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refactor: Major refactor of CKMMatrix

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jstoobysmith 2024-04-29 08:13:52 -04:00
parent fe63fc9994
commit ff89c3f79d
9 changed files with 1096 additions and 1512 deletions
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158
HepLean/FlavorPhysics/CKMMatrix/Basic.lean
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@ -14,7 +14,7 @@ noncomputable section
@[simp]
def phaseShiftMatrix (a b c : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
![![ cexp (I * a), 0, 0], ![0, cexp (I * b), 0], ![0, 0, cexp (I * c)]]
![![cexp (I * a), 0, 0], ![0, cexp (I * b), 0], ![0, 0, cexp (I * c)]]
lemma phaseShiftMatrix_one : phaseShiftMatrix 0 0 0 = 1 := by
ext i j
@ -266,122 +266,58 @@ abbrev VtsAbs := VAbs 2 1
@[simp]
abbrev VtbAbs := VAbs 2 2
lemma VAbs_ge_zero (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : 0 ≤ VAbs i j V := by
obtain ⟨V, hV⟩ := Quot.exists_rep V
rw [← hV]
exact Complex.abs.nonneg _
lemma VAbs_sum_sq_row_eq_one (V : Quotient CKMMatrixSetoid) (i : Fin 3) :
(VAbs i 0 V) ^ 2 + (VAbs i 1 V) ^ 2 + (VAbs i 2 V) ^ 2 = 1 := by
obtain ⟨V, hV⟩ := Quot.exists_rep V
subst hV
have hV := V.prop
rw [mem_unitaryGroup_iff] at hV
have ht := congrFun (congrFun hV i) i
simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
rw [mul_conj, mul_conj, mul_conj] at ht
repeat rw [← Complex.sq_abs] at ht
rw [← ofReal_inj]
simpa using ht
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
subst hV
have hV := V.prop
rw [mem_unitaryGroup_iff'] at hV
have ht := congrFun (congrFun hV i) i
simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
rw [mul_comm, mul_conj, mul_comm, mul_conj, mul_comm, mul_conj] at ht
repeat rw [← Complex.sq_abs] at ht
rw [← ofReal_inj]
simpa using ht
lemma VAbs_leq_one (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : VAbs i j V ≤ 1 := by
have h := VAbs_sum_sq_row_eq_one V i
fin_cases j
change VAbs i 0 V ≤ 1
nlinarith
change VAbs i 1 V ≤ 1
nlinarith
change VAbs i 2 V ≤ 1
nlinarith
lemma VAbs_thd_eq_one_fst_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV : VAbs i 2 V = 1) :
VAbs i 0 V = 0 := by
have h := VAbs_sum_sq_row_eq_one V i
rw [hV] at h
simp at h
nlinarith
lemma VAbs_thd_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV : VAbs i 2 V = 1) :
VAbs i 1 V = 0 := by
have h := VAbs_sum_sq_row_eq_one V i
rw [hV] at h
simp at h
nlinarith
lemma VAbs_fst_col_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
(hV : VAbs 0 i V = 1) : VAbs 1 i V = 0 := by
have h := VAbs_sum_sq_col_eq_one V i
rw [hV] at h
simp at h
nlinarith
lemma VAbs_fst_col_eq_one_thd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
(hV : VAbs 0 i V = 1) : VAbs 2 i V = 0 := by
have h := VAbs_sum_sq_col_eq_one V i
rw [hV] at h
simp at h
nlinarith
lemma VAbs_thd_neq_one_fst_snd_sq_neq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
(hV : VAbs i 2 V ≠ 1) : (VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2) ≠ 0 := by
have h1 : 1 - VAbs i 2 V ^ 2 = VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2 := by
linear_combination - (VAbs_sum_sq_row_eq_one V i)
rw [← h1]
by_contra h
have h2 : VAbs i 2 V ^2 = 1 := by
nlinarith
simp only [Fin.isValue, sq_eq_one_iff] at h2
have h3 : 0 ≤ VAbs i 2 V := VAbs_ge_zero i 2 V
have h4 : VAbs i 2 V = 1 := by
nlinarith
exact hV h4
lemma VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
(hV : VAbs i 2 V ≠ 1) : √(VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2) ≠ 0 := by
rw [Real.sqrt_ne_zero (Left.add_nonneg (sq_nonneg (VAbs i 0 V)) (sq_nonneg (VAbs i 1 V)))]
exact VAbs_thd_neq_one_fst_snd_sq_neq_zero hV
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 VudAbs_sq_add_VusAbs_sq : VudAbs V ^ 2 + VusAbs V ^2 = 1 - VubAbs V ^2 := by
linear_combination (VAbs_sum_sq_row_eq_one V 0)
namespace CKMMatrix
open ComplexConjugate
lemma fst_row_snd_row (V : CKMMatrix) : V.1 1 0 * conj (V.1 0 0) + V.1 1 1 * conj (V.1 0 1)
+ V.1 1 2 * conj (V.1 0 2) = 0 := by
have hV := V.prop
rw [mem_unitaryGroup_iff] at hV
have ht := congrFun (congrFun hV 1) 0
simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
exact ht
section ratios
-- A
def Rubud (V : CKMMatrix) : ℂ := [V]ub / [V]ud
scoped[CKMMatrix] notation (name := ub_ud_ratio) "[" V "]ub|ud" => Rubud V
-- B
def Rusud (V : CKMMatrix) : ℂ := [V]us / [V]ud
scoped[CKMMatrix] notation (name := us_ud_ratio) "[" V "]us|ud" => Rusud V
def Rudus (V : CKMMatrix) : ℂ := [V]ud / [V]us
scoped[CKMMatrix] notation (name := ud_us_ratio) "[" V "]ud|us" => Rudus V
def Rubus (V : CKMMatrix) : ℂ := [V]ub / [V]us
scoped[CKMMatrix] notation (name := ub_us_ratio) "[" V "]ub|us" => Rubus V
-- D
def Rcdcb (V : CKMMatrix) : ℂ := [V]cd / [V]cb
scoped[CKMMatrix] notation (name := cd_cb_ratio) "[" V "]cd|cb" => Rcdcb V
lemma Rcdcb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cd = Rcdcb V * [V]cb := by
rw [Rcdcb]
exact (div_mul_cancel₀ (V.1 1 0) h).symm
-- C'
def Rcscb (V : CKMMatrix) : ℂ := [V]cs / [V]cb
scoped[CKMMatrix] notation (name := cs_cb_ratio) "[" V "]cs|cb" => Rcscb V
lemma Rcscb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cs = Rcscb V * [V]cb := by
rw [Rcscb]
exact (div_mul_cancel₀ [V]cs h).symm
lemma fst_row_thd_row (V : CKMMatrix) : V.1 2 0 * conj (V.1 0 0) + V.1 2 1 * conj (V.1 0 1)
+ V.1 2 2 * conj (V.1 0 2) = 0 := by
have hV := V.prop
rw [mem_unitaryGroup_iff] at hV
have ht := congrFun (congrFun hV 2) 0
simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
exact ht
def Rtb!cbud (V : CKMMatrix) : ℂ := conj [V]tb / ([V]cb * [V]ud)
scoped[CKMMatrix] notation (name := tb_cb_ud_ratio) "[" V "]tb*|cb|ud" => Rtb!cbud V
def Rtb!cbus (V : CKMMatrix) : ℂ := conj [V]tb / ([V]cb * [V]us)
scoped[CKMMatrix] notation (name := tb_cb_us_ratio) "[" V "]tb*|cb|us" => Rtb!cbus V
end ratios
end CKMMatrix

51
HepLean/FlavorPhysics/CKMMatrix/Invariants.lean Normal file
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@ -0,0 +1,51 @@
/-
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 Mathlib.Analysis.SpecialFunctions.Complex.Arg
open Matrix Complex
open ComplexConjugate
open CKMMatrix
noncomputable section
namespace Invariant
@[simps!]
def jarlskogℂCKM (V : CKMMatrix) : ℂ :=
[V]us * [V]cb * conj [V]ub * conj [V]cs
lemma jarlskogℂCKM_equiv (V U : CKMMatrix) (h : V ≈ U) :
jarlskogℂCKM V = jarlskogℂCKM U := by
obtain ⟨a, b, c, e, f, g, h⟩ := h
change V = phaseShiftApply U a b c e f g at h
rw [h]
simp only [jarlskogℂCKM, Fin.isValue, phaseShiftApply.ub,
phaseShiftApply.us, phaseShiftApply.cb, phaseShiftApply.cs]
simp [← exp_conj, conj_ofReal, exp_add, exp_neg]
have ha : cexp (↑a * I) ≠ 0 := exp_ne_zero _
have hb : cexp (↑b * I) ≠ 0 := exp_ne_zero _
have hf : cexp (↑f * I) ≠ 0 := exp_ne_zero _
have hg : cexp (↑g * I) ≠ 0 := exp_ne_zero _
field_simp
ring
@[simp]
def jarlskogℂ : Quotient CKMMatrixSetoid → ℂ :=
Quotient.lift jarlskogℂCKM jarlskogℂCKM_equiv
def VusVubVcdSq (V : Quotient CKMMatrixSetoid) : ℝ :=
VusAbs V ^ 2 * VubAbs V ^ 2 * VcbAbs V ^2 / (VudAbs V ^ 2 + VusAbs V ^2)
def mulExpδ₃ (V : Quotient CKMMatrixSetoid) : ℂ :=
jarlskogℂ V + VusVubVcdSq V
end Invariant
end

176
HepLean/FlavorPhysics/CKMMatrix/Jarlskog.lean
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@ -1,176 +0,0 @@
/-
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.Ratios
import HepLean.FlavorPhysics.CKMMatrix.StandardParameters
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
open Matrix Complex
open ComplexConjugate
open CKMMatrix
noncomputable section
@[simps!]
def jarlskogComplexCKM (V : CKMMatrix) : ℂ :=
[V]us * [V]cb * conj [V]ub * conj [V]cs
lemma jarlskogComplexCKM_equiv (V U : CKMMatrix) (h : V ≈ U) :
jarlskogComplexCKM V = jarlskogComplexCKM U := by
obtain ⟨a, b, c, e, f, g, h⟩ := h
change V = phaseShiftApply U a b c e f g at h
rw [h]
simp only [jarlskogComplexCKM, Fin.isValue, phaseShiftApply.ub,
phaseShiftApply.us, phaseShiftApply.cb, phaseShiftApply.cs]
simp [← exp_conj, conj_ofReal, exp_add, exp_neg]
have ha : cexp (↑a * I) ≠ 0 := exp_ne_zero _
have hb : cexp (↑b * I) ≠ 0 := exp_ne_zero _
have hf : cexp (↑f * I) ≠ 0 := exp_ne_zero _
have hg : cexp (↑g * I) ≠ 0 := exp_ne_zero _
field_simp
ring
def inv₁ (V : Quotient CKMMatrixSetoid) : ℝ :=
VusAbs V ^ 2 * VubAbs V ^ 2 * VcbAbs V ^2 /(VudAbs V ^ 2 + VusAbs V ^2)
lemma inv₁_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
inv₁ ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
Real.sin θ₁₂ ^ 2 * Real.cos θ₁₃ ^ 2 * Real.sin θ₁₃ ^ 2 * Real.sin θ₂₃ ^ 2 := by
simp only [inv₁, VusAbs, VAbs, VAbs', Fin.isValue, sP, standardParameterizationAsMatrix,
neg_mul, Quotient.lift_mk, cons_val', cons_val_one, head_cons,
empty_val', cons_val_fin_one, cons_val_zero, _root_.map_mul, VubAbs, cons_val_two, tail_cons,
VcbAbs, VudAbs, Complex.abs_ofReal]
by_cases hx : Real.cos θ₁₃ ≠ 0
·
rw [Complex.abs_exp]
simp
rw [_root_.abs_of_nonneg h1, _root_.abs_of_nonneg h3, _root_.abs_of_nonneg h2,
_root_.abs_of_nonneg h4]
simp [sq]
ring_nf
nth_rewrite 2 [Real.sin_sq θ₁₂]
ring_nf
field_simp
ring
· simp at hx
rw [hx]
simp
@[simp]
def jarlskogComplex : Quotient CKMMatrixSetoid → ℂ :=
Quotient.lift jarlskogComplexCKM jarlskogComplexCKM_equiv
-- bad name
def expδ₁₃ (V : Quotient CKMMatrixSetoid) : ℂ :=
jarlskogComplex V + inv₁ V
lemma expδ₁₃_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
expδ₁₃ ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
sin θ₁₂ * cos θ₁₃ ^ 2 * sin θ₂₃ * sin θ₁₃ * cos θ₁₂ * cos θ₂₃ * cexp (I * δ₁₃) := by
rw [expδ₁₃]
rw [inv₁_sP _ _ _ _ h1 h2 h3 h4 ]
simp only [expδ₁₃, jarlskogComplex, sP, standardParameterizationAsMatrix, neg_mul,
Quotient.lift_mk, jarlskogComplexCKM, Fin.isValue, cons_val', cons_val_one, head_cons,
empty_val', cons_val_fin_one, cons_val_zero, cons_val_two, tail_cons, _root_.map_mul, ←
exp_conj, map_neg, conj_I, conj_ofReal, neg_neg, map_sub]
simp
ring_nf
rw [exp_neg]
have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _
field_simp
lemma expδ₁₃_sP_V (V : CKMMatrix) (δ₁₃ : ℝ) :
expδ₁₃ ⟦sP (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 expδ₁₃_sP _ _ _ _ ?_ ?_ ?_ ?_
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 expδ₁₃_eq_zero (V : CKMMatrix) (δ₁₃ : ℝ) :
expδ₁₃ ⟦sP (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]
simp
rw [expδ₁₃_sP_V]
simp
have h1 := exp_ne_zero (I * δ₁₃)
simp_all
aesop
lemma inv₂_V_arg (V : CKMMatrix) (δ₁₃ : ℝ)
(h1 : expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
cexp (arg (expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * I) =
cexp (δ₁₃ * I) := by
have h1a := expδ₁₃_sP_V V δ₁₃
have habs : Complex.abs (expδ₁₃ ⟦sP (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 [h1a]
simp [abs_exp]
rw [complexAbs_sin_θ₁₃, complexAbs_cos_θ₁₃, complexAbs_sin_θ₁₂, complexAbs_cos_θ₁₂,
complexAbs_sin_θ₂₃, complexAbs_cos_θ₂₃]
have h2 : expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
Complex.abs (expδ₁₃ ⟦sP (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 (expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
have habs_neq_zero : (Complex.abs (expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
simp
exact h1
rw [← mul_right_inj' habs_neq_zero]
rw [← h2]
def δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := arg (expδ₁₃ V)
theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
V ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧) := by
obtain ⟨δ₁₃', hδ₃⟩ := exists_standardParameterization V
have hSV := (Quotient.eq.mpr (hδ₃))
by_cases h : expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
have h1 := inv₂_V_arg V δ₁₃' h
have h2 := eq_phases_sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
(δ₁₃ ⟦V⟧) (by rw [← h1, ← hSV, δ₁₃])
rw [h2] at hδ₃
exact hδ₃
simp at h
have h1 : δ₁₃ ⟦V⟧ = 0 := by
rw [hSV, δ₁₃, h]
simp
rw [h1]
rw [expδ₁₃_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 )
end

