feat: Some simple extensions of lemmas
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jstoobysmith
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a8e4562363
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7 changed files with 150 additions and 11 deletions
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@ -279,6 +279,16 @@ def finExtractOnePerm (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fin n.succ
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right_inv x := by
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simpa using congrFun (finExtractOnPermHom_inv (σ i) σ.symm) x
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@[simp]
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lemma finExtractOnePerm_apply (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fin n.succ.succ)
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(x : Fin n.succ) : finExtractOnePerm i σ x = predAboveI (σ i)
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(σ ((finExtractOne i).symm (Sum.inr x))) := rfl
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@[simp]
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lemma finExtractOnePerm_symm_apply (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fin n.succ.succ)
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(x : Fin n.succ) : (finExtractOnePerm i σ).symm x = predAboveI (σ.symm (σ i))
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(σ.symm ((finExtractOne (σ i)).symm (Sum.inr x))) := rfl
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/-- The equivalence of types `Fin n.succ.succ ≃ (Fin 1 ⊕ Fin 1) ⊕ Fin n` extracting
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the `i` and `(i.succAbove j)`. -/
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def finExtractTwo {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
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@ -331,4 +341,19 @@ def finMapToEquiv (f1 : Fin n → Fin m) (f2 : Fin m → Fin n)
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left_inv := h'
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right_inv := h
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@[simp]
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lemma finMapToEquiv_apply {f1 : Fin n → Fin m} {f2 : Fin m → Fin n}
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{h : ∀ x, f1 (f2 x) = x} {h' : ∀ x, f2 (f1 x) = x} (x : Fin n) :
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finMapToEquiv f1 f2 h h' x = f1 x := rfl
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@[simp]
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lemma finMapToEquiv_symm_apply {f1 : Fin n → Fin m} {f2 : Fin m → Fin n}
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{h : ∀ x, f1 (f2 x) = x} {h' : ∀ x, f2 (f1 x) = x} (x : Fin m) :
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(finMapToEquiv f1 f2 h h').symm x = f2 x := rfl
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lemma finMapToEquiv_symm_eq {f1 : Fin n → Fin m} {f2 : Fin m → Fin n}
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{h : ∀ x, f1 (f2 x) = x} {h' : ∀ x, f2 (f1 x) = x} :
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(finMapToEquiv f1 f2 h h').symm = finMapToEquiv f2 f1 h' h := by
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rfl
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end HepLean.Fin
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@ -100,6 +100,9 @@ def toIso (m : f ⟶ g) : f ≅ g := {
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simp only [CategoryStruct.comp, Iso.symm_self_id, Iso.refl_hom, Over.id_left, types_id_apply]
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rfl}
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@[simp]
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lemma hom_comp (m : f ⟶ g) (n : g ⟶ h) : (m ≫ n).hom = m.hom ≫ n.hom := by rfl
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end Hom
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section monoidal
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@ -273,12 +276,32 @@ def mk (f : X → C) : OverColor C := Over.mk f
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lemma mk_hom (f : X → C) : (mk f).hom = f := rfl
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open MonoidalCategory
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@[simp]
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lemma mk_left (f : X → C) : (mk f).left = X := rfl
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lemma Hom.fin_ext {n : ℕ} {f g : Fin n → C} (σ σ' : OverColor.mk f ⟶ OverColor.mk g)
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(h : ∀ (i : Fin n), σ.hom.left i = σ'.hom.left i) : σ = σ' := by
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apply Hom.ext
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ext i
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apply h
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@[simp]
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lemma β_hom_toEquiv (f : X → C) (g : Y → C) :
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Hom.toEquiv (β_ (OverColor.mk f) (OverColor.mk g)).hom = Equiv.sumComm X Y := by
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rfl
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@[simp]
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lemma β_inv_toEquiv (f : X → C) (g : Y → C) :
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Hom.toEquiv (β_ (OverColor.mk f) (OverColor.mk g)).inv = Equiv.sumComm Y X := by
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rfl
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@[simp]
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lemma whiskeringLeft_toEquiv (f : X → C) (g : Y → C) (h : Z → C) (σ : OverColor.mk f ⟶ OverColor.mk g) :
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Hom.toEquiv (OverColor.mk h ◁ σ) = (Equiv.refl Z).sumCongr (Hom.toEquiv σ) := by
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simp [MonoidalCategory.whiskerLeft]
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rfl
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end OverColor
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end IndexNotation
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@ -32,6 +32,11 @@ lemma equivToIso_homToEquiv {c : X → C} (e : X ≃ Y) :
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Hom.toEquiv (equivToIso (c := c) e).hom = e := by
