refactor: Lint
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jstoobysmith
parent
1f51e718f2
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62fdab3ace
9 changed files with 37 additions and 1937 deletions
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@ -40,28 +40,30 @@ structure PreTensorStructure (R : Type) [CommSemiring R] where
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/-- The allowed colors of indices.
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For example for a real Lorentz tensor these are `{up, down}`. -/
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Color : Type
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/-- To each color we associate a module. -/
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ColorModule : Color → Type
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/-- A map taking every color to it dual color. -/
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τ : Color → Color
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τ_involutive : Function.Involutive τ
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colorModule_addCommMonoid : ∀ μ, AddCommMonoid (ColorModule μ)
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colorModule_module : ∀ μ, Module R (ColorModule μ)
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/-- The contraction of a vector with a vector wiwth dual color. -/
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contrDual : ∀ μ, ColorModule μ ⊗[R] ColorModule (τ μ) →ₗ[R] R
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namespace PreTensorStructure
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variable (𝓣 : PreTensorStructure R)
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variable {d : ℕ} {X Y Y' Z W : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
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{c c₂ : X → 𝓣.Color} {d : Y → 𝓣.Color} {b : Z → 𝓣.Color}
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{bw : W → 𝓣.Color} {d' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
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{c c₂ : X → 𝓣.Color} {d : Y → 𝓣.Color} {b : Z → 𝓣.Color}
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{bw : W → 𝓣.Color} {d' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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instance : AddCommMonoid (𝓣.ColorModule μ) := 𝓣.colorModule_addCommMonoid μ
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instance : Module R (𝓣.ColorModule μ) := 𝓣.colorModule_module μ
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def Tensor (c : X → 𝓣.Color): Type := ⨂[R] x, 𝓣.ColorModule (c x)
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def Tensor (c : X → 𝓣.Color) : Type := ⨂[R] x, 𝓣.ColorModule (c x)
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instance : AddCommMonoid (𝓣.Tensor c) :=
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PiTensorProduct.instAddCommMonoid fun i => 𝓣.ColorModule (c i)
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@ -153,7 +155,7 @@ lemma mapIso_symm (e : X ≃ Y) (h : c = d ∘ e) :
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apply PiTensorProduct.ext
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apply MultilinearMap.ext
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intro x
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simp [mapIso, LinearMap.compMultilinearMap_apply, LinearEquiv.coe_coe,
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simp [mapIso, LinearMap.compMultilinearMap_apply, LinearEquiv.coe_coe,
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LinearEquiv.symm_apply_apply, PiTensorProduct.reindex_tprod]
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change (PiTensorProduct.reindex R (fun x => 𝓣.ColorModule (c x)) e).symm
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((PiTensorProduct.congr fun y => 𝓣.colorModuleCast _).symm ((PiTensorProduct.tprod R) x)) =
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@ -243,14 +245,13 @@ lemma elimPureTensor_update_left (p : 𝓣.PureTensor c) (q : 𝓣.PureTensor d)
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rfl
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| Sum.inr y => rfl
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def inlPureTensor (p : 𝓣.PureTensor (Sum.elim c d)) : 𝓣.PureTensor c := fun xy => p (Sum.inl xy)
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def inrPureTensor (p : 𝓣.PureTensor (Sum.elim c d)) : 𝓣.PureTensor d := fun xy => p (Sum.inr xy)
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@[simp]
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lemma inlPureTensor_update_left [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim c d)) (x : X)
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(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inl x))) :
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(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inl x))) :
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𝓣.inlPureTensor (Function.update f (Sum.inl x) v1) =Function.update (𝓣.inlPureTensor f) x v1 := by
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funext y
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simp [inlPureTensor, Function.update, Sum.inl.injEq, Sum.elim_inl]
