Simulate Q post Insert induct
QueryImpl.simulateQ_postInsert.induct
Plain-language statement
Induction principle for proj (simulateQ (so.postInsert nx) oa) parametric in a motive OracleComp spec β → m β → Prop. The recursion structure of proj_simulateQ_postInsert is exposed as two cases mirroring OracleComp.inductionOn: in pure x the projected term reduces to pure x, and in query t >>= k it reduces to so t >>= k' for some continua...
Exact Lean statement
@[elab_as_elim]
lemma simulateQ_postInsert.induct [LawfulMonad m] [LawfulMonad n]
(so : QueryImpl spec m) (nx : (t : spec.Domain) → spec.Range t → n α)
(proj : ∀ {γ : Type u}, n γ → m γ)
(hproj_pure : ∀ {γ : Type u} (x : γ), proj (pure x : n γ) = pure x)
(hproj_bind : ∀ {γ δ : Type u} (b : n γ) (f : γ → n δ),
proj (b >>= f) = proj b >>= fun x => proj (f x))
(hproj_apply : ∀ t, proj ((so.postInsert nx) t) = so t)
{motive : OracleComp spec β → m β → Prop}
(h_pure : ∀ (x : β), motive (pure x : OracleComp spec β) (pure x))
(h_query_bind : ∀ (t : spec.Domain) (k : spec.Range t → OracleComp spec β)
(k' : spec.Range t → m β),
(∀ u, motive (k u) (k' u)) → motive (query t >>= k) (so t >>= k'))
(oa : OracleComp spec β) :
motive oa (proj (simulateQ (so.postInsert nx) oa))Formal artifact
Lean source
@[elab_as_elim]lemma simulateQ_postInsert.induct [LawfulMonad m] [LawfulMonad n] (so : QueryImpl spec m) (nx : (t : spec.Domain) → spec.Range t → n α) (proj : ∀ {γ : Type u}, n γ → m γ) (hproj_pure : ∀ {γ : Type u} (x : γ), proj (pure x : n γ) = pure x) (hproj_bind : ∀ {γ δ : Type u} (b : n γ) (f : γ → n δ), proj (b >>= f) = proj b >>= fun x => proj (f x)) (hproj_apply : ∀ t, proj ((so.postInsert nx) t) = so t) {motive : OracleComp spec β → m β → Prop} (h_pure : ∀ (x : β), motive (pure x : OracleComp spec β) (pure x)) (h_query_bind : ∀ (t : spec.Domain) (k : spec.Range t → OracleComp spec β) (k' : spec.Range t → m β), (∀ u, motive (k u) (k' u)) → motive (query t >>= k) (so t >>= k')) (oa : OracleComp spec β) : motive oa (proj (simulateQ (so.postInsert nx) oa)) := by induction oa using OracleComp.inductionOn with | pure x => rw [simulateQ_pure, hproj_pure] exact h_pure x | query_bind t k ih => rw [simulateQ_bind, simulateQ_spec_query, hproj_bind, hproj_apply] exact h_query_bind t k _ ih- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/OracleComp/SimSemantics/QueryImpl/Constructions.lean:283-304
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