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feat: Add proof related to symm and antisymm tensors

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jstoobysmith 2024-10-21 05:38:22 +00:00
parent a0d9d6766a
commit cb0197e127
3 changed files with 37 additions and 5 deletions
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29
HepLean/Tensors/ComplexLorentz/Lemmas.lean
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@ -7,6 +7,9 @@ import HepLean.Tensors.Tree.Elab
import HepLean.Tensors.ComplexLorentz.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basis
import HepLean.Tensors.Tree.NodeIdentities.Basic
import HepLean.Tensors.Tree.NodeIdentities.PermProd
import HepLean.Tensors.Tree.NodeIdentities.PermContr
import HepLean.Tensors.Tree.NodeIdentities.ContrContr
/-!
## Lemmas related to complex Lorentz tensors.
@ -52,6 +55,7 @@ lemma coMetric_expand : {Lorentz.coMetric | μ ν}ᵀ.tensor =
erw [pairIsoSep_tmul]
rfl
/-- The covariant Lorentz metric is symmetric. -/
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]
@ -67,8 +71,9 @@ lemma coMetric_symm : {Lorentz.coMetric | μ ν = Lorentz.coMetric | ν μ}ᵀ :
match i with
| (0 : Fin 2) => rfl
| (1 : Fin 2) => rfl
/-
lemma coMetric_prod_antiSymm (A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V)
set_option maxRecDepth 20000 in
lemma symm_contract_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
@ -77,14 +82,28 @@ lemma coMetric_prod_antiSymm (A : (Lorentz.complexContr ⊗ Lorentz.complexContr
(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))]
nth_rewrite 1 [(contr_tensor_eq (contr_tensor_eq (prod_tensor_eq_snd hs)))]
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
-/
rw [contr_tensor_eq (contr_tensor_eq (prod_perm_left _ _ _ _))]
rw [contr_tensor_eq (perm_contr _ _)]
rw [perm_contr]
rw [perm_tensor_eq (contr_tensor_eq (contr_tensor_eq (prod_perm_right _ _ _ _)))]
rw [perm_tensor_eq (contr_tensor_eq (perm_contr _ _))]
rw [perm_tensor_eq (perm_contr _ _)]
rw [perm_perm]
nth_rewrite 1 [perm_tensor_eq (contr_contr _ _ _)]
rw [perm_perm]
rw [perm_eq_id]
· rfl
· apply OverColor.Hom.ext
rfl
end Fermion
end