refactor: Lint

HepLean.lean

@@ -76,6 +76,7 @@ import HepLean.SpaceTime.LorentzTensor.Fin
 import HepLean.SpaceTime.LorentzTensor.MulActionTensor
 import HepLean.SpaceTime.LorentzTensor.Notation
 import HepLean.SpaceTime.LorentzTensor.Real.Basic
+import HepLean.SpaceTime.LorentzTensor.RisingLowering
 import HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix
 import HepLean.SpaceTime.LorentzVector.Basic
 import HepLean.SpaceTime.LorentzVector.Contraction

HepLean/SpaceTime/LorentzTensor/Basic.lean

@@ -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]
+lemma contrRightAux_comp {V1 V2 V3 V4 V5 : Type} [AddCommMonoid V1] [AddCommMonoid V2]
-    [AddCommMonoid V4] [AddCommMonoid V5] [Module R V1] [Module R V2] [Module R V3] [Module R V2] [Module R V4]
+    [AddCommMonoid V3] [AddCommMonoid V4] [AddCommMonoid V5] [Module R V1] [Module R V3]
-    [Module R V5]
+    [Module R V2] [Module R V4] [Module R V5] (f : V2 ⊗[R] V3 →ₗ[R] R) (g : V4 ⊗[R] V5 →ₗ[R] R) :
-    (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)))
+    contrRightAux (𝓣.contrDual (𝓣.τ (𝓣.τ μ))) ((TensorProduct.congr
-      ((TensorProduct.congr (𝓣.colorModuleCast h) (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) (m)) ⊗ₜ
+    (𝓣.colorModuleCast h) (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) (m)) ⊗ₜ
-      (𝓣.colorModuleCast (𝓣.τ_involutive (𝓣.τ μ)).symm x)) := by
+    (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]

HepLean/SpaceTime/LorentzTensor/Contraction.lean

@@ -347,7 +347,7 @@ lemma contr_equivariant (e : (C ⊕ C) ⊕ P ≃ X)
   simp only [contrAll_rep, LinearMap.comp_id, LinearMap.id_comp]
   have h1 {M N A B : Type} [AddCommMonoid M] [AddCommMonoid N]
       [AddCommMonoid A] [AddCommMonoid B] [Module R M] [Module R N] [Module R A] [Module R B]
-      (f : M →ₗ[R] N) (g : A →ₗ[R] B)  : TensorProduct.map f g
+      (f : M →ₗ[R] N) (g : A →ₗ[R] B) : TensorProduct.map f g
       = TensorProduct.map (LinearMap.id) g ∘ₗ TensorProduct.map f (LinearMap.id) :=
     ext rfl
   rw [h1]

HepLean/SpaceTime/LorentzTensor/MulActionTensor.lean

@@ -124,7 +124,7 @@ lemma rep_mapIso (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) :
 
 @[simp]
 lemma rep_mapIso_apply (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) (x : 𝓣.Tensor cX) :
-    (𝓣.mapIso e h) (g • x) = g • (𝓣.mapIso e h x)  := by
+    (𝓣.mapIso e h) (g • x) = g • (𝓣.mapIso e h x) := by
   trans ((𝓣.rep g) ∘ₗ (𝓣.mapIso e h).toLinearMap) x
   simp
   rfl
@@ -171,7 +171,7 @@ lemma rep_tensoratorEquiv_tmul (g : G) (x : 𝓣.Tensor cX) (y : 𝓣.Tensor cY)
   rfl
 
 lemma rep_tensoratorEquiv_symm (g : G) :
-    (𝓣.tensoratorEquiv cX cY).symm ∘ₗ 𝓣.rep g  = (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) ∘ₗ
+    (𝓣.tensoratorEquiv cX cY).symm ∘ₗ 𝓣.rep g = (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) ∘ₗ
     (𝓣.tensoratorEquiv cX cY).symm.toLinearMap := by
   rw [LinearEquiv.eq_comp_toLinearMap_symm, LinearMap.comp_assoc,
     LinearEquiv.toLinearMap_symm_comp_eq]
@@ -187,9 +187,9 @@ lemma rep_tensoratorEquiv_symm_apply (g : G) (x : 𝓣.Tensor (Sum.elim cX cY))
   rfl
 
