| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Machines.CoinTM1Finite |
| import OAI.Combinatorics.MatchingCount.Tapes.TapeTools |
| import OAI.Combinatorics.MatchingCount.Machines.PhysicalProcess |
| import OAI.Combinatorics.MatchingCount.Graphs.SafeTrace |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.LiteralTape |
| open Turing MachineCost PhysicalStream |
|
|
| variable {Γ Λ σ : Type} [Inhabited Γ] [Inhabited Λ] [Fintype Λ] [Fintype σ] [Inhabited σ] |
| omit [Inhabited Γ] [Fintype σ] [Inhabited σ] in |
| lemma supports_univ (M : Λ→TM1.Stmt Γ Λ σ) : TM1.Supports M Finset.univ := by |
| classical |
| refine ⟨Finset.mem_univ _,fun l _=>?_⟩ |
| induction M l with |
| | move d q ih => exact ih |
| | write f q ih => exact ih |
| | load f q ih => exact ih |
| | branch f q r ih jh => exact ⟨ih,jh⟩ |
| | goto f => exact fun a v=>Finset.mem_univ _ |
| | halt => trivial |
| noncomputable def table (M : Λ→TM1.Stmt Γ Λ σ) := CoinTM1.table M (fun _=>none) |
| noncomputable def states (M : Λ→TM1.Stmt Γ Λ σ) := CoinTM1.states M Finset.univ |
| omit [Inhabited Γ] [Inhabited σ] in |
| lemma supported (M : Λ→TM1.Stmt Γ Λ σ) : PhysicalPost.Supported (table M) (states M) := |
| CoinTM1.table_supported M Finset.univ (supports_univ M) (fun _=>none) (by simp) |
| omit [Inhabited Γ] [Inhabited Λ] [Inhabited σ] in |
| lemma entry_mem (M : Λ→TM1.Stmt Γ Λ σ) (l : Λ) (v : σ) : (Sum.inl l,v)∈states M := |
| CoinTM1.state_mem (CoinTM1.node_entry (Finset.mem_univ _)) |
| omit [Inhabited Λ] [Fintype σ] [Inhabited σ] in |
| lemma Exec.physical {M : Λ→TM1.Stmt Γ Λ σ} {c d : TM1.Cfg Γ Λ σ} {t : ℕ} |
| (h : Exec M c d t) : ∃ k,k≤t*programCost M ∧ ∀ b : Stream, |
| PhysicalPost.execute (table M) k (CoinTM1.trCfg c) b=CoinTM1.trCfg d := by |
| obtain ⟨n,hn,he⟩:=h |
| have hp:=SafeTrace.Path.of_run (safe:=fun _=>True) (M:=M) n c (fun _ _ _ _=>trivial) |
| rw [he] at hp |
| obtain ⟨k,hk,hb⟩:=hp.physical (fun _=>none) (fun _ _=>rfl) (programCost M) |
| (fun l _=>statementCost_le_programCost M l) |
| exact ⟨k,hk.trans (Nat.mul_le_mul_right _ hn),hb⟩ |
| omit [Inhabited Λ] [Fintype σ] [Inhabited σ] in |
| lemma Exec.implements {M : Λ→TM1.Stmt Γ Λ σ} {c d : TM1.Cfg Γ Λ σ} {t : ℕ} |
| (h : Exec M c d t) : ∃ p : Experiment Unit,p.cost≤t*programCost M ∧ |
| PhysicalPost.Implements (table M) (fun _=>CoinTM1.trCfg d) (CoinTM1.trCfg c) p ∧ |
| ∀ f : Unit→ℚ,p.mean f=f () := by |
| obtain ⟨k,hk,hb⟩:=h.physical |
| exact ⟨.delay k (.done ()),by simpa only [Experiment.cost,Nat.add_zero] using hk,.delay _ _ k _ hb (.done ()),fun _=>rfl⟩ |
| end MatchingFPRAS.LiteralTape |
|
|
| end OAI |
|
|
