532 lines
25 KiB
Text
532 lines
25 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.Tree.Basic
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/-!
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## Products and contractions
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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open OverColor
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open HepLean.Fin
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namespace TensorTree
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variable {S : TensorSpecies}
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namespace ContrPair
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variable {n n1 : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n1 → S.C} (q : ContrPair c)
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/-!
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## Left contractions.
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-/
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/-- An equivalence needed to perform contraction. For specified `n` and `n1`
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this reduces to an identity. -/
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def leftContrEquivSuccSucc : Fin (n.succ.succ + n1) ≃ Fin ((n + n1).succ.succ) :=
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(Fin.castOrderIso (by omega)).toEquiv
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/-- An equivalence needed to perform contraction. For specified `n` and `n1`
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this reduces to an identity. -/
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def leftContrEquivSucc : Fin (n.succ + n1) ≃ Fin ((n + n1).succ) :=
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(Fin.castOrderIso (by omega)).toEquiv
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def leftContrI (n1 : ℕ): Fin ((n + n1).succ.succ) := leftContrEquivSuccSucc <| Fin.castAdd n1 q.i
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def leftContrJ (n1 : ℕ) : Fin ((n + n1).succ) := leftContrEquivSucc <| Fin.castAdd n1 q.j
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@[simp]
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lemma leftContrJ_succAbove_leftContrI : (q.leftContrI n1).succAbove (q.leftContrJ n1)
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= leftContrEquivSuccSucc (Fin.castAdd n1 (q.i.succAbove q.j)) := by
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rw [leftContrI, leftContrJ]
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rw [Fin.ext_iff]
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simp only [Fin.succAbove, Nat.succ_eq_add_one, leftContrEquivSucc, RelIso.coe_fn_toEquiv,
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Fin.castOrderIso_apply, leftContrEquivSuccSucc, Fin.coe_cast, Fin.coe_castAdd]
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split_ifs
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<;> rename_i h1 h2
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<;> rw [Fin.lt_def] at h1 h2
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· simp only [Fin.coe_castSucc, Fin.coe_cast, Fin.coe_castAdd]
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· simp_all only [Fin.coe_castSucc, Fin.coe_cast, Fin.coe_castAdd, not_true_eq_false]
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· simp_all only [Fin.coe_castSucc, Fin.coe_cast, Fin.coe_castAdd, not_lt, Fin.val_succ,
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add_right_eq_self, one_ne_zero]
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omega
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· simp only [Fin.val_succ, Fin.coe_cast, Fin.coe_castAdd]
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lemma succAbove_leftContrJ_leftContrI_castAdd (x : Fin n) :
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(q.leftContrI n1).succAbove ((q.leftContrJ n1).succAbove (Fin.castAdd n1 x)) =
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leftContrEquivSuccSucc (Fin.castAdd n1 (q.i.succAbove (q.j.succAbove x))) := by
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rw [Fin.ext_iff]
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simp [leftContrI, leftContrJ, leftContrEquivSuccSucc, Fin.succAbove]
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split_ifs <;> rename_i h1 h2 h3 h4
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<;> rw [Fin.lt_def] at h1 h2 h3 h4
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<;> simp_all [leftContrEquivSucc]
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<;> omega
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lemma succAbove_leftContrJ_leftContrI_natAdd (x : Fin n1) :
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(q.leftContrI n1).succAbove ((q.leftContrJ n1).succAbove (Fin.natAdd n x)) =
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leftContrEquivSuccSucc (Fin.natAdd n.succ.succ x) := by
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rw [Fin.ext_iff]
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simp [leftContrI, leftContrJ, leftContrEquivSuccSucc, Fin.succAbove]
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split_ifs <;> rename_i h1 h2
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<;> rw [Fin.lt_def] at h1 h2
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<;> simp_all [leftContrEquivSucc]
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<;> omega
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def leftContr : ContrPair ((Sum.elim c c1 ∘ (@finSumFinEquiv n.succ.succ n1).symm.toFun) ∘
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leftContrEquivSuccSucc.symm) where
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i := q.leftContrI n1
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j := q.leftContrJ n1
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h := by
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simp only [Nat.succ_eq_add_one, Equiv.toFun_as_coe, leftContrJ_succAbove_leftContrI,
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Function.comp_apply, Equiv.symm_apply_apply, finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl]
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simpa only [leftContrI, Nat.succ_eq_add_one, Equiv.symm_apply_apply,
