| 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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|
