Refactor: Names
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jstoobysmith
parent
ff89c3f79d
commit
490ed0380c
6 changed files with 342 additions and 442 deletions
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@ -16,7 +16,7 @@ open CKMMatrix
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noncomputable section
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-- to be renamed stanParamAsMatrix ...
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def standardParameterizationAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
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def standParamAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
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![![Real.cos θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₃ * exp (-I * δ₁₃)],
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![(-Real.sin θ₁₂ * Real.cos θ₂₃) - (Real.cos θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃)),
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Real.cos θ₁₂ * Real.cos θ₂₃ - Real.sin θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃),
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@ -27,10 +27,10 @@ def standardParameterizationAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ)
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open CKMMatrix
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lemma standardParameterizationAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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((standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by
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lemma standParamAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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((standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by
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funext j i
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simp only [standardParameterizationAsMatrix, neg_mul, Fin.isValue]
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simp only [standParamAsMatrix, neg_mul, Fin.isValue]
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rw [mul_apply]
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have h1 := exp_ne_zero (I * ↑δ₁₃)
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fin_cases j <;> rw [Fin.sum_univ_three]
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@ -87,17 +87,19 @@ lemma standardParameterizationAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ
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rw [sin_sq, sin_sq]
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ring
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def sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix :=
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⟨standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by
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def standParam (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix :=
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⟨standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by
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rw [mem_unitaryGroup_iff']
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exact standardParameterizationAsMatrix_unitary θ₁₂ θ₁₃ θ₂₃ δ₁₃⟩
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exact standParamAsMatrix_unitary θ₁₂ θ₁₃ θ₂₃ δ₁₃⟩
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lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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[sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]t = (conj [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]u ×₃ conj [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) := by
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namespace standParam
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lemma cross_product_t (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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[standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]t =
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(conj [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]u ×₃ conj [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) := by
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have h1 := exp_ne_zero (I * ↑δ₁₃)
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funext i
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fin_cases i
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· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg,
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· simp only [tRow, standParam, standParamAsMatrix, neg_mul, exp_neg,
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Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons,
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head_fin_const, cons_val_one, head_cons, Fin.zero_eta, crossProduct, uRow, cRow,
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LinearMap.mk₂_apply, Pi.conj_apply, _root_.map_mul, map_inv₀, ← exp_conj, conj_I, conj_ofReal,
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@ -106,7 +108,7 @@ lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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ring_nf
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rw [sin_sq]
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ring
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· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue, cons_val',
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· simp only [tRow, standParam, standParamAsMatrix, neg_mul, exp_neg, Fin.isValue, cons_val',
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cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const,
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cons_val_one, head_cons, Fin.mk_one, crossProduct, uRow, cRow, LinearMap.mk₂_apply,
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Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub,
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@ -115,7 +117,7 @@ lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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ring_nf
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rw [sin_sq]
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ring
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· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue,
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· simp only [tRow, standParam, standParamAsMatrix, neg_mul, exp_neg, Fin.isValue,
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cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const,
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cons_val_one, head_cons, Fin.reduceFinMk, crossProduct, uRow, cRow, LinearMap.mk₂_apply,
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Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub,
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@ -125,27 +127,26 @@ lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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rw [sin_sq]
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ring
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lemma eq_sP (U : CKMMatrix) {θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ} (hu : [U]u = [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]u)
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(hc : [U]c = [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) (hU : [U]t = conj [U]u ×₃ conj [U]c) :
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U = sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
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lemma eq_rows (U : CKMMatrix) {θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ} (hu : [U]u = [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]u)
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(hc : [U]c = [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) (hU : [U]t = conj [U]u ×₃ conj [U]c) :
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U = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
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apply ext_Rows hu hc
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rw [hU, sP_cross, hu, hc]
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rw [hU, cross_product_t, hu, hc]
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lemma eq_phases_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ δ₁₃' : ℝ) (h : cexp (δ₁₃ * I) = cexp (δ₁₃' * I)) :
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sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ = sP θ₁₂ θ₁₃ θ₂₃ δ₁₃' := by
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simp [sP, standardParameterizationAsMatrix]
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lemma eq_exp_of_phases (θ₁₂ θ₁₃ θ₂₃ δ₁₃ δ₁₃' : ℝ) (h : cexp (δ₁₃ * I) = cexp (δ₁₃' * I)) :
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standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃' := by
