| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Set MeasureTheory ProbabilityTheory |
| open Set |
| open scoped BigOperators |
| open scoped Classical |
| open Real |
|
|
| namespace CriticalPercolation.FiniteEntropy |
| variable {X : Type*} [Fintype X] |
|
|
| noncomputable def divergence (P R : X → ℝ) : ℝ := |
| ∑ x, P x * Real.log (P x / R x) |
|
|
| lemma mul_log_div_ge_sub {p r : ℝ} (hp : 0 ≤ p) (hr : 0 ≤ r) |
| (hs : 0 < p → 0 < r) : p - r ≤ p * Real.log (p / r) := by |
| rcases eq_or_lt_of_le hp with rfl | hp |
| · simp only [zero_sub, zero_mul] |
| exact neg_nonpos.mpr hr |
| · have hr := hs hp |
| have h := mul_le_mul_of_nonneg_left |
| (Real.log_le_sub_one_of_pos (div_pos hr hp)) hp.le |
| rw [Real.log_div hr.ne' hp.ne'] at h |
| rw [Real.log_div hp.ne' hr.ne'] |
| have he : p * (r / p - 1) = r - p := by field_simp |
| rw [he] at h |
| nlinarith |
|
|
| lemma mass_sub_le_divergence (P R : X → ℝ) (hP : ∀ x, 0 ≤ P x) |
| (hR : ∀ x, 0 ≤ R x) (hs : ∀ x, 0 < P x → 0 < R x) : |
| (∑ x, P x) - (∑ x, R x) ≤ divergence P R := by |
| rw [← Finset.sum_sub_distrib] |
| exact Finset.sum_le_sum fun x _ => mul_log_div_ge_sub (hP x) (hR x) (hs x) |
|
|
| lemma divergence_nonneg (P R : X → ℝ) (hP : ∀ x, 0 ≤ P x) |
| (hR : ∀ x, 0 ≤ R x) (hPsum : ∑ x, P x = 1) (hRsum : ∑ x, R x = 1) |
| (hs : ∀ x, 0 < P x → 0 < R x) : 0 ≤ divergence P R := by |
| simpa only [hPsum, hRsum, sub_self] using mass_sub_le_divergence P R hP hR hs |
|
|
| lemma normalizer_pos (P R : X → ℝ) (_hP : ∀ x, 0 ≤ P x) |
| (hR : ∀ x, 0 ≤ R x) (hPsum : ∑ x, P x = 1) |
| (hs : ∀ x, 0 < P x → 0 < R x) (g : X → ℝ) : |
| 0 < ∑ x, R x * Real.exp (g x) := by |
| obtain ⟨x, hx⟩ : ∃ x, 0 < P x := by |
| by_contra! h |
| have hh := Finset.sum_nonpos (s := Finset.univ) (fun x _ => h x) |
| linarith |
| exact (mul_pos (hs x hx) (Real.exp_pos _)).trans_le |
| (Finset.single_le_sum (fun y _ => mul_nonneg (hR y) (Real.exp_nonneg _)) (Finset.mem_univ x)) |
|
|
| theorem entropy_inequality (P R : X → ℝ) (hP : ∀ x, 0 ≤ P x) |
| (hR : ∀ x, 0 ≤ R x) (hPsum : ∑ x, P x = 1) |
| (hs : ∀ x, 0 < P x → 0 < R x) (g : X → ℝ) : |
| ∑ x, P x * g x ≤ divergence P R + Real.log (∑ x, R x * Real.exp (g x)) := by |
| let Z := ∑ x, R x * Real.exp (g x) |
| have hZ : 0 < Z := normalizer_pos P R hP hR hPsum hs g |
| let R' := fun x => R x * Real.exp (g x) / Z |
| have hR' : ∀ x, 0 ≤ R' x := fun x => div_nonneg |
| (mul_nonneg (hR x) (Real.exp_nonneg _)) hZ.le |
| have hs' : ∀ x, 0 < P x → 0 < R' x := fun x hx => div_pos |
| (mul_pos (hs x hx) (Real.exp_pos _)) hZ |
| have hRsum : ∑ x, R' x = 1 := by |
| simp only [R', ← Finset.sum_div, Z, div_self hZ.ne'] |
| have hdiv := divergence_nonneg P R' hP hR' hPsum hRsum hs' |
