frap/ProgramDerivation.v

(** 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 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.


(* We have generally focused so far on proving that programs meet
 * specifications.  What if we could generate programs from their
 * specifications, in ways that guarantee correctness?  Let's explore that
 * direction, in the tradition of *program derivation* via
 * *stepwise refinement*. *)


(** * The computation monad *)

(* One counterintuitive design choice will be to represent specifications and
 * implementations in the same "language," which is essentially Gallina with the
 * added ability to pick elements nondeterministically from arbitrary sets. *)

(* Specifically, a process producing type [A] is represents as [comp A]. *)
Definition comp (A : Type) := A -> Prop.
(* The computation is represented by the set of legal values it might
 * generate. *)

(* Computations form a monad, with the following two operators. *)
Definition ret {A} (x : A) : comp A := eq x.
(* Note how [eq x] is one way of writing "the singleton set of [x],", using
 * partial application of the two-argument equality predicate [eq]. *)
Definition bind {A B} (c1 : comp A) (c2 : A -> comp B) : comp B :=
  fun b => exists a, c1 a /\ c2 a b.
(* As in some of our earlier examples, [bind] is for sequencing one computation
 * after another.  For this monad, existential quantification provides a natural
 * explanation of sequencing. *)

Definition pick_ {A} (P : A -> Prop) : comp A := P.
(* Here is a convenient wrapper function for injecting an arbitrary set into
 * [comp].  This operator stands for nondeterministic choice of any value in the
 * set. *)

(* Here is the correctness condition, for when [c2] implements [c1].  From left
 * to right in the operands of [refine], we move closer to a concrete
 * implementation. *)
Definition refine {A} (c1 c2 : comp A) :=
  forall a, c2 a -> c1 a.
(* Note how this definition is just subset inclusion, in the right direction. *)

(* Next, we use Coq's *setoid* feature to declare compatibility of our
 * definitions with the [rewrite] tactic.  See the Coq manual on setoids for
 * background on what we are doing and why. *)

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. *)

(* Here we have one of the monad laws, showing how traditional computational
 * reduction is compatible with refinement. *)
Theorem bind_ret : forall A B (v : A) (c2 : A -> comp B),
    refine (bind (ret v) c2) (c2 v).
Proof.
  morphisms.
Qed.

(* Here's an example specific to this monad.  One way to resolve a
 * nondeterministic pick from a set is to replace it with a specific element
 * from the set. *)
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).


(** * Picking a number not in a list *)

(* Let's illustrate the big idea with an example derivation. *)

(* 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.

(** ** Interlude: defining some tactics for key parts of derivation *)

(* We run this next step to hide an evar, so that rewriting isn't too eager to
 * make up values for it. *)
Ltac hide_evars :=
  match goal with
  | [ |- refine _ (?f _ _) ] => set f
  | [ |- refine _ (?f _) ] => set f
  | [ |- refine _ ?f ] => set f
  end.

(* This tactic starts a script that finds a term to refine another. *)
Ltac begin :=
  eexists; simplify; hide_evars.

(* This tactic ends such a derivation, in effect undoing the effect of
 * [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.

(** ** Back to the example *)

(* Let's derive an efficient implementation. *)
Definition implementation : sig (fun f : list nat -> comp nat => forall ls, refine (notInList ls) (f ls)).
  begin.
  rewrite notInList_decompose.
  rewrite listMax_refines.
  rewrite bind_ret.
  rewrite increment_refines.
  finish.
Defined.

(* We can extract the program that we found as a standlone, executable Gallina
 * term. *)
Definition impl := Eval simpl in proj1_sig implementation.

(* We'll locally 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) *)

(* Stepwise refinement can be most satisfying to build objects with multiple
 * methods.  The specification of such an object is often called an abstract
 * data type (ADT), and we studied them (from a verification perspective) in
 * module DataAbstraction.  Let's see how we can build implementations
 * automatically from ADTs.  First, some preliminary definitions. *)

(* Every method inhabits this type, where [state] is the type of private state
 * for the object. *)
Record method_ {state : Type} := {
  MethodName : string;
  MethodBody : state -> nat -> comp (state * nat)
}.
(* A body takes the current state as input and produces the new state as output.
 * Additionally, we have hardcoded both the parameter type and the return type
 * to [nat], for simplicity.  The only wrinkle is that a body result is in the
 * [comp] monad, to let it mix features from specification and
 * implementation. *)

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).

(* Next, this type collects several method definitions, given a list of their
 * names. *)
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.

(* Finally, the definition of an abstract data type, which will apply to both
 * specifications (the classical sense of ADT) and implementations. *)
Record adt {names : list string} := {
  AdtState : Type;
  AdtConstructor : comp AdtState;
  AdtMethods : methods AdtState names
}.
(* An ADT has a state type, one constructor to initialize the state, and a set
 * of methods that may read and write the state. *)

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).

