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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Supplementary Coq material: first-class functions and continuations
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Require Import Frap Program.
(* Next stop in touring the basic Coq ingredients of functional programming and
* proof: functions as first-class data. These days, most trendy programming
* languages contain this feature, though it can't hurt to review; and we'll see
* patterns in specification and proof that are helpful to handle first-class
* functions. *)
(** * Some data fodder for us to compute with later *)
(* Records are a handy way to define datatypes in terms of the named fields that
* each value must contain. *)
Record programming_language := {
Name : string;
PurelyFunctional : bool;
AppearedInYear : nat
}.
(* Here's a quick example of a set of programming languages, which we will use
* below in some example computations. *)
Definition pascal := {|
Name := "Pascal";
PurelyFunctional := false;
AppearedInYear := 1970
|}.
Definition clang := {|
Name := "C";
PurelyFunctional := false;
AppearedInYear := 1972
|}.
Definition gallina := {|
Name := "Gallina";
PurelyFunctional := true;
AppearedInYear := 1989
|}.
Definition haskell := {|
Name := "Haskell";
PurelyFunctional := true;
AppearedInYear := 1990
|}.
Definition ocaml := {|
Name := "OCaml";
PurelyFunctional := false;
AppearedInYear := 1996
|}.
Definition languages := [pascal; clang; gallina; haskell; ocaml].
(** * Classic list functions *)
(* The trio of "map/filter/reduce" are commonly presented as workhorse
* *higher-order functions* for lists. That is, they are functions that take
* functions as arguments. *)
(* [map] runs a function on every position of a list to make a new list. *)
Fixpoint map {A B} (f : A -> B) (ls : list A) : list B :=
match ls with
| nil => nil
| x :: ls' => f x :: map f ls'
end.
Compute map (fun n => n + 2) [1; 3; 8].
(* Note the use of an *anonymous function* above via [fun]. *)
(* [filter] keeps only those elements of a list that pass a Boolean test. *)
Fixpoint filter {A} (f : A -> bool) (ls : list A) : list A :=
match ls with
| nil => nil
| x :: ls' => if f x then x :: filter f ls' else filter f ls'
end.
Compute filter (fun n => if n <=? 3 then true else false) [1; 3; 8].
(* The [if ... then true else false] bit might seem wasteful. Actually, the
* [<=?] operator has a fancy type that needs converting to [bool]. We'll get
* more specific about such types in a future class. *)
(* [fold_left], a relative of "reduce," repeatedly applies a function to all
* elements of a list. *)
Fixpoint fold_left {A B} (f : B -> A -> B) (ls : list A) (acc : B) : B :=
match ls with
| nil => acc
| x :: ls' => fold_left f ls' (f acc x)
end.
Compute fold_left max [1; 3; 8] 0.
(* Another way to see [fold_left] in action: *)
Theorem fold_left3 : forall {A B} (f : B -> A -> B) (x y z : A) (acc : B),
fold_left f [x; y; z] acc = f (f (f acc x) y) z.
Proof.
simplify.
equality.
Qed.
(* Let's use these classics to implement a few simple "database queries" on the
* list of programming languages. Note that each field name from
* [programming_language] is itself a first-class function, for projecting that
* field from any record! *)
Compute map Name languages.
(* names of languages *)
Compute map Name (filter PurelyFunctional languages).
(* names of purely functional languages *)
Compute fold_left max (map AppearedInYear languages) 0.
(* maximum year in which a language appeared *)
Compute fold_left max (map AppearedInYear (filter PurelyFunctional languages)) 0.
(* maximum year in which a purely functional language appeared *)
(* To avoid confusing things, we'll revert to the standard library's (identical)
* versions of these functions for the remainder. *)
Reset map.
