| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.TripleCDFTrial |
| import OAI.Probability.SKBarriers.Parisi.CommonMinimizer |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem continuousAt_small_interval {f : ℝ → ℝ} (hf : ContinuousAt f 0) |
| {b L : ℝ} (hb : f 0 < b) (hL : 0 < L) : |
| ∃ e ∈ Ioo (0:ℝ) L, ∀ x ∈ Icc (0:ℝ) e, f x < b := by |
| obtain ⟨d,hd,hf'⟩ := Metric.eventually_nhds_iff.mp (hf.eventually_lt continuousAt_const hb) |
| refine ⟨min L d/2,⟨by positivity,?_⟩,?_⟩ |
| · have H := min_le_left L d; linarith [lt_min hL hd] |
| · intro x hx |
| apply hf' |
| rw [Real.dist_eq,sub_zero,abs_of_nonneg hx.1] |
| have H := min_le_right L d |
| linarith [hx.2] |
|
|
| theorem cdfArea_antitone {α : ℝ → ℝ} (hm : Monotone α) (ha : ∀ z,0 ≤ α z) |
| {h q : ℝ} (hhq : h ≤ q) : (∫ z in q..1,α z) ≤ ∫ z in h..1,α z := by |
| have H := intervalIntegral.integral_add_adjacent_intervals (hm.intervalIntegrable (μ:=volume) (a:=h) (b:=q)) |
| (hm.intervalIntegrable (μ:=volume) (a:=q) (b:=1)) |
| have H' := intervalIntegral.integral_nonneg (μ:=volume) hhq (fun z _ => ha z) |
| linarith |
|
|
| theorem exists_negative_tripleCDF_coefficient {β : ℝ} (hβ : 0 < β) |
| {h : ℝ} (hh : 0 < h) (hh1 : h ≤ 1) : |
| ∃ μ : ProbabilityMeasure ℝ, (μ : Measure ℝ) (Icc (0:ℝ) 1)=1 ∧ |
| scalarCDFParisi β (cdf (μ : Measure ℝ))=finiteParisiInf β ∧ |
| ∃ s₁ s₂ τ r₀ c : ℝ, 0 < s₁ ∧ s₁ < s₂ ∧ s₂ < h ∧ 0 < τ ∧ |
| 0 < r₀ ∧ r₀ < s₁ ∧ 0 < c ∧ |
| ∀ r q : ℝ, r ∈ Icc (0:ℝ) r₀ → q ∈ Icc h 1 → |
| |q-scalarCDFOverlap β (cdf (μ : Measure ℝ)) q| ≤ τ → |
| tripleCDFTrialCoefficient β (cdf (μ : Measure ℝ)) (s₂-s₁)⁻¹ r s₁ s₂ q ≤ -c := by |
| obtain ⟨μ,hμ,hmin,h0,hχ0,hβA,hstrict,hχ⟩ := exists_scalarCDFParisi_minimizer_locking_data hβ.ne' |
| let α := cdf (μ : Measure ℝ) |
| let A₀ : ℝ := ∫ z in (0:ℝ)..1,α z |
| let Ah : ℝ := ∫ z in h..1,α z |
| have ha (z) : α z ∈ Icc (0:ℝ) 1 := ⟨cdf_nonneg _ _,cdf_le_one _ _⟩ |
| have hAh : 0 ≤ Ah := intervalIntegral.integral_nonneg hh1 (fun z _ => (ha z).1) |
| have hAA : Ah < A₀ := hstrict h hh |
| have hA₀ : 0 < A₀ := hAh.trans_lt hAA |
| have HB : β*A₀ ≤ 1 := by simpa only [abs_of_pos hβ] using hβA |
| have Hprod : β^2*Ah*A₀ < 1 := by |
| have H := mul_lt_mul_of_pos_left hAA (mul_pos (sq_pos_of_pos hβ) hA₀) |
| have H' : (β*A₀)^2 ≤ 1 := pow_le_one₀ (mul_pos hβ hA₀).le HB |
| nlinarith only [H,H'] |
| let g : ℝ → ℝ := fun s => β^2*Ah*(A₀+2*(β^2*s)+4*(β^2*s)^2)-1 |
| have hg : ContinuousAt g 0 := by dsimp [g]; fun_prop |
| have hg0 : g 0 < 0 := by dsimp [g]; nlinarith only [Hprod] |
| obtain ⟨s₂,hs₂,hgs⟩ := continuousAt_small_interval hg hg0 hh |
| have hgs₂ : g s₂ < 0 := hgs s₂ ⟨hs₂.1.le,le_rfl⟩ |
| let s₁ := s₂/2 |
| have hs₁ : 0 < s₁ := half_pos hs₂.1 |
| have hs : s₁ < s₂ := by dsimp [s₁]; linarith [hs₂.1] |
| let K := (s₂-s₁)⁻¹ |
| have hK : 0 ≤ K := inv_nonneg.mpr (sub_pos.mpr hs).le |
| have hm : 0 < α s₁ := cdf_pos_of_zero_mem_support μ h0 hs₁ |
| let C := A₀+2*(β^2*s₂)+4*(β^2*s₂)^2 |