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rfl
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@[simp]
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lemma equivToIso_inv_homToEquiv {c : X → C} (e : X ≃ Y) :
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Hom.toEquiv (equivToIso (c := c) e).inv = e.symm := by
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rfl
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/-- The homomorphism between `c : X → C` and `c ∘ e.symm` as objects in `OverColor C` for an
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equivalence `e`. -/
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def equivToHom {c : X → C} (e : X ≃ Y) : mk c ⟶ mk (c ∘ e.symm) :=
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@ -65,6 +70,17 @@ lemma mkIso_refl_hom {c : X → C} : (mkIso (by rfl : c =c)).hom = 𝟙 _ := by
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rw [mkIso]
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rfl
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lemma mkIso_hom_hom_left {c1 c2 : X → C} (h : c1 = c2) : (mkIso h).hom.hom.left =
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(Equiv.refl X).toFun := by
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rw [mkIso]
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rfl
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@[simp]
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lemma mkIso_hom_hom_left_apply {c1 c2 : X → C} (h : c1 = c2) (x : X) :
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(mkIso h).hom.hom.left x = x := by
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rw [mkIso_hom_hom_left]
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rfl
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@[simp]
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lemma equivToIso_mkIso_hom {c1 c2 : X → C} (h : c1 = c2) :
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Hom.toEquiv (mkIso h).hom = Equiv.refl _ := by
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@ -81,6 +97,17 @@ def equivToHomEq {c : X → C} {c1 : Y → C} (e : X ≃ Y)
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(h : ∀ x, c1 x = (c ∘ e.symm) x := by decide) : mk c ⟶ mk c1 :=
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(equivToHom e).trans (mkIso (funext fun x => (h x).symm)).hom
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@[simp]
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lemma equivToHomEq_hom_left {c : X → C} {c1 : Y → C} (e : X ≃ Y)
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(h : ∀ x, c1 x = (c ∘ e.symm) x := by decide) : (equivToHomEq e h).hom.left =
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e.toFun := by
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rfl
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@[simp]
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lemma equivToHomEq_toEquiv {c : X → C} {c1 : Y → C} (e : X ≃ Y)
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(h : ∀ x, c1 x = (c ∘ e.symm) x := by decide) : Hom.toEquiv (equivToHomEq e h) = e := by
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rfl
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/-- The isomorphism splitting a `mk c` for `Fin 2 → C` into the tensor product of
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the `Fin 1 → C` maps defined by `c 0` and `c 1`. -/
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def fin2Iso {c : Fin 2 → C} : mk c ≅ mk ![c 0] ⊗ mk ![c 1] := by
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@ -163,6 +190,14 @@ def extractTwo {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ)
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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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@[simp]
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lemma extractTwo_hom_left_apply {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) (x : Fin n.succ) :
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(extractTwo i j σ).hom.left x = (finExtractOnePerm ((finExtractOnePerm ((Hom.toEquiv σ).symm i) (Hom.toEquiv σ)).symm j)
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(finExtractOnePerm ((Hom.toEquiv σ).symm i) (Hom.toEquiv σ))) x := by
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simp only [extractTwo, extractOne]
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rfl
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/-- Removes a given `i : Fin n.succ.succ` and `j : Fin n.succ` from a morphism in `OverColor C`.
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This is from and to different (by equivalent) objects to `extractTwo`. -/
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def extractTwoAux {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ)
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@ -231,7 +266,7 @@ lemma extractTwo_finExtractTwo_succ {n : ℕ} (i : Fin n.succ.succ.succ) (j : Fi
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simp only [Nat.succ_eq_add_one, Equiv.apply_symm_apply]
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match k with
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| Sum.inl (Sum.inl 0) =>
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simp
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simp [-OverColor.mk_left]
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| Sum.inl (Sum.inr 0) =>
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simp only [Fin.isValue, finExtractTwo_symm_inl_inr_apply, Sum.map_inl, id_eq]
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have h1 : ((Hom.toEquiv σ) (Fin.succAbove
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@ -5,6 +5,10 @@ Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.TensorSpecies.UnitTensor
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import HepLean.Tensors.TensorSpecies.ContractLemmas
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import HepLean.Tensors.Tree.NodeIdentities.ProdComm
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import HepLean.Tensors.Tree.NodeIdentities.PermProd
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import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
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import HepLean.Tensors.Tree.NodeIdentities.PermContr
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/-!
