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Refactor: Names

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jstoobysmith 2024-04-29 09:22:32 -04:00
parent ff89c3f79d
commit 490ed0380c
6 changed files with 342 additions and 442 deletions
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6
HepLean/FlavorPhysics/CKMMatrix/Basic.lean
View file

@ -273,11 +273,10 @@ open ComplexConjugate
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
@ -289,7 +288,7 @@ 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
@ -298,7 +297,6 @@ lemma Rcdcb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cd = Rcdcb V * [V]cb :=
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

2
HepLean/FlavorPhysics/CKMMatrix/Invariants.lean
View file

@ -43,7 +43,7 @@ def jarlskogℂ : Quotient CKMMatrixSetoid → ℂ :=
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 :=
def mulExpδ₁₃ (V : Quotient CKMMatrixSetoid) : ℂ :=
jarlskogℂ V + VusVubVcdSq V

63
HepLean/FlavorPhysics/CKMMatrix/PhaseFreedom.lean
View file

@ -19,7 +19,7 @@ open ComplexConjugate
section phaseShiftApply
variable (u c t d s b : ℝ)
lemma ud_eq_abs (V : CKMMatrix) (h1 : u + d = - arg [V]ud) :
lemma shift_ud_phase_zero (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]
@ -31,7 +31,7 @@ lemma ud_eq_abs (V : CKMMatrix) (h1 : u + d = - arg [V]ud) :
simp
rfl
lemma us_eq_abs {V : CKMMatrix} (h1 : u + s = - arg [V]us) :
lemma shift_us_phase_zero {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]
@ -43,7 +43,7 @@ lemma us_eq_abs {V : CKMMatrix} (h1 : u + s = - arg [V]us) :
simp
rfl
lemma ub_eq_abs {V : CKMMatrix} (h1 : u + b = - arg [V]ub) :
lemma shift_ub_phase_zero {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]
@ -55,7 +55,7 @@ lemma ub_eq_abs {V : CKMMatrix} (h1 : u + b = - arg [V]ub) :
simp
rfl
lemma cs_eq_abs {V : CKMMatrix} (h1 : c + s = - arg [V]cs) :
lemma shift_cs_phase_zero {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]
@ -67,7 +67,7 @@ lemma cs_eq_abs {V : CKMMatrix} (h1 : c + s = - arg [V]cs) :
simp
rfl
lemma cb_eq_abs {V : CKMMatrix} (h1 : c + b = - arg [V]cb) :
lemma shift_cb_phase_zero {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]
@ -79,7 +79,7 @@ lemma cb_eq_abs {V : CKMMatrix} (h1 : c + b = - arg [V]cb) :
simp
rfl
lemma tb_eq_abs {V : CKMMatrix} (h1 : t + b = - arg [V]tb) :
lemma shift_tb_phase_zero {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]
@ -91,7 +91,7 @@ lemma tb_eq_abs {V : CKMMatrix} (h1 : t + b = - arg [V]tb) :
simp
rfl
lemma cd_eq_neg_abs {V : CKMMatrix} (h1 : c + d = Real.pi - arg [V]cd) :
lemma shift_cd_phase_pi {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]
@ -103,8 +103,8 @@ lemma cd_eq_neg_abs {V : CKMMatrix} (h1 : c + d = Real.pi - arg [V]cd) :
simp
rfl
lemma t_eq_conj {V : CKMMatrix} {τ : ℝ} (hτ : cexp (τ * I) • (conj [V]u ×₃ conj [V]c) = [V]t)
(h1 : τ = - u - c - t - d - s - b) :
lemma shift_cross_product_phase_zero {V : CKMMatrix} {τ : ℝ}
(hτ : cexp (τ * I) • (conj [V]u ×₃ conj [V]c) = [V]t) (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 _ _ _ _ _ _ _
@ -146,16 +146,17 @@ end phaseShiftApply
variable (a b c d e f : ℝ)
-- rename
def UCond₁ (U : CKMMatrix) : Prop := [U]ud = VudAbs ⟦U⟧ ∧ [U]us = VusAbs ⟦U⟧
def fstRowThdColRealCond (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 :=
def ubOnePhaseCond (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)
lemma fstRowThdColRealCond_shift_solution {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) :
b = - τ + arg [V]ud + arg [V]us + arg [V]tb + a ∧
c = - τ + arg [V]cb + arg [V]ud + arg [V]us + a ∧
@ -184,7 +185,8 @@ lemma all_cond_sol {V : CKMMatrix} (h1 : a + d = - arg [V]ud) (h2 : a + e = - a
ring_nf
simp
lemma UCond₃_solv {V : CKMMatrix} (h1 : a + f = - arg [V]ub) (h2 : 0 = - a - b - c - d - e - f)