181
HepLean/FlavorPhysics/CKMMatrix/PhaseFreedom.lean
View file

@ -5,6 +5,7 @@ Authors: Joseph Tooby-Smith
-/
import HepLean.FlavorPhysics.CKMMatrix.Basic
import HepLean.FlavorPhysics.CKMMatrix.Rows
import HepLean.FlavorPhysics.CKMMatrix.Relations
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
@ -14,86 +15,88 @@ open Matrix Complex
noncomputable section
namespace CKMMatrix
open ComplexConjugate
variable (a b c d e f : ℝ)
lemma ud_eq_abs (V : CKMMatrix) (h1 : a + d = - arg [V]ud) :
[phaseShiftApply V a b c d e f]ud = VudAbs ⟦V⟧ := by
section phaseShiftApply
variable (u c t d s b : ℝ)
lemma ud_eq_abs (V : CKMMatrix) (h1 : u + d = - arg [V]ud) :
[phaseShiftApply V u c t d s b]ud = VudAbs ⟦V⟧ := by
rw [phaseShiftApply.ud]
rw [← abs_mul_exp_arg_mul_I [V]ud]
rw [mul_comm, mul_assoc, ← exp_add]
have h2 : ↑(arg (V.1 0 0)) * I + (↑a * I + ↑d * I) = ↑(arg (V.1 0 0) + (a + d)) * I := by
have h2 : ↑(arg (V.1 0 0)) * I + (↑u * I + ↑d * I) = ↑(arg (V.1 0 0) + (u + d)) * I := by
simp [add_assoc]
ring
rw [h2, h1]
simp
rfl
lemma us_eq_abs {V : CKMMatrix} (h1 : a + e = - arg [V]us) :
[phaseShiftApply V a b c d e f]us = VusAbs ⟦V⟧ := by
lemma us_eq_abs {V : CKMMatrix} (h1 : u + s = - arg [V]us) :
[phaseShiftApply V u c t d s b]us = VusAbs ⟦V⟧ := by
rw [phaseShiftApply.us]
rw [← abs_mul_exp_arg_mul_I [V]us]
rw [mul_comm, mul_assoc, ← exp_add]
have h2 : ↑(arg [V]us) * I + (↑a * I + ↑e * I) = ↑(arg [V]us + (a + e)) * I := by
have h2 : ↑(arg [V]us) * I + (↑u * I + ↑s * I) = ↑(arg [V]us + (u + s)) * I := by
simp [add_assoc]
ring
rw [h2, h1]
simp
rfl
lemma ub_eq_abs {V : CKMMatrix} (h1 : a + f = - arg [V]ub) :
[phaseShiftApply V a b c d e f]ub = VubAbs ⟦V⟧ := by
lemma ub_eq_abs {V : CKMMatrix} (h1 : u + b = - arg [V]ub) :
[phaseShiftApply V u c t d s b]ub = VubAbs ⟦V⟧ := by
rw [phaseShiftApply.ub]
rw [← abs_mul_exp_arg_mul_I [V]ub]
rw [mul_comm, mul_assoc, ← exp_add]
have h2 : ↑(arg [V]ub) * I + (↑a * I + ↑f * I) = ↑(arg [V]ub + (a + f)) * I := by
have h2 : ↑(arg [V]ub) * I + (↑u * I + ↑b * I) = ↑(arg [V]ub + (u + b)) * I := by
simp [add_assoc]
ring
rw [h2, h1]
simp
rfl
lemma cs_eq_abs {V : CKMMatrix} (h1 : b + e = - arg [V]cs) :
[phaseShiftApply V a b c d e f]cs = VcsAbs ⟦V⟧ := by
lemma cs_eq_abs {V : CKMMatrix} (h1 : c + s = - arg [V]cs) :
[phaseShiftApply V u c t d s b]cs = VcsAbs ⟦V⟧ := by
rw [phaseShiftApply.cs]
rw [← abs_mul_exp_arg_mul_I [V]cs]
rw [mul_comm, mul_assoc, ← exp_add]
have h2 : ↑(arg [V]cs) * I + (↑b * I + ↑e * I) = ↑(arg [V]cs + (b + e)) * I := by
have h2 : ↑(arg [V]cs) * I + (↑c * I + ↑s * I) = ↑(arg [V]cs + (c + s)) * I := by
simp [add_assoc]
ring
rw [h2, h1]
simp
rfl
lemma cb_eq_abs {V : CKMMatrix} (h1 : b + f = - arg [V]cb) :
[phaseShiftApply V a b c d e f]cb = VcbAbs ⟦V⟧ := by
lemma cb_eq_abs {V : CKMMatrix} (h1 : c + b = - arg [V]cb) :
[phaseShiftApply V u c t d s b]cb = VcbAbs ⟦V⟧ := by
rw [phaseShiftApply.cb]
rw [← abs_mul_exp_arg_mul_I [V]cb]
rw [mul_comm, mul_assoc, ← exp_add]
have h2 : ↑(arg [V]cb) * I + (↑b * I + ↑f * I) = ↑(arg [V]cb + (b + f)) * I := by
have h2 : ↑(arg [V]cb) * I + (↑c * I + ↑b * I) = ↑(arg [V]cb + (c + b)) * I := by
simp [add_assoc]
ring
rw [h2, h1]
simp
rfl
lemma tb_eq_abs {V : CKMMatrix} (h1 : c + f = - arg [V]tb) :
[phaseShiftApply V a b c d e f]tb = VtbAbs ⟦V⟧ := by
lemma tb_eq_abs {V : CKMMatrix} (h1 : t + b = - arg [V]tb) :
[phaseShiftApply V u c t d s b]tb = VtbAbs ⟦V⟧ := by
rw [phaseShiftApply.tb]
rw [← abs_mul_exp_arg_mul_I [V]tb]
rw [mul_comm, mul_assoc, ← exp_add]
have h2 : ↑(arg [V]tb) * I + (↑c * I + ↑f * I) = ↑(arg [V]tb + (c + f)) * I := by
have h2 : ↑(arg [V]tb) * I + (↑t * I + ↑b * I) = ↑(arg [V]tb + (t + b)) * I := by
simp [add_assoc]
ring
rw [h2, h1]
simp
rfl
lemma cd_eq_neg_abs {V : CKMMatrix} (h1 : b + d = Real.pi - arg [V]cd) :
[phaseShiftApply V a b c d e f]cd = - VcdAbs ⟦V⟧ := by
lemma cd_eq_neg_abs {V : CKMMatrix} (h1 : c + d = Real.pi - arg [V]cd) :
[phaseShiftApply V u c t d s b]cd = - VcdAbs ⟦V⟧ := by
rw [phaseShiftApply.cd]
rw [← abs_mul_exp_arg_mul_I [V]cd]
rw [mul_comm, mul_assoc, ← exp_add]
have h2 : ↑(arg [V]cd) * I + (↑b * I + ↑d * I) = ↑(arg [V]cd + (b + d)) * I := by
have h2 : ↑(arg [V]cd) * I + (↑c * I + ↑d * I) = ↑(arg [V]cd + (c + d)) * I := by
simp [add_assoc]
ring
rw [h2, h1]
@ -101,9 +104,9 @@ lemma cd_eq_neg_abs {V : CKMMatrix} (h1 : b + d = Real.pi - arg [V]cd) :
rfl
lemma t_eq_conj {V : CKMMatrix} {τ : ℝ} (hτ : cexp (τ * I) • (conj [V]u ×₃ conj [V]c) = [V]t)
(h1 : τ = - a - b - c - d - e - f) :
[phaseShiftApply V a b c d e f]t =
conj [phaseShiftApply V a b c d e f]u ×₃ conj [phaseShiftApply V a b c d e f]c := by
(h1 : τ = - u - c - t - d - s - b) :
[phaseShiftApply V u c t d s b]t =
conj [phaseShiftApply V u c t d s b]u ×₃ conj [phaseShiftApply V u c t d s b]c := by
change _ = phaseShiftApply.ucCross _ _ _ _ _ _ _
funext i
fin_cases i
@ -138,6 +141,19 @@ lemma t_eq_conj {V : CKMMatrix} {τ : ℝ} (hτ : cexp (τ * I) • (conj [V]u
simp
ring
end phaseShiftApply
variable (a b c d e f : ℝ)
-- rename
def UCond₁ (U : CKMMatrix) : Prop := [U]ud = VudAbs ⟦U⟧ ∧ [U]us = VusAbs ⟦U⟧
∧ [U]cb = VcbAbs ⟦U⟧ ∧ [U]tb = VtbAbs ⟦U⟧ ∧ [U]t = conj [U]u ×₃ conj [U]c
-- rename
def UCond₃ (U : CKMMatrix) : Prop :=
[U]ud = 0 ∧ [U]us = 0 ∧ [U]cb = 0 ∧ [U]ub = 1 ∧ [U]t = conj [U]u ×₃ conj [U]c
∧ [U]cd = - VcdAbs ⟦U⟧ ∧ [U]cs = √(1 - VcdAbs ⟦U⟧ ^ 2)
-- bad name for this lemma
lemma all_cond_sol {V : CKMMatrix} (h1 : a + d = - arg [V]ud) (h2 : a + e = - arg [V]us) (h3 : b + f = - arg [V]cb)
(h4 : c + f = - arg [V]tb) (h5 : τ = - a - b - c - d - e - f) :
@ -168,10 +184,28 @@ lemma all_cond_sol {V : CKMMatrix} (h1 : a + d = - arg [V]ud) (h2 : a + e = - a
ring_nf
simp
-- rename
def UCond₁ (U : CKMMatrix) : Prop := [U]ud = VudAbs ⟦U⟧ ∧ [U]us = VusAbs ⟦U⟧
∧ [U]cb = VcbAbs ⟦U⟧ ∧ [U]tb = VtbAbs ⟦U⟧ ∧ [U]t = conj [U]u ×₃ conj [U]c
lemma UCond₃_solv {V : CKMMatrix} (h1 : a + f = - arg [V]ub) (h2 : 0 = - a - b - c - d - e - f)
(h3 : b + d = Real.pi - arg [V]cd) (h5 : b + e = - arg [V]cs) :
c = - Real.pi + arg [V]cd + arg [V]cs + arg [V]ub + b ∧
d = Real.pi - arg [V]cd - b ∧
e = - arg [V]cs - b ∧
f = - arg [V]ub - a := by
have hf : f = - arg [V]ub - a := by
linear_combination h1
subst hf
have he : e = - arg [V]cs - b := by
linear_combination h5
have hd : d = Real.pi - arg [V]cd - b := by
linear_combination h3
subst he hd
simp_all
ring_nf at h2
have hc : c = - Real.pi + arg [V]cd + arg [V]cs + arg [V]ub + b := by
linear_combination h2
subst hc
ring
-- rename
lemma all_eq_abs (V : CKMMatrix) :
∃ (U : CKMMatrix), V ≈ U ∧ UCond₁ U:= by
obtain ⟨τ, hτ⟩ := V.uRow_cross_cRow_eq_tRow
@ -209,9 +243,100 @@ lemma all_eq_abs (V : CKMMatrix) :
ring
lemma UCond₃_exists {V : CKMMatrix} (hb :¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) (hV : UCond₁ V) :
∃ (U : CKMMatrix), V ≈ U ∧ UCond₃ U:= by
let U : CKMMatrix := phaseShiftApply V 0 0 (- Real.pi + arg [V]cd + arg [V]cs + arg [V]ub)
(Real.pi - arg [V]cd ) (- arg [V]cs) (- arg [V]ub )
use U
have hUV : ⟦U⟧ = ⟦V⟧ := by
simp
symm
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
· simp [not_or] at hb
have h1 : VudAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs, hb]
simp [VAbs] at h1
exact h1
apply And.intro
· simp [not_or] at hb
have h1 : VusAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs, hb]
simp [VAbs] at h1
exact h1
apply And.intro
· simp [not_or] at hb
have h3 := cb_eq_zero_of_ud_us_zero hb
have h1 : VcbAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs, h3]
simp [VAbs] at h1
exact h1
apply And.intro
· have hU1 : [U]ub = VubAbs ⟦V⟧ := by
apply ub_eq_abs _ _ _ _ _ _ _
ring
rw [hU1]
have h1:= (ud_us_neq_zero_iff_ub_neq_one V).mpr.mt hb
simpa using h1
apply And.intro
· have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
apply t_eq_conj _ _ _ _ _ _ hτ.symm
ring
apply And.intro
· rw [hUV]
apply cd_eq_neg_abs _ _ _ _ _ _ _
ring
have hcs : [U]cs = VcsAbs ⟦U⟧ := by
rw [hUV]
apply cs_eq_abs _ _ _ _ _ _ _
ring
rw [hcs, hUV, cs_of_ud_us_zero hb]
lemma cd_of_us_or_ud_neq_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
(hV : UCond₁ V) : [V]cd = (- VtbAbs ⟦V⟧ * VusAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) +
(- VubAbs ⟦V⟧ * VudAbs ⟦V⟧ * VcbAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2 )) * cexp (- arg [V]ub * I)
:= by
have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
rw [cd_of_ud_us_ub_cb_tb hb hτ]
rw [hV.1, hV.2.1, hV.2.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
have h1 : conj [V]ub = VubAbs ⟦V⟧ * cexp (- arg [V]ub * I) := by
nth_rewrite 1 [← abs_mul_exp_arg_mul_I [V]ub]
rw [@RingHom.map_mul]
simp [conj_ofReal, ← exp_conj, VAbs]
rw [h1]
ring_nf
lemma cs_of_us_or_ud_neq_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
(hV : UCond₁ V) : [V]cs = (VtbAbs ⟦V⟧ * VudAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2))
+ (- VubAbs ⟦V⟧ * VusAbs ⟦V⟧ * VcbAbs ⟦V⟧/ (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) * cexp (- arg [V]ub * I)
:= by
have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
rw [cs_of_ud_us_ub_cb_tb hb hτ]
rw [hV.1, hV.2.1, hV.2.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
have h1 : conj [V]ub = VubAbs ⟦V⟧ * cexp (- arg [V]ub * I) := by
nth_rewrite 1 [← abs_mul_exp_arg_mul_I [V]ub]
rw [@RingHom.map_mul]
simp [conj_ofReal, ← exp_conj, VAbs]
rw [h1]
ring_nf
end CKMMatrix
end