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simp [standParam, standParamAsMatrix]
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apply CKMMatrix_ext
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simp
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rw [show exp (I * δ₁₃) = exp (I * δ₁₃') by rw [mul_comm, h, mul_comm]]
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rw [show cexp (-(I * ↑δ₁₃)) = cexp (-(I * ↑δ₁₃')) by rw [exp_neg, exp_neg, mul_comm, h, mul_comm]]
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namespace Invariant
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lemma VusVubVcdSq_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
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open Invariant in
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lemma VusVubVcdSq_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
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(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
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VusVubVcdSq ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
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VusVubVcdSq ⟦standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
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Real.sin θ₁₂ ^ 2 * Real.cos θ₁₃ ^ 2 * Real.sin θ₁₃ ^ 2 * Real.sin θ₂₃ ^ 2 := by
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simp only [VusVubVcdSq, VusAbs, VAbs, VAbs', Fin.isValue, sP, standardParameterizationAsMatrix,
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simp only [VusVubVcdSq, VusAbs, VAbs, VAbs', Fin.isValue, standParam, standParamAsMatrix,
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neg_mul, Quotient.lift_mk, cons_val', cons_val_one, head_cons,
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empty_val', cons_val_fin_one, cons_val_zero, _root_.map_mul, VubAbs, cons_val_two, tail_cons,
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VcbAbs, VudAbs, Complex.abs_ofReal]
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@ -164,12 +165,13 @@ lemma VusVubVcdSq_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Rea
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rw [hx]
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simp
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lemma mulExpδ₃_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
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open Invariant in
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lemma mulExpδ₁₃_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
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(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
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mulExpδ₃ ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
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mulExpδ₁₃ ⟦standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
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sin θ₁₂ * cos θ₁₃ ^ 2 * sin θ₂₃ * sin θ₁₃ * cos θ₁₂ * cos θ₂₃ * cexp (I * δ₁₃) := by
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rw [mulExpδ₃, VusVubVcdSq_sP _ _ _ _ h1 h2 h3 h4 ]
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simp only [jarlskogℂ, sP, standardParameterizationAsMatrix, neg_mul,
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rw [mulExpδ₁₃, VusVubVcdSq_eq _ _ _ _ h1 h2 h3 h4 ]
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simp only [jarlskogℂ, standParam, standParamAsMatrix, neg_mul,
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Quotient.lift_mk, jarlskogℂCKM, Fin.isValue, cons_val', cons_val_one, head_cons,
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empty_val', cons_val_fin_one, cons_val_zero, cons_val_two, tail_cons, _root_.map_mul, ←
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exp_conj, map_neg, conj_I, conj_ofReal, neg_neg, map_sub]
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@ -179,6 +181,7 @@ lemma mulExpδ₃_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Rea
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have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _
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field_simp
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end Invariant
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end standParam
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end
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@ -37,7 +37,7 @@ def C₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₃ V)
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def C₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₂₃ V)
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def δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
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arg (Invariant.mulExpδ₃ V)
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arg (Invariant.mulExpδ₁₃ V)
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section sines
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@ -277,67 +277,7 @@ lemma VtbAbs_eq_C₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VtbAbs V =
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rw [mul_comm]
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exact (mul_div_cancel₀ (VtbAbs V) (VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero ha)).symm
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end VAbs
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namespace Invariant
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lemma mulExpδ₃_sP_inv (V : CKMMatrix) (δ₁₃ : ℝ) :
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mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
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sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧) * cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := by
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refine mulExpδ₃_sP _ _ _ _ ?_ ?_ ?_ ?_
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rw [S₁₂_eq_sin_θ₁₂]
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exact S₁₂_nonneg _
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exact Real.cos_arcsin_nonneg _
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rw [S₂₃_eq_sin_θ₂₃]
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exact S₂₃_nonneg _
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exact Real.cos_arcsin_nonneg _
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lemma mulExpδ₃_eq_zero (V : CKMMatrix) (δ₁₃ : ℝ) :
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mulExpδ₃ ⟦sP (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, VtbAbs_eq_C₂₃_mul_C₁₃,
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← 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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rw [mulExpδ₃_sP_inv]
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simp
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have h1 := exp_ne_zero (I * δ₁₃)
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simp_all
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aesop
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lemma mulExpδ₃_abs (V : CKMMatrix) (δ₁₃ : ℝ) :
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Complex.abs (mulExpδ₃ ⟦sP (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 [mulExpδ₃_sP_inv]
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simp [abs_exp]
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rw [complexAbs_sin_θ₁₃, complexAbs_cos_θ₁₃, complexAbs_sin_θ₁₂, complexAbs_cos_θ₁₂,
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complexAbs_sin_θ₂₃, complexAbs_cos_θ₂₃]
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lemma mulExpδ₃_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
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(h1 : mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
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cexp (arg ( mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ) * I) =
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cexp (δ₁₃ * I) := by
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have h1a := mulExpδ₃_sP_inv V δ₁₃
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have habs := mulExpδ₃_abs V δ₁₃
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have h2 : mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
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Complex.abs (mulExpδ₃ ⟦sP (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 (mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
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have habs_neq_zero : (Complex.abs (mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
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simp
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exact h1
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rw [← mul_right_inj' habs_neq_zero]
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rw [← h2]
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end Invariant
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-- to be moved.
