feat: contr_unit for Complex Lorentz Tensors
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jstoobysmith
parent
14377da3d8
commit
942ee12e60
4 changed files with 201 additions and 2 deletions
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@ -97,6 +97,17 @@ lemma contrCoContraction_basis (i j : Fin 4) :
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refine ite_congr ?h₁ (congrFun rfl) (congrFun rfl)
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simp only [EmbeddingLike.apply_eq_iff_eq, Fin.ext_iff, eq_iff_iff, eq_comm]
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lemma contrCoContraction_basis' (i j : Fin 1 ⊕ Fin 3) :
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contrCoContraction.hom (complexContrBasis i ⊗ₜ complexCoBasis j) =
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if i = j then (1 : ℂ) else 0 := by
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rw [contrCoContraction_hom_tmul]
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simp only [Action.instMonoidalCategory_tensorUnit_V, complexContrBasisFin4, Basis.coe_reindex,
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Function.comp_apply, complexContrBasis_toFin13ℂ, complexCoBasisFin4, complexCoBasis_toFin13ℂ,
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dotProduct_single, mul_one]
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rw [Pi.single_apply]
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refine ite_congr ?h₁ (congrFun rfl) (congrFun rfl)
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exact Eq.propIntro (fun a => id (Eq.symm a)) fun a => id (Eq.symm a)
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/-- The linear map from complexCo ⊗ complexContr to ℂ given by
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summing over components of covariant Lorentz vector and
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contravariant Lorentz vector in the
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@ -126,6 +137,17 @@ lemma coContrContraction_basis (i j : Fin 4) :
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refine ite_congr ?h₁ (congrFun rfl) (congrFun rfl)
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simp only [EmbeddingLike.apply_eq_iff_eq, Fin.ext_iff, eq_iff_iff, eq_comm]
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lemma coContrContraction_basis' (i j : Fin 1 ⊕ Fin 3) :
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coContrContraction.hom (complexCoBasis i ⊗ₜ complexContrBasis j) =
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if i = j then (1 : ℂ) else 0 := by
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rw [coContrContraction_hom_tmul]
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simp only [Action.instMonoidalCategory_tensorUnit_V, complexCoBasisFin4, Basis.coe_reindex,
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Function.comp_apply, complexCoBasis_toFin13ℂ, complexContrBasisFin4, complexContrBasis_toFin13ℂ,
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dotProduct_single, mul_one]
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rw [Pi.single_apply]
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refine ite_congr ?h₁ (congrFun rfl) (congrFun rfl)
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simp only [EmbeddingLike.apply_eq_iff_eq, Fin.ext_iff, eq_iff_iff, eq_comm]
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/-!
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## Symmetry
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@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.LorentzVector.Complex.Two
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import HepLean.SpaceTime.LorentzVector.Complex.Contraction
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/-!
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# Unit for complex Lorentz vectors
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@ -122,6 +123,76 @@ lemma coContrUnit_apply_one : coContrUnit.hom (1 : ℂ) = coContrUnitVal := by
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change coContrUnit.hom.toFun (1 : ℂ) = coContrUnitVal
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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coContrUnit, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
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/-!