 @[simp]
-lemma rep_lid  (g : G)  : TensorProduct.lid R (𝓣.Tensor cX) ∘ₗ
+lemma rep_lid (g : G) : TensorProduct.lid R (𝓣.Tensor cX) ∘ₗ
     (TensorProduct.map (LinearMap.id) (𝓣.rep g)) = (𝓣.rep g) ∘ₗ
-    (TensorProduct.lid R (𝓣.Tensor cX)).toLinearMap  := by
+    (TensorProduct.lid R (𝓣.Tensor cX)).toLinearMap := by
   apply TensorProduct.ext'
   intro r y
   simp

HepLean/SpaceTime/LorentzTensor/Notation.lean

@@ -27,4 +27,7 @@ Further we plan to make easy to define tensors with indices. E.g. `(ψ : Tenᵘ
   For `(ψ : Tenᵘ¹ᵘ²ᵤ₃)`, if one writes e.g. `ψᵤ₁ᵘ²ᵤ₃`, this should correspond to a
   lowering of the first index of `ψ`.
 
+Further, it will be nice if we can have implicit contractions of indices
+  e.g. in Weyl fermions.
+
 -/

HepLean/SpaceTime/LorentzTensor/Real/Basic.lean

@@ -75,10 +75,10 @@ def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
     match μ with
     | .up => LorentzVector.unitUp
     | .down => LorentzVector.unitDown
-  unit_lid μ :=
+  unit_rid μ :=
     match μ with
-    | .up => LorentzVector.unitUp_lid
+    | .up => LorentzVector.unitUp_rid
-    | .down => LorentzVector.unitDown_lid
+    | .down => LorentzVector.unitDown_rid
   metric μ :=
     match μ with
     | realTensor.ColorType.up => asProdLorentzVector

HepLean/SpaceTime/LorentzTensor/RisingLowering.lean

@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
 import HepLean.SpaceTime.LorentzTensor.Basic
-import HepLean.SpaceTime.LorentzTensor.MulActionTensor
 /-!
 
 # Rising and Lowering of indices
@@ -32,10 +31,6 @@ variable {d : ℕ} {X Y Y' Z W C P : Type} [Fintype X] [DecidableEq X] [Fintype
   {cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color}
   {cW : W → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
 
-variable {G H : Type} [Group G] [Group H] [MulActionTensor G 𝓣]
-local infixl:101 " • " => 𝓣.rep
-open MulActionTensor
-
 /-!
 
 ## Properties of the unit
@@ -45,10 +40,10 @@ open MulActionTensor
 /-! TODO: Move -/
 
 lemma unit_lhs_eq (x : 𝓣.ColorModule μ) (y : 𝓣.ColorModule (𝓣.τ μ) ⊗[R] 𝓣.ColorModule μ) :
-    contrLeftAux (𝓣.contrDual μ) (x ⊗ₜ[R] y)  =
+    contrLeftAux (𝓣.contrDual μ) (x ⊗ₜ[R] y) =
     (contrRightAux (𝓣.contrDual (𝓣.τ μ))) ((TensorProduct.comm R _ _) y
     ⊗ₜ[R] (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) x) := by
-  refine  TensorProduct.induction_on y (by simp) ?_ (fun z1 z2 h1 h2 => ?_)
+  refine TensorProduct.induction_on y (by simp) ?_ (fun z1 z2 h1 h2 => ?_)
   intro x1 x2
   simp only [contrRightAux, LinearEquiv.refl_toLinearMap, comm_tmul, colorModuleCast,
     Equiv.cast_symm, LinearEquiv.coe_mk, Equiv.cast_apply, LinearMap.coe_comp, LinearEquiv.coe_coe,
@@ -58,10 +53,9 @@ lemma unit_lhs_eq (x : 𝓣.ColorModule μ) (y : 𝓣.ColorModule (𝓣.τ μ)
   simp [LinearMap.map_add, add_tmul]
   rw [← h1, ← h2, tmul_add, LinearMap.map_add]
 
-
 @[simp]
 lemma unit_lid : (contrRightAux (𝓣.contrDual (𝓣.τ μ))) ((TensorProduct.comm R _ _) (𝓣.unit μ)
-    ⊗ₜ[R] (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) x)  = x := by
+    ⊗ₜ[R] (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm) x) = x := by
   have h1 := 𝓣.unit_rid μ x
   rw [← unit_lhs_eq]
   exact h1
@@ -80,18 +74,17 @@ lemma metric_cast (h : μ = ν) :
   erw [congr_refl_refl]
   simp only [LinearEquiv.refl_apply]
 