lemma ubOnePhaseCond_shift_solution {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 ∧
@ -206,8 +208,8 @@ lemma UCond₃_solv {V : CKMMatrix} (h1 : a + f = - arg [V]ub) (h2 : 0 = - a - b
ring
-- rename
lemma all_eq_abs (V : CKMMatrix) :
∃ (U : CKMMatrix), V ≈ U ∧ UCond₁ U:= by
lemma fstRowThdColRealCond_holds_up_to_equiv (V : CKMMatrix) :
∃ (U : CKMMatrix), V ≈ U ∧ fstRowThdColRealCond U:= by
obtain ⟨τ, hτ⟩ := V.uRow_cross_cRow_eq_tRow
let U : CKMMatrix := phaseShiftApply V
0
@ -225,26 +227,27 @@ lemma all_eq_abs (V : CKMMatrix) :
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
rw [hUV]
apply ud_eq_abs _ _ _ _ _ _ _
apply shift_ud_phase_zero _ _ _ _ _ _ _
ring
apply And.intro
rw [hUV]
apply us_eq_abs
apply shift_us_phase_zero
ring
apply And.intro
rw [hUV]
apply cb_eq_abs
apply shift_cb_phase_zero
ring
apply And.intro
rw [hUV]
apply tb_eq_abs
apply shift_tb_phase_zero
ring
apply t_eq_conj _ _ _ _ _ _ hτ.symm
apply shift_cross_product_phase_zero _ _ _ _ _ _ hτ.symm
ring
lemma UCond₃_exists {V : CKMMatrix} (hb :¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) (hV : UCond₁ V) :
∃ (U : CKMMatrix), V ≈ U ∧ UCond₃ U:= by
lemma ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0))
(hV : fstRowThdColRealCond V) :
∃ (U : CKMMatrix), V ≈ U ∧ ubOnePhaseCond U:= by
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
@ -278,7 +281,7 @@ lemma UCond₃_exists {V : CKMMatrix} (hb :¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) (hV
exact h1
apply And.intro
· have hU1 : [U]ub = VubAbs ⟦V⟧ := by
apply ub_eq_abs _ _ _ _ _ _ _
apply shift_ub_phase_zero _ _ _ _ _ _ _
ring
rw [hU1]
have h1:= (ud_us_neq_zero_iff_ub_neq_one V).mpr.mt hb
@ -287,21 +290,21 @@ lemma UCond₃_exists {V : CKMMatrix} (hb :¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) (hV
· 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
apply shift_cross_product_phase_zero _ _ _ _ _ _ hτ.symm
ring
apply And.intro
· rw [hUV]
apply cd_eq_neg_abs _ _ _ _ _ _ _
apply shift_cd_phase_pi _ _ _ _ _ _ _
ring
have hcs : [U]cs = VcsAbs ⟦U⟧ := by
rw [hUV]
apply cs_eq_abs _ _ _ _ _ _ _
apply shift_cs_phase_zero _ _ _ _ _ _ _
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)) +
lemma cd_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
(hV : fstRowThdColRealCond 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
@ -319,8 +322,8 @@ lemma cd_of_us_or_ud_neq_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us
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))
lemma cs_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
(hV : fstRowThdColRealCond 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

196
HepLean/FlavorPhysics/CKMMatrix/Relations.lean
View file

@ -60,6 +60,8 @@ 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 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)
lemma ud_us_neq_zero_iff_ub_neq_one (V : CKMMatrix) :
[V]ud ≠ 0 ∨ [V]us ≠ 0 ↔ abs [V]ub ≠ 1 := by
@ -105,56 +107,38 @@ lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℂ {V : CKMMatrix} (hb : [V]ud ≠ 0
simp_all
lemma Vabs_sq_add_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
((VudAbs ⟦V⟧ : ℂ) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧)) ≠ 0 := by
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℂ hb
rw [← Complex.sq_abs, ← Complex.sq_abs] at h1
simp [sq] at h1
exact h1
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
lemma fst_row_orthog_snd_row (V : CKMMatrix) :
[V]cd * conj [V]ud + [V]cs * conj [V]us + [V]cb * conj [V]ub = 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
lemma fst_row_orthog_thd_row (V : CKMMatrix) :
[V]td * conj [V]ud + [V]ts * conj [V]us + [V]tb * conj [V]ub = 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) :
