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frap/ProgramDerivation_template.v
Adam Chlipala 7e84adc6bd ProgramDerivation_template
2018-05-06 19:49:10 -04:00

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Require Import Frap.
Require Import Program Setoids.Setoid Classes.RelationClasses Classes.Morphisms Morphisms_Prop.
Require Import Eqdep.
Ltac inv_pair :=
match goal with
| [ H : existT _ _ _ = existT _ _ _ |- _ ] => apply inj_pair2 in H; invert H
end.
(** * The computation monad *)
Definition comp (A : Type) := A -> Prop.
Definition ret {A} (x : A) : comp A := eq x.
Definition bind {A B} (c1 : comp A) (c2 : A -> comp B) : comp B :=
fun b => exists a, c1 a /\ c2 a b.
Definition pick_ {A} (P : A -> Prop) : comp A := P.
Definition refine {A} (c1 c2 : comp A) :=
forall a, c2 a -> c1 a.
Ltac morphisms := unfold refine, impl, pointwise_relation, bind, ret, pick_; hnf; first_order; subst; eauto.
Global Instance refine_PreOrder A : PreOrder (@refine A).
Proof.
constructor; morphisms.
Qed.
Add Parametric Morphism A : (@refine A)
with signature (@refine A) --> (@refine A) ++> impl
as refine_refine.
Proof.
morphisms.
Qed.
Add Parametric Morphism A B : (@bind A B)
with signature (@refine A)
==> (pointwise_relation _ (@refine B))
==> (@refine B)
as refine_bind.
Proof.
morphisms.
Qed.
Add Parametric Morphism A B : (@bind A B)
with signature (flip (@refine A))
==> (pointwise_relation _ (flip (@refine B)))
==> (flip (@refine B))
as refine_bind_flip.
Proof.
morphisms.
Qed.
(** ** OK, back to the details we want to focus on. *)
Theorem bind_ret : forall A B (v : A) (c2 : A -> comp B),
refine (bind (ret v) c2) (c2 v).
Proof.
morphisms.
Qed.
Theorem pick_one : forall {A} {P : A -> Prop} v,
P v
-> refine (pick_ P) (ret v).
Proof.
morphisms.
Qed.
Notation "'pick' x 'where' P" := (pick_ (fun x => P)) (at level 80, x at level 0).
Notation "x <- c1 ; c2" := (bind c1 (fun x => c2)) (at level 81, right associativity).
(** * Some handy tactics *)
Ltac hide_evars :=
match goal with
| [ |- refine _ (?f _ _) ] => set f
| [ |- refine _ (?f _) ] => set f
| [ |- refine _ ?f ] => set f
end.
Ltac begin :=
eexists; simplify; hide_evars.
Ltac finish :=
match goal with
| [ |- refine ?e (?f ?arg1 ?arg2) ] =>
let g := eval pattern arg1, arg2 in e in
match g with
| ?g' _ _ =>
let f' := eval unfold f in f in
unify f' g'; reflexivity
end
| [ |- refine ?e (?f ?arg) ] =>
let g := eval pattern arg in e in
match g with
| ?g' _ =>
let f' := eval unfold f in f in
unify f' g'; reflexivity
end
| [ |- refine ?e ?f ] =>
let f' := eval unfold f in f in
unify f' e; reflexivity
end.
(** * Picking a number not in a list *)
Definition notInList (ls : list nat) :=
pick n where ~In n ls.
Definition implementation : sig (fun f : list nat -> comp nat => forall ls, refine (notInList ls) (f ls)).
Admitted.
(*Definition impl := Eval simpl in proj1_sig implementation.
Transparent max.
Eval compute in impl (1 :: 7 :: 8 :: 2 :: 13 :: 6 :: nil).*)
(** * Abstract data types (ADTs) *)
Record method_ {state : Type} := {
MethodName : string;
MethodBody : state -> nat -> comp (state * nat)
}.
Arguments method_ : clear implicits.
Notation "'method' name [[ self , arg ]] = body" :=
{| MethodName := name;
MethodBody := fun self arg => body |}
(at level 100, self at level 0, arg at level 0).
Inductive methods {state : Type} : list string -> Type :=
| MethodsNil : methods []
| MethodsCons : forall (m : method_ state) {names},
methods names
-> methods (MethodName m :: names).
Arguments methods : clear implicits.
