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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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import Mathlib
import OAI.Combinatorics.MatchingCount.Tapes.RandomWord
namespace OAI
namespace MatchingFPRAS.RandomWord
open Turing BinaryArithmetic
def splitBits (m n : ℕ) : (Fin (m+n) → Bool) ≃ ((Fin m → Bool) × (Fin n → Bool)) :=
(Equiv.arrowCongr finSumFinEquiv.symm (Equiv.refl Bool)).trans
(Equiv.sumArrowEquivProdArrow _ _ _)
def gridBits (m n : ℕ) : (Fin (m*n) → Bool) ≃ (Fin m → Fin n → Bool) :=
(Equiv.arrowCongr finProdFinEquiv.symm (Equiv.refl Bool)).trans (Equiv.curry _ _ _)
def wordTapeEquiv (m : ℕ) : (Fin (m*2+(m+3)) → Bool) ≃
((Fin m → Bool×Bool) × (Fin (m+3) → Bool)) :=
(splitBits (m*2) (m+3)).trans (Equiv.prodCongr
((gridBits m 2).trans (Equiv.piCongrRight fun _=>finTwoArrowEquiv Bool)) (Equiv.refl _))
lemma gridBits_apply (m n : ℕ) (b : Fin (m*n) → Bool) (i : Fin m) (j : Fin n) :
gridBits m n b i j=b (finProdFinEquiv (i,j)) := rfl
lemma splitBits_symm (m n : ℕ) (x : Fin m → Bool) (y : Fin n → Bool) :
(splitBits m n).symm (x,y)=Fin.append x y := by
ext i
refine Fin.addCases (fun i=>?_) (fun j=>?_) i <;> simp [splitBits,Fin.append,finSumFinEquiv]
lemma wordTape_list (m : ℕ) (x : Fin m → Bool×Bool) (y : Fin (m+3) → Bool) :
List.ofFn ((wordTapeEquiv m).symm (x,y))=pairCoins (List.ofFn x)++List.ofFn y := by
let rows : Fin m → Fin 2 → Bool:=fun i=>![ (x i).1,(x i).2 ]
let b : Fin (m*2) → Bool:=(gridBits m 2).symm rows
have hb (i : Fin m) (j : Fin 2) : b (finProdFinEquiv (i,j))=rows i j := by
exact congrFun (congrFun ((gridBits m 2).apply_symm_apply rows) i) j
have he : (wordTapeEquiv m).symm (x,y)=Fin.append b y := splitBits_symm _ _ b y
rw [he,List.ofFn_fin_append]
congr 1
rw [List.ofFn_mul]
have hi (i : Fin m) : (List.ofFn fun j : Fin 2=>b ⟨i*2+j,by omega⟩)=[(x i).1,(x i).2] := by
have he' : (fun j : Fin 2=>b ⟨i*2+j,by omega⟩)=rows i := by
funext j
have hc : finProdFinEquiv (i,j)=(⟨i*2+j,by omega⟩ : Fin (m*2)) := by
apply Fin.ext
change j.val+2*i.val=i.val*2+j.val
omega
exact hc ▸ hb i j
rw [he']
simp [rows,List.ofFn_succ]
simp_rw [hi]
simp [pairCoins,List.map_ofFn,List.flatMap,Function.comp_def]
lemma wordTape_run (m : ℕ) (x : Fin m → Bool×Bool) (y : Fin (m+3) → Bool) :
MatchingFPRAS.run machine (List.replicate m 1++[3]) ((wordTapeEquiv m).symm (x,y))=
cfg none [] (drawnWord (List.ofFn x)++[3]) := by
change execute (List.ofFn ((wordTapeEquiv m).symm (x,y)))
(cfg (some 0) [] (List.replicate m 1++[3]))=_
rw [wordTape_list]
convert execute_word (List.ofFn x) (List.ofFn y) (by simp) using 1
simp
theorem run_word (m : ℕ) (bits : Fin (m*2+(m+3)) → Bool) :
MatchingFPRAS.run machine (List.replicate m 1++[3]) bits=
cfg none [] (drawnWord (List.ofFn (wordTapeEquiv m bits).1)++[3]) := by
have he:=wordTape_run m (wordTapeEquiv m bits).1 (wordTapeEquiv m bits).2
simpa only [Prod.mk.eta,Equiv.symm_apply_apply] using he
def outputCoins (m : ℕ) (bits : Fin (m*2+(m+3)) → Bool) : Fin m → Bool :=
fun i=>(wordTapeEquiv m bits).1 i |>.1
def outputSplit (m : ℕ) : (Fin (m*2+(m+3)) → Bool) ≃
((Fin m → Bool) × ((Fin m → Bool) × (Fin (m+3) → Bool))) :=
(wordTapeEquiv m).trans ((Equiv.prodCongr (Equiv.arrowProdEquivProdArrow _ _ _) (Equiv.refl _)).trans
(Equiv.prodAssoc _ _ _))
lemma outputSplit_fst (m : ℕ) (bits : Fin (m*2+(m+3)) → Bool) :
(outputSplit m bits).1=outputCoins m bits := rfl
theorem outputCoins_fiber (m : ℕ) (x : Fin m → Bool) :
(Finset.univ.filter (fun b=>outputCoins m b=x)).card=2^(2*m+3) := by
classical
let e : ({b : Fin (m*2+(m+3)) → Bool // outputCoins m b=x}) ≃
((Fin m → Bool) × (Fin (m+3) → Bool)) := {
toFun := fun b=>(outputSplit m b.1).2
invFun := fun y=>⟨(outputSplit m).symm (x,y),by
change (outputSplit m ((outputSplit m).symm (x,y))).1=x
rw [Equiv.apply_symm_apply]⟩
left_inv b := by
apply Subtype.ext
apply (outputSplit m).injective
rw [Equiv.apply_symm_apply]
exact Prod.ext b.property.symm rfl
right_inv y := by change (outputSplit m ((outputSplit m).symm (x,y))).2=y; rw [Equiv.apply_symm_apply] }
rw [←Fintype.card_subtype]
rw [Fintype.card_congr e]
simp only [Fintype.card_prod,Fintype.card_fun,Fintype.card_fin,Fintype.card_bool,←pow_add]
congr 1
omega
end MatchingFPRAS.RandomWord
end OAI
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