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finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl] using q.h
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lemma leftContr_map_eq : ((Sum.elim c (OverColor.mk c1).hom ∘ finSumFinEquiv.symm.toFun) ∘ ⇑leftContrEquivSuccSucc.symm) ∘
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(q.leftContr (c1 := c1)).i.succAbove ∘ (q.leftContr (c1 := c1)).j.succAbove =
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Sum.elim (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove)).hom (OverColor.mk c1).hom ∘
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⇑finSumFinEquiv.symm := by
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funext x
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simp only [Nat.succ_eq_add_one, Functor.id_obj, mk_hom, Equiv.toFun_as_coe, Function.comp_apply,
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Functor.const_obj_obj]
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obtain ⟨k, hk⟩ := finSumFinEquiv.surjective x
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subst hk
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match k with
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| Sum.inl k =>
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simp only [finSumFinEquiv_apply_left, finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl,
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Function.comp_apply]
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erw [succAbove_leftContrJ_leftContrI_castAdd]
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simp only [Nat.succ_eq_add_one, Equiv.symm_apply_apply, finSumFinEquiv_symm_apply_castAdd,
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Sum.elim_inl]
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| Sum.inr k =>
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simp only [finSumFinEquiv_apply_right, finSumFinEquiv_symm_apply_natAdd, Sum.elim_inr]
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erw [succAbove_leftContrJ_leftContrI_natAdd]
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simp only [Nat.succ_eq_add_one, Equiv.symm_apply_apply, finSumFinEquiv_symm_apply_natAdd,
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Sum.elim_inr]
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set_option maxHeartbeats 0 in
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lemma contrMap_prod :
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(q.contrMap ▷ S.F.obj (OverColor.mk c1)) ≫ (S.F.μ _ ((OverColor.mk c1))) ≫
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S.F.map (OverColor.equivToIso finSumFinEquiv).hom =
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(S.F.μ ((OverColor.mk c)) ((OverColor.mk c1))) ≫
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S.F.map (OverColor.equivToIso finSumFinEquiv).hom ≫
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S.F.map (OverColor.equivToIso leftContrEquivSuccSucc).hom ≫ q.leftContr.contrMap
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≫ S.F.map (OverColor.mkIso (q.leftContr_map_eq)).hom := by
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ext1
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refine HepLean.PiTensorProduct.induction_tmul (fun p q' => ?_)
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change (S.F.map (equivToIso finSumFinEquiv).hom).hom
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((S.F.μ (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove)) (OverColor.mk c1)).hom
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((q.contrMap.hom (PiTensorProduct.tprod S.k p)) ⊗ₜ[S.k] (PiTensorProduct.tprod S.k) q'))
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= (S.F.map (mkIso _).hom).hom
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(q.leftContr.contrMap.hom
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((S.F.map (equivToIso (@leftContrEquivSuccSucc n n1)).hom).hom
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((S.F.map (equivToIso finSumFinEquiv).hom).hom
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((S.F.μ (OverColor.mk c) (OverColor.mk c1)).hom
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((PiTensorProduct.tprod S.k) p ⊗ₜ[S.k] (PiTensorProduct.tprod S.k) q')))))
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conv_lhs => rw [contrMap, TensorSpecies.contrMap_tprod]
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simp only [TensorSpecies.F_def]
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conv_rhs => rw [lift.obj_μ_tprod_tmul]
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simp only [TensorProduct.smul_tmul, TensorProduct.tmul_smul, map_smul]
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conv_lhs => rw [lift.obj_μ_tprod_tmul]
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change _ = ((lift.obj S.FDiscrete).map (mkIso _).hom).hom
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(q.leftContr.contrMap.hom
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(((lift.obj S.FDiscrete).map (equivToIso leftContrEquivSuccSucc).hom).hom
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(((lift.obj S.FDiscrete).map (equivToIso finSumFinEquiv).hom).hom
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((PiTensorProduct.tprod S.k) _))))
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conv_rhs => rw [lift.map_tprod]
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change _ = ((lift.obj S.FDiscrete).map (mkIso _).hom).hom
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(q.leftContr.contrMap.hom
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(((lift.obj S.FDiscrete).map (equivToIso leftContrEquivSuccSucc).hom).hom
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(
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((PiTensorProduct.tprod S.k) _))))
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conv_rhs => rw [lift.map_tprod]
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change _ = ((lift.obj S.FDiscrete).map (mkIso _).hom).hom
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(q.leftContr.contrMap.hom
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((PiTensorProduct.tprod S.k) _))
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conv_rhs => rw [contrMap, TensorSpecies.contrMap_tprod]
