132 lines
4.7 KiB
Text
132 lines
4.7 KiB
Text
|
/-
|
|||
|
|
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.TensorIndex
|
|||
|
|
import HepLean.SpaceTime.LorentzTensor.IndexNotation.IndexString
|
|||
|
|
import HepLean.SpaceTime.LorentzTensor.EinsteinNotation.Basic
|
|||
|
|
/-!
|
|||
|
|
|
|||
|
|
# Index notation for Einstein tensors
|
|||
|
|
|
|||
|
|
|
|||
|
|
-/
|
|||
|
|
|
|||
|
|
instance : IndexNotation einsteinTensorColor.Color where
|
|||
|
|
charList := {'ᵢ'}
|
|||
|
|
notaEquiv :=
|
|||
|
|
⟨fun _ => ⟨'ᵢ', Finset.mem_singleton.mpr rfl⟩,
|
|||
|
|
fun _ => Unit.unit,
|
|||
|
|
fun _ => rfl,
|
|||
|
|
by
|
|||
|
|
intro c
|
|||
|
|
have hc2 := c.2
|
|||
|
|
simp only [↓Char.isValue, Finset.mem_singleton] at hc2
|
|||
|
|
exact SetCoe.ext (id (Eq.symm hc2))⟩
|
|||
|
|
|
|||
|
|
namespace einsteinTensor
|
|||
|
|
|
|||
|
|
open einsteinTensorColor
|
|||
|
|
open IndexNotation IndexString
|
|||
|
|
open TensorStructure TensorIndex
|
|||
|
|
|
|||
|
|
variable {R : Type} [CommSemiring R] {n m : ℕ}
|
|||
|
|
|
|||
|
|
instance : IndexNotation (einsteinTensor R n).Color := instIndexNotationColorEinsteinTensorColor
|
|||
|
|
instance : DecidableEq (einsteinTensor R n).Color := instDecidableEqColorEinsteinTensorColor
|
|||
|
|
|
|||
|
|
|
|||
|
|
@[simp]
|
|||
|
|
lemma indexNotation_eq_color : @einsteinTensor.instIndexNotationColor R _ n =
|
|||
|
|
instIndexNotationColorEinsteinTensorColor := by
|
|||
|
|
rfl
|
|||
|
|
|
|||
|
|
@[simp]
|
|||
|
|
lemma decidableEq_eq_color : @einsteinTensor.instDecidableEqColor R _ n =
|
|||
|
|
instDecidableEqColorEinsteinTensorColor := by
|
|||
|
|
rfl
|
|||
|
|
|
|||
|
|
@[simp]
|
|||
|
|
lemma einsteinTensor_color : (einsteinTensor R n).Color = einsteinTensorColor.Color := by
|
|||
|
|
rfl
|
|||
|
|
|
|||
|
|
@[simp]
|
|||
|
|
lemma toTensorColor_eq : (einsteinTensor R n).toTensorColor = einsteinTensorColor := by
|
|||
|
|
rfl
|
|||
|
|
|
|||
|
|
|
|||
|
|
/-- The construction of a tensor index from a tensor and a string satisfying conditions
|
|||
|
|
which can be automatically checked. This is a modified version of
|
|||
|
|
`TensorStructure.TensorIndex.mkDualMap` specific to real Lorentz tensors. -/
|
|||
|
|
noncomputable def fromIndexStringColor {R : Type} [CommSemiring R] {cn : Fin n → einsteinTensorColor.Color}
|
|||
|
|
(T : (einsteinTensor R m).Tensor cn) (s : String)
|
|||
|
|
(hs : listCharIsIndexString einsteinTensorColor.Color s.toList = true)
|
|||
|
|
(hn : n = (toIndexList' s hs).length)
|
|||
|
|
(hD : (toIndexList' s hs).withDual = (toIndexList' s hs).withUniqueDual)
|
|||
|
|
(hC : IndexList.ColorCond.bool (toIndexList' s hs))
|
|||
|
|
(hd : TensorColor.ColorMap.DualMap.boolFin'
|
|||
|
|
(toIndexList' s hs).colorMap (cn ∘ Fin.cast hn.symm)) :
|
|||
|
|
(einsteinTensor R m).TensorIndex :=
|
|||
|
|