(** * Sorting, parameterized in a comparison operation *)
(* Another classic family of higher-order functions is for sorting, where we
* typically take a *comparator* as input. Such a function helps us compare
* data elements with each other. Let's do insertion sort as an example. *)
(* Important helper function: take in an assumed-sorted list [ls]; generate a
* new list that is like [ls] but with [new] inserted at the appropriate
* position to maintain sortedness. We use "less than or equal to" test [le] to
* define sortedness. *)
Fixpoint insert {A} (le : A -> A -> bool) (new : A) (ls : list A) : list A :=
match ls with
| [] => [new]
| x :: ls' =>
if le new x then
new :: ls
else
x :: insert le new ls'
end.
(* Now insertion sort is just repeated use of [insert]. *)
Fixpoint insertion_sort {A} (le : A -> A -> bool) (ls : list A) : list A :=
match ls with
| [] => []
| x :: ls' => insert le x (insertion_sort le ls')
end.
(* To help us state our main theorem, we define sortedness. *)
Fixpoint sorted {A} (le : A -> A -> bool) (ls : list A) : bool :=
match ls with
| [] => true
| x1 :: ls' =>
match ls' with
| x2 :: _ => le x1 x2 && sorted le ls'
| [] => true
end
end.
(* [insert] preserves sortedness. Note the crucial hypothesis that comparator
* [le] is *total*: any two elements are related by it, in one order or the
* other. *)
Lemma insert_sorted : forall {A} (le : A -> A -> bool) a,
(forall x y, le x y = false -> le y x = true)
-> forall ls, sorted le ls = true
-> sorted le (insert le a ls) = true.
Proof.
induct ls; simplify; trivial.
cases (le a a0); simplify.
rewrite Heq; simplify.
trivial.
cases ls; simplify.
rewrite H; trivial.
apply andb_true_iff in H0; propositional.
cases (le a a1); simplify.
apply andb_true_iff in H0; propositional.
rewrite H; trivial.
simplify.
rewrite H3, H4; trivial.
rewrite H1; simplify.
trivial.
Qed.
(* Main theorem: [insertion_sort] produces only sorted lists. *)
Theorem insertion_sort_sorted : forall {A} (le : A -> A -> bool),
(forall x y, le x y = false -> le y x = true)
-> forall ls,
sorted le (insertion_sort le ls) = true.
Proof.
induct ls; simplify; trivial.
apply insert_sorted; trivial.
Qed.
(* The other classic requirement of a sorting function is that it return a
* permutation of its input, but we will skip that element here, since it is
* orthogonal to practicing with higher-order functions. *)
(* Let's do a quick example of using [insertion_sort] with a concrete
* comparator. *)
Definition not_introduced_later (l1 l2 : programming_language) : bool :=
if AppearedInYear l1 <=? AppearedInYear l2 then true else false.
Compute insertion_sort
not_introduced_later
[gallina; pascal; clang; ocaml; haskell].
Corollary insertion_sort_languages : forall langs,
sorted not_introduced_later (insertion_sort not_introduced_later langs) = true.
Proof.
simplify.
apply insertion_sort_sorted.
unfold not_introduced_later.
simplify.
cases (AppearedInYear x <=? AppearedInYear y); try equality.
cases (AppearedInYear y <=? AppearedInYear x); try equality.
linear_arithmetic.
Qed.
(** * A language of functions and its interpreter *)
(* Let's now work through an example of a language and its interpreter.
* Specifically, we'll define a language of first-class functions and
* higher-order functions. It would be natural to make our language statically
* typed, but it turns out we need a bit more Coq sophistication to implement a
* proper interpreter for such an embedded language, which we'll postpone for
* module DependentInductiveTypes. Instead, here's a simple "universal type"
* along the lines of dynamically typed languages like Python. *)
Inductive dyn :=
| Bool (b : bool)
| Number (n : nat)
| List (ds : list dyn).
(* Next, we implement dynamic versions of a few handy library functions.