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simp only [TensorProduct.smul_tmul, TensorProduct.tmul_smul, map_smul]
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have hL (a : Fin n.succ.succ) {b : Fin (n + 1 + 1) ⊕ Fin n1}
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(h : b = Sum.inl a) : p a = (S.FDiscrete.map (Discrete.eqToHom (by rw [h]; simp ))).hom
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((lift.discreteSumEquiv S.FDiscrete b)
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(HepLean.PiTensorProduct.elimPureTensor p q' b)) := by
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subst h
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simp only [Nat.succ_eq_add_one, mk_hom, instMonoidalCategoryStruct_tensorObj_hom,
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Sum.elim_inl, eqToHom_refl, Discrete.functor_map_id, Action.id_hom, Functor.id_obj,
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ModuleCat.id_apply]
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rfl
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congr 1
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/- The contraction. -/
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· apply congrArg
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simp only [Monoidal.tensorUnit_obj, Action.instMonoidalCategory_tensorUnit_V,
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Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, Functor.comp_obj,
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Discrete.functor_obj_eq_as, Function.comp_apply, Nat.succ_eq_add_one, mk_hom,
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Equiv.toFun_as_coe, lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl,
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Iso.refl_hom, Action.id_hom, Iso.refl_inv, Functor.id_obj,
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instMonoidalCategoryStruct_tensorObj_hom, LinearEquiv.ofLinear_apply]
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have h1' : ∀ {a a' b c b' c'} (haa' : a = a')
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(_ : b = (S.FDiscrete.map (Discrete.eqToHom (by rw [haa']))).hom b')
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(_ : c = (S.FDiscrete.map (Discrete.eqToHom (by rw [haa']))).hom c'),
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(S.contr.app a).hom (b ⊗ₜ[S.k] c) = (S.contr.app a').hom (b' ⊗ₜ[S.k] c') := by
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intro a a' b c b' c' haa' hbc hcc
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subst haa'
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simp_all
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refine h1' ?_ ?_ ?_
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· simp only [leftContr, Nat.succ_eq_add_one, Equiv.toFun_as_coe, leftContrI,
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Equiv.symm_apply_apply, finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl]
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· erw [ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply]
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simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, equivToIso_homToEquiv,
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LinearEquiv.coe_coe]
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apply hL
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exact Eq.symm ((fun f => (Equiv.apply_eq_iff_eq_symm_apply f).mp) finSumFinEquiv rfl)
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· erw [ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply,
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ModuleCat.id_apply]
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simp only [Discrete.functor_obj_eq_as, Function.comp_apply, AddHom.toFun_eq_coe,
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LinearMap.coe_toAddHom, equivToIso_homToEquiv]
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change _ = (S.FDiscrete.map (Discrete.eqToHom _) ≫ S.FDiscrete.map (Discrete.eqToHom _)).hom _
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rw [← S.FDiscrete.map_comp]
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simp only [eqToHom_trans]
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/- a = q.i.succAbove q.j, d = q.i, b = (finSumFinEquiv.symm (leftContrEquivSuccSucc.symm (q.leftContr.i.succAbove q.leftContr.j))
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h : c (q.i.succAbove q.j) = S.τ (c q.i) -/
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have h1 {a d : Fin n.succ.succ} {b : Fin (n + 1 + 1) ⊕ Fin n1}
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(h1' : b = Sum.inl a) (h2' : c a = S.τ (c d)) :
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(S.FDiscrete.map (Discrete.eqToHom h2')).hom (p a) =
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(S.FDiscrete.map (eqToHom (by subst h1'; simpa using h2'))).hom
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((lift.discreteSumEquiv S.FDiscrete b)
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(HepLean.PiTensorProduct.elimPureTensor p q' b)) := by
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subst h1'
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rfl
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apply h1
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erw [leftContrJ_succAbove_leftContrI]
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simp only [Nat.succ_eq_add_one, Equiv.symm_apply_apply, finSumFinEquiv_symm_apply_castAdd]
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/- The tensor. -/
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· rw [lift.map_tprod]
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conv_lhs => erw [lift.map_tprod]
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apply congrArg
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funext k
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simp only [ Functor.id_obj, mk_hom, Function.comp_apply,
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equivToIso_homToEquiv, lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl,
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Iso.refl_hom, Action.id_hom, Iso.refl_inv, instMonoidalCategoryStruct_tensorObj_hom,
