47 lines
1.3 KiB
Text
47 lines
1.3 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.EinsteinNotation.IndexNotation
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/-!
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# Lemmas regarding Einstein tensors
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-/
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set_option profiler true
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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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/-! TODO: Fix notation here. -/
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set_option maxHeartbeats 0
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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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apply And.intro
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apply And.intro
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· simp only [toTensorColor_eq, indexNotation_eq_color, ColorIndexList.contr,
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fromIndexStringColor_indexList, IndexList.contrIndexList_length]
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decide
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apply And.intro
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· simp only [toTensorColor_eq, indexNotation_eq_color, ColorIndexList.contr,
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fromIndexStringColor_indexList, IndexList.contrIndexList_length]
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rw [IndexList.withUniqueDualInOther_eq_univ_iff_forall]
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intro x
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have h1 : (toIndexList' "ᵢ₁ᵢ₂" (by decide) : IndexList einsteinTensorColor.Color).contrIndexList.length
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= 2 := by
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decide
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sorry
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sorry
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end einsteinTensor
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