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## Contraction of the units
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-/
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/-- Contraction on the right with `contrCoUnit` does nothing. -/
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lemma contr_contrCoUnit (x : complexCo) :
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(λ_ complexCo).hom.hom
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((coContrContraction ▷ complexCo).hom
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((α_ _ _ complexCo).inv.hom
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(x ⊗ₜ[ℂ] contrCoUnit.hom (1 : ℂ)))) = x := by
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obtain ⟨c, hc⟩ := (mem_span_range_iff_exists_fun ℂ).mp (Basis.mem_span complexCoBasis x)
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subst hc
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rw [contrCoUnit_apply_one, contrCoUnitVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.instMonoidalCategory_leftUnitor_hom_hom, Action.instMonoidalCategory_whiskerRight_hom,
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Action.instMonoidalCategory_associator_inv_hom, CategoryTheory.Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
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Finset.sum_singleton, Fin.sum_univ_three, tmul_add, add_tmul, smul_tmul, tmul_smul, map_add,
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_root_.map_smul]
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have h1' (x y : CoeSort.coe complexCo) (z : CoeSort.coe complexContr) :
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(α_ complexCo.V complexContr.V complexCo.V).inv (x ⊗ₜ[ℂ] z ⊗ₜ[ℂ] y) = (x ⊗ₜ[ℂ] z) ⊗ₜ[ℂ] y := rfl
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repeat rw [h1']
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have h1'' ( y : CoeSort.coe complexCo) (z : CoeSort.coe complexCo ⊗[ℂ] CoeSort.coe complexContr) :
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(coContrContraction.hom ▷ complexCo.V) (z ⊗ₜ[ℂ] y) = (coContrContraction.hom z) ⊗ₜ[ℂ] y := rfl
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repeat rw (config := { transparency := .instances }) [h1'']
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repeat rw [coContrContraction_basis']
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simp only [Fin.isValue, leftUnitor, ModuleCat.MonoidalCategory.leftUnitor, ModuleCat.of_coe,
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CategoryTheory.Iso.trans_hom, LinearEquiv.toModuleIso_hom, ModuleCat.ofSelfIso_hom,
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CategoryTheory.Category.comp_id, Action.instMonoidalCategory_tensorUnit_V, ↓reduceIte,
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reduceCtorEq, zero_tmul, map_zero, smul_zero, add_zero, Sum.inr.injEq, one_ne_zero,
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Fin.reduceEq, zero_add, zero_ne_one]
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erw [TensorProduct.lid_tmul, TensorProduct.lid_tmul, TensorProduct.lid_tmul,
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TensorProduct.lid_tmul]
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simp only [Fin.isValue, one_smul]
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repeat rw [add_assoc]
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/-- Contraction on the right with `coContrUnit`. -/
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lemma contr_coContrUnit (x : complexContr) :
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(λ_ complexContr).hom.hom
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((contrCoContraction ▷ complexContr).hom
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((α_ _ _ complexContr).inv.hom
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(x ⊗ₜ[ℂ] coContrUnit.hom (1 : ℂ)))) = x := by
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obtain ⟨c, hc⟩ := (mem_span_range_iff_exists_fun ℂ).mp (Basis.mem_span complexContrBasis x)
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subst hc
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rw [coContrUnit_apply_one, coContrUnitVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.instMonoidalCategory_leftUnitor_hom_hom, Action.instMonoidalCategory_whiskerRight_hom,
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Action.instMonoidalCategory_associator_inv_hom, CategoryTheory.Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
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Finset.sum_singleton, Fin.sum_univ_three, tmul_add, add_tmul, smul_tmul, tmul_smul, map_add,
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_root_.map_smul]
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have h1' (x y : CoeSort.coe complexContr) (z : CoeSort.coe complexCo) :
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(α_ complexContr.V complexCo.V complexContr.V).inv (x ⊗ₜ[ℂ] z ⊗ₜ[ℂ] y) = (x ⊗ₜ[ℂ] z) ⊗ₜ[ℂ] y := rfl
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repeat rw [h1']
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have h1'' ( y : CoeSort.coe complexContr) (z : CoeSort.coe complexContr ⊗[ℂ] CoeSort.coe complexCo) :
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(contrCoContraction.hom ▷ complexContr.V) (z ⊗ₜ[ℂ] y) = (contrCoContraction.hom z) ⊗ₜ[ℂ] y := rfl
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repeat rw (config := { transparency := .instances }) [h1'']
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repeat rw [contrCoContraction_basis']
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simp only [Fin.isValue, Action.instMonoidalCategory_tensorUnit_V, ↓reduceIte, reduceCtorEq,
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zero_tmul, map_zero, smul_zero, add_zero, Sum.inr.injEq, one_ne_zero, Fin.reduceEq, zero_add,
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zero_ne_one]
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erw [TensorProduct.lid_tmul, TensorProduct.lid_tmul, TensorProduct.lid_tmul,
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TensorProduct.lid_tmul]
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simp only [Fin.isValue, one_smul]
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repeat rw [add_assoc]
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end Lorentz
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end
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@ -7,6 +7,7 @@ import HepLean.SpaceTime.WeylFermion.Basic
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import HepLean.SpaceTime.WeylFermion.Contraction
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import Mathlib.LinearAlgebra.TensorProduct.Matrix
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import HepLean.SpaceTime.WeylFermion.Two
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import LLMLean
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/-!
