| 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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