Record adt {names : list string} := {
AdtState : Type;
AdtConstructor : comp AdtState;
AdtMethods : methods AdtState names
}.
Arguments adt : clear implicits.
Notation "'ADT' { 'rep' = state 'and' 'constructor' = constr ms }" :=
{| AdtState := state;
AdtConstructor := constr;
AdtMethods := ms |}.
Notation "'and' m1 'and' .. 'and' mn" :=
(MethodsCons m1 (.. (MethodsCons mn MethodsNil) ..)) (at level 101).
Definition counter := ADT {
rep = nat
and constructor = ret 0
and method "increment1"[[self, arg]] = ret (self + arg, 0)
and method "increment2"[[self, arg]] = ret (self + arg, 0)
and method "value"[[self, _]] = ret (self, self)
}.
(** * ADT refinement *)
Inductive RefineMethods {state1 state2} (R : state1 -> state2 -> Prop)
: forall {names}, methods state1 names -> methods state2 names -> Prop :=
| RmNil : RefineMethods R MethodsNil MethodsNil
| RmCons : forall name names (f1 : state1 -> nat -> comp (state1 * nat))
(f2 : state2 -> nat -> comp (state2 * nat))
(ms1 : methods state1 names) (ms2 : methods state2 names),
(* This condition is the classic "simulation diagram." *)
(forall s1 s2 arg s2' res,
R s1 s2
-> f2 s2 arg (s2', res)
-> exists s1', f1 s1 arg (s1', res)
/\ R s1' s2')
-> RefineMethods R ms1 ms2
-> RefineMethods R (MethodsCons {| MethodName := name; MethodBody := f1 |} ms1)
(MethodsCons {| MethodName := name; MethodBody := f2 |} ms2).
Hint Constructors RefineMethods.
Record adt_refine {names} (adt1 adt2 : adt names) : Prop := {
ArRel : AdtState adt1 -> AdtState adt2 -> Prop;
ArConstructors : forall s2,
AdtConstructor adt2 s2
-> exists s1, AdtConstructor adt1 s1
/\ ArRel s1 s2;
ArMethods : RefineMethods ArRel (AdtMethods adt1) (AdtMethods adt2)
}.
(* We will use this handy tactic to prove ADT refinements. *)
Ltac choose_relation R :=
match goal with
| [ |- adt_refine ?a ?b ] => apply (Build_adt_refine _ a b R)
end; simplify.
(** ** Example: numeric counter *)
Definition split_counter := ADT {
rep = nat * nat
and constructor = ret (0, 0)
and method "increment1"[[self, arg]] = ret ((fst self + arg, snd self), 0)
and method "increment2"[[self, arg]] = ret ((fst self, snd self + arg), 0)
and method "value"[[self, _]] = ret (self, fst self + snd self)
}.
Hint Extern 1 (@eq nat _ _) => simplify; linear_arithmetic.
Theorem split_counter_ok : adt_refine counter split_counter.
Proof.
Admitted.
(** * General refinement strategies *)
Lemma RefineMethods_refl : forall state names (ms : methods state names),
RefineMethods eq ms ms.
Proof.
induct ms.
constructor.
cases m; constructor; first_order.
subst; eauto.
Qed.
Hint Immediate RefineMethods_refl.
Theorem refine_refl : forall names (adt1 : adt names),
adt_refine adt1 adt1.
Proof.
simplify.
choose_relation (@eq (AdtState adt1)); eauto.
Qed.
Lemma RefineMethods_trans : forall state1 state2 state3 names
R1 R2
(ms1 : methods state1 names)
(ms2 : methods state2 names)
(ms3 : methods state3 names),
RefineMethods R1 ms1 ms2
-> RefineMethods R2 ms2 ms3
-> RefineMethods (fun s1 s3 => exists s2, R1 s1 s2 /\ R2 s2 s3) ms1 ms3.
Proof.
induct 1; invert 1; repeat inv_pair; eauto.
econstructor; eauto.
first_order.
eapply H5 in H2; eauto.
first_order.
eapply H in H2; eauto.
first_order.
Qed.
Hint Resolve RefineMethods_trans.
Theorem refine_trans : forall names (adt1 adt2 adt3 : adt names),
adt_refine adt1 adt2
-> adt_refine adt2 adt3
-> adt_refine adt1 adt3.
