Map run simulate Q bind eq of query map eq inv
OracleComp.map_run_simulateQ_bind_eq_of_query_map_eq_inv'
Project documentation
Invariant-gated state-projection theorem for a simulated prefix followed by a stateful continuation.
Exact Lean statement
theorem map_run_simulateQ_bind_eq_of_query_map_eq_inv'
{ι ι' : Type} {spec : OracleSpec ι} {spec' : OracleSpec ι'}
[IsUniformSpec spec']
{σ₁ σ₂ : Type _} {β : Type u}
(impl₁ : QueryImpl spec (StateT σ₁ (OracleComp spec')))
(impl₂ : QueryImpl spec (StateT σ₂ (OracleComp spec')))
(inv : σ₁ → Prop) (proj : σ₁ → σ₂)
(hinv : ∀ t s, inv s →
∀ y ∈ support (m := OracleComp spec') ((impl₁ t).run s), inv y.2)
(hproj : ∀ t s, inv s →
Prod.map id proj <$> (impl₁ t).run s = (impl₂ t).run (proj s))
(oa : OracleComp spec α)
(k₁ : α → StateT σ₁ (OracleComp spec') β)
(k₂ : α → StateT σ₂ (OracleComp spec') β)
(hk : ∀ x s, inv s →
Prod.map id proj <$> (k₁ x).run s = (k₂ x).run (proj s))
(s : σ₁) (hs : inv s) :
Prod.map id proj <$> ((simulateQ impl₁ oa >>= k₁).run s) =
((simulateQ impl₂ oa >>= k₂).run (proj s))Formal artifact
Lean source
theorem map_run_simulateQ_bind_eq_of_query_map_eq_inv' {ι ι' : Type} {spec : OracleSpec ι} {spec' : OracleSpec ι'} [IsUniformSpec spec'] {σ₁ σ₂ : Type _} {β : Type u} (impl₁ : QueryImpl spec (StateT σ₁ (OracleComp spec'))) (impl₂ : QueryImpl spec (StateT σ₂ (OracleComp spec'))) (inv : σ₁ → Prop) (proj : σ₁ → σ₂) (hinv : ∀ t s, inv s → ∀ y ∈ support (m := OracleComp spec') ((impl₁ t).run s), inv y.2) (hproj : ∀ t s, inv s → Prod.map id proj <$> (impl₁ t).run s = (impl₂ t).run (proj s)) (oa : OracleComp spec α) (k₁ : α → StateT σ₁ (OracleComp spec') β) (k₂ : α → StateT σ₂ (OracleComp spec') β) (hk : ∀ x s, inv s → Prod.map id proj <$> (k₁ x).run s = (k₂ x).run (proj s)) (s : σ₁) (hs : inv s) : Prod.map id proj <$> ((simulateQ impl₁ oa >>= k₁).run s) = ((simulateQ impl₂ oa >>= k₂).run (proj s)) := by simp only [StateT.run_bind, map_bind] have hpres := simulateQ_run_preserves_inv_of_query impl₁ inv hinv oa s hs rw [← map_run_simulateQ_eq_of_query_map_eq_inv' impl₁ impl₂ inv proj hinv hproj oa s hs, bind_map_left] exact bind_congr_of_forall_mem_support _ fun x hx => hk x.1 x.2 (hpres x hx)- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/OracleComp/SimSemantics/StateT/StateProjection.lean:146-169
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