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# Units of Weyl fermions
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@ -221,11 +222,108 @@ lemma altRightRightUnit_apply_one : altRightRightUnit.hom (1 : ℂ) = altRightRi
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-/
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lemma contr_leftAltLeftUnitVal (x : leftHanded) :
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/-- Contraction on the right with `altLeftLeftUnit` does nothing. -/
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lemma contr_altLeftLeftUnit (x : leftHanded) :
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(λ_ leftHanded).hom.hom
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(((leftAltContraction) ▷ leftHanded).hom
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((α_ _ _ leftHanded).inv.hom
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(x ⊗ₜ[ℂ] altLeftLeftUnit.hom (1 : ℂ)))) = x := by
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sorry
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obtain ⟨c, hc⟩ := (mem_span_range_iff_exists_fun ℂ).mp (Basis.mem_span leftBasis x)
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subst hc
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rw [altLeftLeftUnit_apply_one, altLeftLeftUnitVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.instMonoidalCategory_leftUnitor_hom_hom, Action.instMonoidalCategory_whiskerRight_hom,
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Action.instMonoidalCategory_associator_inv_hom, CategoryTheory.Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Fin.sum_univ_two, Fin.isValue, tmul_add, add_tmul, smul_tmul, tmul_smul, map_add,
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_root_.map_smul]
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have h1 (x y : leftHanded) (z : altLeftHanded) : (leftAltContraction.hom ▷ leftHanded.V)
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((α_ leftHanded.V altLeftHanded.V leftHanded.V).inv (x ⊗ₜ[ℂ] z ⊗ₜ[ℂ] y)) =
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(leftAltContraction.hom (x ⊗ₜ[ℂ] z)) ⊗ₜ[ℂ] y := rfl
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erw [h1, h1, h1, h1]
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repeat rw [leftAltContraction_basis]
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simp only [Fin.isValue, leftUnitor, ModuleCat.MonoidalCategory.leftUnitor, ModuleCat.of_coe,
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CategoryTheory.Iso.trans_hom, LinearEquiv.toModuleIso_hom, ModuleCat.ofSelfIso_hom,
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CategoryTheory.Category.comp_id, Action.instMonoidalCategory_tensorUnit_V, Fin.val_zero,
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↓reduceIte, Fin.val_one, one_ne_zero, zero_tmul, map_zero, smul_zero, add_zero, zero_ne_one,
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zero_add]
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erw [TensorProduct.lid_tmul, TensorProduct.lid_tmul]
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simp only [Fin.isValue, one_smul]
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/-- Contraction on the right with `leftAltLeftUnit` does nothing. -/
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lemma contr_leftAltLeftUnit (x : altLeftHanded) :
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(λ_ altLeftHanded).hom.hom
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(((altLeftContraction) ▷ altLeftHanded).hom
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((α_ _ _ altLeftHanded).inv.hom
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(x ⊗ₜ[ℂ] leftAltLeftUnit.hom (1 : ℂ)))) = x := by
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obtain ⟨c, hc⟩ := (mem_span_range_iff_exists_fun ℂ).mp (Basis.mem_span altLeftBasis x)
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subst hc
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rw [leftAltLeftUnit_apply_one, leftAltLeftUnitVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.instMonoidalCategory_leftUnitor_hom_hom, Action.instMonoidalCategory_whiskerRight_hom,
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Action.instMonoidalCategory_associator_inv_hom, CategoryTheory.Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Fin.sum_univ_two, Fin.isValue, tmul_add, add_tmul, smul_tmul, tmul_smul, map_add,
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_root_.map_smul]
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have h1 (x y : altLeftHanded) (z : leftHanded) : (altLeftContraction.hom ▷ altLeftHanded.V)
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((α_ altLeftHanded.V leftHanded.V altLeftHanded.V).inv (x ⊗ₜ[ℂ] z ⊗ₜ[ℂ] y)) =
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(altLeftContraction.hom (x ⊗ₜ[ℂ] z)) ⊗ₜ[ℂ] y := rfl
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erw [h1, h1, h1, h1]
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repeat rw [altLeftContraction_basis]
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simp only [Fin.isValue, Action.instMonoidalCategory_tensorUnit_V, Fin.val_zero, ↓reduceIte,
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Fin.val_one, one_ne_zero, zero_tmul, map_zero, smul_zero, add_zero, zero_ne_one, zero_add]
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erw [TensorProduct.lid_tmul, TensorProduct.lid_tmul]
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simp only [Fin.isValue, one_smul]
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/-- Contraction on the right with `altRightRightUnit` does nothing. -/
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lemma contr_altRightRightUnit (x : rightHanded) :
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(λ_ rightHanded).hom.hom
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(((rightAltContraction) ▷ rightHanded).hom
