Plain-language statement
pr_mono when the right side is enriched
Exact Lean statement
lemma pr_le_pr_enrich {f : Prob α} {g : α → Prob β} {p : α × β → Prop} {i : α → Prop}
(ip : ∀ x y, f.prob x ≠ 0 → (g x).prob y ≠ 0 → i x → p (x,y)) :
f.pr i ≤ (f >>= λ x ↦ Prod.mk x <$> g x).pr pFormal artifact
Lean source
lemma pr_le_pr_enrich {f : Prob α} {g : α → Prob β} {p : α × β → Prop} {i : α → Prop} (ip : ∀ x y, f.prob x ≠ 0 → (g x).prob y ≠ 0 → i x → p (x,y)) : f.pr i ≤ (f >>= λ x ↦ Prod.mk x <$> g x).pr p := by simp only [pr_bind]; apply exp_mono; intro x fx; by_cases ix : i x · apply le_of_eq; simp only [ix, if_true]; refine (pr_eq_one.mpr ?_).symm; intro ⟨x',y⟩ pxy contrapose pxy; simp only [not_not, prob_map]; rw [pr_eq_zero]; intro y' n contrapose pxy; simp only [Prod.mk.injEq, not_and, not_forall, not_not, exists_prop] at pxy ⊢ simp only [←pxy.1, pxy.2] at n ⊢; apply ip x y fx n ix · simp only [ix, if_false]; exact pr_nonneg- Project
- debate
- License
- Apache-2.0
- Commit
- de3a6e500ae1
- Source
- Prob/Cond.lean:276-284
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