| 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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|
