ $ congʳ $ isIdentity z ⟩\n x ∙ (z ∙ y) ∎\n }\n\nproblem₃ : ⦃ _ : HasQuixoticBird ⦄ ⦃ _ : HasIdentity ⦄ → HasThrush\nproblem₃ = record\n { T = Q₁ ∙ I\n ; isThrush = λ x y → begin\n Q₁ ∙ I ∙ x ∙ y ≈⟨ isQuixoticBird I x y ⟩\n I ∙ (y ∙ x) ≈⟨ isIdentity $ y ∙ x ⟩\n y ∙ x ∎\n }\n\nproblem₄ : ⦃ _ : HasJaybird ⦄ ⦃ _ : HasThrush ⦄ → HasRobin\nproblem₄ = record\n { R = J ∙ T\n ; isRobin = λ x y z → begin\n J ∙ T ∙ x ∙ y ∙ z ≈⟨ isJaybird T x y z ⟩\n T ∙ x ∙ (T ∙ z ∙ y) ≈⟨ isThrush x $ T ∙ z ∙ y ⟩\n T ∙ z ∙ y ∙ x ≈⟨ congʳ $ isThrush z y ⟩\n y ∙ z ∙ x ∎\n }\n\nproblem₅-C* : ⦃ _ : HasCardinal ⦄ ⦃ _ : HasQuixoticBird ⦄ → HasCardinalOnceRemoved\nproblem₅-C* = record\n { C* = C ∙ (Q₁ ∙ C)\n ; isCardinalOnceRemoved = λ x y z w → begin\n C ∙ (Q₁ ∙ C) ∙ x ∙ y ∙ z ∙ w ≈⟨ congʳ $ congʳ $ isCardinal (Q₁ ∙ C) x y ⟩\n Q₁ ∙ C ∙ y ∙ x ∙ z ∙ w ≈⟨ congʳ $ congʳ $ isQuixoticBird C y x ⟩\n C ∙ (x ∙ y) ∙ z ∙ w ≈⟨ isCardinal (x ∙ y) z w ⟩\n x ∙ y ∙ w ∙ z ∎\n }\n\nproblem₅ : ⦃ _ : HasRobin ⦄ ⦃ _ : HasQuixoticBird ⦄ → HasBluebird\nproblem₅ = record\n { B = C* ∙ Q₁\n ; isBluebird = λ x y z → begin\n C* ∙ Q₁ ∙ x ∙ y ∙ z ≈⟨ isCardinalOnceRemoved Q₁ x y z ⟩\n Q₁ ∙ x ∙ z ∙ y ≈⟨ isQuixoticBird x z y ⟩\n x ∙ (y ∙ z) ∎\n } where\n instance\n hasCardinal = Chapter₁₁.problem₂₁\n hasCardinalOnceRemoved = problem₅-C*\n\nproblem₆ : ⦃ _ : HasJaybird ⦄ ⦃ _ : HasBluebird ⦄ ⦃ _ : HasThrush ⦄\n → ∃[ J₁ ] (∀ x y z w → J₁ ∙ x ∙ y ∙ z ∙ w ≈ y ∙ x ∙ (w ∙ x ∙ z))\nproblem₆ =\n ( B ∙ J ∙ T\n , λ x y z w → begin\n B ∙ J ∙ T ∙ x ∙ y ∙ z ∙ w ≈⟨ (congʳ $ congʳ $ congʳ $ isBluebird J T x) ⟩\n J ∙ (T ∙ x) ∙ y ∙ z ∙ w ≈⟨ isJaybird (T ∙ x) y z w ⟩\n T ∙ x ∙ y ∙ (T ∙ x ∙ w ∙ z) ≈⟨ congʳ $ isThrush x y ⟩\n y ∙ x ∙ (T ∙ x ∙ w ∙ z) ≈⟨ congˡ $ congʳ $ isThrush x w ⟩\n y ∙ x ∙ (w ∙ x ∙ z) ∎\n )\n\nproblem₇ : ⦃ _ : HasCardinal ⦄ ⦃ _ : HasThrush ⦄\n → ∃[ J₁ ] (∀ x y z w → J₁ ∙ x ∙ y ∙ z ∙ w ≈ y ∙ x ∙ (w ∙ x ∙ z))\n → HasMockingbird\nproblem₇ (J₁ , isJ₁) = record\n { M = C ∙ (C ∙ (C ∙ J₁ ∙ T) ∙ T) ∙ T\n ; isMockingbird = λ x → begin\n C ∙ (C ∙ (C ∙ J₁ ∙ T) ∙ T) ∙ T ∙ x ≈⟨ isCardinal _ T x ⟩\n C ∙ (C ∙ J₁ ∙ T) ∙ T ∙ x ∙ T ≈⟨ congʳ $ isCardinal _ T x ⟩\n C ∙ J₁ ∙ T ∙ x ∙ T ∙ T ≈⟨ congʳ $ congʳ $ isCardinal J₁ T x ⟩\n J₁ ∙ x ∙ T ∙ T ∙ T ≈⟨ isJ₁ x T T T ⟩\n T ∙ x ∙ (T ∙ x ∙ T) ≈⟨ isThrush x (T ∙ x ∙ T) ⟩\n T ∙ x ∙ T ∙ x ≈⟨ congʳ $ isThrush x T ⟩\n T ∙ x ∙ x ≈⟨ isThrush x x ⟩\n x ∙ x ∎\n }\n\nmodule _ ⦃ _ : HasJaybird ⦄ ⦃ hasIdentity′ : HasIdentity ⦄ where\n private\n instance\n hasQuixoticBird = problem₂\n\n hasThrush : HasThrush\n hasThrush = problem₃\n\n private\n instance\n _ = hasThrush\n hasRobin = problem₄\n\n hasBluebird : HasBluebird\n hasBluebird = problem₅\n\n private\n instance\n _ = hasBluebird\n hasCardinal = Chapter₁₁.problem₂₁′\n\n hasMockingbird : HasMockingbird\n hasMockingbird = problem₇ problem₆\n\n hasIdentity : HasIdentity\n hasIdentity = hasIdentity′\n\n-- Equality of predicates.\ninfix 4 _≐_\n_≐_ : ∀ {a ℓ} {A : Set a} (P Q : Pred A ℓ) → Set (a ⊔ ℓ)\nP ≐ Q = P ⊆ Q × Q ⊆ P\n\nmodule _ ⦃ _ : Extensional ⦄\n ⦃ _ : HasBluebird ⦄ ⦃ _ : HasMockingbird ⦄ ⦃ _ : HasThrush ⦄\n ⦃ _ : HasIdentity ⦄ ⦃ _ : HasJaybird ⦄ where\n private\n ⟨B,M,T,I⟩ = ⟨ B ∷ M ∷ T ∷ I ∷ [] ⟩\n ⟨J,I⟩ = ⟨ J ∷ I ∷ [] ⟩\n\n module _ where\n private\n instance\n hasCardinal = Chapter₁₁.problem₂₁′\n hasWarbler = Chapter₁₂.problem₇\n\n b : B ∈ ⟨B,M,T,I⟩\n b = [ here refl ]\n\n m : M ∈ ⟨B,M,T,I⟩\n m = [ there (here refl) ]\n\n t : T ∈ ⟨B,M,T,I⟩\n t = [ there (there (here refl)) ]\n\n c : C ∈ ⟨B,M,T,I⟩\n c = b ⟨∙⟩ b ⟨∙⟩ t ⟨∙⟩ (b ⟨∙⟩ b ⟨∙⟩ t) ⟨∙⟩ (b ⟨∙⟩ b ⟨∙⟩ t)\n\n w : W ∈ ⟨B,M,T,I⟩\n w = b ⟨∙⟩ (t ⟨∙⟩ (b ⟨∙⟩ m ⟨∙⟩ (b ⟨∙⟩ b ⟨∙⟩ t))) ⟨∙⟩ (b ⟨∙⟩ b ⟨∙⟩ t)\n\n ⟨J,I⟩⊆⟨B,M,T,I⟩ : ⟨J,I⟩ ⊆ ⟨B,M,T,I⟩\n ⟨J,I⟩⊆⟨B,M,T,I⟩ [ here x≈J ] = subst′\n (trans x≈J $ ext′ $ ext′ $ ext′ $ ext′ $ trans\n (isJaybird _ _ _ _)\n (sym $ isJaybird ⦃ problem₁′ ⦄ _ _ _ _))\n (b ⟨∙⟩ (b ⟨∙⟩ c) ⟨∙⟩ (w ⟨∙⟩ (b ⟨∙⟩ c ⟨∙⟩ (b ⟨∙⟩ (b ⟨∙⟩ b ⟨∙⟩ b)))))\n ⟨J,I⟩⊆⟨B,M,T,I⟩ [ there (here x≈I) ] = [ there (there (there (here x≈I))) ]\n ⟨J,I⟩⊆⟨B,M,T,I⟩ (x∈⟨J,I⟩ ⟨∙⟩ y∈⟨J,I⟩ ∣ xy≈z) = ⟨J,I⟩⊆⟨B,M,T,I⟩ x∈⟨J,I⟩ ⟨∙⟩ ⟨J,I⟩⊆⟨B,M,T,I⟩ y∈⟨J,I⟩ ∣ xy≈z\n\n module _ where\n private\n instance\n hasQuixoticBird = problem₂\n hasRobin = problem₄\n hasCardinal = Chapter₁₁.problem₂₁\n hasCardinalOnceRemoved = problem₅-C*\n\n j : J ∈ ⟨J,I⟩\n j = [ here refl ]\n\n i : I ∈ ⟨J,I⟩\n i = [ there (here refl) ]\n\n q₁ : Q₁ ∈ ⟨J,I⟩\n q₁ = j ⟨∙⟩ i\n\n t : T ∈ ⟨J,I⟩\n t = subst′ (ext′ (ext′ (trans (isThrush _ _) (sym (isThrush ⦃ problem₃ ⦄ _ _))))) $\n j ⟨∙⟩ i ⟨∙⟩ i\n\n r : R ∈ ⟨J,I⟩\n r = j ⟨∙⟩ t\n\n c : C ∈ ⟨J,I⟩\n c = r ⟨∙⟩ r ⟨∙⟩ r\n\n c* : C* ∈ ⟨J,I⟩\n c* = c ⟨∙⟩ (q₁ ⟨∙⟩ c)\n\n b : B ∈ ⟨J,I⟩\n b = subst′ (ext′ (ext′ (ext′ (trans (isBluebird _ _ _) (sym (isBluebird ⦃ problem₅ ⦄ _ _ _)))))) $\n c* ⟨∙⟩ q₁\n\n m : M ∈ ⟨J,I⟩\n m = subst′ (ext′ (trans (isMockingbird _) (sym (isMockingbird ⦃ problem₇ problem₆ ⦄ _)))) $\n c ⟨∙⟩ (c ⟨∙⟩ (c ⟨∙⟩ (b ⟨∙⟩ j ⟨∙⟩ t) ⟨∙⟩ t) ⟨∙⟩ t) ⟨∙⟩ t\n\n ⟨J,I⟩⊇⟨B,M,T,I⟩ : ⟨J,I⟩ ⊇ ⟨B,M,T,I⟩\n ⟨J,I⟩⊇⟨B,M,T,I⟩ [ here x≈B ] = subst′ x≈B b\n ⟨J,I⟩⊇⟨B,M,T,I⟩ [ there (here x≈M) ] = subst′ x≈M m\n ⟨J,I⟩⊇⟨B,M,T,I⟩ [ there (there (here x≈T)) ] = subst′ x≈T t\n ⟨J,I⟩⊇⟨B,M,T,I⟩ [ there (there (there (here x≈I))) ] = [ there (here x≈I) ]\n ⟨J,I⟩⊇⟨B,M,T,I⟩ (x∈⟨B,M,T,I⟩ ⟨∙⟩ y∈⟨B,M,T,I⟩ ∣ xy≈z) = ⟨J,I⟩⊇⟨B,M,T,I⟩ x∈⟨B,M,T,I⟩ ⟨∙⟩ ⟨J,I⟩⊇⟨B,M,T,I⟩ y∈⟨B,M,T,I⟩ ∣ xy≈z\n\n ⟨J,I⟩≐⟨B,M,T,I⟩ : ⟨ J ∷ I ∷ [] ⟩ ≐ ⟨ B ∷ M ∷ T ∷ I ∷ [] ⟩\n ⟨J,I⟩≐⟨B,M,T,I⟩ = (⟨J,I⟩⊆⟨B,M,T,I⟩ , ⟨J,I⟩⊇⟨B,M,T,I⟩)\n", "meta": {"hexsha": "cc4013bfc2885663eb664d40624408c4680d66ae", "size": 8146, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Mockingbird/Problems/Chapter19.agda", "max_stars_repo_name": "splintah/combinatory-logic", "max_stars_repo_head_hexsha": "df8bf877e60b3059532c54a247a36a3d83cd55b0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, 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"1. YES\n2. NO", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.3412328005167754}} {"text": "module nodcap.Base where\n\nopen import Data.Nat as ℕ using (ℕ; suc; zero)\nopen import Data.Pos as ℕ⁺ using (ℕ⁺; suc; _+_)\nopen import Data.List as L using (List; []; _∷_; _++_)\nopen import Data.List.Any using (Any; here; there)\nopen import Data.List.Any.Membership.Propositional using (_∈_; _∼[_]_; bag)\nopen import Data.Product using (_×_; _,_; uncurry; map)\nopen import Data.Sum using (_⊎_; inj₁; inj₂)\nopen import Function using (id; _$_)\nopen import Relation.Binary.PropositionalEquality as P using (_≡_)\n\n-- Types.\n\ndata Type : Set where\n 𝟏 : Type\n ⊥ : Type\n 𝟎 : Type\n ⊤ : Type\n _⊗_ : (A B : Type) → Type\n _⅋_ : (A B : Type) → Type\n _⊕_ : (A B : Type) → Type\n _&_ : (A B : Type) → Type\n ![_]_ : (n : ℕ⁺) (A : Type) → Type\n ?[_]_ : (n : ℕ⁺) (A : Type) → Type\n\n\n-- Duality.\n\n_^ : Type → Type\n𝟏 ^ = ⊥\n⊥ ^ = 𝟏\n𝟎 ^ = ⊤\n⊤ ^ = 𝟎\n(A ⊗ B) ^ = (A ^) ⅋ (B ^)\n(A ⅋ B) ^ = (A ^) ⊗ (B ^)\n(A ⊕ B) ^ = (A ^) & (B ^)\n(A & B) ^ = (A ^) ⊕ (B ^)\n(![ n ] A) ^ = ?[ n ] (A ^)\n(?[ n ] A) ^ = ![ n ] (A ^)\n\n^-inv : (A : Type) → A ^ ^ ≡ A\n^-inv 𝟏 = P.refl\n^-inv ⊥ = P.refl\n^-inv 𝟎 = P.refl\n^-inv ⊤ = P.refl\n^-inv (A ⊗ B) = P.cong₂ _⊗_ (^-inv A) (^-inv B)\n^-inv (A ⅋ B) = P.cong₂ _⅋_ (^-inv A) (^-inv B)\n^-inv (A ⊕ B) = P.cong₂ _⊕_ (^-inv A) (^-inv B)\n^-inv (A & B) = P.cong₂ _&_ (^-inv A) (^-inv B)\n^-inv (![ n ] A) = P.cong ![ n ]_ (^-inv A)\n^-inv (?[ n ] A) = P.cong ?[ n ]_ (^-inv A)\n\n-- Lollipop.\n\n_⊸_ : (A B : Type) → Type\nA ⊸ B = (A ^) ⅋ B\n\n\n-- Polarity.\n\ndata Pos : (A : Type) → Set where\n 𝟎 : Pos 𝟎\n 𝟏 : Pos 𝟏\n _⊗_ : (A B : Type) → Pos (A ⊗ B)\n _⊕_ : (A B : Type) → Pos (A ⊕ B)\n ![_]_ : (n : ℕ⁺) (A : Type) → Pos (![ n ] A)\n\ndata Neg : (A : Type) → Set where\n ⊥ : Neg ⊥\n ⊤ : Neg ⊤\n _⅋_ : (A B : Type) → Neg (A ⅋ B)\n _&_ : (A B : Type) → Neg (A & B)\n ?[_]_ : (n : ℕ⁺) (A : Type) → Neg (?[ n ] A)\n\npol? : (A : Type) → Pos A ⊎ Neg A\npol? 𝟏 = inj₁ 𝟏\npol? ⊥ = inj₂ ⊥\npol? 𝟎 = inj₁ 𝟎\npol? ⊤ = inj₂ ⊤\npol? (A ⊗ B) = inj₁ (A ⊗ B)\npol? (A ⅋ B) = inj₂ (A ⅋ B)\npol? (A ⊕ B) = inj₁ (A ⊕ B)\npol? (A & B) = inj₂ (A & B)\npol? (![ n ] A) = inj₁ (![ n ] A)\npol? (?[ n ] A) = inj₂ (?[ n ] A)\n\n^-posneg : {A : Type} (P : Pos A) → Neg (A ^)\n^-posneg 𝟎 = ⊤\n^-posneg 𝟏 = ⊥\n^-posneg (A ⊗ B) = (A ^) ⅋ (B ^)\n^-posneg (A ⊕ B) = (A ^) & (B ^)\n^-posneg (![ n ] A) = ?[ n ] (A ^)\n\n^-negpos : {A : Type} (N : Neg A) → Pos (A ^)\n^-negpos ⊥ = 𝟏\n^-negpos ⊤ = 𝟎\n^-negpos (A ⅋ B) = (A ^) ⊗ (B ^)\n^-negpos (A & B) = (A ^) ⊕ (B ^)\n^-negpos (?[ n ] A) = ![ n ] (A ^)\n\n\n-- Environments.\n\nEnvironment : Set\nEnvironment = List Type\n\n-- Injectivity.\n\nprivate\n infix 10 _≈_\n\n _≈_ : Type → Type → Set\n A ≈ B = A ≡ B\n\n⊗-inj : {A B C D : Type} → A ⊗ B ≈ C ⊗ D → A ≈ C × B ≈ D\n⊗-inj P.refl = P.refl , P.refl\n\n⅋-inj : {A B C D : Type} → A ⅋ B ≈ C ⅋ D → A ≈ C × B ≈ D\n⅋-inj P.refl = P.refl , P.refl\n\n⊕-inj : {A B C D : Type} → A ⊕ B ≈ C ⊕ D → A ≈ C × B ≈ D\n⊕-inj P.refl = P.refl , P.refl\n\n&-inj : {A B C D : Type} → A & B ≈ C & D → A ≈ C × B ≈ D\n&-inj P.refl = P.refl , P.refl\n\n!-inj : {A B : Type} {m n : ℕ⁺} → ![ m ] A ≈ ![ n ] B → m ≡ n × A ≈ B\n!-inj P.refl = P.refl , P.refl\n\n?-inj : {A B : Type} {m n : ℕ⁺} → ?[ m ] A ≈ ?[ n ] B → m ≡ n × A ≈ B\n?-inj P.refl = P.refl , P.refl\n\n-- This is one of those proofs for which I wish Agda had tactics.\n\n^-inj : {A B : Type} → A ^ ≈ B ^ → A ≈ B\n^-inj {A = 𝟏} {B = 𝟏} p = P.refl\n^-inj {A = 𝟏} {B = ⊥} ()\n^-inj {A = 𝟏} {B = 𝟎} ()\n^-inj {A = 𝟏} {B = ⊤} ()\n^-inj {A = 𝟏} {B = C ⊗ D} ()\n^-inj {A = 𝟏} {B = C ⅋ D} ()\n^-inj {A = 𝟏} {B = C ⊕ D} ()\n^-inj {A = 𝟏} {B = C & D} ()\n^-inj {A = 𝟏} {B = ![ n ] C} ()\n^-inj {A = 𝟏} {B = ?[ n ] C} ()\n^-inj {A = ⊥} {B = 𝟏} ()\n^-inj {A = ⊥} {B = ⊥} p = P.refl\n^-inj {A = ⊥} {B = 𝟎} ()\n^-inj {A = ⊥} {B = ⊤} ()\n^-inj {A = ⊥} {B = C ⊗ D} ()\n^-inj {A = ⊥} {B = C ⅋ D} ()\n^-inj {A = ⊥} {B = C ⊕ D} ()\n^-inj {A = ⊥} {B = C & D} ()\n^-inj {A = ⊥} {B = ![ n ] C} ()\n^-inj {A = ⊥} {B = ?[ n ] C} ()\n^-inj {A = 𝟎} {B = 𝟏} ()\n^-inj {A = 𝟎} {B = ⊥} ()\n^-inj {A = 𝟎} {B = 𝟎} p = P.refl\n^-inj {A = 𝟎} {B = ⊤} ()\n^-inj {A = 𝟎} {B = C ⊗ D} ()\n^-inj {A = 𝟎} {B = C ⅋ D} ()\n^-inj {A = 𝟎} {B = C ⊕ D} ()\n^-inj {A = 𝟎} {B = C & D} ()\n^-inj {A = 𝟎} {B = ![ n ] C} ()\n^-inj {A = 𝟎} {B = ?[ n ] C} ()\n^-inj {A = ⊤} {B = 𝟏} ()\n^-inj {A = ⊤} {B = ⊥} ()\n^-inj {A = ⊤} {B = 𝟎} ()\n^-inj {A = ⊤} {B = ⊤} p = P.refl\n^-inj {A = ⊤} {B = C ⊗ D} ()\n^-inj {A = ⊤} {B = C ⅋ D} ()\n^-inj {A = ⊤} {B = C ⊕ D} ()\n^-inj {A = ⊤} {B = C & D} ()\n^-inj {A = ⊤} {B = ![ n ] C} ()\n^-inj {A = ⊤} {B = ?[ n ] C} ()\n^-inj {A = A ⊗ B} {B = 𝟏} ()\n^-inj {A = A ⊗ B} {B = ⊥} ()\n^-inj {A = A ⊗ B} {B = 𝟎} ()\n^-inj {A = A ⊗ B} {B = ⊤} ()\n^-inj {A = A ⊗ B} {B = C ⊗ D} p = uncurry (P.cong₂ _⊗_) (map ^-inj ^-inj (⅋-inj p))\n^-inj {A = A ⊗ B} {B = C ⅋ D} ()\n^-inj {A = A ⊗ B} {B = C ⊕ D} ()\n^-inj {A = A ⊗ B} {B = C & D} ()\n^-inj {A = A ⊗ B} {B = ![ n ] C} ()\n^-inj {A = A ⊗ B} {B = ?[ n ] C} ()\n^-inj {A = A ⅋ B} {B = 𝟏} ()\n^-inj {A = A ⅋ B} {B = ⊥} ()\n^-inj {A = A ⅋ B} {B = 𝟎} ()\n^-inj {A = A ⅋ B} {B = ⊤} ()\n^-inj {A = A ⅋ B} {B = C ⊗ D} ()\n^-inj {A = A ⅋ B} {B = C ⅋ D} p = uncurry (P.cong₂ _⅋_) (map ^-inj ^-inj (⊗-inj p))\n^-inj {A = A ⅋ B} {B = C ⊕ D} ()\n^-inj {A = A ⅋ B} {B = C & D} ()\n^-inj {A = A ⅋ B} {B = ![ n ] C} ()\n^-inj {A = A ⅋ B} {B = ?