| import Mathlib.Algebra.BigOperators.Fin |
| import Mathlib.Analysis.Real.Sqrt |
| import Mathlib.Tactic.FinCases |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.NormNum |
| import Mathlib.Tactic.Positivity |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| noncomputable section |
| / |
| boolean records whether this is the tagged block. -/ |
| structure CouplingParameters where |
| m : ℝ |
| kappa : ℝ |
| width : Fin 2 → ℝ |
| tagged : Bool |
| m_ge : 100 ≤ m |
| kappa_nonneg : 0 ≤ kappa |
| kappa_le : kappa ≤ 1/100 |
| width_lower : ∀ i, 1/m ≤ width i |
| width_upper : ∀ i, width i ≤ 1/4 |
|
|
| namespace CouplingParameters |
| variable (p : CouplingParameters) |
|
|
| theorem m_pos : 0 < p.m := by linarith [p.m_ge] |
| theorem width_pos (i : Fin 2) : 0 < p.width i := |
| (one_div_pos.mpr p.m_pos).trans_le (p.width_lower i) |
|
|
| def rho (i : Fin 2) : ℝ := p.kappa * Real.sqrt (p.width i / p.m) |
| def tagMass : ℝ := if p.tagged then 1/p.m else 0 |
| def diagonalMass : ℝ := 1 - 2*p.rho 0 - 2*p.rho 1 - p.tagMass |
|
|
| theorem rho_nonneg (i : Fin 2) : 0 ≤ p.rho i := |
| mul_nonneg p.kappa_nonneg (Real.sqrt_nonneg _) |
|
|
| theorem rho_le (i : Fin 2) : p.rho i ≤ p.kappa * p.width i := by |
| apply mul_le_mul_of_nonneg_left _ p.kappa_nonneg |
| apply (Real.sqrt_le_iff).2 |
| refine ⟨(p.width_pos i).le, ?_⟩ |
| apply (div_le_iff₀ p.m_pos).2 |
| have h := (div_le_iff₀ p.m_pos).1 (p.width_lower i) |
| nlinarith [p.width_pos i, mul_nonneg (p.width_pos i).le (sub_nonneg.mpr h)] |
|
|
| theorem rho_div_width_le (i : Fin 2) : p.rho i / p.width i ≤ p.kappa := |
| (div_le_iff₀ (p.width_pos i)).2 (p.rho_le i) |
|
|
| theorem rho_energy (i : Fin 2) : (p.rho i)^2 / p.width i = p.kappa^2 / p.m := by |
| rw [rho, mul_pow, Real.sq_sqrt (div_nonneg (p.width_pos i).le p.m_pos.le)] |
| field_simp [p.m_pos.ne', (p.width_pos i).ne'] |
|
|
| theorem tagMass_nonneg : 0 ≤ p.tagMass := by |
| unfold tagMass |
| split |
| · exact div_nonneg zero_le_one p.m_pos.le |
| · exact le_rfl |
|
|
| theorem tagMass_le : p.tagMass ≤ 1/100 := by |
| unfold tagMass |
| split |
| · exact one_div_le_one_div_of_le (by norm_num) p.m_ge |
| · norm_num |
|
|
| theorem diagonalMass_lower : 98/100 ≤ p.diagonalMass := by |
| have h₀ := (p.rho_le 0).trans (mul_le_mul_of_nonneg_left (p.width_upper 0) p.kappa_nonneg) |
| have h₁ := (p.rho_le 1).trans (mul_le_mul_of_nonneg_left (p.width_upper 1) p.kappa_nonneg) |
| unfold diagonalMass |
| linarith [p.kappa_le, p.tagMass_le] |
|
|
| theorem diagonalMass_pos : 0 < p.diagonalMass := by linarith [p.diagonalMass_lower] |
|
|
| theorem diagonalMass_le : p.diagonalMass ≤ 1 := by |
| unfold diagonalMass |
| linarith [p.rho_nonneg 0, p.rho_nonneg 1, p.tagMass_nonneg] |
|
|
| / |
| In particular equal numerical widths do not identify labels. -/ |
| def replacementMass : Fin 5 → ℝ := |
| ![p.rho 0, p.rho 0, p.rho 1, p.rho 1, p.tagMass] |
|
|
| def leftLength : Fin 5 → ℝ := |
| ![1-p.width 0/2, p.width 0/2, 1-p.width 1/2, p.width 1/2, 1/p.m] |
|
|
| def rightLength : Fin 5 → ℝ := |
| ![p.width 0/2, 1-p.width 0/2, p.width 1/2, 1-p.width 1/2, 1/p.m] |
|
|
| theorem replacementMass_nonneg (i : Fin 5) : 0 ≤ p.replacementMass i := by |
| fin_cases i <;> simp [replacementMass, rho_nonneg, tagMass_nonneg] |
|
|
| theorem mass_sum : p.diagonalMass + ∑ i, p.replacementMass i = 1 := by |
| simp [replacementMass, Fin.sum_univ_succ, diagonalMass] |
| ring |
|
|
| theorem leftLength_pos (i : Fin 5) : 0 < p.leftLength i := by |
| have h₀ := p.width_pos 0 |
| have h₁ := p.width_pos 1 |
| have hu₀ := p.width_upper 0 |
| have hu₁ := p.width_upper 1 |
| have hm := one_div_pos.mpr p.m_pos |
| fin_cases i <;> simp [leftLength] <;> linarith [p.m_pos] |
|
|
| theorem rightLength_pos (i : Fin 5) : 0 < p.rightLength i := by |
| have h₀ := p.width_pos 0 |
| have h₁ := p.width_pos 1 |
| have hu₀ := p.width_upper 0 |
| have hu₁ := p.width_upper 1 |
| have hm := one_div_pos.mpr p.m_pos |
| fin_cases i <;> simp [rightLength] <;> linarith [p.m_pos] |
|
|
| theorem lengths_sum_le (i : Fin 5) : p.leftLength i + p.rightLength i ≤ 1 := by |
| have hm : 1/p.m ≤ 1/100 := one_div_le_one_div_of_le (by norm_num) p.m_ge |
| simp only [one_div] at hm |
| fin_cases i <;> simp [leftLength, rightLength] |
| linarith |
|
|
| end CouplingParameters |
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