TensorStructure.TensorIndex.mkDualMap T ⟨(toIndexList' s hs), hD,
|
|||
|
|
IndexList.ColorCond.iff_bool.mpr hC⟩ hn
|
|||
|
|
(TensorColor.ColorMap.DualMap.boolFin'_DualMap hd)
|
|||
|
|
|
|||
|
|
@[simp]
|
|||
|
|
lemma fromIndexStringColor_indexList {R : Type} [CommSemiring R] {cn : Fin n → einsteinTensorColor.Color}
|
|||
|
|
(T : (einsteinTensor R m).Tensor cn) (s : String)
|
|||
|
|
(hs : listCharIsIndexString einsteinTensorColor.Color s.toList = true)
|
|||
|
|
(hn : n = (toIndexList' s hs).length)
|
|||
|
|
(hD : (toIndexList' s hs).withDual = (toIndexList' s hs).withUniqueDual)
|
|||
|
|
(hC : IndexList.ColorCond.bool (toIndexList' s hs))
|
|||
|
|
(hd : TensorColor.ColorMap.DualMap.boolFin'
|
|||
|
|
(toIndexList' s hs).colorMap (cn ∘ Fin.cast hn.symm)) :
|
|||
|
|
(fromIndexStringColor T s hs hn hD hC hd).toIndexList = toIndexList' s hs := by
|
|||
|
|
rfl
|
|||
|
|
|
|||
|
|
|
|||
|
|
/-- A tactic used to prove `boolFin` for real Lornetz tensors. -/
|
|||
|
|
macro "dualMapTactic" : tactic =>
|
|||
|
|
`(tactic| {
|
|||
|
|
simp only [toTensorColor_eq]
|
|||
|
|
decide })
|
|||
|
|
|
|||
|
|
|
|||
|
|
/-- Notation for the construction of a tensor index from a tensor and a string.
|
|||
|
|
Conditions are checked automatically. -/
|
|||
|
|
notation:20 T "|" S:21 => fromIndexStringColor T S
|
|||
|
|
(by decide)
|
|||
|
|
(by decide) (by decide)
|
|||
|
|
(by decide)
|
|||
|
|
(by dualMapTactic)
|
|||
|
|
|
|||
|
|
/-- A tactics used to prove `colorPropBool` for real Lorentz tensors. -/
|
|||
|
|
macro "prodTactic" : tactic =>
|
|||
|
|
`(tactic| {
|
|||
|
|
apply (ColorIndexList.AppendCond.iff_bool _ _).mpr
|
|||
|
|
change @ColorIndexList.AppendCond.bool einsteinTensorColor
|
|||
|
|
instIndexNotationColorEinsteinTensorColor instDecidableEqColorEinsteinTensorColor _ _
|
|||
|
|
simp only [prod_toIndexList, indexNotation_eq_color, fromIndexStringColor, mkDualMap,
|
|||
|
|
toTensorColor_eq, decidableEq_eq_color]
|
|||
|
|
decide})
|
|||
|
|
|
|||
|
|
lemma mem_fin_list (n : ℕ) (i : Fin n) : i ∈ Fin.list n := by
|
|||
|
|
have h1' : (Fin.list n)[i] = i := Fin.getElem_list _ _
|
|||
|
|
exact h1' ▸ List.getElem_mem _ _ _
|
|||
|
|
|
|||
|
|
instance (n : ℕ) (i : Fin n) : Decidable (i ∈ Fin.list n) :=
|
|||
|
|
isTrue (mem_fin_list n i)
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
/-- The product of Real Lorentz tensors. Conditions on indices are checked automatically. -/
|
|||
|
|
notation:10 T "⊗ᵀ" S:11 => TensorIndex.prod T S (by prodTactic)
|
|||
|
|
/-- An example showing the allowed constructions. -/
|
|||
|
|
example (T : (einsteinTensor R n).Tensor ![Unit.unit, Unit.unit]) : True := by
|
|||
|
|
let _ := T|"ᵢ₂ᵢ₃"
|
|||
|
|
let _ := T|"ᵢ₁ᵢ₂" ⊗ᵀ T|"ᵢ₂ᵢ₁"
|
|||
|
|
let _ := T|"ᵢ₁ᵢ₂" ⊗ᵀ T|"ᵢ₂ᵢ₁" ⊗ᵀ T|"ᵢ₃ᵢ₄"
|
|||
|
|
exact trivial
|
|||
|
|
|
|||
|
|
end einsteinTensor
|