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lemma: Contract with unitTensor

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jstoobysmith 2024-11-15 15:46:08 +00:00
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29
HepLean/Tensors/TensorSpecies/UnitTensor.lean
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@ -6,6 +6,7 @@ Authors: Joseph Tooby-Smith
import HepLean.Tensors.Tree.Elab
import HepLean.Tensors.Tree.NodeIdentities.Basic
import HepLean.Tensors.Tree.NodeIdentities.Congr
import HepLean.Tensors.TensorSpecies.ContractLemmas
/-!
## Units as tensors
@ -37,6 +38,13 @@ lemma unitTensor_congr {c c' : S.C} (h : c = c') : {S.unitTensor c | μ ν}ᵀ.t
change _ = (S.F.map (𝟙 _)).hom (S.unitTensor c)
simp
lemma pairIsoSep_inv_unitTensor (c : S.C) :
(Discrete.pairIsoSep S.FD).inv.hom (S.unitTensor c) =
(S.unit.app (Discrete.mk c)).hom (1 : S.k) := by
simp only [Action.instMonoidalCategory_tensorObj_V, Nat.succ_eq_add_one, Nat.reduceAdd,
unitTensor, Monoidal.tensorUnit_obj, Action.instMonoidalCategory_tensorUnit_V]
erw [Discrete.rep_iso_inv_hom_apply]
lemma unitTensor_eq_dual_perm_eq (c : S.C) : ∀ (x : Fin (Nat.succ 0).succ),
![S.τ c, c] x = (![S.τ (S.τ c), S.τ c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := fun x => by
fin_cases x
@ -105,6 +113,27 @@ lemma dual_unitTensor_eq_perm (c : S.C) : {S.unitTensor (S.τ c) | ν μ}ᵀ.ten
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue]
rfl
lemma contrOneTwoLeft_unitTensor {c1 : S.C}
(x : S.F.obj (OverColor.mk ![c1])) :
contrOneTwoLeft x (S.unitTensor c1) = x := by
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, contrOneTwoLeft,
Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.instMonoidalCategory_leftUnitor_hom_hom, Monoidal.tensorUnit_obj,
Action.instMonoidalCategory_whiskerRight_hom, Functor.comp_obj, Discrete.functor_obj_eq_as,
Function.comp_apply, Action.instMonoidalCategory_associator_inv_hom, Equivalence.symm_inverse,
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj]
erw [pairIsoSep_inv_unitTensor (S := S) (c := c1)]
change (S.F.mapIso (mkIso _)).hom.hom _ = _
rw [Discrete.rep_iso_apply_iff, Discrete.rep_iso_inv_apply_iff]
simpa using S.contr_unit c1 ((OverColor.forgetLiftApp S.FD c1).hom.hom ((S.F.map (OverColor.mkIso (by
funext x; fin_cases x; rfl)).hom).hom x))
lemma one_contr_unitTensor_eq_self {c1 : S.C} (x : S.F.obj (OverColor.mk ![c1])) :
{x | μ ⊗ S.unitTensor c1 | μ ν}ᵀ.tensor = ({x | ν}ᵀ |>
perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by fin_cases x; rfl ))).tensor := by
rw [contr_one_two_left_eq_contrOneTwoLeft, contrOneTwoLeft_unitTensor]
rfl
end TensorSpecies
end