Proof.
simplify.
invert H.
invert H0.
choose_relation (fun s1 s3 => exists s2, ArRel s1 s2 /\ ArRel0 s2 s3); eauto.
apply ArConstructors0 in H.
first_order.
Qed.
(** ** Refining constructors *)
Theorem refine_constructor : forall names state constr1 constr2 (ms : methods _ names),
refine constr1 constr2
-> adt_refine {| AdtState := state;
AdtConstructor := constr1;
AdtMethods := ms |}
{| AdtState := state;
AdtConstructor := constr2;
AdtMethods := ms |}.
Proof.
simplify.
choose_relation (@eq state); eauto.
Qed.
(** ** Refining methods *)
Inductive ReplaceMethod {state} (name : string)
(oldbody newbody : state -> nat -> comp (state * nat))
: forall {names}, methods state names -> methods state names -> Prop :=
| RepmHead : forall names (ms : methods state names),
ReplaceMethod name oldbody newbody
(MethodsCons {| MethodName := name; MethodBody := oldbody |} ms)
(MethodsCons {| MethodName := name; MethodBody := newbody |} ms)
| RepmSkip : forall name' names oldbody' (ms1 ms2 : methods state names),
name' <> name
-> ReplaceMethod name oldbody newbody ms1 ms2
-> ReplaceMethod name oldbody newbody
(MethodsCons {| MethodName := name'; MethodBody := oldbody' |} ms1)
(MethodsCons {| MethodName := name'; MethodBody := oldbody' |} ms2).
Lemma ReplaceMethod_ok : forall state name
(oldbody newbody : state -> nat -> comp (state * nat))
names (ms1 ms2 : methods state names),
(forall s arg, refine (oldbody s arg) (newbody s arg))
-> ReplaceMethod name oldbody newbody ms1 ms2
-> RefineMethods eq ms1 ms2.
Proof.
induct 1.
econstructor; eauto.
unfold refine in *; simplify; subst; eauto.
econstructor; eauto.
simplify; subst; eauto.
Qed.
Hint Resolve ReplaceMethod_ok.
Theorem refine_method : forall state name (oldbody newbody : state -> nat -> comp (state * nat))
names (ms1 ms2 : methods state names) constr,
(forall s arg, refine (oldbody s arg) (newbody s arg))
-> ReplaceMethod name oldbody newbody ms1 ms2
-> adt_refine {| AdtState := state;
AdtConstructor := constr;
AdtMethods := ms1 |}
{| AdtState := state;
AdtConstructor := constr;
AdtMethods := ms2 |}.
Proof.
simplify.
choose_relation (@eq state); eauto.
Qed.
(** ** Representation changes *)
Inductive RepChangeMethods {state1 state2} (absfunc : state2 -> state1)
: forall {names}, methods state1 names -> methods state2 names -> Prop :=
| RchNil :
RepChangeMethods absfunc MethodsNil MethodsNil
| RchCons : forall name names oldbody (ms1 : methods state1 names) (ms2 : methods state2 names),
RepChangeMethods absfunc ms1 ms2
-> RepChangeMethods absfunc
(MethodsCons {| MethodName := name; MethodBody := oldbody |} ms1)
(MethodsCons {| MethodName := name; MethodBody := (fun s arg =>
p <- oldbody (absfunc s) arg;
s' <- pick s' where absfunc s' = fst p;
ret (s', snd p)) |} ms2).
Lemma RepChangeMethods_ok : forall state1 state2 (absfunc : state2 -> state1)
names (ms1 : methods state1 names) (ms2 : methods state2 names),
RepChangeMethods absfunc ms1 ms2
-> RefineMethods (fun s1 s2 => absfunc s2 = s1) ms1 ms2.
Proof.
induct 1; eauto.
econstructor; eauto.
morphisms.
invert H3.
cases x; simplify; subst.
eauto.
Qed.
Hint Resolve RepChangeMethods_ok.
Theorem refine_rep : forall state1 state2 (absfunc : state2 -> state1)
names (ms1 : methods state1 names) (ms2 : methods state2 names)
constr,
RepChangeMethods absfunc ms1 ms2
-> adt_refine {| AdtState := state1;
AdtConstructor := constr;
AdtMethods := ms1 |}
{| AdtState := state2;
AdtConstructor := s0 <- constr; pick s where absfunc s = s0;
AdtMethods := ms2 |}.