655
HepLean/FlavorPhysics/CKMMatrix/Ratios.lean
View file

@ -1,655 +0,0 @@
/-
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.Relations
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom
open Matrix Complex
noncomputable section
namespace CKMMatrix
open ComplexConjugate
-- A
def Rubud (V : CKMMatrix) : ℂ := [V]ub / [V]ud
scoped[CKMMatrix] notation (name := ub_ud_ratio) "[" V "]ub|ud" => Rubud V
-- B
def Rusud (V : CKMMatrix) : ℂ := [V]us / [V]ud
scoped[CKMMatrix] notation (name := us_ud_ratio) "[" V "]us|ud" => Rusud V
def Rudus (V : CKMMatrix) : ℂ := [V]ud / [V]us
scoped[CKMMatrix] notation (name := ud_us_ratio) "[" V "]ud|us" => Rudus V
def Rubus (V : CKMMatrix) : ℂ := [V]ub / [V]us
scoped[CKMMatrix] notation (name := ub_us_ratio) "[" V "]ub|us" => Rubus V
-- D
def Rcdcb (V : CKMMatrix) : ℂ := [V]cd / [V]cb
scoped[CKMMatrix] notation (name := cd_cb_ratio) "[" V "]cd|cb" => Rcdcb V
lemma Rcdcb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cd = Rcdcb V * [V]cb := by
rw [Rcdcb]
exact (div_mul_cancel₀ (V.1 1 0) h).symm
-- C'
def Rcscb (V : CKMMatrix) : ℂ := [V]cs / [V]cb
scoped[CKMMatrix] notation (name := cs_cb_ratio) "[" V "]cs|cb" => Rcscb V
lemma Rcscb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cs = Rcscb V * [V]cb := by
rw [Rcscb]
exact (div_mul_cancel₀ [V]cs h).symm
def Rtb!cbud (V : CKMMatrix) : ℂ := conj [V]tb / ([V]cb * [V]ud)
scoped[CKMMatrix] notation (name := tb_cb_ud_ratio) "[" V "]tb*|cb|ud" => Rtb!cbud V
def Rtb!cbus (V : CKMMatrix) : ℂ := conj [V]tb / ([V]cb * [V]us)
scoped[CKMMatrix] notation (name := tb_cb_us_ratio) "[" V "]tb*|cb|us" => Rtb!cbus V
lemma orthog_fst_snd_row_ratios {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb + [V]cs|cb * conj ([V]us|ud) + conj ([V]ub|ud) = 0 := by
have h1 : ([V]cd * conj ([V]ud) + [V]cs * conj ([V]us)
+ [V]cb * conj ([V]ub)) / ([V]cb * conj ([V]ud)) = 0 := by
rw [fst_row_snd_row V ]
simp only [Fin.isValue, zero_div]
rw [← div_add_div_same, ← div_add_div_same] at h1
rw [mul_div_mul_comm, mul_div_mul_comm, mul_div_mul_comm] at h1
rw [div_self, div_self] at h1
change Rcdcb V * 1 + _ + _ = _ at h1
have h2 : (starRingEnd ℂ) (V.1 0 2) / (starRingEnd ℂ) (V.1 0 0) = conj (Rubud V) := by
simp [Rubud]
have h3 : ((starRingEnd ℂ) (V.1 0 1) / (starRingEnd ℂ) (V.1 0 0)) = conj (Rusud V) := by
simp [Rusud]
rw [h2, h3] at h1
simp at h1
exact h1
exact ha
simpa using hb
lemma orthog_fst_snd_row_ratios_cb_us {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb * conj ([V]ud|us) + [V]cs|cb + conj ([V]ub|us) = 0 := by
have h1 : ([V]cd * conj ([V]ud) + [V]cs * conj ([V]us)
+ [V]cb * conj ([V]ub)) / ([V]cb * conj ([V]us)) = 0 := by
rw [fst_row_snd_row V ]
simp only [Fin.isValue, zero_div]
rw [← div_add_div_same, ← div_add_div_same] at h1
rw [mul_div_mul_comm, mul_div_mul_comm, mul_div_mul_comm] at h1
rw [div_self, div_self] at h1
change _ + _ * 1 + _ = _ at h1
have h2 : (starRingEnd ℂ) (V.1 0 2) / (starRingEnd ℂ) (V.1 0 1) = conj (Rubus V) := by
simp [Rubus]
have h3 : ((starRingEnd ℂ) (V.1 0 0) / (starRingEnd ℂ) (V.1 0 1)) = conj (Rudus V) := by
simp [Rudus]
rw [h2, h3] at h1
simp at h1
exact h1
exact ha
simpa using hb
def R' (V : CKMMatrix) : ℂ := [V]cs|cb * (1 + normSq [V]us|ud) + conj ([V]ub|ud) * [V]us|ud
def R's (V : CKMMatrix) : ℂ := [V]cd|cb * (normSq [V]ud|us + 1) + conj ([V]ub|us) * [V]ud|us
lemma R'_eq (V : CKMMatrix) :
R' V = [V]cs|cb * (1 + normSq ([V]us|ud)) + conj ([V]ub|ud) * [V]us|ud := by rfl
lemma R's_eq (V : CKMMatrix) :
R's V = [V]cd|cb * (normSq ([V]ud|us) + 1) + conj ([V]ub|us) * [V]ud|us := by rfl
lemma one_plus_normSq_Rusud_neq_zero_ℝ (V : CKMMatrix):
1 + normSq ([V]us|ud) ≠ 0 := by
have h1 : 0 ≤ (normSq ([V]us|ud)) := normSq_nonneg ([V]us|ud)
have h2 : 0 < 1 + normSq ([V]us|ud) := by linarith
by_contra hn
have h3 := lt_of_lt_of_eq h2 hn
simp at h3
lemma normSq_Rudus_plus_one_neq_zero_ℝ (V : CKMMatrix):
normSq ([V]ud|us) + 1 ≠ 0 := by
have h1 : 0 ≤ (normSq ([V]ud|us)) := normSq_nonneg ([V]ud|us)
have h2 : 0 < normSq ([V]ud|us) + 1 := by linarith
by_contra hn
have h3 := lt_of_lt_of_eq h2 hn
simp at h3
lemma one_plus_normSq_Rusud_neq_zero_ℂ (V : CKMMatrix):
1 + (normSq ([V]us|ud) : ℂ) ≠ 0 := by
by_contra hn
have h1 := one_plus_normSq_Rusud_neq_zero_ℝ V
simp at h1
rw [← ofReal_inj] at h1
simp_all only [ofReal_add, ofReal_one, ofReal_zero, not_true_eq_false]
lemma normSq_Rudus_plus_one_neq_zero_ℂ (V : CKMMatrix):
(normSq ([V]ud|us) : ℂ) + 1 ≠ 0 := by
by_contra hn
have h1 := normSq_Rudus_plus_one_neq_zero_ℝ V
simp at h1
rw [← ofReal_inj] at h1
simp_all only [ofReal_add, ofReal_one, ofReal_zero, not_true_eq_false]
lemma Rcscb_of_R' (V : CKMMatrix) :
[V]cs|cb = (R' V - conj [V]ub|ud * [V]us|ud) / (1 + normSq [V]us|ud) := by
have h2 : R' V - conj ([V]ub|ud) * [V]us|ud = [V]cs|cb * (1 + normSq [V]us|ud) := by
linear_combination R'_eq V
rw [h2]
rw [mul_div_cancel_right₀]
exact one_plus_normSq_Rusud_neq_zero_ℂ V
lemma Rcdcb_of_R's (V : CKMMatrix) :
[V]cd|cb = (R's V - conj [V]ub|us * [V]ud|us) / (normSq [V]ud|us + 1) := by
have h2 : R's V - conj ([V]ub|us) * [V]ud|us = [V]cd|cb * (normSq [V]ud|us + 1) := by
linear_combination R's_eq V
rw [h2]
rw [mul_div_cancel_right₀]
exact normSq_Rudus_plus_one_neq_zero_ℂ V
lemma Rcdcb_R'_orthog {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb + (conj [V]ub|ud + R' V * conj [V]us|ud ) / (1 + normSq [V]us|ud) = 0 := by
have h1 := orthog_fst_snd_row_ratios hb ha
rw [Rcscb_of_R'] at h1
rw [div_mul_eq_mul_div] at h1
rw [add_assoc] at h1
rw [div_add'] at h1
have h2 : (R' V - conj [V]ub|ud * [V]us|ud) * conj [V]us|ud +
conj [V]ub|ud * (1 + normSq [V]us|ud) = conj [V]ub|ud + R' V * conj [V]us|ud := by
rw [← mul_conj]
ring
rw [h2] at h1
exact h1
exact one_plus_normSq_Rusud_neq_zero_ℂ V
lemma Rcdcb_of_R' {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb = - (conj [V]ub|ud + R' V * conj [V]us|ud ) / (1 + normSq [V]us|ud) := by
linear_combination (Rcdcb_R'_orthog hb ha)
lemma Rcscb_R's_orthog {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0) :
[V]cs|cb + (conj [V]ub|us + R's V * conj [V]ud|us ) / (normSq [V]ud|us + 1) = 0 := by
have h1 := orthog_fst_snd_row_ratios_cb_us hb ha
rw [Rcdcb_of_R's] at h1
rw [div_mul_eq_mul_div] at h1
rw [add_assoc, add_comm [V]cs|cb, ← add_assoc] at h1
rw [div_add', add_comm] at h1
have h2 : (R's V - conj [V]ub|us * [V]ud|us) * conj [V]ud|us +
(starRingEnd ℂ) [V]ub|us * ((normSq [V]ud|us) + 1)= conj [V]ub|us + R's V * conj [V]ud|us := by
rw [← mul_conj]
ring
rw [h2] at h1
exact h1
exact normSq_Rudus_plus_one_neq_zero_ℂ V
lemma Rcscb_of_R's {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0) :
[V]cs|cb = - (conj [V]ub|us + R's V * conj [V]ud|us ) / (normSq [V]ud|us + 1) := by
linear_combination (Rcscb_R's_orthog hb ha)
lemma R'_of_Rcscb_Rcdcb {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0) :
R' V = [V]cs|cb - [V]us|ud * [V]cd|cb := by
rw [Rcdcb_of_R' hb ha, Rcscb_of_R']
have h1 := one_plus_normSq_Rusud_neq_zero_ℂ V
have h2 : conj [V]ud ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
rw [Rubud, map_div₀, Rusud, map_div₀, map_div₀]
simp
rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
field_simp
ring
lemma R's_of_Rcscb_Rcdcb {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0) :
R's V = [V]cd|cb - [V]ud|us * [V]cs|cb := by
rw [Rcscb_of_R's hb ha, Rcdcb_of_R's]
have h1 := normSq_Rudus_plus_one_neq_zero_ℂ V
have h2 : conj [V]us ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
rw [Rubus, map_div₀, Rudus, map_div₀, map_div₀]
simp
rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
field_simp
ring
lemma R'_of_Rtb!cbud {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
R' V = cexp (τ * I) * [V]tb*|cb|ud := by
have h1 : cexp (- τ * I) = conj (cexp (τ * I)) := by
rw [← exp_conj]
simp only [neg_mul, _root_.map_mul, conj_I, mul_neg]
rw [conj_ofReal]
have h2 : [V]tb*|cb|ud = cexp (- τ * I) * R' V := by
rw [h1, R'_of_Rcscb_Rcdcb hb ha]
have h1 := congrFun hτ 2
simp [crossProduct, tRow, uRow, cRow] at h1
apply congrArg conj at h1
simp at h1
rw [Rtb!cbud, Rcscb, Rusud, Rcdcb, h1]
field_simp
ring
rw [h2, ← mul_assoc, ← exp_add]
simp
lemma R's_of_Rtb!cbus {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
R's V = - cexp (τ * I) * [V]tb*|cb|us := by
have h1 : cexp (- τ * I) = conj (cexp (τ * I)) := by
rw [← exp_conj]
simp only [neg_mul, _root_.map_mul, conj_I, mul_neg]
rw [conj_ofReal]
have h2 : [V]tb*|cb|us = - cexp (- τ * I) * R's V := by
rw [h1, R's_of_Rcscb_Rcdcb hb ha]
have h1 := congrFun hτ 2
simp [crossProduct, tRow, uRow, cRow] at h1
apply congrArg conj at h1
simp at h1
rw [Rtb!cbus, Rcscb, Rudus, Rcdcb, h1]
field_simp
ring
rw [h2]
simp
rw [← mul_assoc, ← exp_add]
simp
lemma over_normSq_Rusud (V : CKMMatrix) (hb : [V]ud ≠ 0) (a : ℂ) : a / (1 + normSq [V]us|ud) =
(a * normSq [V]ud) / (normSq [V]ud + normSq [V]us) := by
rw [Rusud]
field_simp
rw [one_add_div]
field_simp
simp
exact hb
lemma over_normSq_Rudus (V : CKMMatrix) (hb : [V]us ≠ 0) (a : ℂ) : a / (normSq [V]ud|us + 1) =
(a * normSq [V]us) / (normSq [V]ud + normSq [V]us) := by
rw [Rudus]
field_simp
rw [div_add_one]
field_simp
simp
exact hb
lemma cd_of_ud_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cd = - (conj [V]ub * [V]ud * [V]cb + cexp (τ * I) * conj [V]tb * conj [V]us) /
(normSq [V]ud + normSq [V]us) := by
obtain hτ2 := R'_of_Rtb!cbud hb ha hτ
rw [Rcdcb_mul_cb ha]
rw [Rcdcb_of_R' hb ha, hτ2]
rw [over_normSq_Rusud V hb, div_mul_eq_mul_div]
congr 1
rw [normSq_eq_conj_mul_self, Rubud, map_div₀, Rusud, map_div₀, Rtb!cbud]
have h1 : conj [V]ud ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
ring
lemma cs_of_ud_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 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 +
cexp (τ * I) * conj [V]tb * conj [V]ud ) /(normSq [V]ud + normSq [V]us) := by