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## Metrics in tensor trees
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@ -28,6 +32,14 @@ def metricTensor (S : TensorSpecies) (c : S.C) : S.F.obj (OverColor.mk ![c, c])
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variable {S : TensorSpecies}
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lemma metricTensor_congr {c c' : S.C} (h : c = c') : {S.metricTensor c | μ ν}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl ))
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{S.metricTensor c' | μ ν}ᵀ).tensor := by
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subst h
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change _ = (S.F.map (𝟙 _)).hom (S.metricTensor c)
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simp
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lemma pairIsoSep_inv_metricTensor (c : S.C) :
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(Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor c) =
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(S.metric.app (Discrete.mk c)).hom (1 : S.k) := by
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@ -48,12 +60,9 @@ lemma contr_metric_braid_unit (c : S.C) : (((S.FD.obj (Discrete.mk c)) ◁
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(OverColor.Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor (S.τ c)))))))) =
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(β_ (S.FD.obj (Discrete.mk (S.τ c))) (S.FD.obj (Discrete.mk c))).hom.hom
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((S.unit.app (Discrete.mk c)).hom (1 : S.k)) := by
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have hx : Function.Injective (β_ (S.FD.obj (Discrete.mk c)) (S.FD.obj (Discrete.mk (S.τ c))) ).hom.hom := by
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change Function.Injective (β_ (S.FD.obj (Discrete.mk c)).V (S.FD.obj (Discrete.mk (S.τ c))).V ).hom
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exact (β_ (S.FD.obj (Discrete.mk c)).V (S.FD.obj (Discrete.mk (S.τ c))).V ).toLinearEquiv.toEquiv.injective
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apply hx
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apply (β_ _ _).toLinearEquiv.toEquiv.injective
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rw [pairIsoSep_inv_metricTensor, pairIsoSep_inv_metricTensor]
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rw [S.contr_metric c]
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erw [S.contr_metric c]
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change _ = (β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).inv.hom
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((β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).hom.hom _)
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rw [Discrete.rep_iso_inv_hom_apply]
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@ -66,16 +75,53 @@ lemma metricTensor_contr_dual_metricTensor_perm_cond (c : S.C) : ∀ (x : Fin (N
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· rfl
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· rfl
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/-- The contraction of a metric tensor with its dual gives the unit. -/
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/-- The contraction of a metric tensor with its dual via the inner indices gives the unit. -/
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lemma metricTensor_contr_dual_metricTensor_eq_unit (c : S.C) :
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{S.metricTensor c | μ ν ⊗ S.metricTensor (S.τ c) | ν ρ}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
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(metricTensor_contr_dual_metricTensor_perm_cond c)) {S.unitTensor c | μ ρ}ᵀ).tensor := by
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{S.metricTensor c | μ ν ⊗ S.metricTensor (S.τ c) | ν ρ}ᵀ.tensor = ({S.unitTensor c | μ ρ}ᵀ |>
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perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
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(metricTensor_contr_dual_metricTensor_perm_cond c))).tensor := by
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rw [contr_two_two_inner, contr_metric_braid_unit, Discrete.pairIsoSep_β]
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change (S.F.map _ ≫ S.F.map _ ).hom _ = _
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rw [← S.F.map_comp]
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rfl
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/-- The contraction of a metric tensor with its dual via the outer indices gives the unit. -/
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lemma metricTensor_contr_dual_metricTensor_outer_eq_unit (c : S.C) :
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{S.metricTensor c | ν μ ⊗ S.metricTensor (S.τ c) | ρ ν}ᵀ.tensor = ({S.unitTensor c | μ ρ}ᵀ |>
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perm (OverColor.equivToHomEq
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(finMapToEquiv ![1, 0] ![1, 0]) (fun x => by fin_cases x <;> rfl))).tensor := by
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conv_lhs =>
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rw [contr_tensor_eq <| prod_tensor_eq_fst <| metricTensor_congr (S.τ_involution c).symm]
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rw [contr_tensor_eq <| prod_comm _ _ _ _]
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rw [perm_contr_congr 2 1 (by rfl) (by rfl)]
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rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_right _ _ _ _]
