Plain-language statement
pr_mono when the left side is enriched
Exact Lean statement
lemma pr_enrich_le_pr {f : Prob α} {g : α → Prob β} {p : α × β → Prop} {i : α → Prop}
(pi : ∀ x y, f.prob x ≠ 0 → (g x).prob y ≠ 0 → p (x,y) → i x) :
(f >>= λ x ↦ Prod.mk x <$> g x).pr p ≤ f.pr iFormal artifact
Lean source
lemma pr_enrich_le_pr {f : Prob α} {g : α → Prob β} {p : α × β → Prop} {i : α → Prop} (pi : ∀ x y, f.prob x ≠ 0 → (g x).prob y ≠ 0 → p (x,y) → i x) : (f >>= λ x ↦ Prod.mk x <$> g x).pr p ≤ f.pr i := by simp only [pr_bind]; apply exp_mono; intro x fx; by_cases ix : i x · simp only [ix, if_true]; apply pr_le_one · apply le_of_eq; simp only [ix, if_false, pr_eq_zero, prob_map]; intro ⟨x',y⟩ pxy contrapose pxy; simp only [not_not] at pxy ⊢; rw [pr_eq_zero]; intro y' n contrapose ix; simp only [not_not] simp only [Prod.mk.injEq, not_and, not_forall, not_not, exists_prop] at ix simp only [←ix.1, ix.2] at n pxy; exact pi x y fx n pxy- Project
- debate
- License
- Apache-2.0
- Commit
- de3a6e500ae1
- Source
- Prob/Cond.lean:264-273
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