refactor: Lint
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jstoobysmith
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1 changed files with 85 additions and 85 deletions
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@ -8,7 +8,6 @@ import HepLean.Tensors.Tree.Basic
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## Products and contractions
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-/
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open IndexNotation
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@ -81,7 +80,6 @@ lemma succAbove_leftContrJ_leftContrI_natAdd (x : Fin n1) :
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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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@ -92,8 +90,9 @@ def leftContr : ContrPair ((Sum.elim c c1 ∘ (@finSumFinEquiv n.succ.succ n1).s
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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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lemma leftContr_map_eq : ((Sum.elim c (OverColor.mk c1).hom ∘ finSumFinEquiv.symm.toFun) ∘
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⇑leftContrEquivSuccSucc.symm) ∘ (q.leftContr (c1 := c1)).i.succAbove ∘
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(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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@ -137,12 +136,15 @@ lemma sum_inr_succAbove_leftContrI_leftContrJ (k : Fin n1) : finSumFinEquiv.sym
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Equiv.symm_apply_apply, finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl]
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erw [succAbove_leftContrJ_leftContrI_natAdd]
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simp
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lemma contrMap_prod_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c).left) → CoeSort.coe (S.FDiscrete.obj { as := (OverColor.mk c).hom i }))
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(q' : (i : (𝟭 Type).obj (OverColor.mk c1).left) → CoeSort.coe (S.FDiscrete.obj { as := (OverColor.mk c1).hom i })):
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lemma contrMap_prod_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c).left) →
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CoeSort.coe (S.FDiscrete.obj { as := (OverColor.mk c).hom i }))
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(q' : (i : (𝟭 Type).obj (OverColor.mk c1).left) →
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CoeSort.coe (S.FDiscrete.obj { as := (OverColor.mk c1).hom i })) :
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(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 (by exact leftContr_map_eq q)).hom).hom
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= (S.F.map (mkIso (by simpa using leftContr_map_eq q)).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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@ -206,11 +208,10 @@ lemma contrMap_prod_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c).left) → C
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exact Eq.symm ((fun f => (Equiv.apply_eq_iff_eq_symm_apply f).mp) finSumFinEquiv rfl)
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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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change _ = (S.FDiscrete.map (Discrete.eqToHom _) ≫
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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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@ -235,9 +236,10 @@ lemma contrMap_prod_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c).left) → C
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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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(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
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(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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@ -276,9 +278,8 @@ 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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q.leftContr.h
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(perm (OverColor.equivToIso ContrPair.leftContrEquivSuccSucc).hom (prod t t1)))).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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@ -366,12 +367,11 @@ lemma rightContr_map_eq : ((Sum.elim c1 (OverColor.mk c).hom ∘ finSumFinEquiv.
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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 sum_inl_succAbove_rightContrI_rightContrJ (k : Fin n1) : (@finSumFinEquiv n1 n.succ.succ).symm
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((q.rightContr (c1 := c1)).i.succAbove
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((q.rightContr (c1 := c1)).j.succAbove
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(((@finSumFinEquiv n1 n) (Sum.inl k))))) =
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Sum.map id (q.i.succAbove ∘ q.j.succAbove) (finSumFinEquiv.symm (finSumFinEquiv (Sum.inl k))) := by
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((q.rightContr (c1 := c1)).j.succAbove (((@finSumFinEquiv n1 n) (Sum.inl k))))) =
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Sum.map id (q.i.succAbove ∘ q.j.succAbove)
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(finSumFinEquiv.symm (finSumFinEquiv (Sum.inl k))) := by
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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 [succAbove_rightContrJ_rightContrI_castAdd]
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@ -379,25 +379,26 @@ lemma sum_inl_succAbove_rightContrI_rightContrJ (k : Fin n1): (@finSumFinEquiv n
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lemma sum_inr_succAbove_rightContrI_rightContrJ (k : Fin n) : (@finSumFinEquiv n1 n.succ.succ).symm
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((q.rightContr (c1 := c1)).i.succAbove
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((q.rightContr (c1 := c1)).j.succAbove
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(
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(finSumFinEquiv (Sum.inr k))))) =
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Sum.map id (q.i.succAbove ∘ q.j.succAbove) (finSumFinEquiv.symm (finSumFinEquiv (Sum.inr k))):= by
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((q.rightContr (c1 := c1)).j.succAbove ((finSumFinEquiv (Sum.inr k))))) =
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Sum.map id (q.i.succAbove ∘ q.j.succAbove)
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(finSumFinEquiv.symm (finSumFinEquiv (Sum.inr k))) := by
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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 [succAbove_rightContrJ_rightContrI_natAdd]
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simp
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lemma prod_contrMap_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c1).left) → CoeSort.coe (S.FDiscrete.obj { as := (OverColor.mk c1).hom i }))
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(q' : (i : (𝟭 Type).obj (OverColor.mk c).left) → CoeSort.coe (S.FDiscrete.obj { as := (OverColor.mk c).hom i })):
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lemma prod_contrMap_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c1).left) →
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CoeSort.coe (S.FDiscrete.obj { as := (OverColor.mk c1).hom i }))
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(q' : (i : (𝟭 Type).obj (OverColor.mk c).left) →
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CoeSort.coe (S.FDiscrete.obj { as := (OverColor.mk c).hom i })) :
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(S.F.map (equivToIso finSumFinEquiv).hom).hom
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((S.F.μ (OverColor.mk c1) (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove))).hom
