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refactor: Lint

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jstoobysmith 2024-07-30 16:31:38 -04:00
parent 6d8ac0054d
commit a97cb62379
8 changed files with 87 additions and 80 deletions
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36
HepLean/SpaceTime/LorentzTensor/Basic.lean
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@ -44,6 +44,7 @@ def contrLeftAux {V1 V2 V3 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCom
TensorProduct.map (f) (LinearEquiv.refl R V3).toLinearMap
∘ₗ (TensorProduct.assoc R _ _ _).symm.toLinearMap
/-- An auxillary function to contract the vector space `V1` and `V2` in `(V3 ⊗[R] V1) ⊗[R] V2`. -/
def contrRightAux {V1 V2 V3 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
[Module R V1] [Module R V2] [Module R V3] (f : V1 ⊗[R] V2 →ₗ[R] R) :
(V3 ⊗[R] V1) ⊗[R] V2 →ₗ[R] V3 :=
@ -51,7 +52,6 @@ def contrRightAux {V1 V2 V3 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCo
TensorProduct.map (LinearEquiv.refl R V3).toLinearMap f ∘ₗ
(TensorProduct.assoc R _ _ _).toLinearMap
/-- An auxillary function to contract the vector space `V1` and `V2` in
`V4 ⊗[R] V1 ⊗[R] V2 ⊗[R] V3`. -/
def contrMidAux {V1 V2 V3 V4 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
@ -60,30 +60,28 @@ def contrMidAux {V1 V2 V3 V4 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddC
(TensorProduct.map (LinearEquiv.refl R V4).toLinearMap (contrLeftAux f)) ∘ₗ
(TensorProduct.assoc R _ _ _).toLinearMap
lemma contrRightAux_comp {V1 V2 V3 V4 V5 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3]
[AddCommMonoid V4] [AddCommMonoid V5] [Module R V1] [Module R V2] [Module R V3] [Module R V2] [Module R V4]
[Module R V5]
(f : V2 ⊗[R] V3 →ₗ[R] R) (g : V4 ⊗[R] V5 →ₗ[R] R) :
lemma contrRightAux_comp {V1 V2 V3 V4 V5 : Type} [AddCommMonoid V1] [AddCommMonoid V2]
[AddCommMonoid V3] [AddCommMonoid V4] [AddCommMonoid V5] [Module R V1] [Module R V3]
[Module R V2] [Module R V4] [Module R V5] (f : V2 ⊗[R] V3 →ₗ[R] R) (g : V4 ⊗[R] V5 →ₗ[R] R) :
(contrRightAux f ∘ₗ TensorProduct.map (LinearMap.id : V1 ⊗[R] V2 →ₗ[R] V1 ⊗[R] V2)
(contrRightAux g)) =
(contrRightAux g) ∘ₗ TensorProduct.map (contrMidAux f) LinearMap.id
∘ₗ (TensorProduct.assoc R _ _ _).symm.toLinearMap := by
apply TensorProduct.ext'
intro x y
refine TensorProduct.induction_on x (by simp) ?_ (fun x z h1 h2 =>
refine TensorProduct.induction_on x (by simp) ?_ (fun x z h1 h2 =>
by simp [add_tmul, LinearMap.map_add, h1, h2])
intro x1 x2
refine TensorProduct.induction_on y (by simp) ?_ (fun x z h1 h2 =>
refine TensorProduct.induction_on y (by simp) ?_ (fun x z h1 h2 =>
by simp [add_tmul, tmul_add, LinearMap.map_add, h1, h2])
intro y x5
refine TensorProduct.induction_on y (by simp) ?_ (fun x z h1 h2 =>
refine TensorProduct.induction_on y (by simp) ?_ (fun x z h1 h2 =>
by simp [add_tmul, tmul_add, LinearMap.map_add, h1, h2])
intro x3 x4
simp [contrRightAux, contrMidAux, contrLeftAux]
erw [TensorProduct.map_tmul]
simp only [LinearMapClass.map_smul, LinearMap.id_coe, id_eq, mk_apply, rid_tmul]