(* Here's one quick example, of a counter with duplicate "increment" methods. *)

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)
}.

(* This example hasn't used the power of the [comp] monad, but we will get
 * there later. *)


(** * ADT refinement *)

(* What does it mean to take sound implementation steps from an ADT toward an
 * efficient implementation?  We formalize refinement for ADTs as well.  The key
 * principle will be *simulation*, very similarly to how we used the idea for
 * compiler verification. *)

(* For a "before" state type [state1] and an "after" state type [state2], we
 * require choice of a simulation relation [R].  This next inductive relation
 * captures when all methods are pairwise compatible with [R], between the
 * "before" and "after" ADTs. *)
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.

(* When does [adt2] refine [adt1]?  When there exists a simulation relation,
 * with respect to which the constructors and methods all satisfy the usual
 * simulation diagram. *)
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 *)

(* Let's refine the previous counter spec into an implementation that maintains
 * two separate counters and adds them on demand. *)

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.

(* Here is why the new implementation is correct. *)
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 *)

(* ADT refinement forms a preorder, as the next two theorems show. *)

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.

(* Note the use of relation composition for [refine_trans]. *)

(** ** Refining constructors *)

(* One way to refine an ADT is to perform [comp]-level refinement within its
 * constructor definition. *)

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 *)

(* Conceptually quite similar, refining within methods requires more syntactic
 * framework. *)

(* This relation captures the process of replacing [oldbody] of method [name]
 * with [newbody]. *)
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).

(* Let's skip ahead to the next [Theorem]. *)

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.

(* It is OK to replace a method body if the new refines the old as a [comp]. *)
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 *)

(* Some of the most interesting refinements select new data structures.  That
 * is, they pick new state types.  Here we formalize that idea in terms of
 * existence of an *abstraction function* from the new state type to the old. *)

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).
(* Interestingly, we managed to rewrite all method bodies automatically, to be
 * compatible with a new data structure!  The catch is that our language of
 * method bodies is inherently noncomputational.  We leave nontrivial work for
 * ourselves, in further refinement of method bodies to remove "pick"
 * operations.  Note how the generic method template above relies on "pick"
 * operations to invert abstraction functions. *)

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 | ].
(* Don't be thrown off by the [refine] tactic used here.  It is not related to
 * our notion of refinement!  See module SubsetTypes for an explanation of
 * it. *)

Ltac refine_finish := apply refine_refl.

(** ** Example: the numeric counter again *)

(* Let's generate the two-counter variant through the process of finding a
 * proof, in contrast to theorem [split_counter_ok], which started with the full
 * code of the transformed ADT. *)

Definition derived_counter : sig (adt_refine counter).
  unfold counter; eexists.
  refine_rep (fun p => fst p + snd p).

  refine_constructor.
  rewrite bind_ret.
  rewrite (pick_one (0, 0)).
  finish.
  equality.

  refine_method "increment1".
  rewrite bind_ret; simplify.
  rewrite (pick_one (fst s + arg, snd s)).
  rewrite bind_ret; simplify.
  finish.
  simplify; linear_arithmetic.

  refine_method "increment2".
  rewrite bind_ret; simplify.
  rewrite (pick_one (fst s, snd s + arg)).
  rewrite bind_ret; simplify.
  finish.
  simplify; linear_arithmetic.

  refine_method "value".
  rewrite bind_ret; simplify.
  rewrite (pick_one s).
  rewrite bind_ret; simplify.
  finish.
  equality.

  refine_finish.
Defined.
  
Eval simpl in proj1_sig derived_counter.


(** * Another refinement strategy: introducing a cache (a.k.a. finite differencing) *)

(* Some methods begin life as expensive computations, such that it pays off to
 * precompute their values.  A generic refinement strategy follows this plan by
 * introducing *caches*. *)

(* Here, [name] names a method whose body leaves the state alone and returns the
 * result of [func] applied to that state. *)
Inductive CachingMethods {state} (name : string) (func : state -> nat)
  : forall {names}, methods state names -> methods (state * nat) names -> Prop :=
| CmNil :
    CachingMethods name func MethodsNil MethodsNil

(* Here is how we rewrite [name] itself.  We are extending state with an extra
 * cache of [func]'s value, so it is legal just to return that cache. *)
| 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)

(* Any other method now picks up an obligation to recompute the cache value
 * whenever changing the state.  We express that recomputation with a pick, to
 * be refined later into efficient logic. *)
| 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 *)

(* Let's work out an example of caching. *)

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).
  unfold nats; eexists.

  refine_cache "sum".

  refine_constructor.
  rewrite bind_ret.
  finish.

  refine_method "add".
  rewrite bind_ret; simplify.
  rewrite (pick_one (arg + snd s)).
  rewrite bind_ret.
  finish.
  equality.

  refine_finish.
Defined.

Eval simpl in proj1_sig optimized_nats.