127 lines
3.1 KiB
Text
127 lines
3.1 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.SpaceTime.LorentzTensor.IndexNotation.GetDual
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import Mathlib.Algebra.Order.Ring.Nat
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import Mathlib.Data.Finset.Sort
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/-!
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# With dual
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We define the finite sets of indices in an index list which have a dual
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-/
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namespace IndexNotation
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namespace IndexList
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variable {X : Type} [IndexNotation X] [Fintype X] [DecidableEq X]
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variable (l l2 l3 : IndexList X)
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/-!
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## Finsets on which getDual? and getDualInOther? are some.
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-/
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/-- The finite set of indices of an index list which have a dual in that index list. -/
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def withDual : Finset (Fin l.length) :=
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Finset.filter (fun i => (l.getDual? i).isSome) Finset.univ
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/-- The finite set of indices of an index list which have a dual in another provided index list. -/
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def withDualInOther : Finset (Fin l.length) :=
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Finset.filter (fun i => (l.getDualInOther? l2 i).isSome) Finset.univ
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/-!
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## Basic properties of withDual
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-/
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@[simp]
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lemma withDual_isSome (i : l.withDual) : (l.getDual? i).isSome := by
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simpa [withDual] using i.2
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@[simp]
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lemma mem_withDual_iff_isSome (i : Fin l.length) : i ∈ l.withDual ↔ (l.getDual? i).isSome := by
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simp [withDual]
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lemma not_mem_withDual_iff_isNone (i : Fin l.length) :
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i ∉ l.withDual ↔ (l.getDual? i).isNone := by
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simp only [mem_withDual_iff_isSome, Bool.not_eq_true, Option.not_isSome,
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Option.isNone_iff_eq_none]
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lemma mem_withDual_iff_exists : i ∈ l.withDual ↔ ∃ j, l.AreDualInSelf i j := by
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simp [withDual, Finset.mem_filter, Finset.mem_univ, getDual?]
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exact Fin.isSome_find_iff
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/-!
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## Basic properties of withDualInOther
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-/
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@[simp]
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lemma mem_withInDualOther_iff_isSome (i : Fin l.length) :
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i ∈ l.withDualInOther l2 ↔ (l.getDualInOther? l2 i).isSome := by
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simp only [withDualInOther, getDualInOther?, Finset.mem_filter, Finset.mem_univ, true_and]
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lemma mem_withInDualOther_iff_exists :
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i ∈ l.withDualInOther l2 ↔ ∃ (j : Fin l2.length), l.AreDualInOther l2 i j := by
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simp [withDualInOther, Finset.mem_filter, Finset.mem_univ, getDualInOther?]
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exact Fin.isSome_find_iff
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/-!
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## Append properties of withDual
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-/
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lemma withDual_append_eq_disjSum : (l ++ l2).withDual =
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Equiv.finsetCongr appendEquiv
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((l.withDual ∪ l.withDualInOther l2).disjSum
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(l2.withDual ∪ l2.withDualInOther l)) := by
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ext i
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obtain ⟨k, hk⟩ := appendEquiv.surjective i
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subst hk
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match k with
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| Sum.inl k =>
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simp
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| Sum.inr k =>
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simp
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/-!
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## Append properties of withDualInOther
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-/
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lemma append_withDualInOther_eq :
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(l ++ l2).withDualInOther l3 =
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Equiv.finsetCongr appendEquiv ((l.withDualInOther l3).disjSum (l2.withDualInOther l3)) := by
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rw [Finset.ext_iff]
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intro i
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obtain ⟨k, hk⟩ := appendEquiv.surjective i
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subst hk
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match k with
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| Sum.inl k =>
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simp
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| Sum.inr k =>
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simp
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lemma withDualInOther_append_eq : l.withDualInOther (l2 ++ l3) =
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l.withDualInOther l2 ∪ l.withDualInOther l3 := by
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ext i
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simp
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end IndexList
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end IndexNotation
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