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f205175 | 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 64 | import Mathlib
import OAI.Probability.CriticalPercolation.Entropy.Divergence
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
open scoped Classical
open Real
open Set Real
open scoped Topology
namespace CriticalPercolation.FiniteEntropy
lemma mul_log_div_scaled {x c p : ℝ} (hx : 0 ≤ x) (hc : 0 < c) (hp : 0 < p) :
c*((x/c)*Real.log ((x/c)/p)) = x*Real.log (x/p)-x*Real.log c := by
rcases eq_or_lt_of_le hx with rfl | hx
· simp
· rw [Real.log_div (div_pos hx hc).ne' hp.ne', Real.log_div hx.ne' hc.ne',
Real.log_div hx.ne' hp.ne']
field_simp
ring
lemma binaryDivergence_mass_split {l r p : ℝ} (hl : 0 ≤ l) (hr : 0 ≤ r)
(hp : p ∈ Set.Ioo 0 1) :
(l+r)*binaryDivergence (r/(l+r)) p +(l+r)*Real.log (l+r) =
l*Real.log l+r*Real.log r-l*Real.log (1-p)-r*Real.log p := by
by_cases hzero : l+r = 0
· have hl0 : l = 0 := by linarith
have hr0 : r = 0 := by linarith
simp [hl0, hr0]
· have hpos : 0 < l+r := lt_of_le_of_ne (add_nonneg hl hr) (Ne.symm hzero)
have hcomp : 1-r/(l+r) = l/(l+r) := by field_simp; ring
have h1 := mul_log_div_scaled hr hpos hp.1
have h2 := mul_log_div_scaled hl hpos (sub_pos.mpr hp.2)
unfold binaryDivergence
rw [hcomp, mul_add, h1, h2, mul_log_div_eq hp.1.ne',
mul_log_div_eq (sub_pos.mpr hp.2).ne']
ring
variable {X : Type*} [Fintype X]
lemma divergence_scale_reference (P R : X → ℝ) (hP : ∀ x, 0 ≤ P x)
(hs : ∀ x, 0 < P x → 0 < R x) {c : ℝ} (hc : 0 < c) :
divergence P (fun x => c*R x) = divergence P R - (∑ x, P x)*Real.log c := by
have ht (x : X) : P x*Real.log (P x/(c*R x)) =
P x*Real.log (P x/R x)-P x*Real.log c := by
rcases eq_or_lt_of_le (hP x) with hz | hp
· simp [← hz]
· have hr := hs x hp
rw [Real.log_div hp.ne' (mul_pos hc hr).ne', Real.log_mul hc.ne' hr.ne',
Real.log_div hp.ne' hr.ne']
ring
simp only [divergence, ht, Finset.sum_sub_distrib, ← Finset.sum_mul]
end CriticalPercolation.FiniteEntropy
end
end
end OAI
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