refactor: Lint
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jstoobysmith
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3 changed files with 45 additions and 35 deletions
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@ -278,6 +278,12 @@ lemma smul_equiv {T₁ T₂ : 𝓣.TensorIndex} (h : T₁ ≈ T₂) (r : R) : r
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-/
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/-- The condition on tensors with indices for their addition to exists.
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This condition states that the the indices of one tensor are exact permutations of indices
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of another after contraction. This includes the id of the index and the color.
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This condition is general enough to allow addition of e.g. `ψᵤ₁ᵤ₂ + φᵤ₂ᵤ₁`, but
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will NOT allow e.g. `ψᵤ₁ᵤ₂ + φᵘ²ᵤ₁`. -/
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def AddCond (T₁ T₂ : 𝓣.TensorIndex) : Prop :=
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T₁.index.PermContr T₂.index
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@ -304,6 +310,7 @@ lemma rel_left {T₁ T₁' T₂ : 𝓣.TensorIndex} (h : AddCond T₁ T₂) (h'
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lemma rel_right {T₁ T₂ T₂' : 𝓣.TensorIndex} (h : AddCond T₁ T₂) (h' : T₂ ≈ T₂') :
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AddCond T₁ T₂' := h.trans h'.1
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/-- The equivalence between indices after contraction given a `AddCond`. -/
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@[simp]
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def toEquiv {T₁ T₂ : 𝓣.TensorIndex} (h : AddCond T₁ T₂) :
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Fin T₁.contr.index.1.length ≃ Fin T₂.contr.index.1.length := h.to_PermContr.toEquiv
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@ -321,6 +328,7 @@ def add (T₁ T₂ : 𝓣.TensorIndex) (h : AddCond T₁ T₂) :
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index := T₂.index.contr
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tensor := (𝓣.mapIso h.toEquiv h.toEquiv_colorMap T₁.contr.tensor) + T₂.contr.tensor
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/-- Notation for addition of tensor indices. -/
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notation:71 T₁ "+["h"]" T₂:72 => add T₁ T₂ h
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namespace AddCond
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@ -342,11 +350,11 @@ lemma of_add_right {T₁ T₂ T₃ : 𝓣.TensorIndex} {h' : AddCond T₂ T₃}
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AddCond T₁ T₂ := h.of_add_right'.trans h'.symm
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lemma of_add_left {T₁ T₂ T₃ : 𝓣.TensorIndex} {h' : AddCond T₁ T₂}
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(h : AddCond (T₁ +[h'] T₂) T₃) : AddCond T₂ T₃ :=
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(h : AddCond (T₁ +[h'] T₂) T₃) : AddCond T₂ T₃ :=
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(of_add_right' h.symm).symm
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lemma of_add_left' {T₁ T₂ T₃ : 𝓣.TensorIndex} {h' : AddCond T₁ T₂}
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(h : AddCond (T₁ +[h'] T₂) T₃) : AddCond T₁ T₃ :=
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(h : AddCond (T₁ +[h'] T₂) T₃) : AddCond T₁ T₃ :=
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(of_add_right h.symm).symm
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lemma add_left_of_add_right {T₁ T₂ T₃ : 𝓣.TensorIndex} {h' : AddCond T₂ T₃}
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@ -358,7 +366,7 @@ lemma add_right_of_add_left {T₁ T₂ T₃ : 𝓣.TensorIndex} {h' : AddCond T
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(h : AddCond (T₁ +[h'] T₂) T₃) : AddCond T₁ (T₂ +[of_add_left h] T₃) :=
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(add_left (of_add_left h) (of_add_left' h).symm).symm
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lemma add_comm {T₁ T₂ : 𝓣.TensorIndex} (h : AddCond T₁ T₂) :
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lemma add_comm {T₁ T₂ : 𝓣.TensorIndex} (h : AddCond T₁ T₂) :
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AddCond (T₁ +[h] T₂) (T₂ +[h.symm] T₁) := by
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apply add_right
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apply add_left
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@ -368,7 +376,7 @@ end AddCond
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@[simp]