[ n ] C} ()\n^-inj {A = A ⊕ B} {B = 𝟏} ()\n^-inj {A = A ⊕ B} {B = ⊥} ()\n^-inj {A = A ⊕ B} {B = 𝟎} ()\n^-inj {A = A ⊕ B} {B = ⊤} ()\n^-inj {A = A ⊕ B} {B = C ⊗ D} ()\n^-inj {A = A ⊕ B} {B = C ⅋ D} ()\n^-inj {A = A ⊕ B} {B = C ⊕ D} p = uncurry (P.cong₂ _⊕_) (map ^-inj ^-inj (&-inj p))\n^-inj {A = A ⊕ B} {B = C & D} ()\n^-inj {A = A ⊕ B} {B = ![ n ] C} ()\n^-inj {A = A ⊕ B} {B = ?[ n ] C} ()\n^-inj {A = A & B} {B = 𝟏} ()\n^-inj {A = A & B} {B = ⊥} ()\n^-inj {A = A & B} {B = 𝟎} ()\n^-inj {A = A & B} {B = ⊤} ()\n^-inj {A = A & B} {B = C ⊗ D} ()\n^-inj {A = A & B} {B = C ⅋ D} ()\n^-inj {A = A & B} {B = C ⊕ D} ()\n^-inj {A = A & B} {B = C & D} p = uncurry (P.cong₂ _&_) (map ^-inj ^-inj (⊕-inj p))\n^-inj {A = A & B} {B = ![ n ] C} ()\n^-inj {A = A & B} {B = ?[ n ] C} ()\n^-inj {A = ![ m ] A} {B = 𝟏} ()\n^-inj {A = ![ m ] A} {B = ⊥} ()\n^-inj {A = ![ m ] A} {B = 𝟎} ()\n^-inj {A = ![ m ] A} {B = ⊤} ()\n^-inj {A = ![ m ] A} {B = C ⊗ D} ()\n^-inj {A = ![ m ] A} {B = C ⅋ D} ()\n^-inj {A = ![ m ] A} {B = C ⊕ D} ()\n^-inj {A = ![ m ] A} {B = C & D} ()\n^-inj {A = ![ m ] A} {B = ![ n ] C} p = uncurry (P.cong₂ ![_]_) (map id ^-inj (?-inj p))\n^-inj {A = ![ m ] A} {B = ?[ n ] C} ()\n^-inj {A = ?[ m ] A} {B = 𝟏} ()\n^-inj {A = ?[ m ] A} {B = ⊥} ()\n^-inj {A = ?[ m ] A} {B = 𝟎} ()\n^-inj {A = ?[ m ] A} {B = ⊤} ()\n^-inj {A = ?[ m ] A} {B = C ⊗ D} ()\n^-inj {A = ?[ m ] A} {B = C ⅋ D} ()\n^-inj {A = ?[ m ] A} {B = C ⊕ D} ()\n^-inj {A = ?[ m ] A} {B = C & D} ()\n^-inj {A = ?[ m ] A} {B = ![ n ] C} ()\n^-inj {A = ?[ m ] A} {B = ?[ n ] C} p = uncurry (P.cong₂ ?[_]_) (map id ^-inj (!-inj p))\n", "meta": {"hexsha": "dac1bd6e59c53b7a21b7783a55448d9a928843a4", "size": 7588, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "src/cpnd1/nodcap/Base.agda", "max_stars_repo_name": "wenkokke/nodcap", "max_stars_repo_head_hexsha": "fb5e78d6182276e4d93c4c0e0d563b6b027bc5c2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2018-09-05T08:58:11.000Z", "max_stars_repo_stars_event_max_datetime": "2019-09-24T20:16:35.000Z", "max_issues_repo_path": "src/cpnd1/nodcap/Base.agda", "max_issues_repo_name": "pepijnkokke/nodcap", "max_issues_repo_head_hexsha": "fb5e78d6182276e4d93c4c0e0d563b6b027bc5c2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/cpnd1/nodcap/Base.agda", "max_forks_repo_name": "pepijnkokke/nodcap", "max_forks_repo_head_hexsha": "fb5e78d6182276e4d93c4c0e0d563b6b027bc5c2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-09-05T08:58:13.000Z", "max_forks_repo_forks_event_max_datetime": "2018-09-05T08:58:13.000Z", "avg_line_length": 32.0168776371, "max_line_length": 90, "alphanum_fraction": 0.3223510807, "num_tokens": 3974, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7401743620390162, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.34123280051677535}} {"text": "{-# OPTIONS --safe #-}\n\nmodule Definition.Typed.Consequences.NeTypeEq where\n\nopen import Definition.Untyped\nopen import Definition.Typed\nopen import Definition.Typed.Properties\nopen import Definition.Typed.Weakening\nopen import Definition.Typed.Consequences.Syntactic\nopen import Definition.Typed.Consequences.Injectivity\nopen import Definition.Typed.Consequences.Substitution\n\nopen import Tools.Product\nimport Tools.PropositionalEquality as PE\n\n-- to be moved in Untyped\n\ntypelevel-injectivity : ∀ {r r' l l'} → [ r , l ] PE.≡ [ r' , l' ] → r PE.≡ r' × l PE.≡ l'\ntypelevel-injectivity PE.refl = PE.refl , PE.refl\n\n-- Helper function for the same variable instance of a context have equal types.\nvarTypeEq′ : ∀ {n R rR T rT Γ} → n ∷ R ^ rR ∈ Γ → n ∷ T ^ rT ∈ Γ → R PE.≡ T × rR PE.≡ rT\nvarTypeEq′ here here = PE.refl , PE.refl\nvarTypeEq′ (there n∷R) (there n∷T) with varTypeEq′ n∷R n∷T\n... | PE.refl , PE.refl = PE.refl , PE.refl\n\n-- The same variable instance of a context have equal types.\nvarTypeEq : ∀ {x A B rA rB Γ} → Γ ⊢ A ^ rA → Γ ⊢ B ^ rB\n → x ∷ A ^ rA ∈ Γ\n → x ∷ B ^ rB ∈ Γ\n → Γ ⊢ A ≡ B ^ rA × rA PE.≡ rB\nvarTypeEq A B x∷A x∷B with varTypeEq′ x∷A x∷B\n... | PE.refl , PE.refl = refl A , PE.refl\n\n\n\n-- The same neutral term have equal types.\n-- to use this with different relevances rA rB we need unicity of relevance for types\nneTypeEq : ∀ {t A B rA lA lA' Γ} → Neutral t → Γ ⊢ t ∷ A ^ [ rA , lA ] → Γ ⊢ t ∷ B ^ [ rA , lA' ] →\n lA PE.≡ lA' × Γ ⊢ A ≡ B ^ [ rA , lA ]\nneTypeEq (var x) (var x₁ x₂) (var x₃ x₄) =\n let V , e = varTypeEq (syntacticTerm (var x₃ x₂)) (syntacticTerm (var x₃ x₄)) x₂ x₄\n _ , el = typelevel-injectivity e\n in el , V \nneTypeEq (∘ₙ neT) (t∷A ∘ⱼ t∷A₁) (t∷B ∘ⱼ t∷B₁) with neTypeEq neT t∷A t∷B\n... | e , q = let _ , _ , _ , elG , w = injectivity q\n in PE.cong _ elG , substTypeEq w (genRefl t∷A₁)\nneTypeEq (natrecₙ neT) (natrecⱼ x t∷A t∷A₁ t∷A₂) (natrecⱼ x₁ t∷B t∷B₁ t∷B₂) =\n PE.refl , refl (substType x₁ t∷B₂)\nneTypeEq Emptyrecₙ (Emptyrecⱼ x t∷A) (Emptyrecⱼ x₁ t∷B) =\n PE.refl , refl x₁\nneTypeEq (Idₙ X) (Idⱼ Y Y₁ Y₂) (Idⱼ Z Z₁ Z₂) =\n let e , q = neTypeEq X Y Z\n el = next-inj e\n in e , PE.subst (λ l → _ ⊢ _ ≡ SProp l ^ _) el (refl (Ugenⱼ (wfTerm Y) ) )\nneTypeEq (Idℕₙ X) (Idⱼ Y Y₁ Y₂) (Idⱼ Z Z₁ Z₂) =\n let e , q = neTypeEq X Y₁ Z₁\n el = ιinj e\n in PE.cong next el , PE.subst (λ l → _ ⊢ _ ≡ SProp l ^ _) el (refl (Ugenⱼ (wfTerm Y) ) )\nneTypeEq (Idℕ0ₙ X) (Idⱼ Y Y₁ Y₂) (Idⱼ Z Z₁ Z₂) =\n let e , q = neTypeEq X Y₂ Z₂\n el = ιinj e\n in PE.cong next el , PE.subst (λ l → _ ⊢ _ ≡ SProp l ^ _) el (refl (Ugenⱼ (wfTerm Y) ) )\nneTypeEq (IdℕSₙ X) (Idⱼ Y Y₁ Y₂) (Idⱼ Z Z₁ Z₂) =\n let e , q = neTypeEq X Y₂ Z₂\n el = ιinj e\n in PE.cong next el , PE.subst (λ l → _ ⊢ _ ≡ SProp l ^ _) el (refl (Ugenⱼ (wfTerm Y) ) )\nneTypeEq (IdUₙ X) (Idⱼ Y Y₁ Y₂) (Idⱼ Z Z₁ Z₂) =\n let e , q = neTypeEq X Y₁ Z₁\n el = ιinj e\n in PE.cong next el , PE.subst (λ l → _ ⊢ _ ≡ SProp l ^ _) el (refl (Ugenⱼ (wfTerm Y) ) )\nneTypeEq (IdUℕₙ X) (Idⱼ Y Y₁ Y₂) (Idⱼ Z Z₁ Z₂) =\n let e , q = neTypeEq X Y₂ Z₂\n el = ιinj e\n in PE.cong next el , PE.subst (λ l → _ ⊢ _ ≡ SProp l ^ _) el (refl (Ugenⱼ (wfTerm Y) ) )\nneTypeEq (IdUΠₙ X) (Idⱼ Y Y₁ Y₂) (Idⱼ Z Z₁ Z₂) =\n let e , q = neTypeEq X Y₂ Z₂\n el = ιinj e\n in PE.cong next el , PE.subst (λ l → _ ⊢ _ ≡ SProp l ^ _) el (refl (Ugenⱼ (wfTerm Y) ) )\nneTypeEq X (castⱼ Y Y₁ Y₂ Y₃) (castⱼ Z Z₁ Z₂ Z₃) = PE.refl , refl (univ Y₁) \nneTypeEq x (conv t∷A x₁) t∷B = \n let e , q = neTypeEq x t∷A t∷B\n in e , trans (sym x₁) q \nneTypeEq x t∷A (conv t∷B x₃) =\n let e , q = neTypeEq x t∷A t∷B\n in e , trans q (PE.subst (λ l → _ ⊢ _ ≡ _ ^ [ _ , l ]) (PE.sym e) x₃) \n\n\nnatTypeEq : ∀ {A rA lA Γ} → Γ ⊢ ℕ ∷ A ^ [ rA , lA ] → rA PE.≡ ! × lA PE.≡ ι ¹ × Γ ⊢ A ≡ U ⁰ ^ [ ! , ι ¹ ]\nnatTypeEq (ℕⱼ x) = PE.refl , PE.refl , refl (univ (univ 0<1 x))\nnatTypeEq (conv X x) = let eqrA , eqlA , eqAU = natTypeEq X in eqrA , eqlA ,\n trans (sym (PE.subst (λ l → _ ⊢ _ ≡ _ ^ [ _ , l ] ) eqlA (PE.subst (λ r → _ ⊢ _ ≡ _ ^ [ r , _ ]) eqrA x))) eqAU \n\nemptyTypeEq : ∀ {A rA lA Γ l} → Γ ⊢ Empty l ∷ A ^ [ rA , lA ] →\n rA PE.≡ ! × lA PE.≡ next l × Γ ⊢ A ≡ SProp l ^ [ ! , next l ]\nemptyTypeEq (Emptyⱼ x) = PE.refl , PE.refl , refl (Ugenⱼ x) \nemptyTypeEq (conv X x) = let eqrA , eqlA , eqAU = emptyTypeEq X in eqrA , eqlA , \n trans (sym (PE.subst (λ l → _ ⊢ _ ≡ _ ^ [ _ , l ] ) eqlA (PE.subst (λ r → _ ⊢ _ ≡ _ ^ [ r , _ ]) eqrA x))) eqAU \n\n", "meta": {"hexsha": "3ba649135ab3977eff47b402088c244446d6924d", "size": 4429, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Definition/Typed/Consequences/NeTypeEq.agda", "max_stars_repo_name": "CoqHott/logrel-mltt", "max_stars_repo_head_hexsha": "e0eeebc4aa5ed791ce3e7c0dc9531bd113dfcc04", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-06-21T08:39:01.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-17T16:13:53.000Z", "max_issues_repo_path": "Definition/Typed/Consequences/NeTypeEq.agda", "max_issues_repo_name": "CoqHott/logrel-mltt", "max_issues_repo_head_hexsha": "e0eeebc4aa5ed791ce3e7c0dc9531bd113dfcc04", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Definition/Typed/Consequences/NeTypeEq.agda", "max_forks_repo_name": "CoqHott/logrel-mltt", "max_forks_repo_head_hexsha": "e0eeebc4aa5ed791ce3e7c0dc9531bd113dfcc04", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2022-01-26T14:55:51.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-15T19:42:19.000Z", "avg_line_length": 44.29, "max_line_length": 114, "alphanum_fraction": 0.5813953488, "num_tokens": 2085, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3410480492395055}} {"text": "-- Andreas, 2017-10-04, issue #2752, report and test case by nad\n--\n-- Problem was: instance does not distribute into mutual blocks.\n\nopen import Agda.Builtin.List\nopen import Agda.Builtin.Size\n\nmutual\n\n data Rose (i : Size) (A : Set) : Set where\n node : List (Rose′ i A) → Rose i A\n\n data Rose′ (i : Size) (A : Set) : Set where\n delay : {j : Size< i} → Rose j A → Rose′ i A\n\nrecord Map (F : Set → Set) : Set₁ where\n field\n map : {A B : Set} → (A → B) → F A → F B\n\nopen Map ⦃ … ⦄ public\n\ninstance\n\n Map-List : Map List\n Map.map Map-List = λ where\n f [] → []\n f (x ∷ xs) → f x ∷ map f xs\n\ninstance\n\n mutual\n\n Map-Rose : ∀ {i} → Map (Rose i)\n Map.map Map-Rose f (node xs) = node (map (map f) xs)\n\n Map-Rose′ : ∀ {i} → Map (Rose′ i)\n Map.map Map-Rose′ f (delay t) = delay (map f t)\n\n-- Was: unresolved instance arguments.\n\n-- Should succeed.\n", "meta": {"hexsha": "58fffb185baab7e1a62edcbdb245d559832ef2d2", "size": 878, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/Succeed/Issue2752.agda", "max_stars_repo_name": "shlevy/agda", "max_stars_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2019-10-29T09:40:30.000Z", "max_stars_repo_stars_event_max_datetime": "2020-09-20T00:28:57.000Z", "max_issues_repo_path": "test/Succeed/Issue2752.agda", "max_issues_repo_name": "shlevy/agda", "max_issues_repo_head_hexsha": "ed8ac6f4062ea8a20fa0f62d5db82d4e68278338", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2018-11-14T15:31:44.000Z", "max_issues_repo_issues_event_max_datetime": "2019-04-01T19:39:26.000Z", "max_forks_repo_path": "test/Succeed/Issue2752.agda", "max_forks_repo_name": "Agda-zh/agda", "max_forks_repo_head_hexsha": "231d6ad8e77b67ff8c4b1cb35a6c31ccd988c3e9", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-03-05T20:02:38.000Z", "max_forks_repo_forks_event_max_datetime": "2019-03-05T20:02:38.000Z", "avg_line_length": 20.9047619048, "max_line_length": 64, "alphanum_fraction": 0.5797266515, "num_tokens": 306, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043816, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.341048041414993}} {"text": "module empty where\n\nopen import level\n\n----------------------------------------------------------------------\n-- datatypes\n----------------------------------------------------------------------\n\ndata ⊥ {ℓ : Level} : Set ℓ where\n\n----------------------------------------------------------------------\n-- syntax\n----------------------------------------------------------------------\n\n----------------------------------------------------------------------\n-- theorems\n----------------------------------------------------------------------\n⊥-elim : ∀{ℓ} → ⊥ {ℓ} → ∀{ℓ'}{P : Set ℓ'} → P\n⊥-elim ()\n\n", "meta": {"hexsha": "fd2cb85734e977f589c421f3bb3bd74135151bf0", "size": 593, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "empty.agda", "max_stars_repo_name": "heades/AUGL", "max_stars_repo_head_hexsha": "b33c6a59d664aed46cac8ef77d34313e148fecc2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "empty.agda", "max_issues_repo_name": "heades/AUGL", "max_issues_repo_head_hexsha": "b33c6a59d664aed46cac8ef77d34313e148fecc2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "empty.agda", "max_forks_repo_name": "heades/AUGL", "max_forks_repo_head_hexsha": "b33c6a59d664aed46cac8ef77d34313e148fecc2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.2380952381, "max_line_length": 70, "alphanum_fraction": 0.1517706577, "num_tokens": 83, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3409323905448783}} {"text": "------------------------------------------------------------------------\n-- The Agda standard library\n--\n-- An irrelevant version of ⊥-elim\n------------------------------------------------------------------------\n\n{-# OPTIONS --without-K --safe #-}\n\nmodule Data.Empty.Irrelevant where\n\nopen import Data.Empty hiding (⊥-elim)\n\n⊥-elim : ∀ {w} {Whatever : Set w} → .