| 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 | |