| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.CDFConditionalStability |
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|
| namespace OAI |
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| section |
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| noncomputable section |
| open scoped NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem scalarCDF_susceptibility_square_le_one (β : ℝ) (α : StieltjesFunction ℝ) |
| (ha : ∀ z,α z∈Icc (0:ℝ) 1) (h1 : α 1=1) {r u : ℝ} |
| (hr : 0≤r) (hru : r<u) (hu : u≤1) |
| (hΓ : ∀ s∈Icc r u,scalarCDFOverlap β α s=s) : |
| β^2*scalarCDFSusceptibilitySquareAverage β α r≤1 := by |
| let d (n : ℕ) : ℝ := (u-r)/((n:ℝ)+1) |
| have hd (n : ℕ) : 0<d n := div_pos (sub_pos.mpr hru) (by positivity) |
| have hdle (n : ℕ) : d n≤u-r := by |
| dsimp [d] |
| apply div_le_self (sub_nonneg.mpr hru.le) |
| linarith [Nat.cast_nonneg (α:=ℝ) n] |
| have H (n : ℕ) : β^2*scalarCDFSusceptibilitySquareAverage β α r≤1+2*(β^2*d n) := by |
| have hqin : r+d n∈Icc r u := ⟨by linarith [hd n],by linarith [hdle n]⟩ |
| have G := scalarCDF_conditional_stability β α ha h1 hr hqin.1 (hqin.2.trans hu) |
| rw [hΓ (r+d n) hqin,hΓ r ⟨le_rfl,hru.le⟩] at G |
| have G' : d n*(β^2*scalarCDFSusceptibilitySquareAverage β α r)≤d n*(1+2*(β^2*d n)) := by |
| nlinarith [G] |
| exact (mul_le_mul_iff_right₀ (hd n)).mp (by simpa only [mul_comm] using G') |
| have ht : Tendsto d atTop (𝓝 0) := by |
| simpa only [d,mul_zero,mul_one_div] using |
| (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜:=ℝ)).const_mul (u-r) |
| have Hlim : Tendsto (fun n => 1+2*(β^2*d n)) atTop (𝓝 1) := by |
| simpa only [mul_zero,add_zero] using ((ht.const_mul (β^2)).const_mul 2).const_add 1 |
| exact ge_of_tendsto Hlim (Eventually.of_forall H) |
|
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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