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a2ee19a | 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 | import OAI.Probability.RadialTransition.WindowLipschitz
namespace OAI
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
def AdmissibleFamily (_t A : ℝ) (b S : ℕ → ℝ) : Prop :=
(∀ n, 0 ≤ b n) ∧ (∀ n, 1 ≤ S n) ∧ Summable b ∧ ∑' n, b n ≤ A
lemma admissible_row {t A : ℝ} {b S : ℕ → ℝ}
(ht : 0 ≤ t) (had : AdmissibleFamily t A b S)
(hrow : 2197*t*bumpIntegral⁻¹*A ≤ (t^2/8)/8) :
2197*t*bumpIntegral⁻¹*(∑' n, b n) ≤ (t^2/8)/8 :=
(mul_le_mul_of_nonneg_left had.2.2.2 (by have := bumpIntegral_pos; positivity)).trans hrow
lemma admissible_pressure_convex {t A : ℝ} {b S : ℕ → ℝ}
(ht : 1 ≤ t) (had : AdmissibleFamily t A b S)
(hrow : 2197*t*bumpIntegral⁻¹*A ≤ (t^2/8)/8) :
ConvexOn ℝ positiveParameters (onParameters (grandPressure (familyPotential t b S) t)) := by
have ht0 : 0 < t := by linarith
have hr := admissible_row ht0.le had hrow
exact convexOn_grandPressure (continuous_familyPotential had.1 had.2.1 had.2.2.1 t)
(bounded_familyPotential had.1 had.2.1 had.2.2.1 t)
(stable_family_of_master ht0.le had.1 had.2.1 had.2.2.1 had.2.2.1 (fun _ => le_rfl) hr) ht0
(fun r h => familyPotential_nonpos_outer had.1 had.2.1 ht h)
lemma family_controls {t A : ℝ} (ht : 1 ≤ t) (hA : 0 ≤ A) (htA : 6912*A ≤ t)
(hrow : 2197*t*bumpIntegral⁻¹*A ≤ (t^2/8)/8) :
∃ U C C₀ : ℝ, 1 ≤ U ∧ 1 ≤ C ∧ 0 < C₀ ∧
∀ b S : ℕ → ℝ, AdmissibleFamily t A b S → ∀ a ∈ Ioc (0:ℝ) A,
(∀ β h, 0 < β → ∃ u, ActiveShift (grandPressure (familyPotential t b S) t) a β h u) ∧
(∀ x ∈ phaseWindow (1/2), ∃ u ∈ Icc (0:ℝ) U,
ActiveShift (grandPressure (familyPotential t b S) t) a (x 0) (x 1) u) ∧
(∀ x ∈ phaseWindow (1/4), ∀ y ∈ phaseWindow (1/4),
|onParameters (idealAttraction (grandPressure (familyPotential t b S) t) a) x-
onParameters (idealAttraction (grandPressure (familyPotential t b S) t) a) y| ≤ C*dist x y) ∧
(∀ x ∈ phaseWindow (1/4),
0 ≤ onParameters (idealAttraction (grandPressure (familyPotential t b S) t) a) x-
onParameters (grandPressure (familyPotential t b S) t) x ∧
onParameters (idealAttraction (grandPressure (familyPotential t b S) t) a) x-
onParameters (grandPressure (familyPotential t b S) t) x ≤ C₀*a) := by
have ht0 : 0 < t := by linarith
obtain ⟨U,hU,hcut⟩ := uniform_ideal_cutoff (by norm_num : (0:ℝ)<1/4)
(by norm_num : (1/4:ℝ)≤2) (by norm_num : (0:ℝ)≤4) hA
let M := Real.exp (t^2+t*(4+2*A*U))/t
have hM : 0 < M := by dsimp [M]; positivity
refine ⟨U,1+8*M,1+U^2,hU,by linarith,by positivity,?_⟩
intro b S had a ha
have hr := admissible_row ht0.le had hrow
have hta : 6912*a ≤ t := (mul_le_mul_of_nonneg_left ha.2 (by norm_num)).trans htA
have hQ := admissible_pressure_convex ht had hrow