* Notice that they have arbitrary default behavior when passed improperly typed
* arguments. *)
Definition dmap (f : dyn -> dyn) (x : dyn) : dyn :=
match x with
| List ds => List (map f ds)
| _ => x
end.
Definition dfilter (f : dyn -> dyn) (x : dyn) : dyn :=
match x with
List ds => List (filter (fun arg => match f arg with
| Bool b => b
| _ => false
end) ds)
| _ => x
end.
Definition disZero (x : dyn) : dyn :=
match x with
| Number 0 => Bool true
| Number _ => Bool false
| _ => x
end.
Definition dnot (x : dyn) : dyn :=
match x with
| Bool b => Bool (negb b)
| x => x
end.
(* Here's our syntax-tree type for functions (transformations). *)
Inductive xform :=
| Identity
| Compose (xf1 xf2 : xform)
| Map (xf1 : xform)
| Filter (xf1 : xform)
| ConstantBool (b : bool)
| ConstantNumber (n : nat)
| IsZero
| Not.
(* And here's our simple interpreter. *)
Fixpoint transform (xf : xform) : dyn -> dyn :=
match xf with
| Identity => id (* from the Coq standard library *)
| Compose f1 f2 => compose (transform f1) (transform f2)
(* ditto for [compose] *)
| Map f => dmap (transform f)
| Filter f => dfilter (transform f)
| ConstantBool b => fun _ => Bool b
| ConstantNumber n => fun _ => Number n
| IsZero => disZero
| Not => dnot
end.
Compute transform (Map IsZero) (List [Number 0; Number 1; Number 2; Number 0; Number 3]).
Compute transform (Filter IsZero) (List [Number 0; Number 1; Number 2; Number 0; Number 3]).
(* Here's a grab bag of optimizations of our programs. *)
Fixpoint optimize (xf : xform) : xform :=
match xf with
| Compose xf1 xf2 =>
match optimize xf1, optimize xf2 with
| Identity, xf2' => xf2'
| xf1', Identity => xf1'
| Not, Not => Identity
| Map xf1', Map xf2' => Map (Compose xf1' xf2')
| Not, ConstantBool b => ConstantBool (negb b)
| IsZero, ConstantNumber 0 => ConstantBool true
| IsZero, ConstantNumber (S _) => ConstantBool false
| xf1', xf2' => Compose xf1' xf2'
end
| Map xf1 =>
match optimize xf1 with
| Identity => Identity
| xf1' => Map xf1'
end
| Filter xf1 =>
match optimize xf1 with
| ConstantBool true => Identity
| xf1' => Filter xf1'
end
| _ => xf
end.
(* This tactic turns out to work well to prove our optimizations correct. We'll
* have to wait for module IntroToProofScripting to understand better what is
* going on. *)
Ltac optimize_ok :=
simplify; unfold compose, dmap in *;
repeat match goal with
| [ H : forall x : dyn, _ = _ |- _ ] => rewrite H
end;
repeat match goal with
| [ H : forall x : dyn, _ = _ |- _ ] => rewrite <- H
end; auto.
(* Now, a few useful alegbraic properties of our wrapper functions. *)
Lemma dnot_dnot : forall d, dnot (dnot d) = d.
Proof.
induct d; simplify; trivial.
SearchRewrite (negb (negb _)).
rewrite negb_involutive.
trivial.
Qed.
Hint Rewrite dnot_dnot.
Lemma map_identity : forall A (f : A -> A) (ls : list A),
(forall x, x = f x)
-> map f ls = ls.
Proof.
induct ls; simplify; equality.
Qed.
Hint Rewrite map_identity map_map using assumption.
Lemma map_same : forall A B (f1 f2 : A -> B) ls,
(forall x, f1 x = f2 x)
-> map f1 ls = map f2 ls.
Proof.
induct ls; simplify; equality.
Qed.
Lemma List_map_same : forall A (f1 f2 : A -> dyn) ls,
(forall x, f1 x = f2 x)
-> List (map f1 ls) = List (map f2 ls).
