Refactor: Metrics as complex lorentz tensors

HepLean/Tensors/ComplexLorentz/Metrics/Lemmas.lean

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+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.Tensors.ComplexLorentz.Metrics.Basis
+import HepLean.Tensors.ComplexLorentz.Basis
+/-!
+
+## Metrics and basis expansions
+
+-/
+open IndexNotation
+open CategoryTheory
+open MonoidalCategory
+open Matrix
+open MatrixGroups
+open Complex
+open TensorProduct
+open IndexNotation
+open CategoryTheory
+open TensorTree
+open OverColor.Discrete
+noncomputable section
+
+namespace complexLorentzTensor
+
+/-- The map to color one gets when multiplying left and right metrics. -/
+def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL] ![Color.upR, Color.upR]) ∘
+  finSumFinEquiv.symm
+
+lemma leftMetric_mul_rightMetric : {εL | α α' ⊗ εR | β β'}ᵀ.tensor
+    = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    - basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
+    - basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
+  rw [prod_tensor_eq_fst (leftMetric_expand_tree)]
+  rw [prod_tensor_eq_snd (rightMetric_expand_tree)]
+  rw [prod_add_both]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_prod _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_tensor_eq <| prod_smul _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_smul _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_eq_one _ _ (by simp)]
+  rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_prod _ _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_basisVector_tree _ _]
+  rw [← add_assoc]
+  simp only [add_tensor, smul_tensor, tensorNode_tensor]
+  change _ = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    +- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
+    +- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)
+  congr 1
+  congr 1
+  congr 1
+  all_goals
+    congr
+    funext x
+    fin_cases x <;> rfl
+
+lemma leftMetric_mul_rightMetric_tree :
+    {εL | α α' ⊗ εR | β β'}ᵀ.tensor
+    = (TensorTree.add (tensorNode
+        (basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1))) <|
+      TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
+        (basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)))) <|
+      TensorTree.add (TensorTree.smul (-1 : ℂ) (tensorNode
+        (basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)))) <|
+      (tensorNode
+        (basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)))).tensor := by
+  rw [leftMetric_mul_rightMetric]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor,
+    smul_tensor, neg_smul, one_smul]
+  rfl
+
+end complexLorentzTensor