⊥ → Whatever\n⊥-elim ()\n", "meta": {"hexsha": "3cc0d65669156dee5eede5eab1aa0114b08b569a", "size": 386, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "test/asset/agda-stdlib-1.0/Data/Empty/Irrelevant.agda", "max_stars_repo_name": "omega12345/agda-mode", "max_stars_repo_head_hexsha": "0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2020-10-07T12:07:53.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-10T21:41:32.000Z", "max_issues_repo_path": "test/asset/agda-stdlib-1.0/Data/Empty/Irrelevant.agda", "max_issues_repo_name": "omega12345/agda-mode", "max_issues_repo_head_hexsha": "0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "test/asset/agda-stdlib-1.0/Data/Empty/Irrelevant.agda", "max_forks_repo_name": "omega12345/agda-mode", "max_forks_repo_head_hexsha": "0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-11-04T06:54:45.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-04T06:54:45.000Z", "avg_line_length": 25.7333333333, "max_line_length": 72, "alphanum_fraction": 0.3989637306, "num_tokens": 80, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3409323905448782}} {"text": "{-# OPTIONS --safe --warning=error #-}\n\nopen import Numbers.Naturals.Semiring\nopen import Numbers.Naturals.Order\nopen import Sets.FinSet.Definition\nopen import LogicalFormulae\nopen import Numbers.Naturals.WithK\nopen import Sets.FinSet.Lemmas\n\nmodule Sets.FinSetWithK where\n\nprivate\n sgEq : {l m : _} {L : Set l} → {pr : L → Set m} → {a b : Sg L pr} → (underlying a ≡ underlying b) → ({c : L} → (r s : pr c) → r ≡ s) → (a ≡ b)\n sgEq {l} {m} {L} {prop} {(a , b1)} {(.a , b)} refl pr2 with pr2 {a} b b1\n sgEq {l} {m} {L} {prop} {(a , b1)} {(.a , .b1)} refl pr2 | refl = refl\n\nfinNotEqualsRefl : {n : ℕ} {a b : FinSet (succ n)} → (p1 p2 : FinNotEquals a b) → p1 ≡ p2\nfinNotEqualsRefl {.1} {.fzero} {.(fsucc fzero)} (fne2 .fzero .(fsucc fzero) (inl (refl ,, refl))) (fne2 .fzero .(fsucc fzero) (inl (refl ,, refl))) = refl\nfinNotEqualsRefl {.1} {.fzero} {.(fsucc fzero)} (fne2 .fzero .(fsucc fzero) (inl (refl ,, refl))) (fne2 .fzero .(fsucc fzero) (inr (() ,, snd)))\nfinNotEqualsRefl {.1} {.(fsucc fzero)} {.fzero} (fne2 .(fsucc fzero) .fzero (inr (refl ,, refl))) (fne2 .(fsucc fzero) .fzero (inl (() ,, snd)))\nfinNotEqualsRefl {.1} {.(fsucc fzero)} {.fzero} (fne2 .(fsucc fzero) .fzero (inr (refl ,, refl))) (fne2 .(fsucc fzero) .fzero (inr (refl ,, refl))) = refl\nfinNotEqualsRefl {.(succ (succ _))} {.fzero} {.(fsucc a)} (fneN .fzero .(fsucc a) (inl (inl (refl ,, (a , refl))))) (fneN .fzero .(fsucc a) (inl (inl (refl ,, (.a , refl))))) = refl\nfinNotEqualsRefl {.(succ (succ _))} {.fzero} {.(fsucc a)} (fneN .fzero .(fsucc a) (inl (inl (refl ,, (a , refl))))) (fneN .fzero .(fsucc a) (inl (inr ((a₁ , ()) ,, snd))))\nfinNotEqualsRefl {.(succ (succ _))} {.fzero} {.(fsucc a)} (fneN .fzero .(fsucc a) (inl (inl (refl ,, (a , refl))))) (fneN .fzero .(fsucc a) (inr ((fst ,, snd) , b))) = exFalso q\n where\n p : fzero ≡ fsucc fst\n p = _&_&_.one b\n q : False\n q with p\n ... | ()\nfinNotEqualsRefl {.(succ (succ _))} {.(fsucc a)} {.fzero} (fneN .(fsucc a) .fzero (inl (inr ((a , refl) ,, refl)))) (fneN .(fsucc a) .fzero (inl (inl (() ,, snd))))\nfinNotEqualsRefl {.(succ (succ _))} {.(fsucc a)} {.fzero} (fneN .(fsucc a) .fzero (inl (inr ((a , refl) ,, refl)))) (fneN .(fsucc a) .fzero (inl (inr ((.a , refl) ,, refl)))) = refl\nfinNotEqualsRefl {.(succ (succ _))} {.(fsucc a)} {.fzero} (fneN .(fsucc a) .fzero (inl (inr ((a , refl) ,, refl)))) (fneN .(fsucc a) .fzero (inr ((fst ,, snd) , b))) = exFalso q\n where\n p : fzero ≡ fsucc snd\n p = _&_&_.two b\n q : False\n q with p\n ... | ()\nfinNotEqualsRefl {.(succ (succ _))} {a} {b} (fneN a b (inr (record { fst = fst ; snd = snd₁ } , snd))) (fneN .a .b (inl (inl (fst1 ,, snd₂)))) = exFalso q\n where\n p : fzero ≡ fsucc fst\n p = transitivity (equalityCommutative fst1) (_&_&_.one snd)\n q : False\n q with p\n ... | ()\nfinNotEqualsRefl {.(succ (succ _))} {a} {b} (fneN a b (inr (record { fst = fst ; snd = snd1 } , snd))) (fneN .a .b (inl (inr (fst₁ ,, snd2)))) = exFalso q\n where\n p : fzero ≡ fsucc snd1\n p = transitivity (equalityCommutative snd2) (_&_&_.two snd)\n q : False\n q with p\n ... | ()\nfinNotEqualsRefl {.(succ (succ _))} {.fzero} {b} (fneN fzero b (inr (record { fst = fst ; snd = snd₁ } , snd))) (fneN .fzero .b (inr ((fst1 ,, snd₂) , b1))) = exFalso q\n where\n p : fzero ≡ fsucc fst1\n p = _&_&_.one b1\n q : False\n q with p\n ... | ()\nfinNotEqualsRefl {.(succ (succ _))} {.(fsucc a)} {.fzero} (fneN (fsucc a) fzero (inr (record { fst = fst ; snd = snd₁ } , snd))) (fneN .(fsucc a) .fzero (inr ((fst₁ ,, snd2) , b1))) = exFalso q\n where\n p : fzero ≡ fsucc snd2\n p = _&_&_.two b1\n q : False\n q with p\n ... | ()\nfinNotEqualsRefl {.(succ (succ _))} {.(fsucc a)} {.(fsucc b)} (fneN (fsucc a) (fsucc b) (inr (record { fst = fst ; snd = snd1 } , snd))) (fneN .(fsucc a) .(fsucc b) (inr ((fst1 ,, snd2) , b1))) = ans\n where\n t : a ≡ fst1\n t = fsuccInjective (_&_&_.one b1)\n t' : a ≡ fst\n t' = fsuccInjective (_&_&_.one snd)\n u : b ≡ snd1\n u = fsuccInjective (_&_&_.two snd)\n u' : b ≡ snd2\n u' = fsuccInjective (_&_&_.two b1)\n equality : {c : FinSet (succ (succ _)) && FinSet (succ (succ _))} → (r1 s : (fsucc a ≡ fsucc (_&&_.fst c)) & (fsucc b ≡ fsucc (_&&_.snd c)) & FinNotEquals (_&&_.fst c) (_&&_.snd c)) → r1 ≡ s\n equality record { one = refl ; two = refl ; three = q } record { one = refl ; two = refl ; three = q' } = applyEquality (λ t → record { one = refl ; two = refl ; three = t }) (finNotEqualsRefl q q')\n r : (fst ,, snd1) ≡ (fst1 ,, snd2)\n r rewrite equalityCommutative t | equalityCommutative t' | equalityCommutative u | equalityCommutative u' = refl\n ans : fneN (fsucc a) (fsucc b) (inr ((fst ,, snd1) , snd)) ≡ fneN (fsucc a) (fsucc b) (inr ((fst1 ,, snd2) , b1))\n ans = applyEquality (λ t → fneN (fsucc a) (fsucc b) (inr t)) (sgEq r equality)\n", "meta": {"hexsha": "77bac63511cd8339a26d2c871e0ed50c5d47b37f", "size": 4841, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "Sets/FinSetWithK.agda", "max_stars_repo_name": "Smaug123/agdaproofs", "max_stars_repo_head_hexsha": "0f4230011039092f58f673abcad8fb0652e6b562", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2019-08-08T12:44:19.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-28T06:04:15.000Z", "max_issues_repo_path": "Sets/FinSetWithK.agda", "max_issues_repo_name": "Smaug123/agdaproofs", "max_issues_repo_head_hexsha": "0f4230011039092f58f673abcad8fb0652e6b562", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 14, "max_issues_repo_issues_event_min_datetime": "2019-01-06T21:11:59.000Z", "max_issues_repo_issues_event_max_datetime": "2020-04-11T11:03:39.000Z", "max_forks_repo_path": "Sets/FinSetWithK.agda", "max_forks_repo_name": "Smaug123/agdaproofs", "max_forks_repo_head_hexsha": "0f4230011039092f58f673abcad8fb0652e6b562", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-11-29T13:23:07.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-29T13:23:07.000Z", "avg_line_length": 57.630952381, "max_line_length": 202, "alphanum_fraction": 0.5622805206, "num_tokens": 2028, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6926419958239133, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.340910172646503}} {"text": "-- Axiomatic embedding of guarded recursion in Agda\nmodule guarded-recursion.embedding where\n\nopen import guarded-recursion.prelude\n renaming (O to zero; S to suc)\nopen Coe\n\nmodule M\n (▹_ : ∀ {a} → Type_ a → Type_ a)\n\n (▸ : ∀ {a} → ▹ (Type_ a) → Type_ a)\n\n (next : ∀ {a} {A : Type_ a} → A → ▹ A)\n\n (▸-rule : ∀ {a} {A : Type_ a} → ▸ (next A) ≡ ▹ A)\n\n (fix : ∀ {a} {A : Type_ a} → (▹ A → A) → A)\n (fix-rule : ∀ {a} {A : Type_ a} {f : ▹ A → A} → fix f ≡ f (next (fix f)))\n\n (_⊛′_ : ∀ {a b} {A : Type_ a} {B : Type_ b} → ▹ (A → B) → ▹ A → ▹ B)\n (_⊛_ : ∀ {a b} {A : Type_ a} {B : A → Type_ b}\n → ▹ ((x : A) → B x) → (x : ▹ A) → ▸ (next B ⊛′ x))\n\n (fix-uniq : ∀ {a} {A : Type_ a} (u : A) f → u ≡ f (next u) → u ≡ fix f)\n\n (next⊛next : ∀ {a b} {A : Type_ a} {B : Type_ b} (f : A → B) (x : A)\n → next f ⊛′ next x ≡ next (f x))\n\n where\n\n roll▸ : ∀ {a} {A : Type_ a} → ▹ A → ▸ (next A)\n roll▸ = coe! ▸-rule\n\n un▸ : ∀ {a} {A : Type_ a} → ▸ (next A) → ▹ A\n un▸ = coe ▸-rule\n\n ▹Fix : ∀ {a} → Type_ a → Type_ a\n ▹Fix X = (▹ X → X) → X\n\n ▹Endo : ∀ {a} → Type_ a → Type_ a\n ▹Endo X = ▹ X → X\n\n μ : ∀ {a} → Fix (Type_ a)\n μ F = fix (F ∘ ▸)\n\n un : ∀ {a f} → fix {A = Type_ a} f → f (next (fix f))\n un = coe fix-rule\n\n unμ : ∀ {a} f → μ {a} f → f (▹ μ f)\n unμ {a} f x rewrite ! (▸-rule {A = μ f}) = un x\n\n roll : ∀ {a f} → f (next (fix f)) → fix {A = Type_ a} f\n roll = coe! fix-rule\n\n μ-rule : ∀ {a} f → μ {a} f ≡ f (▹ μ f)\n μ-rule f = fix-rule ∙ ap f (▸-rule {A = μ f})\n\n rollμ : ∀ {a} f → f (▹ μ f) → μ {a} f\n rollμ f = coe! (μ-rule f)\n\n un₁ : ∀ {a b} {A : Type_ a} {f x} → fix {A = A → Type_ b} f x → f (next (fix f)) x\n un₁ = coe₁ fix-rule\n\n roll₁ : ∀ {a b} {A : Type_ a} {f x} → f (next (fix f)) x → fix {A = A → Type_ b} f x\n roll₁ = coe₁! fix-rule\n\n un₂ : ∀ {a b} {A : Type_ a} {B : Type_ b} {c f x y} → fix {A = A → B → Type_ c} f x y → f (next (fix f)) x y\n un₂ = coe₂ fix-rule\n\n roll₂ : ∀ {a b} {A : Type_ a} {B : Type_ b} {c f x y} → f (next (fix f)) x y → fix {A = A → B → Type_ c} f x y\n roll₂ = coe₂! fix-rule\n\n map▹ : ∀ {a b} {A : Type_ a} {B : Type_ b} → (A → B) → ▹ A → ▹ B\n map▹ f ▹x = next f ⊛′ ▹x\n\n {-\n alternatively\n _⊛′′_ : ∀ {a b} {A : Type_ a} {B : A → Type_ b} → ▹ ((x : A) → B x) → (x : A) → ▹ (B x)\n ▹f ⊛′′ x = map▹ (λ f → f x) ▹f\n -}\n\n {-\n alternatively\n _$_ : ∀ {a b} {A : Type_ a} (B : A → Type_ b) → ▹ A → ▹ (Type_ b)\n f $ ▹x = map▹ f ▹x\n -}\n\n ▹^ : ∀ {a} → ℕ → Type_ a → Type_ a\n ▹^ zero A = A\n ▹^ (suc n) A = ▹ ▹^ n A\n\n next^ : ∀ {a} {A : Type_ a} n → A → ▹^ n A\n next^ zero x = x\n next^ (suc n) x = next (next^ n x)\n\n map▹^ : ∀ {a b} {A : Type_ a} {B : Type_ b} n → (A → B) → ▹^ n A → ▹^ n B\n map▹^ zero f = f\n map▹^ (suc n) f = map▹ (map▹^ n f)\n\n\n module SimpleStream where\n F : Type → Type → Type\n F A X = A × X\n\n S : Type → Type\n S A = μ (F A)\n\n μ₁F' : ∀ {a} {A : Type_ a} → ((A → ▹ Type) → A → Type) → (▹(A → Type) → A → Type)\n μ₁F' F self = F (λ x → (self ⊛′ next x))\n\n μ₁F : ∀ {a} {A : Type_ a} → ((A → Type) → A → Type) → (▹(A → Type) → A → Type)\n μ₁F F self = F (λ x → ▸ (self ⊛′ next x))\n\n μ₁ : ∀ {a} {A : Type_ a} → ((A → Type) → A → Type) → A → Type\n μ₁ F = fix (μ₁F F)\n\n module μId where\n μid : Type\n μid = μ id\n\n μid-rule : μid ≡ ▹ μid\n μid-rule = fix-rule ∙ ▸-rule {A = μ id}\n\n ω : μid\n ω = fix (rollμ id)\n\n module CoNat where\n Coℕ : Type\n Coℕ = μ Maybe\n\n rollNat : Maybe (▹ Coℕ) → Coℕ\n rollNat = rollμ Maybe\n\n ze : Coℕ\n ze = rollNat nothing\n\n su : ▹ Coℕ → Coℕ\n su x = rollNat (just x)\n\n su′ : Coℕ → Coℕ\n su′ = su ∘ next\n\n ω : Coℕ\n ω = fix su\n\n module Neg where\n {- data X : Type where\n rollX : Fix X\n : (X → X) → X\n -}\n X : Type\n X = μ Endo\n\n rollX : Endo (▹ X) → X\n -- : (▹ X → ▹ X) → X\n rollX = rollμ Endo\n\n rollX′ : ▹(Endo X) → X\n -- : ▹(X → X) → X\n rollX′ = rollX ∘ _⊛′_\n\n unX : X → Endo (▹ X)\n unX = unμ Endo\n\n -- δ = λ x → x x\n δ : X → ▹ X\n δ = λ x → (unX x) (next x)\n\n module Neg' where\n {- data X : Type where\n c : Fix X\n : ((X → X) → X) → X\n -}\n X : Type\n X = μ Fix\n\n rollX : Fix (▹ X) → X\n rollX = rollμ Fix\n\n unX : X → Fix (▹ X)\n unX = unμ Fix\n\n module μ₁Id where\n -- μ₁id = ▹∘▹∘…∘▹\n -- μ₁id A = ▹ (▹ … (▹ A))\n μ₁id : Type → Type\n μ₁id = μ₁ id\n\n betterfix₁ : ∀ {a} {A : Type_ a} {x : A} (F : Endo (A → Type)) → (▹ μ₁ F x → μ₁F F (next (μ₁ F)) x) → μ₁ F x\n betterfix₁ {a} {A} {x} F f = fix helper\n where helper : _ → _\n helper self = roll₁ (f self)\n\n ▹ω-inh' : ∀ {A : Type} {x : A} (F : Endo (A → Type)) → (▸ (next (μ₁ F) ⊛′ next x) → μ₁F F (next (μ₁ F)) x) → μ₁ F x\n ▹ω-inh' {A} {x} F f = fix helper\n where helper : _ → _\n helper self = roll₁ (f (coe! (ap ▸ (next⊛next (μ₁ F) x)) (roll▸ self)))\n\n ▹ω-inh : ∀ {A} → μ₁id A\n -- ▹ω-inh {A} = fix λ self → roll₁ (coe! (ap ▸ (next⊛next μ₁id A)) (roll▸ self))\n ▹ω-inh {A} = betterfix₁ id (λ self → coe! (ap ▸ (next⊛next μ₁id A)) (roll▸ self))\n\n -- ▹ω-inh {A} = fix λ self → {!!} -- (coe! (ap ▸ (next⊛next μ₁idω A)) (roll▸ self))\n\n fix2 : ∀ {a} {A : Type_ a} → (▹ A → A) → A\n fix2 f = fix (f ∘ next ∘ f)\n\n fix≡fix2 : ∀ {a} {A : Type_ a} (f : ▹ A → A) → fix f ≡ fix2 f\n fix≡fix2 f = fix-uniq (fix f) (f ∘ next ∘ f) (fix-rule ∙ ap (f ∘ next) fix-rule)\n\n module Streams where\n F : Type → Type → Type\n F A X = A × X\n\n -- S : Type → Type\n -- S A = μ (F A)\n\n F^ : ℕ → Type → Type → Type\n F^ n A X = A × ▹^ n X\n\n S^ : ℕ → Type → Type\n S^ n A = μ (F^ n A)\n\n S : Type → Type\n S = S^ 0\n\n S₂ = S^ 1\n\n unS : ∀ {A} → S A → F A (▹ S A)\n unS = unμ (F _)\n\n rollS : ∀ {A} → F A (▹ S A) → S A\n rollS = rollμ (F _)\n\n unS^ : ∀ {A} n → S^ n A → F^ n A (▹ S^ n A)\n unS^ n = unμ (F^ n _)\n\n rollS^ : ∀ {A} n → F^ n A (▹ S^ n A) → S^ n A\n rollS^ n = rollμ (F^ n _)\n\n hd : ∀ {A} → S A → A\n hd = fst ∘ unS\n\n tl : ∀ {A} → S A → ▹ S A\n tl = snd ∘ unS\n\n cons : ∀ {A} n → A → ▹^ n (▹ (S^ n A)) → S^ n A\n cons n x xs = rollS^ n (x , xs)\n\n infixr 4 _∷_\n _∷_ : ∀ {A} → A → ▹ (S A) → S A\n _∷_ = cons 0\n\n infixr 4 _∷₂_\n _∷₂_ : ∀ {A} → A → ▹^ 2 (S₂ A) → S₂ A\n x ∷₂ xs = roll (x , map▹ roll▸ xs)\n\n repeatS : ∀ {A} → A → S A\n repeatS x = fix λ x… → x ∷ x…\n\n module MapS {A B : Type} (f : A → B) where\n mapSf : ▹(S A → S B) → S A → S B\n mapSf self s = f (hd s) ∷ self ⊛′ tl s\n\n mapS : S A → S B\n mapS = fix mapSf\n\n mapS2f : ▹(S A → S B) → S A → S B\n mapS2f self s = f (hd s) ∷ map▹ (λ s' → f (hd s') ∷ self ⊛′ tl s') (tl s)\n\n mapS2f' : ▹(S A → S B) → S A → S B\n mapS2f' self = mapSf (next (mapSf self))\n\n mapS2f≡mapS2f' : mapS2f ≡ mapS2f'\n mapS2f≡mapS2f' = idp\n\n mapS2 : S A → S B\n mapS2 = fix mapS2f\n\n mapS2' : S A → S B\n mapS2' = fix mapS2f'\n\n mapS2≡mapS2' : mapS2 ≡ mapS2'\n mapS2≡mapS2' = idp\n\n mapS2'' : S A → S B\n mapS2'' = fix2 mapSf\n\n mapS2≡mapS2'' : mapS2 ≡ mapS2''\n mapS2≡mapS2'' = idp\n\n mapS≡mapS2 : mapS ≡ mapS2\n mapS≡mapS2 = fix≡fix2 mapSf\n\n open MapS\n\n group2 : S ℕ → ▹ S₂ ℕ²\n group2 = fix λ self s → map▹ (λ tls → (hd s , hd tls) ∷₂ self ⊛′ tl tls) (tl s)\n\n ‼ : ∀ {A} → (n : ℕ) → S A → ▹^ n A\n ‼ zero = hd\n ‼ (suc n) = map▹ (‼ n) ∘ tl\n\n toFun : ∀ {A} → S A → (n : ℕ) → ▹^ n A\n toFun s n = ‼ n s\n\n fromFun : ∀ {A} → (ℕ → A) → S A\n fromFun {A} = fix λ self (f : ℕ → A) → f 0 ∷ self ⊛′ next (f ∘ suc)\n\n nats : S ℕ\n nats = fix λ self → 0 ∷ map▹ (mapS suc) self\n\n nats2 : S ℕ\n nats2 = fix λ self → 0 ∷ map▹ (mapS2 suc) self\n\n nats≡nats2 : nats ≡ nats2\n nats≡nats2 rewrite mapS≡mapS2 suc = idp\n\n arrow : ▹ ℕ\n arrow = ‼ 1 nats\n\n module Sim\n {A : Type}\n (ℛ : A → A → Type)\n (ℛ-refl : Reflexive ℛ)\n where\n ≈F : ▹(S A × S A → Type) → S A × S A → Type\n ≈F X (xs , ys) = ℛ (hd xs) (hd ys) × ▸ ((map▹ curry X ⊛′ (tl xs)) ⊛′ tl ys)\n\n _≈_ : S A × S A → Type\n _≈_ = fix ≈F\n\n ≈-tail : ∀ {xs ys : S A} → _≈_ (xs , ys) → ▸ ((map▹ curry (next _≈_) ⊛′ tl xs) ⊛′ tl ys)\n ≈-tail pf = snd (un₁ pf)\n\n {- Does not work yet\n ≈-refl : Reflexive (curry _≈_)\n ≈-refl {x} = (fix λ pf x → roll₁ {f = ≈F} (ℛ-refl , helper pf x)) x\n where helper' : _ → _ → _\n helper' pf x = map▹ (λ f → f x) pf\n helper : _ → _ → _\n helper pf x = let r = helper' pf x in {!roll▸ r!}\n -}\n\n module DelayedStreams where\n data F (A : Type) (X : Type) : Type where\n done : F A X\n skip : X → F A X\n yield : A → X → F A X\n\n mapF : ∀ {A B X Y} → (A → B) → (X → Y) → F A X → F B Y\n mapF f g done = done\n mapF f g (skip x) = skip (g x)\n mapF f g (yield a x) = yield (f a) (g x)\n\n S : Type → Type\n S A = μ (F A)\n\n unS : ∀ {A} → S A → F A (▹ S A)\n unS = mapF id un▸ ∘ un\n\n rollS : ∀ {A} → F A (▹ S A) → S A\n rollS = roll ∘ mapF id roll▸\n\n unfoldS : ∀ {A X} → (X → F A (▹ X)) → X → S A\n unfoldS coalg = fix λ self x → rollS (mapF id (λ x′ → self ⊛′ x′) (coalg x))\n\n repeatS : ∀ {A} → A → S A\n repeatS x = fix λ self → rollS (yield x self)\n\n neverS : ∀ {A} → S A\n neverS = fix λ self → rollS (skip self)\n\n -- Co-algebra style...\n mapS : ∀ {A B} → (A → B) → S A → S B\n mapS {A} {B} f = unfoldS (mapF f id ∘ unS)\n\n filterF : ∀ {A X} → (A → 𝟚) → F A X → F A X\n filterF f done = done\n filterF f (skip xs) = skip xs\n filterF f (yield x xs) = if f x then yield x xs\n else skip xs\n\n filterS : ∀ {A} → (A → 𝟚) → S A → S A\n filterS f = unfoldS (filterF f ∘ unS)\n\nmodule FuelBased where\n fix : ∀ {a} {A : Type_ a} → ℕ → (A → A) → A\n fix zero f = STUCK where postulate STUCK : _\n fix (suc n) f = f (fix n f)\n\n fix-rule : ∀ {a} {A : Type_ a} (n : ℕ) {f : A → A} → fix n f ≡ f (fix n f)\n fix-rule zero = ThisIsUnsafeButPlease.trustMe\n fix-rule (suc n) {f} = ap f (fix-rule n)\n\n fix-uniq : ∀ {a} {A : Type_ a} (n : ℕ) (u : A) f → u ≡ f u → u ≡ fix n f\n fix-uniq zero u f pf = ThisIsUnsafeButPlease.trustMe\n fix-uniq (suc n) u f pf = pf ∙ ap f (fix-uniq n u f pf)\n\n module I (n : ℕ) = M id id id idp (fix n) (fix-rule n) id id\n (fix-uniq n) (λ _ _ → idp)\n\nmodule HiddenFix {a} {A : Type_ a} (f : A → A) where\n -- This definition is not intended to termination-check.\n -- Use with care it's really easy to make the type-checker loop.\n {-# TERMINATING #-}\n fix : Hidden A\n fix = hide f (reveal fix)\n\n fix-rule : reveal fix ≡ f (reveal fix)\n fix-rule = idp {a} {A} {reveal fix}\n\n -- This definition is not intended to termination-check.\n -- Use with care it's really easy to make the type-checker loop.\n {-# TERMINATING #-}\n fix-uniq : (u : A) → u ≡ f u → u ≡ reveal fix\n fix-uniq u pf = pf ∙ ap f (fix-uniq u pf) ∙ ! fix-rule\n\nmodule Test where\n open HiddenFix\n open M id id id idp (reveal ∘ fix) (λ {_} {_} {f} → fix-rule f) id id\n (λ {_} {_} u f → fix-uniq f u) (λ _ _ → idp) public\n open Streams\n two : map▹ hd (tl nats) ≡ 1\n two = idp\n\n-- -}\n-- -}\n-- -}\n", "meta": {"hexsha": "032b45bdd2bd28a740e94c7aeee78fa902481476", "size": 12239, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "guarded-recursion/embedding.agda", "max_stars_repo_name": "np/guarded-recursion", "max_stars_repo_head_hexsha": "9fba7d89d8b27e9bb08c27df802608b5fff769e0", 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YES\n2. YES", "lm_q1_score": 0.6370307806984443, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.34087416973852114}} {"text": "open import OutsideIn.Prelude\nopen import OutsideIn.X\nmodule OutsideIn.Inference.ConstraintGen(x : X) where\n import OutsideIn.Constraints as C\n import OutsideIn.TypeSchema as TS\n import OutsideIn.Expressions as E\n import OutsideIn.Environments as V\n open X(x) renaming (funType to _⟶_; appType to _··_)\n open C(x)\n open TS(x)\n open E(x)\n open V(x)\n open import Data.Vec\n\n private module PlusN-m n = Monad (PlusN-is-monad {n})\n module PlusN-f n = Functor (Monad.is-functor (PlusN-is-monad {n}))\n module TypeSchema-f {n} = Functor (type-schema-is-functor {n})\n module Type-f = Functor (type-is-functor)\n module QC-f = Functor (qconstraint-is-functor)\n module Exp-f₁ {tv} {r} = Functor (expression-is-functor₁ {tv} {r})\n module Exp-f₂ {ev} {r} = Functor (expression-is-functor₂ {ev} {r})\n module Constraint-f {s} = Functor (constraint-is-functor {s}) \n module Vec-f {n} = Functor (vec-is-functor {n}) \n open Monad (type-is-monad) using () renaming (unit to TVar)\n\n\n private \n\n upindex : {X : Set → Set}{tv : Set} → ⦃ is-functor : Functor X ⦄ → X tv → X (Ⓢ tv)\n upindex ⦃ is-functor ⦄ e = suc <$> e\n where open Functor (is-functor)\n\n _↑c : {tv : Set}{s : Strata} → (Constraint tv s) → (Constraint (Ⓢ tv) s)\n _↑c {s = s} = upindex ⦃ constraint-is-functor {s} ⦄\n\n _↑e : {ev tv : Set}{r : Shape} → (Expression ev tv r) → (Expression ev (Ⓢ tv) r)\n _↑e = upindex ⦃ expression-is-functor₂ ⦄\n\n _↑a : {ev tv : Set}{r : Shape} → (Alternatives ev tv r) → (Alternatives ev (Ⓢ tv) r)\n _↑a = upindex ⦃ alternatives-is-functor₂ ⦄\n\n _↑t : {tv : Set} → (Type tv) → (Type (Ⓢ tv))\n _↑t = upindex ⦃ type-is-functor ⦄ \n\n _↑q : {tv : Set} → (QConstraint tv) → (QConstraint (Ⓢ tv))\n _↑q = upindex ⦃ qconstraint-is-functor ⦄ \n \n infixr 7 _↑e\n infixr 7 _↑c\n infixr 7 _↑t\n infixr 7 _↑a\n\n applyAll : ∀{tv}(n : ℕ) → Type tv → Type (tv ⨁ n)\n applyAll zero x = x\n applyAll (suc n) x = applyAll n ((x ↑t) ·· (TVar zero))\n\n funType : ∀{tv}{n} → Vec (Type tv) n → Type tv → Type tv\n funType [] t = t\n funType (x ∷ xs) t = x ⟶ (funType xs t) \n\n upType : ∀ {n}{tv} → Type tv → Type (tv ⨁ n)\n upType {n} t = Type-f.map (Monad.unit (PlusN-is-monad {n})) t \n upGamma : ∀ {n}{ev}{tv} → Environment ev tv → Environment ev (tv ⨁ n)\n upGamma {n} Γ = TypeSchema-f.map (Monad.unit (PlusN-is-monad {n})) ∘ Γ \n upExp : ∀ {n}{ev}{tv}{r} → Expression ev tv r → Expression ev (tv ⨁ n) r\n upExp {n} e = Exp-f₂.map (Monad.unit (PlusN-is-monad {n})) e\n upAlts : ∀ {n}{ev}{tv}{r} → Alternatives ev tv r → Alternatives ev (tv ⨁ n) r\n upAlts {n} e = Functor.map alternatives-is-functor₂ (Monad.unit (PlusN-is-monad {n})) e\n\n mutual\n\n syntax alternativeConstraintGen Γ α₀ α₁ alt C = Γ ►′ alt ∶ α₀ ⟶ α₁ ↝ C\n data alternativeConstraintGen {ev : Set}{tv : Set}(Γ : Environment ev tv)(α₀ α₁ : Type tv) : \n {r : Shape} → Alternative ev tv r → Constraint tv Extended → Set where\n Simple : ∀ {r}{n}{v : Name ev (Datacon n)}{e : Expression _ _ r}{a}{τs}{T}{C} \n → let δ = TVar zero\n in Γ v ≡ DC∀ a · τs ⟶ T\n → addAll (Vec-f.map (_↑t) τs) (upGamma {a} Γ ↑Γ) ► (upExp {a} e ↑e) ∶ δ ↝ C \n → Γ ►′ v →′ e ∶ α₀ ⟶ α₁ ↝ Ⅎ′ a · (Ⅎ δ ∼′ (upType {a} α₁ ↑t) ∧′ C) ∧′ applyAll a (TVar T) ∼′ upType {a} α₀\n GADT : ∀ {r}{n}{v : Name ev (Datacon n)}{e : Expression _ _ r}{a}{b}{Q}{τs}{T}{C} \n → let δ = TVar zero\n in Γ v ≡ DC∀′ a , b · Q ⇒ τs ⟶ T\n → addAll (Vec-f.map (_↑t) τs) (upGamma {b} (upGamma {a} Γ) ↑Γ) \n ► (upExp {b} (upExp {a} e) ↑e) ∶ δ ↝ C \n → Γ ►′ v →′ e ∶ α₀ ⟶ α₁ ↝ Ⅎ′ a · Ⅎ′ b · (Imp′ Q (Ⅎ (C ∧′ δ ∼′ (upType {b} (upType {a} α₁) ↑t))))\n ∧′ upType {b} (upType {a} α₀) ∼′ Type-f.map (PlusN-m.unit b) (applyAll a (TVar T))\n \n syntax alternativesConstraintGen Γ α₀ α₁ alts C = Γ ►► alts ∶ α₀ ⟶ α₁ ↝ C\n data alternativesConstraintGen {ev : Set}{tv : Set}(Γ : Environment ev tv)(α₀ α₁ : Type tv) : \n {r : Shape} → Alternatives ev tv r → Constraint tv Extended → Set where\n NoAlternative : Γ ►► esac ∶ α₀ ⟶ α₁ ↝ ε′ \n AnAlternative : ∀ {r₁ r₂}{a : Alternative _ _ r₁}{as : Alternatives _ _ r₂}{C₁}{C₂} \n → Γ ►′ a ∶ α₀ ⟶ α₁ ↝ C₁\n → Γ ►► as ∶ α₀ ⟶ α₁ ↝ C₂ \n → Γ ►► a ∣ as ∶ α₀ ⟶ α₁ ↝ C₂ \n\n syntax constraintGen a c b d = a ► b ∶ c ↝ d\n data constraintGen {ev : Set}{tv : Set}\n (Γ : Environment ev tv)(τ : Type tv) : {r : Shape} → \n Expression ev tv r → Constraint tv Extended → Set where\n VarCon₁ : ∀ {v}{n}{q}{t} \n → Γ (N v) ≡ ∀′ n · q ⇒ t \n → Γ ► Var (N v) ∶ τ ↝ Ⅎ′ n · QC q ∧′ upType {n} τ ∼′ t\n VarCon₂ : ∀ {n}{d}{a}{τs : Vec _ n}{k}\n → Γ (DC d) ≡ DC∀ a · τs ⟶ k\n → Γ ► Var (DC d) ∶ τ ↝ Ⅎ′ a · upType {a} τ ∼′ funType τs (applyAll a (TVar k))\n VarCon₃ : ∀ {n}{d}{a}{b}{Q}{τs : Vec _ n}{k}\n → Γ (DC d) ≡ DC∀′ a , b · Q ⇒ τs ⟶ k\n → Γ ► Var (DC d) ∶ τ ↝ Ⅎ′ a · Ⅎ′ b ·\n QC Q ∧′ upType {b} (upType {a} τ) ∼′ funType τs (upType {b} (applyAll a (TVar k))) \n App : ∀ {r₁}{r₂}{e₁ : Expression _ _ r₁}{e₂ : Expression _ _ r₂}{C₁}{C₂}\n → let α₀ = TVar zero\n α₁ = TVar (suc zero)\n α₂ = TVar (suc (suc zero))\n in upGamma {3} Γ ► upExp {3} e₁ ∶ α₀ ↝ C₁\n → upGamma {3} Γ ► upExp {3} e₂ ∶ α₁ ↝ C₂\n → Γ ► e₁ · e₂ ∶ τ ↝ Ⅎ Ⅎ Ⅎ C₁ ∧′ C₂ ∧′ α₀ ∼′ (α₁ ⟶ α₂) ∧′ upType {3} τ ∼′ α₂\n Abs : ∀ {r}{e : Expression _ _ r}{C}\n → let α₀ = TVar zero\n α₁ = TVar (suc zero)\n in ⟨ ∀′ 0 · ε ⇒ α₀ ⟩, upGamma {2} Γ ► upExp {2} e ∶ α₁ ↝ C\n → Γ ► λ′ e ∶ τ ↝ Ⅎ Ⅎ C ∧′ upType {2} τ ∼′ (α₀ ⟶ α₁)\n Let : ∀{r₁}{r₂}{x : Expression _ _ r₁}{y : Expression _ _ r₂}{C₁}{C₂}\n → let α₀ = TVar zero\n α₁ = TVar (suc zero)\n in upGamma {2} Γ ► upExp {2} x ∶ α₀ ↝ C₁ \n → ⟨ ∀′ 0 · ε ⇒ α₀ ⟩, upGamma {2} Γ ► upExp {2} y ∶ α₁ ↝ C₂\n → Γ ► let₁ x in′ y ∶ τ ↝ Ⅎ Ⅎ C₁ ∧′ C₂ ∧′ upType {2} τ ∼′ α₁\n LetA : ∀{r₁}{r₂}{x : Expression _ _ r₁}{y : Expression _ _ r₂}{t}{C₁}{C₂}\n → let α₀ = TVar zero\n α₁ = TVar (suc zero)\n in upGamma {2} Γ ► upExp {2} x ∶ α₀ ↝ C₁ \n → ⟨ ∀′ 0 · ε ⇒ α₀ ⟩, upGamma {2} Γ ► upExp {2} y ∶ α₁ ↝ C₂\n → Γ ► let₂ x ∷ t in′ y ∶ τ ↝ Ⅎ Ⅎ C₁ ∧′ C₂ ∧′ upType {2} τ ∼′ α₁ ∧′ upType {2} t ∼′ α₀\n GLetA : ∀{n}{r₁}{r₂}{x : Expression _ _ r₁}{y : Expression _ _ r₂}{Q}{t}{C}{C₂}\n → let α₀ = upType {n} (TVar zero)\n α₁ = upType {n} (TVar (suc zero))\n up2 = PlusN-f.map n (PlusN-m.unit 2)\n in upGamma {n} (upGamma {2} Γ) ► Exp-f₂.map up2 x ∶ α₀ ↝ C \n → upGamma {n} (upGamma {2} (⟨ ∀′ n · Q ⇒ t ⟩, Γ)) ► upExp {n} (upExp {2} y) ∶ α₁ ↝ C₂\n → Γ ► let₃ n · x ∷ Q ⇒ t in′ y ∶ τ ↝ Ⅎ Ⅎ Ⅎ′ n · Imp′ (QC-f.map up2 Q) (C ∧′ α₀ ∼′ Type-f.map up2 t)\n ∧′ C₂ ∧′ upType {n} (upType {2} τ) ∼′ α₁\n Case : ∀{r₁}{r₂}{x : Expression _ _ r₁}{alts : Alternatives _ _ r₂}{C₁}{C₂}\n → let α₀ = TVar zero\n α₁ = TVar (suc zero)\n in upGamma {2} Γ ► upExp {2} x ∶ α₀ ↝ C₁\n → upGamma {2} Γ ►► upAlts {2} alts ∶ α₀ ⟶ α₁ ↝ C₂ \n → Γ ► case x of alts ∶ τ ↝ Ⅎ Ⅎ C₁ ∧′ C₂ \n\n\n genConstraint : {ev : Set}{tv : Set}{r : Shape}\n (Γ : Environment ev tv)(e : Expression ev tv r)(τ : Type tv) → ∃ (λ C → Γ ► e ∶ τ ↝ C)\n genConstraintAlternative : {ev : Set}{tv : Set}{r : Shape} (Γ : Environment ev tv)(a : Alternative ev tv r)(α₀ α₁ : Type tv)\n → ∃ (λ C → Γ ►′ a ∶ α₀ ⟶ α₁ ↝ C)\n genConstraintAlternative Γ (n →′ e) α₀ α₁ with Γ n | inspect Γ n \n ... | DC∀ a · τs ⟶ k | iC p with genConstraint (addAll (Vec-f.map _↑t τs) (upGamma {a} Γ ↑Γ)) (upExp {a} e ↑e) (TVar zero)\n ... | C , p₂ = _ , Simple p p₂\n genConstraintAlternative Γ (n →′ e) α₀ α₁ \n | DC∀′ a , b · Q ⇒ τs ⟶ k | iC p with genConstraint (addAll (Vec-f.map _↑t τs) (upGamma {b} (upGamma {a} Γ) ↑Γ)) \n (upExp {b} (upExp {a} e) ↑e) \n (TVar zero) \n ... | C , p₂ = _ , GADT p p₂\n genConstraintAlternatives : {ev : Set}{tv : Set}{r : Shape} (Γ : Environment ev tv)(a : Alternatives ev tv r)(α₀ α₁ : Type tv)\n → ∃ (λ C → Γ ►► a ∶ α₀ ⟶ α₁ ↝ C)\n genConstraintAlternatives Γ esac α₀ α₁ = _ , NoAlternative\n genConstraintAlternatives Γ (a ∣ as) α₀ α₁ with genConstraintAlternative Γ a α₀ α₁ | genConstraintAlternatives Γ as α₀ α₁ \n ... | C₁ , p₁ | C₂ , p₂ = _ , AnAlternative p₁ p₂\n genConstraint Γ (Var (N v)) τ with Γ (N v) | inspect Γ (N v)\n ... | ∀′ n · q ⇒ t | iC prf = _ , VarCon₁ prf \n genConstraint Γ (Var (DC d)) τ with Γ (DC d) | inspect Γ (DC d)\n ... | DC∀ a · τs ⟶ k | iC prf = _ , VarCon₂ prf \n ... | DC∀′ a , b · q ⇒ τs ⟶ k | iC prf = _ , VarCon₃ prf \n genConstraint Γ (e₁ · e₂) τ with genConstraint (upGamma {3} Γ) (upExp {3} e₁) (TVar zero) \n | genConstraint (upGamma {3} Γ) (upExp {3} e₂) (TVar (suc zero))\n ... | C₁ , p₁ | C₂ , p₂ = _ , App p₁ p₂\n genConstraint Γ (λ′ e′) τ with genConstraint (⟨ ∀′ 0 · ε ⇒ TVar zero ⟩, upGamma {2} Γ) (upExp {2} e′) (TVar (suc zero)) \n ... | C , p = _ , Abs p\n genConstraint Γ (let₁ x in′ y) τ with genConstraint (upGamma {2} Γ) (upExp {2} x) (TVar zero) \n | genConstraint (⟨ ∀′ 0 · ε ⇒ TVar zero ⟩, upGamma {2} Γ) (upExp {2} y) (TVar (suc zero)) \n ... | C₁ , p₁ | C₂ , p₂ = _ , Let p₁ p₂\n genConstraint Γ (let₂ x ∷ t in′ y) τ with genConstraint (upGamma {2} Γ) (upExp {2} x) (TVar zero) \n | genConstraint (⟨ ∀′ 0 · ε ⇒ TVar zero ⟩, upGamma {2} Γ) (upExp {2} y) (TVar (suc zero)) \n ... | C₁ , p₁ | C₂ , p₂ = _ , LetA p₁ p₂\n genConstraint Γ (let₃ n · x ∷ Q ⇒ t in′ y) τ with genConstraint (upGamma {n} (upGamma {2} Γ)) \n (Exp-f₂.map (PlusN-f.map n (PlusN-m.unit 2)) x) \n (upType {n} (TVar zero)) \n | genConstraint (upGamma {n} (upGamma {2} (⟨ ∀′ n · Q ⇒ t ⟩, Γ)))\n (upExp {n} (upExp {2} y))\n (upType {n} (TVar (suc zero))) \n ... | C₁ , p₁ | C₂ , p₂ = _ , GLetA p₁ p₂\n genConstraint Γ (case x of alts) τ with genConstraint (upGamma {2} Γ) (upExp {2} x) (TVar zero) \n | genConstraintAlternatives (upGamma {2} Γ) (upAlts {2} alts) (TVar zero) (TVar (suc zero))\n ... | C₁ , p₁ | C₂ , p₂ = _ , Case p₁ p₂ \n", "meta": {"hexsha": "42d7ba3674d2136ea21e6eb9fe43ab419426ea6d", "size": 11447, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "OutsideIn/Inference/ConstraintGen.agda", "max_stars_repo_name": "liamoc/outside-in", "max_stars_repo_head_hexsha": "fc1fc1bba2af95806d9075296f9ed1074afa4c24", "max_stars_repo_licenses": 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4290, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.34085860891317254}} {"text": "{-# OPTIONS --without-K --rewriting #-}\n\nopen import lib.Basics\nopen import lib.types.Empty\nopen import lib.types.Sigma\nopen import lib.types.Paths\n\nmodule lib.types.Pi where\n\ninstance\n Π-level : ∀ {i j} {A : Type i} {B : A → Type j} {n : ℕ₋₂}\n → ((x : A) → has-level n (B x)) → has-level n (Π A B)\n Π-level {n = ⟨-2⟩} p = has-level-in ((λ x → contr-center (p x)) , lemma)\n where abstract lemma = λ f → λ= (λ x → contr-path (p x) (f x))\n Π-level {n = S n} p = has-level-in lemma where\n abstract\n lemma = λ f g →\n equiv-preserves-level λ=-equiv {{Π-level (λ x → has-level-apply (p x) (f x) (g x))}}\n\n Πi-level : ∀ {i j} {A : Type i} {B : A → Type j} {n : ℕ₋₂}\n → ((x : A) → has-level n (B x)) → has-level n ({x : A} → B x)\n Πi-level {A = A} {B} p = equiv-preserves-level e {{Π-level p}} where\n\n e : Π A B ≃ ({x : A} → B x)\n fst e f {x} = f x\n is-equiv.g (snd e) f x = f\n is-equiv.f-g (snd e) _ = idp\n is-equiv.g-f (snd e) _ = idp\n is-equiv.adj (snd e) _ = idp\n\n\n{- Equivalences in a Π-type -}\nΠ-emap-l : ∀ {i j k} {A : Type i} {B : Type j} (P : B → Type k)\n → (e : A ≃ B) → Π A (P ∘ –> e) ≃ Π B P\nΠ-emap-l {A = A} {B = B} P e = equiv f g f-g g-f where\n f : Π A (P ∘ –> e) → Π B P\n f u b = transport P (<–-inv-r e b) (u (<– e b))\n\n g : Π B P → Π A (P ∘ –> e)\n g v a = v (–> e a)\n\n abstract\n f-g : ∀ v → f (g v) == v\n f-g v = λ= λ b → to-transp (apd v (<–-inv-r e b))\n\n g-f : ∀ u → g (f u) == u\n g-f u = λ= λ a → to-transp $ transport (λ p → u _ == _ [ P ↓ p ])\n (<–-inv-adj e a)\n (↓-ap-in P (–> e)\n (apd u $ <–-inv-l e a))\n\nΠ-emap-r : ∀ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k}\n → (∀ x → B x ≃ C x) → Π A B ≃ Π A C\nΠ-emap-r {A = A} {B = B} {C = C} k = equiv f g f-g g-f\n where f : Π A B → Π A C\n f c x = –> (k x) (c x)\n\n g : Π A C → Π A B\n g d x = <– (k x) (d x)\n\n abstract\n f-g : ∀ d → f (g d) == d\n f-g d = λ= (λ x → <–-inv-r (k x) (d x))\n\n g-f : ∀ c → g (f c) == c\n g-f c = λ= (λ x → <–-inv-l (k x) (c x))\n\n{-\nfavonia: This part is not used.\n\nmodule _ {i₀ i₁ j₀ j₁} {A₀ : Type i₀} {A₁ : Type i₁}\n {B₀ : A₀ → Type j₀} {B₁ : A₁ → Type j₁} where\n Π-emap : (u : A₀ ≃ A₁) (v : ∀ a → B₀ (<– u a) ≃ B₁ a) → Π A₀ B₀ ≃ Π A₁ B₁\n Π-emap u v = Π A₀ B₀ ≃⟨ Π-emap-l _ (u ⁻¹) ⁻¹ ⟩\n Π A₁ (B₀ ∘ <– u) ≃⟨ Π-emap-r v ⟩\n Π A₁ B₁ ≃∎\n\n Π-emap' : (u : A₀ ≃ A₁) (v : ∀ a → B₀ a ≃ B₁ (–> u a)) → Π A₀ B₀ ≃ Π A₁ B₁\n Π-emap' u v = Π A₀ B₀ ≃⟨ Π-emap-r v ⟩\n Π A₀ (B₁ ∘ –> u) ≃⟨ Π-emap-l _ u ⟩\n Π A₁ B₁ ≃∎\n-}\n\n\n{- Coversions between functions with implicit and explicit arguments -}\n\nexpose-equiv : ∀ {i j} {A : Type i} {B : A → Type j}\n → ({x : A} → B x) ≃ ((x : A) → B x)\nexpose-equiv = (λ f a → f {a}) , is-eq\n _\n (λ f {a} → f a)\n (λ _ → idp)\n (λ _ → idp)\n\n\n\n{- Dependent paths in a Π-type -}\nmodule _ {i j k} {A : Type i} {B : A → Type j} {C : (a : A) → B a → Type k}\n where\n\n ↓-Π-in : {x x' : A} {p : x == x'} {u : Π (B x) (C x)} {u' : Π (B x') (C x')}\n → ({t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → u t == u' t' [ uncurry C ↓ pair= p q ])\n → (u == u' [ (λ x → Π (B x) (C x)) ↓ p ])\n ↓-Π-in {p = idp} f = λ= (λ x → f (idp {a = x}))\n\n ↓-Π-out : {x x' : A} {p : x == x'} {u : Π (B x) (C x)} {u' : Π (B x') (C x')}\n → (u == u' [ (λ x → Π (B x) (C x)) ↓ p ])\n → ({t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → u t == u' t' [ uncurry C ↓ pair= p q ])\n ↓-Π-out {p = idp} q idp = app= q _\n\n ↓-Π-β : {x x' : A} {p : x == x'} {u : Π (B x) (C x)} {u' : Π (B x') (C x')}\n → (f : {t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → u t == u' t' [ uncurry C ↓ pair= p q ])\n → {t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → ↓-Π-out (↓-Π-in f) q == f q\n ↓-Π-β {p = idp} f idp = app=-β (λ x → f (idp {a = x})) _\n\n ↓-Π-η : {x x' : A} {p : x == x'} {u : Π (B x) (C x)} {u' : Π (B x') (C x')}\n → (q : (u == u' [ (λ x → Π (B x) (C x)) ↓ p ]))\n → ↓-Π-in (↓-Π-out q) == q\n ↓-Π-η {p = idp} q = ! (λ=-η q)\n\n ↓-Π-equiv : {x x' : A} {p : x == x'} {u : Π (B x) (C x)} {u' : Π (B x') (C x')}\n → ({t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → u t == u' t' [ uncurry C ↓ pair= p q ])\n ≃ (u == u' [ (λ x → Π (B x) (C x)) ↓ p ])\n ↓-Π-equiv {p = idp} = equiv ↓-Π-in ↓-Π-out ↓-Π-η\n (λ u → <– (ap-equiv expose-equiv _ _)\n (λ= (λ t → <– (ap-equiv expose-equiv _ _)\n (λ= (λ t' → λ= (↓-Π-β u))))))\n\n{- Dependent paths in a Π-type where the codomain is not dependent on anything\n\nRight now, this is defined in terms of the previous one. Maybe it’s a good idea,\nmaybe not.