refactor: Index notation

HepLean/SpaceTime/LorentzTensor/IndexNotation/Indices/Relations.lean

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+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.SpaceTime.LorentzTensor.IndexNotation.Indices.Color
+import HepLean.SpaceTime.LorentzTensor.IndexNotation.Indices.UniqueDualInOther
+import HepLean.SpaceTime.LorentzTensor.Basic
+import Init.Data.List.Lemmas
+/-!
+
+# Index lists with color conditions
+
+Here we consider `IndexListColor` which is a subtype of all lists of indices
+on those where the elements to be contracted have consistent colors with respect to
+a Tensor Color structure.
+
+-/
+
+namespace IndexNotation
+
+
+namespace ColorIndexList
+
+variable {𝓒 : TensorColor} [IndexNotation 𝓒.Color] [Fintype 𝓒.Color] [DecidableEq 𝓒.Color]
+
+variable (l l' : ColorIndexList 𝓒)
+
+/-!
+
+## Reindexing
+
+-/
+
+def Reindexing : Prop := l.length = l'.length ∧
+  ∀ (h : l.length = l'.length), l.colorMap = l'.colorMap ∘ Fin.cast h ∧
+    Option.map (Fin.cast h) ∘ l.getDual? = l'.getDual? ∘ Fin.cast h
+
+/-!
+
+## Permutation
+
+Test whether two `ColorIndexList`s are permutations of each other.
+To prevent choice problems, this has to be done after contraction.
+
+-/
+
+def ContrPerm : Prop := l.contr.length = l'.contr.length ∧
+  l.contr.withUniqueDualInOther l'.contr.toIndexList = Finset.univ ∧
+  l'.contr.colorMap ∘ Subtype.val ∘ (l.contr.getDualInOtherEquiv l'.contr.toIndexList)
+  = l.contr.colorMap ∘ Subtype.val
+
+
+
+end ColorIndexList
+
+end IndexNotation