Proof.
simplify.
f_equal.
apply map_same; assumption.
Qed.
Lemma filter_same : forall A (f1 f2 : A -> bool) ls,
(forall x, f1 x = f2 x)
-> filter f1 ls = filter f2 ls.
Proof.
induct ls; simplify; try equality.
rewrite H.
cases (f2 a); simplify; equality.
Qed.
Lemma List_filter_same : forall (f1 f2 : dyn -> bool) ls,
(forall x, f1 x = f2 x)
-> List (filter f1 ls) = List (filter f2 ls).
Proof.
simplify.
f_equal.
apply filter_same; assumption.
Qed.
Hint Resolve List_map_same List_filter_same : core.
Hint Extern 5 (_ = match _ with Bool _ => _ | _ => _ end) =>
match goal with
| [ H : forall x : dyn, _ |- _ ] => rewrite <- H
end : core.
Lemma filter_ident : forall A (f : A -> bool) ls,
(forall x, f x = true)
-> filter f ls = ls.
Proof.
induct ls; simplify; try equality.
rewrite H.
equality.
Qed.
Theorem optimize_ok : forall xf x, transform (optimize xf) x = transform xf x.
Proof.
induct xf; simplify; try equality.
{
cases (optimize xf1); optimize_ok;
(cases (optimize xf2); optimize_ok).
cases x; simplify; trivial.
cases n; trivial.
}
{
cases (optimize xf); optimize_ok;
(cases x; optimize_ok).
}
{
cases (optimize xf); optimize_ok;
(cases x; optimize_ok);
repeat match goal with
| [ |- context[match ?E with _ => _ end] ] => cases E; simplify; trivial
end; auto.
rewrite filter_ident; trivial.
intro.
rewrite <- IHxf.
trivial.
}
Qed.
(** ** Are these really optimizations? Can they ever grow a term's size? *)
Fixpoint size (xf : xform) : nat :=
match xf with
| Identity
| Not
| IsZero
| ConstantBool _
| ConstantNumber _ => 1
| Compose xf1 xf2 => 1 + size xf1 + size xf2
| Map xf
| Filter xf => 1 + size xf
end.
(* Answer: no! *)
Theorem optimize_optimizes : forall xf, size (optimize xf) <= size xf.
Proof.
induct xf; simplify; try linear_arithmetic;
repeat match goal with
| [ |- context[match ?E with _ => _ end] ] =>
cases E; simplify; try linear_arithmetic
end.
Qed.
(** ** More interestingly, the same is true of the action of these
transformations on concrete values! *)
Fixpoint sum (ls : list nat) : nat :=
match ls with
| nil => 0
| x :: ls' => x + sum ls'
end.
Fixpoint dsize (d : dyn) : nat :=
match d with
| Bool _
| Number _ => 1
| List ds => 1 + sum (map dsize ds)
end.
(* Some lemmas first, and then the main theorem result *)
Lemma dsize_positive : forall d, 1 <= dsize d.
Proof.
induct d; simplify; linear_arithmetic.
Qed.
Hint Immediate dsize_positive : core.
Lemma sum_map_monotone : forall A (f1 f2 : A -> nat) ds,
(forall x, f1 x <= f2 x)
-> sum (map f1 ds) <= sum (map f2 ds).
Proof.
induct ds; simplify; try linear_arithmetic; propositional.
specialize (H a).
linear_arithmetic.
Qed.
Lemma neverGrow_filter : forall f ds,
sum (map dsize (filter f ds)) <= sum (map dsize ds).
Proof.
induct ds; simplify; try linear_arithmetic.
cases (f a); simplify; linear_arithmetic.
Qed.
Theorem neverGrow : forall xf x,
dsize (transform xf x) <= dsize x.
Proof.
induct xf; simplify; try linear_arithmetic.
unfold id.
trivial.
unfold compose.
eauto using le_trans.
unfold dmap.
cases x; trivial.
simplify.
