feat: Update perm_contr two FIn 2
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jstoobysmith
parent
7358807980
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d2d75e4d36
3 changed files with 61 additions and 12 deletions
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@ -212,6 +212,7 @@ def finExtractTwo {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
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(Equiv.sumCongr (Equiv.refl (Fin 1)) (finExtractOne j)).trans <|
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(Equiv.sumAssoc (Fin 1) (Fin 1) (Fin n)).symm
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@[simp]
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lemma finExtractTwo_apply_fst {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
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finExtractTwo i j i = Sum.inl (Sum.inl 0) := by
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@ -242,4 +243,10 @@ lemma finExtractTwo_symm_inl_inl_apply {n : ℕ} (i : Fin n.succ.succ) (j : Fin
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rw [finExtractTwo]
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simp
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@[simp]
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lemma finExtractTwo_apply_snd {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
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finExtractTwo i j (i.succAbove j) = Sum.inl (Sum.inr 0) := by
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rw [← Equiv.eq_symm_apply]
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simp
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end HepLean.Fin
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@ -98,22 +98,25 @@ lemma extractOne_homToEquiv {n : ℕ} (i : Fin n.succ.succ)
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Hom.toEquiv (extractOne i σ) = (finExtractOnePerm ((Hom.toEquiv σ).symm i) (Hom.toEquiv σ)) := by
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rfl
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def extractTwo {n : ℕ} (i : Fin n.succ.succ.succ) (j : Fin n.succ.succ)
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{c1 c2 : Fin n.succ.succ.succ → C} (σ : mk c1 ⟶ mk c2) :
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def extractTwo {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ)
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{c1 c2 : Fin n.succ.succ → C} (σ : mk c1 ⟶ mk c2) :
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mk (c1 ∘ Fin.succAbove ((Hom.toEquiv σ).symm i) ∘
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Fin.succAbove (((Hom.toEquiv (extractOne i σ))).symm j)) ⟶
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mk (c2 ∘ Fin.succAbove i ∘ Fin.succAbove j) :=
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equivToHomEq (Equiv.refl _) (by simp) ≫ extractOne j (extractOne i σ) ≫
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equivToHomEq (Equiv.refl _) (by simp)
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match n with
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| 0 => equivToHomEq (Equiv.refl _) (by simp)
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| Nat.succ n =>
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equivToHomEq (Equiv.refl _) (by simp) ≫ extractOne j (extractOne i σ) ≫
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equivToHomEq (Equiv.refl _) (by simp)
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def extractTwoAux {n : ℕ} (i : Fin n.succ.succ.succ) (j : Fin n.succ.succ)
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{c c1 : Fin n.succ.succ.succ → C} (σ : mk c ⟶ mk c1) :
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def extractTwoAux {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ)
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{c c1 : Fin n.succ.succ → C} (σ : mk c ⟶ mk c1) :
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mk ((c ∘ ⇑(finExtractTwo ((Hom.toEquiv σ).symm i) ((Hom.toEquiv (extractOne i σ)).symm j)).symm) ∘ Sum.inr) ⟶
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mk ((c1 ∘ ⇑(finExtractTwo i j).symm) ∘ Sum.inr) :=
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equivToHomEq (Equiv.refl _) (by simp) ≫ extractTwo i j σ ≫ equivToHomEq (Equiv.refl _) (by simp)
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def extractTwoAux' {n : ℕ} (i : Fin n.succ.succ.succ) (j : Fin n.succ.succ)
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{c c1 : Fin n.succ.succ.succ → C} (σ : mk c ⟶ mk c1) :
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def extractTwoAux' {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ)
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{c c1 : Fin n.succ.succ → C} (σ : mk c ⟶ mk c1) :
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mk ((c ∘ ⇑(finExtractTwo ((Hom.toEquiv σ).symm i) ((Hom.toEquiv (extractOne i σ)).symm j)).symm) ∘ Sum.inl) ⟶
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mk ((c1 ∘ ⇑(finExtractTwo i j).symm) ∘ Sum.inl) :=
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equivToHomEq (Equiv.refl _) (by
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@ -138,7 +141,7 @@ def extractTwoAux' {n : ℕ} (i : Fin n.succ.succ.succ) (j : Fin n.succ.succ)
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erw [Equiv.apply_eq_iff_eq]
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exact (Fin.succAbove_ne i j).symm)
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lemma extractTwo_finExtractTwo {n : ℕ} (i : Fin n.succ.succ.succ) (j : Fin n.succ.succ)
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lemma extractTwo_finExtractTwo_succ {n : ℕ} (i : Fin n.succ.succ.succ) (j : Fin n.succ.succ)
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{c c1 : Fin n.succ.succ.succ → C} (σ : mk c ⟶ mk c1) :
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σ ≫ (equivToIso (HepLean.Fin.finExtractTwo i j)).hom ≫ (mkSum (c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom =
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(equivToIso (HepLean.Fin.finExtractTwo ((Hom.toEquiv σ).symm i) (((Hom.toEquiv (extractOne i σ))).symm j))).hom
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@ -218,6 +221,45 @@ lemma extractTwo_finExtractTwo {n : ℕ} (i : Fin n.succ.succ.succ) (j : Fin n.