lemma Vcd_mul_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 )
linear_combination (V.fst_row_orthog_snd_row )
lemma Vcs_conj_Vus (V : CKMMatrix) :
lemma Vcs_mul_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]
linear_combination (V.fst_row_orthog_snd_row )
end orthogonal
@ -174,7 +158,7 @@ lemma VAbs_thd_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV
section crossProduct
lemma conj_Vtb_other {V : CKMMatrix} {τ : ℝ}
lemma conj_Vtb_cross_product {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
@ -189,36 +173,34 @@ lemma conj_Vtb_other {V : CKMMatrix} {τ : ℝ}
end crossProduct
lemma conj_Vtb_Vud_other {V : CKMMatrix} {τ : ℝ}
lemma conj_Vtb_mul_Vud {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]
rw [conj_Vtb_cross_product 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 [h2, V.Vcd_mul_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}
lemma conj_Vtb_mul_Vus {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]
rw [conj_Vtb_cross_product 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 [h2, V.Vcs_mul_conj_Vus]
rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
simp
ring
@ -229,7 +211,7 @@ lemma cs_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
[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]
rw [conj_Vtb_mul_Vud hτ]
field_simp
ring
@ -238,47 +220,12 @@ lemma cd_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
[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]
rw [conj_Vtb_mul_Vus 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
@ -287,7 +234,6 @@ lemma VAbs_ge_zero (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : 0 ≤ VAbs i
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
@ -301,6 +247,25 @@ lemma VAbs_leq_one (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : VAbs i j V ≤
end individual
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
section columns
@ -360,9 +325,9 @@ lemma cs_of_ud_us_zero {V : CKMMatrix} (ha : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) :
exact hn.trans h0
exact VAbs_ge_zero _ _ ⟦V⟧
end columns
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)
@ -388,47 +353,6 @@ lemma cb_tb_neq_zero_iff_ub_neq_one (V : CKMMatrix) :
have h2 : ¬ 0 ≤ ( -1 : ℝ) := by simp
exact h2 h1
lemma tb_neq_zero_of_cb_zero_ud_us_neq_zero {V : CKMMatrix} (h : [V]cb = 0)
(h1 : [V]ud ≠ 0 ∨ [V]us ≠ 0) : [V]tb ≠ 0 := by
rw [ud_us_neq_zero_iff_ub_neq_one] at h1
rw [← cb_tb_neq_zero_iff_ub_neq_one] at h1
simp_all
lemma normSq_tb_of_cb_zero {V : CKMMatrix} (h : [V]cb = 0) : normSq [V]tb =
normSq [V]ud + normSq [V]us := by
have h1 := fst_row_normalized_normSq V
rw [← thd_col_normalized_normSq V] at h1
rw [h] at h1
simp at h1
linear_combination -(1 * h1)
lemma div_td_of_cb_zero_ud_us_neq_zero {V : CKMMatrix} (h : [V]cb = 0)
(h1 : [V]ud ≠ 0 ∨ [V]us ≠ 0) (a : ℂ) : a / [V]tb =
(a * conj [V]tb) / (normSq [V]ud + normSq [V]us) := by
have h2 := tb_neq_zero_of_cb_zero_ud_us_neq_zero h h1
have h3 : conj [V]tb ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr h2
have h1 : a / [V]tb = a * conj [V]tb / normSq [V]tb := by
rw [normSq_eq_conj_mul_self]
field_simp
ring
rw [normSq_tb_of_cb_zero h] at h1
simp at h1
exact h1
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
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℂ hb
rw [← Complex.sq_abs, ← Complex.sq_abs] at h1
simp [sq] at h1
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
@ -443,31 +367,7 @@ lemma VAbs_fst_col_eq_one_thd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
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 columns
end CKMMatrix
end

61
HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/Basic.lean
View file

@ -16,7 +16,7 @@ open CKMMatrix
noncomputable section
-- to be renamed stanParamAsMatrix ...
def standardParameterizationAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
def standParamAsMatrix (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),
@ -27,10 +27,10 @@ def standardParameterizationAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ)
open CKMMatrix
lemma standardParameterizationAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
((standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by
lemma standParamAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
((standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by
funext j i
simp only [standardParameterizationAsMatrix, neg_mul, Fin.isValue]
simp only [standParamAsMatrix, neg_mul, Fin.isValue]
rw [mul_apply]
have h1 := exp_ne_zero (I * ↑δ₁₃)
fin_cases j <;> rw [Fin.sum_univ_three]
@ -87,17 +87,19 @@ lemma standardParameterizationAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ
rw [sin_sq, sin_sq]
ring
def sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix :=
⟨standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by
def standParam (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix :=
⟨standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by
rw [mem_unitaryGroup_iff']
exact standardParameterizationAsMatrix_unitary θ₁₂ θ₁₃ θ₂₃ δ₁₃⟩
exact standParamAsMatrix_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
namespace standParam
lemma cross_product_t (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
[standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]t =
(conj [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]u ×₃ conj [standParam 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,
· simp only [tRow, standParam, standParamAsMatrix, 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,
@ -106,7 +108,7 @@ lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
ring_nf
rw [sin_sq]
ring
· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue, cons_val',
· simp only [tRow, standParam, standParamAsMatrix, 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,
@ -115,7 +117,7 @@ lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
ring_nf
rw [sin_sq]
ring
· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue,
· simp only [tRow, standParam, standParamAsMatrix, 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,
@ -125,27 +127,26 @@ lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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
lemma eq_rows (U : CKMMatrix) {θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ} (hu : [U]u = [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]u)
(hc : [U]c = [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) (hU : [U]t = conj [U]u ×₃ conj [U]c) :
U = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
apply ext_Rows hu hc
rw [hU, sP_cross, hu, hc]
rw [hU, cross_product_t, 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]
lemma eq_exp_of_phases (θ₁₂ θ₁₃ θ₂₃ δ₁₃ δ₁₃' : ℝ) (h : cexp (δ₁₃ * I) = cexp (δ₁₃' * I)) :
standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃' := by
simp [standParam, standParamAsMatrix]
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)
open Invariant in
lemma VusVubVcdSq_eq (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⟧ =
VusVubVcdSq ⟦standParam 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,
simp only [VusVubVcdSq, VusAbs, VAbs, VAbs', Fin.isValue, standParam, standParamAsMatrix,
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]
@ -164,12 +165,13 @@ lemma VusVubVcdSq_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Rea
rw [hx]
simp
lemma mulExpδ₃_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
open Invariant in
lemma mulExpδ₁₃_eq (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⟧ =
mulExpδ₁₃ ⟦standParam 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,
rw [mulExpδ₁₃, VusVubVcdSq_eq _ _ _ _ h1 h2 h3 h4 ]
simp only [jarlskogℂ, standParam, standParamAsMatrix, 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]
@ -179,6 +181,7 @@ lemma mulExpδ₃_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Rea
