refactor: Index notation

HepLean/SpaceTime/LorentzTensor/IndexNotation/TensorIndex.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.IndexListColor
+import HepLean.SpaceTime.LorentzTensor.Basic
+/-!
+
+# The structure of a tensor with a string of indices
+
+-/
+
+namespace TensorStructure
+open TensorColor
+open IndexNotation
+
+variable {R : Type} [CommSemiring R] (𝓣 : TensorStructure R)
+
+variable {d : ℕ} {X Y Y' Z W : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
+  [Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
+  {cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color}
+  {cW : W → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν η : 𝓣.Color}
+
+variable [IndexNotation 𝓣.Color] [Fintype 𝓣.Color] [DecidableEq 𝓣.Color]
+
+structure TensorIndex (cn : Fin n → 𝓣.Color) where
+  tensor : 𝓣.Tensor cn
+  index : IndexListColor 𝓣.toTensorColor
+  nat_eq : n = index.1.length
+  quot_eq : 𝓣.colorQuot ∘ index.1.colorMap ∘ Fin.cast nat_eq = 𝓣.colorQuot ∘ cn
+
+namespace TensorIndex
+
+variable {𝓣 : TensorStructure R} [IndexNotation 𝓣.Color] [Fintype 𝓣.Color] [DecidableEq 𝓣.Color]
+variable {n m : ℕ} {cn : Fin n → 𝓣.Color} {cm : Fin m → 𝓣.Color} (T : TensorIndex 𝓣 cn)
+section noncomputable
+
+def smul (r : R) : TensorIndex 𝓣 cn := ⟨r • T.tensor, T.index, T.nat_eq, T.quot_eq⟩
+
+end
+
+end TensorIndex
+
+end TensorStructure