File size: 2,596 Bytes
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 | 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
|