have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _
field_simp
end Invariant
end standParam
end

456
HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/StandardParameters.lean
View file

@ -37,7 +37,7 @@ 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)
arg (Invariant.mulExpδ₁₃ V)
section sines
@ -277,67 +277,7 @@ 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
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) :
lemma VubAbs_of_cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos (θ₁₃ V) = 0) :
VubAbs V = 1 := by
rw [θ₁₃, Real.cos_arcsin, ← VubAbs_eq_S₁₃, Real.sqrt_eq_zero] at h1
have h2 : VubAbs V ^ 2 = 1 := by linear_combination -(1 * h1)
@ -352,128 +292,6 @@ lemma cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos (θ₁₃
rw [_root_.abs_of_nonneg (VAbs_ge_zero 0 2 V)]
exact VAbs_leq_one 0 2 V
open CKMMatrix
section zeroEntries
variable (a b c d e f : ℝ)
lemma sP_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₃ ⟦V⟧) = 0) :
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
have hS13 := congrArg ofReal (S₁₃_of_Vub_one (cos_θ₁₃_zero h))
simp [← S₁₃_eq_ℂsin_θ₁₃] at hS13
have hC12 := congrArg ofReal (C₁₂_of_Vub_one (cos_θ₁₃_zero h))
simp [← C₁₂_eq_ℂcos_θ₁₂] at hC12
have hS12 := congrArg ofReal (S₁₂_of_Vub_one (cos_θ₁₃_zero h))
simp [← S₁₂_eq_ℂsin_θ₁₂] at hS12
use 0, 0, 0, δ₁₃, 0, -δ₁₃
simp [sP, standardParameterizationAsMatrix, h, phaseShift, hS13, hC12, hS12]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
rfl
rfl
lemma sP_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₂ ⟦V⟧) = 0) :
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, δ₁₃, δ₁₃, -δ₁₃, 0, - δ₁₃
have hb := exp_ne_zero (I * δ₁₃)
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
change _ = _ + _ * 0
simp
field_simp
ring
ring
field_simp
ring
change _ = _ + _ * 0
simp
field_simp
ring
ring
field_simp
ring
lemma sP_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₂₃ ⟦V⟧) = 0) :
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, δ₁₃, 0, 0, 0, - δ₁₃
have hb := exp_ne_zero (I * δ₁₃)
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
ring_nf
change _ = _ + _ * 0
simp
ring
field_simp
ring
lemma sP_sin_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₃ ⟦V⟧) = 0) :
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, 0, 0, 0, 0, 0
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
apply Or.inr
rfl
lemma sP_sin_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₂ ⟦V⟧) = 0) :
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, δ₁₃, δ₁₃, 0, -δ₁₃, - δ₁₃
have hb := exp_ne_zero (I * δ₁₃)
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
change _ = _ + _ * 0
simp
ring
field_simp
ring
field_simp
ring
change _ = _ + _ * 0
simp
ring
field_simp
ring
field_simp
ring
lemma sP_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₂₃ ⟦V⟧) = 0) :
sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, 0, δ₁₃, 0, 0, - δ₁₃
have hb := exp_ne_zero (I * δ₁₃)
simp [sP, standardParameterizationAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
change _ = _ + _ * 0
simp
ring
ring
field_simp
ring
lemma Vs_zero_iff_cos_sin_zero (V : CKMMatrix) :
VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0
↔ Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨
@ -483,11 +301,182 @@ lemma Vs_zero_iff_cos_sin_zero (V : CKMMatrix) :
rw [C₁₂, C₁₃, C₂₃, S₁₂_eq_sin_θ₁₂, S₂₃_eq_sin_θ₂₃, S₁₃_eq_sin_θ₁₃]
aesop
end VAbs
end zeroEntries
lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
(hV : UCond₁ V) : V = sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (- arg [V]ub) := by
namespace standParam
open Invariant
lemma mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) :
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧) * cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := by
refine mulExpδ₁₃_eq _ _ _ _ ?_ ?_ ?_ ?_
rw [S₁₂_eq_sin_θ₁₂]
exact S₁₂_nonneg _
exact Real.cos_arcsin_nonneg _
rw [S₂₃_eq_sin_θ₂₃]
exact S₂₃_nonneg _
exact Real.cos_arcsin_nonneg _
lemma mulExpδ₁₃_on_param_eq_zero_iff (V : CKMMatrix) (δ₁₃ : ℝ) :
