177 lines
7.6 KiB
Text
177 lines
7.6 KiB
Text
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.FlavorPhysics.CKMMatrix.Basic
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import HepLean.FlavorPhysics.CKMMatrix.Rows
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import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom
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import HepLean.FlavorPhysics.CKMMatrix.Ratios
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import HepLean.FlavorPhysics.CKMMatrix.StandardParameters
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import Mathlib.Analysis.SpecialFunctions.Complex.Arg
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open Matrix Complex
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open ComplexConjugate
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open CKMMatrix
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noncomputable section
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@[simps!]
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def jarlskogComplexCKM (V : CKMMatrix) : ℂ :=
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[V]us * [V]cb * conj [V]ub * conj [V]cs
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lemma jarlskogComplexCKM_equiv (V U : CKMMatrix) (h : V ≈ U) :
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jarlskogComplexCKM V = jarlskogComplexCKM U := by
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obtain ⟨a, b, c, e, f, g, h⟩ := h
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change V = phaseShiftApply U a b c e f g at h
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rw [h]
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simp only [jarlskogComplexCKM, Fin.isValue, phaseShiftApply.ub,
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phaseShiftApply.us, phaseShiftApply.cb, phaseShiftApply.cs]
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simp [← exp_conj, conj_ofReal, exp_add, exp_neg]
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have ha : cexp (↑a * I) ≠ 0 := exp_ne_zero _
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have hb : cexp (↑b * I) ≠ 0 := exp_ne_zero _
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have hf : cexp (↑f * I) ≠ 0 := exp_ne_zero _
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have hg : cexp (↑g * I) ≠ 0 := exp_ne_zero _
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field_simp
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ring
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def inv₁ (V : Quotient CKMMatrixSetoid) : ℝ :=
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VusAbs V ^ 2 * VubAbs V ^ 2 * VcbAbs V ^2 /(VudAbs V ^ 2 + VusAbs V ^2)
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lemma inv₁_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
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(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
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inv₁ ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
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Real.sin θ₁₂ ^ 2 * Real.cos θ₁₃ ^ 2 * Real.sin θ₁₃ ^ 2 * Real.sin θ₂₃ ^ 2 := by
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simp only [inv₁, VusAbs, VAbs, VAbs', Fin.isValue, sP, standardParameterizationAsMatrix,
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neg_mul, Quotient.lift_mk, cons_val', cons_val_one, head_cons,
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empty_val', cons_val_fin_one, cons_val_zero, _root_.map_mul, VubAbs, cons_val_two, tail_cons,
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VcbAbs, VudAbs, Complex.abs_ofReal]
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by_cases hx : Real.cos θ₁₃ ≠ 0
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·
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rw [Complex.abs_exp]
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simp
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rw [_root_.abs_of_nonneg h1, _root_.abs_of_nonneg h3, _root_.abs_of_nonneg h2,
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_root_.abs_of_nonneg h4]
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simp [sq]
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ring_nf
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nth_rewrite 2 [Real.sin_sq θ₁₂]
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ring_nf
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field_simp
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ring
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· simp at hx
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rw [hx]
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simp
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@[simp]
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def jarlskogComplex : Quotient CKMMatrixSetoid → ℂ :=
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Quotient.lift jarlskogComplexCKM jarlskogComplexCKM_equiv
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-- bad name
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def expδ₁₃ (V : Quotient CKMMatrixSetoid) : ℂ :=
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jarlskogComplex V + inv₁ V
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lemma expδ₁₃_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
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(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
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expδ₁₃ ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
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sin θ₁₂ * cos θ₁₃ ^ 2 * sin θ₂₃ * sin θ₁₃ * cos θ₁₂ * cos θ₂₃ * cexp (I * δ₁₃) := by
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rw [expδ₁₃]
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rw [inv₁_sP _ _ _ _ h1 h2 h3 h4 ]
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simp only [expδ₁₃, jarlskogComplex, sP, standardParameterizationAsMatrix, neg_mul,
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Quotient.lift_mk, jarlskogComplexCKM, Fin.isValue, cons_val', cons_val_one, head_cons,
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empty_val', cons_val_fin_one, cons_val_zero, cons_val_two, tail_cons, _root_.map_mul, ←
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exp_conj, map_neg, conj_I, conj_ofReal, neg_neg, map_sub]
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simp
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ring_nf
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rw [exp_neg]
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have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _
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field_simp
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lemma expδ₁₃_sP_V (V : CKMMatrix) (δ₁₃ : ℝ) :
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expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