Search (S _ <= S _).
apply le_n_S.
apply sum_map_monotone.
trivial.
unfold dfilter.
cases x; trivial.
simplify.
apply le_n_S.
apply neverGrow_filter.
apply dsize_positive.
apply dsize_positive.
unfold disZero.
cases x; trivial.
cases n; trivial.
unfold dnot.
cases x; trivial.
Qed.
(** * Combinators for syntax-tree transformations *)
(* Let's reprise the imperative language from the end of Interpreters. *)
Inductive arith : Set :=
| Const (n : nat)
| Var (x : var)
| Plus (e1 e2 : arith)
| Minus (e1 e2 : arith)
| Times (e1 e2 : arith).
Definition valuation := fmap var nat.
Fixpoint interp (e : arith) (v : valuation) : nat :=
match e with
| Const n => n
| Var x =>
match v $? x with
| None => 0
| Some n => n
end
| Plus e1 e2 => interp e1 v + interp e2 v
| Minus e1 e2 => interp e1 v - interp e2 v
| Times e1 e2 => interp e1 v * interp e2 v
end.
Inductive cmd :=
| Skip
| Assign (x : var) (e : arith)
| Sequence (c1 c2 : cmd)
| Repeat (e : arith) (body : cmd).
Fixpoint selfCompose {A} (f : A -> A) (n : nat) : A -> A :=
match n with
| O => fun x => x
| S n' => fun x => selfCompose f n' (f x)
end.
Lemma selfCompose_extensional : forall {A} (f g : A -> A) n x,
(forall y, f y = g y)
-> selfCompose f n x = selfCompose g n x.
Proof.
induct n; simplify; try equality.
rewrite H.
apply IHn.
trivial.
Qed.
Fixpoint exec (c : cmd) (v : valuation) : valuation :=
match c with
| Skip => v
| Assign x e => v $+ (x, interp e v)
| Sequence c1 c2 => exec c2 (exec c1 v)
| Repeat e body => selfCompose (exec body) (interp e v) v
end.
Fixpoint seqself (c : cmd) (n : nat) : cmd :=
match n with
| O => Skip
| S n' => Sequence c (seqself c n')
end.
(* Now consider a more abstract way of describing optimizations concisely.
* We package tree-rewriting functions of two kinds, in records. *)
Record rule := {
OnCommand : cmd -> cmd;
OnExpression : arith -> arith
}.
(* Such a strategy can be applied *bottom-up* in a syntax tree. *)
Fixpoint bottomUp (r : rule) (c : cmd) : cmd :=
match c with
| Skip => r.(OnCommand) Skip
| Assign x e => r.(OnCommand) (Assign x (r.(OnExpression) e))
| Sequence c1 c2 => r.(OnCommand) (Sequence (bottomUp r c1) (bottomUp r c2))
| Repeat e body => r.(OnCommand) (Repeat (r.(OnExpression) e) (bottomUp r body))
end.
(* Here are a few handy *combinators* for building [rule]s. *)
Definition crule (f : cmd -> cmd) : rule :=
{| OnCommand := f; OnExpression := fun e => e |}.
Definition erule (f : arith -> arith) : rule :=
{| OnCommand := fun c => c; OnExpression := f |}.
Definition andThen (r1 r2 : rule) : rule :=
{| OnCommand := compose r2.(OnCommand) r1.(OnCommand);
OnExpression := compose r2.(OnExpression) r1.(OnExpression) |}.
(* Two basic examples of rules *)
Definition plus0 := erule (fun e =>
match e with
| Plus e' (Const 0) => e'
| _ => e
end).
Definition unrollLoops := crule (fun c =>
match c with
| Repeat (Const n) body => seqself body n
| _ => c
end).
(* Let's see what effects they have on simple examples. *)
Compute bottomUp plus0
(Sequence (Assign "x" (Plus (Var "x") (Const 0)))
(Assign "y" (Var "x"))).
