feat: lemmas relating to index notation

HepLean/Tensors/ComplexLorentz/Lemmas.lean

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+/-
+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.Tensors.Tree.Elab
+import HepLean.Tensors.ComplexLorentz.Basic
+import Mathlib.LinearAlgebra.TensorProduct.Basis
+/-!
+
+## 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 Fermion
+
+lemma coMetric_expand : {Lorentz.coMetric | μ ν}ᵀ.tensor =
+    (PiTensorProduct.tprod ℂ (fun i => Fin.cases (Lorentz.complexCoBasis (Sum.inl 0))
+        (fun i => Fin.cases (Lorentz.complexCoBasis (Sum.inl 0)) (fun i => i.elim0) i) i) :
+        complexLorentzTensor.F.obj (OverColor.mk ![Color.down, Color.down]))
+    - (PiTensorProduct.tprod ℂ (fun i => Fin.cases (Lorentz.complexCoBasis (Sum.inr 0))
+        (fun i => Fin.cases (Lorentz.complexCoBasis (Sum.inr 0)) (fun i => i.elim0) i) i) :
+        complexLorentzTensor.F.obj (OverColor.mk ![Color.down, Color.down]))
+    - (PiTensorProduct.tprod ℂ (fun i => Fin.cases (Lorentz.complexCoBasis (Sum.inr 1))
+        (fun i => Fin.cases (Lorentz.complexCoBasis (Sum.inr 1)) (fun i => i.elim0) i) i) :
+        complexLorentzTensor.F.obj (OverColor.mk ![Color.down, Color.down]))
+    - (PiTensorProduct.tprod ℂ (fun i => Fin.cases (Lorentz.complexCoBasis (Sum.inr 2))
+        (fun i => Fin.cases (Lorentz.complexCoBasis (Sum.inr 2)) (fun i => i.elim0) i) i) :
+        complexLorentzTensor.F.obj (OverColor.mk ![Color.down, Color.down])) := by
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
+    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
+    Functor.id_obj, Fin.isValue]
+  erw [Lorentz.coMetric_apply_one, Lorentz.coMetricVal_expand_tmul]
+  simp only [Fin.isValue, map_sub]
+  congr 1
+  congr 1
+  congr 1
+  all_goals
+    erw [pairIsoSep_tmul]
+    rfl
+
+lemma coMetric_symm : {Lorentz.coMetric | μ ν = Lorentz.coMetric | ν μ}ᵀ := by
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, perm_tensor]
+  rw [coMetric_expand]
+  simp only [TensorStruct.F, Nat.succ_eq_add_one, Nat.reduceAdd, Functor.id_obj, Fin.isValue,
+    map_sub]
+  congr 1
+  congr 1
+  congr 1
+  all_goals
+    erw [OverColor.lift.map_tprod]
+    apply congrArg
+    funext i
+    match i with
+    | (0 : Fin 2) => rfl
+    | (1 : Fin 2) => rfl
+
+open TensorTree in
+
+lemma coMetric_prod_antiSymm (A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V)
+    (S : (Lorentz.complexCo ⊗ Lorentz.complexCo).V)
+    (hA : (twoNodeE complexLorentzTensor Color.up Color.up A).tensor =
+      (TensorTree.neg (perm
+      (OverColor.equivToHomEq (Equality.finMapToEquiv ![1, 0] ![1, 0]) (by decide))
+      (twoNodeE complexLorentzTensor  Color.up  Color.up A))).tensor)
+    (hs : {S | μ ν = S | ν μ}ᵀ) : {A | μ ν ⊗ S | μ ν}ᵀ.tensor = 0 := by
+  have h1 : {A | μ ν ⊗ S | μ ν}ᵀ.tensor = - {A | μ ν ⊗ S | μ ν}ᵀ.tensor := by
+    nth_rewrite 1 [contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_fst hA))]
+    rw [contr_tensor_eq (contr_tensor_eq (neg_fst_prod _ _))]
+    rw [contr_tensor_eq (neg_contr _)]
+    rw [neg_contr]
+    rw [neg_tensor]
+    apply congrArg
+    sorry
+    sorry
+
+end Fermion
+
+end