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sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
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* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := by
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refine expδ₁₃_sP _ _ _ _ ?_ ?_ ?_ ?_
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rw [S₁₂_eq_sin_θ₁₂]
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exact S₁₂_nonneg _
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exact Real.cos_arcsin_nonneg _
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rw [S₂₃_eq_sin_θ₂₃]
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exact S₂₃_nonneg _
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exact Real.cos_arcsin_nonneg _
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lemma expδ₁₃_eq_zero (V : CKMMatrix) (δ₁₃ : ℝ) :
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expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
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VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 := by
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rw [VudAbs_eq_C₁₂_mul_C₁₃, VubAbs_eq_S₁₃, VusAbs_eq_S₁₂_mul_C₁₃, VcbAbs_eq_S₂₃_mul_C₁₃, VtbAbs_eq_C₂₃_mul_C₁₃,
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← ofReal_inj,
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← ofReal_inj, ← ofReal_inj, ← ofReal_inj, ← ofReal_inj]
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simp only [ofReal_mul]
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rw [← S₁₃_eq_ℂsin_θ₁₃, ← S₁₂_eq_ℂsin_θ₁₂, ← S₂₃_eq_ℂsin_θ₂₃,
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← C₁₃_eq_ℂcos_θ₁₃, ← C₂₃_eq_ℂcos_θ₂₃,← C₁₂_eq_ℂcos_θ₁₂]
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simp
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rw [expδ₁₃_sP_V]
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simp
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have h1 := exp_ne_zero (I * δ₁₃)
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simp_all
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aesop
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lemma inv₂_V_arg (V : CKMMatrix) (δ₁₃ : ℝ)
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(h1 : expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
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cexp (arg (expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * I) =
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cexp (δ₁₃ * I) := by
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have h1a := expδ₁₃_sP_V V δ₁₃
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have habs : Complex.abs (expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) =
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sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
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* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) := by
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rw [h1a]
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simp [abs_exp]
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rw [complexAbs_sin_θ₁₃, complexAbs_cos_θ₁₃, complexAbs_sin_θ₁₂, complexAbs_cos_θ₁₂,
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complexAbs_sin_θ₂₃, complexAbs_cos_θ₂₃]
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have h2 : expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
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Complex.abs (expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
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rw [habs, h1a]
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ring_nf
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nth_rewrite 1 [← abs_mul_exp_arg_mul_I (expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
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have habs_neq_zero : (Complex.abs (expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
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simp
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exact h1
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rw [← mul_right_inj' habs_neq_zero]
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rw [← h2]
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def δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := arg (expδ₁₃ V)
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theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
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V ≈ sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧) := by
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obtain ⟨δ₁₃', hδ₃⟩ := exists_standardParameterization V
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have hSV := (Quotient.eq.mpr (hδ₃))
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by_cases h : expδ₁₃ ⟦sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
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have h1 := inv₂_V_arg V δ₁₃' h
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have h2 := eq_phases_sP (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
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(δ₁₃ ⟦V⟧) (by rw [← h1, ← hSV, δ₁₃])
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rw [h2] at hδ₃
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exact hδ₃
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simp at h
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have h1 : δ₁₃ ⟦V⟧ = 0 := by
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rw [hSV, δ₁₃, h]
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simp
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rw [h1]
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rw [expδ₁₃_eq_zero, Vs_zero_iff_cos_sin_zero] at h
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cases' h with h h
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exact phaseShiftEquivRelation.trans hδ₃ (sP_cos_θ₁₂_eq_zero δ₁₃' h )
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cases' h with h h
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exact phaseShiftEquivRelation.trans hδ₃ (sP_cos_θ₁₃_eq_zero δ₁₃' h )
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cases' h with h h
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exact phaseShiftEquivRelation.trans hδ₃ (sP_cos_θ₂₃_eq_zero δ₁₃' h )
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cases' h with h h
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exact phaseShiftEquivRelation.trans hδ₃ (sP_sin_θ₁₂_eq_zero δ₁₃' h )
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cases' h with h h
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exact phaseShiftEquivRelation.trans hδ₃ (sP_sin_θ₁₃_eq_zero δ₁₃' h )
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exact phaseShiftEquivRelation.trans hδ₃ (sP_sin_θ₂₃_eq_zero δ₁₃' h )
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end
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