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refactor: Lint

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jstoobysmith 2024-11-15 10:44:42 +00:00
parent 9763e1240b
commit 05b4d134ec
9 changed files with 51 additions and 40 deletions
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17
HepLean/Tensors/TensorSpecies/MetricTensor.lean
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@ -33,22 +33,22 @@ def metricTensor (S : TensorSpecies) (c : S.C) : S.F.obj (OverColor.mk ![c, c])
variable {S : TensorSpecies}
lemma metricTensor_congr {c c' : S.C} (h : c = c') : {S.metricTensor c | μ ν}ᵀ.tensor =
(perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl ))
(perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl))
{S.metricTensor c' | μ ν}ᵀ).tensor := by
subst h
change _ = (S.F.map (𝟙 _)).hom (S.metricTensor c)
simp
lemma pairIsoSep_inv_metricTensor (c : S.C) :
(Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor c) =
(S.metric.app (Discrete.mk c)).hom (1 : S.k) := by
simp [metricTensor]
simp only [Action.instMonoidalCategory_tensorObj_V, Nat.succ_eq_add_one, Nat.reduceAdd,
metricTensor, Monoidal.tensorUnit_obj, Action.instMonoidalCategory_tensorUnit_V]
erw [Discrete.rep_iso_inv_hom_apply]
/-- Contraction of a metric tensor with a metric tensor gives the unit.
Like `S.contr_metric` but with the braiding appearing on the side of the unit. -/
lemma contr_metric_braid_unit (c : S.C) : (((S.FD.obj (Discrete.mk c)) ◁
lemma contr_metric_braid_unit (c : S.C) : (((S.FD.obj (Discrete.mk c)) ◁
(λ_ (S.FD.obj (Discrete.mk (S.τ c)))).hom).hom
(((S.FD.obj (Discrete.mk c)) ◁ ((S.contr.app (Discrete.mk c)) ▷
(S.FD.obj (Discrete.mk (S.τ c))))).hom
@ -63,12 +63,13 @@ lemma contr_metric_braid_unit (c : S.C) : (((S.FD.obj (Discrete.mk c)) ◁
apply (β_ _ _).toLinearEquiv.toEquiv.injective
rw [pairIsoSep_inv_metricTensor, pairIsoSep_inv_metricTensor]
erw [S.contr_metric c]
change _ = (β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).inv.hom
change _ = (β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).inv.hom
((β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).hom.hom _)
rw [Discrete.rep_iso_inv_hom_apply]
lemma metricTensor_contr_dual_metricTensor_perm_cond (c : S.C) : ∀ (x : Fin (Nat.succ 0).succ),
((Sum.elim ![c, c] ![S.τ c, S.τ c] ∘ ⇑finSumFinEquiv.symm) ∘ Fin.succAbove 1 ∘ Fin.succAbove 1) x =
((Sum.elim ![c, c] ![S.τ c, S.τ c] ∘ ⇑finSumFinEquiv.symm) ∘
Fin.succAbove 1 ∘ Fin.succAbove 1) x =
(![S.τ c, c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := by
intro x
fin_cases x
@ -81,12 +82,12 @@ lemma metricTensor_contr_dual_metricTensor_eq_unit (c : S.C) :
perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
(metricTensor_contr_dual_metricTensor_perm_cond c))).tensor := by
rw [contr_two_two_inner, contr_metric_braid_unit, Discrete.pairIsoSep_β]
change (S.F.map _ ≫ S.F.map _ ).hom _ = _
change (S.F.map _ ≫ S.F.map _).hom _ = _
rw [← S.F.map_comp]
rfl
/-- The contraction of a metric tensor with its dual via the outer indices gives the unit. -/
lemma metricTensor_contr_dual_metricTensor_outer_eq_unit (c : S.C) :
lemma metricTensor_contr_dual_metricTensor_outer_eq_unit (c : S.C) :
{S.metricTensor c | ν μ ⊗ S.metricTensor (S.τ c) | ρ ν}ᵀ.tensor = ({S.unitTensor c | μ ρ}ᵀ |>
perm (OverColor.equivToHomEq
(finMapToEquiv ![1, 0] ![1, 0]) (fun x => by fin_cases x <;> rfl))).tensor := by