have hτ2 := R'_of_Rtb!cbud hb ha hτ
rw [Rcscb_mul_cb ha]
rw [Rcscb_of_R', hτ2]
rw [over_normSq_Rusud V hb, div_mul_eq_mul_div]
congr 1
rw [normSq_eq_conj_mul_self, Rusud, Rtb!cbud, Rubud, map_div₀]
have h1 : conj [V]ud ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
ring
lemma cd_of_us_neq_zero {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cd =
- (conj [V]ub * [V]ud * [V]cb + cexp (τ * I) * conj [V]tb * conj [V]us) /
(normSq [V]ud + normSq [V]us) := by
have hτ2 := R's_of_Rtb!cbus hb ha hτ
rw [Rcdcb_mul_cb ha]
rw [Rcdcb_of_R's, hτ2]
rw [over_normSq_Rudus V hb, div_mul_eq_mul_div]
congr 1
rw [normSq_eq_conj_mul_self, Rubus, map_div₀, Rudus, Rtb!cbus]
have h1 : conj [V]us ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
ring
lemma cs_of_us_neq_zero {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 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 +
cexp (τ * I) * conj [V]tb * conj [V]ud ) / (normSq [V]ud + normSq [V]us) := by
have hτ2 := R's_of_Rtb!cbus hb ha hτ
rw [Rcscb_mul_cb ha]
rw [Rcscb_of_R's hb ha, hτ2]
rw [over_normSq_Rudus V hb, div_mul_eq_mul_div]
congr 1
rw [normSq_eq_conj_mul_self, Rudus, map_div₀, Rtb!cbus, Rubus, map_div₀]
have h1 : conj [V]us ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
ring
lemma cd_of_us_or_ud_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cd = - (conj [V]ub * [V]ud * [V]cb + cexp (τ * I) * conj [V]tb * conj [V]us) /
(normSq [V]ud + normSq [V]us) := by
cases' hb with hb hb
exact cd_of_ud_neq_zero hb ha hτ
exact cd_of_us_neq_zero hb ha hτ
lemma cs_of_us_or_ud_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb ≠ 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 +
cexp (τ * I) * conj [V]tb * conj [V]ud ) / (normSq [V]ud + normSq [V]us) := by
cases' hb with hb hb
exact cs_of_ud_neq_zero hb ha hτ
exact cs_of_us_neq_zero hb ha hτ
lemma cd_of_us_or_ud_neq_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb ≠ 0)
(hV : UCond₁ V) : [V]cd = (- VtbAbs ⟦V⟧ * VusAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) +
(- VubAbs ⟦V⟧ * VudAbs ⟦V⟧ * VcbAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2 )) * cexp (- arg [V]ub * I)
:= by
have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
rw [cd_of_us_or_ud_neq_zero hb ha hτ]
rw [hV.1, hV.2.1, hV.2.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
have h1 : conj [V]ub = VubAbs ⟦V⟧ * cexp (- arg [V]ub * I) := by
nth_rewrite 1 [← abs_mul_exp_arg_mul_I [V]ub]
rw [@RingHom.map_mul]
simp [conj_ofReal, ← exp_conj, VAbs]
rw [h1]
ring_nf
lemma cs_of_us_or_ud_neq_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb ≠ 0)
(hV : UCond₁ V) : [V]cs = (VtbAbs ⟦V⟧ * VudAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2))
+ (- VubAbs ⟦V⟧ * VusAbs ⟦V⟧ * VcbAbs ⟦V⟧/ (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) * cexp (- arg [V]ub * I)
:= by
have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
rw [cs_of_us_or_ud_neq_zero hb ha hτ]
rw [hV.1, hV.2.1, hV.2.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
have h1 : conj [V]ub = VubAbs ⟦V⟧ * cexp (- arg [V]ub * I) := by
nth_rewrite 1 [← abs_mul_exp_arg_mul_I [V]ub]
rw [@RingHom.map_mul]
simp [conj_ofReal, ← exp_conj, VAbs]
rw [h1]
ring_nf
lemma cd_of_cb_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
{κ : ℝ} (hκ : [V]u = cexp (κ * I) • (conj [V]c ×₃ conj [V]t)) :
[V]cd = - cexp (κ * I) * conj [V]tb * conj [V]us / (normSq [V]ud + normSq [V]us) := by
have h2 : [V]cd = - cexp (κ * I) * conj [V]us / [V]tb := by
have h1 := congrFun hκ 1
simp [crossProduct, tRow, uRow, cRow] at h1
rw [ha] at h1
apply congrArg conj at h1
simp at h1
rw [h1]
have hx := tb_neq_zero_of_cb_zero_ud_us_neq_zero ha hb
field_simp
have h1 : conj (cexp (↑κ * I)) = cexp (- κ * I) := by
simp [← exp_conj, conj_ofReal]
rw [h1]
rw [← mul_assoc]
rw [← exp_add]
simp
rw [div_td_of_cb_zero_ud_us_neq_zero ha hb] at h2
rw [h2]
congr 1
ring
lemma cs_of_cb_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
{κ : ℝ} (hκ1 : [V]u = cexp (κ * I) • (conj [V]c ×₃ conj [V]t)) :
[V]cs = cexp (κ * I) * conj [V]tb * conj [V]ud / (normSq [V]ud + normSq [V]us) := by
have h2 : [V]cs = cexp (κ * I) * conj [V]ud / [V]tb := by
have h1 := congrFun hκ1 0
simp [crossProduct, tRow, uRow, cRow] at h1
rw [ha] at h1
apply congrArg conj at h1
simp at h1
rw [h1]
have hx := tb_neq_zero_of_cb_zero_ud_us_neq_zero ha hb
field_simp
have h1 : conj (cexp (↑κ * I)) = cexp (- κ * I) := by
rw [← exp_conj]
simp only [neg_mul, _root_.map_mul, conj_I, mul_neg]
rw [conj_ofReal]
rw [h1]
rw [← mul_assoc]
rw [← exp_add]
simp
rw [div_td_of_cb_zero_ud_us_neq_zero ha hb] at h2
rw [h2]
congr 1
ring
lemma cd_of_cb_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
(hV : UCond₁ V) {κ : ℝ} (hκ1 : [V]u = cexp (κ * I) • (conj [V]c ×₃ conj [V]t)) :
[V]cd = (- VtbAbs ⟦V⟧ * VusAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) * cexp (κ * I) := by
rw [cd_of_cb_zero hb ha hκ1]
rw [hV.1, hV.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
ring_nf
simp
lemma cs_of_cb_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
(hV : UCond₁ V) {κ : ℝ} (hκ1 : [V]u = cexp (κ * I) • (conj [V]c ×₃ conj [V]t)) :
[V]cs = (VtbAbs ⟦V⟧ * VudAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) * cexp (κ * I) := by
rw [cs_of_cb_zero hb ha hκ1]
rw [hV.1, hV.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
ring_nf
def UCond₂ (U : CKMMatrix) : Prop := [U]ud = VudAbs ⟦U⟧ ∧ [U]us = VusAbs ⟦U⟧
∧ [U]ub = VubAbs ⟦U⟧ ∧ [U]t = conj [U]u ×₃ conj [U]c
∧ [U]cd = (- VtbAbs ⟦U⟧ * VusAbs ⟦U⟧ / (VudAbs ⟦U⟧ ^2 + VusAbs ⟦U⟧ ^2)) ∧
[U]cs = (VtbAbs ⟦U⟧ * VudAbs ⟦U⟧ / (VudAbs ⟦U⟧ ^2 + VusAbs ⟦U⟧ ^2))
lemma UCond₂_solv {V : CKMMatrix} (h1 : a + d = 0) (h2 : a + e = 0) (h3 : a + f = - arg [V]ub)
(h4 : 0 = - a - b - c - d - e - f) (h5 : b + d = - κ) (h6 : b + e = - κ) :
b = - κ + a ∧
c = κ + arg [V]ub + a ∧
d = - a ∧
e = - a ∧
f = - arg [V]ub - a := by
have hd : d = - a := by
linear_combination h1
subst hd
have he : e = - a := by
linear_combination h2
subst he
simp_all
have hb : b = - κ + a := by
linear_combination h6
subst hb
simp_all
have hf : f = - arg [V]ub - a := by
linear_combination h3
subst hf
simp_all
ring_nf at h4
have hc : c = κ + arg [V]ub + a := by
linear_combination h4
simp_all
lemma UCond₂_exists {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
(hV : UCond₁ V) : ∃ (U : CKMMatrix), V ≈ U ∧ UCond₂ U:= by
obtain ⟨κ, hκ⟩ := V.cRow_cross_tRow_eq_uRow
let U : CKMMatrix := phaseShiftApply V 0 (- κ) (κ + arg [V]ub) 0 0 (- arg [V]ub)
use U
have hUV : ⟦U⟧ = ⟦V⟧ := by
simp
symm
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
· rw [hUV]
apply ud_eq_abs _ _ _ _ _ _ _
rw [hV.1, arg_ofReal_of_nonneg]
simp
exact Complex.abs.nonneg _
apply And.intro
· rw [hUV]
apply us_eq_abs _ _ _ _ _ _ _
rw [hV.2.1, arg_ofReal_of_nonneg]
simp
exact Complex.abs.nonneg _
apply And.intro
· rw [hUV]
apply ub_eq_abs _ _ _ _ _ _ _
ring
apply And.intro
· have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
apply t_eq_conj _ _ _ _ _ _ hτ.symm
ring
apply And.intro
· rw [phaseShiftApply.cd]
rw [cd_of_cb_zero_UCond hb ha hV hκ]
rw [mul_comm, mul_assoc, ← exp_add, hUV]
simp
· rw [phaseShiftApply.cs]
rw [cs_of_cb_zero_UCond hb ha hV hκ]
rw [mul_comm, mul_assoc, ← exp_add, hUV]
simp
section zero
def UCond₃ (U : CKMMatrix) : Prop :=
[U]ud = 0 ∧ [U]us = 0 ∧ [U]cb = 0 ∧
[U]ub = 1 ∧ [U]t = conj [U]u ×₃ conj [U]c
∧ [U]cd = - VcdAbs ⟦U⟧ ∧ [U]cs = √(1 - VcdAbs ⟦U⟧ ^ 2)
lemma UCond₃_solv {V : CKMMatrix} (h1 : a + f = - arg [V]ub) (h2 : 0 = - a - b - c - d - e - f)
(h3 : b + d = Real.pi - arg [V]cd) (h5 : b + e = - arg [V]cs) :
c = - Real.pi + arg [V]cd + arg [V]cs + arg [V]ub + b ∧
d = Real.pi - arg [V]cd - b ∧
e = - arg [V]cs - b ∧
f = - arg [V]ub - a := by
have hf : f = - arg [V]ub - a := by
linear_combination h1
subst hf
have he : e = - arg [V]cs - b := by
linear_combination h5
have hd : d = Real.pi - arg [V]cd - b := by
linear_combination h3
subst he hd
simp_all
ring_nf at h2
have hc : c = - Real.pi + arg [V]cd + arg [V]cs + arg [V]ub + b := by
linear_combination h2
subst hc
ring
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
symm
rw [Real.sqrt_eq_iff_sq_eq]
simp [not_or] at ha
rw [cb_eq_zero_of_ud_us_zero ha] at h1
simp at h1
simp [VAbs]
linear_combination h1
simp
rw [@abs_le]
have h1 := VAbs_leq_one 1 0 ⟦V⟧
have h0 := VAbs_ge_zero 1 0 ⟦V⟧
simp_all
have hn : -1 ≤ (0 : ℝ) := by simp
exact hn.trans h0
exact VAbs_ge_zero _ _ ⟦V⟧
lemma UCond₃_exists {V : CKMMatrix} (hb :¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) (hV : UCond₁ V) :
∃ (U : CKMMatrix), V ≈ U ∧ UCond₃ U:= by
let U : CKMMatrix := phaseShiftApply V 0 0 (- Real.pi + arg [V]cd + arg [V]cs + arg [V]ub)
(Real.pi - arg [V]cd ) (- arg [V]cs) (- arg [V]ub )
use U
have hUV : ⟦U⟧ = ⟦V⟧ := by
simp
symm
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
· simp [not_or] at hb
have h1 : VudAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs, hb]
simp [VAbs] at h1
exact h1
apply And.intro
· simp [not_or] at hb
have h1 : VusAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs, hb]
simp [VAbs] at h1
exact h1
apply And.intro
· simp [not_or] at hb
have h3 := cb_eq_zero_of_ud_us_zero hb
have h1 : VcbAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs, h3]
simp [VAbs] at h1
exact h1
apply And.intro
· have hU1 : [U]ub = VubAbs ⟦V⟧ := by
apply ub_eq_abs _ _ _ _ _ _ _
ring
rw [hU1]
have h1:= (ud_us_neq_zero_iff_ub_neq_one V).mpr.mt hb
simpa using h1
apply And.intro
· have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
apply t_eq_conj _ _ _ _ _ _ hτ.symm
ring
apply And.intro
· rw [hUV]
apply cd_eq_neg_abs _ _ _ _ _ _ _
ring
have hcs : [U]cs = VcsAbs ⟦U⟧ := by
rw [hUV]
apply cs_eq_abs _ _ _ _ _ _ _
ring
rw [hcs, hUV, cs_of_ud_us_zero hb]
end zero
end CKMMatrix
end