-
-
 @[simp]
-lemma metric_contrRight_unit (μ : 𝓣.Color) (x : 𝓣.ColorModule μ ) :
+lemma metric_contrRight_unit (μ : 𝓣.Color) (x : 𝓣.ColorModule μ) :
-    (contrRightAux (𝓣.contrDual μ)) (𝓣.metric μ  ⊗ₜ[R]
+    (contrRightAux (𝓣.contrDual μ)) (𝓣.metric μ ⊗ₜ[R]
     ((contrRightAux (𝓣.contrDual (𝓣.τ μ)))
       (𝓣.metric (𝓣.τ μ) ⊗ₜ[R] (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm x)))) = x := by
   change (contrRightAux (𝓣.contrDual μ) ∘ₗ TensorProduct.map (LinearMap.id)
       (contrRightAux (𝓣.contrDual (𝓣.τ μ)))) (𝓣.metric μ
       ⊗ₜ[R] 𝓣.metric (𝓣.τ μ) ⊗ₜ[R] (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm x)) = _
   rw [contrRightAux_comp]
-  simp
+  simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul,
+    map_tmul, LinearMap.id_coe, id_eq]
   rw [𝓣.metric_dual]
   simp only [unit_lid]
 
@@ -101,13 +94,19 @@ lemma metric_contrRight_unit (μ : 𝓣.Color) (x : 𝓣.ColorModule μ ) :
 
 -/
 
-def dualizeSymm  (μ : 𝓣.Color) : 𝓣.ColorModule (𝓣.τ μ) →ₗ[R] 𝓣.ColorModule μ :=
+/-- Takes a vector with index with dual color to a vector with index the underlying color.
-  contrRightAux (𝓣.contrDual μ)  ∘ₗ
+  Obtained by contraction with the metric. -/
+def dualizeSymm (μ : 𝓣.Color) : 𝓣.ColorModule (𝓣.τ μ) →ₗ[R] 𝓣.ColorModule μ :=
+  contrRightAux (𝓣.contrDual μ) ∘ₗ
     TensorProduct.lTensorHomToHomLTensor R _ _ _ (𝓣.metric μ ⊗ₜ[R] LinearMap.id)
 
-def dualizeFun  (μ : 𝓣.Color) : 𝓣.ColorModule μ →ₗ[R] 𝓣.ColorModule (𝓣.τ μ) :=
+/-- Takes a vector to a vector with the dual color index.
+  Obtained by contraction with the metric. -/
+def dualizeFun (μ : 𝓣.Color) : 𝓣.ColorModule μ →ₗ[R] 𝓣.ColorModule (𝓣.τ μ) :=
   𝓣.dualizeSymm (𝓣.τ μ) ∘ₗ (𝓣.colorModuleCast (𝓣.τ_involutive μ).symm).toLinearMap
 
+/-- Equivalence between the module with a color `μ` and the module with color
+  `𝓣.τ μ` obtained by contraction with the metric. -/
 def dualizeModule (μ : 𝓣.Color) : 𝓣.ColorModule μ ≃ₗ[R] 𝓣.ColorModule (𝓣.τ μ) := by
   refine LinearEquiv.ofLinear (𝓣.dualizeFun μ) (𝓣.dualizeSymm μ) ?_ ?_
   · apply LinearMap.ext
@@ -121,6 +120,7 @@ def dualizeModule (μ : 𝓣.Color) : 𝓣.ColorModule μ ≃ₗ[R] 𝓣.ColorMo
       Function.comp_apply, lTensorHomToHomLTensor_apply, LinearMap.id_coe, id_eq,
       metric_contrRight_unit]
 
+/-- Dualizes the color of all indicies of a tensor by contraction with the metric. -/
 def dualizeAll : 𝓣.Tensor cX ≃ₗ[R] 𝓣.Tensor (𝓣.τ ∘ cX) := by
   refine LinearEquiv.ofLinear
     (PiTensorProduct.map (fun x => (𝓣.dualizeModule (cX x)).toLinearMap))
@@ -132,7 +132,8 @@ def dualizeAll : 𝓣.Tensor cX ≃ₗ[R] 𝓣.Tensor (𝓣.τ ∘ cX) := by
       simp [map_add, add_tmul, hx]
       simp_all only [Function.comp_apply, LinearMap.coe_comp, LinearMap.id_coe, id_eq])
     intro rx fx
-    simp
+    simp only [Function.comp_apply, PiTensorProduct.tprodCoeff_eq_smul_tprod,
+      LinearMapClass.map_smul, LinearMap.coe_comp, LinearMap.id_coe, id_eq]
     apply congrArg
     change (PiTensorProduct.map _)
       ((PiTensorProduct.map _) ((PiTensorProduct.tprod R) fx)) = _
@@ -158,6 +159,8 @@ lemma dualize_cond' (e : C ⊕ P ≃ X) :
 
