| import Mathlib |
| import OAI.GroupTheory.SimpleOvergroups.EnlargementEffective |
| import OAI.GroupTheory.SimpleOvergroups.TwoEnlargements |
|
|
| namespace OAI |
|
|
|
|
| noncomputable section |
| open scoped Classical |
| namespace SimpleFInftyOvergroups.SemanticEncoding |
| open BoundedEncoding EffectiveNames NamedAction SourceEnlargement |
|
|
| namespace IdempotentNames |
| abbrev Index (k : Type) [CommRing k] := BinaryPrefixOperators.idempotents k |
| instance {k : Type} [CommRing k] [Finite k] : Fintype (Index k) := |
| (BinaryPrefixOperators.idempotents_finite k).fintype |
| instance {k : Type} [CommRing k] [Finite k] : Primcodable (Index k) := |
| FiniteAlgebraicEnvelopes.Effect.finitePrimcodable _ |
| |
| variable {k : Type} [CommRing k] |
| def name (p : Index k) : Enlargement.Coeff.Name k := |
| (BinaryPrefixOperators.Arithmetic.Lists.value_surjective p.val).choose |
| @[simp] theorem value_name (p : Index k) : Enlargement.Coeff.value (name p) = p.val := |
| (BinaryPrefixOperators.Arithmetic.Lists.value_surjective p.val).choose_spec |
| theorem idempotent (p : Index k) : |
| Enlargement.Coeff.value (name p) * Enlargement.Coeff.value (name p) = |
| Enlargement.Coeff.value (name p) := by |
| simpa only [value_name] using BinaryPrefixOperators.idempotents_are_idempotent k p.property |
| end IdempotentNames |
| |
| namespace SemanticEnlargement |
| variable {G X V Y K L k : Type} [Group V] [MulAction V Y] [Nonempty Y] |
| [CommRing k] [Primcodable k] [Primcodable K] [Primcodable L] |
| {m : ℕ} (N : Operations G X) (C : ActionCalculus V Y K L m) |
| (I : Interpretation N C) |
| |
| local instance restrictedSMul : SMul (enlargedGroup k V) (enlargedSet k Y) := |
| (SourceEnlargement.enlargedAction k V Y).toSMul |
| |
| abbrev Word := Enlargement.Word (IdempotentNames.Index k) G |
| abbrev Point := Enlargement.Point (IdempotentNames.Index k) G X |
| |
| abbrev ambient := Enlargement.interpretation N (IdempotentNames.name (k:=k)) I IdempotentNames.idempotent |
| |
| def calculus : ActionCalculus (enlargedGroup k V) (enlargedSet k Y) (ℕ × ℕ) ℕ m where |
| group := |
| { ball := fun b w => (Enlargement.calculus (k:=k) C).group.ball b w.val |
| mono := fun _ _ h _ hw => (Enlargement.calculus (k:=k) C).group.mono h hw |
| key := fun w => (Enlargement.calculus (k:=k) C).group.key w.val } |
| point := (Enlargement.calculus (k:=k) C).point |
| one_mem := (Enlargement.calculus (k:=k) C).one_mem |
| inv_mem := (Enlargement.calculus (k:=k) C).inv_mem |
| mul_mem := (Enlargement.calculus (k:=k) C).mul_mem |
| smul_mem := (Enlargement.calculus (k:=k) C).smul_mem |
| inv_congr := (Enlargement.calculus (k:=k) C).inv_congr |
| mul_congr := (Enlargement.calculus (k:=k) C).mul_congr |
| smul_congr := (Enlargement.calculus (k:=k) C).smul_congr |
|
|
| omit [Nonempty Y] [Primcodable k] [Primcodable K] [Primcodable L] in |
| theorem group_mem (w : Word (k:=k) (G:=G)) : |
| Enlargement.groupValue N (IdempotentNames.name (k:=k)) I IdempotentNames.idempotent w ∈ enlargedGroup k V := by |
| induction w with |
| | nil => exact (enlargedGroup k V).one_mem |
| | cons a w ih => |
| apply (enlargedGroup k V).mul_mem _ ih |
| change SourceEnlargement.unitMap _ _ _ ∈ _ |
| simp only [IdempotentNames.value_name] |
| exact Subgroup.subset_closure ⟨a.1, (I.group a.2).val, rfl⟩ |
|
|
| def groupValue (w : Word (k:=k) (G:=G)) : enlargedGroup k V := |
| ⟨Enlargement.groupValue N IdempotentNames.name I IdempotentNames.idempotent w,group_mem N C I w⟩ |
| def pointValue (x : Point (k:=k) (G:=G) (X:=X)) : enlargedSet k Y := |
| Enlargement.pointValue N IdempotentNames.name I IdempotentNames.idempotent x |
|
|
| omit [Nonempty Y] [Primcodable k] [Primcodable K] [Primcodable L] in |
| @[simp] theorem groupValue_nil : groupValue N C I ([] : Word (k:=k) (G:=G)) = 1 := rfl |
| omit [Nonempty Y] [Primcodable k] [Primcodable K] [Primcodable L] in |
| @[simp] theorem groupValue_append (u v : Word (k:=k) (G:=G)) : |
| groupValue N C I (u++v) = groupValue N C I u * groupValue N C I v := by |
| apply Subtype.ext |
| exact Enlargement.groupValue_append _ _ _ _ _ _ |
| omit [Nonempty Y] [Primcodable k] [Primcodable K] [Primcodable L] in |
| @[simp] theorem groupValue_inverse (u : Word (k:=k) (G:=G)) : |