end TensorStructure
/-- An initial structure specifying a tensor system (e.g. a system in which you can
@ -110,7 +108,8 @@ structure TensorStructure (R : Type) [CommSemiring R] where
/-- The unit of the contraction. -/
unit : (μ : Color) → ColorModule (τ μ) ⊗[R] ColorModule μ
/-- The unit is a right identity. -/
unit_rid : ∀ μ (x : ColorModule μ), TensorStructure.contrLeftAux (contrDual μ) (x ⊗ₜ[R] unit μ) = x
unit_rid : ∀ μ (x : ColorModule μ),
TensorStructure.contrLeftAux (contrDual μ) (x ⊗ₜ[R] unit μ) = x
/-- The metric for a given color. -/
metric : (μ : Color) → ColorModule μ ⊗[R] ColorModule μ
/-- The metric contracted with its dual is the unit. -/
@ -166,7 +165,7 @@ def colorModuleCast (h : μ = ν) : 𝓣.ColorModule μ ≃ₗ[R] 𝓣.ColorModu
def colorRel (μ ν : 𝓣.Color) : Prop := μ = ν ∨ μ = 𝓣.τ ν
/-- An equivalence relation on colors which is true if the two colors are equal or are duals. -/
def colorEquivRel : Equivalence 𝓣.colorRel where
lemma colorRel_equivalence : Equivalence 𝓣.colorRel where
refl := by
intro x
left
@ -195,7 +194,7 @@ def colorEquivRel : Equivalence 𝓣.colorRel where
/-- The structure of a setoid on colors, two colors are related if they are equal,
or dual. -/
instance colorSetoid : Setoid 𝓣.Color := ⟨𝓣.colorRel, 𝓣.colorEquivRel⟩
instance colorSetoid : Setoid 𝓣.Color := ⟨𝓣.colorRel, 𝓣.colorRel_equivalence⟩
/-- A map taking a color to its equivalence class in `colorSetoid`. -/
def colorQuot (μ : 𝓣.Color) : Quotient 𝓣.colorSetoid :=
@ -579,14 +578,15 @@ lemma contrDual_symm' (μ : 𝓣.Color) (x : 𝓣.ColorModule (𝓣.τ μ))
congr
simp [colorModuleCast]
lemma contrDual_symm_contrRightAux (h : ν = η):
lemma contrDual_symm_contrRightAux (h : ν = η) :
(𝓣.colorModuleCast h) ∘ₗ contrRightAux (𝓣.contrDual μ) =
contrRightAux (𝓣.contrDual (𝓣.τ (𝓣.τ μ))) ∘ₗ
(TensorProduct.congr (TensorProduct.congr (𝓣.colorModuleCast h) (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm))
(TensorProduct.congr (
TensorProduct.congr (𝓣.colorModuleCast h) (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm))
(𝓣.colorModuleCast ((𝓣.τ_involutive (𝓣.τ μ)).symm))).toLinearMap := by
apply TensorProduct.ext'
intro x y
refine TensorProduct.induction_on x (by simp) ?_ ?_
refine TensorProduct.induction_on x (by simp) ?_ ?_
· intro x z
simp [contrRightAux]
congr
@ -598,9 +598,9 @@ lemma contrDual_symm_contrRightAux (h : ν = η):
lemma contrDual_symm_contrRightAux_apply_tmul (h : ν = η)
(m : 𝓣.ColorModule ν ⊗[R] 𝓣.ColorModule μ) (x : 𝓣.ColorModule (𝓣.τ μ)) :
𝓣.colorModuleCast h (contrRightAux (𝓣.contrDual μ) (m ⊗ₜ[R] x)) =
contrRightAux (𝓣.contrDual (𝓣.τ (𝓣.τ μ)))
((TensorProduct.congr (𝓣.colorModuleCast h) (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) (m)) ⊗ₜ
(𝓣.colorModuleCast (𝓣.τ_involutive (𝓣.τ μ)).symm x)) := by
contrRightAux (𝓣.contrDual (𝓣.τ (𝓣.τ μ))) ((TensorProduct.congr
(𝓣.colorModuleCast h) (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) (m)) ⊗ₜ
(𝓣.colorModuleCast (𝓣.τ_involutive (𝓣.τ μ)).symm x)) := by
trans ((𝓣.colorModuleCast h) ∘ₗ contrRightAux (𝓣.contrDual μ)) (m ⊗ₜ[R] x)
rfl
rw [contrDual_symm_contrRightAux]