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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.PhysicalProgram
import OAI.Combinatorics.MatchingCount.Tapes.Codec
namespace OAI
namespace MatchingFPRAS.PhysicalPost.Program
open Turing PhysicalStream
lemma run_tape (P : Program) (input : List Symbol) (n : ℕ) (b : Bits n) :
(MatchingFPRAS.run P.machine input b).tape=
(execute P.table n (P.initial (Tape.mk₁ input)) (extend b)).tape := by
let : Inhabited {q //q∈P.states}:=⟨⟨P.entry,P.entry_mem⟩⟩
rw [machine,restrictedMachine_run]
exact (congrArg Cfg.tape (execute_lift P.table P.states P.supported n
⟨some default,Tape.mk₁ input⟩ (extend b))).symm
lemma run_halted (P : Program) (input : List Symbol) (n : ℕ) (b : Bits n)
(h : (execute P.table n (P.initial (Tape.mk₁ input)) (extend b)).state=none) :
(MatchingFPRAS.run P.machine input b).state=none := by
let : Inhabited {q //q∈P.states}:=⟨⟨P.entry,P.entry_mem⟩⟩
have he:=congrArg Cfg.state (execute_lift P.table P.states P.supported n
⟨some default,Tape.mk₁ input⟩ (extend b))
have hs : (execute (restrict P.table P.states P.supported) n
⟨some default,Tape.mk₁ input⟩ (extend b)).state=none := by
exact Option.map_eq_none_iff.mp (he.symm.trans h)
rw [machine,restrictedMachine_run]
change ((execute (restrict P.table P.states P.supported) n
⟨some default,Tape.mk₁ input⟩ (extend b)).state.map enumeration)=none
rw [hs,Option.map_none]
lemma Realizes.run_all {P : Program} {input : List Symbol} {f : Experiment (Tape Symbol)} {N : ℕ}
(hP : P.Realizes (Tape.mk₁ input) f N) (R : Tape Symbol→Prop) (hf : f.All R)
(n : ℕ) (hn : N≤n) (b : Bits n) :
(MatchingFPRAS.run P.machine input b).state=none ∧
R (MatchingFPRAS.run P.machine input b).tape := by
obtain ⟨p,hp,hm,hc⟩:=hP
have hex:=hp.padded_execution (fun _=>rfl) n (hc.trans hn) (extend b)
have ht : (MatchingFPRAS.run P.machine input b).tape=(p.run (extend b)).1 := by
rw [run_tape,hex];rfl
refine ⟨run_halted P input n b (by rw [hex];rfl),?_⟩
rw [ht]
exact p.all_run R (Experiment.all_of_mean_eq hm R hf) (extend b)
lemma Realizes.run_mean {P : Program} {input : List Symbol} {f : Experiment (Tape Symbol)} {N : ℕ}
(hP : P.Realizes (Tape.mk₁ input) f N) (w : Tape Symbol→ℚ)
(n : ℕ) (hn : N≤n) :
average (fun b : Bits n=>w (MatchingFPRAS.run P.machine input b).tape)=f.mean w := by
obtain ⟨p,hp,hm,hc⟩:=hP
simp_rw [run_tape]
exact (hp.fair_law (fun _=>rfl) n (hc.trans hn) (fun c=>w c.tape)).trans (hm w)
end MatchingFPRAS.PhysicalPost.Program
end OAI
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