405
HepLean/FlavorPhysics/CKMMatrix/Relations.lean
View file

@ -16,29 +16,50 @@ noncomputable section
namespace CKMMatrix
open ComplexConjugate
lemma fst_row_normalized_abs (V : CKMMatrix) :
abs [V]ud ^ 2 + abs [V]us ^ 2 + abs [V]ub ^ 2 = 1 := by
have h1 := VAbs_sum_sq_row_eq_one ⟦V⟧ 0
simp [VAbs] at h1
exact h1
section rows
lemma VAbs_sum_sq_row_eq_one (V : Quotient CKMMatrixSetoid) (i : Fin 3) :
(VAbs i 0 V) ^ 2 + (VAbs i 1 V) ^ 2 + (VAbs i 2 V) ^ 2 = 1 := by
obtain ⟨V, hV⟩ := Quot.exists_rep V
subst hV
have hV := V.prop
rw [mem_unitaryGroup_iff] at hV
have ht := congrFun (congrFun hV i) i
simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
rw [mul_conj, mul_conj, mul_conj] at ht
repeat rw [← Complex.sq_abs] at ht
rw [← ofReal_inj]
simpa using ht
lemma fst_row_normalized_abs (V : CKMMatrix) : abs [V]ud ^ 2 + abs [V]us ^ 2 + abs [V]ub ^ 2 = 1 :=
VAbs_sum_sq_row_eq_one ⟦V⟧ 0
lemma snd_row_normalized_abs (V : CKMMatrix) : abs [V]cd ^ 2 + abs [V]cs ^ 2 + abs [V]cb ^ 2 = 1 :=
VAbs_sum_sq_row_eq_one ⟦V⟧ 1
lemma thd_row_normalized_abs (V : CKMMatrix) : abs [V]td ^ 2 + abs [V]ts ^ 2 + abs [V]tb ^ 2 = 1 :=
VAbs_sum_sq_row_eq_one ⟦V⟧ 2
lemma fst_row_normalized_normSq (V : CKMMatrix) :
normSq [V]ud + normSq [V]us + normSq [V]ub = 1 := by
have h1 := V.fst_row_normalized_abs
repeat rw [Complex.sq_abs] at h1
exact h1
lemma snd_row_normalized_abs (V : CKMMatrix) :
abs [V]cd ^ 2 + abs [V]cs ^ 2 + abs [V]cb ^ 2 = 1 := by
have h1 := VAbs_sum_sq_row_eq_one ⟦V⟧ 1
simp [VAbs] at h1
exact h1
repeat rw [← Complex.sq_abs]
exact V.fst_row_normalized_abs
lemma snd_row_normalized_normSq (V : CKMMatrix) :
normSq [V]cd + normSq [V]cs + normSq [V]cb = 1 := by
have h1 := V.snd_row_normalized_abs
repeat rw [Complex.sq_abs] at h1
exact h1
repeat rw [← Complex.sq_abs]
exact V.snd_row_normalized_abs
lemma thd_row_normalized_normSq (V : CKMMatrix) :
normSq [V]td + normSq [V]ts + normSq [V]tb = 1 := by
repeat rw [← Complex.sq_abs]
exact V.thd_row_normalized_abs
-- rename
lemma normSq_Vud_plus_normSq_Vus (V : CKMMatrix) :
normSq [V]ud + normSq [V]us = 1 - normSq [V]ub := by
linear_combination (fst_row_normalized_normSq V)
lemma ud_us_neq_zero_iff_ub_neq_one (V : CKMMatrix) :
[V]ud ≠ 0 ∨ [V]us ≠ 0 ↔ abs [V]ub ≠ 1 := by
@ -60,6 +81,242 @@ lemma ud_us_neq_zero_iff_ub_neq_one (V : CKMMatrix) :
have h2 : ¬ 0 ≤ ( -1 : ℝ) := by simp
exact h2 h1
lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℝ {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
normSq [V]ud + normSq [V]us ≠ 0 := by
rw [normSq_Vud_plus_normSq_Vus V]
rw [ud_us_neq_zero_iff_ub_neq_one] at hb
by_contra hn
rw [← Complex.sq_abs] at hn
have h2 : Complex.abs (V.1 0 2) ^2 = 1 := by
linear_combination -(1 * hn)
simp at h2
cases' h2 with h2 h2
exact hb h2
have h3 := Complex.abs.nonneg [V]ub
rw [h2] at h3
have h2 : ¬ 0 ≤ ( -1 : ℝ) := by simp
exact h2 h3
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
simp at h1
rw [← ofReal_inj] at h1
simp_all
section orthogonal
lemma fst_row_snd_row (V : CKMMatrix) : V.1 1 0 * conj (V.1 0 0) + V.1 1 1 * conj (V.1 0 1)
+ V.1 1 2 * conj (V.1 0 2) = 0 := by
have hV := V.prop
rw [mem_unitaryGroup_iff] at hV
have ht := congrFun (congrFun hV 1) 0
simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
exact ht
lemma fst_row_thd_row (V : CKMMatrix) : V.1 2 0 * conj (V.1 0 0) + V.1 2 1 * conj (V.1 0 1)
+ V.1 2 2 * conj (V.1 0 2) = 0 := by
have hV := V.prop
rw [mem_unitaryGroup_iff] at hV
have ht := congrFun (congrFun hV 2) 0
simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
exact ht
lemma Vcd_conj_Vud (V : CKMMatrix) :
[V]cd * conj [V]ud = -[V]cs * conj [V]us - [V]cb * conj [V]ub := by
linear_combination (V.fst_row_snd_row )
lemma Vcs_conj_Vus (V : CKMMatrix) :
[V]cs * conj [V]us = - [V]cd * conj [V]ud - [V]cb * conj [V]ub := by
linear_combination (V.fst_row_snd_row )
lemma orthog_fst_snd_row_ratios {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb + [V]cs|cb * conj ([V]us|ud) + conj ([V]ub|ud) = 0 := by
suffices h1 : ([V]cd * conj [V]ud + [V]cs * conj [V]us
+ [V]cb * conj [V]ub) / ([V]cb * conj [V]ud) = 0 from
by
rw [← h1, Rubud, Rusud, Rcdcb, Rcscb]
have : conj [V]ud ≠ 0 := star_eq_zero.mp.mt hb
field_simp
ring
simp only [fst_row_snd_row V, Fin.isValue, zero_div]
lemma orthog_fst_snd_row_ratios_cb_us {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb * conj [V]ud|us + [V]cs|cb + conj ([V]ub|us) = 0 := by
suffices h1 : ([V]cd * conj [V]ud + [V]cs * conj [V]us
+ [V]cb * conj [V]ub) / ([V]cb * conj [V]us) = 0 from
by
rw [← h1, Rubus, Rudus, Rcdcb, Rcscb]
have : conj [V]us ≠ 0 := star_eq_zero.mp.mt hb
field_simp
ring
simp only [fst_row_snd_row V, Fin.isValue, zero_div]
end orthogonal
lemma VAbs_thd_eq_one_fst_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV : VAbs i 2 V = 1) :
VAbs i 0 V = 0 := by
have h := VAbs_sum_sq_row_eq_one V i
rw [hV] at h
simp at h
nlinarith
lemma VAbs_thd_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV : VAbs i 2 V = 1) :
VAbs i 1 V = 0 := by
have h := VAbs_sum_sq_row_eq_one V i
rw [hV] at h
simp at h
nlinarith
section crossProduct
lemma conj_Vtb_other {V : CKMMatrix} {τ : ℝ}
(hτ : [V]t = cexp (τ * I) • (conj [V]u ×₃ conj [V]c)) :
conj [V]tb = cexp (- τ * I) * ([V]cs * [V]ud - [V]us * [V]cd) := by
have h1 := congrFun hτ 2
simp [crossProduct, tRow, uRow, cRow] at h1
apply congrArg conj at h1
simp at h1
rw [h1]
simp only [← exp_conj, _root_.map_mul, conj_ofReal, conj_I, mul_neg, Fin.isValue, neg_mul]
field_simp
ring_nf
simp
end crossProduct
lemma conj_Vtb_Vud_other {V : CKMMatrix} {τ : ℝ}
(hτ : [V]t = cexp (τ * I) • (conj [V]u ×₃ conj [V]c)) :
cexp (τ * I) * conj [V]tb * conj [V]ud =
[V]cs * (normSq [V]ud + normSq [V]us) + [V]cb * conj [V]ub * [V]us := by
rw [conj_Vtb_other hτ]
simp [exp_neg]
have h1 := exp_ne_zero (τ * I)
field_simp
have h2 : cexp (τ * I) * ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]ud = cexp (τ * I) * [V]cs
* [V]ud * conj [V]ud - cexp (τ * I) * [V]us * ([V]cd * conj [V]ud) := by
ring
rw [h2, V.Vcd_conj_Vud]
rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
simp
ring
-- cexp (τ * I) * conj [V]tb * conj [V]us
lemma conj_Vtb_Vus_other {V : CKMMatrix} {τ : ℝ}
(hτ : [V]t = cexp (τ * I) • (conj [V]u ×₃ conj [V]c)) :
cexp (τ * I) * conj [V]tb * conj [V]us =
- ([V]cd * (normSq [V]ud + normSq [V]us) + [V]cb * conj ([V]ub) * [V]ud) := by
rw [conj_Vtb_other hτ]
simp [exp_neg]
have h1 := exp_ne_zero (τ * I)
field_simp
have h2 : cexp (τ * I) * ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]us = cexp (τ * I) * ([V]cs
* conj [V]us) * [V]ud - cexp (τ * I) * [V]us * [V]cd * conj [V]us := by
ring
rw [h2, V.Vcs_conj_Vus]
rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
simp
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 +
cexp (τ * I) * conj [V]tb * conj [V]ud) / (normSq [V]ud + normSq [V]us) := by
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℂ h
rw [conj_Vtb_Vud_other hτ]
field_simp
ring
lemma cd_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]cd = - (conj [V]ub * [V]ud * [V]cb + cexp (τ * I) * conj [V]tb * conj [V]us) /
(normSq [V]ud + normSq [V]us) := by
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℂ h
rw [conj_Vtb_Vus_other hτ]
field_simp
ring
end rows
section ratios
lemma one_plus_normSq_Rusud_neq_zero_ℝ (V : CKMMatrix):
1 + normSq ([V]us|ud) ≠ 0 := by
have h1 : 0 ≤ (normSq ([V]us|ud)) := normSq_nonneg ([V]us|ud)
have h2 : 0 < 1 + normSq ([V]us|ud) := by linarith
by_contra hn
have h3 := lt_of_lt_of_eq h2 hn
simp at h3
lemma normSq_Rudus_plus_one_neq_zero_ℝ (V : CKMMatrix):
normSq ([V]ud|us) + 1 ≠ 0 := by
have h1 : 0 ≤ (normSq ([V]ud|us)) := normSq_nonneg ([V]ud|us)
have h2 : 0 < normSq ([V]ud|us) + 1 := by linarith
by_contra hn
have h3 := lt_of_lt_of_eq h2 hn
simp at h3
lemma one_plus_normSq_Rusud_neq_zero_ℂ (V : CKMMatrix):
1 + (normSq ([V]us|ud) : ℂ) ≠ 0 := by
by_contra hn
have h1 := one_plus_normSq_Rusud_neq_zero_ℝ V
simp at h1
rw [← ofReal_inj] at h1
simp_all only [ofReal_add, ofReal_one, ofReal_zero, not_true_eq_false]
lemma normSq_Rudus_plus_one_neq_zero_ℂ (V : CKMMatrix):
(normSq ([V]ud|us) : ℂ) + 1 ≠ 0 := by
by_contra hn
have h1 := normSq_Rudus_plus_one_neq_zero_ℝ V
simp at h1
rw [← ofReal_inj] at h1
simp_all only [ofReal_add, ofReal_one, ofReal_zero, not_true_eq_false]
end ratios
section individual
lemma VAbs_ge_zero (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : 0 ≤ VAbs i j V := by
obtain ⟨V, hV⟩ := Quot.exists_rep V
rw [← hV]
exact Complex.abs.nonneg _
lemma VAbs_leq_one (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : VAbs i j V ≤ 1 := by
have h := VAbs_sum_sq_row_eq_one V i
fin_cases j
change VAbs i 0 V ≤ 1
nlinarith
change VAbs i 1 V ≤ 1
nlinarith
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
subst hV
have hV := V.prop
rw [mem_unitaryGroup_iff'] at hV
have ht := congrFun (congrFun hV i) i
simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
rw [mul_comm, mul_conj, mul_comm, mul_conj, mul_comm, mul_conj] at ht
repeat rw [← Complex.sq_abs] at ht
rw [← ofReal_inj]
simpa using ht
lemma thd_col_normalized_abs (V : CKMMatrix) :
abs [V]ub ^ 2 + abs [V]cb ^ 2 + abs [V]tb ^ 2 = 1 := by
have h1 := VAbs_sum_sq_col_eq_one ⟦V⟧ 2
@ -72,6 +329,43 @@ lemma thd_col_normalized_normSq (V : CKMMatrix) :
repeat rw [Complex.sq_abs] at h1
exact h1
lemma cb_eq_zero_of_ud_us_zero {V : CKMMatrix} (h : [V]ud = 0 ∧ [V]us = 0) :
[V]cb = 0 := by
have h1 := fst_row_normalized_abs V
rw [← thd_col_normalized_abs V] at h1
simp [h] at h1
rw [add_assoc] at h1
simp at h1
rw [add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)] at h1
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
symm
rw [Real.sqrt_eq_iff_sq_eq]
simp [not_or] at ha
rw [cb_eq_zero_of_ud_us_zero ha] at h1
simp at h1
simp [VAbs]
linear_combination h1
simp
rw [@abs_le]
have h1 := VAbs_leq_one 1 0 ⟦V⟧
have h0 := VAbs_ge_zero 1 0 ⟦V⟧
simp_all
have hn : -1 ≤ (0 : ℝ) := by simp
exact hn.trans h0
exact VAbs_ge_zero _ _ ⟦V⟧
end columns
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
@ -100,16 +394,6 @@ lemma tb_neq_zero_of_cb_zero_ud_us_neq_zero {V : CKMMatrix} (h : [V]cb = 0)
rw [← cb_tb_neq_zero_iff_ub_neq_one] at h1
simp_all
lemma cb_eq_zero_of_ud_us_zero {V : CKMMatrix} (h : [V]ud = 0 ∧ [V]us = 0) :
[V]cb = 0 := by
have h1 := fst_row_normalized_abs V
rw [← thd_col_normalized_abs V] at h1
simp [h] at h1
rw [add_assoc] at h1
simp at h1
rw [add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)] at h1
simp at h1
exact h1.1
@ -135,32 +419,6 @@ lemma div_td_of_cb_zero_ud_us_neq_zero {V : CKMMatrix} (h : [V]cb = 0)
simp at h1
exact h1
lemma normSq_Vud_plus_normSq_Vus (V : CKMMatrix) :
normSq [V]ud + normSq [V]us = 1 - normSq [V]ub := by
linear_combination (fst_row_normalized_normSq V)
lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℝ {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
normSq [V]ud + normSq [V]us ≠ 0 := by
rw [normSq_Vud_plus_normSq_Vus V]
rw [ud_us_neq_zero_iff_ub_neq_one] at hb
by_contra hn
rw [← Complex.sq_abs] at hn
have h2 : Complex.abs (V.1 0 2) ^2 = 1 := by
linear_combination -(1 * hn)
simp at h2
cases' h2 with h2 h2
exact hb h2
have h3 := Complex.abs.nonneg [V]ub
rw [h2] at h3
have h2 : ¬ 0 ≤ ( -1 : ℝ) := by simp
exact h2 h3
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
simp at h1
rw [← ofReal_inj] at h1
simp_all
lemma Vabs_sq_add_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
((VAbs 0 0 ⟦V⟧ : ℂ) * ↑(VAbs 0 0 ⟦V⟧) + ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) ≠ 0 := by
@ -170,5 +428,46 @@ lemma Vabs_sq_add_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
exact h1
lemma VAbs_fst_col_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
(hV : VAbs 0 i V = 1) : VAbs 1 i V = 0 := by
have h := VAbs_sum_sq_col_eq_one V i
rw [hV] at h
simp at h
nlinarith
lemma VAbs_fst_col_eq_one_thd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
(hV : VAbs 0 i V = 1) : VAbs 2 i V = 0 := by
have h := VAbs_sum_sq_col_eq_one V i
rw [hV] at h
simp at h
nlinarith
lemma VAbs_thd_neq_one_fst_snd_sq_neq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
(hV : VAbs i 2 V ≠ 1) : (VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2) ≠ 0 := by
have h1 : 1 - VAbs i 2 V ^ 2 = VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2 := by
linear_combination - (VAbs_sum_sq_row_eq_one V i)
rw [← h1]
by_contra h
have h2 : VAbs i 2 V ^2 = 1 := by
nlinarith
simp only [Fin.isValue, sq_eq_one_iff] at h2
have h3 : 0 ≤ VAbs i 2 V := VAbs_ge_zero i 2 V
have h4 : VAbs i 2 V = 1 := by
nlinarith
exact hV h4
lemma VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
(hV : VAbs i 2 V ≠ 1) : √(VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2) ≠ 0 := by
rw [Real.sqrt_ne_zero (Left.add_nonneg (sq_nonneg (VAbs i 0 V)) (sq_nonneg (VAbs i 1 V)))]
exact VAbs_thd_neq_one_fst_snd_sq_neq_zero hV
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 VudAbs_sq_add_VusAbs_sq : VudAbs V ^ 2 + VusAbs V ^2 = 1 - VubAbs V ^2 := by
linear_combination (VAbs_sum_sq_row_eq_one V 0)
end CKMMatrix
end

3
HepLean/FlavorPhysics/CKMMatrix/Rows.lean
View file

@ -283,9 +283,6 @@ lemma uRow_cross_cRow_eq_tRow (V : CKMMatrix) :
rw [hx, hτ]
def uRow₁₂ (V : CKMMatrix) : Fin 2 → ℂ := ![[V]ud, [V]us]
def cRow₁₂ (V : CKMMatrix) : Fin 2 → ℂ := ![[V]cd, [V]cs]

