refactor: Remove double empty lines

HepLean/FlavorPhysics/CKMMatrix/Basic.lean

@@ -17,12 +17,10 @@ related by phase shifts.
 The notation `[V]ud` etc can be used for the elements of a CKM matrix, and
 `[V]ud|us` etc for the ratios of elements.
 
-
 -/
 
 open Matrix Complex
 
-
 noncomputable section
 
 /-- Given three real numbers `a b c` the complex matrix with `exp (I * a)` etc on the
@@ -107,7 +105,6 @@ lemma phaseShiftRelation_trans {U V W : unitaryGroup (Fin 3) ℂ} :
   rw [phaseShiftMatrix_mul]
   rw [add_comm k e, add_comm l f, add_comm m g]
 
-
 lemma phaseShiftRelation_equiv : Equivalence PhaseShiftRelation where
   refl := phaseShiftRelation_refl
   symm := phaseShiftRelation_symm
@@ -270,7 +267,6 @@ lemma tb (V : CKMMatrix) (a b c d e f : ℝ) :
 
 end phaseShiftApply
 
-
 /-- The aboslute value of the `(i,j)`th element of `V`. -/
 @[simp]
 def VAbs' (V : unitaryGroup (Fin 3) ℂ) (i j : Fin 3) : ℝ := Complex.abs (V i j)
@@ -341,11 +337,9 @@ abbrev VtsAbs := VAbs 2 1
 @[simp]
 abbrev VtbAbs := VAbs 2 2
 
-
 namespace CKMMatrix
 open ComplexConjugate
 
-
 section ratios
 
 /-- The ratio of the `ub` and `ud` elements of a CKM matrix. -/
@@ -394,7 +388,6 @@ lemma Rcscb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cs = Rcscb V * [V]cb :=
 
 end ratios
 
-
 end CKMMatrix
 
 end

HepLean/FlavorPhysics/CKMMatrix/Invariants.lean

@@ -15,17 +15,14 @@ this equivalence.
 
 Of note, this file defines the complex jarlskog invariant.
 
-
 -/
 open Matrix Complex
 open ComplexConjugate
 open CKMMatrix
 
-
 noncomputable section
 namespace Invariant
 
-
 /-- The complex jarlskog invariant for a CKM matrix. -/
 @[simps!]
 def jarlskogℂCKM (V : CKMMatrix) : ℂ :=
@@ -62,6 +59,5 @@ standard parameterization. -/
 def mulExpδ₁₃ (V : Quotient CKMMatrixSetoid) : ℂ :=
   jarlskogℂ V + VusVubVcdSq V
 
-
 end Invariant
 end

HepLean/FlavorPhysics/CKMMatrix/PhaseFreedom.lean

@@ -19,11 +19,9 @@ In this file we define two sets of conditions on the CKM matrices
 and `ubOnePhaseCond` which we show can be satisfied by any CKM matrix up to equivalence as long as
 the ub element as absolute value 1.
 
-
 -/
 open Matrix Complex
 
-
 noncomputable section
 namespace CKMMatrix
 open ComplexConjugate
@@ -260,7 +258,6 @@ lemma fstRowThdColRealCond_holds_up_to_equiv (V : CKMMatrix) :
   apply shift_cross_product_phase_zero _ _ _ _ _ _ hτ.symm
   ring
 
-
 lemma ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0))
     (hV : FstRowThdColRealCond V)  :
     ∃ (U : CKMMatrix), V ≈ U ∧ ubOnePhaseCond U:= by

HepLean/FlavorPhysics/CKMMatrix/Relations.lean

@@ -16,7 +16,6 @@ matrix.
 
 open Matrix Complex
 
-
 noncomputable section
 
 namespace CKMMatrix
@@ -121,9 +120,6 @@ lemma VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid}
   rw [← ud_us_neq_zero_iff_ub_neq_one V] at hV
   simpa [← Complex.sq_abs] using (normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hV)
 
-
-
-
 lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℂ  {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
     (normSq [V]ud : ℂ) + normSq [V]us ≠ 0 := by
   have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hb
@@ -131,7 +127,6 @@ lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℂ  {V : CKMMatrix} (hb : [V]ud ≠ 0
   rw [← ofReal_inj] at h1
   simp_all
 
-
 lemma Vabs_sq_add_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
     ((VudAbs ⟦V⟧ : ℂ) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧)) ≠ 0 := by
   have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℂ hb
@@ -230,7 +225,6 @@ lemma conj_Vtb_mul_Vus {V : CKMMatrix} {τ : ℝ}
   simp only [Fin.isValue, neg_mul]
   ring
 
-
 lemma cs_of_ud_us_ub_cb_tb {V : CKMMatrix}  (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
     {τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
     [V]cs = (- conj [V]ub * [V]us  * [V]cb +
@@ -249,10 +243,8 @@ lemma cd_of_ud_us_ub_cb_tb {V : CKMMatrix}  (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
   field_simp
   ring
 
-
 end rows
 
-
 section individual
 
 lemma VAbs_ge_zero  (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : 0 ≤ VAbs i j V := by
@@ -270,12 +262,10 @@ lemma VAbs_leq_one (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : VAbs i j V ≤
   change VAbs i 2 V ≤ 1
   nlinarith
 
