feat: Add statement for prod contr on right
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jstoobysmith
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511fe92cef
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1 changed files with 124 additions and 10 deletions
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@ -196,7 +196,7 @@ lemma contrMap_prod :
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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
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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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@ -215,9 +215,14 @@ lemma contrMap_prod :
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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 [lift.discreteFunctorMapEqIso]
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repeat erw [ModuleCat.id_apply]
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simp
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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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change _ = (S.FDiscrete.map (eqToHom _)).hom
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((lift.discreteSumEquiv S.FDiscrete
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(finSumFinEquiv.symm
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@ -229,12 +234,43 @@ lemma contrMap_prod :
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(leftContrEquivSuccSucc.symm
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(q.leftContr.i.succAbove
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(q.leftContr.j.succAbove k))))))
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sorry
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/- l = k, l' = (finSumFinEquiv.symm (leftContrEquivSuccSucc.symm (q.leftContr.i.succAbove (q.leftContr.j.succAbove k)))),
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-/
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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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rw [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.inr k =>
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simp [finSumFinEquiv_apply_left, finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl, leftContr]
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rw [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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/-!
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@ -242,6 +278,84 @@ lemma contrMap_prod :
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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
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<;> simp_all
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<;> omega
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lemma succAbove_rightContrJ_rightContrI_natAdd (x : Fin n) :
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(q.rightContrI n1).succAbove ((q.rightContrJ n1).succAbove (Fin.natAdd n1 x)) =
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(Fin.natAdd n1 ((q.i.succAbove) (q.j.succAbove 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 h3 h4
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<;> rw [Fin.lt_def] at h1 h2 h3 h4
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<;> simp_all
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<;> omega
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def rightContr : ContrPair ((Sum.elim c1 c ∘ (@finSumFinEquiv n1 n.succ.succ).symm.toFun)) where
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i := q.rightContrI n1
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j := q.rightContrJ n1
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h := by
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simp only [Nat.add_eq, Nat.succ_eq_add_one, Equiv.toFun_as_coe,
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rightContrJ_succAbove_rightContrI, Function.comp_apply, finSumFinEquiv_symm_apply_natAdd,
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Sum.elim_inr]
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simpa [rightContrI] using q.h
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lemma rightContr_map_eq : ((Sum.elim c1 (OverColor.mk c).hom ∘ finSumFinEquiv.symm.toFun)) ∘
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(q.rightContr (c1 := c1)).i.succAbove ∘ (q.rightContr (c1 := c1)).j.succAbove =
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Sum.elim (OverColor.mk c1).hom (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove)).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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erw [succAbove_rightContrJ_rightContrI_castAdd]
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simp only [Nat.succ_eq_add_one, finSumFinEquiv_symm_apply_castAdd, 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_rightContrJ_rightContrI_natAdd]
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simp only [finSumFinEquiv_symm_apply_natAdd, Sum.elim_inr, Function.comp_apply]
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lemma prod_contr :
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(S.F.obj (OverColor.mk c1) ◁ q.contrMap) ≫ (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 c1)) ((OverColor.mk c))) ≫
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S.F.map (OverColor.equivToIso finSumFinEquiv).hom ≫
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q.rightContr.contrMap ≫ S.F.map (OverColor.mkIso (q.rightContr_map_eq)).hom:= by
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sorry
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end ContrPair
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theorem contr_prod {n n1 : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n1 → S.C} {i : Fin n.succ.succ}
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