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9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Computability.DegreeRigidity.Syntax.BoundedSupport
namespace OAI
namespace TuringRigidity.BoundedForcing
open RecursiveNames TransitiveNameModel BoundedSetTheory AtomicForcing
universe u
namespace CertificateCode
def slots : ℕ → ℕ → (ℕ → ℕ) → Formula
| 0, d, _v => .equal d d
| n+1, d, v => .conj (slots n d v) (.member (v n) d)
theorem eval_slots (n d : ℕ) (v : ℕ → ℕ) (env : ℕ → ZFSet.{u}) :
(slots n d v).Eval env ↔ ∀ i, i < n → env (v i) ∈ env d := by
induction n with
| zero => simp [slots,Formula.Eval]
| succ n ih =>
simp only [slots,Formula.Eval,ih]
constructor
· rintro ⟨h,hlast⟩ i hi
rcases Nat.lt_succ_iff_lt_or_eq.mp hi with hi|rfl
· exact h i hi
· exact hlast
· intro h; exact ⟨fun i hi => h i (by omega),h n (by omega)⟩
def matrix (φ : Formula) (ci oi pi : ℕ) (v : ℕ → ℕ) : Formula :=
.conj (.transitive 2) (.conj (AtomicFormula.product 2 1)
(.conj (AtomicFormula.graph 2 (ci+3) 1 (oi+3) 0)
(.conj (slots (bound φ+1) 2 (fun i => v i+3))
(code φ 2 (ci+3) 1 (oi+3) 0
(fun i => truncate (bound φ+1) v i+3) (pi+3)))))
def certificate (φ : Formula) (ci oi pi : ℕ) (v : ℕ → ℕ) : SigmaFormula :=
.existsSet (.existsSet (.existsSet (.bounded (matrix φ ci oi pi v))))
end CertificateCode
variable {c : ZFSet.{u}} [Preorder (Conditions c)]
theorem eval_certificate_matrix (φ : Formula) (d k f o : ZFSet.{u})
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(e : ℕ → Name (Conditions c)) (p : Conditions c)
(env : ℕ → ZFSet.{u}) (ci oi pi : ℕ) (v : ℕ → ℕ)
(hc : env ci = c) (ho' : env oi = o) (hp : env pi = label c p)
(hv : ∀ i, env (v i) = (e i).encode (label c)) :
(CertificateCode.matrix φ ci oi pi v).Eval (cons f (cons k (cons d env))) ↔
Transitive d ∧ k = ZFSet.prod d d ∧ Graph d c o f ∧
(∀ i, i < bound φ+1 → (e i).encode (label c) ∈ d) ∧ Forces e φ p := by
have query (hd : Transitive d) (hk : k = ZFSet.prod d d) (hf : Graph d c o f)
(hs : ∀ i, i < bound φ+1 → (e i).encode (label c) ∈ d) :
(code φ 2 (ci+3) 1 (oi+3) 0
(fun i => truncate (bound φ+1) v i+3) (pi+3)).Eval
(cons f (cons k (cons d env))) ↔ Forces e φ p := by
let e' := truncate (bound φ+1) e
have he' (i) : (e' i).encode (label c) ∈ d := by
dsimp only [e',truncate]
split
· exact hs i (by assumption)
· exact hs 0 (by omega)
have ht (i) : env (truncate (bound φ+1) v i) = (e' i).encode (label c) := by
dsimp only [e',truncate]
split <;> exact hv _
exact (eval_code φ hd ho hf e' he' p (cons f (cons k (cons d env)))
2 (ci+3) 1 (oi+3) 0 (fun i => truncate (bound φ+1) v i+3) (pi+3)
⟨rfl,hc,hk,ho',rfl,hp⟩ ht).trans
(forces_congr φ e' e (fun i hi => ite_eq_left (by omega)) p)
simp only [CertificateCode.matrix,Formula.Eval,Formula.eval_transitive,
AtomicFormula.eval_product,CertificateCode.eval_slots,cons_zero,cons_succ,hv]
constructor
· rintro ⟨hd,hk,hgraph,hs,hquery⟩
have hf : Graph d c o f := by
have h := (AtomicFormula.eval_graph 2 (ci+3) 1 (oi+3) 0 _ hk).mp hgraph
simpa only [cons_zero,cons_succ,hc,ho'] using h
exact ⟨hd,hk,hf,hs,(query hd hk hf hs).mp hquery⟩
· rintro ⟨hd,hk,hf,hs,hq⟩
refine ⟨hd,hk,?_,hs,(query hd hk hf hs).mpr hq⟩
apply (AtomicFormula.eval_graph 2 (ci+3) 1 (oi+3) 0 _ hk).mpr
simpa only [cons_zero,cons_succ,hc,ho'] using hf
theorem realize_boundedCertificate (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : Separation M) (hR : SigmaReplacement M) (hI : Infinity M)
{o : ZFSet.{u}}
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(φ : Formula) (e : ℕ → Name (Conditions c)) (p : Conditions c)
(env : ℕ → ZFSet.{u}) (henv : ∀ i, env i ∈ M)
(ci oi pi : ℕ) (v : ℕ → ℕ)
(hc : env ci = c) (ho' : env oi = o) (hp : env pi = label c p)
(hv : ∀ i, env (v i) = (e i).encode (label c)) :
(CertificateCode.certificate φ ci oi pi v).Realize M env ↔ Forces e φ p := by
have matrix (d k f : ZFSet.{u}) (hd : d ∈ M) (hk : k ∈ M) (hf : f ∈ M) :=
(Formula.absolute (CertificateCode.matrix φ ci oi pi v) M hM
(cons f (cons k (cons d env))) (by
intro i; rcases i with _|i; exact hf
rcases i with _|i; exact hk
rcases i with _|i; exact hd
exact henv i)).trans (eval_certificate_matrix φ d k f o ho e p env ci oi pi v hc ho' hp hv)
change (∃ d ∈ M, ∃ k ∈ M, ∃ f ∈ M, _) ↔ _
constructor
· rintro ⟨d,hd,k,hk,f,hf,hm⟩
exact ((matrix d k f hd hk hf).mp hm).2.2.2.2
· intro hforce
have he (i) : (e i).encode (label c) ∈ M := hv i ▸ henv (v i)
obtain ⟨d,hd,hdT,_,hs⟩ := finite_container M hM hP hU hS hR hI
(fun i => (e i).encode (label c)) he (bound φ+1)
have hk := product_mem M hM hP hU hPow hS hd hd
obtain ⟨f,hf,hfG⟩ := internal_atomic_graph M hM hP hU hPow hS hd
(hc ▸ henv ci) (ho' ▸ henv oi)
exact ⟨d,hd,_,hk,f,hf,(matrix d _ f hd hk hf).mpr ⟨hdT,rfl,hfG,hs,hforce⟩⟩
end TuringRigidity.BoundedForcing
end OAI
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