82 lines
2.6 KiB
Text
82 lines
2.6 KiB
Text
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.Tree.Basic
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/-!
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## Basic node identities
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This file contains the basic node identities for tensor trees.
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More compliciated identities appear in there own files.
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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open OverColor
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open HepLean.Fin
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open TensorProduct
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namespace TensorTree
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variable {S : TensorStruct}
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/-!
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## Equality of constructors.
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-/
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informal_lemma constVecNode_eq_vecNode where
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math :≈ "A constVecNode has equal tensor to the vecNode with the map evaluated at 1."
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deps :≈ [``constVecNode, ``vecNode]
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informal_lemma constTwoNode_eq_twoNode where
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math :≈ "A constTwoNode has equal tensor to the twoNode with the map evaluated at 1."
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deps :≈ [``constTwoNode, ``twoNode]
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informal_lemma constThreeNode_eq_threeNode where
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math :≈ "A constThreeNode has equal tensor to the threeNode with the map evaluated at 1."
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deps :≈ [``constThreeNode, ``threeNode]
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/-!
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## Negation
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-/
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@[simp]
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lemma neg_neg (t : TensorTree S c) : (neg (neg t)).tensor = t.tensor := by
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simp only [neg_tensor, _root_.neg_neg]
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@[simp]
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lemma neg_fst_prod {c1 : Fin n → S.C} {c2 : Fin m → S.C} (T1 : TensorTree S c1)
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(T2 : TensorTree S c2) :
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(prod (neg T1) T2).tensor = (neg (prod T1 T2)).tensor := by
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simp only [prod_tensor, Functor.id_obj, Action.instMonoidalCategory_tensorObj_V,
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Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, neg_tensor, neg_tmul, map_neg]
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@[simp]
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lemma neg_snd_prod {c1 : Fin n → S.C} {c2 : Fin m → S.C} (T1 : TensorTree S c1)
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(T2 : TensorTree S c2) :
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(prod T1 (neg T2)).tensor = (neg (prod T1 T2)).tensor := by
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simp only [prod_tensor, Functor.id_obj, Action.instMonoidalCategory_tensorObj_V,
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Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, neg_tensor, tmul_neg, map_neg]
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@[simp]
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lemma neg_contr {n : ℕ} {c : Fin n.succ.succ → S.C} {i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = S.τ (c i)}
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(t : TensorTree S c) : (contr i j h (neg t)).tensor = (neg (contr i j h t)).tensor := by
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simp only [Nat.succ_eq_add_one, contr_tensor, neg_tensor, map_neg]
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lemma neg_perm {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
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(σ : (OverColor.mk c) ⟶ (OverColor.mk c1)) (t : TensorTree S c) :
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(perm σ (neg t)).tensor = (neg (perm σ t)).tensor := by
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simp only [perm_tensor, neg_tensor, map_neg]
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end TensorTree
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