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lemma add_index (T₁ T₂ : 𝓣.TensorIndex) (h : AddCond T₁ T₂) :
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(add T₁ T₂ h).index = T₂.index.contr := rfl
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(add T₁ T₂ h).index = T₂.index.contr := rfl
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@[simp]
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lemma add_tensor (T₁ T₂ : 𝓣.TensorIndex) (h : AddCond T₁ T₂) :
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@ -377,7 +385,7 @@ lemma add_tensor (T₁ T₂ : 𝓣.TensorIndex) (h : AddCond T₁ T₂) :
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/-- Scalar multiplication commutes with addition. -/
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lemma smul_add (r : R) (T₁ T₂ : 𝓣.TensorIndex) (h : AddCond T₁ T₂) :
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r • (T₁ +[h] T₂) = r • T₁ +[h] r • T₂ := by
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r • (T₁ +[h] T₂) = r • T₁ +[h] r • T₂ := by
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refine ext _ _ rfl ?_
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simp [add]
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rw [tensor_eq_of_eq (smul_contr r T₁), tensor_eq_of_eq (smul_contr r T₂)]
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@ -391,39 +399,20 @@ lemma add_hasNoContr (T₁ T₂ : 𝓣.TensorIndex) (h : AddCond T₁ T₂) :
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@[simp]
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lemma contr_add (T₁ T₂ : 𝓣.TensorIndex) (h : AddCond T₁ T₂) :
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(T₁ +[h] T₂).contr = T₁ +[h] T₂ :=
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contr_of_hasNoContr (T₁ +[h] T₂) (add_hasNoContr T₁ T₂ h)
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contr_of_hasNoContr (T₁ +[h] T₂) (add_hasNoContr T₁ T₂ h)
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@[simp]
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lemma contr_add_tensor (T₁ T₂ : 𝓣.TensorIndex) (h : AddCond T₁ T₂) :
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(T₁ +[h] T₂).contr.tensor =
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𝓣.mapIso (Fin.castOrderIso (by rw [index_eq_of_eq (contr_add T₁ T₂ h)])).toEquiv
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(index_eq_colorMap_eq (index_eq_of_eq (contr_add T₁ T₂ h))) (T₁ +[h] T₂).tensor :=
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(index_eq_colorMap_eq (index_eq_of_eq (contr_add T₁ T₂ h))) (T₁ +[h] T₂).tensor :=
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tensor_eq_of_eq (contr_add T₁ T₂ h)
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open AddCond in
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lemma add_assoc' {T₁ T₂ T₃ : 𝓣.TensorIndex} {h' : AddCond T₂ T₃} (h : AddCond T₁ (T₂ +[h'] T₃)) :
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T₁ +[h] (T₂ +[h'] T₃) = T₁ +[h'.of_add_right h] T₂ +[h'.add_left_of_add_right h] T₃ := by
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refine ext _ _ ?_ ?_
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simp
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simp only [add_index, add_tensor, contr_index, toEquiv, contr_add_tensor, map_add, mapIso_mapIso]
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rw [_root_.add_assoc]
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congr
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rw [← PermContr.toEquiv_contr_eq, ← PermContr.toEquiv_contr_eq]
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rw [PermContr.toEquiv_trans, PermContr.toEquiv_trans, PermContr.toEquiv_trans]
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simp only [IndexListColor.contr_contr]
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simp only [IndexListColor.contr_contr]
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rw [← PermContr.toEquiv_contr_eq, PermContr.toEquiv_trans]
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simp only [IndexListColor.contr_contr]
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open AddCond in
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lemma add_assoc {T₁ T₂ T₃ : 𝓣.TensorIndex} {h' : AddCond T₁ T₂} (h : AddCond (T₁ +[h'] T₂) T₃) :
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T₁ +[h'] T₂ +[h] T₃ = T₁ +[h'.add_right_of_add_left h] (T₂ +[h'.of_add_left h] T₃) := by
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rw [add_assoc']
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lemma add_comm {T₁ T₂ : 𝓣.TensorIndex} (h : AddCond T₁ T₂) : T₁ +[h] T₂ ≈ T₂ +[h.symm] T₁ := by
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apply And.intro h.add_comm
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intro h
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simp
