Advantage le expected Query Slack plus prob Event bad
QueryImpl.Stateful.advantage_le_expectedQuerySlack_plus_probEvent_bad
Plain-language statement
State-dependent ε-perturbed identical-until-bad with output bad flag.
Exact Lean statement
theorem advantage_le_expectedQuerySlack_plus_probEvent_bad
(h₀ h₁ : QueryImpl.Stateful unifSpec E (σ × Bool))
(s_init : σ)
(chargedQuery : E.Domain → Prop) [DecidablePred chargedQuery]
(querySlack : σ → ℝ≥0∞)
(h_step_tv_charged : ∀ (t : E.Domain), chargedQuery t → ∀ (s : σ),
ENNReal.ofReal (tvDist ((h₀ t).run (s, false)) ((h₁ t).run (s, false))) ≤
querySlack s)
(h_step_eq_uncharged : ∀ (t : E.Domain), ¬ chargedQuery t → ∀ (p : σ × Bool),
(h₀ t).run p = (h₁ t).run p)
(h_mono₀ : ∀ (t : E.Domain) (p : σ × Bool), p.2 = true →
∀ z ∈ support ((h₀ t).run p), z.2.2 = true)
(A : OracleComp E Bool) {queryBudget : ℕ}
(h_bound : OracleComp.IsQueryBoundP A chargedQuery queryBudget) :
ENNReal.ofReal (h₀.advantage (s_init, false) h₁ (s_init, false) A)
≤ expectedQuerySlack h₀ chargedQuery querySlack A queryBudget (s_init, false)
+ Pr[fun z : Bool × σ × Bool => z.2.2 = true |
(simulateQ h₀ A).run (s_init, false)]Formal artifact
Lean source
theorem advantage_le_expectedQuerySlack_plus_probEvent_bad (h₀ h₁ : QueryImpl.Stateful unifSpec E (σ × Bool)) (s_init : σ) (chargedQuery : E.Domain → Prop) [DecidablePred chargedQuery] (querySlack : σ → ℝ≥0∞) (h_step_tv_charged : ∀ (t : E.Domain), chargedQuery t → ∀ (s : σ), ENNReal.ofReal (tvDist ((h₀ t).run (s, false)) ((h₁ t).run (s, false))) ≤ querySlack s) (h_step_eq_uncharged : ∀ (t : E.Domain), ¬ chargedQuery t → ∀ (p : σ × Bool), (h₀ t).run p = (h₁ t).run p) (h_mono₀ : ∀ (t : E.Domain) (p : σ × Bool), p.2 = true → ∀ z ∈ support ((h₀ t).run p), z.2.2 = true) (A : OracleComp E Bool) {queryBudget : ℕ} (h_bound : OracleComp.IsQueryBoundP A chargedQuery queryBudget) : ENNReal.ofReal (h₀.advantage (s_init, false) h₁ (s_init, false) A) ≤ expectedQuerySlack h₀ chargedQuery querySlack A queryBudget (s_init, false) + Pr[fun z : Bool × σ × Bool => z.2.2 = true | (simulateQ h₀ A).run (s_init, false)] := by set sim₀ : ProbComp Bool := (simulateQ h₀ A).run' (s_init, false) with hsim₀_def set sim₁ : ProbComp Bool := (simulateQ h₁ A).run' (s_init, false) with hsim₁_def have hrun₀ : h₀.runProb (s_init, false) A = sim₀ := by simp [runProb, run, hsim₀_def] have hrun₁ : h₁.runProb (s_init, false) A = sim₁ := by simp [runProb, run, hsim₁_def] have h_adv_le_tv : h₀.advantage (s_init, false) h₁ (s_init, false) A ≤ tvDist sim₀ sim₁ := by rw [QueryImpl.Stateful.advantage, ProbComp.boolDistAdvantage, hrun₀, hrun₁] exact abs_probOutput_toReal_sub_le_tvDist sim₀ sim₁ have h_bridge : ENNReal.ofReal (tvDist sim₀ sim₁) ≤ expectedQuerySlack h₀ chargedQuery querySlack A queryBudget (s_init, false) + Pr[fun z : Bool × σ × Bool => z.2.2 = true | (simulateQ h₀ A).run (s_init, false)] := by rw [hsim₀_def, hsim₁_def] exact ofReal_tvDist_simulateQ_le_expectedQuerySlack_plus_probEvent_output_bad h₀ h₁ chargedQuery querySlack h_step_tv_charged h_step_eq_uncharged h_mono₀ A h_bound s_init exact (ENNReal.ofReal_le_ofReal h_adv_le_tv).trans h_bridge- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/StateSeparating/IdenticalUntilBad.lean:31-68
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