105 lines
3.6 KiB
Text
105 lines
3.6 KiB
Text
/-
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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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## Products and contractions
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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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namespace TensorTree
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variable {S : TensorSpecies}
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namespace ContrPair
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variable {n n1 : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n1 → S.C} (q : ContrPair c)
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/-!
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## Left contractions.
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-/
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/-- An equivalence needed to perform contraction. For specified `n` and `n1`
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this reduces to an identity. -/
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def leftContrEquivSuccSucc : Fin (n.succ.succ + n1) ≃ Fin ((n + n1).succ.succ) :=
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(Fin.castOrderIso (by omega)).toEquiv
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/-- An equivalence needed to perform contraction. For specified `n` and `n1`
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this reduces to an identity. -/
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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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def leftContrI (n1 : ℕ): Fin ((n + n1).succ.succ) := leftContrEquivSuccSucc <| Fin.castAdd n1 q.i
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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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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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Fin.castOrderIso_apply, leftContrEquivSuccSucc, Fin.coe_cast, Fin.coe_castAdd]
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split_ifs
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<;> rename_i h1 h2
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<;> rw [Fin.lt_def] at h1 h2
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· simp only [Fin.coe_castSucc, Fin.coe_cast, Fin.coe_castAdd]
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· simp_all only [Fin.coe_castSucc, Fin.coe_cast, Fin.coe_castAdd, not_true_eq_false]
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· simp_all only [Fin.coe_castSucc, Fin.coe_cast, Fin.coe_castAdd, not_lt, Fin.val_succ,
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add_right_eq_self, one_ne_zero]
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omega
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· simp only [Fin.val_succ, Fin.coe_cast, Fin.coe_castAdd]
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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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j := q.leftContrJ n1
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h := by
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simp only [Nat.succ_eq_add_one, Equiv.toFun_as_coe, leftContrJ_succAbove_leftContrI,
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Function.comp_apply, Equiv.symm_apply_apply, finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl]
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simpa only [leftContrI, Nat.succ_eq_add_one, Equiv.symm_apply_apply,
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finSumFinEquiv_symm_apply_castAdd, Sum.elim_inl] using q.h
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lemma leftContr_map_eq : ((Sum.elim c (OverColor.mk c1).hom ∘ finSumFinEquiv.symm.toFun) ∘ ⇑leftContrEquivSuccSucc.symm) ∘
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(q.leftContr (c1 := c1)).i.succAbove ∘ (q.leftContr (c1 := c1)).j.succAbove =
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Sum.elim (OverColor.mk (c ∘ q.i.succAbove ∘ q.j.succAbove)).hom (OverColor.mk c1).hom ∘
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⇑finSumFinEquiv.symm := by
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funext x
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simp
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sorry
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lemma contrMap_prod :
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(q.contrMap ▷ S.F.obj (OverColor.mk c1)) ≫ (S.F.μ _ ((OverColor.mk c1))) ≫
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S.F.map (OverColor.equivToIso finSumFinEquiv).hom =
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(S.F.μ ((OverColor.mk c)) ((OverColor.mk c1))) ≫
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S.F.map (OverColor.equivToIso finSumFinEquiv).hom ≫
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S.F.map (OverColor.equivToIso leftContrEquivSuccSucc).hom ≫ q.leftContr.contrMap
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≫ S.F.map (OverColor.mkIso (q.leftContr_map_eq)).hom := by
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sorry
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/-!
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## Right contractions.
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-/
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end ContrPair
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theorem contr_prod {n n1 : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n1 → S.C} {i : Fin n.succ.succ}
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{j : Fin n.succ} (hij : c (i.succAbove j) = S.τ (c i))
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(t : TensorTree S c) (t1 : TensorTree S c1) :
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(prod t t1).tensor = sorry :=by
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sorry
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end TensorTree
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