/- Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved. Released under Apache 2.0 license. Authors: Joseph Tooby-Smith -/ import HepLean.FlavorPhysics.CKMMatrix.Basic import HepLean.FlavorPhysics.CKMMatrix.Rows import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom import HepLean.FlavorPhysics.CKMMatrix.Invariants import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic import Mathlib.Analysis.SpecialFunctions.Complex.Arg /-! # Standard parameters for the CKM Matrix Given a CKM matrix `V` we can extract four real numbers `θ₁₂`, `θ₁₃`, `θ₂₃` and `δ₁₃`. 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 open CKMMatrix noncomputable section /-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₂` in the standard parameterization. --/ def S₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := VusAbs V / (√ (VudAbs V ^ 2 + VusAbs V ^ 2)) /-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₃` in the standard parameterization. --/ def S₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := VubAbs V /-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the standard parameterization. --/ def S₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := if VubAbs V = 1 then VcdAbs V else VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) /-- Given a CKM matrix `V` the real number corresponding to `θ₁₂` in the standard parameterization. --/ def θ₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₂ V) /-- Given a CKM matrix `V` the real number corresponding to `θ₁₃` in the standard parameterization. --/ def θ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₃ V) /-- Given a CKM matrix `V` the real number corresponding to `θ₂₃` in the standard parameterization. --/ def θ₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₂₃ V) /-- Given a CKM matrix `V` the real number corresponding to `cos θ₁₂` in the standard parameterization. --/ def C₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₂ V) /-- Given a CKM matrix `V` the real number corresponding to `cos θ₁₃` in the standard parameterization. --/ def C₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₃ V) /-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the standard parameterization. --/ def C₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₂₃ V) /-- Given a CKM matrix `V` the real number corresponding to the phase `δ₁₃` in the standard parameterization. --/ def δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := arg (Invariant.mulExpδ₁₃ V) section sines lemma S₁₂_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₁₂ V := by rw [S₁₂, div_nonneg_iff] apply Or.inl apply (And.intro (VAbs_ge_zero 0 1 V) (Real.sqrt_nonneg (VudAbs V ^ 2 + VusAbs V ^ 2))) lemma S₁₃_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₁₃ V := VAbs_ge_zero 0 2 V lemma S₂₃_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₂₃ V := by by_cases ha : VubAbs V = 1 rw [S₂₃, if_pos ha] exact VAbs_ge_zero 1 0 V rw [S₂₃, if_neg ha, @div_nonneg_iff] apply Or.inl apply And.intro (VAbs_ge_zero 1 2 V) (Real.sqrt_nonneg (VudAbs V ^ 2 + VusAbs V ^ 2)) lemma S₁₂_leq_one (V : Quotient CKMMatrixSetoid) : S₁₂ V ≤ 1 := by rw [S₁₂, @div_le_one_iff] by_cases h1 : √(VudAbs V ^ 2 + VusAbs V ^ 2) = 0 simp [h1] have h2 := le_iff_eq_or_lt.mp (Real.sqrt_nonneg (VudAbs V ^ 2 + VusAbs V ^ 2)) have h3 : 0 < √(VudAbs V ^ 2 + VusAbs V ^ 2) := by cases' h2 with h2 h2 simp_all exact h2 apply Or.inl simp_all rw [Real.le_sqrt (VAbs_ge_zero 0 1 V) (le_of_lt h3)] simp only [Fin.isValue, le_add_iff_nonneg_left] exact sq_nonneg (VAbs 0 0 V) lemma S₁₃_leq_one (V : Quotient CKMMatrixSetoid) : S₁₃ V ≤ 1 := VAbs_leq_one 0 2 V lemma S₂₃_leq_one (V : Quotient CKMMatrixSetoid) : S₂₃ V ≤ 1 := by by_cases ha : VubAbs V = 1 rw [S₂₃, if_pos ha] exact VAbs_leq_one 1 0 V rw [S₂₃, if_neg ha, @div_le_one_iff] by_cases h1 : √(VudAbs V ^ 2 + VusAbs V ^ 2) = 0 simp [h1] have h2 := le_iff_eq_or_lt.mp (Real.sqrt_nonneg (VudAbs V ^ 2 + VusAbs V ^ 2)) have h3 : 0 < √(VudAbs V ^ 2 + VusAbs V ^ 2) := by cases' h2 with h2 h2 simp_all exact h2 apply Or.inl simp_all rw [Real.le_sqrt (VAbs_ge_zero 1 2 V) (le_of_lt h3)] rw [VudAbs_sq_add_VusAbs_sq, ← VcbAbs_sq_add_VtbAbs_sq] simp only [Fin.isValue, VcbAbs, VtbAbs, le_add_iff_nonneg_right] exact sq_nonneg (VAbs 2 2 V) lemma S₁₂_eq_sin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₂ V) = S₁₂ V := Real.sin_arcsin (le_trans (by simp) (S₁₂_nonneg V)) (S₁₂_leq_one V) lemma S₁₃_eq_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₃ V) = S₁₃ V := Real.sin_arcsin (le_trans (by simp) (S₁₃_nonneg V)) (S₁₃_leq_one V) lemma S₂₃_eq_sin_θ₂₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₂₃ V) = S₂₃ V := Real.sin_arcsin (le_trans (by simp) (S₂₃_nonneg V)) (S₂₃_leq_one V) lemma S₁₂_eq_ℂsin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.sin (θ₁₂ V) = S₁₂ V := (ofReal_sin _).symm.trans (congrArg ofReal (S₁₂_eq_sin_θ₁₂ V)) lemma S₁₃_eq_ℂsin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.sin (θ₁₃ V) = S₁₃ V := (ofReal_sin _).symm.trans (congrArg ofReal (S₁₃_eq_sin_θ₁₃ V)) lemma S₂₃_eq_ℂsin_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.sin (θ₂₃ V) = S₂₃ V := (ofReal_sin _).symm.trans (congrArg ofReal (S₂₃_eq_sin_θ₂₃ V)) lemma complexAbs_sin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.sin (θ₁₂ V)) = sin (θ₁₂ V):= by rw [S₁₂_eq_ℂsin_θ₁₂, Complex.abs_ofReal, ofReal_inj, abs_eq_self] exact S₁₂_nonneg _ lemma complexAbs_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.sin (θ₁₃ V)) = sin (θ₁₃ V):= by rw [S₁₃_eq_ℂsin_θ₁₃, Complex.abs_ofReal, ofReal_inj, abs_eq_self] exact S₁₃_nonneg _ lemma complexAbs_sin_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.sin (θ₂₃ V)) = sin (θ₂₃ V):= by rw [S₂₃_eq_ℂsin_θ₂₃, Complex.abs_ofReal, ofReal_inj, abs_eq_self] exact S₂₃_nonneg _ lemma S₁₂_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₁₂ V = 0 := by have h1 : 1 - VubAbs V ^ 2 = VudAbs V ^ 2 + VusAbs V ^ 2 := by linear_combination - (VAbs_sum_sq_row_eq_one V 0) simp [S₁₂, ← h1, ha] lemma S₁₃_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₁₃ V = 1 := by rw [S₁₃, ha] lemma S₂₃_of_Vub_eq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₂₃ V = VcdAbs V := by rw [S₂₃, if_pos ha] lemma S₂₃_of_Vub_neq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) : S₂₃ V = VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := by rw [S₂₃, if_neg ha] end sines section cosines lemma C₁₂_eq_ℂcos_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₁₂ V) = C₁₂ V := by simp [C₁₂] lemma C₁₃_eq_ℂcos_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₁₃ V) = C₁₃ V := by simp [C₁₃] lemma C₂₃_eq_ℂcos_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₂₃ V) = C₂₃ V := by