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760f90e | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 | import OAI.Computability.RationalHitting.MatrixComputation
namespace OAI
namespace RationalHitting.StackLanguage
lemma range_flatMap_to_finRange {A : Type} (n : ℕ) (f : ℕ → List A) :
(List.range n).flatMap f=(List.finRange n).flatMap (fun i => f i.val) := by
rw [←finRange_map_val,List.flatMap_map]
namespace Poly
variable {A B : Type} [Coding A] [Coding B]
theorem tabulateUp {n : A → ℕ} {f : A×Unary → B}
(hn : Poly (fun a => Unary.mk (n a))) (hf : Poly f) :
Poly (fun a => (List.range (n a)).map (fun i => f (a,⟨i⟩))) :=
hf.tabulate.comp (hn.pair (show Poly (fun a : A => a) from id))
theorem flatMapUp {n : A → ℕ} {f : A×Unary → List B}
(hn : Poly (fun a => Unary.mk (n a))) (hf : Poly f) :
Poly (fun a => (List.range (n a)).flatMap (fun i => f (a,⟨i⟩))) :=
(listFlatten.comp (tabulateUp hn hf)).of_eq (by intro a;rw [List.flatMap_def])
theorem wordRange {n : A → ℕ} {f : A×Unary → BitWord}
(hn : Poly (fun a => Unary.mk (n a))) (hf : Poly f) :
Poly (fun a => BitWord.mk ((List.range (n a)).flatMap (fun i => (f (a,⟨i⟩)).val))) :=
(wordConcat.comp (tabulateUp hn hf)).of_eq (by intro a;simp [List.flatMap_map])
end Poly
end RationalHitting.StackLanguage
namespace RationalHitting.StackLanguage
namespace Runtime
abbrev MatrixWordArgs := (DenseSquare×DenseSquare)×ℤ
def matrixWord (A B : DenseSquare) (den : ℤ) : BitWord :=
⟨(List.range A.dim).flatMap fun i =>
(List.range A.dim).flatMap fun j =>
encodeRat (Rat.divInt (cramer A B i j).val.det den)⟩
@[fun_prop] theorem matrixWord_poly : Poly (fun a : MatrixWordArgs => matrixWord a.1.1 a.1.2 a.2) := by
have hc : Poly (fun a : (MatrixWordArgs×Unary)×Unary =>
cramer a.1.1.1.1 a.1.1.1.2 a.1.2.val a.2.val) :=
cramer_poly.comp (show Poly (fun a : (MatrixWordArgs×Unary)×Unary =>
(a.1.1.1,(a.1.2,a.2))) by fun_prop)
have hv := Poly.rationalEncode.comp ((Poly.denseDet.comp hc).pair
(show Poly (fun a : (MatrixWordArgs×Unary)×Unary => a.1.1.2) by fun_prop))
have hr := Poly.wordRange (Poly.denseDim.comp (show Poly (fun a : MatrixWordArgs×Unary => a.1.1.1) by fun_prop)) hv
exact Poly.wordRange (Poly.denseDim.comp (show Poly (fun a : MatrixWordArgs => a.1.1) by fun_prop)) hr
lemma matrixWord_of {d : ℕ} (A B : Matrix (Fin d) (Fin d) ℤ) (den : ℤ) :
(matrixWord (DenseSquare.of A) (DenseSquare.of B) den).val=
(List.finRange d).flatMap (fun i => (List.finRange d).flatMap (fun j =>
encodeRat (Rat.divInt (A.updateCol i (fun r => B r j)).det den))) := by
unfold matrixWord
change (List.range d).flatMap _ = _
rw [range_flatMap_to_finRange]
apply List.flatMap_congr
intro i hi
rw [range_flatMap_to_finRange]
apply List.flatMap_congr
intro j hj
change Fin d at j
exact cramer_encode_of A B den i j
abbrev TupleWordArgs := (Params×GridPoint)×DenseSquare×ℤ
def tupleWord (p : Params) (x : GridPoint) (A : DenseSquare) (den : ℤ) : BitWord :=
⟨(List.range p.n).flatMap fun i => (matrixWord A (matrix p x ⟨i+1⟩) den).val⟩
@[fun_prop] theorem tupleWord_poly : Poly (fun a : TupleWordArgs => tupleWord a.1.1 a.1.2 a.2.1 a.2.2) := by
have hm : Poly (fun a : TupleWordArgs×Unary => matrix a.1.1.1 a.1.1.2 (a.2+1)) := by fun_prop
have hv := matrixWord_poly.comp
(((show Poly (fun a : TupleWordArgs×Unary => a.1.2.1) by fun_prop).pair hm).pair
(show Poly (fun a : TupleWordArgs×Unary => a.1.2.2) by fun_prop))
exact Poly.wordRange (show Poly (fun a : TupleWordArgs => a.1.1.1) by fun_prop) hv
lemma tupleWord_eq (n s : ℕ) (x : GridPoint) (den : ℤ) :
(tupleWord (Params.of n s) x
(matrix (Params.of n s) x 0) den).val =
encodeTuple (ExecutableGenerator.tuple (ExecutableGenerator.normalized
(ExecutableGenerator.inputTables n (Parameters.N n s) (Parameters.exponent n s) (gridFn x)) den)) := by
unfold tupleWord encodeTuple
change (List.range n).flatMap _ = _
rw [range_flatMap_to_finRange]
apply List.flatMap_congr
intro a ha
let U := Generator.U n (Parameters.N n s) (Parameters.exponent n s) (gridFn x)
have hzero : matrix (Params.of n s) x 0=DenseSquare.of (U 0) := matrix_eq n s x 0