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 := by
rw [VudAbs_eq_C₁₂_mul_C₁₃, VubAbs_eq_S₁₃, VusAbs_eq_S₁₂_mul_C₁₃, VcbAbs_eq_S₂₃_mul_C₁₃, VtbAbs_eq_C₂₃_mul_C₁₃,
← ofReal_inj,
← ofReal_inj, ← ofReal_inj, ← ofReal_inj, ← ofReal_inj]
simp only [ofReal_mul]
rw [← S₁₃_eq_ℂsin_θ₁₃, ← S₁₂_eq_ℂsin_θ₁₂, ← S₂₃_eq_ℂsin_θ₂₃,
← C₁₃_eq_ℂcos_θ₁₃, ← C₂₃_eq_ℂcos_θ₂₃,← C₁₂_eq_ℂcos_θ₁₂]
rw [mulExpδ₁₃_on_param_δ₁₃]
simp
have h1 := exp_ne_zero (I * δ₁₃)
simp_all
aesop
lemma mulExpδ₁₃_on_param_abs (V : CKMMatrix) (δ₁₃ : ℝ) :
Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) =
sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) := by
rw [mulExpδ₁₃_on_param_δ₁₃]
simp [abs_exp]
rw [complexAbs_sin_θ₁₃, complexAbs_cos_θ₁₃, complexAbs_sin_θ₁₂, complexAbs_cos_θ₁₂,
complexAbs_sin_θ₂₃, complexAbs_cos_θ₂₃]
lemma mulExpδ₁₃_on_param_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
(h1 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
cexp (arg ( mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ) * I) =
cexp (δ₁₃ * I) := by
have h1a := mulExpδ₁₃_on_param_δ₁₃ V δ₁₃
have habs := mulExpδ₁₃_on_param_abs V δ₁₃
have h2 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
rw [habs, h1a]
ring_nf
nth_rewrite 1 [← abs_mul_exp_arg_mul_I (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
have habs_neq_zero : (Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
simp
exact h1
rw [← mul_right_inj' habs_neq_zero]
rw [← h2]
lemma on_param_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₃ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
have hS13 := congrArg ofReal (S₁₃_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
simp [← S₁₃_eq_ℂsin_θ₁₃] at hS13
have hC12 := congrArg ofReal (C₁₂_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
simp [← C₁₂_eq_ℂcos_θ₁₂] at hC12
have hS12 := congrArg ofReal (S₁₂_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
simp [← S₁₂_eq_ℂsin_θ₁₂] at hS12
use 0, 0, 0, δ₁₃, 0, -δ₁₃
simp [standParam, standParamAsMatrix, h, phaseShift, hS13, hC12, hS12]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
rfl
rfl
lemma on_param_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₂ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, δ₁₃, δ₁₃, -δ₁₃, 0, - δ₁₃
have hb := exp_ne_zero (I * δ₁₃)
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
change _ = _ + _ * 0
simp
field_simp
ring
ring
field_simp
ring
change _ = _ + _ * 0
simp
field_simp
ring
ring
field_simp
ring
lemma on_param_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₂₃ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, δ₁₃, 0, 0, 0, - δ₁₃
have hb := exp_ne_zero (I * δ₁₃)
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
ring_nf
change _ = _ + _ * 0
simp
ring
field_simp
ring
lemma on_param_sin_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₃ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, 0, 0, 0, 0, 0
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
apply Or.inr
rfl
lemma on_param_sin_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₂ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, δ₁₃, δ₁₃, 0, -δ₁₃, - δ₁₃
have hb := exp_ne_zero (I * δ₁₃)
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
change _ = _ + _ * 0
simp
ring
field_simp
ring
field_simp
ring
change _ = _ + _ * 0
simp
ring
field_simp
ring
field_simp
ring
lemma on_param_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₂₃ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, 0, δ₁₃, 0, 0, - δ₁₃
have hb := exp_ne_zero (I * δ₁₃)
simp [standParam, standParamAsMatrix, h, phaseShift, exp_neg]
funext i j
fin_cases i <;> fin_cases j <;>
simp [mul_apply, Fin.sum_univ_three, mul_apply, Fin.sum_univ_three]
apply Or.inr
rfl
change _ = _ + _ * 0
simp
ring
ring
field_simp
ring
lemma eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
(hV : fstRowThdColRealCond V) : V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (- arg [V]ub) := by
have hb' : VubAbs ⟦V⟧ ≠ 1 := by
rw [ud_us_neq_zero_iff_ub_neq_one] at hb
simp [VAbs, hb]
@ -497,18 +486,18 @@ lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