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LinearEquiv.ofLinear_apply, Equiv.toFun_as_coe, equivToIso_mkIso_hom, Equiv.refl_symm,
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Functor.mapIso_hom, eqToIso.hom, Functor.mapIso_inv, eqToIso.inv]
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conv_rhs => repeat erw [ModuleCat.id_apply]
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simp only [Nat.succ_eq_add_one, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
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LinearEquiv.coe_coe]
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have h1 (l : (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove)).left ⊕ (OverColor.mk c1).left)
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(l' : Fin n.succ.succ ⊕ Fin n1)
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(h : Sum.elim c c1 l' = Sum.elim (c ∘ q.i.succAbove ∘ q.j.succAbove) c1 l)
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(h' : l' = (Sum.map (q.i.succAbove ∘ q.j.succAbove) id l))
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: (lift.discreteSumEquiv S.FDiscrete l)
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(HepLean.PiTensorProduct.elimPureTensor (fun k => p (q.i.succAbove (q.j.succAbove k))) q' l) =
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(S.FDiscrete.map (eqToHom (by simp [h] ))).hom
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((lift.discreteSumEquiv S.FDiscrete l')
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(HepLean.PiTensorProduct.elimPureTensor p q' l')) := by
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subst h'
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match l with
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| Sum.inl l =>
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simp only [Nat.succ_eq_add_one, instMonoidalCategoryStruct_tensorObj_hom, mk_hom,
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Sum.elim_inl, Function.comp_apply, Functor.id_obj, Sum.map_inl, eqToHom_refl,
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Discrete.functor_map_id, Action.id_hom, ModuleCat.id_apply]
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rfl
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| Sum.inr l =>
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simp only [Nat.succ_eq_add_one, instMonoidalCategoryStruct_tensorObj_hom, mk_hom,
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Sum.elim_inr, Functor.id_obj, Function.comp_apply, Sum.map_inr, id_eq, eqToHom_refl,
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Discrete.functor_map_id, Action.id_hom, ModuleCat.id_apply]
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rfl
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refine h1 _ _ ?_ ?_
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· simpa using Discrete.eqToIso.proof_1
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(Hom.toEquiv_comp_inv_apply (mkIso (leftContr_map_eq q)).hom k)
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· obtain ⟨k, hk⟩ := finSumFinEquiv.surjective k
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subst hk
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erw [Equiv.symm_apply_apply]
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match k with
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| Sum.inl k =>
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simp only [Nat.succ_eq_add_one, Equiv.toFun_as_coe, leftContr, finSumFinEquiv_apply_left,
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Sum.map_inl, Function.comp_apply]
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erw [Equiv.refl_apply, Equiv.refl_apply]
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erw [succAbove_leftContrJ_leftContrI_castAdd]
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simp only [Nat.succ_eq_add_one, Equiv.invFun_as_coe, Equiv.symm_apply_apply,
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finSumFinEquiv_symm_apply_castAdd]
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| Sum.inr k =>
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simp [finSumFinEquiv_apply_left, finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl, leftContr]
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erw [Equiv.refl_apply, Equiv.refl_apply]
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erw [succAbove_leftContrJ_leftContrI_natAdd]
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simp only [Nat.succ_eq_add_one, Equiv.invFun_as_coe, Equiv.symm_apply_apply,
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finSumFinEquiv_symm_apply_natAdd]
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lemma contr_prod
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(t : TensorTree S c) (t1 : TensorTree S c1) :
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(prod (contr q.i q.j q.h t) t1).tensor = ((perm (OverColor.mkIso q.leftContr_map_eq).hom
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(contr (q.leftContrI n1) (q.leftContrJ n1)
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q.leftContr.h (
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perm (OverColor.equivToIso ContrPair.leftContrEquivSuccSucc).hom (prod t t1)
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))).tensor) := by
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simp only [contr_tensor, perm_tensor, prod_tensor]
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change ((q.contrMap ▷ S.F.obj (OverColor.mk c1)) ≫ (S.F.μ _ ((OverColor.mk c1))) ≫
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S.F.map (OverColor.equivToIso finSumFinEquiv).hom).hom (t.tensor ⊗ₜ[S.k] t1.tensor) = _
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rw [contrMap_prod]
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simp only [Nat.succ_eq_add_one, Functor.id_obj, mk_hom, Action.instMonoidalCategory_tensorObj_V,
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Functor.const_obj_obj, Equiv.toFun_as_coe, Action.comp_hom, Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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ModuleCat.coe_comp, Function.comp_apply]
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apply congrArg
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apply congrArg
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rfl
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/-!
