reactor: Removal of double spaces
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jstoobysmith
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ce92e1d649
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13f62a50eb
64 changed files with 550 additions and 546 deletions
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@ -12,7 +12,7 @@ import Mathlib.Analysis.SpecialFunctions.Complex.Arg
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/-!
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# Standard parameters for the CKM Matrix
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Given a CKM matrix `V` we can extract four real numbers `θ₁₂`, `θ₁₃`, `θ₂₃` and `δ₁₃`.
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Given a CKM matrix `V` we can extract four real numbers `θ₁₂`, `θ₁₃`, `θ₂₃` and `δ₁₃`.
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These, when used in the standard parameterization return `V` up to equivalence.
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This leads to the theorem `standParam.exists_for_CKMatrix` which says that up to equivalence every
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@ -26,15 +26,15 @@ open CKMMatrix
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noncomputable section
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/-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₂` in the
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standard parameterization. --/
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standard parameterization. --/
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def S₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := VusAbs V / (√ (VudAbs V ^ 2 + VusAbs V ^ 2))
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/-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₃` in the
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standard parameterization. --/
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standard parameterization. --/
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def S₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := VubAbs V
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/-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the
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standard parameterization. --/
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standard parameterization. --/
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def S₂₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
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if VubAbs V = 1 then VcdAbs V
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else VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2)
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@ -56,7 +56,7 @@ standard parameterization. --/
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def C₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₂ V)
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/-- Given a CKM matrix `V` the real number corresponding to `cos θ₁₃` in the
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standard parameterization. --/
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standard parameterization. --/
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def C₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₃ V)
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/-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the
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@ -64,7 +64,7 @@ standard parameterization. --/
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def C₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₂₃ V)
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/-- Given a CKM matrix `V` the real number corresponding to the phase `δ₁₃` in the
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standard parameterization. --/
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standard parameterization. --/
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def δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
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arg (Invariant.mulExpδ₁₃ V)
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@ -336,7 +336,7 @@ namespace standParam
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open Invariant
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lemma mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) :
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mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
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mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
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sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
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* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := by
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refine mulExpδ₁₃_eq _ _ _ _ ?_ ?_ ?_ ?_
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@ -348,11 +348,11 @@ lemma mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) :
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exact Real.cos_arcsin_nonneg _
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lemma mulExpδ₁₃_on_param_eq_zero_iff (V : CKMMatrix) (δ₁₃ : ℝ) :
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mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
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mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
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VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 := by
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rw [VudAbs_eq_C₁₂_mul_C₁₃, VubAbs_eq_S₁₃, VusAbs_eq_S₁₂_mul_C₁₃, VcbAbs_eq_S₂₃_mul_C₁₃,
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VtbAbs_eq_C₂₃_mul_C₁₃, ← ofReal_inj,
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← ofReal_inj, ← ofReal_inj, ← ofReal_inj, ← ofReal_inj]
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← ofReal_inj, ← ofReal_inj, ← ofReal_inj, ← ofReal_inj]
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simp only [ofReal_mul]
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rw [← S₁₃_eq_ℂsin_θ₁₃, ← S₁₂_eq_ℂsin_θ₁₂, ← S₂₃_eq_ℂsin_θ₂₃,
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← C₁₃_eq_ℂcos_θ₁₃, ← C₂₃_eq_ℂcos_θ₂₃,← C₁₂_eq_ℂcos_θ₁₂]
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@ -364,7 +364,7 @@ lemma mulExpδ₁₃_on_param_eq_zero_iff (V : CKMMatrix) (δ₁₃ : ℝ) :
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aesop
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lemma mulExpδ₁₃_on_param_abs (V : CKMMatrix) (δ₁₃ : ℝ) :
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Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) =
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Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) =
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sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
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* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) := by
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rw [mulExpδ₁₃_on_param_δ₁₃]
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@ -373,19 +373,19 @@ lemma mulExpδ₁₃_on_param_abs (V : CKMMatrix) (δ₁₃ : ℝ) :
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complexAbs_sin_θ₂₃, complexAbs_cos_θ₂₃]
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lemma mulExpδ₁₃_on_param_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
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(h1 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
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cexp (arg ( mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ) * I) =
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(h1 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
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cexp (arg ( mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ) * I) =
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cexp (δ₁₃ * I) := by
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have h1a := mulExpδ₁₃_on_param_δ₁₃ V δ₁₃
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have habs := mulExpδ₁₃_on_param_abs V δ₁₃
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have h2 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = Complex.abs
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(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
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have h2 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = Complex.abs
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(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
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rw [habs, h1a]
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ring_nf
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nth_rewrite 1 [← abs_mul_exp_arg_mul_I (mulExpδ₁₃
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⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
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⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
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have habs_neq_zero :
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(Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
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(Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
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simp only [ne_eq, ofReal_eq_zero, map_eq_zero]
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exact h1
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rw [← mul_right_inj' habs_neq_zero]
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@ -393,7 +393,7 @@ lemma mulExpδ₁₃_on_param_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
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lemma on_param_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₃ ⟦V⟧) = 0) :
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standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
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have hS13 := congrArg ofReal (S₁₃_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
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have hS13 := congrArg ofReal (S₁₃_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
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simp [← S₁₃_eq_ℂsin_θ₁₃] at hS13
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have hC12 := congrArg ofReal (C₁₂_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
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simp [← C₁₂_eq_ℂcos_θ₁₂] at hC12
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@ -638,8 +638,8 @@ theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
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V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧) := by
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obtain ⟨δ₁₃', hδ₃⟩ := exists_δ₁₃ V
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have hSV := (Quotient.eq.mpr (hδ₃))
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by_cases h : Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
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have h2 := eq_exp_of_phases (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
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by_cases h : Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
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have h2 := eq_exp_of_phases (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
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(δ₁₃ ⟦V⟧) (by rw [← mulExpδ₁₃_on_param_neq_zero_arg V δ₁₃' h, ← hSV, δ₁₃, Invariant.mulExpδ₁₃])
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rw [h2] at hδ₃
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exact hδ₃
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