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refactor: Simps

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jstoobysmith 2024-11-09 18:06:48 +00:00
parent a7142ef99b
commit e963be5ef8
5 changed files with 20 additions and 18 deletions
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12
HepLean/Lorentz/SL2C/Basic.lean
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@ -97,14 +97,15 @@ open Lorentz in
lemma toMatrix_apply_contrMod (M : SL(2, ℂ)) (v : ContrMod 3) :
(toMatrix M) *ᵥ v = ContrMod.toSelfAdjoint.symm
((toLinearMapSelfAdjointMatrix M) (ContrMod.toSelfAdjoint v)) := by
simp [toMatrix, LinearMap.toMatrix_apply, ContrMod.mulVec]
simp only [ContrMod.mulVec, toMatrix, MonoidHom.coe_mk, OneHom.coe_mk]
obtain ⟨a, ha⟩ := ContrMod.toSelfAdjoint.symm.surjective v
subst ha
rw [LinearEquiv.apply_symm_apply]
simp [ContrMod.toSelfAdjoint]
change ContrMod.toFin1dℝEquiv.symm ((
((LinearMap.toMatrix PauliMatrix.σSAL PauliMatrix.σSAL) (toLinearMapSelfAdjointMatrix M)))
*ᵥ (((Finsupp.linearEquivFunOnFinite ℝ ℝ (Fin 1 ⊕ Fin 3)) (PauliMatrix.σSAL.repr a)))) = _
simp only [ContrMod.toSelfAdjoint, LinearEquiv.trans_symm, LinearEquiv.symm_symm,
LinearEquiv.trans_apply]
change ContrMod.toFin1dℝEquiv.symm ((
((LinearMap.toMatrix PauliMatrix.σSAL PauliMatrix.σSAL) (toLinearMapSelfAdjointMatrix M)))
*ᵥ (((Finsupp.linearEquivFunOnFinite ℝ ℝ (Fin 1 ⊕ Fin 3)) (PauliMatrix.σSAL.repr a)))) = _
apply congrArg
erw [LinearMap.toMatrix_mulVec_repr]
rfl
@ -136,7 +137,6 @@ lemma toLorentzGroup_eq_σSAL (M : SL(2, ℂ)) :
PauliMatrix.σSAL PauliMatrix.σSAL (toLinearMapSelfAdjointMatrix M) := by
rfl
lemma toLinearMapSelfAdjointMatrix_basis (i : Fin 1 ⊕ Fin 3) :
toLinearMapSelfAdjointMatrix M (PauliMatrix.σSAL i) =
∑ j, (toLorentzGroup M).1 j i •