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## Right contractions.
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-/
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def rightContrI (n1 : ℕ): Fin ((n1 + n).succ.succ) := Fin.natAdd n1 q.i
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def rightContrJ (n1 : ℕ) : Fin ((n1 + n).succ) := Fin.natAdd n1 q.j
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@[simp]
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lemma rightContrJ_succAbove_rightContrI : (q.rightContrI n1).succAbove (q.rightContrJ n1)
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= (Fin.natAdd n1 (q.i.succAbove q.j)) := by
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rw [rightContrI, rightContrJ]
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rw [Fin.ext_iff]
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simp only [Fin.succAbove, Nat.succ_eq_add_one, Fin.coe_natAdd]
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split_ifs
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<;> rename_i h1 h2
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<;> rw [Fin.lt_def] at h1 h2
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· simp only [Fin.coe_castSucc, Fin.coe_natAdd]
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· simp_all only [Fin.coe_castSucc, Fin.coe_natAdd, add_lt_add_iff_left, not_true_eq_false]
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· simp_all only [Fin.coe_castSucc, Fin.coe_natAdd, add_lt_add_iff_left, not_lt, Fin.val_succ,
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add_right_eq_self, one_ne_zero]
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omega
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· simp only [Fin.val_succ, Fin.coe_natAdd]
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omega
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lemma succAbove_rightContrJ_rightContrI_castAdd (x : Fin n1) :
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(q.rightContrI n1).succAbove ((q.rightContrJ n1).succAbove (Fin.castAdd n x)) =
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(Fin.castAdd n.succ.succ x) := by
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rw [Fin.ext_iff]
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simp [rightContrI, rightContrJ, Fin.succAbove]
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split_ifs <;> rename_i h1 h2
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<;> rw [Fin.lt_def] at h1 h2
|
||
<;> simp_all
|
||
<;> omega
|
||
|
||
lemma succAbove_rightContrJ_rightContrI_natAdd (x : Fin n) :
|
||
(q.rightContrI n1).succAbove ((q.rightContrJ n1).succAbove (Fin.natAdd n1 x)) =
|
||
(Fin.natAdd n1 ((q.i.succAbove) (q.j.succAbove x))) := by
|
||
rw [Fin.ext_iff]
|
||
simp [rightContrI, rightContrJ, Fin.succAbove]
|
||
split_ifs <;> rename_i h1 h2 h3 h4
|
||
<;> rw [Fin.lt_def] at h1 h2 h3 h4
|
||
<;> simp_all
|
||
<;> omega
|
||
|
||
def rightContr : ContrPair ((Sum.elim c1 c ∘ (@finSumFinEquiv n1 n.succ.succ).symm.toFun)) where
|
||
i := q.rightContrI n1
|
||
j := q.rightContrJ n1
|
||
h := by
|
||
simp only [Nat.add_eq, Nat.succ_eq_add_one, Equiv.toFun_as_coe,
|
||
rightContrJ_succAbove_rightContrI, Function.comp_apply, finSumFinEquiv_symm_apply_natAdd,
|
||
Sum.elim_inr]
|
||