184
HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/Basic.lean Normal file
View file

@ -0,0 +1,184 @@
/-
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 Mathlib.Analysis.SpecialFunctions.Complex.Arg
open Matrix Complex
open ComplexConjugate
open CKMMatrix
noncomputable section
-- to be renamed stanParamAsMatrix ...
def standardParameterizationAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
![![Real.cos θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₃ * exp (-I * δ₁₃)],
![(-Real.sin θ₁₂ * Real.cos θ₂₃) - (Real.cos θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃)),
Real.cos θ₁₂ * Real.cos θ₂₃ - Real.sin θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃),
Real.sin θ₂₃ * Real.cos θ₁₃],
![Real.sin θ₁₂ * Real.sin θ₂₃ - Real.cos θ₁₂ * Real.sin θ₁₃ * Real.cos θ₂₃ * exp (I * δ₁₃),
(-Real.cos θ₁₂ * Real.sin θ₂₃) - (Real.sin θ₁₂ * Real.sin θ₁₃ * Real.cos θ₂₃ * exp (I * δ₁₃)),
Real.cos θ₂₃ * Real.cos θ₁₃]]
open CKMMatrix
lemma standardParameterizationAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
((standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by
funext j i
simp only [standardParameterizationAsMatrix, neg_mul, Fin.isValue]
rw [mul_apply]
have h1 := exp_ne_zero (I * ↑δ₁₃)
fin_cases j <;> rw [Fin.sum_univ_three]
simp only [Fin.zero_eta, Fin.isValue, conjTranspose_apply, cons_val', cons_val_zero, empty_val',
cons_val_fin_one, star_mul', RCLike.star_def, conj_ofReal, cons_val_one, head_cons, star_sub,
star_neg, ← exp_conj, _root_.map_mul, conj_I, neg_mul, cons_val_two, tail_cons, head_fin_const]
simp [conj_ofReal]
rw [exp_neg ]
fin_cases i <;> simp
· ring_nf
field_simp
rw [sin_sq, sin_sq, sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq, sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq]
ring
simp only [Fin.mk_one, Fin.isValue, conjTranspose_apply, cons_val', cons_val_one, head_cons,
empty_val', cons_val_fin_one, cons_val_zero, star_mul', RCLike.star_def, conj_ofReal, star_sub,
← exp_conj, _root_.map_mul, conj_I, neg_mul, cons_val_two, tail_cons, head_fin_const, star_neg]
simp [conj_ofReal]
rw [exp_neg]
fin_cases i <;> simp
· ring_nf
field_simp
rw [sin_sq, sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq, sin_sq, sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq]
ring
simp only [Fin.reduceFinMk, Fin.isValue, conjTranspose_apply, cons_val', cons_val_two, tail_cons,
head_cons, empty_val', cons_val_fin_one, cons_val_zero, star_mul', RCLike.star_def, conj_ofReal,
← exp_conj, map_neg, _root_.map_mul, conj_I, neg_mul, neg_neg, cons_val_one, head_fin_const]
simp [conj_ofReal]
rw [exp_neg]
fin_cases i <;> simp
· ring_nf
rw [sin_sq]
ring
· ring_nf
rw [sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq, sin_sq]
ring
def sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix :=
⟨standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by
rw [mem_unitaryGroup_iff']
exact standardParameterizationAsMatrix_unitary θ₁₂ θ₁₃ θ₂₃ δ₁₃⟩
lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
[sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]t = (conj [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]u ×₃ conj [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) := by
have h1 := exp_ne_zero (I * ↑δ₁₃)
funext i
fin_cases i
· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg,
Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons,
head_fin_const, cons_val_one, head_cons, Fin.zero_eta, crossProduct, uRow, cRow,
LinearMap.mk₂_apply, Pi.conj_apply, _root_.map_mul, map_inv₀, ← exp_conj, conj_I, conj_ofReal,
inv_inv, map_sub, map_neg]
field_simp
ring_nf
rw [sin_sq]
ring
· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue, cons_val',
cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const,
cons_val_one, head_cons, Fin.mk_one, crossProduct, uRow, cRow, LinearMap.mk₂_apply,
Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub,
map_neg]
field_simp
ring_nf
rw [sin_sq]
ring
· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue,
cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const,
cons_val_one, head_cons, Fin.reduceFinMk, crossProduct, uRow, cRow, LinearMap.mk₂_apply,
Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub,
map_neg]
field_simp
ring_nf
rw [sin_sq]
ring
lemma eq_sP (U : CKMMatrix) {θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ} (hu : [U]u = [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]u)
(hc : [U]c = [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) (hU : [U]t = conj [U]u ×₃ conj [U]c) :
U = sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
apply ext_Rows hu hc
rw [hU, sP_cross, hu, hc]
lemma eq_phases_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ δ₁₃' : ℝ) (h : cexp (δ₁₃ * I) = cexp (δ₁₃' * I)) :
sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ = sP θ₁₂ θ₁₃ θ₂₃ δ₁₃' := by
simp [sP, standardParameterizationAsMatrix]
apply CKMMatrix_ext
simp
rw [show exp (I * δ₁₃) = exp (I * δ₁₃') by rw [mul_comm, h, mul_comm]]
rw [show cexp (-(I * ↑δ₁₃)) = cexp (-(I * ↑δ₁₃')) by rw [exp_neg, exp_neg, mul_comm, h, mul_comm]]
namespace Invariant
lemma VusVubVcdSq_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
VusVubVcdSq ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
Real.sin θ₁₂ ^ 2 * Real.cos θ₁₃ ^ 2 * Real.sin θ₁₃ ^ 2 * Real.sin θ₂₃ ^ 2 := by
simp only [VusVubVcdSq, VusAbs, VAbs, VAbs', Fin.isValue, sP, standardParameterizationAsMatrix,
neg_mul, Quotient.lift_mk, cons_val', cons_val_one, head_cons,
empty_val', cons_val_fin_one, cons_val_zero, _root_.map_mul, VubAbs, cons_val_two, tail_cons,
VcbAbs, VudAbs, Complex.abs_ofReal]
by_cases hx : Real.cos θ₁₃ ≠ 0
· rw [Complex.abs_exp]
simp
rw [_root_.abs_of_nonneg h1, _root_.abs_of_nonneg h3, _root_.abs_of_nonneg h2,
_root_.abs_of_nonneg h4]
simp [sq]
ring_nf
nth_rewrite 2 [Real.sin_sq θ₁₂]
ring_nf
field_simp
ring
· simp at hx
rw [hx]
simp
lemma mulExpδ₃_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
mulExpδ₃ ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
sin θ₁₂ * cos θ₁₃ ^ 2 * sin θ₂₃ * sin θ₁₃ * cos θ₁₂ * cos θ₂₃ * cexp (I * δ₁₃) := by
rw [mulExpδ₃, VusVubVcdSq_sP _ _ _ _ h1 h2 h3 h4 ]
simp only [jarlskogℂ, sP, standardParameterizationAsMatrix, neg_mul,
Quotient.lift_mk, jarlskogℂCKM, Fin.isValue, cons_val', cons_val_one, head_cons,
empty_val', cons_val_fin_one, cons_val_zero, cons_val_two, tail_cons, _root_.map_mul, ←
exp_conj, map_neg, conj_I, conj_ofReal, neg_neg, map_sub]
simp
ring_nf
rw [exp_neg]
have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _
field_simp
end Invariant
end

795
HepLean/FlavorPhysics/CKMMatrix/StandardParameters.lean → HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/StandardParameters.lean
View file