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(predAboveI ((Hom.toEquiv σ).symm i) ((Hom.toEquiv σ).symm (i.succAbove j))) x hn'
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exact hy
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lemma extractTwo_finExtractTwo {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ)
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{c c1 : Fin n.succ.succ → C} (σ : mk c ⟶ mk c1) :
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σ ≫ (equivToIso (HepLean.Fin.finExtractTwo i j)).hom ≫ (mkSum (c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom =
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(equivToIso (HepLean.Fin.finExtractTwo ((Hom.toEquiv σ).symm i) (((Hom.toEquiv (extractOne i σ))).symm j))).hom
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≫ (mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo ((Hom.toEquiv σ).symm i) (((Hom.toEquiv (extractOne i σ))).symm j)).symm)).hom
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≫ ((extractTwoAux' i j σ) ⊗ (extractTwoAux i j σ)) := by
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match n with
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| 0 =>
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apply IndexNotation.OverColor.Hom.ext
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ext x
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simp [CategoryStruct.comp,extractTwoAux', extractTwoAux, mkSum,equivToIso, Hom.toIso]
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change ((finExtractTwo i j) (σ.hom.left x)) = Sum.map (Equiv.refl _) (Equiv.refl _) _
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simp
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change (finExtractTwo i j) ((Hom.toEquiv σ) x) = ((finExtractTwo ((Hom.toEquiv σ).symm i)
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((finExtractOnePerm ((Hom.toEquiv σ).symm i) (Hom.toEquiv σ)).symm j)) x)
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obtain ⟨k, hk⟩ := (Hom.toEquiv σ).symm.surjective x
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subst hk
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simp
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have hk : k = i ∨ k = i.succAbove j := by
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match i, j, k with
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| (0 : Fin 2), (0 : Fin 1), (0 : Fin 2) => exact Or.intro_left (0 = Fin.succAbove 0 0) rfl
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| (0 : Fin 2), (0 : Fin 1), (1 : Fin 2) => exact Or.inr rfl
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| (1 : Fin 2), (0 : Fin 1), (0 : Fin 2) => exact Or.inr rfl
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| (1 : Fin 2), (0 : Fin 1), (1 : Fin 2) => exact Or.intro_left (1 = Fin.succAbove 1 0) rfl
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rcases hk with hk | hk
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subst hk
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simp
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subst hk
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simp
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rw [← Equiv.symm_apply_eq]
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simp [finExtractOnePerm, finExtractOnPermHom]
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erw [Equiv.apply_symm_apply]
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rw [succsAbove_predAboveI]
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rfl
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simp
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erw [Equiv.apply_eq_iff_eq]
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exact (Fin.succAbove_ne i j).symm
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| Nat.succ n => exact extractTwo_finExtractTwo_succ i j σ
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/-- The isomorphism between a `Fin 1 ⊕ Fin 1 → C` satisfying the condition
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`c (Sum.inr 0) = τ (c (Sum.inl 0))`
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and an object in the image of `contrPair`. -/
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@ -381,6 +381,7 @@ def contrMap {n : ℕ} (c : Fin n.succ.succ → S.C)
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(tensorHom (S.contr.app (Discrete.mk (c i))) (𝟙 _)) ≫
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(MonoidalCategory.leftUnitor _).hom
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/-- Contraction commutes with `S.F.map σ` on removing corresponding indices from `σ`. -/
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lemma contrMap_naturality {n : ℕ} {c c1 : Fin n.succ.succ.succ → S.C}
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{i : Fin n.succ.succ.succ} {j : Fin n.succ.succ} {h : c1 (i.succAbove j) = S.τ (c1 i)}
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(σ : (OverColor.mk c) ⟶ (OverColor.mk c1)) :
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@ -413,9 +414,6 @@ lemma contrMap_naturality {n : ℕ} {c c1 : Fin n.succ.succ.succ → S.C}
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simp only [Nat.succ_eq_add_one, extractOne_homToEquiv, Monoidal.tensorUnit_obj,
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Monoidal.tensorUnit_map, Category.comp_id]
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end TensorStruct
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/-- A syntax tree for tensor expressions. -/
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@ -622,6 +620,8 @@ lemma neg_perm {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
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open OverColor
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/-- Permuting indices, and then contracting is equivalent to contracting and then permuting,
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once care is taking about ensuring one is contracting the same idices. -/
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lemma perm_contr {n : ℕ} {c : Fin n.succ.succ.succ → S.C} {c1 : Fin n.succ.succ.succ → S.C}
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{i : Fin n.succ.succ.succ} {j : Fin n.succ.succ}
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{h : c1 (i.succAbove j) = S.τ (c1 i)} (t : TensorTree S c)
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