| have he : divergence P R' = divergence P R - (∑ x, P x * g x) + Real.log Z := by |
| have hterm (x : X) : P x * Real.log (P x / R' x) = |
| P x * Real.log (P x / R x) - P x * g x + P x * Real.log Z := by |
| rcases eq_or_lt_of_le (hP x) with hx | hx |
| · simp [← hx] |
| · have hr := hs x hx |
| simp only [R', Real.log_div hx.ne' (ne_of_gt (hs' x hx)), |
| Real.log_div (mul_pos hr (Real.exp_pos _)).ne' hZ.ne', |
| Real.log_mul hr.ne' (Real.exp_ne_zero _), Real.log_exp, |
| Real.log_div hx.ne' hr.ne'] |
| ring |
| simp only [divergence, hterm, Finset.sum_add_distrib, Finset.sum_sub_distrib, |
| ← Finset.sum_mul, hPsum, one_mul] |
| rw [he] at hdiv |
| linarith |
| |
| end CriticalPercolation.FiniteEntropy |
| |
| open Set Real |
| open scoped Topology |
| |
| namespace CriticalPercolation.FiniteEntropy |
| |
| noncomputable def binaryDivergence (r p : ℝ) : ℝ := |
| r * Real.log (r / p) + (1-r) * Real.log ((1-r)/(1-p)) |
| |
| lemma mul_log_div_le {r p : ℝ} (hr : 0 ≤ r) (hp : 0 < p) : |
| r * Real.log (r/p) ≤ r * (r/p - 1) := by |
| rcases eq_or_lt_of_le hr with rfl | hr |
| · simp |
| · exact mul_le_mul_of_nonneg_left (Real.log_le_sub_one_of_pos (div_pos hr hp)) hr.le |
| |
| theorem binaryDivergence_upper {p r : ℝ} (hp : p ∈ Ioo 0 1) (hr : r ∈ Icc 0 1) : |
| binaryDivergence r p ≤ (r-p)^2 / (p*(1-p)) := by |
| have hq : 0 < 1-p := sub_pos.mpr hp.2 |
| have h1 := mul_log_div_le hr.1 hp.1 |
| have h2 := mul_log_div_le (sub_nonneg.mpr hr.2) hq |
| have he : r * (r/p-1) + (1-r)*((1-r)/(1-p)-1) = (r-p)^2/(p*(1-p)) := by |
| field_simp [hp.1.ne', hq.ne'] |
| ring |
| exact (add_le_add h1 h2).trans_eq he |
| |
| lemma mul_log_div_eq {r p : ℝ} (hp : p ≠ 0) : |
| r * Real.log (r / p) = r * Real.log r - r * Real.log p := by |
| by_cases hr : r = 0 |
| · simp [hr] |
| · rw [Real.log_div hr hp] |
| ring |
| |
| noncomputable def binaryRemainder (p r : ℝ) : ℝ := |
| r * Real.log r + (1-r) * Real.log (1-r) - |
| r * Real.log p - (1-r)*Real.log (1-p) - 2*(r-p)^2 |
| |
| lemma continuous_binaryRemainder (p : ℝ) : Continuous (binaryRemainder p) := by |
| unfold binaryRemainder |
| fun_prop |
| |
| lemma binaryRemainder_eq {p : ℝ} (hp : p ∈ Ioo 0 1) (r : ℝ) : |
| binaryRemainder p r = binaryDivergence r p - 2*(r-p)^2 := by |
| unfold binaryRemainder binaryDivergence |
| rw [mul_log_div_eq hp.1.ne', mul_log_div_eq (sub_pos.mpr hp.2).ne'] |
| ring |
| |
| lemma hasDerivAt_binaryRemainder (p : ℝ) {r : ℝ} (hr : r ∈ Ioo 0 1) : |
| HasDerivAt (binaryRemainder p) |
| (Real.log r - Real.log (1-r) - Real.log p + Real.log (1-p) - 4*(r-p)) r := by |
| have hid := hasDerivAt_id r |
| have ho := (hasDerivAt_const r (1:ℝ)).sub hid |
| have hmul := (Real.hasDerivAt_mul_log hr.1.ne').add |
| ((Real.hasDerivAt_mul_log (sub_pos.mpr hr.2).ne').comp r ho) |