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lemma cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos (θ₁₃ V) = 0) :
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lemma VubAbs_of_cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos (θ₁₃ V) = 0) :
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VubAbs V = 1 := by
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rw [θ₁₃, Real.cos_arcsin, ← VubAbs_eq_S₁₃, Real.sqrt_eq_zero] at h1
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have h2 : VubAbs V ^ 2 = 1 := by linear_combination -(1 * h1)
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@ -352,128 +292,6 @@ lemma cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos (θ₁₃
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rw [_root_.abs_of_nonneg (VAbs_ge_zero 0 2 V)]
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exact VAbs_leq_one 0 2 V
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open CKMMatrix
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section zeroEntries
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variable (a b c d e f : ℝ)
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lemma sP_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₃ ⟦V⟧) = 0) :
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sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
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have hS13 := congrArg ofReal (S₁₃_of_Vub_one (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 (cos_θ₁₃_zero h))
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simp [← C₁₂_eq_ℂcos_θ₁₂] at hC12
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have hS12 := congrArg ofReal (S₁₂_of_Vub_one (cos_θ₁₃_zero h))
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simp [← S₁₂_eq_ℂsin_θ₁₂] at hS12
|
||||
use 0, 0, 0, δ₁₃, 0, -δ₁₃
|
||||
simp [sP, standardParameterizationAsMatrix, h, phaseShift, hS13, hC12, hS12]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
rfl
|
||||
rfl
|
||||
|
||||
lemma sP_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₂ ⟦V⟧) = 0) :
|
||||
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
|
||||
use 0, δ₁₃, δ₁₃, -δ₁₃, 0, - δ₁₃
|
||||
have hb := exp_ne_zero (I * δ₁₃)
|
||||
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
field_simp
|
||||
ring
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
field_simp
|
||||
ring
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
||||
lemma sP_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₂₃ ⟦V⟧) = 0) :
|
||||
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
|
||||
use 0, δ₁₃, 0, 0, 0, - δ₁₃
|
||||
have hb := exp_ne_zero (I * δ₁₃)
|
||||
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
ring_nf
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
||||
lemma sP_sin_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₃ ⟦V⟧) = 0) :
|
||||
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
|
||||
use 0, 0, 0, 0, 0, 0
|
||||
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
apply Or.inr
|
||||
rfl
|
||||
|
||||
lemma sP_sin_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₂ ⟦V⟧) = 0) :
|
||||
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
|
||||
use 0, δ₁₃, δ₁₃, 0, -δ₁₃, - δ₁₃
|
||||
have hb := exp_ne_zero (I * δ₁₃)
|
||||
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
||||
|
||||
lemma sP_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₂₃ ⟦V⟧) = 0) :
|
||||
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
|
||||
use 0, 0, δ₁₃, 0, 0, - δ₁₃
|
||||
have hb := exp_ne_zero (I * δ₁₃)
|
||||
