refactor: Move tensors & index notation

HepLean/Tensors/EinsteinNotation/Lemmas.lean

@@ -0,0 +1,41 @@
+/-
+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.Tensors.EinsteinNotation.IndexNotation
+/-!
+
+# Lemmas regarding Einstein tensors
+
+This file is currently a stub.
+
+-/
+namespace einsteinTensor
+
+open einsteinTensorColor
+open IndexNotation IndexString
+open TensorStructure TensorIndex
+
+variable {R : Type} [CommSemiring R] {n m : ℕ}
+/-
+lemma swap_eq_transpose (T : (einsteinTensor R n).Tensor ![Unit.unit, Unit.unit]) :
+    (T|"ᵢ₁ᵢ₂") ≈ ((toMatrix.symm (toMatrix T).transpose)|"ᵢ₂ᵢ₁") := by
+  refine Rel.of_withDual_empty ?_ ?_ ?_ ?_
+  · apply And.intro
+    simp only [toTensorColor_eq, indexNotation_eq_color, ColorIndexList.contr, fromIndexStringColor,
+      mkDualMap, decidableEq_eq_color]
+    decide
+    simp only [toTensorColor_eq, indexNotation_eq_color, fromIndexStringColor, mkDualMap,
+      ColorIndexList.colorMap', decidableEq_eq_color, ColorIndexList.contr]
+    decide
+  · simp only [toTensorColor_eq, indexNotation_eq_color, fromIndexStringColor, mkDualMap,
+      decidableEq_eq_color]
+    decide
+  · simp only [toTensorColor_eq, indexNotation_eq_color, fromIndexStringColor, mkDualMap,
+      ColorIndexList.colorMap', decidableEq_eq_color]
+    decide
+  simp [fromIndexStringColor, mkDualMap]-/
+
+end einsteinTensor