Run simulate Q to Query Impl of State Query Impl
ProbResponder.run_simulateQ_toQueryImpl_ofStateQueryImpl
Plain-language statement
The stateful-responder probability bridge: running an adversary against the responder built from a StateT σ ProbComp handler (ofStateQueryImpl impl) is exactly the evaluation distribution of running it against impl itself , the responder's SPMF program-run is 𝒟 of the ProbComp program-run, jointly in the returned value and the final sta...
Exact Lean statement
theorem run_simulateQ_toQueryImpl_ofStateQueryImpl {ι₀ : Type}
{spec₀ : OracleSpec.{0, 0} ι₀} {σ α : Type}
(impl : QueryImpl spec₀ (StateT σ ProbComp)) (oa : OracleComp spec₀ α) (s : σ) :
(simulateQ (ofStateQueryImpl impl).toQueryImpl oa).run s =
𝒟[(simulateQ impl oa).run s]Formal artifact
Lean source
theorem run_simulateQ_toQueryImpl_ofStateQueryImpl {ι₀ : Type} {spec₀ : OracleSpec.{0, 0} ι₀} {σ α : Type} (impl : QueryImpl spec₀ (StateT σ ProbComp)) (oa : OracleComp spec₀ α) (s : σ) : (simulateQ (ofStateQueryImpl impl).toQueryImpl oa).run s = 𝒟[(simulateQ impl oa).run s] := by induction oa generalizing s with | pure x => change (simulateQ (ofStateQueryImpl impl).toQueryImpl (pure x)).run s = 𝒟[(simulateQ impl (pure x)).run s] rw [simulateQ_pure, simulateQ_pure, StateT.run_pure, StateT.run_pure, evalDist_pure] | queryBind t k ih => have hqb : ∀ {r : Type → Type} [Monad r] [LawfulMonad r] (impl' : QueryImpl spec₀ r), simulateQ impl' (OracleComp.queryBind t k) = impl' t >>= fun u => simulateQ impl' (k u) := by intro r _ _ impl' rw [show OracleComp.queryBind t k = (liftM (spec₀.query t) : OracleComp spec₀ (spec₀.Range t)) >>= k from rfl, simulateQ_bind, simulateQ_spec_query] rw [hqb, hqb, StateT.run_bind, StateT.run_bind, evalDist_bind] exact bind_congr fun p => ih p.1 p.2- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/OracleComp/Coinductive/Responder.lean:198-217
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