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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Algorithm.AlgorithmLaw
import OAI.Combinatorics.MatchingCount.Machines.Wrapper
import OAI.Combinatorics.MatchingCount.Machines.PhysicalProgramSeal
namespace OAI
namespace MatchingFPRAS.LiteralPhysical
open Turing PhysicalPost PhysicalStream LiteralAlgorithm
open scoped Classical
def ProperTape (G : GraphInput) (T : Tape Symbol) : Prop :=
∃ q : ℚ,0≤q ∧ T.right₀=ListBlank.mk (encodeRat q) ∧ (Z G=0→q=0)
def AccurateTape (G : GraphInput) (ε : ℚ) (T : Tape Symbol) : Prop :=
∃ q : ℚ,T.right₀=ListBlank.mk (encodeRat q) ∧ (1-ε)*(Z G:ℚ)≤q ∧ q≤(1+ε)*(Z G:ℚ)
lemma properTape_of_word (G : GraphInput) (as : List Symbol) (h : Proper G as) : ProperTape G (Tape.mk₁ as) := by
obtain ⟨q,hq,rfl,hz⟩:=h
exact ⟨q,hq,by simp [Tape.mk₁,Tape.mk₂],hz⟩
lemma accurateTape_of_word (G : GraphInput) (ε : ℚ) (as : List Symbol) (h : AccurateWord G ε as) :
AccurateTape G ε (Tape.mk₁ as) := by
obtain ⟨q,rfl,hq⟩:=h
exact ⟨q,by simp [Tape.mk₁,Tape.mk₂],hq⟩
lemma mainProgram_all (G : GraphInput) (ε δ : ℚ) (he : 0<ε) (he1 : ε<1) (hd : 0≤δ)
(n : ℕ) (hn : wrapTime algorithmRealizer (encodeInput G ε δ)≤n) (b : Bits n) :
(MatchingFPRAS.run mainProgram.machine (encodeInput G ε δ) b).state=none ∧
ProperTape G (MatchingFPRAS.run mainProgram.machine (encodeInput G ε δ) b).tape := by
apply (mainProgram_realizes G ε δ).run_all _ _ n hn b
rw [Experiment.all_map]
exact (algorithm_proper G ε δ he he1 hd).mono (properTape_of_word G)
lemma mainProgram_outputs (G : GraphInput) (ε δ : ℚ) (he : 0<ε) (he1 : ε<1) (hd : 0≤δ)
(n : ℕ) (hn : wrapTime algorithmRealizer (encodeInput G ε δ)≤n) :
(∀ b : Bits n,∃ q : ℚ,0≤q ∧ Outputs mainProgram.machine (encodeInput G ε δ) b q) ∧
(Z G=0→∀ b : Bits n,Outputs mainProgram.machine (encodeInput G ε δ) b 0) := by
constructor
· intro b
obtain ⟨hh,q,hq,ht,_⟩:=mainProgram_all G ε δ he he1 hd n hn b
exact ⟨q,hq,hh,ht⟩
· intro hz b
obtain ⟨hh,q,_hq,ht,hzq⟩:=mainProgram_all G ε δ he he1 hd n hn b
rw [hzq hz] at ht
exact ⟨hh,ht⟩
lemma average_good (A : RandomMachine) (G : GraphInput) (ε δ : ℚ) (n : ℕ)
(hh : ∀ b : Bits n,(MatchingFPRAS.run A (encodeInput G ε δ) b).state=none) :
average (fun b : Bits n=>if AccurateTape G ε (MatchingFPRAS.run A (encodeInput G ε δ) b).tape then 1 else 0)=
((goodTapes A G ε δ n).card:ℚ)/(2:ℚ)^n := by
have he (b : Bits n) : AccurateTape G ε (MatchingFPRAS.run A (encodeInput G ε δ) b).tape ↔
∃ q : ℚ,Outputs A (encodeInput G ε δ) b q ∧ (1-ε)*(Z G:ℚ)≤q ∧ q≤(1+ε)*(Z G:ℚ) := by
constructor
· rintro ⟨q,ht,hq⟩;exact ⟨q,⟨hh b,ht⟩,hq⟩
· rintro ⟨q,⟨_,ht⟩,hq⟩;exact ⟨q,ht,hq⟩
simp_rw [he]
simp only [average,goodTapes,Finset.sum_boole]
lemma mainProgram_success (G : GraphInput) (ε δ : ℚ) (he : 0<ε) (he1 : ε<1) (hd : 0<δ)
(n : ℕ) (hn : wrapTime algorithmRealizer (encodeInput G ε δ)≤n) :
1-δ≤((goodTapes mainProgram.machine G ε δ n).card:ℚ)/(2:ℚ)^n := by
have hs : 1-δ≤(algorithm (encodeInput G ε δ)).mean
(fun as=>if AccurateTape G ε (Tape.mk₁ as) then 1 else 0) := by
apply (algorithm_success G ε δ he he1 hd).trans (Experiment.mean_mono _ _ _ ?_)
exact (Experiment.all_true _).mono (by
intro as _
by_cases h : AccurateWord G ε as
· simp only [ite_eq_left h,ite_eq_left (accurateTape_of_word G ε as h),le_refl]
· simp only [ite_eq_right h];split_ifs <;> norm_num)
have hm:=(mainProgram_realizes G ε δ).run_mean (fun T=>if AccurateTape G ε T then 1 else 0) n hn
rw [Experiment.mean_map] at hm
rw [←hm,average_good mainProgram.machine G ε δ n
(fun b=>(mainProgram_all G ε δ he he1 hd.le n hn b).1)] at hs
exact hs
end MatchingFPRAS.LiteralPhysical
end OAI
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