refactor: Move dual to minkowskiMatrix

HepLean/SpaceTime/MinkowskiMetric.lean

@@ -89,6 +89,58 @@ lemma inr_i_inr_i (i : Fin d) : @minkowskiMatrix d (Sum.inr i) (Sum.inr i) = -1
   simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
   simp_all only [diagonal_apply_eq, Sum.elim_inr]
 
+
+variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
+
+/-- The dual of a matrix with respect to the Minkowski metric. -/
+def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
+
+@[simp]
+lemma dual_id : @dual d 1 = 1 := by
+  simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
+
+@[simp]
+lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
+  simp only [dual, transpose_mul]
+  trans η * Λ'ᵀ * (η * η) * Λᵀ * η
+  · noncomm_ring [minkowskiMatrix.sq]
+  · noncomm_ring
+
+@[simp]
+lemma dual_dual : dual (dual Λ) = Λ := by
+  simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
+  trans (η * η) * Λ * (η * η)
+  · noncomm_ring
+  · noncomm_ring [minkowskiMatrix.sq]
+
+@[simp]
+lemma dual_eta : @dual d η = η := by
+  simp only [dual, eq_transpose]
+  noncomm_ring [minkowskiMatrix.sq]
+
+@[simp]
+lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
+  simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
+  noncomm_ring
+
+@[simp]
+lemma det_dual : (dual Λ).det = Λ.det := by
+  simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]
+  group
+  norm_cast
+  simp
+
+lemma dual_apply (μ ν : Fin 1 ⊕ Fin d) :
+    dual Λ μ ν = η μ μ * Λ ν μ * η ν ν := by
+  simp only [dual, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
+    diagonal_mul, transpose_apply, diagonal_apply_eq]
+
+lemma dual_apply_minkowskiMatrix (μ ν : Fin 1 ⊕ Fin d) :
+    dual Λ μ ν * η ν ν = η μ μ * Λ ν μ := by
+  rw [dual_apply, mul_assoc]
+  simp
+
+
 end minkowskiMatrix
 
 /-!
@@ -238,54 +290,6 @@ section matrices
 
 variable (Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
 
-/-- The dual of a matrix with respect to the Minkowski metric. -/
-def dual : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ := η * Λᵀ * η
-
-@[simp]
-lemma dual_id : @dual d 1 = 1 := by
-  simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq
-
-@[simp]
-lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
-  simp only [dual, transpose_mul]
-  trans η * Λ'ᵀ * (η * η) * Λᵀ * η
-  · noncomm_ring [minkowskiMatrix.sq]
-  · noncomm_ring
-
-@[simp]
-lemma dual_dual : dual (dual Λ) = Λ := by
-  simp only [dual, transpose_mul, transpose_transpose, eq_transpose]
-  trans (η * η) * Λ * (η * η)
-  · noncomm_ring
-  · noncomm_ring [minkowskiMatrix.sq]
-
-@[simp]
-lemma dual_eta : @dual d η = η := by
-  simp only [dual, eq_transpose]
-  noncomm_ring [minkowskiMatrix.sq]
-
-@[simp]
-lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
-  simp only [dual, transpose_transpose, transpose_mul, eq_transpose]
-  noncomm_ring
-
-@[simp]
-lemma det_dual : (dual Λ).det = Λ.det := by
-  simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]
-  group
-  norm_cast
-  simp
-
-lemma dual_apply (μ ν : Fin 1 ⊕ Fin d) :
-    dual Λ μ ν = η μ μ * Λ ν μ * η ν ν := by
-  simp only [dual, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
-    diagonal_mul, transpose_apply, diagonal_apply_eq]
-
-lemma dual_apply_minkowskiMatrix (μ ν : Fin 1 ⊕ Fin d) :
-    dual Λ μ ν * η ν ν = η μ μ * Λ ν μ := by
-  rw [dual_apply, mul_assoc]
-  simp
-
 @[simp]
 lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ := by
   simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, dual, minkowskiLinearForm_apply,