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rw [perm_tensor_eq <| perm_contr_congr 2 1 (by rfl) (by rfl)]
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rw [perm_perm]
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rw [perm_tensor_eq <| contr_swap _ _]
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rw [perm_perm]
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erw [perm_tensor_eq <| metricTensor_contr_dual_metricTensor_eq_unit _]
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rw [perm_perm]
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rw [perm_tensor_eq <| dual_unitTensor_eq_perm _]
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rw [perm_perm]
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apply perm_congr _ rfl
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apply OverColor.Hom.fin_ext
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intro i
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simp only [Functor.id_obj, mk_hom, Function.comp_apply, Equiv.refl_symm, Equiv.coe_refl, id_eq,
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Fin.zero_eta, Matrix.cons_val_zero, List.pmap.eq_1, ContrPair.contrSwapHom,
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extractOne_homToEquiv, Category.assoc, Hom.hom_comp, Over.comp_left, equivToHomEq_hom_left,
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Equiv.toFun_as_coe, types_comp_apply, finMapToEquiv_apply, mkIso_hom_hom_left_apply]
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rw [extractTwo_hom_left_apply]
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simp only [mk_left, braidPerm_toEquiv, Equiv.symm_trans_apply, Equiv.symm_symm,
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Equiv.sumComm_symm, Equiv.sumComm_apply, finExtractOnePerm_symm_apply, Equiv.trans_apply,
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Equiv.symm_apply_apply, Sum.swap_swap, Equiv.apply_symm_apply, finExtractOne_symm_inr_apply,
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Fin.zero_succAbove, List.pmap.eq_2, Fin.mk_one, List.pmap.eq_1, Matrix.cons_val_one,
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Matrix.head_cons, extractTwo_hom_left_apply, permProdRight_toEquiv, equivToHomEq_toEquiv,
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Equiv.sumCongr_refl, Equiv.refl_trans, Equiv.symm_trans_self, Equiv.refl_symm, Equiv.refl_apply,
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predAboveI_succAbove, finExtractOnePerm_apply]
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fin_cases i
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· decide
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· decide
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end TensorSpecies
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end
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@ -140,7 +140,7 @@ lemma perm_eq_iff_eq_perm {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
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apply Hom.ext
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ext x
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change (Hom.toEquiv σ) ((Hom.toEquiv σ).symm x) = x
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simp
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simp [-OverColor.mk_left]
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rw [h1]
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simp
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@ -35,6 +35,11 @@ def permProdLeft := (equivToIso finSumFinEquiv).inv ≫ σ ▷ OverColor.mk c2
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def permProdRight := (equivToIso finSumFinEquiv).inv ≫ OverColor.mk c2 ◁ σ
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≫ (equivToIso finSumFinEquiv).hom
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@[simp]
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lemma permProdRight_toEquiv : Hom.toEquiv (permProdRight c2 σ) = finSumFinEquiv.symm.trans
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(((Equiv.refl (Fin n2)).sumCongr (Hom.toEquiv σ)).trans finSumFinEquiv) := by
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simp [permProdRight]
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/-- When a `prod` acts on a `perm` node in the left entry, the `perm` node can be moved through
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the `prod` node via right-whiskering. -/
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theorem prod_perm_left (t : TensorTree S c) (t2 : TensorTree S c2) :
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@ -31,6 +31,11 @@ def braidPerm : OverColor.mk (Sum.elim c2 c ∘ ⇑finSumFinEquiv.symm) ⟶
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(β_ (OverColor.mk c2) (OverColor.mk c)).hom
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≫ (equivToIso finSumFinEquiv).hom
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@[simp]
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lemma braidPerm_toEquiv : Hom.toEquiv (braidPerm c c2) = finSumFinEquiv.symm.trans
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((Equiv.sumComm (Fin n2) (Fin n)).trans finSumFinEquiv) := by
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simp [braidPerm]
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lemma finSumFinEquiv_comp_braidPerm :
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(equivToIso finSumFinEquiv).hom ≫ braidPerm c c2 =
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(β_ (OverColor.mk c2) (OverColor.mk c)).hom
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