Prob Event le tsum prob Output mul cost of mem support
probEvent_le_tsum_probOutput_mul_cost_of_mem_support
Plain-language statement
First-moment / Markov bound (support-restricted cost). Variant of probEvent_le_tsum_probOutput_mul_cost whose c ≥ 1 hypothesis need only hold on the support of mx.
Exact Lean statement
lemma probEvent_le_tsum_probOutput_mul_cost_of_mem_support
(mx : m α) (p : α → Prop) (c : α → ℝ≥0∞)
(hc : ∀ x ∈ support mx, p x → 1 ≤ c x) :
Pr[ p | mx] ≤ ∑' x : α, Pr[= x | mx] * c xFormal artifact
Lean source
lemma probEvent_le_tsum_probOutput_mul_cost_of_mem_support (mx : m α) (p : α → Prop) (c : α → ℝ≥0∞) (hc : ∀ x ∈ support mx, p x → 1 ≤ c x) : Pr[ p | mx] ≤ ∑' x : α, Pr[= x | mx] * c x := by have := Classical.decPred p rw [probEvent_eq_tsum_ite mx p] refine ENNReal.tsum_le_tsum fun x => ?_ by_cases hx : x ∈ support mx · split_ifs with hp · calc Pr[= x | mx] = Pr[= x | mx] * 1 := (mul_one _).symm _ ≤ Pr[= x | mx] * c x := by gcongr exact hc x hx hp · exact zero_le · rw [probOutput_eq_zero_of_not_mem_support hx] simp- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/EvalDist/Defs/Basic.lean:833-848
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