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((PiTensorProduct.tprod S.k) p ⊗ₜ[S.k] (q.contrMap.hom (PiTensorProduct.tprod S.k q')))) =
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(S.F.map (mkIso (by exact (rightContr_map_eq q))).hom).hom
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(S.F.map (mkIso (by simpa using (rightContr_map_eq q))).hom).hom
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(q.rightContr.contrMap.hom
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(((S.F.map (equivToIso finSumFinEquiv).hom).hom
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((S.F.μ (OverColor.mk c1) (OverColor.mk c)).hom ((PiTensorProduct.tprod S.k) p ⊗ₜ[S.k] (PiTensorProduct.tprod S.k) q'))))) := by
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((S.F.μ (OverColor.mk c1) (OverColor.mk c)).hom
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((PiTensorProduct.tprod S.k) p ⊗ₜ[S.k] (PiTensorProduct.tprod S.k) q'))))) := by
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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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@ -438,11 +439,12 @@ lemma prod_contrMap_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c1).left) →
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ModuleCat.id_apply]
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rfl
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apply hL
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simp [rightContr, rightContrI]
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· erw [ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply, 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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simp only [Nat.succ_eq_add_one, rightContr, Nat.add_eq, Equiv.toFun_as_coe, rightContrI,
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finSumFinEquiv_symm_apply_natAdd]
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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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change _ = (S.FDiscrete.map (Discrete.eqToHom _) ≫
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S.FDiscrete.map (Discrete.eqToHom _)).hom _
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rw [← S.FDiscrete.map_comp]
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simp
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have h1 {a d : Fin n.succ.succ} {b : Fin n1 ⊕ Fin (n + 1 + 1) }
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@ -467,14 +469,15 @@ lemma prod_contrMap_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c1).left) →
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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 [Nat.succ_eq_add_one, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
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simp only [Nat.succ_eq_add_one, Nat.add_eq, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
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LinearEquiv.coe_coe]
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have h1 (l : (OverColor.mk c1).left ⊕ (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove)).left)
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(l' :Fin n1 ⊕ Fin n.succ.succ)
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(h : Sum.elim c1 c l' = Sum.elim c1 (c ∘ q.i.succAbove ∘ q.j.succAbove) l)
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(h' : l' = (Sum.map id (q.i.succAbove ∘ q.j.succAbove) l))
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: (lift.discreteSumEquiv S.FDiscrete l)
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(HepLean.PiTensorProduct.elimPureTensor p (fun k => q' (q.i.succAbove (q.j.succAbove k))) l) =
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(h' : l' = (Sum.map id (q.i.succAbove ∘ q.j.succAbove) l)) :
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(lift.discreteSumEquiv S.FDiscrete l)
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(HepLean.PiTensorProduct.elimPureTensor p
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(fun k => q' (q.i.succAbove (q.j.succAbove k))) 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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@ -511,9 +514,7 @@ lemma prod_contrMap :
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lemma prod_contr (t1 : TensorTree S c1) (t : TensorTree S c) :
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(prod t1 (contr q.i q.j q.h t)).tensor = ((perm (OverColor.mkIso q.rightContr_map_eq).hom
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(contr (q.rightContrI n1) (q.rightContrJ n1)
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q.rightContr.h (
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(prod t1 t)
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))).tensor) := by
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q.rightContr.h (prod t1 t))).tensor) := by
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simp only [contr_tensor, perm_tensor, prod_tensor]
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change ((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).hom (t1.tensor ⊗ₜ[S.k] t.tensor) = _
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@ -531,21 +532,20 @@ 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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{j : Fin n.succ} (hij : c (i.succAbove j) = S.τ (c i))
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(t : TensorTree S c) (t1 : TensorTree S c1) :
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(prod (contr i j hij t) t1).tensor = ((perm (OverColor.mkIso (ContrPair.mk i j hij).leftContr_map_eq).hom
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(prod (contr i j hij t) t1).tensor =
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((perm (OverColor.mkIso (ContrPair.mk i j hij).leftContr_map_eq).hom
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(contr ((ContrPair.mk i j hij).leftContrI n1) ((ContrPair.mk i j hij).leftContrJ n1)
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(ContrPair.mk i j hij).leftContr.h (
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perm (OverColor.equivToIso ContrPair.leftContrEquivSuccSucc).hom (prod t t1)
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))).tensor) :=
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perm (OverColor.equivToIso ContrPair.leftContrEquivSuccSucc).hom (prod t t1)))).tensor) :=
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(ContrPair.mk i j hij).contr_prod t t1
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theorem prod_contr {n n1 : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n1 → S.C} {i : Fin n.succ.succ}
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{j : Fin n.succ} (hij : c (i.succAbove j) = S.τ (c i))
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(t1 : TensorTree S c1) (t : TensorTree S c) :
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(prod t1 (contr i j hij t)).tensor = ((perm (OverColor.mkIso (ContrPair.mk i j hij).rightContr_map_eq).hom
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(prod t1 (contr i j hij t)).tensor =
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((perm (OverColor.mkIso (ContrPair.mk i j hij).rightContr_map_eq).hom
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(contr ((ContrPair.mk i j hij).rightContrI n1) ((ContrPair.mk i j hij).rightContrJ n1)
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(ContrPair.mk i j hij).rightContr.h (
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(prod t1 t)
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))).tensor) :=
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(ContrPair.mk i j hij).rightContr.h (prod t1 t))).tensor) :=
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(ContrPair.mk i j hij).prod_contr t1 t
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end TensorTree
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