feat: Important fix to ContrContr

HepLean/Tensors/ComplexLorentz/PauliMatrices/Basic.lean

@@ -4,7 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
 import HepLean.Tensors.Tree.NodeIdentities.ProdAssoc
+import HepLean.Tensors.Tree.NodeIdentities.ProdComm
 import HepLean.Tensors.Tree.NodeIdentities.ProdContr
+import HepLean.Tensors.Tree.NodeIdentities.ContrContr
+import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
+import HepLean.Tensors.Tree.NodeIdentities.PermContr
 import HepLean.Tensors.Tree.NodeIdentities.Congr
 /-!
 
@@ -75,4 +79,61 @@ lemma tensorNode_pauliContrDown : {pauliContrDown | μ α β}ᵀ.tensor =
     Fermion.altRightMetric | β β'}ᵀ.tensor := by
   rfl
 
+/-!
+
+## Basic equalities
+
+-/
+
+set_option maxRecDepth 5000 in
+/-- A rearanging of `pauliCoDown` to place the pauli matrices on the right. -/
+lemma pauliCoDown_eq_metric_mul_pauliCo :
+    {pauliCoDown | μ α' β' = Fermion.altLeftMetric | α α' ⊗
+    Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β}ᵀ := by
+  conv =>
+    lhs
+    rw [tensorNode_pauliCoDown]
+    rw [contr_tensor_eq <| contr_prod _ _ _]
+    rw [perm_contr]
+    erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _]
+    rw [perm_tensor_eq <| contr_congr 1 2]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_tensor_eq <| contr_congr 1 2]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_congr 1 2]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_assoc' _ _ _ _ _ _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 1 2]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_congr 1 2]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_comm  _ _ _ _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_congr 4 1]
+    rw [perm_perm]
+  conv =>
+    rhs
+    rw [perm_tensor_eq <| contr_swap _ _]
+    rw [perm_perm]
+    erw [perm_tensor_eq <| contr_congr 4 1]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_tensor_eq <| contr_swap _ _]
+    erw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_congr 4 1]
+    rw [perm_perm]
+  apply perm_congr _ rfl
+  decide
+
+
 end complexLorentzTensor