 /-! TODO: Show that `dualize` is equivariant with respect to the group action. -/
 
+/-- Given an equivalence `C ⊕ P ≃ X` dualizes those indices of a tensor which correspond to
+  `C` whilst leaving the indices `P` invariant. -/
 def dualize (e : C ⊕ P ≃ X) : 𝓣.Tensor cX ≃ₗ[R]
     𝓣.Tensor (Sum.elim (𝓣.τ ∘ cX ∘ e ∘ Sum.inl) (cX ∘ e ∘ Sum.inr) ∘ e.symm) :=
   𝓣.mapIso e.symm (𝓣.dualize_cond e) ≪≫ₗ

HepLean/SpaceTime/LorentzVector/Contraction.lean

@@ -115,37 +115,32 @@ lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : Lorentz
 
 /-- The unit of the contraction of contravariant Lorentz vector and a
   covariant Lorentz vector. -/
-def unitUp : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
+def unitUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
-  ∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
+  ∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
 
-lemma unitUp_lid (x : LorentzVector d) :
+lemma unitUp_rid (x : LorentzVector d) :
-    TensorProduct.rid ℝ _
+    TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x := by
-    (TensorProduct.map (LinearEquiv.refl ℝ _).toLinearMap
+  simp only [unitUp, LinearEquiv.refl_toLinearMap]
-    (contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap)
+  rw [tmul_sum]
-    ((TensorProduct.assoc ℝ _ _ _) (unitUp ⊗ₜ[ℝ] x))) = x := by
+  simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,
-  simp only [LinearEquiv.refl_toLinearMap, unitUp]
+    Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,
-  rw [sum_tmul]
+    LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
-  simp only [Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
+    contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul, map_sum, decomp_stdBasis']
-    Finset.sum_singleton, map_add, assoc_tmul, map_sum, map_tmul, LinearMap.id_coe, id_eq,
-    LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, comm_tmul,
-    contrUpDown_stdBasis_left, rid_tmul, decomp_stdBasis']
 
 /-- The unit of the contraction of covariant Lorentz vector and a
   contravariant Lorentz vector. -/
-def unitDown : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
+def unitDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
-  ∑ i, CovariantLorentzVector.stdBasis i ⊗ₜ[ℝ] LorentzVector.stdBasis i
+  ∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
 
-lemma unitDown_lid (x : CovariantLorentzVector d) :
+lemma unitDown_rid (x : CovariantLorentzVector d) :
-    TensorProduct.rid ℝ _
+    TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x := by
-    (TensorProduct.map (LinearEquiv.refl ℝ _).toLinearMap
+  simp only [unitDown, LinearEquiv.refl_toLinearMap]
-    (contrDownUp ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap)
+  rw [tmul_sum]
-    ((TensorProduct.assoc ℝ _ _ _) (unitDown ⊗ₜ[ℝ] x))) = x := by
+  simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,
-  simp only [LinearEquiv.refl_toLinearMap, unitDown]
+    Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
-  rw [sum_tmul]
+    Finset.sum_singleton, map_add, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
-  simp only [contrDownUp, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,
+    assoc_symm_tmul, map_tmul, comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq,
-    Fin.isValue, Finset.sum_singleton, map_add, assoc_tmul, map_sum, map_tmul, LinearMap.id_coe,
+    lid_tmul, map_sum, CovariantLorentzVector.decomp_stdBasis']
-    id_eq, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, comm_tmul,
-    contrUpDown_stdBasis_right, rid_tmul, CovariantLorentzVector.decomp_stdBasis']
 
 /-!
 