simpa [rightContrI] using q.h
|
||
|
||
lemma rightContr_map_eq : ((Sum.elim c1 (OverColor.mk c).hom ∘ finSumFinEquiv.symm.toFun)) ∘
|
||
(q.rightContr (c1 := c1)).i.succAbove ∘ (q.rightContr (c1 := c1)).j.succAbove =
|
||
Sum.elim (OverColor.mk c1).hom (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove)).hom ∘
|
||
⇑finSumFinEquiv.symm := by
|
||
funext x
|
||
simp only [Nat.succ_eq_add_one, Functor.id_obj, mk_hom, Equiv.toFun_as_coe, Function.comp_apply,
|
||
Functor.const_obj_obj]
|
||
obtain ⟨k, hk⟩ := finSumFinEquiv.surjective x
|
||
subst hk
|
||
match k with
|
||
| Sum.inl k =>
|
||
simp only [finSumFinEquiv_apply_left, finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl]
|
||
erw [succAbove_rightContrJ_rightContrI_castAdd]
|
||
simp only [Nat.succ_eq_add_one, finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl]
|
||
| Sum.inr k =>
|
||
simp only [finSumFinEquiv_apply_right, finSumFinEquiv_symm_apply_natAdd, Sum.elim_inr]
|
||
erw [succAbove_rightContrJ_rightContrI_natAdd]
|
||
simp only [finSumFinEquiv_symm_apply_natAdd, Sum.elim_inr, Function.comp_apply]
|
||
|
||
set_option maxHeartbeats 0 in
|
||
lemma prod_contrMap :
|
||
(S.F.obj (OverColor.mk c1) ◁ q.contrMap) ≫ (S.F.μ ((OverColor.mk c1)) _) ≫
|
||
S.F.map (OverColor.equivToIso finSumFinEquiv).hom =
|
||
(S.F.μ ((OverColor.mk c1)) ((OverColor.mk c))) ≫
|
||
S.F.map (OverColor.equivToIso finSumFinEquiv).hom ≫
|
||
q.rightContr.contrMap ≫ S.F.map (OverColor.mkIso (q.rightContr_map_eq)).hom := by
|
||
ext1
|
||
refine HepLean.PiTensorProduct.induction_tmul (fun p q' => ?_)
|
||
change (S.F.map (equivToIso finSumFinEquiv).hom).hom
|
||
((S.F.μ (OverColor.mk c1) (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove))).hom
|
||
((PiTensorProduct.tprod S.k) p ⊗ₜ[S.k] (q.contrMap.hom (PiTensorProduct.tprod S.k q')))) =
|
||
(S.F.map (mkIso _).hom).hom
|
||
(q.rightContr.contrMap.hom
|
||
(((S.F.map (equivToIso finSumFinEquiv).hom ).hom
|
||
((S.F.μ (OverColor.mk c1) (OverColor.mk c)).hom ((PiTensorProduct.tprod S.k) p ⊗ₜ[S.k] (PiTensorProduct.tprod S.k) q')))))
|
||
conv_lhs => rw [contrMap, TensorSpecies.contrMap_tprod]
|
||
simp only [TensorSpecies.F_def]
|
||
conv_rhs => rw [lift.obj_μ_tprod_tmul]
|
||
simp only [TensorProduct.smul_tmul, TensorProduct.tmul_smul, map_smul]
|
||
conv_lhs => rw [lift.obj_μ_tprod_tmul]
|
||
conv_rhs => erw [lift.map_tprod]
|
||
conv_rhs => erw [contrMap, TensorSpecies.contrMap_tprod]
|
||
simp only [TensorProduct.smul_tmul, TensorProduct.tmul_smul, map_smul]
|
||
congr 1
|
||
/- The contraction. -/
|
||
· apply congrArg
|
||
simp only [Monoidal.tensorUnit_obj, Action.instMonoidalCategory_tensorUnit_V,
|
||
Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
|
||
Action.FunctorCategoryEquivalence.functor_obj_obj, Functor.comp_obj,
|
||
Discrete.functor_obj_eq_as, Function.comp_apply, Nat.succ_eq_add_one, mk_hom,
|
||
Equiv.toFun_as_coe, lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl,
|
||
Iso.refl_hom, Action.id_hom, Iso.refl_inv, Functor.id_obj,
|
||
instMonoidalCategoryStruct_tensorObj_hom, LinearEquiv.ofLinear_apply]
|
||
have h1' : ∀ {a a' b c b' c'} (haa' : a = a')
|
||
(_ : b = (S.FDiscrete.map (Discrete.eqToHom (by rw [haa']))).hom b')
|
||
(_ : c = (S.FDiscrete.map (Discrete.eqToHom (by rw [haa']))).hom c'),
|
||
