diff options
Diffstat (limited to 'lib/Impure/ImpMonads.v')
-rw-r--r-- | lib/Impure/ImpMonads.v | 148 |
1 files changed, 148 insertions, 0 deletions
diff --git a/lib/Impure/ImpMonads.v b/lib/Impure/ImpMonads.v new file mode 100644 index 00000000..f01a2755 --- /dev/null +++ b/lib/Impure/ImpMonads.v @@ -0,0 +1,148 @@ +(** 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. |