| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Machines.PhysicalProgram |
| import OAI.Combinatorics.MatchingCount.Tapes.Codec |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.PhysicalPost.Program |
| open Turing PhysicalStream |
|
|
| lemma Realizes.congr {P : Program} {input : Tape Symbol} {f g : Experiment (Tape Symbol)} {N : ℕ} |
| (h : P.Realizes input f N) (he : ∀ w,f.mean w=g.mean w) : P.Realizes input g N := by |
| obtain ⟨p,hp,hm,hc⟩:=h |
| exact ⟨p,hp,fun w=>(hm w).trans (he w),hc⟩ |
| lemma Realizes.mono {P : Program} {input : Tape Symbol} {f : Experiment (Tape Symbol)} {N S : ℕ} |
| (h : P.Realizes input f N) (hn : N≤S) : P.Realizes input f S := by |
| obtain ⟨p,hp,hm,hc⟩:=h |
| exact ⟨p,hp,hm,hc.trans hn⟩ |
| lemma seq_realizes_map {A : Type} (P R : Program) (input : Tape Symbol) (f : Experiment A) |
| (enc : A→Tape Symbol) (g : A→Experiment (Tape Symbol)) (N S : ℕ) |
| (hP : P.Realizes input (f.map enc) N) (hR : f.All (fun a=>R.Realizes (enc a) (g a) S)) : |
| (P.seq R).Realizes input (f.bind g) (N+S) := by |
| have hR' : (f.map enc).All (fun T=>R.Realizes T (R.earlyTapes S T) S) := by |
| rw [Experiment.all_map] |
| exact hR.mono (fun a ha=>⟨R.earlyTapes S (enc a),ha.early.1,fun _=>rfl,earlyTapes_cost R S (enc a)⟩) |
| apply (seq_realizes P R input (f.map enc) (R.earlyTapes S) N S hP hR').congr |
| intro w |
| rw [Experiment.mean_bind,Experiment.mean_map,Experiment.mean_bind] |
| exact f.mean_congr_all _ _ (hR.mono (fun a ha=>ha.early.2 w)) |
| end MatchingFPRAS.PhysicalPost.Program |
|
|
| end OAI |
|
|