apply add_nonneg (sq_nonneg _) (sq_nonneg _)
simp at h1
have hx := Vabs_sq_add_neq_zero hb
refine eq_sP V ?_ ?_ hV.2.2.2.2
refine eq_rows V ?_ ?_ hV.2.2.2.2
funext i
fin_cases i
simp only [uRow, Fin.isValue, Fin.zero_eta, cons_val_zero, sP, standardParameterizationAsMatrix,
simp only [uRow, Fin.isValue, Fin.zero_eta, cons_val_zero, standParam, standParamAsMatrix,
ofReal_cos, ofReal_sin, ofReal_neg, mul_neg, neg_mul, neg_neg, cons_val', empty_val',
cons_val_fin_one, cons_val_one, head_cons, cons_val_two, tail_cons]
rw [hV.1, VudAbs_eq_C₁₂_mul_C₁₃ ⟦V⟧]
simp [C₁₂, C₁₃]
simp [uRow, sP, standardParameterizationAsMatrix]
simp [uRow, standParam, standParamAsMatrix]
rw [hV.2.1, VusAbs_eq_S₁₂_mul_C₁₃ ⟦V⟧, ← S₁₂_eq_sin_θ₁₂ ⟦V⟧, C₁₃]
simp
simp [uRow, sP, standardParameterizationAsMatrix]
simp [uRow, standParam, standParamAsMatrix]
nth_rewrite 1 [← abs_mul_exp_arg_mul_I (V.1 0 2)]
rw [show Complex.abs (V.1 0 2) = VubAbs ⟦V⟧ from rfl]
rw [VubAbs_eq_S₁₃, ← S₁₃_eq_sin_θ₁₃ ⟦V⟧]
@ -517,8 +506,8 @@ lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
simp
funext i
fin_cases i
simp [cRow, sP, standardParameterizationAsMatrix]
rw [cd_of_us_or_ud_neq_zero_UCond hb hV]
simp [cRow, standParam, standParamAsMatrix]
rw [cd_of_fstRowThdColRealCond hb hV]
rw [S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧, S₁₂, C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_eq_Vud_div_sqrt hb']
rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, S₂₃_of_Vub_neq_one hb', C₂₃_eq_ℂcos_θ₂₃ ⟦V⟧,
C₂₃_of_Vub_neq_one hb', S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃]
@ -527,50 +516,52 @@ lemma UCond₁_eq_sP {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
simp [sq]
field_simp
ring_nf
simp [cRow, sP, standardParameterizationAsMatrix]
simp [cRow, standParam, standParamAsMatrix]
rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₂₃_eq_ℂcos_θ₂₃ ⟦V⟧, S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧,
S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧]
rw [C₁₂_eq_Vud_div_sqrt hb', C₂₃_of_Vub_neq_one hb', S₁₂, S₁₃, S₂₃_of_Vub_neq_one hb']
rw [cs_of_us_or_ud_neq_zero_UCond hb hV]
rw [cs_of_fstRowThdColRealCond hb hV]
field_simp
rw [h1]
simp [sq]
field_simp
ring_nf
simp [cRow, sP, standardParameterizationAsMatrix]
simp [cRow, standParam, standParamAsMatrix]
rw [hV.2.2.1]
rw [VcbAbs_eq_S₂₃_mul_C₁₃ ⟦V⟧, S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, C₁₃]
simp
lemma UCond₃_eq_sP {V : CKMMatrix} (hV : UCond₃ V) : V = sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
lemma eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : ubOnePhaseCond V) :
V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
have h1 : VubAbs ⟦V⟧ = 1 := by
simp [VAbs]
rw [hV.2.2.2.1]
simp
refine eq_sP V ?_ ?_ hV.2.2.2.2.1
refine eq_rows V ?_ ?_ hV.2.2.2.2.1
funext i
fin_cases i
simp [uRow, sP, standardParameterizationAsMatrix]
simp [uRow, standParam, standParamAsMatrix]
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.1]
simp
simp [uRow, sP, standardParameterizationAsMatrix]
simp [uRow, standParam, standParamAsMatrix]
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.2.1]
simp
simp [uRow, sP, standardParameterizationAsMatrix]
simp [uRow, standParam, standParamAsMatrix]
rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃]
simp [VAbs]
rw [hV.2.2.2.1]
simp
funext i
fin_cases i
simp [cRow, sP, standardParameterizationAsMatrix]
simp [cRow, standParam, standParamAsMatrix]
rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, S₂₃_of_Vub_eq_one h1]
rw [S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧, S₁₂_of_Vub_one h1]
rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_of_Vub_one h1]
rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃_of_Vub_one h1]
rw [hV.2.2.2.2.2.1]
simp
simp [cRow, sP, standardParameterizationAsMatrix]
simp [cRow, standParam, standParamAsMatrix]
rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, S₂₃_of_Vub_eq_one h1]
rw [S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧, S₁₂_of_Vub_one h1]
rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_of_Vub_one h1]
@ -580,13 +571,14 @@ lemma UCond₃_eq_sP {V : CKMMatrix} (hV : UCond₃ V) : V = sP (θ₁₂ ⟦V
rw [θ₂₃, Real.cos_arcsin]
simp at h3
rw [h3, S₂₃_of_Vub_eq_one h1, hV.2.2.2.2.2.2]
simp [cRow, sP, standardParameterizationAsMatrix]
simp [cRow, standParam, standParamAsMatrix]
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.2.2.1]
simp
theorem exists_standardParameterization_δ₁₃ (V : CKMMatrix) :
∃ (δ₃ : ℝ), V ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ := by
obtain ⟨U, hU⟩ := all_eq_abs V
theorem exists_δ₁₃ (V : CKMMatrix) :
∃ (δ₃ : ℝ), V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ := by
obtain ⟨U, hU⟩ := fstRowThdColRealCond_holds_up_to_equiv V
have hUV : ⟦U⟧ = ⟦V⟧ := (Quotient.eq.mpr (phaseShiftEquivRelation.symm hU.1))
by_cases ha : [V]ud ≠ 0 ∨ [V]us ≠ 0
· have haU : [U]ud ≠ 0 ∨ [U]us ≠ 0 := by -- should be much simplier
@ -598,7 +590,7 @@ theorem exists_standardParameterization_δ₁₃ (V : CKMMatrix) :
rw [hUV] at hna
simp [VAbs] at hna
simp_all
have hU' := UCond₁_eq_sP haU hU.2
have hU' := eq_standParam_of_fstRowThdColRealCond haU hU.2
rw [hU'] at hU
use (- arg ([U]ub))
rw [← hUV]
@ -610,23 +602,22 @@ theorem exists_standardParameterization_δ₁₃ (V : CKMMatrix) :
simp [VAbs]
exact ha
simpa [not_or, VAbs] using h1
have ⟨U2, hU2⟩ := UCond₃_exists haU hU.2
have ⟨U2, hU2⟩ := ubOnePhaseCond_hold_up_to_equiv_of_ub_one haU hU.2
have hUVa2 : V ≈ U2 := phaseShiftEquivRelation.trans hU.1 hU2.1
have hUV2 : ⟦U2⟧ = ⟦V⟧ := (Quotient.eq.mpr (phaseShiftEquivRelation.symm hUVa2))
have hx := UCond₃_eq_sP hU2.2
have hx := eq_standParam_of_ubOnePhaseCond hU2.2
use 0
rw [← hUV2, ← hx]
exact hUVa2
open Invariant in
theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
V ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧) := by
obtain ⟨δ₁₃', hδ₃⟩ := exists_standardParameterization_δ₁₃ V
V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧) := by
obtain ⟨δ₁₃', hδ₃⟩ := exists_δ₁₃ V
have hSV := (Quotient.eq.mpr (hδ₃))
by_cases h : Invariant.mulExpδ₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
have h1 := Invariant.mulExpδ₃_neq_zero_arg V δ₁₃' h
have h2 := eq_phases_sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
(δ₁₃ ⟦V⟧) (by rw [← h1, ← hSV, δ₁₃, Invariant.mulExpδ₃])
by_cases h : Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
have h2 := eq_exp_of_phases (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
(δ₁₃ ⟦V⟧) (by rw [← mulExpδ₁₃_on_param_neq_zero_arg V δ₁₃' h, ← hSV, δ₁₃, Invariant.mulExpδ₁₃])
rw [h2] at hδ₃
exact hδ₃
simp at h
@ -634,22 +625,27 @@ theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
rw [hSV, δ₁₃, h]
simp
rw [h1]
rw [mulExpδ₃_eq_zero, Vs_zero_iff_cos_sin_zero] at h
cases' h with h h
exact phaseShiftEquivRelation.trans hδ₃ (sP_cos_θ₁₂_eq_zero δ₁₃' h )
cases' h with h h
exact phaseShiftEquivRelation.trans hδ₃ (sP_cos_θ₁₃_eq_zero δ₁₃' h )
cases' h with h h
exact phaseShiftEquivRelation.trans hδ₃ (sP_cos_θ₂₃_eq_zero δ₁₃' h )
cases' h with h h
exact phaseShiftEquivRelation.trans hδ₃ (sP_sin_θ₁₂_eq_zero δ₁₃' h )
cases' h with h h
exact phaseShiftEquivRelation.trans hδ₃ (sP_sin_θ₁₃_eq_zero δ₁₃' h )
exact phaseShiftEquivRelation.trans hδ₃ (sP_sin_θ₂₃_eq_zero δ₁₃' h )
rw [mulExpδ₁₃_on_param_eq_zero_iff, Vs_zero_iff_cos_sin_zero] at h
refine phaseShiftEquivRelation.trans hδ₃ ?_
rcases h with h | h | h | h | h | h
exact on_param_cos_θ₁₂_eq_zero δ₁₃' h
exact on_param_cos_θ₁₃_eq_zero δ₁₃' h
exact on_param_cos_θ₂₃_eq_zero δ₁₃' h
exact on_param_sin_θ₁₂_eq_zero δ₁₃' h
exact on_param_sin_θ₁₃_eq_zero δ₁₃' h
exact on_param_sin_θ₂₃_eq_zero δ₁₃' h
lemma exists_standardParameterization (V : CKMMatrix) :
∃ (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ), V ≈ sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
theorem exists_for_CKMatrix (V : CKMMatrix) :
∃ (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ), V ≈ standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
use θ₁₂ ⟦V⟧, θ₁₃ ⟦V⟧, θ₂₃ ⟦V⟧, δ₁₃ ⟦V⟧
exact eq_standardParameterization_δ₃ V
end standParam
open CKMMatrix
end