Proof.
simplify.
choose_relation (fun s1 s2 => absfunc s2 = s1); eauto.
Qed.
(** ** Tactics for easy use of those refinement principles *)
Ltac refine_rep f := eapply refine_trans; [ apply refine_rep with (absfunc := f);
repeat (apply RchNil
|| refine (RchCons _ _ _ _ _ _ _)) | cbv beta ].
Ltac refine_constructor := eapply refine_trans; [ apply refine_constructor; hide_evars | ].
Ltac refine_method nam := eapply refine_trans; [ eapply refine_method with (name := nam); [
| repeat (refine (RepmHead _ _ _ _ _)
|| (refine (RepmSkip _ _ _ _ _ _ _ _ _ _); [ equality | ])) ];
cbv beta; simplify; hide_evars | ].
Ltac refine_finish := apply refine_refl.
(** ** Example: the numeric counter again *)
Definition derived_counter : sig (adt_refine counter).
Admitted.
Eval simpl in proj1_sig derived_counter.
(** * Another refinement strategy: introducing a cache (a.k.a. finite differencing) *)
Inductive CachingMethods {state} (name : string) (func : state -> nat)
: forall {names}, methods state names -> methods (state * nat) names -> Prop :=
| CmNil :
CachingMethods name func MethodsNil MethodsNil
| CmCached : forall names (ms1 : methods state names) (ms2 : methods _ names),
CachingMethods name func ms1 ms2
-> CachingMethods name func
(MethodsCons {| MethodName := name; MethodBody := (fun s _ => ret (s, func s)) |} ms1)
(MethodsCons {| MethodName := name; MethodBody := (fun s arg => ret (s, snd s)) |} ms2)
| CmDefault : forall name' names oldbody (ms1 : methods state names) (ms2 : methods _ names),
name' <> name
-> CachingMethods name func ms1 ms2
-> CachingMethods name func
(MethodsCons {| MethodName := name'; MethodBody := oldbody |} ms1)
(MethodsCons {| MethodName := name'; MethodBody := (fun s arg =>
p <- oldbody (fst s) arg;
new_cache <- pick c where (func (fst s) = snd s -> func (fst p) = c);
ret ((fst p, new_cache), snd p)) |} ms2).
Lemma CachingMethods_ok : forall state name (func : state -> nat)
names (ms1 : methods state names) (ms2 : methods (state * nat) names),
CachingMethods name func ms1 ms2
-> RefineMethods (fun s1 s2 => fst s2 = s1 /\ snd s2 = func s1) ms1 ms2.
Proof.
induct 1; eauto.
econstructor; eauto.
unfold ret, bind.
simplify; first_order; subst.
invert H1.
rewrite H2.
eauto.
econstructor; eauto.
unfold ret, bind.
simplify; first_order; subst.
invert H5.
unfold pick_ in H4.
cases x; simplify.
eauto.
Qed.
Hint Resolve CachingMethods_ok.
Theorem refine_cache : forall state name (func : state -> nat)
names (ms1 : methods state names) (ms2 : methods (state * nat) names)
constr,
CachingMethods name func ms1 ms2
-> adt_refine {| AdtState := state;
AdtConstructor := constr;
AdtMethods := ms1 |}
{| AdtState := state * nat;
AdtConstructor := s0 <- constr; ret (s0, func s0);
AdtMethods := ms2 |}.
Proof.
simplify.
choose_relation (fun s1 s2 => fst s2 = s1 /\ snd s2 = func s1); eauto.
unfold bind, ret in *.
first_order; subst.
simplify; eauto.
Qed.
Ltac refine_cache nam := eapply refine_trans; [ eapply refine_cache with (name := nam);
repeat (apply CmNil
|| refine (CmCached _ _ _ _ _ _)
|| (refine (CmDefault _ _ _ _ _ _ _ _ _); [ equality | ])) | ].
(** ** An example with lists of numbers *)
Definition sum := fold_right plus 0.
Definition nats := ADT {
rep = list nat
and constructor = ret []
and method "add"[[self, n]] = ret (n :: self, 0)
and method "sum"[[self, _]] = ret (self, sum self)
}.
Definition optimized_nats : sig (adt_refine nats).
Admitted.
Eval simpl in proj1_sig optimized_nats.
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