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@ -262,14 +263,14 @@ lemma inlPureTensor_update_left [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Su
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@[simp]
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lemma inrPureTensor_update_left [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim c d)) (x : X)
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(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inl x))) :
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(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inl x))) :
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𝓣.inrPureTensor (Function.update f (Sum.inl x) v1) = (𝓣.inrPureTensor f) := by
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funext x
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simp [inrPureTensor, Function.update]
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@[simp]
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lemma inrPureTensor_update_right [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim c d)) (y : Y)
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(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inr y))) :
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(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inr y))) :
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𝓣.inrPureTensor (Function.update f (Sum.inr y) v1) =Function.update (𝓣.inrPureTensor f) y v1 := by
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funext y
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simp [inrPureTensor, Function.update, Sum.inl.injEq, Sum.elim_inl]
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@ -281,7 +282,7 @@ lemma inrPureTensor_update_right [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (S
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@[simp]
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lemma inlPureTensor_update_right [DecidableEq (X ⊕ Y)] (f : 𝓣.PureTensor (Sum.elim c d)) (y : Y)
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(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inr y))) :
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(v1 : 𝓣.ColorModule (Sum.elim c d (Sum.inr y))) :
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𝓣.inlPureTensor (Function.update f (Sum.inr y) v1) = (𝓣.inlPureTensor f) := by
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funext x
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simp [inlPureTensor, Function.update]
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@ -345,7 +346,7 @@ def tensoratorEquiv (c : X → 𝓣.Color) (d : Y → 𝓣.Color) : 𝓣.Tensor
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LinearMap.id ((PiTensorProduct.tprod R) p)
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rw [PiTensorProduct.lift.tprod]
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simp [elimPureTensorMulLin, elimPureTensor]
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change (PiTensorProduct.tprod R) _ = _
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change (PiTensorProduct.tprod R) _ = _
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apply congrArg
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funext x
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match x with
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@ -354,7 +355,7 @@ def tensoratorEquiv (c : X → 𝓣.Color) (d : Y → 𝓣.Color) : 𝓣.Tensor
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(by
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apply tensorProd_piTensorProd_ext
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intro p q
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simp [tensorator, tensoratorSymm]
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simp [tensorator, tensoratorSymm]
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change (PiTensorProduct.lift 𝓣.domCoprod) ((PiTensorProduct.lift (𝓣.elimPureTensorMulLin p)) ((PiTensorProduct.tprod R) q)) =_
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rw [PiTensorProduct.lift.tprod]
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simp [elimPureTensorMulLin]
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@ -381,9 +382,9 @@ lemma tensoratorEquiv_mapIso_cond {e : X ≃ Y} {e' : Z ≃ Y} {e'' : W ≃ X}
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@[simp]
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lemma tensoratorEquiv_mapIso (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
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(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') (h'' : bW = c ∘ e'') :
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(TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) ≪≫ₗ (𝓣.tensoratorEquiv c d)
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(TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) ≪≫ₗ (𝓣.tensoratorEquiv c d)
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= (𝓣.tensoratorEquiv bW b)
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≪≫ₗ (𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'' )) := by
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≪≫ₗ (𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'')) := by
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apply LinearEquiv.toLinearMap_inj.mp
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apply tensorProd_piTensorProd_ext