have hact := family_ideal_global_active ht ha.1.le hta had.1 had.2.1 had.2.2.1 hr
have hP := convexOn_idealAttraction hQ hact
have hcut' (x : Parameter) (hx : x ∈ phaseWindow (1/2)) :
∃ u ∈ Icc (0:ℝ) U, ActiveShift (grandPressure (familyPotential t b S) t) a (x 0) (x 1) u := by
obtain ⟨hβ,hh⟩ := phaseWindow_half_bounds hx
obtain ⟨κ,hκ,hc,hbound⟩ := hcut t ht htA (x 0) ⟨by linarith [hβ.1],by linarith [hβ.2]⟩
(x 1) (by have := abs_le.mp hh; constructor <;> linarith) a ⟨ha.1.le,ha.2⟩
exact family_ideal_exists_active ht (by linarith [hβ.1]) ha.1.le had.1 had.2.1 had.2.2.1 hr hκ hc hU hbound
have hbound (x : Parameter) (hx : x ∈ phaseWindow (1/2)) :
|onParameters (idealAttraction (grandPressure (familyPotential t b S) t) a) x| ≤ M := by
obtain ⟨hβ,hh⟩ := phaseWindow_half_bounds hx
have hβ0 : 0 < x 0 := by linarith [hβ.1]
obtain ⟨u,hub,hu⟩ := hcut' x hx
have hs := stable_family_of_master ht0.le had.1 had.2.1 had.2.2.1 had.2.2.1 (fun _ => le_rfl) hr
have hφ := continuous_familyPotential had.1 had.2.1 had.2.2.1 t
have hbd := bounded_familyPotential had.1 had.2.1 had.2.2.1 t
have hn := (grandPressure_spec hφ hbd hs ht0 hβ0.le
(fun r h => familyPotential_nonpos_outer had.1 had.2.1 ht h) (x 1)).2.1
have hP0 := ideal_nonneg_of_active hu hn
change |idealAttraction (grandPressure (familyPotential t b S) t) a (x 0) (x 1)| ≤ M
rw [abs_of_nonneg hP0,idealAttraction_eq_of_active hu]
have hp := (grandPressure_spec hφ hbd hs ht0 hβ0.le
(fun r h => familyPotential_nonpos_outer had.1 had.2.1 ht h) (x 1+x 0*a*u)).2.2
have hmul : x 0*a*u ≤ 2*A*U :=
mul_le_mul (mul_le_mul (show x 0≤2 by linarith [hβ.2]) ha.2 ha.1.le (by norm_num))
hub.2 hub.1 (by positivity)
have hexp : x 0*(t^2/2)+t*(x 1+x 0*a*u) ≤ t^2+t*(4+2*A*U) := by
have hh' := (abs_le.mp hh).2
have hbmul := mul_le_mul_of_nonneg_right (show x 0≤2 by linarith [hβ.2]) (sq_nonneg t)
have hhmul := mul_le_mul_of_nonneg_left hh' ht0.le
nlinarith [mul_le_mul_of_nonneg_left hmul ht0.le]
have hd := div_le_div_of_nonneg_right (Real.exp_le_exp.mpr hexp) ht0.le
dsimp [attractionBranch,M]
have ha0 := ha.1.le
have hsub : 0 ≤ x 0*a/2*u^2 := by positivity
linarith
refine ⟨hact,hcut',?_,?_⟩
· intro x hx y hy
exact (window_lipschitz hM.le hP hbound x hx y hy).trans
(mul_le_mul_of_nonneg_right (by linarith : 8*M≤1+8*M) dist_nonneg)
· intro x hx
obtain ⟨u,hub,hu⟩ := hcut' x (phaseWindow_mono (by norm_num) hx)
have hβ := (phaseWindow_quarter_bounds hx).1
have hh := ideal_increment_bound hQ ha.1 (by linarith [hβ.1]) hu hub
refine ⟨hh.1,hh.2.trans ?_⟩
have hm := mul_le_mul_of_nonneg_right (show x 0≤2 by linarith [hβ.2]) (mul_nonneg ha.1.le (sq_nonneg U))
nlinarith [ha.1]
end RadialTransition
end
end OAI
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