reactor: Removal of double spaces
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jstoobysmith
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64 changed files with 550 additions and 546 deletions
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@ -28,7 +28,7 @@ def lorentzAlgebra : LieSubalgebra ℝ (Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin
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namespace lorentzAlgebra
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open minkowskiMatrix
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lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
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lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
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have h1 := A.2
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erw [mem_skewAdjointMatricesLieSubalgebra] at h1
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simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h1
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@ -39,7 +39,7 @@ lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1
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simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h
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lemma mem_iff {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ} :
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A ∈ lorentzAlgebra ↔ Aᵀ * η = - η * A :=
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A ∈ lorentzAlgebra ↔ Aᵀ * η = - η * A :=
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Iff.intro (fun h => transpose_eta ⟨A, h⟩) (fun h => mem_of_transpose_eta_eq_eta_mul_self h)
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lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
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@ -52,7 +52,7 @@ lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
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rw [minkowskiMatrix.sq]
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all_goals noncomm_ring
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· nth_rewrite 2 [h]
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trans (η * η) * Aᵀ * η
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trans (η * η) * Aᵀ * η
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rw [minkowskiMatrix.sq]
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all_goals noncomm_ring
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@ -80,7 +80,7 @@ end lorentzAlgebra
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@[simps!]
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instance lorentzVectorAsLieRingModule : LieRingModule lorentzAlgebra (LorentzVector 3) where
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bracket Λ x := Λ.1.mulVec x
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bracket Λ x := Λ.1.mulVec x
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add_lie Λ1 Λ2 x := by
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simp [add_mulVec]
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lie_add Λ x1 x2 := by
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@ -30,7 +30,7 @@ We start studying the properties of matrices which preserve `ηLin`.
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These matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.
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-/
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variable {d : ℕ}
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variable {d : ℕ}
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open minkowskiMetric in
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/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/
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@ -74,7 +74,7 @@ lemma mem_iff_on_right : Λ ∈ LorentzGroup d ↔
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rw [← dual_mulVec_right, mulVec_mulVec]
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exact h x y
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lemma mem_iff_dual_mul_self : Λ ∈ LorentzGroup d ↔ dual Λ * Λ = 1 := by
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lemma mem_iff_dual_mul_self : Λ ∈ LorentzGroup d ↔ dual Λ * Λ = 1 := by
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rw [mem_iff_on_right, matrix_eq_id_iff]
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exact forall_comm
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@ -145,7 +145,7 @@ namespace LorentzGroup
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open minkowskiMetric
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variable {Λ Λ' : LorentzGroup d}
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variable {Λ Λ' : LorentzGroup d}
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lemma coe_inv : (Λ⁻¹).1 = Λ.1⁻¹:= by
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refine (inv_eq_left_inv ?h).symm
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@ -172,7 +172,7 @@ def toGL : LorentzGroup d →* GL (Fin 1 ⊕ Fin d) ℝ where
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map_one' := by
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simp
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rfl
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map_mul' x y := by
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map_mul' x y := by
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simp only [lorentzGroupIsGroup, _root_.mul_inv_rev, coe_inv]
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ext
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rfl
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@ -99,7 +99,7 @@ lemma toMatrix_mulVec (u v : FuturePointing d) (x : LorentzVector d) :
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open minkowskiMatrix LorentzVector in
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@[simp]
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lemma toMatrix_apply (u v : FuturePointing d) (μ ν : Fin 1 ⊕ Fin d) :
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lemma toMatrix_apply (u v : FuturePointing d) (μ ν : Fin 1 ⊕ Fin d) :
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(toMatrix u v) μ ν = η μ μ * (⟪e μ, e ν⟫ₘ + 2 * ⟪e ν, u⟫ₘ * ⟪e μ, v⟫ₘ
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- ⟪e μ, u + v⟫ₘ * ⟪e ν, u + v⟫ₘ / (1 + ⟪u, v.1.1⟫ₘ)) := by
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rw [matrix_apply_stdBasis (toMatrix u v) μ ν, toMatrix_mulVec]
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@ -27,7 +27,7 @@ variable (Λ : LorentzGroup d)
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open LorentzVector
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open minkowskiMetric
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/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
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/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
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def IsOrthochronous : Prop := 0 ≤ timeComp Λ
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lemma IsOrthochronous_iff_futurePointing :
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@ -62,7 +62,7 @@ lemma not_orthochronous_iff_le_zero :
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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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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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/-- An auxillary function used in the definition of `orthchroMapReal`. -/
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@ -75,7 +75,7 @@ lemma stepFunction_continuous : Continuous stepFunction := by
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<;> intro a ha
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rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
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rw [ha]
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simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
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simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
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have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
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exact Eq.symm (if_neg h1)
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rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
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@ -31,7 +31,7 @@ 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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local notation "ℤ₂" => Multiplicative (ZMod 2)
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instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
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@ -51,7 +51,7 @@ def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
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/-- The continuous map taking a Lorentz matrix to its determinant. -/
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def detContinuous : C(𝓛 d, ℤ₂) :=
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ContinuousMap.comp coeForℤ₂ {
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ContinuousMap.comp coeForℤ₂ {
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toFun := fun Λ => ⟨Λ.1.det, Or.symm (LorentzGroup.det_eq_one_or_neg_one _)⟩,
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continuous_toFun := by
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refine Continuous.subtype_mk ?_ _
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@ -65,7 +65,7 @@ lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
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intro h
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simp [detContinuous] at h
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cases' det_eq_one_or_neg_one Λ with h1 h1
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<;> cases' det_eq_one_or_neg_one Λ' with h2 h2
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<;> cases' det_eq_one_or_neg_one Λ' with h2 h2
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<;> simp_all [h1, h2, h]
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rw [← toMul_zero, @Equiv.apply_eq_iff_eq] at h
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· change (0 : Fin 2) = (1 : Fin 2) at h
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@ -47,7 +47,7 @@ instance (d : ℕ) (μ : RealLorentzTensor.Colors) : Fintype (RealLorentzTensor.