(S.contr.app a).hom (b ⊗ₜ[S.k] c) = (S.contr.app a').hom (b' ⊗ₜ[S.k] c') := by
|
||
intro a a' b c b' c' haa' hbc hcc
|
||
subst haa'
|
||
simp_all
|
||
refine h1' ?_ ?_ ?_
|
||
· simp only [Nat.add_eq, rightContr, Nat.succ_eq_add_one, Equiv.toFun_as_coe, rightContrI,
|
||
finSumFinEquiv_symm_apply_natAdd, Sum.elim_inr]
|
||
· erw [ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply]
|
||
simp
|
||
have hL (a : Fin n.succ.succ) {b : Fin n1 ⊕ Fin n.succ.succ}
|
||
(h : b = Sum.inr a) : q' a = (S.FDiscrete.map (Discrete.eqToHom (by rw [h]; simp ))).hom
|
||
((lift.discreteSumEquiv S.FDiscrete b)
|
||
(HepLean.PiTensorProduct.elimPureTensor p q' b)) := by
|
||
subst h
|
||
simp only [Nat.succ_eq_add_one, mk_hom, instMonoidalCategoryStruct_tensorObj_hom,
|
||
Sum.elim_inl, eqToHom_refl, Discrete.functor_map_id, Action.id_hom, Functor.id_obj,
|
||
ModuleCat.id_apply]
|
||
rfl
|
||
apply hL
|
||
simp [rightContr, rightContrI]
|
||
· erw [ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply]
|
||
simp only [Discrete.functor_obj_eq_as, Function.comp_apply, AddHom.toFun_eq_coe,
|
||
LinearMap.coe_toAddHom, equivToIso_homToEquiv]
|
||
change _ = (S.FDiscrete.map (Discrete.eqToHom _) ≫ S.FDiscrete.map (Discrete.eqToHom _)).hom _
|
||
rw [← S.FDiscrete.map_comp]
|
||
simp
|
||
have h1 {a d : Fin n.succ.succ} {b : Fin n1 ⊕ Fin (n + 1 + 1) }
|
||
(h1' : b = Sum.inr a) (h2' : c a = S.τ (c d)) :
|
||
(S.FDiscrete.map (Discrete.eqToHom h2')).hom (q' a) =
|
||
(S.FDiscrete.map (eqToHom (by subst h1'; simpa using h2'))).hom
|
||
((lift.discreteSumEquiv S.FDiscrete b)
|
||
(HepLean.PiTensorProduct.elimPureTensor p q' b)) := by
|
||
subst h1'
|
||
rfl
|
||
apply h1
|
||
erw [rightContrJ_succAbove_rightContrI]
|
||
simp only [finSumFinEquiv_symm_apply_natAdd, Nat.succ_eq_add_one]
|
||
/- The tensor. -/
|
||
· rw [lift.map_tprod]
|
||
conv_lhs => erw [lift.map_tprod]
|
||
apply congrArg
|
||
funext k
|
||
simp only [ Functor.id_obj, mk_hom, Function.comp_apply,
|
||
equivToIso_homToEquiv, lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl,
|
||
Iso.refl_hom, Action.id_hom, Iso.refl_inv, instMonoidalCategoryStruct_tensorObj_hom,
|
||
LinearEquiv.ofLinear_apply, Equiv.toFun_as_coe, equivToIso_mkIso_hom, Equiv.refl_symm,
|
||
Functor.mapIso_hom, eqToIso.hom, Functor.mapIso_inv, eqToIso.inv]
|
||
conv_rhs => repeat erw [ModuleCat.id_apply]
|
||
simp [Nat.succ_eq_add_one, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
|
||
LinearEquiv.coe_coe]
|
||
have h1 (l : (OverColor.mk c1).left ⊕ (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove)).left)
|
||
(l' :Fin n1 ⊕ Fin n.succ.succ )
|
||
(h : Sum.elim c1 c l' = Sum.elim c1 (c ∘ q.i.succAbove ∘ q.j.succAbove) l)
|
||
(h' : l' = (Sum.map id (q.i.succAbove ∘ q.j.succAbove) l))
|
||
: (lift.discreteSumEquiv S.FDiscrete l)
|
||
(HepLean.PiTensorProduct.elimPureTensor p (fun k => q' (q.i.succAbove (q.j.succAbove k))) l) =
|
||
(S.FDiscrete.map (eqToHom (by simp [h] ))).hom
|
||
((lift.discreteSumEquiv S.FDiscrete l')
|
||
(HepLean.PiTensorProduct.elimPureTensor p q' l')) := by
|
||
subst h'
|
||
match l with
|
||
| Sum.inl l =>
|
||
simp only [Nat.succ_eq_add_one, instMonoidalCategoryStruct_tensorObj_hom, mk_hom,
|
||
Sum.elim_inl, Function.comp_apply, Functor.id_obj, Sum.map_inl, eqToHom_refl,
|
||
Discrete.functor_map_id, Action.id_hom, ModuleCat.id_apply]
|
||
rfl
|