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
ring
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
||||
|
||||
lemma Vs_zero_iff_cos_sin_zero (V : CKMMatrix) :
|
||||
VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0
|
||||
↔ Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨
|
||||
|
|
@ -483,11 +301,182 @@ lemma Vs_zero_iff_cos_sin_zero (V : CKMMatrix) :
|
|||
rw [C₁₂, C₁₃, C₂₃, S₁₂_eq_sin_θ₁₂, S₂₃_eq_sin_θ₂₃, S₁₃_eq_sin_θ₁₃]
|
||||
aesop
|
||||
|
||||
end VAbs
|
||||
|
||||
end zeroEntries
|
||||
|
||||
lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
|
||||
(hV : UCond₁ V) : V = sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (- arg [V]ub) := by
|
||||
namespace standParam
|
||||
open Invariant
|
||||
|
||||
lemma mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) :
|
||||
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
|
||||
sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧) * cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := by
|
||||
refine mulExpδ₁₃_eq _ _ _ _ ?_ ?_ ?_ ?_
|
||||
rw [S₁₂_eq_sin_θ₁₂]
|
||||
exact S₁₂_nonneg _
|
||||
exact Real.cos_arcsin_nonneg _
|
||||
rw [S₂₃_eq_sin_θ₂₃]
|
||||
exact S₂₃_nonneg _
|
||||
exact Real.cos_arcsin_nonneg _
|
||||
|
||||
lemma mulExpδ₁₃_on_param_eq_zero_iff (V : CKMMatrix) (δ₁₃ : ℝ) :
|
||||
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
|
||||
VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 := by
|
||||
rw [VudAbs_eq_C₁₂_mul_C₁₃, VubAbs_eq_S₁₃, VusAbs_eq_S₁₂_mul_C₁₃, VcbAbs_eq_S₂₃_mul_C₁₃, VtbAbs_eq_C₂₃_mul_C₁₃,
|
||||
← ofReal_inj,
|
||||
← ofReal_inj, ← ofReal_inj, ← ofReal_inj, ← ofReal_inj]
|
||||
simp only [ofReal_mul]
|
||||
rw [← S₁₃_eq_ℂsin_θ₁₃, ← S₁₂_eq_ℂsin_θ₁₂, ← S₂₃_eq_ℂsin_θ₂₃,
|
||||
← C₁₃_eq_ℂcos_θ₁₃, ← C₂₃_eq_ℂcos_θ₂₃,← C₁₂_eq_ℂcos_θ₁₂]
|
||||
rw [mulExpδ₁₃_on_param_δ₁₃]
|
||||
simp
|
||||
have h1 := exp_ne_zero (I * δ₁₃)
|
||||
simp_all
|
||||
aesop
|
||||
|
||||
lemma mulExpδ₁₃_on_param_abs (V : CKMMatrix) (δ₁₃ : ℝ) :
|
||||
Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) =
|
||||
sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
|
||||
* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) := by
|
||||
rw [mulExpδ₁₃_on_param_δ₁₃]
|
||||
simp [abs_exp]
|
||||
rw [complexAbs_sin_θ₁₃, complexAbs_cos_θ₁₃, complexAbs_sin_θ₁₂, complexAbs_cos_θ₁₂,
|
||||
complexAbs_sin_θ₂₃, complexAbs_cos_θ₂₃]
|
||||
|
||||
lemma mulExpδ₁₃_on_param_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
|
||||
(h1 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
|
||||
cexp (arg ( mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ) * I) =
|
||||
cexp (δ₁₃ * I) := by
|
||||
have h1a := mulExpδ₁₃_on_param_δ₁₃ V δ₁₃
|
||||
have habs := mulExpδ₁₃_on_param_abs V δ₁₃
|
||||
have h2 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
|
||||
Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
|
||||
rw [habs, h1a]
|
||||
ring_nf
|
||||
nth_rewrite 1 [← abs_mul_exp_arg_mul_I (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
|
||||
have habs_neq_zero : (Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
|
||||
simp
|
||||
exact h1
|
||||
rw [← mul_right_inj' habs_neq_zero]
|
||||
rw [← h2]
|
||||
|
||||
lemma on_param_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₃ ⟦V⟧) = 0) :
|
||||
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
|
||||
have hS13 := congrArg ofReal (S₁₃_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
|
||||
simp [← S₁₃_eq_ℂsin_θ₁₃] at hS13
|
||||
have hC12 := congrArg ofReal (C₁₂_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
|
||||
simp [← C₁₂_eq_ℂcos_θ₁₂] at hC12