| groupValue N C I (Enlargement.inverse N u) = (groupValue N C I u)⁻¹ := by |
| apply Subtype.ext |
| exact Enlargement.groupValue_inverse _ _ _ _ _ |
| omit [Nonempty Y] [Primcodable k] [Primcodable K] [Primcodable L] in |
| @[simp] theorem groupValue_singleton (p : IdempotentNames.Index k) (g : G) : |
| groupValue N C I [(p,g)] = SourceEnlargement.j k p (I.group g).val := by |
| apply Subtype.ext |
| simp [groupValue, |
| Enlargement.groupValue, Enlargement.rootValue, SourceEnlargement.j, SourceEnlargement.map] |
| rfl |
|
|
| omit [Nonempty Y] [Primcodable k] [Primcodable K] [Primcodable L] in |
| @[simp] theorem pointValue_act (g : Word (k:=k) (G:=G)) (x : Point (k:=k) (G:=G) (X:=X)) : |
| pointValue N C I (g++x.1,x.2) = groupValue N C I g • pointValue N C I x := |
| Enlargement.pointValue_act _ _ _ _ _ _ |
| omit [Nonempty Y] [Primcodable k] [Primcodable K] [Primcodable L] in |
| @[simp] theorem pointValue_basis (x : X) : |
| pointValue N C I (([] : Word (k:=k) (G:=G)),x) = pointEmbedding k (I.point x).val := by |
| simp only [pointValue, Enlargement.pointValue, Enlargement.groupValue_nil, one_smul, |
| pointEmbedding] |
|
|
| def interpretation : Interpretation (Enlargement.operations (P:=IdempotentNames.Index k) N) |
| (calculus (k:=k) C) where |
| group w := ⟨groupValue N C I w,Enlargement.gcost N w,Enlargement.group_mem N IdempotentNames.name I IdempotentNames.idempotent w⟩ |
| point x := ⟨pointValue N C I x, Enlargement.xcost N x, |
| Enlargement.point_mem N IdempotentNames.name I IdempotentNames.idempotent x⟩ |
| group_cost _ := rfl |
| point_cost _ := rfl |
| one := WithBound.ext' (groupValue_nil N C I) rfl |
| mul u v := WithBound.ext' (groupValue_append N C I u v) (Enlargement.gcost_append N u v) |
| inv u := WithBound.ext' (groupValue_inverse N C I u) (Enlargement.gcost_inverse N u) |
| act g x := WithBound.ext' (pointValue_act N C I g x) |
| (by change Enlargement.xcost N (g++x.1,x.2) = Enlargement.gcost N g + Enlargement.xcost N x |
| simp [Enlargement.xcost,Nat.add_assoc]) |
|
|
| theorem groupKey_injective (hg : Function.Injective C.group.key) : |
| Function.Injective (calculus (k:=k) C).group.key := by |
| intro u v h |
| apply Subtype.ext |
| apply Units.ext |
| apply MonoidAlgebra.coeff_injective |
| apply Finsupp.mapDomain_injective hg |
| exact congrArg Prod.fst (Enlargement.pairedKey_injective h) |
|
|
| theorem pointKey_injective (hx : Function.Injective C.point.key) : |
| Function.Injective (calculus (k:=k) C).point.key := by |
| intro u v h |
| exact Finsupp.mapDomain_injective hx (Enlargement.arrayKey_injective h) |
|
|
| omit [Nonempty Y] [Primcodable k] [Primcodable K] [Primcodable L] in |
| theorem groupValue_surjective (hg : Function.Surjective (fun g => (I.group g).val)) : |
| Function.Surjective (groupValue (k:=k) N C I) := by |
| intro w |
| have h : ∃ u, (groupValue N C I u).val = w.val := by |
| apply Subgroup.closure_induction (p:=fun w _ => ∃ u, (groupValue N C I u).val = w) (hx:=w.property) |
| · rintro _ ⟨p,g,rfl⟩ |
| obtain ⟨a,rfl⟩ := hg g |
| exact ⟨[(p,a)],congrArg Subtype.val (groupValue_singleton N C I p a)⟩ |
| · exact ⟨[],rfl⟩ |
| · rintro _ _ _ _ ⟨u,rfl⟩ ⟨v,rfl⟩ |
| exact ⟨u++v,congrArg Subtype.val (groupValue_append N C I u v)⟩ |
| · rintro _ _ ⟨u,rfl⟩ |
| exact ⟨Enlargement.inverse N u,congrArg Subtype.val (groupValue_inverse N C I u)⟩ |
| obtain ⟨u,hu⟩ := h |
| exact ⟨u,Subtype.ext hu⟩ |
|
|
| variable [Primcodable G] [Primcodable X] |
| omit [Primcodable G] [Primcodable X] in |
| theorem partialGroupKey_full (f : G → Part K) (w : Word (k:=k) (G:=G)) {d : ℕ} |
| (hw : Enlargement.gcost N w ≤ d) (hf : ∀ g, N.gcost g ≤ d → I.gkey g ∈ f g) : |
| (interpretation N C I).gkey w ∈ Enlargement.partialGroupKey N IdempotentNames.name f w := |
| Enlargement.partialGroupKey_full _ _ _ _ _ _ hw hf |
|
|
| omit [Primcodable G] [Primcodable X] in |
| theorem partialPointKey_full (f : X → Part L) (x : Point (k:=k) (G:=G) (X:=X)) {d : ℕ} |
| (hx : Enlargement.xcost N x ≤ d) (hf : ∀ y, N.xcost y ≤ d → I.xkey y ∈ f y) : |
| (interpretation N C I).xkey x ∈ Enlargement.partialPointKey N IdempotentNames.name f x := |
| Enlargement.partialPointKey_full _ _ _ _ _ _ hx hf |
|
|
| end SemanticEnlargement |
| end SimpleFInftyOvergroups.SemanticEncoding |
| end |
|
|
| end OAI |
|
|
