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refactor: LInt

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This commit is contained in:
jstoobysmith 2024-11-09 18:12:05 +00:00
parent e963be5ef8
commit d058f41689
11 changed files with 38 additions and 50 deletions
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8
HepLean/Lorentz/RealVector/Modules.lean
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@ -86,7 +86,6 @@ lemma toFin1dℝ_eq_val (ψ : ContrMod d) : ψ.toFin1dℝ = ψ.val := by rfl
@[simps!]
def stdBasis : Basis (Fin 1 ⊕ Fin d) ℝ (ContrMod d) := Basis.ofEquivFun toFin1dℝEquiv
@[simp]
lemma stdBasis_toFin1dℝEquiv_apply_same (μ : Fin 1 ⊕ Fin d) :
toFin1dℝEquiv (stdBasis μ) μ = 1 := by
@ -121,7 +120,7 @@ lemma stdBasis_apply (μ ν : Fin 1 ⊕ Fin d) : (stdBasis μ).val ν = if μ =
/-- Decomposition of a contrvariant Lorentz vector into the standard basis. -/
lemma stdBasis_decomp (v : ContrMod d) : v = ∑ i, v.toFin1dℝ i • stdBasis i := by
apply toFin1dℝEquiv.injective
simp only [map_sum, _root_.map_smul]
simp only [map_sum, _root_.map_smul]
funext μ
rw [Fintype.sum_apply μ fun c => toFin1dℝEquiv v c • toFin1dℝEquiv (stdBasis c)]
change _ = ∑ x : Fin 1 ⊕ Fin d, toFin1dℝEquiv v x • (toFin1dℝEquiv (stdBasis x) μ)
@ -189,7 +188,6 @@ def toSpace (v : ContrMod d) : EuclideanSpace ℝ (Fin d) := v.val ∘ Sum.inr
-/
/-- The representation of the Lorentz group acting on `ContrℝModule d`. -/
def rep : Representation ℝ (LorentzGroup d) (ContrMod d) where
toFun g := Matrix.toLinAlgEquiv stdBasis g
@ -237,7 +235,6 @@ lemma toSelfAdjoint_apply_coe (x : ContrMod 3) : (toSelfAdjoint x).1 =
rw [toSelfAdjoint_apply]
rfl
lemma toSelfAdjoint_stdBasis (i : Fin 1 ⊕ Fin 3) :
toSelfAdjoint (stdBasis i) = PauliMatrix.σSAL i := by
rw [toSelfAdjoint_apply]
@ -352,11 +349,10 @@ lemma stdBasis_apply (μ ν : Fin 1 ⊕ Fin d) : (stdBasis μ).val ν = if μ =
refine ite_congr ?h₁ (congrFun rfl) (congrFun rfl)
exact Eq.propIntro (fun a => id (Eq.symm a)) fun a => id (Eq.symm a)
/-- Decomposition of a covariant Lorentz vector into the standard basis. -/
lemma stdBasis_decomp (v : CoMod d) : v = ∑ i, v.toFin1dℝ i • stdBasis i := by
apply toFin1dℝEquiv.injective
simp only [map_sum, _root_.map_smul]
simp only [map_sum, _root_.map_smul]
funext μ
rw [Fintype.sum_apply μ fun c => toFin1dℝEquiv v c • toFin1dℝEquiv (stdBasis c)]
change _ = ∑ x : Fin 1 ⊕ Fin d, toFin1dℝEquiv v x • (toFin1dℝEquiv (stdBasis x) μ)