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