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ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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
/-- One block of the actual coupling, with two distinct width labels. The
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]
/-- 0/1 are top/bottom of width 0, 2/3 of width 1, 4 is the tag.
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
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