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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Parisi.CDFConditionalStability
namespace OAI
section
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)
end SK.Analytic
end
end
end OAI
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