feat: Lemmas regarding bispinors

HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean

@@ -4,6 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
 import HepLean.Tensors.ComplexLorentz.PauliLower
+import HepLean.Tensors.Tree.NodeIdentities.ProdContr
+import HepLean.Tensors.Tree.NodeIdentities.Congr
+import HepLean.Tensors.Tree.NodeIdentities.ProdAssoc
 /-!
 
 ## Bispinors
@@ -33,6 +36,54 @@ lemma tensorNode_contrBispinorUp (p : complexContr) :
     (tensorNode (contrBispinorUp p)).tensor = {p | μ ⊗ pauliCo | μ α β}ᵀ.tensor := by
   rw [contrBispinorUp, tensorNode_tensor]
 
+/-- An up-bispinor is equal to `pauliCo | μ α β ⊗ p | μ`-/
+lemma contrBispinorUp_eq_pauliCo_self (p : complexContr) :
+    {contrBispinorUp p | α β = pauliCo | μ α β ⊗ p | μ}ᵀ := by
+  rw [tensorNode_contrBispinorUp]
+  conv_lhs =>
+    rw [contr_tensor_eq <| prod_comm _ _ _ _]
+    rw [perm_contr]
+    rw [perm_tensor_eq <| contr_swap _ _]
+    rw [perm_perm]
+  apply perm_congr
+  · apply OverColor.Hom.ext
+    ext x
+    match x with
+    | (0 : Fin 2) => rfl
+    | (1 : Fin 2)  => rfl
+  · rfl
+
+set_option maxRecDepth 2000 in
+lemma altRightMetric_contr_contrBispinorUp_assoc (p : complexContr) :
+    {Fermion.altRightMetric | β β' ⊗ contrBispinorUp p | α β =
+    Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β ⊗ p | μ}ᵀ := by
+  conv_lhs =>
+    rw [contr_tensor_eq <| prod_tensor_eq_snd <| contrBispinorUp_eq_pauliCo_self _]
+    rw [contr_tensor_eq <| prod_perm_right  _ _ _ _]
+    rw [perm_contr]
+    rw [perm_tensor_eq <| contr_tensor_eq <| prod_contr _ _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_assoc _ _ _ _ _ _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+  conv_rhs =>
+    rw [perm_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    erw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_contr _ _ _]
+    rw [perm_perm]
+  apply perm_congr (_) rfl
+  · apply OverColor.Hom.fin_ext
+    intro i
+    fin_cases i
+    exact rfl
+    exact rfl
+
 /-- A bispinor `pₐₐ` created from a lorentz vector `p^μ`. -/
 def contrBispinorDown (p : complexContr) :=
   {Fermion.altLeftMetric | α α' ⊗ Fermion.altRightMetric | β β' ⊗
@@ -45,9 +96,70 @@ lemma tensorNode_contrBispinorDown (p : complexContr) :
     Fermion.altRightMetric | β β' ⊗ (contrBispinorUp p) | α β}ᵀ.tensor := by
   rw [contrBispinorDown, tensorNode_tensor]
 
+set_option maxRecDepth 10000 in
+lemma contrBispinorDown_eq_metric_contr_contrBispinorUp  (p : complexContr) :
+    {contrBispinorDown p | α' β' = Fermion.altLeftMetric | α α' ⊗
+    (Fermion.altRightMetric | β β' ⊗ contrBispinorUp p | α β)}ᵀ := by
+  rw [tensorNode_contrBispinorDown]
+  conv_lhs =>
+    rw [contr_tensor_eq <| contr_tensor_eq <| prod_assoc' _ _ _ _ _ _]
+    rw [contr_tensor_eq <| perm_contr _ _]
+    rw [perm_contr]
+    rw [perm_tensor_eq <| contr_contr _ _ _]
+    rw [perm_perm]
+  conv_rhs =>
+    rw [perm_tensor_eq <| contr_tensor_eq <| prod_contr _ _ _ ]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+  apply perm_congr
+  · apply OverColor.Hom.ext
+    ext x
+    match x with
+    | (0 : Fin 2) => rfl
+    | (1 : Fin 2) => rfl
+  · rfl
+
+set_option maxHeartbeats 400000 in
+set_option maxRecDepth 2000 in
+lemma contrBispinorDown_eq_contr_with_self (p : complexContr) :
+    {contrBispinorDown p | α' β' = (Fermion.altLeftMetric | α α' ⊗
+    (Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β)) ⊗ p | μ}ᵀ := by
+  rw [contrBispinorDown_eq_metric_contr_contrBispinorUp]
+  conv_lhs =>
+    rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| altRightMetric_contr_contrBispinorUp_assoc _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_right _ _ _ _]
+    rw [perm_tensor_eq <| perm_contr _ _ ]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_tensor_eq <| prod_contr _ _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <|
+        prod_assoc _ _ _ _ _ _]
+    rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+  conv =>
+    rhs
+    rw [perm_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    erw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
+    rw [perm_tensor_eq <| perm_contr _ _]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_contr _ _ _]
+    rw [perm_perm]
+  apply congrArg
+  apply congrFun
+  apply congrArg
+  apply OverColor.Hom.fin_ext
+  intro i
+  fin_cases i
+  exact rfl
+  exact rfl
+
 /-- Expansion of a `contrBispinorDown` into the original contravariant tensor nested
   between pauli matrices and metrics. -/
-lemma contrBispinorDown_full_nested (p : complexContr) :
+lemma contrBispinorDown_eq_metric_mul_self_mul_pauli (p : complexContr) :
     {contrBispinorDown p | α β}ᵀ.tensor = {Fermion.altLeftMetric | α α' ⊗
     Fermion.altRightMetric | β β' ⊗ (p | μ ⊗ pauliCo | μ α β)}ᵀ.tensor := by
   conv =>