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frap/ProgramDerivation.v
Adam Chlipala 4505284871 ProgramDerivation: refine_method and refine_rep
2018-05-05 14:40:51 -04:00

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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 16: Deriving Programs from Specifications
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/
* Some material borrowed from Fiat <http://plv.csail.mit.edu/fiat/> *)
Require Import FrapWithoutSets.
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; hnf; first_order.
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.
Theorem bind_ret : forall A B (v : A) (c2 : A -> comp B),
refine (bind (ret v) c2) (c2 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).
(** * Picking a number not in a list *)
(* A specification of what it means to choose a number that is not in a
* particular list *)
Definition notInList (ls : list nat) :=
pick n where ~In n ls.
(* We can use a simple property to justify a decomposition of the original
* spec. *)
Theorem notInList_decompose : forall ls,
refine (notInList ls) (upper <- pick upper where forall n, In n ls -> upper >= n;
pick beyond where beyond > upper).
Proof.
simplify.
unfold notInList, refine, bind, pick_, not.
first_order.
apply H in H0.
linear_arithmetic.
Qed.
(* A simple traversal will find the maximum list element, which is a good upper
* bound. *)
Definition listMax := fold_right max 0.
(* ...and we can prove it! *)
Theorem listMax_upperBound : forall init ls,
forall n, In n ls -> fold_right max init ls >= n.
Proof.
induct ls; simplify; propositional.
linear_arithmetic.
apply IHls in H0.
linear_arithmetic.
Qed.
(* Now we restate that result as a computation refinement. *)
Theorem listMax_refines : forall ls,
refine (pick upper where forall n, In n ls -> upper >= n) (ret (listMax ls)).
Proof.
unfold refine, pick_, ret; simplify; subst.
apply listMax_upperBound; assumption.
Qed.
(* An easy way to find a number higher than another: add 1! *)
Theorem increment_refines : forall n,
refine (pick higher where higher > n) (ret (n + 1)).
Proof.
unfold refine, pick_, ret; simplify; subst.
linear_arithmetic.
Qed.
Ltac begin :=
eexists; simplify;
(* We run this next step to hide an evar, so that rewriting isn't too eager to
* make up values for it. *)
match goal with
| [ |- refine _ (?f _) ] => set f
end.
Ltac finish :=
match goal with
| [ |- 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
end.
(* Let's derive an efficient implementation. *)
Theorem implementation : { f : list nat -> comp nat | forall ls, refine (notInList ls) (f ls) }.
Proof.
begin.
rewrite notInList_decompose.
rewrite listMax_refines.
setoid_rewrite increment_refines.
(* ^-- Different tactic here to let us rewrite under a binder! *)
rewrite bind_ret.
finish.
Defined.
(* We can extract the program that we found as a standlone, executable Gallina
* term. *)
Definition impl := Eval simpl in proj1_sig implementation.
Print impl.
(* We'll temporarily expose the definition of [max], so we can compute neatly
* here. *)
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.
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.
Notation "'method' name [[ self , arg ]] = body" :=
{| MethodName := name;
MethodBody := fun self arg => body |}
(at level 100, self at level 0, arg at level 0).
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).
(** * 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),
(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) := {
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)
}.
Ltac choose_relation R :=
match goal with
| [ |- adt_refine ?a ?b ] => apply (Build_adt_refine _ a b R)
end; simplify.
(** ** Example: numeric counter *)
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)
}.
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.
choose_relation (fun n p => n = fst p + snd p).
unfold ret in *; subst.
eauto.
repeat constructor; simplify; unfold ret in *; subst;
match goal with
| [ H : (_, _) = (_, _) |- _ ] => invert H
end; eauto.
Grab Existential Variables.
exact 0.
Qed.
(** * 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 X.
invert X0.
choose_relation (fun s1 s3 => exists s2, ArRel0 s1 s2 /\ ArRel1 s2 s3); eauto.
apply ArConstructors1 in H.
first_order.
Qed.
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.
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).
Theorem 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.
Inductive RepChangeMethods {state1 state2} (R : state1 -> state2 -> Prop)
: forall {names}, methods state1 names -> methods state2 names -> Prop :=
| RchNil :
RepChangeMethods R MethodsNil MethodsNil
| RchCons : forall name names oldbody (ms1 : methods state1 names) (ms2 : methods state2 names),
RepChangeMethods R ms1 ms2
-> RepChangeMethods R
(MethodsCons {| MethodName := name; MethodBody := oldbody |} ms1)
(MethodsCons {| MethodName := name; MethodBody := (fun s arg =>
pick s'_res where
forall s0, R s0 s
-> exists s', oldbody s0 arg (s', snd s'_res)
/\ R s' (fst s'_res)) |} ms2).
Lemma RepChangeMethods_ok : forall state1 state2 (R : state1 -> state2 -> Prop)
names (ms1 : methods state1 names) (ms2 : methods state2 names),
RepChangeMethods R ms1 ms2
-> RefineMethods R ms1 ms2.
Proof.
induct 1; eauto.
Qed.
Hint Resolve RepChangeMethods_ok.
Theorem refine_rep : forall state1 state2 (R : state1 -> state2 -> Prop)
names (ms1 : methods state1 names) (ms2 : methods state2 names)
constr,
RepChangeMethods R ms1 ms2
-> adt_refine {| AdtState := state1;
AdtConstructor := constr;
AdtMethods := ms1 |}
{| AdtState := state2;
AdtConstructor := s0 <- constr; pick s where R s0 s;
AdtMethods := ms2 |}.
Proof.
simplify.
choose_relation R; eauto.
Qed.
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