| convert ((hmul.sub (hid.mul_const (Real.log p))).sub |
| (ho.mul_const (Real.log (1-p)))).sub |
| (((hid.sub_const p).pow 2).const_mul 2) using 1 <;> |
| first | rfl | (simp only [id_eq]; ring) |
| |
| lemma hasDerivAt_binaryRemainder_deriv (p : ℝ) {r : ℝ} (hr : r ∈ Ioo 0 1) : |
| HasDerivAt |
| (fun r => Real.log r - Real.log (1-r) - Real.log p + Real.log (1-p) - 4*(r-p)) |
| (r⁻¹ + (1-r)⁻¹ - 4) r := by |
| have hid := hasDerivAt_id r |
| have ho := (hasDerivAt_const r (1:ℝ)).sub hid |
| convert (((((Real.hasDerivAt_log hr.1.ne').sub |
| ((Real.hasDerivAt_log (sub_pos.mpr hr.2).ne').comp r ho)).sub_const |
| (Real.log p)).add_const (Real.log (1-p))).sub |
| ((hid.sub_const p).const_mul 4)) using 1 <;> first | rfl | ring |
| |
| lemma binary_second_deriv_nonneg {r : ℝ} (hr : r ∈ Ioo 0 1) : |
| 0 ≤ r⁻¹ + (1-r)⁻¹ - 4 := by |
| have hq : 0 < 1-r := sub_pos.mpr hr.2 |
| have he : r⁻¹ + (1-r)⁻¹ - 4 = (2*r-1)^2 / (r*(1-r)) := by |
| field_simp [hr.1.ne', hq.ne'] |
| ring |
| rw [he] |
| exact div_nonneg (sq_nonneg _) (mul_pos hr.1 hq).le |
| |
| lemma convexOn_binaryRemainder (p : ℝ) : ConvexOn ℝ (Icc 0 1) (binaryRemainder p) := by |
| apply convexOn_of_hasDerivWithinAt2_nonneg (convex_Icc (0:ℝ) 1) |
| (continuous_binaryRemainder p).continuousOn |
| (f' := fun r => Real.log r - Real.log (1-r) - Real.log p + Real.log (1-p) - 4*(r-p)) |
| (f'' := fun r => r⁻¹ + (1-r)⁻¹ - 4) |
| · intro r hr |
| rw [interior_Icc] at hr |
| exact (hasDerivAt_binaryRemainder p hr).hasDerivWithinAt |
| · intro r hr |
| rw [interior_Icc] at hr |
| exact (hasDerivAt_binaryRemainder_deriv p hr).hasDerivWithinAt |
| · intro r hr |
| rw [interior_Icc] at hr |
| exact binary_second_deriv_nonneg hr |
|
|
| theorem binaryDivergence_lower {p r : ℝ} (hp : p ∈ Ioo 0 1) (hr : r ∈ Icc 0 1) : |
| 2*(r-p)^2 ≤ binaryDivergence r p := by |
| have hconv := convexOn_binaryRemainder p |
| have hder : HasDerivAt (binaryRemainder p) 0 p := by |
| simpa using hasDerivAt_binaryRemainder p hp |
| have hmin : binaryRemainder p p ≤ binaryRemainder p r := by |
| rcases lt_trichotomy p r with h | rfl | h |
| · have hs := hconv.le_slope_of_hasDerivAt ⟨hp.1.le, hp.2.le⟩ hr h hder |
| rw [slope_def_field] at hs |
| have ht := (le_div_iff₀ (sub_pos.mpr h)).mp hs |
| simpa using ht |
| · rfl |
| · have hs := hconv.slope_le_of_hasDerivAt hr ⟨hp.1.le, hp.2.le⟩ h hder |
| rw [slope_def_field] at hs |
| have ht := (div_le_iff₀ (sub_pos.mpr h)).mp hs |
| linarith |
| rw [binaryRemainder_eq hp, binaryRemainder_eq hp] at hmin |
| have hz : binaryDivergence p p = 0 := by |
| simp [binaryDivergence, hp.1.ne', (sub_pos.mpr hp.2).ne'] |
| rw [hz] at hmin |
| nlinarith |
|
|
| end CriticalPercolation.FiniteEntropy |
|
|
| end |
| end |
|
|
| end OAI |
|
|
