| 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 |
|
|