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((α_ _ _ rightHanded).inv.hom
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(x ⊗ₜ[ℂ] altRightRightUnit.hom (1 : ℂ)))) = x := by
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obtain ⟨c, hc⟩ := (mem_span_range_iff_exists_fun ℂ).mp (Basis.mem_span rightBasis x)
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subst hc
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rw [altRightRightUnit_apply_one, altRightRightUnitVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.instMonoidalCategory_leftUnitor_hom_hom, Action.instMonoidalCategory_whiskerRight_hom,
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Action.instMonoidalCategory_associator_inv_hom, CategoryTheory.Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Fin.sum_univ_two, Fin.isValue, tmul_add, add_tmul, smul_tmul, tmul_smul, map_add,
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_root_.map_smul]
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have h1 (x y : rightHanded) (z : altRightHanded) : (rightAltContraction.hom ▷ rightHanded.V)
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((α_ rightHanded.V altRightHanded.V rightHanded.V).inv (x ⊗ₜ[ℂ] z ⊗ₜ[ℂ] y)) =
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(rightAltContraction.hom (x ⊗ₜ[ℂ] z)) ⊗ₜ[ℂ] y := rfl
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erw [h1, h1, h1, h1]
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repeat rw [rightAltContraction_basis]
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simp only [Fin.isValue, Action.instMonoidalCategory_tensorUnit_V, Fin.val_zero, ↓reduceIte,
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Fin.val_one, one_ne_zero, zero_tmul, map_zero, smul_zero, add_zero, zero_ne_one, zero_add]
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erw [TensorProduct.lid_tmul, TensorProduct.lid_tmul]
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simp only [Fin.isValue, one_smul]
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/-- Contraction on the right with `rightAltRightUnit` does nothing. -/
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lemma contr_rightAltRightUnit (x : altRightHanded) :
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(λ_ altRightHanded).hom.hom
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(((altRightContraction) ▷ altRightHanded).hom
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((α_ _ _ altRightHanded).inv.hom
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(x ⊗ₜ[ℂ] rightAltRightUnit.hom (1 : ℂ)))) = x := by
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obtain ⟨c, hc⟩ := (mem_span_range_iff_exists_fun ℂ).mp (Basis.mem_span altRightBasis x)
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subst hc
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rw [rightAltRightUnit_apply_one, rightAltRightUnitVal_expand_tmul]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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Action.instMonoidalCategory_leftUnitor_hom_hom, Action.instMonoidalCategory_whiskerRight_hom,
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Action.instMonoidalCategory_associator_inv_hom, CategoryTheory.Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Fin.sum_univ_two, Fin.isValue, tmul_add, add_tmul, smul_tmul, tmul_smul, map_add,
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_root_.map_smul]
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have h1 (x y : altRightHanded) (z : rightHanded) : (altRightContraction.hom ▷ altRightHanded.V)
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((α_ altRightHanded.V rightHanded.V altRightHanded.V).inv (x ⊗ₜ[ℂ] z ⊗ₜ[ℂ] y)) =
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(altRightContraction.hom (x ⊗ₜ[ℂ] z)) ⊗ₜ[ℂ] y := rfl
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erw [h1, h1, h1, h1]
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repeat rw [altRightContraction_basis]
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simp only [Fin.isValue, Action.instMonoidalCategory_tensorUnit_V, Fin.val_zero, ↓reduceIte,
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Fin.val_one, one_ne_zero, zero_tmul, map_zero, smul_zero, add_zero, zero_ne_one, zero_add]
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erw [TensorProduct.lid_tmul, TensorProduct.lid_tmul]
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simp only [Fin.isValue, one_smul]
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end
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end Fermion
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@ -167,6 +167,14 @@ def complexLorentzTensor : TensorSpecies where
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| Color.downR => Fermion.altRightContraction_tmul_symm
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| Color.up => Lorentz.contrCoContraction_tmul_symm
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| Color.down => Lorentz.coContrContraction_tmul_symm
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contr_unit := fun c =>
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match c with
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| Color.upL => Fermion.contr_altLeftLeftUnit
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| Color.downL => Fermion.contr_leftAltLeftUnit
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| Color.upR => Fermion.contr_altRightRightUnit
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| Color.downR => Fermion.contr_rightAltRightUnit
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| Color.up => Lorentz.contr_coContrUnit
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| Color.down => Lorentz.contr_contrCoUnit
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instance : DecidableEq complexLorentzTensor.C := Fermion.instDecidableEqColor
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