|
||||
have hS12 := congrArg ofReal (S₁₂_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
|
||||
simp [← S₁₂_eq_ℂsin_θ₁₂] at hS12
|
||||
use 0, 0, 0, δ₁₃, 0, -δ₁₃
|
||||
simp [standParam, standParamAsMatrix, h, phaseShift, hS13, hC12, hS12]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
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
|
||||
have hb := exp_ne_zero (I * δ₁₃)
|
||||
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
field_simp
|
||||
ring
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
field_simp
|
||||
ring
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
||||
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
|
||||
have hb := exp_ne_zero (I * δ₁₃)
|
||||
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
ring_nf
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
||||
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
|
||||
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
apply Or.inr
|
||||
rfl
|
||||
|
||||
lemma on_param_sin_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₂ ⟦V⟧) = 0) :
|
||||
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
|
||||
use 0, δ₁₃, δ₁₃, 0, -δ₁₃, - δ₁₃
|
||||
have hb := exp_ne_zero (I * δ₁₃)
|
||||
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
||||
|
||||
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
|
||||
have hb := exp_ne_zero (I * δ₁₃)
|
||||
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
|
||||
funext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
|
||||
apply Or.inr
|
||||
rfl
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
ring
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
||||
|
||||
|
||||
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
|
||||
rw [ud_us_neq_zero_iff_ub_neq_one] at hb
|
||||
simp [VAbs, hb]
|
||||
|
|
@ -497,18 +486,18 @@ lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
|
|||
apply add_nonneg (sq_nonneg _) (sq_nonneg _)
|
||||
simp at h1
|
||||
have hx := Vabs_sq_add_neq_zero hb
|
||||
refine eq_sP V ?_ ?_ hV.2.2.2.2
|
||||
refine eq_rows V ?_ ?_ hV.2.2.2.2
|
||||
funext i
|
||||
fin_cases i
|
||||
simp only [uRow, Fin.isValue, Fin.zero_eta, cons_val_zero, sP, standardParameterizationAsMatrix,
|
||||
simp only [uRow, Fin.isValue, Fin.zero_eta, cons_val_zero, standParam, standParamAsMatrix,
|
||||
ofReal_cos, ofReal_sin, ofReal_neg, mul_neg, neg_mul, neg_neg, cons_val', empty_val',
|
||||
cons_val_fin_one, cons_val_one, head_cons, cons_val_two, tail_cons]
|
||||
rw [hV.1, VudAbs_eq_C₁₂_mul_C₁₃ ⟦V⟧]
|
||||
simp [C₁₂, C₁₃]
|
||||
simp [uRow, sP, standardParameterizationAsMatrix]
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
rw [hV.2.1, VusAbs_eq_S₁₂_mul_C₁₃ ⟦V⟧, ← S₁₂_eq_sin_θ₁₂ ⟦V⟧, C₁₃]
|
||||
simp
|
||||
simp [uRow, sP, standardParameterizationAsMatrix]
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
nth_rewrite 1 [← abs_mul_exp_arg_mul_I (V.1 0 2)]
|
||||
rw [show Complex.abs (V.1 0 2) = VubAbs ⟦V⟧ from rfl]
|
||||
rw [VubAbs_eq_S₁₃, ← S₁₃_eq_sin_θ₁₃ ⟦V⟧]
|
||||
|
|
@ -517,8 +506,8 @@ lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
|
|||
simp
|
||||
funext i
|
||||
fin_cases i
|
||||
simp [cRow, sP, standardParameterizationAsMatrix]
|
||||
rw [cd_of_us_or_ud_neq_zero_UCond hb hV]
|
||||
simp [cRow, standParam, standParamAsMatrix]
|
||||
rw [cd_of_fstRowThdColRealCond hb hV]
|
||||
rw [S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧, S₁₂, C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_eq_Vud_div_sqrt hb']
|
||||
rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, S₂₃_of_Vub_neq_one hb', C₂₃_eq_ℂcos_θ₂₃ ⟦V⟧,