\n-}\nmodule _ {i j k} {A : Type i} {B : A → Type j} {C : Type k} {x x' : A}\n {p : x == x'} {u : B x → C} {u' : B x' → C} where\n\n ↓-app→cst-in :\n ({t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → u t == u' t')\n → (u == u' [ (λ x → B x → C) ↓ p ])\n ↓-app→cst-in f = ↓-Π-in (λ q → ↓-cst-in (f q))\n\n ↓-app→cst-out :\n (u == u' [ (λ x → B x → C) ↓ p ])\n → ({t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → u t == u' t')\n ↓-app→cst-out r q = ↓-cst-out (↓-Π-out r q)\n\n ↓-app→cst-β :\n (f : ({t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → u t == u' t'))\n → {t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → ↓-app→cst-out (↓-app→cst-in f) q == f q\n ↓-app→cst-β f q =\n ↓-app→cst-out (↓-app→cst-in f) q\n =⟨ idp ⟩\n ↓-cst-out (↓-Π-out (↓-Π-in (λ qq → ↓-cst-in (f qq))) q)\n =⟨ ↓-Π-β (λ qq → ↓-cst-in (f qq)) q |in-ctx\n ↓-cst-out ⟩\n ↓-cst-out (↓-cst-in {p = pair= p q} (f q))\n =⟨ ↓-cst-β (pair= p q) (f q) ⟩\n f q =∎\n\n{- favonia: these lemmas are not used anywhere\n\n{- Similar to above, with domain being the identity function. -}\n{- These lemmas were in homotopy.FunctionOver and in different conventions. -}\n\nmodule _ {i j} {A B : Type i} {C : Type j}\n {u : A → C} {v : B → C} where\n\n ↓-idf→cst-in : ∀ (p : A == B)\n → u == v ∘ coe p\n → u == v [ (λ x → x → C) ↓ p ]\n ↓-idf→cst-in idp q = q\n\n ↓-idf→cst-ua-in : ∀ (e : A ≃ B)\n → u == v ∘ –> e\n → u == v [ (λ x → x → C) ↓ ua e ]\n ↓-idf→cst-ua-in e q = ↓-idf→cst-in (ua e) (q ∙ ap (v ∘_) (λ= λ a → ! (coe-β e a)))\n\n ↓-idf→cst-in' : ∀ (p : A == B)\n → u ∘ coe! p == v\n → u == v [ (λ x → x → C) ↓ p ]\n ↓-idf→cst-in' idp q = q\n\n ↓-idf→cst-ua-in' : ∀ (e : A ≃ B)\n → u ∘ <– e == v\n → u == v [ (λ x → x → C) ↓ ua e ]\n ↓-idf→cst-ua-in' e q = ↓-idf→cst-in' (ua e) (ap (u ∘_) (λ= λ a → coe!-β e a) ∙ q)\n\nmodule _ {i j} {A B : Type i} {C : Type j}\n {u : C → A} {v : C → B} where\n\n ↓-cst→idf-in : ∀ (p : A == B)\n → coe p ∘ u == v\n → u == v [ (λ x → C → x) ↓ p ]\n ↓-cst→idf-in idp q = q\n\n ↓-cst→idf-ua-in : ∀ (e : A ≃ B)\n → –> e ∘ u == v\n → u == v [ (λ x → C → x) ↓ ua e ]\n ↓-cst→idf-ua-in e q = ↓-cst→idf-in (ua e) (ap (_∘ u) (λ= λ a → coe-β e a) ∙ q)\n\n ↓-cst→idf-in' : ∀ (p : A == B)\n → u == coe! p ∘ v\n → u == v [ (λ x → C → x) ↓ p ]\n ↓-cst→idf-in' idp q = q\n\n ↓-cst→idf-ua-in' : ∀ (e : A ≃ B)\n → u == <– e ∘ v\n → u == v [ (λ x → C → x) ↓ ua e ]\n ↓-cst→idf-ua-in' e q = ↓-cst→idf-in' (ua e) (q ∙ ap (_∘ v) (λ= λ a → ! (coe!-β e a)))\n-}\n\n{- Dependent paths in an arrow type -}\nmodule _ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k}\n {x x' : A} {p : x == x'} {u : B x → C x} {u' : B x' → C x'} where\n\n ↓-→-in :\n ({t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → u t == u' t' [ C ↓ p ])\n → (u == u' [ (λ x → B x → C x) ↓ p ])\n ↓-→-in f = ↓-Π-in (λ q → ↓-cst2-in p q (f q))\n\n ↓-→-out :\n (u == u' [ (λ x → B x → C x) ↓ p ])\n → ({t : B x} {t' : B x'} (q : t == t' [ B ↓ p ])\n → u t == u' t' [ C ↓ p ])\n ↓-→-out r q = ↓-cst2-out p q (↓-Π-out r q)\n\n{- Transport form of dependent path in an arrow type -}\nmodule _ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k} where\n\n ↓-→-from-transp : {x x' : A} {p : x == x'}\n {u : B x → C x} {u' : B x' → C x'}\n → transport C p ∘ u == u' ∘ transport B p\n → u == u' [ (λ x → B x → C x) ↓ p ]\n ↓-→-from-transp {p = idp} q = q\n\n ↓-→-to-transp : {x x' : A} {p : x == x'}\n {u : B x → C x} {u' : B x' → C x'}\n → u == u' [ (λ x → B x → C x) ↓ p ]\n → transport C p ∘ u == u' ∘ transport B p\n ↓-→-to-transp {p = idp} q = q\n\n-- Dependent paths in a Π-type where the domain is constant\nmodule _ {i j k} {A : Type i} {B : Type j} {C : A → B → Type k} where\n\n ↓-Π-cst-app-in : {x x' : A} {p : x == x'}\n {u : (b : B) → C x b} {u' : (b : B) → C x' b}\n → ((b : B) → u b == u' b [ (λ x → C x b) ↓ p ])\n → (u == u' [ (λ x → (b : B) → C x b) ↓ p ])\n ↓-Π-cst-app-in {p = idp} f = λ= f\n\n ↓-Π-cst-app-out : {x x' : A} {p : x == x'}\n {u : (b : B) → C x b} {u' : (b : B) → C x' b}\n → (u == u' [ (λ x → (b : B) → C x b) ↓ p ])\n → ((b : B) → u b == u' b [ (λ x → C x b) ↓ p ])\n ↓-Π-cst-app-out {p = idp} q = app= q\n\nsplit-ap2 : ∀ {i j k} {A : Type i} {B : A → Type j} {C : Type k} (f : Σ A B → C)\n {x y : A} (p : x == y)\n {u : B x} {v : B y} (q : u == v [ B ↓ p ])\n → ap f (pair= p q) == ↓-app→cst-out (apd (curry f) p) q\nsplit-ap2 f idp idp = idp\n\n{-\nInteraction of [apd] with function composition.\nThe basic idea is that [apd (g ∘ f) p == apd g (apd f p)] but the version here\nis well-typed. Note that we assume a propositional equality [r] between\n[apd f p] and [q].\n-}\napd-∘ : ∀ {i j k} {A : Type i} {B : A → Type j} {C : (a : A) → B a → Type k}\n (g : {a : A} → Π (B a) (C a)) (f : Π A B) {x y : A} (p : x == y)\n {q : f x == f y [ B ↓ p ]} (r : apd f p == q)\n → apd (g ∘ f) p == ↓-apd-out C r (apd↓ g q)\napd-∘ g f idp idp = idp\n\n{- When [g] is nondependent, it’s much simpler -}\napd-∘' : ∀ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k}\n (g : {a : A} → B a → C a) (f : Π A B) {x y : A} (p : x == y)\n → apd (g ∘ f) p == ap↓ g (apd f p)\napd-∘' g f idp = idp\n\n∘'-apd : ∀ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k}\n (g : {a : A} → B a → C a) (f : Π A B) {x y : A} (p : x == y)\n → ap↓ g (apd f p) == apd (g ∘ f) p\n∘'-apd g f idp = idp\n\n{- And when [f] is nondependent, it’s also a bit simpler -}\napd-∘'' : ∀ {i j k} {A : Type i} {B : Type j} {C : (b : B) → Type k}\n (g : Π B C) (f : A → B) {x y : A} (p : x == y)\n {q : f x == f y} (r : ap f p == q)\n → apd (g ∘ f) p == ↓-ap-out= C f p r (apd g q) --(apd↓ g q)\napd-∘'' g f idp idp = idp\n\n\n{- 2-dimensional coherence conditions -}\n\n-- lhs :\n-- ∀ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k} {f g : Π A B}\n-- {x y : A} {p : x == y} {u : f x == g x} {v : f y == g y}\n-- (k : (u ◃ apd g p) == (apd f p ▹ v))\n-- (h : {a : A} → B a → C a)\n-- → ap h u ◃ apd (h ∘ g) p == ap↓ h (u ◃ apd g p)\n\n-- rhs :\n-- ∀ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k} {f g : Π A B}\n-- {x y : A} {p : x == y} {u : f x == g x} {v : f y == g y}\n-- (k : (u ◃ apd g p) == (apd f p ▹ v))\n-- (h : {a : A} → B a → C a)\n-- → ap↓ h (apd f p ▹ v) == apd (h ∘ f) p ▹ ap h v\n\n-- ap↓-↓-=-in :\n-- ∀ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k} {f g : Π A B}\n-- {x y : A} {p : x == y} {u : f x == g x} {v : f y == g y}\n-- (k : (u ◃ apd g p) == (apd f p ▹ v))\n-- (h : {a : A} → B a → C a)\n-- → ap↓ (λ {a} → ap (h {a = a})) (↓-=-in {p = p} {u = u} {v = v} k)\n-- == ↓-=-in (lhs {f = f} {g = g} k h ∙ ap (ap↓ (λ {a} → h {a = a})) k\n-- ∙ rhs {f = f} {g = g} k h)\n\n{-\nCommutation of [ap↓ (ap h)] and [↓-swap!]. This is \"just\" J, but it’s not as\neasy as it seems.\n-}\n\n-- module Ap↓-swap! {i j k ℓ} {A : Type i} {B : Type j} {C : Type k}\n-- {D : Type ℓ} (h : C → D) (f : A → C) (g : B → C)\n-- {a a' : A} {p : a == a'} {b b' : B} {q : b == b'}\n-- (r : f a == g b') (s : f a' == g b)\n-- (t : r == s ∙ ap g q [ (λ x → f x == g b') ↓ p ])\n-- where\n\n-- lhs : ap h (ap f p ∙' s) == ap (h ∘ f) p ∙' ap h s\n-- lhs = ap-∙' h (ap f p) s ∙ (ap (λ u → u ∙' ap h s) (∘-ap h f p))\n\n-- rhs : ap h (s ∙ ap g q) == ap h s ∙ ap (h ∘ g) q\n-- rhs = ap-∙ h s (ap g q) ∙ (ap (λ u → ap h s ∙ u) (∘-ap h g q))\n\n-- β : ap↓ (ap h) (↓-swap! f g r s t) ==\n-- lhs ◃ ↓-swap! (h ∘ f) (h ∘ g) (ap h r) (ap h s) (ap↓ (ap h) t ▹ rhs)\n-- β with a | a' | p | b | b' | q | r | s | t\n-- β | a | .a | idp | b | .b | idp | r | s | t = coh r s t where\n\n-- T : {x x' : C} (r s : x == x') (t : r == s ∙ idp) → Type _\n-- T r s t =\n-- ap (ap h) (∙'-unit-l s ∙ ! (∙-unit-r s) ∙ ! t) ==\n-- (ap-∙' h idp s ∙ idp)\n-- ∙\n-- (∙'-unit-l (ap h s) ∙\n-- ! (∙-unit-r (ap h s)) ∙\n-- !\n-- (ap (ap h) t ∙'\n-- (ap-∙ h s idp ∙ idp)))\n\n-- coh' : {x x' : C} {r s : x == x'} (t : r == s) → T r s (t ∙ ! (∙-unit-r s))\n-- coh' {r = idp} {s = .idp} idp = idp\n\n-- coh : {x x' : C} (r s : x == x') (t : r == s ∙ idp) → T r s t\n-- coh r s t = transport (λ t → T r s t) (coh2 t (∙-unit-r s)) (coh' (t ∙ ∙-unit-r s)) where\n\n-- coh2 : ∀ {i} {A : Type i} {x y z : A} (p : x == y) (q : y == z) → (p ∙ q) ∙ ! q == p\n-- coh2 idp idp = idp\n\n\n-- module _ {i j k} {A : Type i} {B B' : Type j} {C : Type k} (f : A → C) (g' : B' → B) (g : B → C) where\n\n-- abc : {a a' : A} {p : a == a'} {c c' : B'} {q' : c == c'} {q : g' c == g' c'}\n-- (r : f a == g (g' c')) (s : f a' == g (g' c))\n-- (t : q == ap g' q')\n-- (α : r == s ∙ ap g q [ (λ x → f x == g (g' c')) ↓ p ])\n-- → {!(↓-swap! f g r s α ▹ ?) ∙'2ᵈ ?!} == ↓-swap! f (g ∘ g') {p = p} {q = q'} r s (α ▹ ap (λ u → s ∙ u) (ap (ap g) t ∙ ∘-ap g g' q'))\n-- abc = {!!}\n\n{- Functoriality of application and function extensionality -}\n\n∙-app= : ∀ {i j} {A : Type i} {B : A → Type j} {f g h : Π A B}\n (α : f == g) (β : g == h)\n → α ∙ β == λ= (λ x → app= α x ∙ app= β x)\n∙-app= idp β = λ=-η β\n\n∙-λ= : ∀ {i j} {A : Type i} {B : A → Type j} {f g h : Π A B}\n (α : f ∼ g) (β : g ∼ h)\n → λ= α ∙ λ= β == λ= (λ x → α x ∙ β x)\n∙-λ= α β = ∙-app= (λ= α) (λ= β)\n ∙ ap λ= (λ= (λ x → ap (λ w → w ∙ app= (λ= β) x) (app=-β α x)\n ∙ ap (λ w → α x ∙ w) (app=-β β x)))\n", "meta": {"hexsha": "f7969b418d30211ded8ad2ddd93d12fd6b81e4b5", "size": 14124, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "core/lib/types/Pi.agda", "max_stars_repo_name": "timjb/HoTT-Agda", "max_stars_repo_head_hexsha": "66f800adef943afdf08c17b8ecfba67340fead5e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "core/lib/types/Pi.agda", "max_issues_repo_name": "timjb/HoTT-Agda", "max_issues_repo_head_hexsha": "66f800adef943afdf08c17b8ecfba67340fead5e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "core/lib/types/Pi.agda", "max_forks_repo_name": "timjb/HoTT-Agda", "max_forks_repo_head_hexsha": "66f800adef943afdf08c17b8ecfba67340fead5e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.7569620253, "max_line_length": 138, "alphanum_fraction": 0.3740441801, "num_tokens": 6854, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7090191460821871, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3406685820202251}} {"text": "{-# OPTIONS --without-K #-}\n\nopen import Base\nopen import Homotopy.PushoutDef\nopen import Homotopy.VanKampen.Guide\n\nmodule Homotopy.VanKampen.SplitCode {i} (d : pushout-diag i)\n (l : legend i (pushout-diag.C d)) (a₁ : pushout-diag.A d) where\n\n open pushout-diag d\n open legend l\n\n open import Homotopy.Truncation\n open import Homotopy.PathTruncation\n\n -- Definition.