feat: Important fix to ContrContr

HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean

@@ -87,5 +87,57 @@ lemma tensorNode_coBispinorDown (p : complexCo) :
     ⊗ coBispinorUp p | α β}ᵀ.tensor := by
   rw [coBispinorDown, tensorNode_tensor]
 
+/-!
+
+## Basic equalities.
+
+-/
+
+lemma contrBispinorDown_expand (p : complexContr) :
+    {contrBispinorDown p | α β}ᵀ.tensor = {Fermion.altLeftMetric | α α' ⊗ Fermion.altRightMetric | β β' ⊗
+    (pauliCo | μ α β ⊗ p | μ)}ᵀ.tensor := by
+  rw [tensorNode_contrBispinorDown p]
+  rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_contrBispinorUp p]
+
+set_option maxRecDepth 5000 in
+lemma contrBispinorDown_eq_pauliCoDown_contr (p : complexContr) :
+    {contrBispinorDown p | α β = pauliCoDown | μ α β ⊗ p | μ}ᵀ  := by
+  conv =>
+    rhs
+    rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_fst <| pauliCoDown_eq_metric_mul_pauliCo]
+    rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_left _ _ _ _]
+    rw [perm_tensor_eq <| perm_contr_congr 2 2]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
+    rw [perm_tensor_eq <| perm_contr_congr 2 2]
+    rw [perm_perm]
+    apply (perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _).trans
+    rw [perm_tensor_eq <| contr_tensor_eq <|  contr_tensor_eq <| contr_prod _ _ _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
+    rw [perm_tensor_eq <| perm_contr_congr 2 2]
+    rw [perm_perm]
+    erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_assoc' _ _ _ _ _ _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_contr_congr 0 4]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
+    rw [perm_tensor_eq <| perm_contr_congr 2 2]
+    rw [perm_perm]
+  conv =>
+    lhs
+    rw [contrBispinorDown_expand p]
+    rw [contr_tensor_eq <| contr_tensor_eq <| prod_contr _ _ _]
+    rw [contr_tensor_eq <| perm_contr_congr 0 3]
+    rw [perm_contr_congr 1 2]
+    apply (perm_tensor_eq <| contr_tensor_eq <| contr_contr _ _ _).trans
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_contr _ _ _]
+    rw [perm_perm]
+  apply perm_congr _ rfl
+  decide
+
+
+
+
 end complexLorentzTensor
 end