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intro p q
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@ -399,7 +400,7 @@ lemma tensoratorEquiv_mapIso_apply (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
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(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') (h'' : bW = c ∘ e'')
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(x : 𝓣.Tensor bW ⊗[R] 𝓣.Tensor b) :
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(𝓣.tensoratorEquiv c d) ((TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) x) =
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(𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'' )) ((𝓣.tensoratorEquiv bW b) x) := by
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(𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'')) ((𝓣.tensoratorEquiv bW b) x) := by
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trans ((TensorProduct.congr (𝓣.mapIso e'' h'') (𝓣.mapIso e' h')) ≪≫ₗ (𝓣.tensoratorEquiv c d)) x
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rfl
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rw [tensoratorEquiv_mapIso]
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@ -409,9 +410,9 @@ lemma tensoratorEquiv_mapIso_apply (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
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lemma tensoratorEquiv_mapIso_tmul (e : X ≃ Y) (e' : Z ≃ Y) (e'' : W ≃ X)
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(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') (h'' : bW = c ∘ e'')
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(x : 𝓣.Tensor bW) (y : 𝓣.Tensor b) :
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(x : 𝓣.Tensor bW) (y : 𝓣.Tensor b) :
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(𝓣.tensoratorEquiv c d) ((𝓣.mapIso e'' h'' x) ⊗ₜ[R] (𝓣.mapIso e' h' y)) =
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(𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'' )) ((𝓣.tensoratorEquiv bW b) (x ⊗ₜ y)) := by
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(𝓣.mapIso (Equiv.sumCongr e'' e') (𝓣.tensoratorEquiv_mapIso_cond h h' h'')) ((𝓣.tensoratorEquiv bW b) (x ⊗ₜ y)) := by
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rw [← tensoratorEquiv_mapIso_apply]
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rfl
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exact e
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@ -430,14 +431,14 @@ def decompEmbedSet (f : Y ↪ X) :
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Equiv.sumCongr ((Equiv.subtypeEquivRight (by simp))) <|
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(Function.Embedding.toEquivRange f).symm
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def decompEmbedColorLeft (c : X → 𝓣.Color) (f : Y ↪ X) : {x // x ∈ (Finset.image f Finset.univ)ᶜ} → 𝓣.Color :=
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def decompEmbedColorLeft (c : X → 𝓣.Color) (f : Y ↪ X) : {x // x ∈ (Finset.image f Finset.univ)ᶜ} → 𝓣.Color :=
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(c ∘ (decompEmbedSet f).symm) ∘ Sum.inl
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def decompEmbedColorRight (c : X → 𝓣.Color) (f : Y ↪ X) : Y → 𝓣.Color :=
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def decompEmbedColorRight (c : X → 𝓣.Color) (f : Y ↪ X) : Y → 𝓣.Color :=
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(c ∘ (decompEmbedSet f).symm) ∘ Sum.inr
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lemma decompEmbed_cond (c : X → 𝓣.Color) (f : Y ↪ X) : c =
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(Sum.elim (𝓣.decompEmbedColorLeft c f) (𝓣.decompEmbedColorRight c f)) ∘ decompEmbedSet f := by
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lemma decompEmbed_cond (c : X → 𝓣.Color) (f : Y ↪ X) : c =
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(Sum.elim (𝓣.decompEmbedColorLeft c f) (𝓣.decompEmbedColorRight c f)) ∘ decompEmbedSet f := by
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simpa [decompEmbedColorLeft, decompEmbedColorRight] using (Equiv.comp_symm_eq _ _ _).mp rfl
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/-- Decomposes a tensor into a tensor product based on an embedding. -/
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@ -446,7 +447,6 @@ def decompEmbed (f : Y ↪ X) :
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(𝓣.mapIso (decompEmbedSet f) (𝓣.decompEmbed_cond c f)) ≪≫ₗ
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(𝓣.tensoratorEquiv (𝓣.decompEmbedColorLeft c f) (𝓣.decompEmbedColorRight c f)).symm
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/-!
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## Contraction
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@ -457,7 +457,7 @@ def pairProd : 𝓣.Tensor c ⊗[R] 𝓣.Tensor c₂ →ₗ[R]
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⨂[R] x, 𝓣.ColorModule (c x) ⊗[R] 𝓣.ColorModule (c₂ x) :=
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TensorProduct.lift (
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PiTensorProduct.map₂ (fun x =>
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TensorProduct.mk R (𝓣.ColorModule (c x)) (𝓣.ColorModule (c₂ x)) ))
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TensorProduct.mk R (𝓣.ColorModule (c x)) (𝓣.ColorModule (c₂ x))))