Compute bottomUp unrollLoops
(Repeat (Plus (Const 2) (Const 0))
(Sequence (Assign "x" (Plus (Var "x") (Const 0)))
(Assign "y" (Var "x")))).
Compute bottomUp (andThen plus0 unrollLoops)
(Repeat (Plus (Const 2) (Const 0))
(Sequence (Assign "x" (Plus (Var "x") (Const 0)))
(Assign "y" (Var "x")))).
(* Here is a good semantic correctness notion for rules. *)
Definition correct (r : rule) :=
(forall c v, exec (r.(OnCommand) c) v = exec c v)
/\ (forall e v, interp (r.(OnExpression) e) v = interp e v).
(* Some theorems for proving correctness of our combinators *)
Theorem crule_correct : forall f,
(forall c v, exec (f c) v = exec c v)
-> correct (crule f).
Proof.
first_order.
Qed.
Theorem erule_correct : forall f,
(forall e v, interp (f e) v = interp e v)
-> correct (erule f).
Proof.
first_order.
Qed.
Theorem andThen_correct : forall r1 r2,
correct r1
-> correct r2
-> correct (andThen r1 r2).
Proof.
unfold andThen; first_order; simplify; eauto using eq_trans.
Qed.
(* A bottom-up traversal with a correct rule is also correct. *)
Theorem bottomUp_correct : forall r,
correct r
-> forall c v, exec (bottomUp r c) v = exec c v.
Proof.
unfold correct; induct c; simplify; propositional.
rewrite H0.
trivial.
rewrite H0.
simplify.
equality.
rewrite H0.
simplify.
equality.
rewrite H0.
simplify.
rewrite H1.
apply selfCompose_extensional.
trivial.
Qed.
(* A twist: we can also package bottom-up traversal as a rule in its own right,
* which can then be used in other bottom-up traversals! *)
Definition rBottomUp (r : rule) : rule :=
{| OnCommand := bottomUp r;
OnExpression := r.(OnExpression) |}.
Theorem rBottomUp_correct : forall r,
correct r
-> correct (rBottomUp r).
Proof.
unfold correct; simplify; propositional.
apply bottomUp_correct.
unfold correct; propositional.
Qed.
(* This example demonstrates how this kind of nested traversal can find
* additional optimizations. Watch as the program shrinks while we ratchet up
* the level of nesting. *)
Definition zzz := Assign "x" (Plus (Plus (Plus (Var "x") (Const 0)) (Const 0)) (Const 0)).
Compute bottomUp plus0 zzz.
Compute bottomUp (rBottomUp plus0) zzz.
Compute bottomUp (rBottomUp (rBottomUp plus0)) zzz.
Compute bottomUp (rBottomUp (rBottomUp (rBottomUp plus0))) zzz.
(** * Motivating continuations with search problems *)
(* We're getting into advanced material here, so it may often make sense to stop
* before this point in a class presentation. But, if you want to learn about
* one of the classic cool features of functional programming.... *)
(* One fascinating flavor of first-class functions is *continuations*, which are
* essentially functions that are meant to be called on the *results* of other
* functions. To motivate the idea, let's first develop a somewhat slow
* function. We'll switch to a continuation-based version to see the
* benefit. *)
(* Here's a simple way to compute all lists that can be formed by dropping zero
* or more elements out of some original list. *)
Fixpoint allSublists {A} (ls : list A) : list (list A) :=
match ls with
| [] => [[]]
| x :: ls' =>
let lss := allSublists ls' in
lss ++ map (fun ls'' => x :: ls'') lss
end.
Compute allSublists [1; 2; 3].
(* This is the main function we want to define. It looks for a sublist whose
* sum matches some target. *)
Fixpoint sublistSummingTo (ns : list nat) (target : nat) : option (list nat) :=
match filter (fun ns' => if sum ns' ==n target then true else false) (allSublists ns) with
| ns' :: _ => Some ns'
| [] => None
end.