@ -6,161 +6,194 @@ Authors: Joseph Tooby-Smith
import HepLean.FlavorPhysics.CKMMatrix.Basic
import HepLean.FlavorPhysics.CKMMatrix.Rows
import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom
import HepLean.FlavorPhysics.CKMMatrix.Ratios
import HepLean.FlavorPhysics.CKMMatrix.Invariants
import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
open Matrix Complex
open ComplexConjugate
open CKMMatrix
noncomputable section
def S₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := VusAbs V / (√ (VudAbs V ^ 2 + VusAbs V ^ 2))
def S₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := VubAbs V
lemma VubAbs_eq_S₁₃ (V : Quotient CKMMatrixSetoid) : VubAbs V = S₁₃ V := rfl
def θ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₃ V)
lemma S₁₃_eq_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₃ V) = S₁₃ V := by
refine Real.sin_arcsin ?_ ?_
have h1 := VAbs_ge_zero 0 2 V
rw [← VubAbs_eq_S₁₃]
simp
linarith
rw [← VubAbs_eq_S₁₃]
exact (VAbs_leq_one 0 2 V)
lemma S₁₃_eq_ℂsin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.sin (θ₁₃ V) = S₁₃ V := by
rw [← S₁₃_eq_sin_θ₁₃]
simp
lemma complexAbs_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.sin (θ₁₃ V)) =
sin (θ₁₃ V):= by
rw [S₁₃_eq_ℂsin_θ₁₃, ← VubAbs_eq_S₁₃]
rw [Complex.abs_ofReal]
simp
exact VAbs_ge_zero 0 2 V
lemma S₁₃_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₁₃ V = 1 := by
rw [← VubAbs_eq_S₁₃, ha]
def C₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₃ V)
lemma C₁₃_eq_ℂcos_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₁₃ V) = C₁₃ V := by
simp [C₁₃]
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
exact Real.cos_arcsin_nonneg _
lemma 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
rw [_root_.abs_of_nonneg (VAbs_ge_zero 0 2 V)]
exact VAbs_leq_one 0 2 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 C₁₃_of_Vub_eq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : C₁₃ V = 0 := by
rw [C₁₃, θ₁₃, Real.cos_arcsin, ← VubAbs_eq_S₁₃, ha]
simp
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]
/-- If VubAbs V = 1 this will give zero.-/
def S₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := VusAbs V / (√ (VudAbs V ^ 2 + VusAbs V ^ 2))
def S₂₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
if VubAbs V = 1 then VcdAbs V
else VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2)
def θ₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₂ V)
lemma S₁₂_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₁₂ V := by
rw [S₁₂]
rw [@div_nonneg_iff]
apply Or.inl
apply And.intro
exact VAbs_ge_zero 0 1 V
exact Real.sqrt_nonneg (VudAbs V ^ 2 + VusAbs V ^ 2)
def θ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₃ V)
def θ₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₂₃ V)
def C₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₂ V)
def C₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₃ V)
def C₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₂₃ V)
def δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
arg (Invariant.mulExpδ₃ 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]
rw [@div_le_one_iff]
rw [S₁₂, @div_le_one_iff]
by_cases h1 : √(VudAbs V ^ 2 + VusAbs V ^ 2) = 0
simp [h1]
have h2 := Real.sqrt_nonneg (VudAbs V ^ 2 + VusAbs V ^ 2)
rw [le_iff_eq_or_lt] at h2
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]
rw [Real.le_sqrt (VAbs_ge_zero 0 1 V) (le_of_lt h3)]
simp
exact sq_nonneg (VAbs 0 0 V)
exact VAbs_ge_zero 0 1 V
exact le_of_lt h3
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
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 S₁₂_eq_ℂsin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.sin (θ₁₂ V) = S₁₂ V := by
rw [← S₁₂_eq_sin_θ₁₂]
simp
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 complexAbs_sin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.sin (θ₁₂ V)) =
sin (θ₁₂ V):= by
rw [S₁₂_eq_ℂsin_θ₁₂, Complex.abs_ofReal]
simp
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
rw [S₁₂]
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 [ha]
simp
simp [S₁₂, ← h1, ha]
def C₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₂ V)
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
exact Real.cos_arcsin_nonneg _
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]
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
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
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]
@ -173,7 +206,43 @@ lemma C₁₂_eq_Vud_div_sqrt {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠
exact VAbs_thd_neq_one_fst_snd_sq_neq_zero ha
exact (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V)))
theorem VusAbs_eq_S₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VusAbs V = S₁₂ V * C₁₃ V := by
--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 VAbs_thd_neq_one_fst_snd_sq_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
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) (VAbs_thd_neq_one_sqrt_fst_snd_sq_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)
@ -188,107 +257,7 @@ theorem VusAbs_eq_S₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VusAbs V
have h2 := VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero ha
exact (mul_div_cancel₀ (VusAbs V) h2).symm
theorem 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, ← VubAbs_eq_S₁₃, ha]
simp
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) (VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero ha)).symm
def S₂₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
if VubAbs V = 1 then
VcdAbs V
else
VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2)
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]
rw [@div_nonneg_iff]
apply Or.inl
apply And.intro
exact VAbs_ge_zero 1 2 V
exact Real.sqrt_nonneg (VudAbs V ^ 2 + VusAbs V ^ 2)
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]
rw [@div_le_one_iff]
by_cases h1 : √(VudAbs V ^ 2 + VusAbs V ^ 2) = 0
simp [h1]
have h2 := Real.sqrt_nonneg (VudAbs V ^ 2 + VusAbs V ^ 2)
rw [le_iff_eq_or_lt] at h2
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]
rw [VudAbs_sq_add_VusAbs_sq, ← VcbAbs_sq_add_VtbAbs_sq]
simp
exact sq_nonneg (VAbs 2 2 V)
exact VAbs_ge_zero 1 2 V
exact le_of_lt h3
def θ₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₂₃ 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 S₂₃_eq_ℂsin_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.sin (θ₂₃ V) = S₂₃ V := by
rw [← S₂₃_eq_sin_θ₂₃]
simp
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]
simp
exact S₂₃_nonneg _
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]
def C₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₂₃ V)
lemma C₂₃_eq_ℂcos_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₂₃ V) = C₂₃ V := by
simp [C₂₃]
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
exact Real.cos_arcsin_nonneg _
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 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 VAbs_thd_neq_one_fst_snd_sq_neq_zero ha
exact (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V)))
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
@ -308,146 +277,86 @@ lemma VtbAbs_eq_C₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VtbAbs V =
rw [mul_comm]
exact (mul_div_cancel₀ (VtbAbs V) (VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero ha)).symm
end VAbs
lemma S₁₂_mul_C₂₃_of_Vud_neq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) :
S₁₂ V * C₂₃ V = VusAbs V * VtbAbs V / (VudAbs V ^ 2 + VusAbs V ^ 2) := by
rw [S₁₂, C₂₃_of_Vub_neq_one ha]
rw [@div_mul_div_comm]
rw [Real.mul_self_sqrt]
exact (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V)))
lemma C₁₂_mul_S₂₃_mul_S₁₃_of_Vud_neq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) :
C₁₂ V * S₂₃ V * S₁₃ V = VudAbs V * VcbAbs V * VubAbs V / (VudAbs V ^ 2 + VusAbs V ^ 2) := by
rw [C₁₂_eq_Vud_div_sqrt ha, S₂₃_of_Vub_neq_one ha, S₁₃]
rw [@div_mul_div_comm]
rw [Real.mul_self_sqrt (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V)))]
exact (mul_div_right_comm (VudAbs V * VcbAbs V) (VubAbs V) (VudAbs V ^ 2 + VusAbs V ^ 2)).symm
namespace Invariant
lemma mulExpδ₃_sP_inv (V : CKMMatrix) (δ₁₃ : ℝ) :
mulExpδ₃ ⟦sP (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 mulExpδ₃_sP _ _ _ _ ?_ ?_ ?_ ?_
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 mulExpδ₃_eq_zero (V : CKMMatrix) (δ₁₃ : ℝ) :
mulExpδ₃ ⟦sP (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 [mulExpδ₃_sP_inv]
simp
have h1 := exp_ne_zero (I * δ₁₃)
simp_all
aesop
lemma mulExpδ₃_abs (V : CKMMatrix) (δ₁₃ : ℝ) :
Complex.abs (mulExpδ₃ ⟦sP (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 [mulExpδ₃_sP_inv]
simp [abs_exp]
rw [complexAbs_sin_θ₁₃, complexAbs_cos_θ₁₃, complexAbs_sin_θ₁₂, complexAbs_cos_θ₁₂,
complexAbs_sin_θ₂₃, complexAbs_cos_θ₂₃]
lemma mulExpδ₃_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
(h1 : mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
cexp (arg ( mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ) * I) =
cexp (δ₁₃ * I) := by
have h1a := mulExpδ₃_sP_inv V δ₁₃
have habs := mulExpδ₃_abs V δ₁₃
have h2 : mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
Complex.abs (mulExpδ₃ ⟦sP (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 (mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
have habs_neq_zero : (Complex.abs (mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
simp
exact h1
rw [← mul_right_inj' habs_neq_zero]
rw [← h2]
end Invariant
-- to be moved.
lemma 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
rw [_root_.abs_of_nonneg (VAbs_ge_zero 0 2 V)]
exact VAbs_leq_one 0 2 V
def standardParameterizationAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
![![Real.cos θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₃ * exp (-I * δ₁₃)],
