| import OAI.Probability.RadialTransition.WindowLipschitz |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory Set Filter |
| namespace RadialTransition |
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| def AdmissibleFamily (_t A : ℝ) (b S : ℕ → ℝ) : Prop := |
| (∀ n, 0 ≤ b n) ∧ (∀ n, 1 ≤ S n) ∧ Summable b ∧ ∑' n, b n ≤ A |
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| 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 |
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| 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) |
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| 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] |
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| end RadialTransition |
| end |
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| end OAI |
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