Compute sublistSummingTo [1; 2; 3] 6.
Compute sublistSummingTo [1; 2; 3] 5.
Compute sublistSummingTo [1; 2; 3] 7.
(* This function will be handy to generate some test cases. *)
Fixpoint countingDown (from : nat) :=
match from with
| O => []
| S from' => from' :: countingDown from'
end.
Compute countingDown 10.
(* This one is pretty slow! There are quite a few sublists of
* [countingDown 18], you know. *)
Time Compute sublistSummingTo (countingDown 18) 1.
(* Can we set things up so that we can avoid generating *all* sublists, instead
* checking each one for the right sum, as it is generated? And can we do it in
* a *generic* way, where we still have sublists calculation that isn't
* specialized to any particular acceptance condition? Continuations provide a
* nice ingredient! *)
(* This variant of [allSublists] takes a while to digest. Both of the new
* arguments are continuations. *)
Fixpoint allSublistsK {A R} (ls : list A)
(* First, notice new type parameter [R], for "result."
* The function will return a value of this type. *)
(failed : unit -> R)
(* If no acceptable sublist is found, return the result of calling this
* function. [unit] is the degenerate standard-library type inhabited
* only by [tt]. *)
(found : list A -> (unit -> R) -> R)
(* Whenever an acceptable sublist is found, return the result of
* calling this function on it. The 2nd argument is a failure
* continuation, just like our own [failed]. That is, when [found]
* "doesn't like" the list, it returns the result of calling the
* function we pass to it. See below for why this is a perfect
* plumbing strategy. *)
: R :=
match ls with
| [] =>
found [] failed
(* [ls] is empty? Then the only sublist is [[]], which we should send to
* our success continuation [found] for vetting. *)
| x :: ls' =>
(* [ls] is nonempty? Let's proceed to finding all sublists of [ls']. *)
allSublistsK ls'
failed
(* Any failure here bubbles up to a failure in the original
* call. *)
(fun sol failed' =>
(* Any success here should first be passed on to the
* original success continuation [found]. *)
found sol (fun _ =>
(* However, if [found] doesn't like [sol], then
* maybe it likes [x :: sol]! Note how we
* customize the failure continuation passed to
* [found], to implement a kind of backtracking
* search, interleaved with generation of
* candidates. *)
found (x :: sol) failed'))
end.
(* Now it is easy to define a variant of [sublistSummingTo], where result type
* [R] gets instantiated as [option (list nat)]. *)
Definition sublistSummingToK (ns : list nat) (target : nat) : option (list nat) :=
allSublistsK ns
(fun _ => None)
(* Failure continuation: return None. *)
(fun sol failed =>
if sum sol ==n target then Some sol else failed tt)
(* Success continuation: check if sum is right, if so returning
* [Some]. *).
Time Compute sublistSummingToK (countingDown 18) 1.
(* Significantly faster now! We avoid materializing the full list of sublists,
* before starting to filter them. We will return below to proof of this
* function, which is irksomely involved. *)
(** * The classics in continuation-passing style *)
(* We can rewrite the classic list higher-order functions in
* *continuation-passing style*, where they return answers by calling
* continuations rather than just returning normally. This style might be
* familiar from, e.g., how *asynchronous programming* works in JavaScript. *)
(* Notice how, not only does [mapK] have a CPS (continuation-passing style)
* type, but its function argument also has a CPS type. *)
Fixpoint mapK {A B R} (f : A -> (B -> R) -> R) (ls : list A) (k : list B -> R) : R :=
match ls with
| nil => k nil
| x :: ls' => f x (fun x' => mapK f ls' (fun ls'' => k (x' :: ls'')))
end.