||
| Sum.inr l =>
|
||
simp only [Nat.succ_eq_add_one, instMonoidalCategoryStruct_tensorObj_hom, mk_hom,
|
||
Sum.elim_inr, Functor.id_obj, Function.comp_apply, Sum.map_inr, id_eq, eqToHom_refl,
|
||
Discrete.functor_map_id, Action.id_hom, ModuleCat.id_apply]
|
||
rfl
|
||
refine h1 _ _ ?_ ?_
|
||
· simpa using Discrete.eqToIso.proof_1
|
||
(Hom.toEquiv_comp_inv_apply (mkIso (rightContr_map_eq q)).hom k)
|
||
· obtain ⟨k, hk⟩ := finSumFinEquiv.surjective k
|
||
subst hk
|
||
erw [Equiv.symm_apply_apply]
|
||
match k with
|
||
| Sum.inl k =>
|
||
simp only [Nat.succ_eq_add_one, rightContr, Nat.add_eq, Equiv.toFun_as_coe,
|
||
finSumFinEquiv_apply_left, Sum.map_inl, id_eq]
|
||
erw [Equiv.refl_apply, Equiv.refl_apply]
|
||
rw [succAbove_rightContrJ_rightContrI_castAdd]
|
||
simp only [Nat.succ_eq_add_one, Equiv.invFun_as_coe, finSumFinEquiv_symm_apply_castAdd]
|
||
| Sum.inr k =>
|
||
simp only [Nat.succ_eq_add_one, rightContr, Nat.add_eq, Equiv.toFun_as_coe,
|
||
finSumFinEquiv_apply_right, Sum.map_inr, Function.comp_apply]
|
||
erw [Equiv.refl_apply, Equiv.refl_apply]
|
||
rw [succAbove_rightContrJ_rightContrI_natAdd]
|
||
simp only [Equiv.invFun_as_coe, finSumFinEquiv_symm_apply_natAdd]
|
||
|
||
|
||
lemma prod_contr (t1 : TensorTree S c1) (t : TensorTree S c) :
|
||
(prod t1 (contr q.i q.j q.h t)).tensor = ((perm (OverColor.mkIso q.rightContr_map_eq).hom
|
||
(contr (q.rightContrI n1) (q.rightContrJ n1)
|
||
q.rightContr.h (
|
||
(prod t1 t)
|
||
))).tensor) := by
|
||
simp only [contr_tensor, perm_tensor, prod_tensor]
|
||
change ( (S.F.obj (OverColor.mk c1) ◁ q.contrMap) ≫ (S.F.μ ((OverColor.mk c1)) _) ≫
|
||
S.F.map (OverColor.equivToIso finSumFinEquiv).hom).hom (t1.tensor ⊗ₜ[S.k] t.tensor) = _
|
||
rw [prod_contrMap]
|
||
simp only [Nat.succ_eq_add_one, Functor.id_obj, mk_hom, Action.instMonoidalCategory_tensorObj_V,
|
||
Functor.const_obj_obj, Equiv.toFun_as_coe, Action.comp_hom, Equivalence.symm_inverse,
|
||
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
|
||
ModuleCat.coe_comp, Function.comp_apply]
|
||
apply congrArg
|
||
apply congrArg
|
||
rfl
|
||
|
||
end ContrPair
|
||
|
||
theorem contr_prod {n n1 : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n1 → S.C} {i : Fin n.succ.succ}
|
||
{j : Fin n.succ} (hij : c (i.succAbove j) = S.τ (c i))
|
||
(t : TensorTree S c) (t1 : TensorTree S c1) :
|
||
(prod (contr i j hij t) t1).tensor = ((perm (OverColor.mkIso (ContrPair.mk i j hij).leftContr_map_eq).hom
|
||
(contr ((ContrPair.mk i j hij).leftContrI n1) ((ContrPair.mk i j hij).leftContrJ n1)
|
||
(ContrPair.mk i j hij).leftContr.h (
|
||
perm (OverColor.equivToIso ContrPair.leftContrEquivSuccSucc).hom (prod t t1)
|
||
))).tensor) :=
|
||
(ContrPair.mk i j hij).contr_prod t t1
|
||
|
||
theorem prod_contr {n n1 : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n1 → S.C} {i : Fin n.succ.succ}
|
||
{j : Fin n.succ} (hij : c (i.succAbove j) = S.τ (c i))
|
||
(t1 : TensorTree S c1) (t : TensorTree S c) :
|
||
(prod t1 (contr i j hij t)).tensor = ((perm (OverColor.mkIso (ContrPair.mk i j hij).rightContr_map_eq).hom
|
||
(contr ((ContrPair.mk i j hij).rightContrI n1) ((ContrPair.mk i j hij).rightContrJ n1)
|
||
(ContrPair.mk i j hij).rightContr.h (
|
||
(prod t1 t)
|
||
))).tensor) :=
|
||
(ContrPair.mk i j hij).prod_contr t1 t
|
||
|
||
end TensorTree
|