simp [C₂₃] lemma complexAbs_cos_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.cos (θ₁₂ V)) = cos (θ₁₂ V):= by rw [C₁₂_eq_ℂcos_θ₁₂, Complex.abs_ofReal] simp only [ofReal_inj, abs_eq_self] exact Real.cos_arcsin_nonneg _ lemma complexAbs_cos_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.cos (θ₁₃ V)) = cos (θ₁₃ V):= by rw [C₁₃_eq_ℂcos_θ₁₃, Complex.abs_ofReal] simp only [ofReal_inj, abs_eq_self] exact Real.cos_arcsin_nonneg _ lemma complexAbs_cos_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.cos (θ₂₃ V)) = cos (θ₂₃ V):= by rw [C₂₃_eq_ℂcos_θ₂₃, Complex.abs_ofReal] simp only [ofReal_inj, abs_eq_self] exact Real.cos_arcsin_nonneg _ lemma S₁₂_sq_add_C₁₂_sq (V : Quotient CKMMatrixSetoid) : S₁₂ V ^ 2 + C₁₂ V ^ 2 = 1 := by rw [← S₁₂_eq_sin_θ₁₂ V, C₁₂] exact Real.sin_sq_add_cos_sq (θ₁₂ V) lemma S₁₃_sq_add_C₁₃_sq (V : Quotient CKMMatrixSetoid) : S₁₃ V ^ 2 + C₁₃ V ^ 2 = 1 := by rw [← S₁₃_eq_sin_θ₁₃ V, C₁₃] exact Real.sin_sq_add_cos_sq (θ₁₃ V) lemma S₂₃_sq_add_C₂₃_sq (V : Quotient CKMMatrixSetoid) : S₂₃ V ^ 2 + C₂₃ V ^ 2 = 1 := by rw [← S₂₃_eq_sin_θ₂₃ V, C₂₃] exact Real.sin_sq_add_cos_sq (θ₂₃ V) lemma C₁₂_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : C₁₂ V = 1 := by rw [C₁₂, θ₁₂, Real.cos_arcsin, S₁₂_of_Vub_one ha] simp lemma C₁₃_of_Vub_eq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : C₁₃ V = 0 := by rw [C₁₃, θ₁₃, Real.cos_arcsin, S₁₃, ha] simp --rename lemma C₁₂_eq_Vud_div_sqrt {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) : C₁₂ V = VudAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := by rw [C₁₂, θ₁₂, Real.cos_arcsin, S₁₂, div_pow, Real.sq_sqrt] rw [one_sub_div] simp only [VudAbs, Fin.isValue, VusAbs, add_sub_cancel_right] rw [Real.sqrt_div] rw [Real.sqrt_sq] exact VAbs_ge_zero 0 0 V exact sq_nonneg (VAbs 0 0 V) exact VAbsub_neq_zero_Vud_Vus_neq_zero ha exact (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V))) --rename lemma C₁₃_eq_add_sq (V : Quotient CKMMatrixSetoid) : C₁₃ V = √ (VudAbs V ^ 2 + VusAbs V ^ 2) := by rw [C₁₃, θ₁₃, Real.cos_arcsin, S₁₃] have h1 : 1 - VubAbs V ^ 2 = VudAbs V ^ 2 + VusAbs V ^ 2 := by linear_combination - (VAbs_sum_sq_row_eq_one V 0) rw [h1] lemma C₂₃_of_Vub_neq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) : C₂₃ V = VtbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := by rw [C₂₃, θ₂₃, Real.cos_arcsin, S₂₃_of_Vub_neq_one ha, div_pow, Real.sq_sqrt] rw [VudAbs_sq_add_VusAbs_sq, ← VcbAbs_sq_add_VtbAbs_sq] rw [one_sub_div] simp only [VcbAbs, Fin.isValue, VtbAbs, add_sub_cancel_left] rw [Real.sqrt_div (sq_nonneg (VAbs 2 2 V))] rw [Real.sqrt_sq (VAbs_ge_zero 2 2 V)] rw [VcbAbs_sq_add_VtbAbs_sq, ← VudAbs_sq_add_VusAbs_sq ] exact VAbsub_neq_zero_Vud_Vus_neq_zero ha exact (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V))) end cosines section VAbs -- rename to VudAbs_standard_param lemma VudAbs_eq_C₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VudAbs V = C₁₂ V * C₁₃ V := by by_cases ha : VubAbs V = 1 change VAbs 0 0 V = C₁₂ V * C₁₃ V rw [VAbs_thd_eq_one_fst_eq_zero ha] rw [C₁₃, θ₁₃, Real.cos_arcsin, S₁₃, ha] simp only [one_pow, sub_self, Real.sqrt_zero, mul_zero] rw [C₁₂_eq_Vud_div_sqrt ha, C₁₃, θ₁₃, Real.cos_arcsin, S₁₃] have h1 : 1 - VubAbs V ^ 2 = VudAbs V ^ 2 + VusAbs V ^ 2 := by linear_combination - (VAbs_sum_sq_row_eq_one V 0) rw [h1, mul_comm] exact (mul_div_cancel₀ (VudAbs V) (VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero ha)).symm lemma VusAbs_eq_S₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VusAbs V = S₁₂ V * C₁₃ V := by rw [C₁₃, θ₁₃, Real.cos_arcsin, S₁₂, S₁₃] have h1 : 1 - VubAbs V ^ 2 = VudAbs V ^ 2 + VusAbs V ^ 2 := by linear_combination - (VAbs_sum_sq_row_eq_one V 0) rw [h1] rw [mul_comm] by_cases ha : VubAbs V = 1 rw [ha] at h1 simp only [one_pow, sub_self, Fin.isValue] at h1 rw [← h1] simp only [Real.sqrt_zero, div_zero, mul_zero] exact VAbs_thd_eq_one_snd_eq_zero ha have h2 := VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero ha exact (mul_div_cancel₀ (VusAbs V) h2).symm lemma VubAbs_eq_S₁₃ (V : Quotient CKMMatrixSetoid) : VubAbs V = S₁₃ V := rfl lemma VcbAbs_eq_S₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VcbAbs V = S₂₃ V * C₁₃ V := by by_cases ha : VubAbs V = 1 rw [C₁₃_of_Vub_eq_one ha] simp only [VcbAbs, Fin.isValue, mul_zero] exact VAbs_fst_col_eq_one_snd_eq_zero ha rw [S₂₃_of_Vub_neq_one ha, C₁₃_eq_add_sq] rw [mul_comm] exact (mul_div_cancel₀ (VcbAbs V) (VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero ha)).symm lemma VtbAbs_eq_C₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VtbAbs V = C₂₃ V * C₁₃ V := by by_cases ha : VubAbs V = 1 rw [C₁₃_of_Vub_eq_one ha] simp only [VtbAbs, Fin.isValue, mul_zero] exact VAbs_fst_col_eq_one_thd_eq_zero ha rw [C₂₃_of_Vub_neq_one ha, C₁₃_eq_add_sq] rw [mul_comm] exact (mul_div_cancel₀ (VtbAbs V) (VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero ha)).symm lemma VubAbs_of_cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos (θ₁₃ V) = 0) : VubAbs V = 1 := by rw [θ₁₃, Real.cos_arcsin, ← VubAbs_eq_S₁₃, Real.sqrt_eq_zero] at h1 have h2 : VubAbs V ^ 2 = 1 := by linear_combination -(1 * h1) simp at h2 cases' h2 with h2 h2 exact h2 have h3 := VAbs_ge_zero 0 2 V rw [h2] at h3 simp at h3 linarith simp only [VubAbs, Fin.isValue, sub_nonneg, sq_le_one_iff_abs_le_one] rw [_root_.abs_of_nonneg (VAbs_ge_zero 0 2 V)] exact VAbs_leq_one 0 2 V 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 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦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] rw [C₁₂, C₁₃, C₂₃, S₁₂_eq_sin_θ₁₂, S₂₃_eq_sin_θ₂₃, S₁₃_eq_sin_θ₁₃] aesop end VAbs 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 only [mul_eq_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, ofReal_zero] 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 only [ne_eq, ofReal_eq_zero, map_eq_zero] 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 only [mul_zero, add_zero, neg_inj] field_simp ring ring_nf field_simp ring_nf change _ = _ + _ * 0 simp only [mul_zero, add_zero] field_simp ring 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 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 only [mul_zero, add_zero] 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 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 only [mul_zero, add_zero, neg_inj] ring field_simp ring_nf field_simp ring_nf change _ = _ + _ * 0 simp only [mul_zero, add_zero, neg_inj] ring_nf field_simp 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 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 only [mul_zero, add_zero, neg_inj] ring 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 rw [ud_us_neq_zero_iff_ub_neq_one] at hb