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refactor: Delete examples file

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jstoobysmith 2024-10-22 10:41:14 +00:00
parent 18709d4e32
commit 91a171b3ff
4 changed files with 11 additions and 83 deletions
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25
HepLean/Tensors/ComplexLorentz/Lemmas.lean
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@ -12,7 +12,6 @@ import HepLean.Tensors.Tree.NodeIdentities.PermContr
import HepLean.Tensors.Tree.NodeIdentities.ProdComm
import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
import HepLean.Tensors.Tree.NodeIdentities.ContrContr
import LLMLean
/-!
## Lemmas related to complex Lorentz tensors.
@ -33,15 +32,14 @@ noncomputable section
namespace Fermion
example : 0 < complexLorentzTensor.repDim (![Color.down] 0):= by decide
/-- The vectors forming a basis of
`complexLorentzTensor.F.obj (OverColor.mk ![Color.down, Color.down])`.
Not proved it is a basis yet. -/
def coCoBasis (b : Fin 4 × Fin 4) :
complexLorentzTensor.F.obj (OverColor.mk ![Color.down, Color.down]) :=
PiTensorProduct.tprod ℂ (fun i => Fin.cases (Lorentz.complexCoBasisFin4 b.1)
(fun i => Fin.cases (Lorentz.complexCoBasisFin4 b.2) (fun i => i.elim0) i) i)
lemma coCoBasis_eval (e1 e2 : Fin (complexLorentzTensor.repDim Color.down)) (i : Fin 4 × Fin 4) :
complexLorentzTensor.castFin0ToField
((complexLorentzTensor.evalMap 0 e2) ((complexLorentzTensor.evalMap 0 e1) (coCoBasis i))) =
@ -49,7 +47,7 @@ lemma coCoBasis_eval (e1 e2 : Fin (complexLorentzTensor.repDim Color.down)) (i :
simp only [coCoBasis]
have h1 := @TensorSpecies.evalMap_tprod complexLorentzTensor _ (![Color.down, Color.down]) 0 e1
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, Functor.id_obj,
OverColor.mk_hom, Function.comp_apply, cons_val_zero, Fin.cases_zero, Fin.cases_succ] at h1
OverColor.mk_hom, Function.comp_apply, cons_val_zero, Fin.cases_zero, Fin.cases_succ] at h1
erw [h1]
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, Functor.id_obj, OverColor.mk_hom,
Fin.cases_zero, Fin.cases_succ, _root_.map_smul, smul_eq_mul]
@ -60,21 +58,18 @@ lemma coCoBasis_eval (e1 e2 : Fin (complexLorentzTensor.repDim Color.down)) (i :
erw [complexLorentzTensor.castFin0ToField_tprod]
simp only [Fin.isValue, mul_one]
change (Lorentz.complexCoBasisFin4.repr (Lorentz.complexCoBasisFin4 i.1)) e1 *
(Lorentz.complexCoBasisFin4.repr (Lorentz.complexCoBasisFin4 i.2)) e2 = _
(Lorentz.complexCoBasisFin4.repr (Lorentz.complexCoBasisFin4 i.2)) e2 = _
simp only [Basis.repr_self]
rw [Finsupp.single_apply, Finsupp.single_apply]
rw [@ite_zero_mul_ite_zero]
simp
simp only [mul_one]
congr
simp_all only [Fin.isValue, Fin.succAbove_zero, Fin.zero_succAbove, eq_iff_iff]
obtain ⟨fst, snd⟩ := i
simp_all only [Fin.isValue, Prod.mk.injEq]
lemma coMetric_expand : {Lorentz.coMetric | μ ν}ᵀ.tensor =
coCoBasis (0, 0)
- coCoBasis (1, 1)
- coCoBasis (2, 2)
- coCoBasis (3, 3):= by
coCoBasis (0, 0) - coCoBasis (1, 1) - coCoBasis (2, 2) - coCoBasis (3, 3) := 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]
@ -108,7 +103,7 @@ lemma coMetric_symm : {Lorentz.coMetric | μ ν = Lorentz.coMetric | ν μ}ᵀ :
| (0 : Fin 2) => rfl
| (1 : Fin 2) => rfl
lemma coMetric_0_0_field : {Lorentz.coMetric | 0 0}ᵀ.field = 1 := by
lemma coMetric_0_0_field : {Lorentz.coMetric | 0 0}ᵀ.field = 1 := by
rw [field, eval_tensor, eval_tensor, coMetric_expand]
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue,
Function.comp_apply, Fin.succ_zero_eq_one, cons_val_one, head_cons, Fin.ofNat'_zero,
@ -184,9 +179,7 @@ lemma symm_contr_antiSymm {S : (Lorentz.complexCo ⊗ Lorentz.complexCo).V}
{A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
(hA : {A | μ ν = - (A | ν μ)}ᵀ) (hs : {S | μ ν = S | ν μ}ᵀ) :
{S | μ ν ⊗ A | μ ν}ᵀ.tensor = 0 := by
rw [contr_rank_2_symm']
rw [perm_tensor]
rw [antiSymm_contr_symm hA hs]
rw [contr_rank_2_symm', perm_tensor, antiSymm_contr_symm hA hs]
rfl
end Fermion