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feat: Writing toDualRep in terms of contrOneTwoLeft

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This commit is contained in:
jstoobysmith 2024-11-18 06:07:03 +00:00
parent 5d21f74062
commit 2c26584cfc
5 changed files with 49 additions and 13 deletions
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21
HepLean/Tensors/TensorSpecies/ContractLemmas.lean
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@ -29,13 +29,11 @@ variable {S : TensorSpecies}
def contrOneTwoLeft {c1 c2 : S.C}
(x : S.F.obj (OverColor.mk ![c1])) (y : S.F.obj (OverColor.mk ![S.τ c1, c2])) :
S.F.obj (OverColor.mk ![c2]) :=
(S.F.map (OverColor.mkIso (by funext x; fin_cases x; rfl)).hom).hom <|
(OverColor.forgetLiftApp S.FD c2).inv.hom <|
(OverColor.forgetLiftAppCon S.FD c2).inv.hom <|
(λ_ (S.FD.obj (Discrete.mk c2))).hom.hom <|
((S.contr.app (Discrete.mk c1)) ▷ (S.FD.obj (Discrete.mk (c2 )))).hom <|
(α_ _ _ (S.FD.obj (Discrete.mk (c2 )))).inv.hom <|
(OverColor.forgetLiftApp S.FD c1).hom.hom ((S.F.map (OverColor.mkIso (by
funext x; fin_cases x; rfl)).hom).hom x) ⊗ₜ
(OverColor.forgetLiftAppCon S.FD c1).hom.hom (x) ⊗ₜ
(OverColor.Discrete.pairIsoSep S.FD).inv.hom y
@[simp]
@ -72,19 +70,20 @@ lemma contrOneTwoLeft_tprod_eq {c1 c2 : S.C}
(fy : (i : (𝟭 Type).obj (OverColor.mk ![S.τ c1, c2]).left)
→ CoeSort.coe (S.FD.obj { as := (OverColor.mk ![S.τ c1, c2]).hom i })) :
contrOneTwoLeft (PiTensorProduct.tprod S.k fx) (PiTensorProduct.tprod S.k fy) =
(S.F.map (OverColor.mkIso (by funext x; fin_cases x; rfl)).hom).hom
((OverColor.forgetLiftApp S.FD c2).inv.hom (
((OverColor.forgetLiftAppCon S.FD c2).inv.hom (
((S.contr.app (Discrete.mk c1)).hom (fx (0 : Fin 1) ⊗ₜ fy (0 : Fin 2)) •
fy (1 : Fin 2)))) := by
rw [contrOneTwoLeft]
apply congrArg
apply congrArg
rw [Discrete.pairIsoSep_inv_tprod S.FD fy]
rw [Discrete.pairIsoSep_inv_tprod S.FD fy, OverColor.forgetLiftAppCon]
change (S.contr.app { as := c1 }).hom (_ ⊗ₜ[S.k] fy (0 : Fin 2)) • fy (1 : Fin 2) = _
congr
simp
simp only [Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
Action.FunctorCategoryEquivalence.functor_obj_obj, Nat.succ_eq_add_one, Nat.reduceAdd,
Iso.trans_hom, Functor.mapIso_hom, Action.comp_hom, mk_left, Functor.id_obj, mk_hom,
ModuleCat.coe_comp, Function.comp_apply, LinearMap.id_coe, id_eq, Fin.isValue]
rw [forgetLiftApp_hom_hom_apply_eq]
simp [S.F_def]
simp only [mk_left, Functor.id_obj, Fin.isValue]
erw [OverColor.lift.map_tprod]
congr
funext x
@ -107,6 +106,7 @@ lemma contr_one_two_left_eq_contrOneTwoLeft_tprod {c1 c2 : S.C} (x : S.F.obj (Ov
subst hy
conv_rhs =>
rw [contrOneTwoLeft_tprod_eq]
rw [OverColor.forgetLiftAppCon_inv_apply_expand]
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, Fin.succAbove_zero, mk_left,
Functor.id_obj, mk_hom, contr_tensor, prod_tensor, Action.instMonoidalCategory_tensorObj_V,
Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
@ -184,7 +184,6 @@ lemma contr_one_two_left_eq_contrOneTwoLeft {c1 c2 : S.C} (x : S.F.obj (OverColo
simpa using contr_one_two_left_eq_contrOneTwoLeft_tprod (PiTensorProduct.tprod S.k fx)
(PiTensorProduct.tprod S.k fy) fx fy
/-- Expands the inner contraction of two 2-tensors which are
tprods in terms of basic categorical
constructions and fields of the tensor species. -/