Prob Event bind le prob Event convex
probEvent_bind_le_probEvent_convex
Plain-language statement
Convex prefix-event split for a bind. The off-prefix tail bound ε is charged only on the mass outside p, giving Pr[p] + (1 - Pr[p]) * ε.
Exact Lean statement
lemma probEvent_bind_le_probEvent_convex [MonadLiftT m SPMF] [LawfulMonadLiftT m SPMF]
[MonadLiftT m SetM] [EvalDistCompatible m]
{mx : m α} {my : α → m β} {q : β → Prop} {p : α → Prop} {ε : ENNReal}
(h : ∀ x ∈ support mx, ¬ p x → Pr[ q | my x] ≤ ε) :
Pr[ q | mx >>= my] ≤ Pr[ p | mx] + (1 - Pr[ p | mx]) * εFormal artifact
Lean source
lemma probEvent_bind_le_probEvent_convex [MonadLiftT m SPMF] [LawfulMonadLiftT m SPMF] [MonadLiftT m SetM] [EvalDistCompatible m] {mx : m α} {my : α → m β} {q : β → Prop} {p : α → Prop} {ε : ENNReal} (h : ∀ x ∈ support mx, ¬ p x → Pr[ q | my x] ≤ ε) : Pr[ q | mx >>= my] ≤ Pr[ p | mx] + (1 - Pr[ p | mx]) * ε := by classical have hsplit : Pr[ q | mx >>= my] ≤ Pr[ p | mx] + Pr[ fun x ↦ ¬ p x | mx] * ε := by rw [probEvent_bind_eq_tsum, probEvent_eq_tsum_indicator (p := p), probEvent_eq_tsum_indicator (p := fun x ↦ ¬ p x)] calc ∑' x, Pr[= x | mx] * Pr[ q | my x] ≤ ∑' x, ({x | p x}.indicator (Pr[= · | mx]) x + {x | ¬ p x}.indicator (fun x ↦ Pr[= x | mx] * ε) x) := by refine ENNReal.tsum_le_tsum fun x ↦ ?_ by_cases hp : p x · refine le_trans (mul_le_mul' le_rfl probEvent_le_one) ?_ simp [hp] · by_cases hx : x ∈ support mx · refine le_trans (mul_le_mul' le_rfl (h x hx hp)) ?_ simp [hp] · simp [probOutput_eq_zero_of_not_mem_support hx] _ = (∑' x, {x | p x}.indicator (Pr[= · | mx]) x) + ∑' x, {x | ¬ p x}.indicator (fun x ↦ Pr[= x | mx] * ε) x := ENNReal.tsum_add _ = (∑' x, {x | p x}.indicator (Pr[= · | mx]) x) + (∑' x, {x | ¬ p x}.indicator (Pr[= · | mx]) x) * ε := by rw [← ENNReal.tsum_mul_right] refine congrArg _ (tsum_congr fun x ↦ ?_) by_cases hp : p x <;> simp [Set.indicator, hp] have hle_one : Pr[ p | mx] + Pr[ fun x ↦ ¬ p x | mx] ≤ 1 := by rw [probEvent_eq_tsum_indicator (p := p), probEvent_eq_tsum_indicator (p := fun x ↦ ¬ p x), ← ENNReal.tsum_add] refine le_trans (ENNReal.tsum_le_tsum fun x ↦ ?_) (tsum_probOutput_le_one (mx := mx)) by_cases hp : p x <;> simp [Set.indicator, hp] refine le_trans hsplit (add_le_add le_rfl (mul_le_mul' ?_ le_rfl)) exact ENNReal.le_sub_of_add_le_left probEvent_ne_top hle_one- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/EvalDist/Monad/Basic.lean:345-379
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