feat: Important fix to ContrContr

HepLean/Tensors/ComplexLorentz/Basic.lean

@@ -209,5 +209,17 @@ lemma basis_contr (c : complexLorentzTensor.C) (i : Fin (complexLorentzTensor.re
   | Color.up => Lorentz.contrCoContraction_basis _ _
   | Color.down => Lorentz.coContrContraction_basis _ _
 
+instance  {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
+    DecidableEq (OverColor.mk c).left := instDecidableEqFin n
+
+instance  {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
+    Fintype (OverColor.mk c).left := Fin.fintype n
+
+instance {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
+    {c1 : Fin m → complexLorentzTensor.C} (σ σ' : OverColor.mk c ⟶ OverColor.mk c1) :
+    Decidable (σ = σ') :=
+  decidable_of_iff _ (OverColor.Hom.ext_iff σ σ')
+
+
 end complexLorentzTensor
 end

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

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