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lemma mkPiAlgebra_equiv (e : X ≃ Y) :
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(PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R X R)) =
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@ -476,7 +476,7 @@ def contrAll' : 𝓣.Tensor c ⊗[R] 𝓣.Tensor (𝓣.τ ∘ c) →ₗ[R] R :=
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(𝓣.pairProd)
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lemma contrAll'_mapIso_cond {e : X ≃ Y} (h : c = d ∘ e) :
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𝓣.τ ∘ d = (𝓣.τ ∘ c) ∘ ⇑e.symm := by
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𝓣.τ ∘ d = (𝓣.τ ∘ c) ∘ ⇑e.symm := by
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subst h
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ext1 x
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simp
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@ -485,7 +485,7 @@ lemma contrAll'_mapIso_cond {e : X ≃ Y} (h : c = d ∘ e) :
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lemma contrAll'_mapIso (e : X ≃ Y) (h : c = d ∘ e) :
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𝓣.contrAll' ∘ₗ
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(TensorProduct.congr (𝓣.mapIso e h) (LinearEquiv.refl R _)).toLinearMap =
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𝓣.contrAll' ∘ₗ (TensorProduct.congr (LinearEquiv.refl R _)
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𝓣.contrAll' ∘ₗ (TensorProduct.congr (LinearEquiv.refl R _)
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(𝓣.mapIso e.symm (𝓣.contrAll'_mapIso_cond h))).toLinearMap := by
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apply TensorProduct.ext'
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refine fun x ↦
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@ -537,7 +537,7 @@ lemma contrAll'_mapIso (e : X ≃ Y) (h : c = d ∘ e) :
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@[simp]
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lemma contrAll'_mapIso_tmul (e : X ≃ Y) (h : c = d ∘ e) (x : 𝓣.Tensor c) (y : 𝓣.Tensor (𝓣.τ ∘ d)) :
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𝓣.contrAll' ((𝓣.mapIso e h) x ⊗ₜ[R] y) = 𝓣.contrAll' (x ⊗ₜ[R] (𝓣.mapIso e.symm (𝓣.contrAll'_mapIso_cond h) y)) := by
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change (𝓣.contrAll' ∘ₗ
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change (𝓣.contrAll' ∘ₗ
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(TensorProduct.congr (𝓣.mapIso e h) (LinearEquiv.refl R _)).toLinearMap) (x ⊗ₜ[R] y) = _
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rw [contrAll'_mapIso]
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rfl
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@ -562,7 +562,7 @@ lemma contrAll_mapIso_right_cond {e : X ≃ Y} {e' : Z ≃ Y}
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@[simp]
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lemma contrAll_mapIso_right_tmul (e : X ≃ Y) (e' : Z ≃ Y)
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(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') (x : 𝓣.Tensor c) (z : 𝓣.Tensor b) :
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(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') (x : 𝓣.Tensor c) (z : 𝓣.Tensor b) :
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𝓣.contrAll e h (x ⊗ₜ[R] (𝓣.mapIso e' h' z)) =
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𝓣.contrAll (e.trans e'.symm) (𝓣.contrAll_mapIso_right_cond h h') (x ⊗ₜ[R] z) := by
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rw [contrAll, contrAll]
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@ -572,7 +572,7 @@ lemma contrAll_mapIso_right_tmul (e : X ≃ Y) (e' : Z ≃ Y)
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@[simp]
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lemma contrAll_comp_mapIso_right (e : X ≃ Y) (e' : Z ≃ Y)
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(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') : 𝓣.contrAll e h ∘ₗ
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(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = d ∘ e') : 𝓣.contrAll e h ∘ₗ
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(TensorProduct.congr (LinearEquiv.refl R (𝓣.Tensor c)) (𝓣.mapIso e' h')).toLinearMap
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= 𝓣.contrAll (e.trans e'.symm) (𝓣.contrAll_mapIso_right_cond h h') := by
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apply TensorProduct.ext'
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@ -587,7 +587,7 @@ lemma contrAll_mapIso_left_cond {e : X ≃ Y} {e' : Z ≃ X}
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@[simp]
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lemma contrAll_mapIso_left_tmul {e : X ≃ Y} {e' : Z ≃ X}
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(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = c ∘ e') (x : 𝓣.Tensor b) (y : 𝓣.Tensor d) :
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(h : c = 𝓣.τ ∘ d ∘ e) (h' : b = c ∘ e') (x : 𝓣.Tensor b) (y : 𝓣.Tensor d) :
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𝓣.contrAll e h ((𝓣.mapIso e' h' x) ⊗ₜ[R] y) =
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𝓣.contrAll (e'.trans e) (𝓣.contrAll_mapIso_left_cond h h') (x ⊗ₜ[R] y) := by
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rw [contrAll, contrAll]
