refactor: Remove double empty lines
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@ -44,7 +44,6 @@ def LorentzGroup (d : ℕ) : Set (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ
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{Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ |
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∀ (x y : LorentzVector d), ⟪Λ *ᵥ x, Λ *ᵥ y⟫ₘ = ⟪x, y⟫ₘ}
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namespace LorentzGroup
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/-- Notation for the Lorentz group. -/
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scoped[LorentzGroup] notation (name := lorentzGroup_notation) "𝓛" => LorentzGroup
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@ -140,7 +140,6 @@ lemma toMatrix_continuous (u : FuturePointing d) : Continuous (toMatrix u) := by
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exact FuturePointing.metric_continuous _
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exact fun x => FuturePointing.one_add_metric_non_zero u x
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lemma toMatrix_in_lorentzGroup (u v : FuturePointing d) : (toMatrix u v) ∈ LorentzGroup d:= by
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intro x y
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rw [toMatrix_mulVec, toMatrix_mulVec, genBoost, genBoostAux₁, genBoostAux₂]
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@ -161,7 +160,6 @@ lemma toLorentz_continuous (u : FuturePointing d) : Continuous (toLorentz u) :=
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refine Continuous.subtype_mk ?_ (fun x => toMatrix_in_lorentzGroup u x)
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exact toMatrix_continuous u
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lemma toLorentz_joined_to_1 (u v : FuturePointing d) : Joined 1 (toLorentz u v) := by
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obtain ⟨f, _⟩ := FuturePointing.isPathConnected.joinedIn u trivial v trivial
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use ContinuousMap.comp ⟨toLorentz u, toLorentz_continuous u⟩ f
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@ -180,8 +178,6 @@ lemma isProper (u v : FuturePointing d) : IsProper (toLorentz u v) :=
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end genBoost
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end LorentzGroup
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end
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@ -17,10 +17,8 @@ matrices.
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-/
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noncomputable section
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open Matrix
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open Complex
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open ComplexConjugate
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@ -66,7 +64,6 @@ lemma not_orthochronous_iff_le_zero :
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rw [IsOrthochronous_iff_ge_one]
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linarith
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/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
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def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ ,
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Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
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@ -11,10 +11,8 @@ We define the give a series of lemmas related to the determinant of the lorentz
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-/
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noncomputable section
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open Matrix
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open Complex
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open ComplexConjugate
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@ -31,7 +29,6 @@ lemma det_eq_one_or_neg_one (Λ : 𝓛 d) : Λ.1.det = 1 ∨ Λ.1.det = -1 := by
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simp [det_mul, det_dual] at h1
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exact mul_self_eq_one_iff.mp h1
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local notation "ℤ₂" => Multiplicative (ZMod 2)
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instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
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@ -76,7 +73,6 @@ lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
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· intro h
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simp [detContinuous, h]
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/-- The representation taking a lorentz matrix to its determinant. -/
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@[simps!]
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def detRep : 𝓛 d →* ℤ₂ where
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@ -54,8 +54,6 @@ def SO3ToLorentz : SO(3) →* LorentzGroup 3 where
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apply Subtype.eq
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simp [Matrix.fromBlocks_multiply]
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end LorentzGroup
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end
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