@@ -175,6 +170,7 @@ end LorentzVector
 
 namespace minkowskiMatrix
 open LorentzVector
+open TensorStructure
 open scoped minkowskiMatrix
 variable {d : ℕ}
 
@@ -188,37 +184,39 @@ def asProdCovariantLorentzVector : CovariantLorentzVector d ⊗[ℝ] CovariantLo
 
 @[simp]
 lemma contrLeft_asProdLorentzVector (x : CovariantLorentzVector d) :
-    contrDualLeftAux contrDownUp (x ⊗ₜ[ℝ] asProdLorentzVector) =
+    contrLeftAux contrDownUp (x ⊗ₜ[ℝ] asProdLorentzVector) =
     ∑ μ, ((η μ μ * x μ) • LorentzVector.stdBasis μ) := by
   simp only [asProdLorentzVector]
   rw [tmul_sum]
   rw [map_sum]
   refine Finset.sum_congr rfl (fun μ _ => ?_)
-  simp only [contrDualLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap, tmul_smul, map_smul,
+  simp only [contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap, tmul_smul, map_smul,
     LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
     comm_tmul, contrUpDown_stdBasis_right, LinearMap.id_coe, id_eq, lid_tmul]
   exact smul_smul (η μ μ) (x μ) (e μ)
 
 @[simp]
 lemma contrLeft_asProdCovariantLorentzVector (x : LorentzVector d) :
-    contrDualLeftAux contrUpDown (x ⊗ₜ[ℝ] asProdCovariantLorentzVector) =
+    contrLeftAux contrUpDown (x ⊗ₜ[ℝ] asProdCovariantLorentzVector) =
     ∑ μ, ((η μ μ * x μ) • CovariantLorentzVector.stdBasis μ) := by
   simp only [asProdCovariantLorentzVector]
   rw [tmul_sum]
   rw [map_sum]
   refine Finset.sum_congr rfl (fun μ _ => ?_)
-  simp only [contrDualLeftAux, LinearEquiv.refl_toLinearMap, tmul_smul, LinearMapClass.map_smul,
+  simp only [contrLeftAux, LinearEquiv.refl_toLinearMap, tmul_smul, LinearMapClass.map_smul,
     LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul,
     contrUpDown_stdBasis_left, LinearMap.id_coe, id_eq, lid_tmul]
   exact smul_smul (η μ μ) (x μ) (CovariantLorentzVector.stdBasis μ)
 
 @[simp]
 lemma asProdLorentzVector_contr_asCovariantProdLorentzVector :
-    (contrDualMidAux (contrUpDown) (asProdLorentzVector ⊗ₜ[ℝ] asProdCovariantLorentzVector))
+    (contrMidAux (contrUpDown) (asProdLorentzVector ⊗ₜ[ℝ] asProdCovariantLorentzVector))
-    = @unitUp d := by
+    = TensorProduct.comm ℝ _ _ (@unitUp d) := by
-  simp only [contrDualMidAux, LinearEquiv.refl_toLinearMap, asProdLorentzVector, LinearMap.coe_comp,
+  simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asProdLorentzVector, LinearMap.coe_comp,
     LinearEquiv.coe_coe, Function.comp_apply]
   rw [sum_tmul, map_sum, map_sum, unitUp]
+  simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
+    Finset.sum_singleton, map_add, comm_tmul, map_sum]
   refine Finset.sum_congr rfl (fun μ _ => ?_)
   rw [← tmul_smul, TensorProduct.assoc_tmul]
   simp only [map_tmul, LinearMap.id_coe, id_eq, contrLeft_asProdCovariantLorentzVector]
@@ -239,11 +237,13 @@ lemma asProdLorentzVector_contr_asCovariantProdLorentzVector :
 
 @[simp]
 lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
-    (contrDualMidAux (contrDownUp) (asProdCovariantLorentzVector ⊗ₜ[ℝ] asProdLorentzVector))
+    (contrMidAux (contrDownUp) (asProdCovariantLorentzVector ⊗ₜ[ℝ] asProdLorentzVector))
-    = @unitDown d := by
+    = TensorProduct.comm ℝ _ _ (@unitDown d) := by
-  simp only [contrDualMidAux, LinearEquiv.refl_toLinearMap, asProdCovariantLorentzVector,
+  simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asProdCovariantLorentzVector,
     LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
   rw [sum_tmul, map_sum, map_sum, unitDown]
+  simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
+    Finset.sum_singleton, map_add, comm_tmul, map_sum]
   refine Finset.sum_congr rfl (fun μ _ => ?_)
   rw [smul_tmul, TensorProduct.assoc_tmul]
   simp only [tmul_smul, LinearMapClass.map_smul, map_tmul, LinearMap.id_coe, id_eq,