| 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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|
