feat: Expansion lemmas for units

HepLean/SpaceTime/WeylFermion/Unit.lean

@@ -28,6 +28,14 @@ open CategoryTheory.MonoidalCategory
 def leftAltLeftUnitVal : (leftHanded ⊗ altLeftHanded).V :=
   leftAltLeftToMatrix.symm 1
 
+/-- Expansion of `leftAltLeftUnitVal` into the basis. -/
+lemma leftAltLeftUnitVal_expand_tmul : leftAltLeftUnitVal =
+    leftBasis 0 ⊗ₜ[ℂ] altLeftBasis 0 + leftBasis 1 ⊗ₜ[ℂ] altLeftBasis 1 := by
+  simp only [Action.instMonoidalCategory_tensorObj_V, leftAltLeftUnitVal, Fin.isValue]
+  erw [leftAltLeftToMatrix_symm_expand_tmul]
+  simp only [Fin.sum_univ_two, Fin.isValue, one_apply_eq, one_smul, ne_eq, zero_ne_one,
+    not_false_eq_true, one_apply_ne, zero_smul, add_zero, one_ne_zero, zero_add]
+
 /-- The left-alt-left unit `δᵃₐ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded `,
   manifesting the invariance under the `SL(2,ℂ)` action. -/
 def leftAltLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded where
@@ -55,10 +63,23 @@ def leftAltLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded
     apply congrArg
     simp
 
+lemma leftAltLeftUnit_apply_one : leftAltLeftUnit.hom (1 : ℂ) = leftAltLeftUnitVal := by
+  change leftAltLeftUnit.hom.toFun (1 : ℂ) = leftAltLeftUnitVal
+  simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
+    leftAltLeftUnit, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
+
 /-- The alt-left-left unit `δₐᵃ` as an element of `(altLeftHanded ⊗ leftHanded).V`. -/
 def altLeftLeftUnitVal : (altLeftHanded ⊗ leftHanded).V :=
   altLeftLeftToMatrix.symm 1
 
+/-- Expansion of `altLeftLeftUnitVal` into the basis. -/
+lemma altLeftLeftUnitVal_expand_tmul : altLeftLeftUnitVal =
+    altLeftBasis 0 ⊗ₜ[ℂ] leftBasis 0 + altLeftBasis 1 ⊗ₜ[ℂ] leftBasis 1 := by
+  simp only [Action.instMonoidalCategory_tensorObj_V, altLeftLeftUnitVal, Fin.isValue]
+  erw [altLeftLeftToMatrix_symm_expand_tmul]
+  simp only [Fin.sum_univ_two, Fin.isValue, one_apply_eq, one_smul, ne_eq, zero_ne_one,
+    not_false_eq_true, one_apply_ne, zero_smul, add_zero, one_ne_zero, zero_add]
+
 /-- The alt-left-left unit `δₐᵃ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded `,
   manifesting the invariance under the `SL(2,ℂ)` action. -/
 def altLeftLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded where
@@ -87,11 +108,24 @@ def altLeftLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded
     simp only [mul_one, ← transpose_mul, SpecialLinearGroup.det_coe, isUnit_iff_ne_zero, ne_eq,
       one_ne_zero, not_false_eq_true, mul_nonsing_inv, transpose_one]
 
+lemma altLeftLeftUnit_apply_one : altLeftLeftUnit.hom (1 : ℂ) = altLeftLeftUnitVal := by
+  change altLeftLeftUnit.hom.toFun (1 : ℂ) = altLeftLeftUnitVal
+  simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
+    altLeftLeftUnit, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
+
 /-- The right-alt-right unit `δ^{dot a}_{dot a}` as an element of
   `(rightHanded ⊗ altRightHanded).V`. -/
 def rightAltRightUnitVal : (rightHanded ⊗ altRightHanded).V :=
   rightAltRightToMatrix.symm 1
 
+/-- Expansion of `rightAltRightUnitVal` into the basis. -/
+lemma rightAltRightUnitVal_expand_tmul : rightAltRightUnitVal =
+    rightBasis 0 ⊗ₜ[ℂ] altRightBasis 0 + rightBasis 1 ⊗ₜ[ℂ] altRightBasis 1 := by
+  simp only [Action.instMonoidalCategory_tensorObj_V, rightAltRightUnitVal, Fin.isValue]
+  erw [rightAltRightToMatrix_symm_expand_tmul]
+  simp only [Fin.sum_univ_two, Fin.isValue, one_apply_eq, one_smul, ne_eq, zero_ne_one,
+    not_false_eq_true, one_apply_ne, zero_smul, add_zero, one_ne_zero, zero_add]
+
 /-- The right-alt-right unit `δ^{dot a}_{dot a}` as a morphism
   `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ altRightHanded`, manifesting
   the invariance under the `SL(2,ℂ)` action. -/
@@ -126,11 +160,24 @@ def rightAltRightUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ altRightHa
     rw [@conjTranspose_nonsing_inv]
     simp
 
+lemma rightAltRightUnit_apply_one : rightAltRightUnit.hom (1 : ℂ) = rightAltRightUnitVal := by
+  change rightAltRightUnit.hom.toFun (1 : ℂ) = rightAltRightUnitVal
+  simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
+    rightAltRightUnit, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
+
 /-- The alt-right-right unit `δ_{dot a}^{dot a}` as an element of
   `(rightHanded ⊗ altRightHanded).V`. -/
 def altRightRightUnitVal : (altRightHanded ⊗ rightHanded).V :=
   altRightRightToMatrix.symm 1
 
+/-- Expansion of `altRightRightUnitVal` into the basis. -/
+lemma altRightRightUnitVal_expand_tmul : altRightRightUnitVal =
+    altRightBasis 0 ⊗ₜ[ℂ] rightBasis 0 + altRightBasis 1 ⊗ₜ[ℂ] rightBasis 1 := by
+  simp only [Action.instMonoidalCategory_tensorObj_V, altRightRightUnitVal, Fin.isValue]
+  erw [altRightRightToMatrix_symm_expand_tmul]
+  simp only [Fin.sum_univ_two, Fin.isValue, one_apply_eq, one_smul, ne_eq, zero_ne_one,
+    not_false_eq_true, one_apply_ne, zero_smul, add_zero, one_ne_zero, zero_add]
+
 /-- The alt-right-right unit `δ_{dot a}^{dot a}` as a morphism
   `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ rightHanded`, manifesting
   the invariance under the `SL(2,ℂ)` action. -/
@@ -163,5 +210,22 @@ def altRightRightUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ rightHa
     rw [@conjTranspose_nonsing_inv]
     simp
 
+lemma altRightRightUnit_apply_one : altRightRightUnit.hom (1 : ℂ) = altRightRightUnitVal := by
+  change altRightRightUnit.hom.toFun (1 : ℂ) = altRightRightUnitVal
+  simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
+    altRightRightUnit, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
+
+/-!
+
+## Contraction of the units
+
+-/
+
+lemma contr_leftAltLeftUnitVal (x : leftHanded) :
+    (λ_ leftHanded).hom.hom
+    (((leftAltContraction) ▷ leftHanded).hom
+    ((α_ _ _ leftHanded).inv.hom
+    (x ⊗ₜ[ℂ] altLeftLeftUnit.hom (1 : ℂ)))) = x := by
+  sorry
 end
 end Fermion