(** * Theorems about State effects *)
(* begin hide *)
Program
Setoid
Morphisms
RelationClasses.
Basics.Basics
Basics.Category
Basics.CategoryKleisli
Basics.Monad
Core.ITreeDefinition
Core.KTree
Core.KTreeFacts
Eq.Eq
Eq.UpToTaus
Indexed.Sum
Interp.Interp
Interp.InterpFacts
Interp.RecursionFacts
Events.State.
Open Scope itree_scope.
(* end hide *)
Definition _interp_state {E F S R}
(f : E ~> stateT S (itree F)) (ot : itreeF E R _)
: stateT S (itree F) R := fun s =>
match ot with
| RetF r => Ret (s, r)
| TauF t => Tau (interp_state f t s)
| VisF e k => f _ e s >>= (fun sx => Tau (interp_state f (k (snd sx)) (fst sx)))
end.
Instance eq_itree_interp_state {E F S R} (h : E ~> Monads.stateT S (itree F)) :
Proper (eq_itree eq ==> eq ==> eq_itree eq)
(@interp_state _ _ _ _ _ _ h R).

Lemma interp_state_bind {E F : Type -> Type} {A B S : Type}
(f : forall T, E T -> S -> itree F (S * T)%type)
(t : itree E A) (k : A -> itree E B)
(s : S) :
(interp_state f (t >>= k) s)
≅
(interp_state f t s >>= fun st => interp_state f (k (snd st)) (fst st)).