have hsucc : matrix (Params.of n s) x ⟨a.val+1⟩=DenseSquare.of (U a.succ) := matrix_eq n s x a.succ
trans (matrixWord (DenseSquare.of (U 0)) (DenseSquare.of (U a.succ)) den).val
· exact congrArg₂ (fun A B => (matrixWord A B den).val) hzero hsucc
rw [matrixWord_of]
apply List.flatMap_congr
intro i hi
apply List.flatMap_congr
intro j hj
congr 1
rw [ExecutableGenerator.tuple_normalized,ExecutableGenerator.normalizedEntry,
DeterminantAlgorithm.determinant_eq,DeterminantAlgorithm.denote_tabulate,
ExecutableGenerator.inputTables_denote,ExecutableGenerator.inputTables_denote]
end Runtime
end RationalHitting.StackLanguage
namespace RationalHitting.StackLanguage
namespace Runtime
def point (p : Params) (x : GridPoint) : List BitWord :=
let A := matrix p x 0
let den := A.val.det
if den=0 then [] else [tupleWord p x A den]
@[fun_prop] theorem point_poly : Poly (fun a : Params×GridPoint => point a.1 a.2) := by
have hf : Poly (fun a : TupleWordArgs =>
if a.2.2=0 then ([] : List BitWord) else [tupleWord a.1.1 a.1.2 a.2.1 a.2.2]) := by
apply Poly.ite
· exact Poly.intZero.comp (show Poly (fun a : TupleWordArgs => a.2.2) by fun_prop)
· exact Poly.const []
· exact Poly.listSingleton.comp tupleWord_poly
have hm : Poly (fun a : Params×GridPoint => matrix a.1 a.2 0) := by fun_prop
exact hf.comp ((show Poly (fun a : Params×GridPoint => a) from Poly.id).pair
(hm.pair (Poly.denseDet.comp hm)))
lemma matrix_det_eq (n s : ℕ) (x : GridPoint) (a : Fin (n+1)) :
(matrix (Params.of n s) x ⟨a.val⟩).val.det=
(Generator.U n (Parameters.N n s) (Parameters.exponent n s) (gridFn x) a).det := by
apply congrArg Matrix.det
funext i j
exact entry_eq n _ _ (gridFn x) a i j
lemma point_eq (n s : ℕ) (x : GridPoint) :
point (Params.of n s) x=
((ExecutableGenerator.atPoint n (Parameters.N n s) (Parameters.exponent n s) (gridFn x)).toList.map
(fun X => BitWord.mk (encodeTuple X))) := by
unfold point ExecutableGenerator.atPoint
simp only [DeterminantAlgorithm.determinant_eq,ExecutableGenerator.inputTables_denote]
have hd : (matrix (Params.of n s) x 0).val.det=
(Generator.U n (Parameters.N n s) (Parameters.exponent n s) (gridFn x) 0).det :=
matrix_det_eq n s x 0
rw [hd]
split_ifs with hz
· rfl
· simp only [Option.toList_some,List.map_cons,List.map_nil,List.cons.injEq, and_true]
apply congrArg BitWord.mk
exact tupleWord_eq n s x _
end Runtime
end RationalHitting.StackLanguage
namespace RationalHitting.StackLanguage
lemma map_filterMap_toList {A B C : Type} (l : List A) (f : A → Option B) (g : B → C) :
(l.filterMap f).map g=l.flatMap (fun a => (f a).toList.map g) := by
induction l with
| nil => rfl
| cons a l ih => cases h : f a <;> simp [h,ih]
namespace Runtime
def words (p : Params) : List BitWord :=
(List.range p.H).flatMap fun t => (List.range p.H).flatMap fun q =>
(List.range p.H).flatMap fun r => point p (⟨t+1⟩,⟨q+1⟩,⟨r+1⟩)
@[fun_prop] theorem words_poly : Poly words := by
have hp : Poly (fun a : ((Params×Unary)×Unary)×Unary =>
point a.1.1.1 (a.1.1.2+1,a.1.2+1,a.2+1)) := by fun_prop
have h1 := Poly.flatMapUp (show Poly (fun a : (Params×Unary)×Unary => a.1.1.2.2.2.2) by fun_prop) hp
have h2 := Poly.flatMapUp (show Poly (fun a : Params×Unary => a.1.2.2.2.2) by fun_prop) h1
exact Poly.flatMapUp (show Poly (fun a : Params => a.2.2.2.2) by fun_prop) h2
lemma gridFn_fin {H : ℕ} (t p r : Fin H) :
gridFn (⟨t.val+1⟩,⟨p.val+1⟩,⟨r.val+1⟩)=fun i => (![t,p,r] i).val+1 := by
funext i;fin_cases i <;> rfl
lemma words_eq (n s : ℕ) : words (Params.of n s)=
(ExecutableGenerator.list n s).tuples.map (fun X => BitWord.mk (encodeTuple X)) := by
unfold words
rw [show (Params.of n s).H=Parameters.H n s from rfl]
simp only [ExecutableGenerator.list,ExecutableGenerator.output]
rw [map_filterMap_toList]
simp only [Generator.grid,List.flatMap_assoc,List.flatMap_map]
rw [range_flatMap_to_finRange]
apply List.flatMap_congr
intro t ht
rw [range_flatMap_to_finRange]
apply List.flatMap_congr
intro p hp
rw [range_flatMap_to_finRange]
apply List.flatMap_congr
intro r hr
rw [point_eq,gridFn_fin]
end Runtime
end RationalHitting.StackLanguage
end OAI
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