Le prob Output bind of forall support
SignatureAlg.le_probOutput_bind_of_forall_support
Plain-language statement
If every value x in the support of gen satisfies Pr[= a | f x] ≥ 1 - δ, then the overall probability satisfies Pr[= a | gen >>= f] ≥ 1 - δ. This reduces a "for all keys" completeness statement to per-key bounds.
Exact Lean statement
lemma le_probOutput_bind_of_forall_support {α β : Type} {a : β} {δ : ℝ≥0∞} (gen : ProbComp α)
(f : α → ProbComp β) (h : ∀ x, x ∈ support gen → 1 - δ ≤ Pr[= a | f x]) :
1 - δ ≤ Pr[= a | gen >>= f]Formal artifact
Lean source
lemma le_probOutput_bind_of_forall_support {α β : Type} {a : β} {δ : ℝ≥0∞} (gen : ProbComp α) (f : α → ProbComp β) (h : ∀ x, x ∈ support gen → 1 - δ ≤ Pr[= a | f x]) : 1 - δ ≤ Pr[= a | gen >>= f] := by rw [probOutput_bind_eq_tsum] calc 1 - δ = ∑' x, Pr[= x | gen] * (1 - δ) := by rw [ENNReal.tsum_mul_right, tsum_probOutput_of_liftM_PMF, one_mul] _ ≤ ∑' x, Pr[= x | gen] * Pr[= a | f x] := by refine ENNReal.tsum_le_tsum fun x => ?_ by_cases hx : x ∈ support gen · gcongr; exact h x hx · simp [probOutput_eq_zero_of_not_mem_support hx]- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/CryptoFoundations/SignatureAlg.lean:131-141
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