|
||||
C₂₃_of_Vub_neq_one hb', S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃]
|
||||
|
|
@ -527,50 +516,52 @@ lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
|
|||
simp [sq]
|
||||
field_simp
|
||||
ring_nf
|
||||
simp [cRow, sP, standardParameterizationAsMatrix]
|
||||
simp [cRow, standParam, standParamAsMatrix]
|
||||
rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₂₃_eq_ℂcos_θ₂₃ ⟦V⟧, S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧,
|
||||
S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧]
|
||||
rw [C₁₂_eq_Vud_div_sqrt hb', C₂₃_of_Vub_neq_one hb', S₁₂, S₁₃, S₂₃_of_Vub_neq_one hb']
|
||||
rw [cs_of_us_or_ud_neq_zero_UCond hb hV]
|
||||
rw [cs_of_fstRowThdColRealCond hb hV]
|
||||
field_simp
|
||||
rw [h1]
|
||||
simp [sq]
|
||||
field_simp
|
||||
ring_nf
|
||||
simp [cRow, sP, standardParameterizationAsMatrix]
|
||||
simp [cRow, standParam, standParamAsMatrix]
|
||||
rw [hV.2.2.1]
|
||||
rw [VcbAbs_eq_S₂₃_mul_C₁₃ ⟦V⟧, S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, C₁₃]
|
||||
simp
|
||||
|
||||
lemma UCond₃_eq_sP {V : CKMMatrix} (hV : UCond₃ V) : V = sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
|
||||
|
||||
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
|
||||
simp [VAbs]
|
||||
rw [hV.2.2.2.1]
|
||||
simp
|
||||
refine eq_sP V ?_ ?_ hV.2.2.2.2.1
|
||||
refine eq_rows V ?_ ?_ hV.2.2.2.2.1
|
||||
funext i
|
||||
fin_cases i
|
||||
simp [uRow, sP, standardParameterizationAsMatrix]
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.1]
|
||||
simp
|
||||
simp [uRow, sP, standardParameterizationAsMatrix]
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.2.1]
|
||||
simp
|
||||
simp [uRow, sP, standardParameterizationAsMatrix]
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃]
|
||||
simp [VAbs]
|
||||
rw [hV.2.2.2.1]
|
||||
simp
|
||||
funext i
|
||||
fin_cases i
|
||||
simp [cRow, sP, standardParameterizationAsMatrix]
|
||||
simp [cRow, standParam, standParamAsMatrix]
|
||||
rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, S₂₃_of_Vub_eq_one h1]
|
||||
rw [S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧, S₁₂_of_Vub_one h1]
|
||||
rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_of_Vub_one h1]
|
||||
rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃_of_Vub_one h1]
|
||||
rw [hV.2.2.2.2.2.1]
|
||||
simp
|
||||
simp [cRow, sP, standardParameterizationAsMatrix]
|
||||
simp [cRow, standParam, standParamAsMatrix]
|
||||
rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, S₂₃_of_Vub_eq_one h1]
|
||||
rw [S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧, S₁₂_of_Vub_one h1]
|
||||
rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_of_Vub_one h1]
|
||||
|
|
@ -580,13 +571,14 @@ lemma UCond₃_eq_sP {V : CKMMatrix} (hV : UCond₃ V) : V = sP (θ₁₂ ⟦V
|
|||
rw [θ₂₃, Real.cos_arcsin]
|
||||
simp at h3
|
||||
rw [h3, S₂₃_of_Vub_eq_one h1, hV.2.2.2.2.2.2]
|
||||
simp [cRow, sP, standardParameterizationAsMatrix]
|
||||
simp [cRow, standParam, standParamAsMatrix]
|
||||
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.2.2.1]
|
||||
simp
|
||||
|
||||
theorem exists_standardParameterization_δ₁₃ (V : CKMMatrix) :
|
||||
∃ (δ₃ : ℝ), V ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ := by
|
||||
obtain ⟨U, hU⟩ := all_eq_abs V
|
||||
|
||||
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
|
||||
have hUV : ⟦U⟧ = ⟦V⟧ := (Quotient.eq.mpr (phaseShiftEquivRelation.symm hU.1))
|
||||
by_cases ha : [V]ud ≠ 0 ∨ [V]us ≠ 0
|
||||
· have haU : [U]ud ≠ 0 ∨ [U]us ≠ 0 := by -- should be much simplier
|
||||
|
|
