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e5a97c6 | 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.CoinTM1
import OAI.Combinatorics.MatchingCount.Machines.PhysicalRestriction
namespace OAI
namespace MatchingFPRAS.CoinTM1
open Turing
open scoped Classical
variable {Γ Λ σ : Type} [Inhabited Λ] [Fintype σ]
noncomputable def nodes (M : Λ → TM1.Stmt Γ Λ σ) (S : Finset Λ) : Finset (Node Γ Λ σ) :=
S.disjSum (S.biUnion (fun l=>TM1.stmts₁ (M l)))
noncomputable def states (M : Λ → TM1.Stmt Γ Λ σ) (S : Finset Λ) : Finset (State Γ Λ σ) :=
(nodes M S) ×ˢ Finset.univ
omit [Inhabited Λ] [Fintype σ] in
lemma node_entry {M : Λ → TM1.Stmt Γ Λ σ} {S : Finset Λ} {l : Λ} (h : l∈S) :
Sum.inl l∈nodes M S := by simp only [nodes,Finset.inl_mem_disjSum]; exact h
omit [Inhabited Λ] [Fintype σ] in
lemma node_pending {M : Λ → TM1.Stmt Γ Λ σ} {S : Finset Λ} {q : TM1.Stmt Γ Λ σ}
(h : some q∈TM1.stmts M S) : Sum.inr q∈nodes M S := by
simpa only [nodes,Finset.inr_mem_disjSum,TM1.stmts,Finset.some_mem_insertNone] using h
omit [Inhabited Λ] in
lemma state_mem {M : Λ → TM1.Stmt Γ Λ σ} {S : Finset Λ} {q : Node Γ Λ σ} {v : σ}
(h : q∈nodes M S) : (q,v)∈states M S := by
simp only [states,Finset.mem_product,Finset.mem_univ,and_true]; exact h
lemma aux_supported {M : Λ → TM1.Stmt Γ Λ σ} {S : Finset Λ}
(hs : TM1.Supports M S) (a : Γ) (q : TM1.Stmt Γ Λ σ)
(hq : some q∈TM1.stmts M S) (v : σ) (q' : State Γ Λ σ) (act : TM0.Stmt Γ)
(he : aux a q v=some (q',act)) : q'∈states M S := by
classical
induction q generalizing v q' act with
| move d q ih =>
simp only [aux,Option.some.injEq,Prod.mk.injEq] at he
rcases he with ⟨rfl,rfl⟩
apply state_mem
apply node_pending
exact TM1.stmts_trans (by simp only [TM1.stmts₁,Finset.mem_insert]; exact Or.inr TM1.stmts₁_self) hq
| write f q ih =>
simp only [aux,Option.some.injEq,Prod.mk.injEq] at he
rcases he with ⟨rfl,rfl⟩
apply state_mem
apply node_pending
exact TM1.stmts_trans (by simp only [TM1.stmts₁,Finset.mem_insert]; exact Or.inr TM1.stmts₁_self) hq
| load f q ih =>
exact ih (TM1.stmts_trans (by simp only [TM1.stmts₁,Finset.mem_insert]; exact Or.inr TM1.stmts₁_self) hq) _ _ _ he
| branch f q r ihq ihr =>
simp only [aux] at he
split_ifs at he with hf
· exact ihq (TM1.stmts_trans (by simp only [TM1.stmts₁,Finset.mem_insert,Finset.mem_union]; exact Or.inr (Or.inl TM1.stmts₁_self)) hq) _ _ _ he
· exact ihr (TM1.stmts_trans (by simp only [TM1.stmts₁,Finset.mem_insert,Finset.mem_union]; exact Or.inr (Or.inr TM1.stmts₁_self)) hq) _ _ _ he
| goto f =>
simp only [aux,Option.some.injEq,Prod.mk.injEq] at he
rcases he with ⟨rfl,rfl⟩
exact state_mem (node_entry (TM1.stmts_supportsStmt hs hq a v))
| halt => simp [aux] at he
theorem table_supported (M : Λ → TM1.Stmt Γ Λ σ) (S : Finset Λ) (hs : TM1.Supports M S)
(coin : Coins Λ) (hc : ∀ l∈S,∀ f,coin l=some f→∀ b,f b∈S) :
PhysicalPost.Supported (table M coin) (states M S) := by
intro q hq a b q' act he
have hn : q.1∈nodes M S := (Finset.mem_product.mp hq).1
rcases q with ⟨l,v⟩
cases l with
| inl l =>
have hl : l∈S := by simpa only [nodes,Finset.inl_mem_disjSum] using hn
simp only [table] at he
split at he
· apply aux_supported hs a (M l) _ v q' act he
simp only [TM1.stmts,Finset.some_mem_insertNone,Finset.mem_biUnion]
exact ⟨l,hl,TM1.stmts₁_self⟩
· rename_i f hf
simp only [Option.some.injEq,Prod.mk.injEq] at he
rcases he with ⟨rfl,rfl⟩
exact state_mem (node_entry (hc l hl f hf b))
| inr q =>
apply aux_supported hs a q _ v q' act he
simpa only [nodes,Finset.inr_mem_disjSum,TM1.stmts,Finset.some_mem_insertNone] using hn
end MatchingFPRAS.CoinTM1
end OAI
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