| have hC : 0 ≤ C := by exact add_nonneg (add_nonneg hA₀.le (mul_nonneg (by norm_num) (mul_nonneg (sq_nonneg β) hs₂.1.le))) (mul_nonneg (by norm_num) (sq_nonneg _)) |
| let R : ℝ → ℝ := fun t => β^2*(Ah+t)*C-1 |
| let F : ℝ → ℝ := fun t => β^2*K*t+β^2*α s₁*R t |
| have hR0 : R 0 < 0 := by simpa only [R,C,g,add_zero] using hgs₂ |
| have hF0 : F 0 < 0 := by |
| dsimp [F] |
| simpa only [mul_zero,zero_add] using mul_neg_of_pos_of_neg (mul_pos (sq_pos_of_pos hβ) hm) hR0 |
| let c₀ := -F 0/2 |
| have hc₀ : 0 < c₀ := by dsimp [c₀]; linarith |
| have hFc : F 0 < -c₀ := by dsimp [c₀]; linarith |
| have hF : ContinuousAt F 0 := by dsimp [F,R]; fun_prop |
| obtain ⟨τ,hτ,hFτ⟩ := continuousAt_small_interval hF hFc zero_lt_one |
| have HT : F τ < -c₀ := hFτ τ ⟨hτ.1.le,le_rfl⟩ |
| have hRτ : R τ < 0 := by |
| have HP : 0 ≤ β^2*K*τ := mul_nonneg (mul_nonneg (sq_nonneg β) hK) hτ.1.le |
| have HB : 0 < β^2*α s₁ := mul_pos (sq_pos_of_pos hβ) hm |
| dsimp only [F] at HT |
| nlinarith |
| have hE : ContinuousAt (tripleTimeCommonError β K) 0 := by |
| unfold tripleTimeCommonError |
| fun_prop |
| have hE0 : tripleTimeCommonError β K 0 < c₀/2 := by |
| simp only [tripleTimeCommonError,mul_zero,zero_pow (by decide : 2≠0),add_zero,Real.sqrt_zero] |
| linarith |
| obtain ⟨r₀,hr₀,hEr⟩ := continuousAt_small_interval hE hE0 hs₁ |
| refine ⟨μ,hμ,hmin,s₁,s₂,τ,r₀,c₀/2,hs₁,hs,hs₂.2,hτ.1,hr₀.1,hr₀.2, |
| half_pos hc₀,?_⟩ |
| intro r q hr hq he |
| have hq' : q ∈ Icc (0:ℝ) 1 := ⟨hh.le.trans hq.1,hq.2⟩ |
| have Hdiff : α q*(q-scalarCDFOverlap β α q) ≤ τ := by |
| calc |
| _ ≤ α q*τ := mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans he) (ha q).1 |
| _ ≤ τ := mul_le_of_le_one_left hτ.1.le (ha q).2 |
| have Hmean : scalarCDFSusceptibilityAverage β α q ≤ Ah+τ := by |
| have H := hχ q hq' |
| have HA := cdfArea_antitone α.mono (fun z => (ha z).1) hq.1 |
| change scalarCDFSusceptibilityAverage β α q=(∫ z in q..1,α z)+α q*(q-scalarCDFOverlap β α q) at H |
| dsimp only [Ah] at ⊢ |
| linarith |
| have Hresponse : tripleCDFResponse β α s₂ q ≤ R τ := by |
| unfold tripleCDFResponse |
| change β^2*scalarCDFSusceptibilityAverage β α q* |
| (scalarCDFHessian β α 0 1 0+2*(β^2*s₂)+4*(β^2*s₂)^2)-1 ≤ R τ |
| rw [hχ0] |
| exact sub_le_sub_right (mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_left Hmean (sq_nonneg β)) hC) 1 |
| have Hcoefficient : tripleCDFTrialCoefficient β α K r s₁ s₂ q ≤ F τ+tripleTimeCommonError β K r := by |
| unfold tripleCDFTrialCoefficient |
| rw [max_eq_right (Hresponse.trans hRτ.le),add_zero] |
| have HA := hχ q hq' |
| change scalarCDFSusceptibilityAverage β α q=(∫ z in q..1,α z)+α q*(q-scalarCDFOverlap β α q) at HA |
| have H1 := mul_le_mul_of_nonneg_left Hdiff (mul_nonneg (sq_nonneg β) hK) |
| have H2 := mul_le_mul_of_nonneg_left Hresponse (mul_pos (sq_pos_of_pos hβ) hm).le |
| dsimp only [F] |
| rw [HA,add_sub_cancel_left] |
| nlinarith only [H1,H2] |
| have Her := hEr r hr |
| exact Hcoefficient.trans (le_of_lt (by linarith)) |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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