![(-Real.sin θ₁₂ * Real.cos θ₂₃) - (Real.cos θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃)),
Real.cos θ₁₂ * Real.cos θ₂₃ - Real.sin θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃),
Real.sin θ₂₃ * Real.cos θ₁₃],
![Real.sin θ₁₂ * Real.sin θ₂₃ - Real.cos θ₁₂ * Real.sin θ₁₃ * Real.cos θ₂₃ * exp (I * δ₁₃),
(-Real.cos θ₁₂ * Real.sin θ₂₃) - (Real.sin θ₁₂ * Real.sin θ₁₃ * Real.cos θ₂₃ * exp (I * δ₁₃)),
Real.cos θ₂₃ * Real.cos θ₁₃]]
open CKMMatrix
lemma standardParameterizationAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
((standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by
funext j i
simp only [standardParameterizationAsMatrix, neg_mul, Fin.isValue]
rw [mul_apply]
have h1 := exp_ne_zero (I * ↑δ₁₃)
fin_cases j <;> rw [Fin.sum_univ_three]
simp only [Fin.zero_eta, Fin.isValue, conjTranspose_apply, cons_val', cons_val_zero, empty_val',
cons_val_fin_one, star_mul', RCLike.star_def, conj_ofReal, cons_val_one, head_cons, star_sub,
star_neg, ← exp_conj, _root_.map_mul, conj_I, neg_mul, cons_val_two, tail_cons, head_fin_const]
simp [conj_ofReal]
rw [exp_neg ]
fin_cases i <;> simp
· ring_nf
field_simp
rw [sin_sq, sin_sq, sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq, sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq]
ring
simp only [Fin.mk_one, Fin.isValue, conjTranspose_apply, cons_val', cons_val_one, head_cons,
empty_val', cons_val_fin_one, cons_val_zero, star_mul', RCLike.star_def, conj_ofReal, star_sub,
← exp_conj, _root_.map_mul, conj_I, neg_mul, cons_val_two, tail_cons, head_fin_const, star_neg]
simp [conj_ofReal]
rw [exp_neg]
fin_cases i <;> simp
· ring_nf
field_simp
rw [sin_sq, sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq, sin_sq, sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq]
ring
simp only [Fin.reduceFinMk, Fin.isValue, conjTranspose_apply, cons_val', cons_val_two, tail_cons,
head_cons, empty_val', cons_val_fin_one, cons_val_zero, star_mul', RCLike.star_def, conj_ofReal,
← exp_conj, map_neg, _root_.map_mul, conj_I, neg_mul, neg_neg, cons_val_one, head_fin_const]
simp [conj_ofReal]
rw [exp_neg]
fin_cases i <;> simp
· ring_nf
rw [sin_sq]
ring
· ring_nf
rw [sin_sq]
ring
· ring_nf
field_simp
rw [sin_sq, sin_sq]
ring
def sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix :=
⟨standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by
rw [mem_unitaryGroup_iff']
exact standardParameterizationAsMatrix_unitary θ₁₂ θ₁₃ θ₂₃ δ₁₃⟩
lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
[sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]t = (conj [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]u ×₃ conj [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) := by
have h1 := exp_ne_zero (I * ↑δ₁₃)
funext i
fin_cases i
· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg,
Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons,
head_fin_const, cons_val_one, head_cons, Fin.zero_eta, crossProduct, uRow, cRow,
LinearMap.mk₂_apply, Pi.conj_apply, _root_.map_mul, map_inv₀, ← exp_conj, conj_I, conj_ofReal,
inv_inv, map_sub, map_neg]
field_simp
ring_nf
rw [sin_sq]
ring
· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue, cons_val',
cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const,
cons_val_one, head_cons, Fin.mk_one, crossProduct, uRow, cRow, LinearMap.mk₂_apply,
Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub,
map_neg]
field_simp
ring_nf
rw [sin_sq]
ring
· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue,
cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const,
cons_val_one, head_cons, Fin.reduceFinMk, crossProduct, uRow, cRow, LinearMap.mk₂_apply,
Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub,
map_neg]
field_simp
ring_nf
rw [sin_sq]
ring
lemma eq_sP (U : CKMMatrix) {θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ} (hu : [U]u = [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]u)
(hc : [U]c = [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) (hU : [U]t = conj [U]u ×₃ conj [U]c) :
U = sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
apply ext_Rows hu hc
rw [hU, sP_cross, hu, hc]
lemma eq_phases_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ δ₁₃' : ℝ) (h : cexp (δ₁₃ * I) = cexp (δ₁₃' * I)) :
sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ = sP θ₁₂ θ₁₃ θ₂₃ δ₁₃' := by
simp [sP, standardParameterizationAsMatrix]
apply CKMMatrix_ext
simp
rw [show exp (I * δ₁₃) = exp (I * δ₁₃') by rw [mul_comm, h, mul_comm]]
rw [show cexp (-(I * ↑δ₁₃)) = cexp (-(I * ↑δ₁₃')) by rw [exp_neg, exp_neg, mul_comm, h, mul_comm]]
section zeroEntries
variable (a b c d e f : ℝ)
@ -575,15 +484,9 @@ lemma Vs_zero_iff_cos_sin_zero (V : CKMMatrix) :
aesop
end zeroEntries
lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb ≠ 0)
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
have hb' : VubAbs ⟦V⟧ ≠ 1 := by
rw [ud_us_neq_zero_iff_ub_neq_one] at hb
@ -615,7 +518,7 @@ lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V
funext i
fin_cases i
simp [cRow, sP, standardParameterizationAsMatrix]
rw [cd_of_us_or_ud_neq_zero_UCond hb ha hV]
rw [cd_of_us_or_ud_neq_zero_UCond 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]
@ -628,7 +531,7 @@ lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V
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 ha hV]
rw [cs_of_us_or_ud_neq_zero_UCond hb hV]
field_simp
rw [h1]
simp [sq]
@ -639,75 +542,6 @@ lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V
rw [VcbAbs_eq_S₂₃_mul_C₁₃ ⟦V⟧, S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, C₁₃]
simp
lemma UCond₂_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
(hV : UCond₂ V) : V = sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
have h23 : S₂₃ ⟦V⟧ = 0 := by
have h1 := VcbAbs_eq_S₂₃_mul_C₁₃ ⟦V⟧
simp [VAbs] at h1
rw [ha] at h1
simp at h1
cases' h1 with h1 h1
exact h1
have h2 : abs [V]ud = 0 := by
change VudAbs ⟦V⟧ = 0
rw [VudAbs_eq_C₁₂_mul_C₁₃, h1]
simp
have h2b : abs [V]us = 0 := by
change VusAbs ⟦V⟧ = 0
rw [VusAbs_eq_S₁₂_mul_C₁₃, h1]
simp
simp_all
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 _)
have hx := Vabs_sq_add_neq_zero hb
simp at h1
have hb' : VubAbs ⟦V⟧ ≠ 1 := by
rw [ud_us_neq_zero_iff_ub_neq_one] at hb
simp [VAbs, hb]
refine eq_sP V ?_ ?_ hV.2.2.2.1
funext i
fin_cases i
simp [uRow, sP, standardParameterizationAsMatrix]
rw [hV.1, VudAbs_eq_C₁₂_mul_C₁₃ ⟦V⟧, C₁₂, C₁₃]
simp
simp [uRow, sP, standardParameterizationAsMatrix]
rw [hV.2.1, VusAbs_eq_S₁₂_mul_C₁₃ ⟦V⟧, ← S₁₂_eq_sin_θ₁₂ ⟦V⟧, C₁₃]
simp
simp [uRow, sP, standardParameterizationAsMatrix]
rw [hV.2.2.1, VubAbs_eq_S₁₃, S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧]
funext i
fin_cases i
simp [cRow, sP, standardParameterizationAsMatrix]
rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, h23]
simp
rw [C₂₃_eq_ℂcos_θ₂₃ ⟦V⟧, C₂₃_of_Vub_neq_one hb']
rw [S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧, S₁₂]
rw [hV.2.2.2.2.1]
field_simp
rw [h1]
simp [sq]
field_simp
ring_nf
simp
simp [cRow, sP, standardParameterizationAsMatrix]
rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, h23]
simp
rw [C₂₃_eq_ℂcos_θ₂₃ ⟦V⟧, C₂₃_of_Vub_neq_one hb']
rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_eq_Vud_div_sqrt hb']
rw [hV.2.2.2.2.2]
field_simp
rw [h1]
simp [sq]
field_simp
ring_nf
simp [cRow, sP, standardParameterizationAsMatrix]
rw [ha, S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, h23]
simp
lemma UCond₃_eq_sP {V : CKMMatrix} (hV : UCond₃ V) : V = sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
have h1 : VubAbs ⟦V⟧ = 1 := by
simp [VAbs]
@ -750,83 +584,72 @@ lemma UCond₃_eq_sP {V : CKMMatrix} (hV : UCond₃ V) : V = sP (θ₁₂ ⟦V
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.2.2.1]
simp
theorem exists_standardParameterization (V : CKMMatrix) :
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
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
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
by_cases hb : [V]cb ≠ 0
have hbU : [U]cb ≠ 0 := by
by_contra hn
simp at hn
have hna : VcbAbs ⟦U⟧ = 0 := by
simp [VAbs]
exact hn
rw [hUV] at hna
simp [VAbs] at hna
simp_all
have hU' := UCond₁_eq_sP haU hbU hU.2
rw [hU'] at hU
use (- arg ([U]ub))
rw [← hUV]
exact hU.1
simp at hb
have hbU : [U]cb = 0 := by
have hna : VcbAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs]
exact hb
simpa [VAbs] using hna
obtain ⟨U2, hU2⟩ := UCond₂_exists haU hbU hU.2
have hUVa2 : V ≈ U2 := phaseShiftEquivRelation.trans hU.1 hU2.1
have hUV2 : ⟦U2⟧ = ⟦V⟧ := (Quotient.eq.mpr (phaseShiftEquivRelation.symm hUVa2))
have haU2 : [U2]ud ≠ 0 ∨ [U2]us ≠ 0 := by
by_contra hn
simp [not_or] at hn
have hna : VudAbs ⟦U2⟧ = 0 ∧ VusAbs ⟦U2⟧ =0 := by
simp [VAbs]
exact hn
rw [hUV2] at hna
simp [VAbs] at hna
simp_all
have hbU2 : [U2]cb = 0 := by
have hna : VcbAbs ⟦U2⟧ = 0 := by
rw [hUV2]
simp [VAbs]
exact hb
simpa [VAbs] using hna
have hU2' := UCond₂_eq_sP haU2 hbU2 hU2.2
use 0
rw [← hUV2, ← hU2']
exact hUVa2
have haU : ¬ ([U]ud ≠ 0 ∨ [U]us ≠ 0) := by
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⟩ := UCond₃_exists 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
use 0
rw [← hUV2, ← hx]
exact hUVa2
· 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' := UCond₁_eq_sP 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⟩ := UCond₃_exists 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
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
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])
rw [h2] at hδ₃
exact hδ₃
simp at h
have h1 : δ₁₃ ⟦V⟧ = 0 := by
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 )
lemma exists_standardParameterization (V : CKMMatrix) :
∃ (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ), V ≈ sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
use θ₁₂ ⟦V⟧, θ₁₃ ⟦V⟧, θ₂₃ ⟦V⟧, δ₁₃ ⟦V⟧
exact eq_standardParameterization_δ₃ V
end