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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.PushMass
namespace OAI
section
section
namespace CriticalPercolation.FiniteEntropy
open scoped Classical
variable {X Y : Type*} [Fintype X] [Fintype Y]
omit [Fintype Y] in
lemma supported_event_entropy (P R : X → ℝ) (hP : ∀ x, 0 ≤ P x)
(hP1 : ∑ x, P x = 1) (hR : ∀ x, 0 < R x) (A : Set X)
(hs : ∀ x, P x ≠ 0 → x ∈ A) :
0 < (∑ x, if x ∈ A then R x else 0) ∧
Real.log (1/(∑ x, if x ∈ A then R x else 0)) ≤ divergence P R := by
let R' : X → ℝ := fun x => if x ∈ A then R x else 0
have hn : ∀ x, 0 ≤ R' x := by
intro x
unfold R'
split_ifs
· exact (hR x).le
· exact le_rfl
have hp : ∀ x, 0 < P x → 0 < R' x := by
intro x hx
simp only [R',ite_eq_left (hs x hx.ne')]
exact hR x
have ho : 0 < ∑ x, R' x := by
have hh := normalizer_pos P R' hP hn hP1 hp (fun _ => 0)
simpa only [Real.exp_zero,mul_one] using hh
refine ⟨ho,?_⟩
have hh := log_sum_inequality P R' hP hn hp
rw [hP1,one_mul] at hh
refine hh.trans_eq ?_
unfold divergence
apply Finset.sum_congr rfl
intro x _
by_cases hx : P x=0
· simp only [hx,zero_mul]
· simp only [R',ite_eq_left (hs x hx)]
omit [Fintype Y] in
lemma pushMass_supported (P : X → ℝ) (f : X → Y) (A : Set Y)
(hs : ∀ x, P x ≠ 0 → f x ∈ A) (y : Y) (hy : pushMass f P y ≠ 0) : y ∈ A := by
by_contra hn
apply hy
unfold pushMass
apply Finset.sum_eq_zero
intro x _
by_cases hx : f x=y
· rw [ite_eq_left hx]
by_contra hp
exact hn (hx ▸ hs x hp)
· exact ite_eq_right hx
end CriticalPercolation.FiniteEntropy
end
end
end OAI
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