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-(** Impure monad for interface with impure code
-*)
-
-
-Require Import Program.
-
-
-Module Type MayReturnMonad.
-
-  Axiom t: Type -> Type.
-
-  Axiom mayRet: forall {A:Type}, t A -> A -> Prop.
-
-  Axiom ret: forall {A}, A -> t A.
-
-  Axiom bind: forall {A B}, (t A) -> (A -> t B) -> t B.
-
-  Axiom mk_annot: forall {A} (k: t A), t { a: A | mayRet k a }.
-
-  Axiom mayRet_ret: forall A (a b:A), 
-     mayRet (ret a) b -> a=b.
-
-  Axiom mayRet_bind: forall A B k1 k2 (b:B),
-     mayRet (bind k1 k2) b -> exists a:A, mayRet k1 a /\ mayRet (k2 a) b.
-
-End MayReturnMonad.
-
-
-
-(** Model of impure computation as predicate *)
-Module PowerSetMonad<: MayReturnMonad.
- 
-   Definition t (A:Type) := A -> Prop.
-
-   Definition mayRet {A:Type} (k: t A) a: Prop := k a.
-
-   Definition ret {A:Type} (a:A) := eq a.
-
-   Definition bind {A B:Type} (k1: t A) (k2: A -> t B) := 
-     fun b => exists a, k1 a /\ k2 a b.
-
-   Definition mk_annot {A} (k: t A) : t { a | mayRet k a } := fun _ => True.
-
-   Lemma mayRet_ret A (a b:A): mayRet (ret a) b -> a=b.
-   Proof.
-     unfold mayRet, ret. firstorder.
-   Qed.
-
-   Lemma mayRet_bind A B k1 k2 (b:B):
-         mayRet (bind k1 k2) b -> exists (a:A), mayRet k1 a /\ mayRet (k2 a) b.
-   Proof.
-     unfold mayRet, bind. 
-     firstorder.
-   Qed.
-
-End PowerSetMonad.
-
-
-(** The identity interpretation *)
-Module IdentityMonad<: MayReturnMonad.
-
-   Definition t (A:Type) := A.
-
-   (* may-return semantics of computations *)
-   Definition mayRet {A:Type} (a b:A): Prop := a=b.
-
-   Definition ret {A:Type} (a:A) := a.
-
-   Definition bind {A B:Type} (k1: A) (k2: A -> B) := k2 k1.
-
-   Definition mk_annot {A} (k: t A) : t { a: A | mayRet k a } 
-    := exist _ k (eq_refl k) .
-
-   Lemma mayRet_ret (A:Type) (a b:A): mayRet (ret a) b -> a=b.
-   Proof.
-     intuition.
-   Qed.
-
-   Lemma mayRet_bind (A B:Type) (k1:t A) k2 (b:B):
-         mayRet (bind k1 k2) b -> exists (a:A), mayRet k1 a /\ mayRet (k2 a) b.
-   Proof.
-     firstorder.
-   Qed.
-
-End IdentityMonad.
-
-
-(** Model of impure computation as state-transformers *)
-Module StateMonad<: MayReturnMonad.
- 
-   Parameter St: Type. (* A global state *)
-
-   Definition t (A:Type) := St -> A * St.
-
-   Definition mayRet {A:Type} (k: t A) a: Prop :=
-     exists s, fst (k s)=a.
-
-   Definition ret {A:Type} (a:A) := fun (s:St) => (a,s).
-
-   Definition bind {A B:Type} (k1: t A) (k2: A -> t B) := 
-     fun s0 => let r := k1 s0 in k2 (fst r) (snd r).
-
-   Program Definition mk_annot {A} (k: t A) : t { a | mayRet k a } := 
-     fun s0 => let r := k s0 in (exist _ (fst r) _, snd r).
-   Obligation 1.
-     unfold mayRet; eauto.
-   Qed.
-
-   Lemma mayRet_ret {A:Type} (a b:A): mayRet (ret a) b -> a=b.
-   Proof.
-     unfold mayRet, ret. firstorder.
-   Qed.
-
-   Lemma mayRet_bind {A B:Type} k1 k2 (b:B):
-         mayRet (bind k1 k2) b -> exists (a:A), mayRet k1 a /\ mayRet (k2 a) b.
-   Proof.
-     unfold mayRet, bind. firstorder eauto.
-   Qed. 
-
-End StateMonad.
-
-(** The deferred interpretation *)
-Module DeferredMonad<: MayReturnMonad.
-
-   Definition t (A:Type) := unit -> A.
-
-   (* may-return semantics of computations *)
-   Definition mayRet {A:Type} (a: t A) (b:A): Prop := a tt=b.
-
-   Definition ret {A:Type} (a:A) : t A := fun _ => a.
-
-   Definition bind {A B:Type} (k1: t A) (k2: A -> t B) : t B := fun _ => k2 (k1 tt) tt.
-
-   Definition mk_annot {A} (k: t A) : t { a: A | mayRet k a } 
-    := fun _ => exist _ (k tt) (eq_refl (k tt)).
-
-   Lemma mayRet_ret (A:Type) (a b: A): mayRet (ret a) b -> a=b.
-   Proof.
-     intuition.
-   Qed.
-
-   Lemma mayRet_bind (A B:Type) (k1:t A) k2 (b:B):
-         mayRet (bind k1 k2) b -> exists (a:A), mayRet k1 a /\ mayRet (k2 a) b.
-   Proof.
-     firstorder.
-   Qed.
-
-End DeferredMonad.