-
 end individual
 
 section columns
 
-
 lemma VAbs_sum_sq_col_eq_one (V : Quotient CKMMatrixSetoid) (i : Fin 3) :
     (VAbs 0 i V) ^ 2 + (VAbs 1 i V) ^ 2 + (VAbs 2 i V) ^ 2 = 1 := by
   obtain ⟨V, hV⟩ := Quot.exists_rep V
@@ -312,7 +302,6 @@ lemma cb_eq_zero_of_ud_us_zero {V : CKMMatrix} (h :  [V]ud = 0 ∧ [V]us = 0) :
   simp at h1
   exact h1.1
 
-
 lemma cs_of_ud_us_zero {V : CKMMatrix} (ha : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) :
     VcsAbs ⟦V⟧ = √(1 - VcdAbs ⟦V⟧ ^ 2) := by
   have h1 := snd_row_normalized_abs V
@@ -336,8 +325,6 @@ lemma VcbAbs_sq_add_VtbAbs_sq (V : Quotient CKMMatrixSetoid) :
     VcbAbs V ^ 2 + VtbAbs V ^ 2 = 1 - VubAbs V ^2 := by
   linear_combination (VAbs_sum_sq_col_eq_one V 2)
 
-
-
 lemma cb_tb_neq_zero_iff_ub_neq_one (V : CKMMatrix) :
     [V]cb ≠ 0 ∨ [V]tb ≠ 0 ↔ abs [V]ub ≠ 1 := by
   have h2 := V.thd_col_normalized_abs

HepLean/FlavorPhysics/CKMMatrix/Rows.lean

@@ -14,7 +14,6 @@ proves some properties between them.
 
 The first row can be extracted as `[V]u` for a CKM matrix `V`.
 
-
 -/
 
 open Matrix Complex
@@ -24,7 +23,6 @@ noncomputable section
 
 namespace CKMMatrix
 
-
 /-- The `u`th row of the CKM matrix. -/
 def uRow (V : CKMMatrix) : Fin 3 → ℂ := ![[V]ud, [V]us, [V]ub]
 

HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/Basic.lean

@@ -16,7 +16,6 @@ four real numbers `θ₁₂`, `θ₁₃`, `θ₂₃` and `δ₁₃`.
 We will show that every CKM matrix can be written within this standard parameterization
 in the file `FlavorPhysics.CKMMatrix.StandardParameters`.
 
-
 -/
 open Matrix Complex
 open ComplexConjugate
@@ -194,7 +193,5 @@ lemma mulExpδ₁₃_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤
   have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _
   field_simp
 
-
-
 end standParam
 end

HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/StandardParameters.lean

@@ -18,7 +18,6 @@ These, when used in the standard parameterization return `V` up to equivalence.
 This leads to the theorem `standParam.exists_for_CKMatrix` which says that up to equivalence every
 CKM matrix can be written using the standard parameterization.
 
-
 -/
 open Matrix Complex
 open ComplexConjugate
@@ -333,7 +332,6 @@ lemma Vs_zero_iff_cos_sin_zero (V : CKMMatrix) :
 
 end VAbs
 
-
 namespace standParam
 open Invariant
 
@@ -409,7 +407,6 @@ lemma on_param_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.c
   rfl
   rfl
 
-
 lemma on_param_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₂ ⟦V⟧) = 0) :
     standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
   use 0, δ₁₃, δ₁₃, -δ₁₃, 0, - δ₁₃
@@ -434,7 +431,6 @@ lemma on_param_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.c
   ring_nf
   field_simp
 
-
 lemma on_param_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₂₃ ⟦V⟧) = 0) :
     standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
   use 0, δ₁₃, 0, 0, 0, - δ₁₃
@@ -451,7 +447,6 @@ lemma on_param_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.c
   ring
   field_simp
 
-
 lemma on_param_sin_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₃ ⟦V⟧) = 0) :
     standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
   use 0, 0, 0, 0, 0, 0
@@ -488,8 +483,6 @@ lemma on_param_sin_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.s
   ring_nf
   field_simp
 
-
-
 lemma on_param_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₂₃ ⟦V⟧) = 0) :
     standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
   use 0, 0, δ₁₃, 0, 0, - δ₁₃
@@ -506,9 +499,6 @@ lemma on_param_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.s
   ring
   field_simp
 
-
-
-
 lemma eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
     (hV : FstRowThdColRealCond V) : V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (- arg [V]ub) := by
   have hb' : VubAbs ⟦V⟧ ≠ 1 := by
@@ -565,7 +555,6 @@ lemma eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0
   rw [VcbAbs_eq_S₂₃_mul_C₁₃ ⟦V⟧, S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, C₁₃]
   simp
 
-
 lemma eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : ubOnePhaseCond V) :
     V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
   have h1 : VubAbs ⟦V⟧ = 1 := by
@@ -610,7 +599,6 @@ lemma eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : ubOnePhaseCond V) :
   rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.2.2.1]
   simp
 
-
 theorem exists_δ₁₃ (V : CKMMatrix) :
     ∃ (δ₃ : ℝ), V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ := by
   obtain ⟨U, hU⟩ := fstRowThdColRealCond_holds_up_to_equiv V
@@ -670,7 +658,6 @@ theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
   exact on_param_sin_θ₁₃_eq_zero δ₁₃' h
   exact on_param_sin_θ₂₃_eq_zero δ₁₃' h
 
-
 theorem exists_for_CKMatrix (V : CKMMatrix) :
     ∃ (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ), V ≈ standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
   use θ₁₂ ⟦V⟧, θ₁₃ ⟦V⟧, θ₂₃ ⟦V⟧, δ₁₃ ⟦V⟧
@@ -678,9 +665,6 @@ theorem exists_for_CKMatrix (V : CKMMatrix) :
 
 end standParam
 
-
 open CKMMatrix
 
-
-
 end