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0dd3af4 | 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 | import Mathlib
import OAI.GroupTheory.SimpleOvergroups.EnlargementEffective
import OAI.GroupTheory.SimpleOvergroups.MatrixEffective
import OAI.GroupTheory.SimpleOvergroups.EffectiveActions
namespace OAI
namespace SimpleFInftyOvergroups.EffectiveNames.Double
open BoundedEncoding NamedAction
noncomputable section
open scoped Classical
variable {A k₀ k₁ k₂ P₁ P₂ : Type*}
[CommRing k₀] [CommRing k₁] [CommRing k₂]
[Primcodable A] [Primcodable k₀] [Primcodable k₁] [Primcodable k₂]
[Finite k₀] [Finite k₁] [Finite k₂]
[Primcodable P₁] [Primcodable P₂] [Finite P₁] [Finite P₂]
{n : ℕ}
abbrev G₀ (k A : Type*) (n : ℕ) := MatrixWords.Word k A n
abbrev X₀ (k A : Type*) (n : ℕ) := MatrixWords.Point k A n
abbrev G₁ (P k A : Type*) (n : ℕ) := Enlargement.Word P (G₀ k A n)
abbrev X₁ (P k A : Type*) (n : ℕ) := Enlargement.Point P (G₀ k A n) (X₀ k A n)
abbrev G₂ (P₁ P₂ k A : Type*) (n : ℕ) := Enlargement.Word P₂ (G₁ P₁ k A n)
abbrev X₂ (P₁ P₂ k A : Type*) (n : ℕ) := Enlargement.Point P₂ (G₁ P₁ k A n) (X₁ P₁ k A n)
abbrev base (grade : List A → ℕ) := MatrixWords.nameOperations (k:=k₀) (n:=n) grade
abbrev first (grade : List A → ℕ) := Enlargement.operations (P:=P₁) (base (k₀:=k₀) (n:=n) grade)
abbrev operations (grade : List A → ℕ) :=
Enlargement.operations (P:=P₂) (first (P₁:=P₁) (k₀:=k₀) (n:=n) grade)
variable (e₁ : P₁ → Enlargement.Coeff.Name k₁) (e₂ : P₂ → Enlargement.Coeff.Name k₂)
variable (grade : List A → ℕ) (omega : List A) (f : List A → Part (List A))
def partialGroupKey : G₂ P₁ P₂ k₀ A n → Part (ℕ × ℕ) :=
Enlargement.partialGroupKey (first grade) e₂
(Enlargement.partialGroupKey (base grade) e₁
(MatrixWords.partialGroupKey (k:=k₀) (n:=n) omega f))
def partialPointKey : X₂ P₁ P₂ k₀ A n → Part ℕ :=
Enlargement.partialPointKey (first grade) e₂
(Enlargement.partialPointKey (base grade) e₁
(MatrixWords.partialPointKey (k:=k₀) (n:=n) omega f))
def test (w : List (Letter (G₂ P₁ P₂ k₀ A n) (X₂ P₁ P₂ k₀ A n))) :
Part (Result (ℕ × ℕ) ℕ) :=
(operations grade).test (partialGroupKey e₁ e₂ grade omega f)
(partialPointKey e₁ e₂ grade omega f) w
variable {m : ℕ} (C : WordCalculus (FreeMonoid A) m)
attribute [local instance] WordCalculus.rawMatrixAction
abbrev calculus := Enlargement.calculus (k:=k₂)
(Enlargement.calculus (k:=k₁) (MatrixWords.calculus (k:=k₀) (n:=n) C))
variable (he₁ : ∀ p, Enlargement.Coeff.value (e₁ p) * Enlargement.Coeff.value (e₁ p) =
Enlargement.Coeff.value (e₁ p))
(he₂ : ∀ p, Enlargement.Coeff.value (e₂ p) * Enlargement.Coeff.value (e₂ p) =
Enlargement.Coeff.value (e₂ p))
abbrev interpret₁ := Enlargement.interpretation (base (MatrixWords.grade C)) e₁
(MatrixWords.interpretation (k:=k₀) (n:=n) C) he₁
abbrev interpretation := Enlargement.interpretation (first (MatrixWords.grade C)) e₂
(interpret₁ (k₀:=k₀) (n:=n) e₁ C he₁) he₂
omit [Finite k₁] [Finite k₂] [Primcodable P₁] [Primcodable P₂] [Finite P₁] [Finite P₂] in
theorem partialGroupKey_full {d : ℕ} (w : G₂ P₁ P₂ k₀ A n)
(hw : (operations (MatrixWords.grade C)).gcost w ≤ d)
(hf : ∀ u, MatrixWords.grade C u ≤ d →
(C.normal (FreeMonoid.ofList u)).toList ∈ f u) :
(interpretation e₁ e₂ C he₁ he₂).gkey w ∈
partialGroupKey e₁ e₂ (MatrixWords.grade C) C.omega.toList f w := by
apply Enlargement.partialGroupKey_full _ _ _ _ _ _ hw
intro g hg
apply Enlargement.partialGroupKey_full _ _ _ _ _ _ hg
intro h hh
exact MatrixWords.partialGroupKey_full C f h hh hf
omit [Finite k₁] [Finite k₂] [Primcodable P₁] [Primcodable P₂] [Finite P₁] [Finite P₂] in
theorem partialPointKey_full {d : ℕ} (x : X₂ P₁ P₂ k₀ A n)
(hx : (operations (MatrixWords.grade C)).xcost x ≤ d)
(hf : ∀ u, MatrixWords.grade C u ≤ d →
(C.normal (FreeMonoid.ofList u)).toList ∈ f u) :
