refactor: Major refactor of CKMMatrix

HepLean/FlavorPhysics/CKMMatrix/Invariants.lean

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+/-
+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