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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 | 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
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