(interpretation e₁ e₂ C he₁ he₂).xkey x ∈
partialPointKey e₁ e₂ (MatrixWords.grade C) C.omega.toList f x := by
apply Enlargement.partialPointKey_full _ _ _ _ _ _ hx
intro y hy
apply Enlargement.partialPointKey_full _ _ _ _ _ _ hy
intro z hz
exact MatrixWords.partialPointKey_full C f z hz hf
omit [Finite k₁] [Finite k₂] [Primcodable P₁] [Primcodable P₂] [Finite P₁] [Finite P₂] in
theorem test_full {d : ℕ} (w : List (Letter (G₂ P₁ P₂ k₀ A n) (X₂ P₁ P₂ k₀ A n)))
(hw : (operations (MatrixWords.grade C)).wordCost w ≤ d)
(hf : ∀ u, MatrixWords.grade C u ≤ d →
(C.normal (FreeMonoid.ofList u)).toList ∈ f u) :
(operations (MatrixWords.grade C)).totalTest
(interpretation e₁ e₂ C he₁ he₂).gkey (interpretation e₁ e₂ C he₁ he₂).xkey w ∈
test e₁ e₂ (MatrixWords.grade C) C.omega.toList f w := by
exact Operations.test_full _ _ _ _ _
(fun g hg => partialGroupKey_full e₁ e₂ f C he₁ he₂ g hg hf)
(fun x hx => partialPointKey_full e₁ e₂ f C he₁ he₂ x hx hf) w hw
local instance : Primcodable (G₂ P₁ P₂ k₀ A n) := inferInstance
local instance : Primcodable (X₂ P₁ P₂ k₀ A n) := inferInstance
theorem primrec_inv_base : Primrec ((base (k₀:=k₀) (n:=n) grade).inv) := MatrixWords.primrec_inverse
omit [Finite k₀] in
theorem primrec_act_base : Primrec₂ ((base (k₀:=k₀) (n:=n) grade).act) :=
(Primrec.list_append.comp Primrec.fst (Primrec.fst.comp Primrec.snd)).pair
(Primrec.snd.comp Primrec.snd)
omit [Finite P₁] in
theorem primrec_inv_first : Primrec ((first (P₁:=P₁) (k₀:=k₀) (n:=n) grade).inv) :=
Enlargement.primrec_inverse _ (primrec_inv_base grade)
omit [Finite P₁] [Finite P₂] in
theorem primrec_plan : Primrec₂ ((operations (P₁:=P₁) (P₂:=P₂) (k₀:=k₀) (n:=n) grade).plan) :=
Enlargement.primrec_plan _ (primrec_inv_first grade)
variable {Param : Type*} [Primcodable Param] {F : Param → List A → Part (List A)}
theorem partrec_partialGroupKey (hF : Partrec₂ F) :
Partrec₂ (fun p : Param => partialGroupKey (k₀:=k₀) (n:=n) e₁ e₂ grade omega (F p)) :=
Enlargement.partrec_partialGroupKey _ e₂ Primrec.list_append (primrec_inv_first grade)
(Enlargement.partrec_partialGroupKey _ e₁ Primrec.list_append (primrec_inv_base grade)
(MatrixWords.partrec_partialGroupKey omega hF))
theorem partrec_partialPointKey (hF : Partrec₂ F) :
Partrec₂ (fun p : Param => partialPointKey (k₀:=k₀) (n:=n) e₁ e₂ grade omega (F p)) :=
Enlargement.partrec_partialPointKey _ e₂ Primrec.list_append (Enlargement.primrec_act _)
(Enlargement.partrec_partialPointKey _ e₁ Primrec.list_append (primrec_act_base grade)
(MatrixWords.partrec_partialPointKey omega hF))
theorem partrec_test (hF : Partrec₂ F) :
Partrec₂ (fun p : Param => test (k₀:=k₀) (n:=n) e₁ e₂ grade omega (F p)) :=
Operations.partrec_test _ (primrec_plan grade).to_comp
(partrec_partialGroupKey e₁ e₂ grade omega hF) (partrec_partialPointKey e₁ e₂ grade omega hF)
omit [Finite P₁] [Finite P₂] in
theorem primrec_gcost (hg : Primrec grade) :
Primrec ((operations (P₁:=P₁) (P₂:=P₂) (k₀:=k₀) (n:=n) grade).gcost) :=
Enlargement.primrec_gcost _ (Enlargement.primrec_gcost _ (MatrixWords.primrec_gcost hg))
omit [Finite P₁] [Finite P₂] in
theorem primrec_xcost (hg : Primrec grade) :
Primrec ((operations (P₁:=P₁) (P₂:=P₂) (k₀:=k₀) (n:=n) grade).xcost) :=
Enlargement.primrec_xcost _ (Enlargement.primrec_gcost _ (MatrixWords.primrec_gcost hg))
(Enlargement.primrec_xcost _ (MatrixWords.primrec_gcost hg) (MatrixWords.primrec_xcost hg))
omit [Finite P₁] [Finite P₂] in
theorem primrec_cost (hg : Primrec grade) :
Primrec ((operations (P₁:=P₁) (P₂:=P₂) (k₀:=k₀) (n:=n) grade).letterCost) :=
(operations grade).primrec_letterCost (primrec_gcost grade hg) (primrec_xcost grade hg)
end
end SimpleFInftyOvergroups.EffectiveNames.Double
end OAI
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