@ -598,7 +590,7 @@ theorem exists_standardParameterization_δ₁₃ (V : CKMMatrix) :
|
|||
rw [hUV] at hna
|
||||
simp [VAbs] at hna
|
||||
simp_all
|
||||
have hU' := UCond₁_eq_sP haU hU.2
|
||||
have hU' := eq_standParam_of_fstRowThdColRealCond haU hU.2
|
||||
rw [hU'] at hU
|
||||
use (- arg ([U]ub))
|
||||
rw [← hUV]
|
||||
|
|
@ -610,23 +602,22 @@ theorem exists_standardParameterization_δ₁₃ (V : CKMMatrix) :
|
|||
simp [VAbs]
|
||||
exact ha
|
||||
simpa [not_or, VAbs] using h1
|
||||
have ⟨U2, hU2⟩ := UCond₃_exists haU hU.2
|
||||
have ⟨U2, hU2⟩ := ubOnePhaseCond_hold_up_to_equiv_of_ub_one haU hU.2
|
||||
have hUVa2 : V ≈ U2 := phaseShiftEquivRelation.trans hU.1 hU2.1
|
||||
have hUV2 : ⟦U2⟧ = ⟦V⟧ := (Quotient.eq.mpr (phaseShiftEquivRelation.symm hUVa2))
|
||||
have hx := UCond₃_eq_sP hU2.2
|
||||
have hx := eq_standParam_of_ubOnePhaseCond hU2.2
|
||||
use 0
|
||||
rw [← hUV2, ← hx]
|
||||
exact hUVa2
|
||||
|
||||
open Invariant in
|
||||
theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
|
||||
V ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧) := by
|
||||
obtain ⟨δ₁₃', hδ₃⟩ := exists_standardParameterization_δ₁₃ V
|
||||
V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧) := by
|
||||
obtain ⟨δ₁₃', hδ₃⟩ := exists_δ₁₃ V
|
||||
have hSV := (Quotient.eq.mpr (hδ₃))
|
||||
by_cases h : Invariant.mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
|
||||
have h1 := Invariant.mulExpδ₃_neq_zero_arg V δ₁₃' h
|
||||
have h2 := eq_phases_sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
|
||||
(δ₁₃ ⟦V⟧) (by rw [← h1, ← hSV, δ₁₃, Invariant.mulExpδ₃])
|
||||
by_cases h : Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
|
||||
have h2 := eq_exp_of_phases (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
|
||||
(δ₁₃ ⟦V⟧) (by rw [← mulExpδ₁₃_on_param_neq_zero_arg V δ₁₃' h, ← hSV, δ₁₃, Invariant.mulExpδ₁₃])
|
||||
rw [h2] at hδ₃
|
||||
exact hδ₃
|
||||
simp at h
|
||||
|
|
@ -634,22 +625,27 @@ theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
|
|||
rw [hSV, δ₁₃, h]
|
||||
simp
|
||||
rw [h1]
|
||||
rw [mulExpδ₃_eq_zero, Vs_zero_iff_cos_sin_zero] at h
|
||||
cases' h with h h
|
||||
exact phaseShiftEquivRelation.trans hδ₃ (sP_cos_θ₁₂_eq_zero δ₁₃' h )
|
||||
cases' h with h h
|
||||
exact phaseShiftEquivRelation.trans hδ₃ (sP_cos_θ₁₃_eq_zero δ₁₃' h )
|
||||
cases' h with h h
|
||||
exact phaseShiftEquivRelation.trans hδ₃ (sP_cos_θ₂₃_eq_zero δ₁₃' h )
|
||||
cases' h with h h
|
||||
exact phaseShiftEquivRelation.trans hδ₃ (sP_sin_θ₁₂_eq_zero δ₁₃' h )
|
||||
cases' h with h h
|
||||
exact phaseShiftEquivRelation.trans hδ₃ (sP_sin_θ₁₃_eq_zero δ₁₃' h )
|
||||
exact phaseShiftEquivRelation.trans hδ₃ (sP_sin_θ₂₃_eq_zero δ₁₃' h )
|
||||
rw [mulExpδ₁₃_on_param_eq_zero_iff, Vs_zero_iff_cos_sin_zero] at h
|
||||
refine phaseShiftEquivRelation.trans hδ₃ ?_
|
||||
rcases h with h | h | h | h | h | h
|
||||
exact on_param_cos_θ₁₂_eq_zero δ₁₃' h
|
||||
exact on_param_cos_θ₁₃_eq_zero δ₁₃' h
|
||||
exact on_param_cos_θ₂₃_eq_zero δ₁₃' h
|
||||
exact on_param_sin_θ₁₂_eq_zero δ₁₃' h
|
||||
exact on_param_sin_θ₁₃_eq_zero δ₁₃' h
|
||||
exact on_param_sin_θ₂₃_eq_zero δ₁₃' h
|
||||
|
||||
lemma exists_standardParameterization (V : CKMMatrix) :
|
||||
∃ (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ), V ≈ sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
|
||||
|
||||
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⟧
|
||||
exact eq_standardParameterization_δ₃ V
|
||||
|
||||
end standParam
|
||||
|
||||
|
||||
open CKMMatrix
|
||||
|
||||
|
||||
|
||||
end
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue