refactor: Simp to simp only ...
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jstoobysmith
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b2ac704d80
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12 changed files with 257 additions and 192 deletions
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@ -39,9 +39,9 @@ lemma contrFin1Fin1_naturality {n : ℕ} {c c1 : Fin n.succ.succ → S.C}
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(perm_contr_cond S h σ)).hom
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≫ ((Discrete.pairτ S.FDiscrete S.τ).map (Discrete.eqToHom (Hom.toEquiv_comp_inv_apply σ i)
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: (Discrete.mk (c ((Hom.toEquiv σ).symm i))) ⟶ (Discrete.mk (c1 i)))) := by
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erw [← CategoryTheory.Iso.eq_comp_inv ]
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erw [← CategoryTheory.Iso.eq_comp_inv]
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rw [CategoryTheory.Category.assoc]
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erw [← CategoryTheory.Iso.inv_comp_eq ]
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erw [← CategoryTheory.Iso.inv_comp_eq]
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ext1
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apply TensorProduct.ext'
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intro x y
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@ -51,7 +51,11 @@ lemma contrFin1Fin1_naturality {n : ℕ} {c c1 : Fin n.succ.succ → S.C}
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extractOne_homToEquiv, Action.Hom.comp_hom, LinearMap.coe_comp]
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trans (S.F.map (extractTwoAux' i j σ)).hom (PiTensorProduct.tprod S.k (fun k =>
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match k with | Sum.inl 0 => x | Sum.inr 0 => (S.FDiscrete.map
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(eqToHom (by simp; erw [perm_contr_cond S h σ]))).hom y))
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(eqToHom (by
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simp only [Nat.succ_eq_add_one, Discrete.functor_obj_eq_as, Function.comp_apply,
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extractOne_homToEquiv, Fin.isValue, mk_hom, finExtractTwo_symm_inl_inr_apply,
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Discrete.mk.injEq]
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erw [perm_contr_cond S h σ]))).hom y))
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· apply congrArg
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have h1' {α :Type} {a b c d : α} (hab : a= b) (hcd : c = d) (h : a = d) : b = c := by
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rw [← hab, hcd]
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@ -77,8 +81,11 @@ lemma contrFin1Fin1_naturality {n : ℕ} {c c1 : Fin n.succ.succ → S.C}
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match i with
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| Sum.inl 0 => rfl
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| Sum.inr 0 =>
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simp [lift.discreteFunctorMapEqIso]
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change ((S.FDiscrete.map (eqToHom _)) ≫ S.FDiscrete.map (eqToHom _)).hom y = ((S.FDiscrete.map (eqToHom _)) ≫ S.FDiscrete.map (eqToHom _)).hom y
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simp only [Nat.succ_eq_add_one, mk_hom, Fin.isValue, Function.comp_apply,
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extractOne_homToEquiv, lift.discreteFunctorMapEqIso, Functor.mapIso_hom, eqToIso.hom,
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Functor.mapIso_inv, eqToIso.inv, Functor.id_obj, Discrete.functor_obj_eq_as,
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LinearEquiv.ofLinear_apply]
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change ((S.FDiscrete.map (eqToHom _)) ≫ S.FDiscrete.map (eqToHom _)).hom y = ((S.FDiscrete.map (eqToHom _)) ≫ S.FDiscrete.map (eqToHom _)).hom y
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rw [← Functor.map_comp, ← Functor.map_comp]
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simp only [Fin.isValue, Nat.succ_eq_add_one, Discrete.functor_obj_eq_as, Function.comp_apply,
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eqToHom_trans]
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@ -89,12 +96,11 @@ lemma contrIso_comm_aux_1 {n : ℕ} {c c1 : Fin n.succ.succ → S.C}
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(σ : (OverColor.mk c) ⟶ (OverColor.mk c1)) :
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((S.F.map σ).hom ≫ (S.F.map (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).hom) ≫
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(S.F.map (mkSum (c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).hom =
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(S.F.map (equivToIso (HepLean.Fin.finExtractTwo ((Hom.toEquiv σ).symm i)
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((HepLean.Fin.finExtractOnePerm ((Hom.toEquiv σ).symm i) (Hom.toEquiv σ)).symm j))).hom).hom ≫ (S.F.map
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(mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo ((Hom.toEquiv σ).symm i)
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(S.F.map (equivToIso (HepLean.Fin.finExtractTwo ((Hom.toEquiv σ).symm i)
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((HepLean.Fin.finExtractOnePerm ((Hom.toEquiv σ).symm i) (Hom.toEquiv σ)).symm j))).hom).hom ≫ (S.F.map
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(mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo ((Hom.toEquiv σ).symm i)
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((HepLean.Fin.finExtractOnePerm ((Hom.toEquiv σ).symm i) (Hom.toEquiv σ)).symm j)).symm)).hom).hom
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≫ (S.F.map (extractTwoAux' i j σ ⊗ extractTwoAux i j σ)).hom
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:= by
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≫ (S.F.map (extractTwoAux' i j σ ⊗ extractTwoAux i j σ)).hom := by
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ext X
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change ((S.F.map σ) ≫ (S.F.map (equivToIso (HepLean.Fin.finExtractTwo i j)).hom) ≫ (S.F.map (mkSum (c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom)).hom X = _
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rw [← Functor.map_comp, ← Functor.map_comp]
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@ -109,14 +115,14 @@ lemma contrIso_comm_aux_2 {n : ℕ} {c c1 : Fin n.succ.succ → S.C}
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(S.F.map (extractTwoAux' i j σ ⊗ extractTwoAux i j σ)).hom ≫
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(S.F.μIso (OverColor.mk ((c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl))
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(OverColor.mk ((c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv.hom =
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(S.F.μIso _ _).inv.hom ≫ (S.F.map (extractTwoAux' i j σ) ⊗ S.F.map (extractTwoAux i j σ)).hom
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:= by
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(S.F.μIso _ _).inv.hom ≫
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(S.F.map (extractTwoAux' i j σ) ⊗ S.F.map (extractTwoAux i j σ)).hom := by
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have h1 : (S.F.map (extractTwoAux' i j σ ⊗ extractTwoAux i j σ)) ≫
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(S.F.μIso (OverColor.mk ((c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl))
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(OverColor.mk ((c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv =
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(S.F.μIso _ _).inv ≫ (S.F.map (extractTwoAux' i j σ) ⊗ S.F.map (extractTwoAux i j σ)) := by
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erw [CategoryTheory.IsIso.comp_inv_eq, CategoryTheory.Category.assoc]
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erw [CategoryTheory.IsIso.eq_inv_comp ]
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erw [CategoryTheory.IsIso.eq_inv_comp]
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exact Eq.symm
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(LaxMonoidalFunctor.μ_natural S.F.toLaxMonoidalFunctor (extractTwoAux' i j σ)
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(extractTwoAux i j σ))
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@ -143,8 +149,9 @@ lemma contrIso_comm_aux_3 {n : ℕ} {c c1 : Fin n.succ.succ → S.C}
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def contrIsoComm {n : ℕ} {c c1 : Fin n.succ.succ → S.C}
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{i : Fin n.succ.succ} {j : Fin n.succ} (σ : (OverColor.mk c) ⟶ (OverColor.mk c1)) :=
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(((Discrete.pairτ S.FDiscrete S.τ).map (Discrete.eqToHom (Hom.toEquiv_comp_inv_apply σ i)
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: (Discrete.mk (c ((Hom.toEquiv σ).symm i))) ⟶ (Discrete.mk (c1 i)))) ⊗ (S.F.map (extractTwo i j σ)))
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(((Discrete.pairτ S.FDiscrete S.τ).map (Discrete.eqToHom (Hom.toEquiv_comp_inv_apply σ i) :
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(Discrete.mk (c ((Hom.toEquiv σ).symm i))) ⟶
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(Discrete.mk (c1 i)))) ⊗ (S.F.map (extractTwo i j σ)))
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lemma contrIso_comm_aux_5 {n : ℕ} {c c1 : Fin n.succ.succ → S.C}
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{i : Fin n.succ.succ} {j : Fin n.succ} (h : c1 (i.succAbove j) = S.τ (c1 i))
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@ -155,8 +162,7 @@ lemma contrIso_comm_aux_5 {n : ℕ} {c c1 : Fin n.succ.succ → S.C}
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((HepLean.Fin.finExtractOnePerm ((Hom.toEquiv σ).symm i) (Hom.toEquiv σ)).symm j)
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(perm_contr_cond S h σ)).hom.hom ⊗ (S.F.map (mkIso (contrIso.proof_1 S c ((Hom.toEquiv σ).symm i)
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((HepLean.Fin.finExtractOnePerm ((Hom.toEquiv σ).symm i) (Hom.toEquiv σ)).symm j))).hom).hom)
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≫ (S.contrIsoComm σ).hom
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:= by
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≫ (S.contrIsoComm σ).hom := by
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erw [← CategoryTheory.MonoidalCategory.tensor_comp (f₁ := (S.F.map (extractTwoAux' i j σ)).hom)]
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rw [contrIso_comm_aux_3 S σ]
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rw [contrFin1Fin1_naturality S h σ]
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