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@ -609,7 +609,7 @@ end PreTensorStructure
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structure TensorStructure (R : Type) [CommSemiring R] extends PreTensorStructure R where
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contrDual_symm : ∀ μ,
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(contrDual μ) ∘ₗ (TensorProduct.comm R (ColorModule (τ μ)) (ColorModule μ)).toLinearMap =
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(contrDual μ) ∘ₗ (TensorProduct.comm R (ColorModule (τ μ)) (ColorModule μ)).toLinearMap =
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(contrDual (τ μ)) ∘ₗ (TensorProduct.congr (LinearEquiv.refl _ _)
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(toPreTensorStructure.colorModuleCast (by rw[toPreTensorStructure.τ_involutive]))).toLinearMap
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@ -621,8 +621,7 @@ variable (𝓣 : TensorStructure R)
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variable {d : ℕ} {X Y Y' Z : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z]
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{c c₂ : X → 𝓣.Color} {d : Y → 𝓣.Color} {b : Z → 𝓣.Color} {d' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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{c c₂ : X → 𝓣.Color} {d : Y → 𝓣.Color} {b : Z → 𝓣.Color} {d' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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end TensorStructure
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@ -642,8 +641,7 @@ variable (𝓣 : GroupTensorStructure R G)
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variable {d : ℕ} {X Y Y' Z : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z]
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{c c₂ : X → 𝓣.Color} {d : Y → 𝓣.Color} {b : Z → 𝓣.Color} {d' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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{c c₂ : X → 𝓣.Color} {d : Y → 𝓣.Color} {b : Z → 𝓣.Color} {d' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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def rep : Representation R G (𝓣.Tensor c) where
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toFun g := PiTensorProduct.map (fun x => 𝓣.repColorModule (c x) g)
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@ -663,12 +661,11 @@ lemma repColorModule_colorModuleCast_apply (h : μ = ν) (g : G) (x : 𝓣.Color
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@[simp]
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lemma repColorModule_colorModuleCast (h : μ = ν) (g : G) :
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(𝓣.repColorModule ν g) ∘ₗ (𝓣.colorModuleCast h).toLinearMap =
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(𝓣.colorModuleCast h).toLinearMap ∘ₗ (𝓣.repColorModule μ g) := by
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(𝓣.colorModuleCast h).toLinearMap ∘ₗ (𝓣.repColorModule μ g) := by
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apply LinearMap.ext
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intro x
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simp [repColorModule_colorModuleCast_apply]
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@[simp]
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lemma rep_mapIso (e : X ≃ Y) (h : c = d ∘ e) (g : G) :
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(𝓣.rep g) ∘ₗ (𝓣.mapIso e h).toLinearMap = (𝓣.mapIso e h).toLinearMap ∘ₗ 𝓣.rep g := by
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@ -710,7 +707,7 @@ lemma rep_tprod (g : G) (f : (i : X) → 𝓣.ColorModule (c i)) :
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-/
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lemma rep_tensoratorEquiv (g : G) :
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(𝓣.tensoratorEquiv c d) ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.rep g ∘ₗ
|
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(𝓣.tensoratorEquiv c d) ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.rep g ∘ₗ
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(𝓣.tensoratorEquiv c d).toLinearMap := by
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||||
apply tensorProd_piTensorProd_ext
|
||||
intro p q
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||||
|
|
@ -733,7 +730,6 @@ lemma rep_tensoratorEquiv_tmul (g : G) (x : 𝓣.Tensor c) (y : 𝓣.Tensor d) :
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|||
nth_rewrite 1 [← rep_tensoratorEquiv_apply]
|
||||
rfl
|
||||
|
||||
|
||||
/-!
|
||||
|
||||
## Group acting on contraction
|
||||
|
|
@ -759,7 +755,7 @@ lemma contrAll_rep {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (
|
|||
simp [contrAll']
|
||||
apply congrArg
|
||||
simp [pairProd]
|
||||
change (PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _) =
|
||||
change (PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _) =
|
||||
(PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _)
|
||||
rw [PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map_tprod,
|
||||
PiTensorProduct.map_tprod]
|
||||
|
|
@ -786,5 +782,4 @@ lemma contrAll_rep_tmul {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃
|
|||
|
||||
end GroupTensorStructure
|
||||
|
||||
|
||||
end
|
||||
|
|
|
|||
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Reference in a new issue