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ProgramDerivation: adding caches
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@ -501,3 +501,106 @@ Definition derived_counter : { counter' | adt_refine counter counter' }.
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Defined.
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Eval simpl in proj1_sig derived_counter.
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(** * Another refinement strategy: introducing a cache (a.k.a. finite differencing) *)
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Inductive CachingMethods {state} (name : string) (func : state -> nat)
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: forall {names}, methods state names -> methods (state * nat) names -> Prop :=
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| CmNil :
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CachingMethods name func MethodsNil MethodsNil
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| CmCached : forall names (ms1 : methods state names) (ms2 : methods _ names),
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CachingMethods name func ms1 ms2
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-> CachingMethods name func
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(MethodsCons {| MethodName := name; MethodBody := (fun s _ => ret (s, func s)) |} ms1)
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(MethodsCons {| MethodName := name; MethodBody := (fun s arg => ret (s, snd s)) |} ms2)
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| CmDefault : forall name' names oldbody (ms1 : methods state names) (ms2 : methods _ names),
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name' <> name
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-> CachingMethods name func ms1 ms2
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-> CachingMethods name func
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(MethodsCons {| MethodName := name'; MethodBody := oldbody |} ms1)
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(MethodsCons {| MethodName := name'; MethodBody := (fun s arg =>
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p <- oldbody (fst s) arg;
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new_cache <- pick c where (func (fst s) = snd s -> func (fst p) = c);
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ret ((fst p, new_cache), snd p)) |} ms2).
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Lemma CachingMethods_ok : forall state name (func : state -> nat)
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names (ms1 : methods state names) (ms2 : methods (state * nat) names),
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CachingMethods name func ms1 ms2
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-> RefineMethods (fun s1 s2 => fst s2 = s1 /\ snd s2 = func s1) ms1 ms2.
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Proof.
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induct 1; eauto.
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econstructor; eauto.
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unfold ret, bind.
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simplify; first_order; subst.
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invert H1.
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rewrite H2.
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eauto.
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econstructor; eauto.
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unfold ret, bind.
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simplify; first_order; subst.
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invert H5.
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unfold pick_ in H4.
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cases x; simplify.
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eauto.
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Qed.
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Hint Resolve CachingMethods_ok.
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Theorem refine_cache : forall state name (func : state -> nat)
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names (ms1 : methods state names) (ms2 : methods (state * nat) names)
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constr,
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CachingMethods name func ms1 ms2
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-> adt_refine {| AdtState := state;
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AdtConstructor := constr;
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AdtMethods := ms1 |}
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{| AdtState := state * nat;
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AdtConstructor := s0 <- constr; ret (s0, func s0);
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AdtMethods := ms2 |}.
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Proof.
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simplify.
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choose_relation (fun s1 s2 => fst s2 = s1 /\ snd s2 = func s1); eauto.
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unfold bind, ret in *.
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first_order; subst.
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simplify; eauto.
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Qed.
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Ltac refine_cache nam := eapply refine_trans; [ eapply refine_cache with (name := nam);
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repeat (apply CmNil
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|| refine (CmCached _ _ _ _ _ _)
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|| (refine (CmDefault _ _ _ _ _ _ _ _ _); [ equality | ])) | ].
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(** ** An example with lists of numbers *)
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Definition sum := fold_right plus 0.
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Definition nats := ADT {
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rep = list nat
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and constructor = ret []
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and method "add"[[self, n]] = ret (n :: self, 0)
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and method "sum"[[self, _]] = ret (self, sum self)
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}.
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Definition optimized_nats : { nats' | adt_refine nats nats' }.
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unfold nats; eexists.
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refine_cache "sum".
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refine_constructor.
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rewrite bind_ret.
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finish.
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refine_method "add".
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rewrite bind_ret; simplify.
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rewrite (pick_one (arg + snd s)).
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rewrite bind_ret.
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finish.
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equality.
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refine_finish.
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Defined.
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Eval simpl in proj1_sig optimized_nats.
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