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simp only [contr_index, add_index, contr_add_tensor, add_tensor, AddCond.toEquiv, map_add,
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mapIso_mapIso]
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rw [_root_.add_comm]
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congr 1
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all_goals
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@ -440,22 +429,43 @@ lemma add_rel_left {T₁ T₁' T₂ : 𝓣.TensorIndex} (h : AddCond T₁ T₂)
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T₁ +[h] T₂ ≈ T₁' +[h.rel_left h'] T₂ := by
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apply And.intro (PermContr.refl _)
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intro h
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simp
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simp only [contr_index, add_index, contr_add_tensor, add_tensor, toEquiv, map_add, mapIso_mapIso,
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PermContr.toEquiv_refl, Equiv.refl_symm, mapIso_refl, LinearEquiv.refl_apply]
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congr 1
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rw [h'.to_eq]
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simp
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simp only [mapIso_mapIso]
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congr 1
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congr 1
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rw [PermContr.toEquiv_symm, ← PermContr.toEquiv_contr_eq, PermContr.toEquiv_trans,
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PermContr.toEquiv_trans, PermContr.toEquiv_trans]
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simp only [IndexListColor.contr_contr]
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/-! TODO: Show that contr add equals add. -/
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/-! TODO: Show that add is associative. -/
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open AddCond in
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lemma add_rel_right {T₁ T₂ T₂' : 𝓣.TensorIndex} (h : AddCond T₁ T₂) (h' : T₂ ≈ T₂') :
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T₁ +[h] T₂ ≈ T₁ +[h.rel_right h'] T₂' :=
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(add_comm _).trans ((add_rel_left _ h').trans (add_comm _))
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open AddCond in
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lemma add_assoc' {T₁ T₂ T₃ : 𝓣.TensorIndex} {h' : AddCond T₂ T₃} (h : AddCond T₁ (T₂ +[h'] T₃)) :
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T₁ +[h] (T₂ +[h'] T₃) = T₁ +[h'.of_add_right h] T₂ +[h'.add_left_of_add_right h] T₃ := by
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refine ext _ _ ?_ ?_
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simp only [add_index, IndexListColor.contr_contr]
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simp only [add_index, add_tensor, contr_index, toEquiv, contr_add_tensor, map_add, mapIso_mapIso]
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rw [_root_.add_assoc]
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congr
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rw [← PermContr.toEquiv_contr_eq, ← PermContr.toEquiv_contr_eq]
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rw [PermContr.toEquiv_trans, PermContr.toEquiv_trans, PermContr.toEquiv_trans]
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simp only [IndexListColor.contr_contr]
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simp only [IndexListColor.contr_contr]
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rw [← PermContr.toEquiv_contr_eq, PermContr.toEquiv_trans]
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simp only [IndexListColor.contr_contr]
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open AddCond in
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lemma add_assoc {T₁ T₂ T₃ : 𝓣.TensorIndex} {h' : AddCond T₁ T₂} (h : AddCond (T₁ +[h'] T₂) T₃) :
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T₁ +[h'] T₂ +[h] T₃ = T₁ +[h'.add_right_of_add_left h] (T₂ +[h'.of_add_left h] T₃) := by
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rw [add_assoc']
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/-! TODO: Show that the product is well defined with respect to Rel. -/
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/-! TODO : Show that addition is well defined with respect to Rel. -/
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end TensorIndex
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end
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