\n module _ where\n private\n data #code-a (a₂ : A) : Set i\n data #code-b (b₂ : B) : Set i\n\n data #code-a a₂ where\n #a : a₁ ≡₀ a₂ → #code-a a₂\n #ba : ∀ n → #code-b (g $ loc n) → f (loc n) ≡₀ a₂ → #code-a a₂\n data #code-b b₂ where\n #ab : ∀ n → #code-a (f $ loc n) → g (loc n) ≡₀ b₂ → #code-b b₂\n\n code-b : B → Set i\n code-b = #code-b\n\n code-a : A → Set i\n code-a = #code-a\n\n postulate -- HIT\n code-a-is-set : ∀ a → is-set (code-a a)\n code-b-is-set : ∀ b → is-set (code-b b)\n\n a : ∀ {a₂} → a₁ ≡₀ a₂ → code-a a₂\n a = #a\n\n infixl 6 a\n syntax a co = ⟧a co\n\n ba : ∀ {a₂} n → code-b (g $ loc n) → f (loc n) ≡₀ a₂ → code-a a₂\n ba = #ba\n\n infixl 6 ba\n syntax ba n co p = co b⟦ n ⟧a p\n\n ab : ∀ {b₂} n → code-a (f $ loc n) → g (loc n) ≡₀ b₂ → code-b b₂\n ab = #ab\n\n infixl 6 ab\n syntax ab n co p = co a⟦ n ⟧b p\n\n postulate -- HIT\n code-a-refl-refl : ∀ n (co : code-a (f $ loc n))\n → co a⟦ n ⟧b refl₀ b⟦ n ⟧a refl₀ ≡ co\n code-b-refl-refl : ∀ n (co : code-b (g $ loc n))\n → co b⟦ n ⟧a refl₀ a⟦ n ⟧b refl₀ ≡ co\n -- The other diriction is fine.\n -- There's no canonical choice between two, I think.\n code-ab-swap : ∀ n₁ co n₂ (r : loc n₁ ≡₀ loc n₂)\n → co a⟦ n₁ ⟧b ap₀ g r b⟦ n₂ ⟧a refl₀\n ≡ co a⟦ n₁ ⟧b refl₀ b⟦ n₁ ⟧a ap₀ f r\n\n -- Dependent recursor.\n code-rec : ∀ {j}\n (P-a : ∀ {a₂} → code-a a₂ → Set j) ⦃ _ : ∀ {a₂} co → is-set $ P-a {a₂} co ⦄\n (P-b : ∀ {b₂} → code-b b₂ → Set j) ⦃ _ : ∀ {b₂} co → is-set $ P-b {b₂} co ⦄\n (h₀-a : ∀ {a₂} (p : _ ≡₀ a₂) → P-a (a p))\n (h₀-ba : ∀ {a₂} n {co} (_ : P-b co) (p : _ ≡₀ a₂)\n → P-a (co b⟦ n ⟧a p))\n (h₀-ab : ∀ {b₂} n {co} (_ : P-a co) (p : _ ≡₀ b₂)\n → P-b (co a⟦ n ⟧b p))\n (h₁-a-rr : ∀ n {co} (pco : P-a co) →\n transport (P-a {f $ loc n}) (code-a-refl-refl n co)\n (h₀-ba n (h₀-ab n pco $ refl₀) $ refl₀)\n ≡ pco)\n (h₁-b-rr : ∀ n {co} (pco : P-b co) →\n transport (P-b {g $ loc n}) (code-b-refl-refl n co)\n (h₀-ab n (h₀-ba n pco $ refl₀) $ refl₀)\n ≡ pco)\n (h₁-ab-s : ∀ n₁ {co} (pco : P-a co) n₂ r →\n transport (P-a {f $ loc n₂}) (code-ab-swap n₁ co n₂ r)\n (h₀-ba n₂ (h₀-ab n₁ pco $ ap₀ g r) $ refl₀)\n ≡ (h₀-ba n₁ (h₀-ab n₁ pco $ refl₀) $ ap₀ f r))\n → (∀ {a₂} (co : code-a a₂) → P-a co)\n × (∀ {b₂} (co : code-b b₂) → P-b co)\n code-rec P-a P-b h₀-a h₀-ba h₀-ab _ _ _ = elim-a , elim-b\n where\n elim-a : ∀ {a₂} (co : code-a a₂) → P-a co\n elim-b : ∀ {b₂} (co : code-b b₂) → P-b co\n elim-a (#a p) = h₀-a p\n elim-a (#ba n co p) = h₀-ba n (elim-b co) p\n elim-b (#ab n co p) = h₀-ab n (elim-a co) p\n\n -- This is actually \"partially dependent\".\n -- P-a and P-b are indexed by the end points.\n -- This is fine because they are fixed in\n -- all transports in 1-cells.\n code-rec-nondep : ∀ {j}\n (P-a : A → Set j) ⦃ _ : ∀ a → is-set $ P-a a ⦄\n (P-b : B → Set j) ⦃ _ : ∀ b → is-set $ P-b b ⦄\n (h₀-a : ∀ {a₂} (p : a₁ ≡₀ a₂) → P-a a₂)\n (h₀-ba : ∀ {a₂} n {co : code-b (g $ loc n)} → P-b (g $ loc n)\n → f (loc n) ≡₀ a₂ → P-a a₂)\n (h₀-ab : ∀ {b₂} n {co : code-a (f $ loc n)} → P-a (f $ loc n)\n → g (loc n) ≡₀ b₂ → P-b b₂)\n (h₁-a-rr : ∀ n {co} (pco : P-a (f $ loc n))\n → h₀-ba n {co a⟦ n ⟧b refl₀} (h₀-ab n {co} pco $ refl₀) refl₀ ≡ pco)\n (h₁-b-rr : ∀ n {co} (pco : P-b (g $ loc n))\n → h₀-ab n {co b⟦ n ⟧a refl₀} (h₀-ba n {co} pco $ refl₀) refl₀ ≡ pco)\n (h₁-ab-s : ∀ n₁ {co : code-a $ f $ loc n₁} (pco : P-a $ f $ loc n₁) n₂ (r : loc n₁ ≡₀ loc n₂) →\n h₀-ba n₂ {co a⟦ n₁ ⟧b ap₀ g r} (h₀-ab n₁ {co} pco $ ap₀ g r) refl₀\n ≡ h₀-ba n₁ {co a⟦ n₁ ⟧b refl₀} (h₀-ab n₁ {co} pco $ refl₀) (ap₀ f r))\n → (∀ {a₂} (co : code-a a₂) → P-a a₂)\n × (∀ {b₂} (co : code-b b₂) → P-b b₂)\n code-rec-nondep P-a P-b h₀-a h₀-ba h₀-ab _ _ _ = elim-a , elim-b\n where\n elim-a : ∀ {a₂} (co : code-a a₂) → P-a a₂\n elim-b : ∀ {b₂} (co : code-b b₂) → P-b b₂\n elim-a (#a p) = h₀-a p\n elim-a (#ba c co p) = h₀-ba c {co} (elim-b co) p\n elim-b (#ab c co p) = h₀-ab c {co} (elim-a co) p\n\n module _ where\n trans-a : ∀ {y z} (q : y ≡ z) (p : a₁ ≡₀ y)\n → transport code-a q (⟧a p) ≡ ⟧a (p ∘₀ proj q)\n trans-a refl p = ap a (! (refl₀-right-unit p))\n\n trans-ba : ∀ {y z} (q : y ≡ z) n co (p : f (loc n) ≡₀ y)\n → transport code-a q (co b⟦ n ⟧a p) ≡ co b⟦ n ⟧a p ∘₀ proj q\n trans-ba refl n co p = ap (ba n co) (! (refl₀-right-unit p))\n\n trans-ab : ∀ {y b} (q : y ≡ b) n co (p : g (loc n) ≡₀ y)\n → transport code-b q (co a⟦ n ⟧b p) ≡ co a⟦ n ⟧b p ∘₀ proj q\n trans-ab refl n co p = ap (ab n co) (! (refl₀-right-unit p))\n\n module _ where\n -- all-paths\n abstract\n code-has-all-cells₂-a : ∀ {a} {x y : code-a a} (p q : x ≡ y) → p ≡ q\n code-has-all-cells₂-a = prop-has-all-paths $ code-a-is-set _ _ _\n\n code-has-all-cells₂-b : ∀ {b} {x y : code-b b} (p q : x ≡ y) → p ≡ q\n code-has-all-cells₂-b = prop-has-all-paths $ code-b-is-set _ _ _\n\n code-a-rec : ∀ {j}\n (P-a : ∀ {a₂} → code-a a₂ → Set j) ⦃ _ : ∀ {a₂} co → is-set $ P-a {a₂} co ⦄\n (h₀-a : ∀ {a₂} (p : _ ≡₀ a₂) → P-a (⟧a p))\n (h₀-ba : ∀ {a₂} n co (p : _ ≡₀ a₂)\n → P-a (co b⟦ n ⟧a p))\n (h₁-a-rr : ∀ n {co} (pco : P-a co) →\n transport (P-a {f $ loc n}) (code-a-refl-refl n co)\n (h₀-ba n (co a⟦ n ⟧b refl₀) $ refl₀)\n ≡ pco)\n → (∀ {a₂} (co : code-a a₂) → P-a co)\n code-a-rec {j} P-a ⦃ P-a-is-set ⦄ h₀-a h₀-ba h₁-a-rr = π₁ $ code-rec\n P-a ⦃ P-a-is-set ⦄\n (λ _ → unit) ⦃ λ _ → unit-is-set ⦄\n h₀-a\n (λ n {co} _ p → h₀-ba n co p)\n (λ _ _ _ → tt)\n h₁-a-rr\n (λ _ _ → prop-has-all-paths unit-is-prop _ _)\n -- FIXME This proof is too ugly.\n (λ n₁ {co} (pco : P-a co) n₂ r →\n transport (P-a {f $ loc n₂}) (code-ab-swap n₁ co n₂ r)\n (h₀-ba n₂ (co a⟦ n₁ ⟧b ap₀ g r) $ refl₀)\n ≡⟨ ! $ h₁-a-rr n₂ _ ⟩\n transport (P-a {f $ loc n₂}) (code-a-refl-refl n₂ _)\n (h₀-ba n₂ (_ a⟦ n₂ ⟧b refl₀) $ refl₀)\n ≡⟨ ap (λ x → transport (P-a {f $ loc n₂}) (code-a-refl-refl n₂ _)\n $ h₀-ba n₂ (_ a⟦ n₂ ⟧b refl₀) $ refl₀) $ code-ab-swap n₁ co n₂ r ⟩\n transport (P-a {f $ loc n₂}) (code-a-refl-refl n₂ _)\n (h₀-ba n₂ (_ a⟦ n₂ ⟧b refl₀) $ refl₀)\n ≡⟨ h₁-a-rr n₂ _ ⟩∎\n h₀-ba n₁ (co a⟦ n₁ ⟧b refl₀) (ap₀ f r)\n ∎)\n\n code-b-rec : ∀ {j}\n (P-b : ∀ {b₂} → code-b b₂ → Set j) ⦃ _ : ∀ {b₂} co → is-set $ P-b {b₂} co ⦄\n (h₀-ab : ∀ {b₂} n co (p : _ ≡₀ b₂)\n → P-b (co a⟦ n ⟧b p))\n (h₁-b-rr : ∀ n {co} (pco : P-b co) →\n transport (P-b {g $ loc n}) (code-b-refl-refl n co)\n (h₀-ab n (co b⟦ n ⟧a refl₀) $ refl₀)\n ≡ pco)\n → (∀ {b₂} (co : code-b b₂) → P-b co)\n code-b-rec {j} P-b ⦃ P-b-is-set ⦄ h₀-ab h₁-b-rr = π₂ $ code-rec\n (λ _ → unit) ⦃ λ _ → unit-is-set ⦄\n P-b ⦃ P-b-is-set ⦄\n (λ _ → tt)\n (λ _ _ _ → tt)\n (λ n {co} _ p → h₀-ab n co p)\n (λ _ _ → prop-has-all-paths unit-is-prop _ _)\n h₁-b-rr\n (λ _ _ _ _ → prop-has-all-paths unit-is-prop _ _)\n\n -- Derived 1-cells.\n module _ where\n abstract\n code-ba-swap : ∀ n₁ co n₂ (r : loc n₁ ≡₀ loc n₂)\n → co b⟦ n₁ ⟧a ap₀ f r a⟦ n₂ ⟧b refl₀\n ≡ co b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b ap₀ g r\n code-ba-swap n₁ co n₂ r =\n co b⟦ n₁ ⟧a ap₀ f r a⟦ n₂ ⟧b refl₀\n ≡⟨ ap (λ x → x b⟦ n₁ ⟧a ap₀ f r a⟦ n₂ ⟧b refl₀) $ ! $ code-b-refl-refl n₁ co ⟩\n co b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b refl₀ b⟦ n₁ ⟧a ap₀ f r a⟦ n₂ ⟧b refl₀\n ≡⟨ ap (λ x → x a⟦ n₂ ⟧b refl₀) $ ! $ code-ab-swap n₁ (co b⟦ n₁ ⟧a refl₀) n₂ r ⟩\n co b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b ap₀ g r b⟦ n₂ ⟧a refl₀ a⟦ n₂ ⟧b refl₀\n ≡⟨ code-b-refl-refl n₂ $ co b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b ap₀ g r ⟩∎\n co b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b ap₀ g r\n ∎\n\n abstract\n -- Old (provable) rules.\n code-a-merge : ∀ {a₂} n p₁ (p₂ : _ ≡₀ a₂)\n → ⟧a p₁ a⟦ n ⟧b refl₀ b⟦ n ⟧a p₂\n ≡ ⟧a p₁ ∘₀ p₂\n code-a-merge {a₂} n p₁ = π₀-extend\n ⦃ λ _ → ≡-is-set $ code-a-is-set a₂ ⦄\n ( λ p₂ →\n ⟧a p₁ a⟦ n ⟧b refl₀ b⟦ n ⟧a proj p₂\n ≡⟨ ! $ trans-ba p₂ n (⟧a p₁ a⟦ n ⟧b refl₀) refl₀ ⟩\n transport code-a p₂ (⟧a p₁ a⟦ n ⟧b refl₀ b⟦ n ⟧a refl₀)\n ≡⟨ ap (transport code-a p₂) $ code-a-refl-refl n (⟧a p₁) ⟩\n transport code-a p₂ (⟧a p₁)\n ≡⟨ trans-a p₂ p₁ ⟩∎\n ⟧a p₁ ∘₀ proj p₂\n ∎)\n\n abstract\n code-ab-merge : ∀ {b₂} n₁ co n₂ p₁ (p₂ : _ ≡₀ b₂)\n → co a⟦ n₁ ⟧b p₁ b⟦ n₂ ⟧a refl₀ a⟦ n₂ ⟧b p₂\n ≡ co a⟦ n₁ ⟧b p₁ ∘₀ p₂\n code-ab-merge {b₂} n₁ co n₂ p₁ = π₀-extend\n ⦃ λ _ → ≡-is-set $ code-b-is-set b₂ ⦄\n ( λ p₂ →\n co a⟦ n₁ ⟧b p₁ b⟦ n₂ ⟧a refl₀ a⟦ n₂ ⟧b proj p₂\n ≡⟨ ! $ trans-ab p₂ n₂ (co a⟦ n₁ ⟧b p₁ b⟦ n₂ ⟧a refl₀) refl₀ ⟩\n transport code-b p₂ (co a⟦ n₁ ⟧b p₁ b⟦ n₂ ⟧a refl₀ a⟦ n₂ ⟧b refl₀)\n ≡⟨ ap (transport code-b p₂) $ code-b-refl-refl n₂ (co a⟦ n₁ ⟧b p₁) ⟩\n transport code-b p₂ (co a⟦ n₁ ⟧b p₁)\n ≡⟨ trans-ab p₂ n₁ co p₁ ⟩∎\n co a⟦ n₁ ⟧b p₁ ∘₀ proj p₂\n ∎)\n\n abstract\n code-ba-merge : ∀ {a₂} n₁ co n₂ p₁ (p₂ : _ ≡₀ a₂)\n → co b⟦ n₁ ⟧a p₁ a⟦ n₂ ⟧b refl₀ b⟦ n₂ ⟧a p₂\n ≡ co b⟦ n₁ ⟧a p₁ ∘₀ p₂\n code-ba-merge {a₂} n₁ co n₂ p₁ = π₀-extend\n ⦃ λ _ → ≡-is-set $ code-a-is-set a₂ ⦄\n ( λ p₂ →\n co b⟦ n₁ ⟧a p₁ a⟦ n₂ ⟧b refl₀ b⟦ n₂ ⟧a proj p₂\n ≡⟨ ! $ trans-ba p₂ n₂ (co b⟦ n₁ ⟧a p₁ a⟦ n₂ ⟧b refl₀) refl₀ ⟩\n transport code-a p₂ (co b⟦ n₁ ⟧a p₁ a⟦ n₂ ⟧b refl₀ b⟦ n₂ ⟧a refl₀)\n ≡⟨ ap (transport code-a p₂) $ code-a-refl-refl n₂ (co b⟦ n₁ ⟧a p₁) ⟩\n transport code-a p₂ (co b⟦ n₁ ⟧a p₁)\n ≡⟨ trans-ba p₂ n₁ co p₁ ⟩∎\n co b⟦ n₁ ⟧a p₁ ∘₀ proj p₂\n ∎)\n\n abstract\n code-a-shift : ∀ {b₂} n₁ p₁ n₂ (r : loc n₁ ≡₀ loc n₂) (p₂ : _ ≡₀ b₂)\n → ⟧a p₁ ∘₀ ap₀ f r a⟦ n₂ ⟧b p₂\n ≡ ⟧a p₁ a⟦ n₁ ⟧b ap₀ g r ∘₀ p₂\n code-a-shift n₁ p₁ n₂ r p₂ =\n ⟧a p₁ ∘₀ ap₀ f r a⟦ n₂ ⟧b p₂\n ≡⟨ ap (λ x → x a⟦ n₂ ⟧b p₂) $ ! $ code-a-merge n₁ p₁ (ap₀ f r) ⟩\n ⟧a p₁ a⟦ n₁ ⟧b refl₀ b⟦ n₁ ⟧a ap₀ f r a⟦ n₂ ⟧b p₂\n ≡⟨ ap (λ x → x a⟦ n₂ ⟧b p₂) $ ! $ code-ab-swap n₁ (⟧a p₁) n₂ r ⟩\n ⟧a p₁ a⟦ n₁ ⟧b ap₀ g r b⟦ n₂ ⟧a refl₀ a⟦ n₂ ⟧b p₂\n ≡⟨ code-ab-merge n₁ (⟧a p₁) n₂ (ap₀ g r) p₂ ⟩∎\n ⟧a p₁ a⟦ n₁ ⟧b ap₀ g r ∘₀ p₂\n ∎\n\n abstract\n code-ba-shift : ∀ {b₂} n₁ co n₂ p₁ n₃ (r : loc n₂ ≡₀ loc n₃) (p₂ : _ ≡₀ b₂)\n → co b⟦ n₁ ⟧a p₁ ∘₀ ap₀ f r a⟦ n₃ ⟧b p₂\n ≡ co b⟦ n₁ ⟧a p₁ a⟦ n₂ ⟧b ap₀ g r ∘₀ p₂\n code-ba-shift n₁ co n₂ p₁ n₃ r p₂ =\n co b⟦ n₁ ⟧a p₁ ∘₀ ap₀ f r a⟦ n₃ ⟧b p₂\n ≡⟨ ap (λ x → x a⟦ n₃ ⟧b p₂) $ ! $ code-ba-merge n₁ co n₂ p₁ (ap₀ f r) ⟩\n co b⟦ n₁ ⟧a p₁ a⟦ n₂ ⟧b refl₀ b⟦ n₂ ⟧a ap₀ f r a⟦ n₃ ⟧b p₂\n ≡⟨ ap (λ x → x a⟦ n₃ ⟧b p₂) $ ! $ code-ab-swap n₂ (co b⟦ n₁ ⟧a p₁) n₃ r ⟩\n co b⟦ n₁ ⟧a p₁ a⟦ n₂ ⟧b ap₀ g r b⟦ n₃ ⟧a refl₀ a⟦ n₃ ⟧b p₂\n ≡⟨ code-ab-merge n₂ (co b⟦ n₁ ⟧a p₁) n₃ (ap₀ g r) p₂ ⟩∎\n co b⟦ n₁ ⟧a p₁ a⟦ n₂ ⟧b ap₀ g r ∘₀ p₂\n ∎\n\n abstract\n code-ab-shift : ∀ {a₂} n₁ co n₂ p₁ n₃ (r : loc n₂ ≡₀ loc n₃) (p₂ : _ ≡₀ a₂)\n → co a⟦ n₁ ⟧b p₁ ∘₀ ap₀ g r b⟦ n₃ ⟧a p₂\n ≡ co a⟦ n₁ ⟧b p₁ b⟦ n₂ ⟧a ap₀ f r ∘₀ p₂\n code-ab-shift n₁ co n₂ p₁ n₃ r p₂ =\n co a⟦ n₁ ⟧b p₁ ∘₀ ap₀ g r b⟦ n₃ ⟧a p₂\n ≡⟨ ap (λ x → x b⟦ n₃ ⟧a p₂) $ ! $ code-ab-merge n₁ co n₂ p₁ (ap₀ g r) ⟩\n co a⟦ n₁ ⟧b p₁ b⟦ n₂ ⟧a refl₀ a⟦ n₂ ⟧b ap₀ g r b⟦ n₃ ⟧a p₂\n ≡⟨ ap (λ x → x b⟦ n₃ ⟧a p₂) $ ! $ code-ba-swap n₂ (co a⟦ n₁ ⟧b p₁) n₃ r ⟩\n co a⟦ n₁ ⟧b p₁ b⟦ n₂ ⟧a ap₀ f r a⟦ n₃ ⟧b refl₀ b⟦ n₃ ⟧a p₂\n ≡⟨ code-ba-merge n₂ (co a⟦ n₁ ⟧b p₁) n₃ (ap₀ f r) p₂ ⟩∎\n co a⟦ n₁ ⟧b p₁ b⟦ n₂ ⟧a ap₀ f r ∘₀ p₂\n ∎\n\n -- Definition of code\n module _ where\n P : Set i\n P = pushout d\n\n -- Basic conversion\n a⇒b : ∀ n → code-a (f $ loc n) → code-b (g $ loc n)\n a⇒b n co = co a⟦ n ⟧b refl₀\n\n b⇒a : ∀ n → code-b (g $ loc n) → code-a (f $ loc n)\n b⇒a n co = co b⟦ n ⟧a refl₀\n\n private\n -- The head of the code\n code-glue-eq-loc : ∀ n → code-a (f $ loc n) ≃ code-b (g $ loc n)\n code-glue-eq-loc n = a⇒b n ,\n iso-is-eq (a⇒b n) (b⇒a n) (code-b-refl-refl n) (code-a-refl-refl n)\n\n abstract\n trans-code-a-r : ∀ n₁ n₂ (r : loc n₁ ≡ loc n₂) (co : code-a $ f $ loc n₁)\n → transport (code-a ◯ f) r co ≡ co a⟦ n₁ ⟧b refl₀ b⟦ n₁ ⟧a proj (ap f r)\n trans-code-a-r n₁ n₂ r co =\n transport (code-a ◯ f) r co\n ≡⟨ ! $ trans-ap code-a f r co ⟩\n transport code-a (ap f r) co\n ≡⟨ ap (transport code-a $ ap f r) $ ! $ code-a-refl-refl n₁ co ⟩\n transport code-a (ap f r) (co a⟦ n₁ ⟧b refl₀ b⟦ n₁ ⟧a refl₀)\n ≡⟨ trans-ba (ap f r) n₁ _ refl₀ ⟩∎\n co a⟦ n₁ ⟧b refl₀ b⟦ n₁ ⟧a proj (ap f r)\n ∎\n\n abstract\n trans-code-b-r : ∀ n₁ n₂ (r : loc n₁ ≡ loc n₂) (co : code-b $ g $ loc n₁)\n → transport (code-b ◯ g) r co ≡ co b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b proj (ap g r)\n trans-code-b-r n₁ n₂ r co =\n transport (code-b ◯ g) r co\n ≡⟨ ! $ trans-ap code-b g r co ⟩\n transport code-b (ap g r) co\n ≡⟨ ap (transport code-b $ ap g r) $ ! $ code-b-refl-refl n₁ co ⟩\n transport code-b (ap g r) (co b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b refl₀)\n ≡⟨ trans-ab (ap g r) n₁ _ refl₀ ⟩∎\n co b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b proj (ap g r)\n ∎\n\n private\n abstract\n code-glue-eq-route : ∀ n₁ n₂ r\n → transport (λ c → code-a (f c) ≃ code-b (g c)) r (code-glue-eq-loc n₁)\n ≡ code-glue-eq-loc n₂\n code-glue-eq-route n₁ n₂ r = equiv-eq $ funext λ co →\n π₁ (transport (λ c → code-a (f c) ≃ code-b (g c)) r (code-glue-eq-loc n₁)) co\n ≡⟨ ap (λ x → x co) $ app-trans\n (λ c → code-a (f c) ≃ code-b (g c))\n (λ c → code-a (f c) → code-b (g c))\n (λ _ → π₁) r (code-glue-eq-loc n₁) ⟩\n transport (λ c → code-a (f c) → code-b (g c)) r (a⇒b n₁) co\n ≡⟨ trans-→ (code-a ◯ f) (code-b ◯ g) r (a⇒b n₁) co ⟩\n transport (code-b ◯ g) r (a⇒b n₁ $ transport (code-a ◯ f) (! r) co)\n ≡⟨ ap (transport (code-b ◯ g) r ◯ a⇒b n₁) $ trans-code-a-r _ _ (! r) co ⟩\n transport (code-b ◯ g) r (co a⟦ n₂ ⟧b refl₀ b⟦ n₂ ⟧a proj (ap f $ ! r) a⟦ n₁ ⟧b refl₀)\n ≡⟨ ap (λ x → transport (code-b ◯ g) r $ x a⟦ n₁ ⟧b refl₀) $ ! $ code-ab-swap _ co _ (proj $ ! r) ⟩\n transport (code-b ◯ g) r (co a⟦ n₂ ⟧b proj (ap g $ ! r) b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b refl₀)\n ≡⟨ ap (transport (code-b ◯ g) r) $ code-b-refl-refl n₁ _ ⟩\n transport (code-b ◯ g) r (co a⟦ n₂ ⟧b proj (ap g $ ! r))\n ≡⟨ trans-code-b-r _ _ r _ ⟩\n co a⟦ n₂ ⟧b proj (ap g $ ! r) b⟦ n₁ ⟧a refl₀ a⟦ n₁ ⟧b proj (ap g r)\n ≡⟨ code-ab-merge n₂ co n₁ _ _ ⟩\n co a⟦ n₂ ⟧b proj (ap g (! r) ∘ ap g r)\n ≡⟨ ap (λ x → co a⟦ n₂ ⟧b proj x) $ concat-ap g (! r) r ⟩\n co a⟦ n₂ ⟧b proj (ap g $ ! r ∘ r)\n ≡⟨ ap (λ x → co a⟦ n₂ ⟧b proj (ap g x)) $ opposite-left-inverse r ⟩∎\n a⇒b n₂ co\n ∎\n\n private\n code-glue-eq : ∀ c → code-a (f c) ≃ code-b (g c)\n code-glue-eq = visit-fiber-rec l\n (λ c → code-a (f c) ≃ code-b (g c))\n ⦃ λ _ → ≃-is-set (code-a-is-set _) (code-b-is-set _) ⦄\n code-glue-eq-loc\n code-glue-eq-route\n\n -- The data type!\n code : P → Set i\n code = pushout-rec-nondep (Set i) code-a code-b (eq-to-path ◯ code-glue-eq)\n\n -- Useful lemma\n abstract\n trans-code-glue-loc : ∀ n₂ co → transport code (glue $ loc n₂) co ≡ a⇒b n₂ co\n trans-code-glue-loc n₂ co =\n transport code (glue $ loc n₂) co\n ≡⟨ ! $ trans-ap (λ X → X) code (glue $ loc n₂) co ⟩\n transport (λ X → X) (ap code $ glue $ loc n₂) co\n ≡⟨ ap (λ x → transport (λ X → X) x co)\n $ pushout-β-glue-nondep (Set i) code-a code-b (eq-to-path ◯ code-glue-eq) (loc n₂) ⟩\n transport (λ X → X) (eq-to-path $ code-glue-eq $ loc n₂) co\n ≡⟨ ap (λ x → transport (λ X → X) (eq-to-path x) co)\n $ visit-fiber-β-loc l\n (λ c → code-a (f c) ≃ code-b (g c))\n ⦃ λ _ → ≃-is-set (code-a-is-set _) (code-b-is-set _) ⦄\n code-glue-eq-loc\n code-glue-eq-route\n n₂ ⟩\n transport (λ X → X) (eq-to-path $ code-glue-eq-loc n₂) co\n ≡⟨ trans-id-eq-to-path (code-glue-eq-loc n₂) co ⟩∎\n a⇒b n₂ co\n ∎\n\n abstract\n trans-code-!glue-loc : ∀ n₂ co → transport code (! $ glue $ loc n₂) co ≡ b⇒a n₂ co\n trans-code-!glue-loc n₂ co = move!-transp-right code (glue $ loc n₂) co (b⇒a n₂ co) $ ! $\n transport code (glue $ loc n₂) (b⇒a n₂ co)\n ≡⟨ trans-code-glue-loc n₂ (b⇒a n₂ co) ⟩\n a⇒b n₂ (b⇒a n₂ co)\n ≡⟨ code-b-refl-refl n₂ co ⟩∎\n co\n ∎\n\n -- Truncation level\n abstract\n code-is-set : ∀ p → is-set $ code p\n code-is-set = pushout-rec\n (λ p → is-set $ code p)\n code-a-is-set\n code-b-is-set\n (λ _ → prop-has-all-paths is-set-is-prop _ _)\n", "meta": {"hexsha": "f2dd1ca0c3be7a01e26e905fb6eaf42e31eff17d", "size": 18199, "ext": "agda", "lang": "Agda", "max_stars_repo_path": "old/Homotopy/VanKampen/SplitCode.agda", "max_stars_repo_name": "UlrikBuchholtz/HoTT-Agda", "max_stars_repo_head_hexsha": "f8fa68bf753d64d7f45556ca09d0da7976709afa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 294, "max_stars_repo_stars_event_min_datetime": "2015-01-09T16:23:23.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T13:54:45.000Z", "max_issues_repo_path": "old/Homotopy/VanKampen/SplitCode.agda", "max_issues_repo_name": "nicolaikraus/HoTT-Agda", "max_issues_repo_head_hexsha": "939a2d83e090fcc924f69f7dfa5b65b3b79fe633", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 31, "max_issues_repo_issues_event_min_datetime": "2015-03-05T20:09:00.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-03T19:15:25.000Z", "max_forks_repo_path": "old/Homotopy/VanKampen/SplitCode.agda", "max_forks_repo_name": "nicolaikraus/HoTT-Agda", "max_forks_repo_head_hexsha": "939a2d83e090fcc924f69f7dfa5b65b3b79fe633", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 50, "max_forks_repo_forks_event_min_datetime": "2015-01-10T01:48:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-14T03:03:25.000Z", "avg_line_length": 41.9331797235, "max_line_length": 112, "alphanum_fraction": 0.4499697786, "num_tokens": 8272, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3406685761117337}} {"text": "{-# OPTIONS --erased-cubical --safe #-}\n\nmodule Music where\n\nopen import Data.Nat using (ℕ; zero; suc; _+_; _*_; _≤_; _≤?_)\nopen import Data.Integer using (ℤ; +_)\nopen import Data.List using (List; foldr; []; _∷_; reverse; sum; map)\nopen import Data.Product using (_×_; _,_)\nopen import Data.Sum using (_⊎_; inj₁; inj₂)\nopen import Data.Vec using (Vec; []; _∷_; replicate; concat; zipWith; toList; _++_; foldr₁; take; drop) renaming (map to vmap)\nopen import Function using (_∘_)\n\nopen import Data.Nat.Properties using (<⇒≤)\nopen import Relation.Nullary using (yes; no)\n\nopen import Relation.Binary.PropositionalEquality using (sym; subst)\n\nopen import Nat\nopen import Note\nopen import Pitch\nopen import Interval\n\n-- A point in the music grid, which can either be a tone,\n-- a continuation of a previous tone, or a rest.\ndata Point : Set where\n tone : Pitch → Point\n hold : Pitch → Point\n rest : Point\n\ndata Melody (n : ℕ) : Set where\n melody : Vec Point n → Melody n\n\nunmelody : {n : ℕ} → Melody n → Vec Point n\nunmelody (melody ps) = ps\n\ninfixr 5 _m++_\n_m++_ : {m n : ℕ} → Melody m → Melody n → Melody (m + n)\nmelody a m++ melody b = melody (a ++ b)\n\nnote→melody : (n : Note) → Melody (noteDuration n)\nnote→melody (tone zero p) = melody []\nnote→melody (tone (suc d) p) = melody (tone p ∷ replicate (hold p))\nnote→melody (rest _) = melody (replicate rest)\n\nnotes→melody : (ns : List Note) → Melody (sum (map noteDuration ns))\nnotes→melody [] = melody []\nnotes→melody (n ∷ ns) = note→melody n m++ notes→melody ns\n\npitches→melody : {n : ℕ} → (d : Duration) → (ps : Vec Pitch n) → Melody (n * d)\npitches→melody d ps = melody (concat (vmap (unmelody ∘ note→melody ∘ tone d) ps))\n\n-- Assumes melody is well-formed in that a held note has the\n-- same pitch as the note before it.\n-- Does not consolidate rests.\nmelody→notes : {n : ℕ} → Melody n → List Note\nmelody→notes (melody m) = (reverse ∘ mn 0 ∘ reverse ∘ toList) m\n where mn : ℕ → List Point → List Note -- c is the number of held points\n mn c [] = []\n mn c (tone p ∷ ps) = tone (suc c) p ∷ mn 0 ps\n mn c (hold _ ∷ ps) = mn (suc c) ps\n mn c (rest ∷ ps) = rest 1 ∷ mn 0 ps\n\ntransposePoint : ℤ → Point → Point\ntransposePoint k (tone p) = tone (transposePitch k p)\ntransposePoint k (hold p) = hold (transposePitch k p)\ntransposePoint k rest = rest\n\ntransposeMelody : {n : ℕ} → ℤ → Melody n → Melody n\ntransposeMelody k = melody ∘ vmap (transposePoint k) ∘ unmelody\n\ndata Chord (n : ℕ) : Set where\n chord : Vec Point n → Chord n\n\nunchord : {n : ℕ} → Chord n → Vec Point n\nunchord (chord ps) = ps\n\n-- We represent music as a v × d grid where v is the number of voices and d is the duration.\n-- The primary representation is as parallel melodies (counterpoint).\ndata Counterpoint (v : ℕ) (d : ℕ): Set where\n cp : Vec (Melody d) v → Counterpoint v d\n\nuncp : {v d : ℕ} → Counterpoint v d → Vec (Melody d) v\nuncp (cp m) = m\n\n-- An alternative representation of music is as a series of chords (harmonic progression).\ndata Harmony (v : ℕ) (d : ℕ): Set where\n harmony : Vec (Chord v) d → Harmony v d\n\nunharmony : {v d : ℕ} → Harmony v d → Vec (Chord v) d\nunharmony (harmony h) = h\n\npitches→harmony : {n : ℕ} (d : Duration) → (ps : Vec Pitch n) → Harmony n d\npitches→harmony zero ps = harmony []\npitches→harmony (suc d) ps = harmony (chord (vmap tone ps) ∷ replicate (chord (vmap hold ps)))\n\npitchPair→Harmony : (d : Duration) → PitchPair → Harmony 2 d\npitchPair→Harmony d (p , q) = pitches→harmony d (p ∷ q ∷ [])\n\npitchInterval→Harmony : (d : Duration) → PitchInterval → Harmony 2 d\npitchInterval→Harmony d = pitchPair→Harmony d ∘ pitchIntervalToPitchPair\n\n{-\npitchIntervalsToCounterpoint : PitchInterval → Counterpoint\npitchIntervalsToCounterpoint = pitchPairToCounterpoint ∘ pitchIntervalToPitchPair\n-}\n\naddEmptyVoice : {v d : ℕ} → Harmony v d → Harmony (suc v) d\naddEmptyVoice (harmony h) = harmony (vmap (chord ∘ (rest ∷_) ∘ unchord) h)\n\ninfixl 5 _+H+_\n_+H+_ : {v d d' : ℕ} → Harmony v d → Harmony v d' → Harmony v (d + d')\nh +H+ h' = harmony (unharmony h ++ unharmony h')\n\nfoldIntoHarmony : {k n : ℕ} (ds : Vec Duration (suc k)) → (pss : Vec (Vec Pitch n) (suc k)) → Harmony n (foldr₁ _+_ ds)\nfoldIntoHarmony (d ∷ []) (ps ∷ []) = pitches→harmony d ps\nfoldIntoHarmony (d ∷ d' ∷ ds) (ps ∷ ps' ∷ pss) = (pitches→harmony d ps) +H+ (foldIntoHarmony (d' ∷ ds) (ps' ∷ pss))\n\n-- matrix transposition\nmtranspose : {A : Set}{m n : ℕ} → Vec (Vec A n) m → Vec (Vec A m) n\nmtranspose [] = replicate []\nmtranspose (xs ∷ xss) = zipWith _∷_ xs (mtranspose xss)\n\ncounterpoint→harmony : {v d : ℕ} → Counterpoint v d → Harmony v d\ncounterpoint→harmony = harmony ∘ vmap chord ∘ mtranspose ∘ vmap unmelody ∘ uncp\n\nharmony→counterpoint : {v d : ℕ} → Harmony v d → Counterpoint v d\nharmony→counterpoint = cp ∘ vmap melody ∘ mtranspose ∘ vmap unchord ∘ unharmony\n\n-- Fix length of a melody, either truncating or padding with rests\nfixLength : {m : ℕ} → (n : ℕ) → Melody m → Melody n\nfixLength {m} n (melody ns) with <-∨-≥ n m\n... | inj₁ n