Plain-language statement
f.exp u = 0 in terms of forall, for nonnegative u
Exact Lean statement
lemma exp_eq_zero_iff {f : Prob α} {u : α → ℝ} (h : ∀ x, f.prob x ≠ 0 → 0 ≤ u x) :
f.exp u = 0 ↔ ∀ x, f.prob x ≠ 0 → u x = 0Formal artifact
Lean source
lemma exp_eq_zero_iff {f : Prob α} {u : α → ℝ} (h : ∀ x, f.prob x ≠ 0 → 0 ≤ u x) : f.exp u = 0 ↔ ∀ x, f.prob x ≠ 0 → u x = 0 := by constructor · intro h; contrapose h; simp only [not_forall] at h; apply ne_of_gt rcases h with ⟨x,px,ux⟩ refine lt_of_lt_of_le ?_ (le_exp_of_cut (fun y ↦ y = x) (f.prob x) (u x) ?_ ?_ ?_ ?_) · apply mul_pos; exact px.symm.lt_of_le (prob_nonneg _); exact (Ne.symm ux).lt_of_le (h _ px) · simp only [pr_eq_prob, le_refl] · intro y _ e; rw [e] · intro y py _; exact h _ py · exact h _ px · exact exp_eq_zero- Project
- debate
- License
- Apache-2.0
- Commit
- de3a6e500ae1
- Source
- Prob/Arith.lean:268-279
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