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e4d9d2b | 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 | import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Tactic
namespace OAI
/-! # The quantitative entropy deficit of a separated branch
This is the scalar rearrangement in the branching argument of Rajala--Sturm,
arXiv:1207.6754, proof of Theorem 1.1, pp. 10--11. The three chord inequalities
retain the curvature error and the separation entropy explicitly.
-/
open Set Filter
open scoped Topology
namespace RegularPointCoordinates
theorem branch_entropy_deficit {t ε A M E U D H L : ℝ}
(ht0 : 0≤t) (ht1 : t≤1)
(hmean : H=(U+D)/2-L)
(hfirst : (t+ε)*E≤ε*M+t*H+A*t*ε*(t+ε))
(hup : (1-t)*U≤(1-t-ε)*E+ε*M+A*ε*(1-t-ε)*(1-t))
(hdown : (1-t)*D≤(1-t-ε)*E+ε*M+A*ε*(1-t-ε)*(1-t)) :
t*(1-t)*L≤ε*(M-E+A*t*(1-t)) := by
have hm : (1-t)*((U+D)/2)≤(1-t-ε)*E+ε*M+A*ε*(1-t-ε)*(1-t) := by
linarith only [hup,hdown]
have h₀ := mul_le_mul_of_nonneg_left hfirst (sub_nonneg.mpr ht1)
have h₁ := mul_le_mul_of_nonneg_left hm ht0
rw [hmean] at h₀
nlinarith only [h₀,h₁]
theorem branch_entropy_deficit_of_lower_bound {t ε A M E U D H L B : ℝ}
(ht0 : 0≤t) (ht1 : t≤1) (hε : 0≤ε) (hB : B≤E)
(hmean : H=(U+D)/2-L)
(hfirst : (t+ε)*E≤ε*M+t*H+A*t*ε*(t+ε))
(hup : (1-t)*U≤(1-t-ε)*E+ε*M+A*ε*(1-t-ε)*(1-t))
(hdown : (1-t)*D≤(1-t-ε)*E+ε*M+A*ε*(1-t-ε)*(1-t)) :
t*(1-t)*L≤ε*(M-B+A*t*(1-t)) := by
exact (branch_entropy_deficit ht0 ht1 hmean hfirst hup hdown).trans
(mul_le_mul_of_nonneg_left (by linarith) hε)
theorem tendsto_branch_entropy_budget (C B A : ℝ) :
Tendsto (fun ε : ℝ => ε*(C-Real.log ε-B+A)) (𝓝[>] 0) (𝓝 0) := by
have hlin : Tendsto (fun ε : ℝ => ε*(C-B+A)) (𝓝[>] 0) (𝓝 0) := by
have hid : Tendsto (fun ε : ℝ => ε) (𝓝[>] 0) (𝓝 0) :=
continuousAt_id.tendsto.mono_left nhdsWithin_le_nhds
simpa using hid.mul_const (C-B+A)
have hlog : Tendsto (fun ε : ℝ => ε*Real.log ε) (𝓝[>] 0) (𝓝 0) := by
simpa using (Real.continuous_mul_log.continuousAt (x:=0)).tendsto.mono_left
(nhdsWithin_le_nhds (s:=Ioi 0))
have h := hlin.sub hlog
simp only [sub_zero] at h
convert h using 1
funext ε
ring
theorem exists_small_branch_budget {c : ℝ} (hc : 0<c) (C B A : ℝ) :
∀ᶠ ε : ℝ in 𝓝[>] 0, ε*(C-Real.log ε-B+A)<c :=
(tendsto_branch_entropy_budget C B A).eventually (Iio_mem_nhds hc)
end RegularPointCoordinates
end OAI
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