File size: 5,510 Bytes
ea76aac | 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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import OAI.Computability.MatchingCount.OptionCases
namespace OAI
section
namespace BinaryMatching.Raw
open StackCode Polynomial
abbrev SearchContext := ℕ×(Matrix ℕ×(Matrix ℤ×ℕ))
abbrev SearchState := SearchContext×Paths
def searchStep (s : SearchState) : SearchState :=
(s.1,pathStep s.1.1 s.1.2.1 s.1.2.2.1 s.2)
def searchInitial (z : SearchContext×ℕ) : SearchState :=
(z.1,initial z.1.1 z.1.2.2.2)
lemma searchStep_iterate (z : SearchContext×ℕ) (j : ℕ) :
searchStep^[j] (searchInitial z)=(z.1,search z.1.1 z.1.2.1 z.1.2.2.1 z.1.2.2.2 j) := by
induction j with
| zero => rfl
| succ j ih => simp only [Function.iterate_succ_apply',ih,searchStep,search]
namespace Compiler
open StackCode.Poly
@[fun_prop] theorem residual_poly :
Poly (fun z : (Matrix ℕ×Matrix ℤ)×(ℕ×ℕ) => residual z.1.1 z.1.2 z.2.1 z.2.2) := by
unfold residual
fun_prop
@[fun_prop] theorem initial_poly : Poly (fun z : ℕ×ℕ => initial z.1 z.2) := by
unfold initial
apply Poly.rawTab Prod.fst (fun z : (ℕ×ℕ)×ℕ => if z.2=z.1.2 then Option.some [z.1.2] else none)
· fun_prop
· have h : Poly (fun z : (ℕ×ℕ)×ℕ => if decide (z.2=z.1.2) then Option.some [z.1.2] else none) := by fun_prop
exact h.congr (fun z => by simp)
@[fun_prop] theorem predecessor_poly :
Poly (fun z : ((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ =>
predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2) := by
unfold predecessor
apply Poly.findWith (fun z => List.range z.1.1.1)
(fun z : (((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ)×ℕ =>
residual z.1.1.1.2.1 z.1.1.1.2.2 z.2 z.1.2 && (nth z.1.1.2 z.2).isSome)
· fun_prop
· have hr := Poly.comp _ (fun z : (((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ)×ℕ =>
((z.1.1.1.2.1,z.1.1.1.2.2),(z.2,z.1.2))) residual_poly (by fun_prop)
have hs : Poly (fun z : (((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ)×ℕ =>
(nth z.1.1.2 z.2).isSome) := by fun_prop
exact Poly.comp _ _ Poly.boolAnd (Poly.pair _ _ hr hs)
def pathEntry (z : ((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ) : Option (List ℕ) :=
match nth z.1.2 z.2 with
| Option.some l => Option.some l
| none => match predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2 with
| none => none
| Option.some u => (nth z.1.2 u).map (fun l => z.2::l)
@[fun_prop] theorem pathEntry_poly : Poly pathEntry := by
have h := Poly.optionCasesWith
(fun z : ((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ => nth z.1.2 z.2)
(fun z => match predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2 with
| none => none
| Option.some u => (nth z.1.2 u).map (fun l => z.2::l))
(fun z => Option.some z.2)
(by fun_prop)
(by
have hi := Poly.optionCasesWith
(fun z : ((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ => predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2)
(fun _ => (none:Option (List ℕ)))
(fun z => (nth z.1.1.2 z.2).map (fun l => z.1.2::l))
predecessor_poly (by fun_prop)
(Poly.optionMapWith (fun z : (((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ)×ℕ => nth z.1.1.2 z.2)
(fun z => z.1.1.2::z.2) (by fun_prop) (by fun_prop))
exact hi.congr (fun z => by cases predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2 <;> rfl))
(by fun_prop)
exact h.congr (fun z => by unfold pathEntry; cases nth z.1.2 z.2 <;> rfl)
@[fun_prop] theorem pathStep_poly :
Poly (fun z : (ℕ×(Matrix ℕ×Matrix ℤ))×Paths => pathStep z.1.1 z.1.2.1 z.1.2.2 z.2) := by
exact Poly.rawTab (fun z => z.1.1) pathEntry (by fun_prop) pathEntry_poly
@[fun_prop] theorem searchStep_poly : Poly searchStep := by
unfold searchStep
apply Poly.pair
· fun_prop
· exact Poly.comp _ (fun s : SearchState => ((s.1.1,(s.1.2.1,s.1.2.2.1)),s.2))
pathStep_poly (by fun_prop)
@[fun_prop] theorem searchInitial_poly : Poly searchInitial := by
unfold searchInitial
apply Poly.pair
· fun_prop
· exact Poly.comp _ (fun z : SearchContext×ℕ => (z.1.1,z.1.2.2.2)) initial_poly (by fun_prop)
@[fun_prop] theorem search_poly : Poly (fun z : SearchContext×ℕ =>
search z.1.1 z.1.2.1 z.1.2.2.1 z.1.2.2.2 z.2) := by
have hi := Poly.iterateFrom searchStep searchStep_poly Prod.snd searchInitial (by fun_prop)
searchInitial_poly (100*(X+1)^3) (by
intro z j hj
rw [searchStep_iterate]
let N := (encode (z.2,searchInitial z)).length
have hctx : (encode z.1).length≤N := by
simp only [N,searchInitial,code_pair_length,code_nat_length]; omega
have hn : z.1.1≤N := by
have := (pair_left_le z.1).trans hctx
simpa only [code_nat_length] using this
have ha : z.1.2.2.2≤N := by
have := (pair_right_le z.1.2.2).trans ((pair_right_le z.1.2).trans ((pair_right_le z.1).trans hctx))
simpa only [code_nat_length] using this
have hk : j≤N := by
simp only [N,code_pair_length,code_nat_length]; omega
have hb := search_code_bound z.1.1 z.1.2.1 z.1.2.2.1 z.1.2.2.2 j
have hm : z.1.1*(2*(4*z.1.2.2.2+3+j*(4*z.1.1+2))+1)≤
N*(2*(4*N+3+N*(4*N+2))+1) := by gcongr
simp only [eval_mul,eval_ofNat,eval_pow,eval_add,eval_X,eval_one]
change (encode (z.1,search z.1.1 z.1.2.1 z.1.2.2.1 z.1.2.2.2 j)).length≤100*(N+1)^3
rw [code_pair_length]
have hbb := hb.trans hm
nlinarith)
have h := Poly.second _ hi
exact h.congr (fun z => by rw [searchStep_iterate])
end Compiler
end BinaryMatching.Raw
end
end OAI
|