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| RealLorentzTensor.Colors.down => instFintypeSum (Fin 1) (Fin d)
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/-- An `IndexValue` is a set of actual values an index can take. e.g. for a
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3-tensor (0, 1, 2). -/
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3-tensor (0, 1, 2). -/
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@[simp]
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def RealLorentzTensor.IndexValue {X : FintypeCat} (d : ℕ ) (c : X → RealLorentzTensor.Colors) :
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Type 0 := (x : X) → RealLorentzTensor.ColorsIndex d (c x)
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@ -107,9 +107,9 @@ def congrColorsDual {μ ν : Colors} (h : μ = τ ν) :
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ColorsIndex d μ ≃ ColorsIndex d ν :=
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(castColorsIndex h).trans dualColorsIndex.symm
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lemma congrColorsDual_symm {μ ν : Colors} (h : μ = τ ν) :
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lemma congrColorsDual_symm {μ ν : Colors} (h : μ = τ ν) :
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(congrColorsDual h).symm =
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@congrColorsDual d _ _ ((Function.Involutive.eq_iff τ_involutive).mp h.symm) := by
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@congrColorsDual d _ _ ((Function.Involutive.eq_iff τ_involutive).mp h.symm) := by
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match μ, ν with
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| Colors.up, Colors.down => rfl
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| Colors.down, Colors.up => rfl
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@ -124,7 +124,7 @@ lemma color_eq_dual_symm {μ ν : Colors} (h : μ = τ ν) : ν = τ μ :=
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-/
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/-- An equivalence of Index values from an equality of color maps. -/
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def castIndexValue {X : FintypeCat} {T S : X → Colors} (h : T = S) :
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def castIndexValue {X : FintypeCat} {T S : X → Colors} (h : T = S) :
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IndexValue d T ≃ IndexValue d S where
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toFun i := (fun μ => castColorsIndex (congrFun h μ) (i μ))
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invFun i := (fun μ => (castColorsIndex (congrFun h μ)).symm (i μ))
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@ -133,7 +133,7 @@ def castIndexValue {X : FintypeCat} {T S : X → Colors} (h : T = S) :
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right_inv i := by
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simp
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lemma indexValue_eq {T₁ T₂ : X → RealLorentzTensor.Colors} (d : ℕ) (h : T₁ = T₂) :
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lemma indexValue_eq {T₁ T₂ : X → RealLorentzTensor.Colors} (d : ℕ) (h : T₁ = T₂) :
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IndexValue d T₁ = IndexValue d T₂ :=
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pi_congr fun a => congrArg (ColorsIndex d) (congrFun h a)
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@ -176,7 +176,7 @@ lemma ext' {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color)
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/-- An equivalence between `X → Fin 1 ⊕ Fin d` and `Y → Fin 1 ⊕ Fin d` given an isomorphism
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between `X` and `Y`. -/
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def congrSetIndexValue (d : ℕ) (f : X ≃ Y) (i : X → Colors) :
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IndexValue d i ≃ IndexValue d (i ∘ f.symm) :=
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IndexValue d i ≃ IndexValue d (i ∘ f.symm) :=
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Equiv.piCongrLeft' _ f
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/-- Given an equivalence of indexing sets, a map on Lorentz tensors. -/
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@ -259,7 +259,7 @@ def inrIndexValue {Tc : X → Colors} {Sc : Y → Colors}
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def sumCommIndexValue {X Y : FintypeCat} (Tc : X → Colors) (Sc : Y → Colors) :
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IndexValue d (sumElimIndexColor Tc Sc) ≃ IndexValue d (sumElimIndexColor Sc Tc) :=
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(congrSetIndexValue d (sumCommFintypeCat X Y) (sumElimIndexColor Tc Sc)).trans
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(castIndexValue ((sumElimIndexColor_symm Sc Tc).symm))
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(castIndexValue ((sumElimIndexColor_symm Sc Tc).symm))
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lemma sumCommIndexValue_inlIndexValue {X Y : FintypeCat} {Tc : X → Colors} {Sc : Y → Colors}
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(i : IndexValue d (sumElimIndexColor Tc Sc)) :
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@ -300,7 +300,7 @@ def unmarkedColor (T : Marked d X n) : X → Colors :=
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T.color ∘ Sum.inl
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/-- The colors of marked indices. -/
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def markedColor (T : Marked d X n) : FintypeCat.of (Σ _ : Fin n, PUnit) → Colors :=
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def markedColor (T : Marked d X n) : FintypeCat.of (Σ _ : Fin n, PUnit) → Colors :=
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T.color ∘ Sum.inr
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/-- The index values restricted to unmarked indices. -/
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