simp [VAbs, hb] have h1 : ofReal (√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) = ofReal ((VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) ) := by rw [Real.mul_self_sqrt ] apply add_nonneg (sq_nonneg _) (sq_nonneg _) simp at h1 have hx := Vabs_sq_add_neq_zero hb 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, 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, standParam, standParamAsMatrix] rw [hV.2.1, VusAbs_eq_S₁₂_mul_C₁₃ ⟦V⟧, ← S₁₂_eq_sin_θ₁₂ ⟦V⟧, C₁₃] simp only [ofReal_mul, ofReal_sin, ofReal_cos] 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⟧] simp only [ofReal_sin, Fin.isValue, mul_eq_mul_left_iff] ring_nf simp only [true_or] funext i fin_cases i 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] field_simp rw [h1] simp [sq] field_simp ring_nf 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_fstRowThdColRealCond hb hV] field_simp rw [h1] simp [sq] field_simp ring_nf simp [cRow, standParam, standParamAsMatrix] rw [hV.2.2.1] 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 simp [VAbs] rw [hV.2.2.2.1] simp refine eq_rows V ?_ ?_ hV.2.2.2.2.1 funext i fin_cases i simp [uRow, standParam, standParamAsMatrix] rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.1] simp only [ofReal_zero, mul_zero] simp [uRow, standParam, standParamAsMatrix] rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.2.1] simp only [ofReal_zero, mul_zero] simp [uRow, standParam, standParamAsMatrix] rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃] simp [VAbs] rw [hV.2.2.2.1] simp only [_root_.map_one, ofReal_one] funext i fin_cases i 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 only [VcdAbs, Fin.isValue, ofReal_zero, zero_mul, neg_zero, ofReal_one, mul_one, one_mul, zero_sub] 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] simp only [Fin.isValue, ofReal_one, one_mul, ofReal_zero, mul_one, VcdAbs, zero_mul, sub_zero] have h3 : (Real.cos (θ₂₃ ⟦V⟧) : ℂ) = √(1 - S₂₃ ⟦V⟧ ^ 2) := by 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, standParam, standParamAsMatrix] 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 have hUV : ⟦U⟧ = ⟦V⟧ := (Quotient.eq.mpr (phaseShiftRelation_equiv.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 by_contra hn simp [not_or] at hn have hna : VudAbs ⟦U⟧ = 0 ∧ VusAbs ⟦U⟧ =0 := by simp [VAbs] exact hn rw [hUV] at hna simp [VAbs] at hna simp_all have hU' := eq_standParam_of_fstRowThdColRealCond haU hU.2 rw [hU'] at hU use (- arg ([U]ub)) rw [← hUV] exact hU.1 · have haU : ¬ ([U]ud ≠ 0 ∨ [U]us ≠ 0) := by -- should be much simplier simp [not_or] at ha have h1 : VudAbs ⟦U⟧ = 0 ∧ VusAbs ⟦U⟧ = 0 := by rw [hUV] simp [VAbs] exact ha simpa [not_or, VAbs] using h1 have ⟨U2, hU2⟩ := ubOnePhaseCond_hold_up_to_equiv_of_ub_one haU hU.2 have hUVa2 : V ≈ U2 := phaseShiftRelation_equiv.trans hU.1 hU2.1 have hUV2 : ⟦U2⟧ = ⟦V⟧ := (Quotient.eq.mpr (phaseShiftRelation_equiv.symm hUVa2)) 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 ≈ 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.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 have h1 : δ₁₃ ⟦V⟧ = 0 := by rw [hSV, δ₁₃, h] simp rw [h1] rw [mulExpδ₁₃_on_param_eq_zero_iff, Vs_zero_iff_cos_sin_zero] at h refine phaseShiftRelation_equiv.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 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