(** * 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).
Instance eutt_interp_state {E F: Type -> Type} {S : Type}
(h : E ~> Monads.stateT S (itree F)) R :
Proper (eutt eq ==> eq ==> eutt eq) (@interp_state E (itree F) S _ _ _ h R).
(* SAZ: These are probably too specialized. *)
Definition state_eq {E S X}
: (stateT S (itree E) X) -> (stateT S (itree E) X) -> Prop :=
fun t1 t2 => forall s, eq_itree eq (t1 s) (t2 s).

Lemma interp_state_iter' {E F } S (f : E ~> stateT S (itree F)) {I A}
(t  : I -> itree E (I + A))
: forall i, state_eq (State.interp_state f (ITree.iter t i))
(Basics.iter (fun i => State.interp_state f (t i)) i).