refactor: Lint
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jstoobysmith
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6 changed files with 55 additions and 37 deletions
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@ -211,8 +211,8 @@ lemma basis_contr_pauliMatrix_basis_tree_expand {n : ℕ} {c : Fin n → complex
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(i.succAbove (j.succAbove k))
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(contr i j h (TensorTree.prod (tensorNode (basisVector c b))
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(constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
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PauliMatrix.asConsTensor))).tensor = (((
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TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 0 0))
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PauliMatrix.asConsTensor))).tensor =
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(((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 0 0))
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(tensorNode (basisVector c' (b' 0 0 0))))).add
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(((TensorTree.smul (contrBasisVectorMul i j (pauliMatrixBasisProdMap b 0 1 1))
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(tensorNode (basisVector c' (b' 0 1 1))))).add
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@ -39,13 +39,17 @@ def leftContrEquivSuccSucc : Fin (n.succ.succ + n1) ≃ Fin ((n + n1).succ.succ)
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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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/-- The new fst index of a contraction obtained on moving a contraction in the LHS of a product
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through the product. -/
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def leftContrI (n1 : ℕ) : Fin ((n + n1).succ.succ) := leftContrEquivSuccSucc <| Fin.castAdd n1 q.i
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/-- The new snd index of a contraction obtained on moving a contraction in the LHS of a product
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through the product. -/
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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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= 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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@ -64,7 +68,8 @@ 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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simp only [Fin.succAbove, leftContrJ, Nat.succ_eq_add_one, leftContrI, leftContrEquivSuccSucc,
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RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply, Fin.coe_cast, Fin.coe_castAdd]
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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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@ -74,12 +79,15 @@ 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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simp only [Fin.succAbove, leftContrJ, Nat.succ_eq_add_one, leftContrI, leftContrEquivSuccSucc,
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RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply, Fin.coe_cast, Fin.coe_natAdd]
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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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/-- The pair of contraction indices obtained on moving a contraction in the LHS of a product
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through the product. -/
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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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@ -114,11 +122,10 @@ lemma leftContr_map_eq : ((Sum.elim c (OverColor.mk c1).hom ∘ finSumFinEquiv.s
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Sum.elim_inr]
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lemma sum_inl_succAbove_leftContrI_leftContrJ (k : Fin n) : finSumFinEquiv.symm
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(leftContrEquivSuccSucc.symm
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((q.leftContr (c1 := c1)).i.succAbove
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((q.leftContr (c1 := c1)).j.succAbove
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(
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(finSumFinEquiv (Sum.inl k)))))) =
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(leftContrEquivSuccSucc.symm
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((q.leftContr (c1 := c1)).i.succAbove
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((q.leftContr (c1 := c1)).j.succAbove
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((finSumFinEquiv (Sum.inl k)))))) =
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Sum.map (q.i.succAbove ∘ q.j.succAbove) id (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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@ -126,12 +133,9 @@ lemma sum_inl_succAbove_leftContrI_leftContrJ (k : Fin n) : finSumFinEquiv.symm
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simp
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lemma sum_inr_succAbove_leftContrI_leftContrJ (k : Fin n1) : finSumFinEquiv.symm
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(leftContrEquivSuccSucc.symm
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((q.leftContr (c1 := c1)).i.succAbove
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((q.leftContr (c1 := c1)).j.succAbove
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(
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(finSumFinEquiv (Sum.inr k)))))) =
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Sum.map (q.i.succAbove ∘ q.j.succAbove) id (Sum.inr k) := by
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(leftContrEquivSuccSucc.symm ((q.leftContr (c1 := c1)).i.succAbove
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((q.leftContr (c1 := c1)).j.succAbove ((finSumFinEquiv (Sum.inr k)))))) =
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Sum.map (q.i.succAbove ∘ q.j.succAbove) id (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_leftContrJ_leftContrI_natAdd]
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@ -144,7 +148,7 @@ lemma contrMap_prod_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c).left) →
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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 simpa using leftContr_map_eq q)).hom).hom
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= (S.F.map (mkIso (by exact 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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@ -163,9 +167,8 @@ lemma contrMap_prod_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c).left) →
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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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(((lift.obj S.FDiscrete).map (equivToIso leftContrEquivSuccSucc).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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@ -298,13 +301,17 @@ lemma contr_prod
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-/
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/-- The new fst index of a contraction obtained on moving a contraction in the RHS of a product
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through the product. -/
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def rightContrI (n1 : ℕ) : Fin ((n1 + n).succ.succ) := Fin.natAdd n1 q.i
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/-- The new snd index of a contraction obtained on moving a contraction in the RHS of a product
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through the product. -/
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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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= (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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@ -323,7 +330,7 @@ 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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simp only [Fin.succAbove, rightContrJ, Nat.succ_eq_add_one, rightContrI, Fin.coe_castAdd]
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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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@ -333,12 +340,14 @@ 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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simp only [Fin.succAbove, rightContrJ, Nat.succ_eq_add_one, rightContrI, Fin.coe_natAdd]
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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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/-- The new pair of indices obtained on moving a contraction in the RHS of a product
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through the product. -/
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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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@ -394,7 +403,7 @@ lemma prod_contrMap_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c1).left) →
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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 simpa using (rightContr_map_eq q))).hom).hom
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(S.F.map (mkIso (by exact (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
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@ -428,7 +437,8 @@ lemma prod_contrMap_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c1).left) →
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· simp only [Nat.add_eq, rightContr, Nat.succ_eq_add_one, Equiv.toFun_as_coe, rightContrI,
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finSumFinEquiv_symm_apply_natAdd, Sum.elim_inr]
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· erw [ModuleCat.id_apply, ModuleCat.id_apply, ModuleCat.id_apply]
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simp
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simp only [Nat.add_eq, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, equivToIso_homToEquiv,
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LinearEquiv.coe_coe]
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have hL (a : Fin n.succ.succ) {b : Fin n1 ⊕ Fin n.succ.succ}
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(h : b = Sum.inr a) : q' a = (S.FDiscrete.map (Discrete.eqToHom (by rw [h]; simp))).hom
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((lift.discreteSumEquiv S.FDiscrete b)
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@ -446,7 +456,7 @@ lemma prod_contrMap_tprod (p : (i : (𝟭 Type).obj (OverColor.mk c1).left) →
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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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simp only [Nat.add_eq, eqToHom_trans]
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have h1 {a d : Fin n.succ.succ} {b : Fin n1 ⊕ Fin (n + 1 + 1) }
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(h1' : b = Sum.inr a) (h2' : c a = S.τ (c d)) :
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(S.FDiscrete.map (Discrete.eqToHom h2')).hom (q' a) =
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@ -535,8 +545,8 @@ theorem contr_prod {n n1 : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n1 → S
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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)))).tensor) :=
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(ContrPair.mk i j hij).leftContr.h
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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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