PhysLean/HepLean/Lorentz/ComplexTensor/Lemmas.lean

/-
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.Lorentz.ComplexTensor.Basis
import HepLean.Tensors.Tree.NodeIdentities.PermProd
/-!

## Lemmas related to complex Lorentz tensors.

-/
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

set_option maxRecDepth 5000 in
lemma antiSymm_contr_symm {A : complexLorentzTensor.F.obj (OverColor.mk ![Color.up, Color.up])}
    {S : complexLorentzTensor.F.obj (OverColor.mk ![Color.down, Color.down])}
    (hA : {A | μ ν = - (A | ν μ)}ᵀ) (hs : {S | μ ν = S | ν μ}ᵀ) :
    {A | μ ν ⊗ S | μ ν = - A | μ ν ⊗ S | μ ν}ᵀ := by
  conv =>
    lhs
    rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_fst <| hA]
    rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| hs]
    rw [contr_tensor_eq <| contr_tensor_eq <| prod_perm_left _ _ _ _]
    rw [contr_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| prod_perm_right _ _ _ _]
    rw [contr_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
    rw [contr_tensor_eq <| perm_contr_congr 1 2]
    rw [perm_contr_congr 0 0]
    rw [perm_tensor_eq <| contr_contr _ _ _]
    rw [perm_perm]
    rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| neg_fst_prod _ _]
    rw [perm_tensor_eq <| contr_tensor_eq <| neg_contr _]
    rw [perm_tensor_eq <| neg_contr _]
  apply perm_congr _ rfl
  decide

end complexLorentzTensor

end