File size: 3,141 Bytes
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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import Mathlib
import OAI.Combinatorics.MatchingCount.Tapes.RandomWordLaw
namespace OAI
namespace MatchingFPRAS.RandomWord
open Turing BinaryArithmetic
lemma draw_forward_suffix (xs : List (Bool×Bool)) (L R : List Symbol) :
execute (pairCoins xs) (cfg (some 0) L (List.replicate xs.length 1++3::R))=
cfg (some 0) ((drawnWord xs).reverse++L) (3::R) := by
induction xs generalizing L with
| nil=>rfl
| cons p xs ih=>
simp only [pairCoins,List.flatMap_cons,List.length_cons,List.replicate_succ,List.cons_append]
change execute ([p.1,p.2]++pairCoins xs) (cfg (some 0) L (1::(List.replicate xs.length 1++3::R)))=_
rw [execute_append,draw_two,ih]
simp [drawnWord,List.reverse_cons,List.append_assoc]
lemma finish_run_suffix (L R : List Symbol) (hL : ∀ b∈L,b≠0)
(xs : List Bool) (hx : xs.length=L.length+3) :
execute xs (cfg (some 0) L (3::R))=cfg none [] (L.reverse++3::R) := by
obtain ⟨ys,b,rfl⟩ : ∃ ys b,xs=ys++[b] := by
have hn : xs.reverse≠[] := by
intro h
have h' : xs.length=0 := by simpa only [List.length_reverse,List.length_nil] using congrArg List.length h
omega
obtain ⟨b,ys,he⟩:=List.exists_cons_of_ne_nil hn
exact ⟨ys.reverse,b,by have h:=congrArg List.reverse he; simpa using h⟩
have hy : ys.length=L.length+2 := by simpa using hx
have he : execute ys (cfg (some 0) L (3::R))=cfg (some 3) [] (L.reverse++3::R) := by
have hr:=reset_run L R 3 (by decide) hL ys hy
cases ys with
| nil=>simp at hy
| cons c ys=>
rw [execute_cons] at hr ⊢
have ht : tick machine (cfg (some 0) L (3::R)) c=tick machine (cfg (some 2) L (3::R)) c := by
simp [tick,machine,Tape.mk₂,Tape.mk']
rw [ht]
exact hr
rw [execute_append,he]
simp [execute,tick,machine,cfg]
lemma execute_word_suffix (xs : List (Bool×Bool)) (tail : List Bool) (R : List Symbol)
(ht : tail.length=xs.length+3) :
execute (pairCoins xs++tail) (cfg (some 0) [] (List.replicate xs.length 1++3::R))=
cfg none [] (drawnWord xs++3::R) := by
rw [execute_append,draw_forward_suffix,List.append_nil]
have hn : ∀ b∈(drawnWord xs).reverse,b≠0 := by
intro b hb
obtain ⟨p,hp,rfl⟩:=List.mem_map.mp (List.mem_reverse.mp hb)
exact bitSymbol_ne_zero _
simpa using finish_run_suffix (drawnWord xs).reverse R hn tail (by simpa [drawnWord] using ht)
lemma run_word_suffix (m : ℕ) (R : List Symbol) (bits : Fin (m*2+(m+3))→Bool) :
run machine (List.replicate m 1++3::R) bits=
cfg none [] ((List.ofFn (outputCoins m bits)).map bitSymbol++3::R) := by
obtain ⟨⟨x,y⟩,rfl⟩:=(wordTapeEquiv m).symm.surjective bits
change execute (List.ofFn ((wordTapeEquiv m).symm (x,y)))
(cfg (some 0) [] (List.replicate m 1++3::R))=_
rw [wordTape_list]
have h:=execute_word_suffix (List.ofFn x) (List.ofFn y) R (by simp)
have ho : outputCoins m ((wordTapeEquiv m).symm (x,y))=(fun i=>(x i).1) := by
funext i
simp only [outputCoins,Equiv.apply_symm_apply]
rw [ho]
simpa only [List.length_ofFn,drawnWord,List.map_ofFn,Function.comp_def] using h
end MatchingFPRAS.RandomWord
end OAI
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