Fixpoint filterK {A R} (f : A -> (bool -> R) -> R) (ls : list A) (k : list A -> R) : R :=
match ls with
| nil => k nil
| x :: ls' => f x (fun b => filterK f ls' (fun ls'' => k (if b then x :: ls'' else ls'')))
end.
Fixpoint fold_leftK {A B R} (f : B -> A -> (B -> R) -> R) (ls : list A) (acc : B) (k : B -> R) : R :=
match ls with
| nil => k acc
| x :: ls' => f acc x (fun x' => fold_leftK f ls' x' k)
end.
(* And CPS versions of the additional functions used in our examples earlier *)
Definition NameK {R} (l : programming_language) (k : string -> R) : R :=
k (Name l).
Definition PurelyFunctionalK {R} (l : programming_language) (k : bool -> R) : R :=
k (PurelyFunctional l).
Definition AppearedInYearK {R} (l : programming_language) (k : nat -> R) : R :=
k (AppearedInYear l).
Definition maxK {R} (n1 n2 : nat) (k : nat -> R) : R :=
k (max n1 n2).
(* The examples from before give the same answers, when suitably translated. *)
Compute mapK NameK languages (fun ls => ls).
Compute filterK PurelyFunctionalK languages (fun ls => mapK NameK ls (fun x => x)).
Compute mapK AppearedInYearK languages (fun ls => fold_leftK maxK ls 0 (fun x => x)).
Compute filterK PurelyFunctionalK languages
(fun ls1 => mapK AppearedInYearK ls1
(fun ls2 => fold_leftK maxK ls2 0 (fun x => x))).
(* We can prove that each such example always gives correct answers, for any
* list of languages. *)
Theorem mapK_ok : forall {A B R} (f : A -> (B -> R) -> R) (f_base : A -> B),
(forall x k, f x k = k (f_base x))
-> forall (ls : list A) (k : list B -> R),
mapK f ls k = k (map f_base ls).
Proof.
induct ls; simplify; try equality.
rewrite H.
apply IHls.
Qed.
Theorem names_ok : forall langs,
mapK NameK langs (fun ls => ls) = map Name langs.
Proof.
simplify.
apply mapK_ok with (f_base := Name).
unfold NameK.
trivial.
Qed.
Theorem filterK_ok : forall {A R} (f : A -> (bool -> R) -> R) (f_base : A -> bool),
(forall x k, f x k = k (f_base x))
-> forall (ls : list A) (k : list A -> R),
filterK f ls k = k (filter f_base ls).
Proof.
induct ls; simplify; try equality.
rewrite H.
apply IHls.
Qed.
Theorem purenames_ok : forall langs,
filterK PurelyFunctionalK langs (fun ls => mapK NameK ls (fun x => x))
= map Name (filter PurelyFunctional langs).
Proof.
simplify.
rewrite filterK_ok with (f_base := PurelyFunctional); trivial.
apply mapK_ok with (f_base := Name); trivial.
Qed.
Theorem fold_leftK_ok : forall {A B R} (f : B -> A -> (B -> R) -> R) (f_base : B -> A -> B),
(forall x acc k, f x acc k = k (f_base x acc))
-> forall (ls : list A) (acc : B) (k : B -> R),
fold_leftK f ls acc k = k (fold_left f_base ls acc).
Proof.
induct ls; simplify; try equality.
rewrite H.
apply IHls.
Qed.
Theorem latest_ok : forall langs,
mapK AppearedInYearK langs (fun ls => fold_leftK maxK ls 0 (fun x => x))
= fold_left max (map AppearedInYear langs) 0.
Proof.
simplify.
rewrite mapK_ok with (f_base := AppearedInYear); trivial.
apply fold_leftK_ok with (f_base := max); trivial.
Qed.
Theorem latestpure_ok : forall langs,
filterK PurelyFunctionalK langs
(fun ls1 => mapK AppearedInYearK ls1
(fun ls2 => fold_leftK maxK ls2 0 (fun x => x)))
= fold_left max (map AppearedInYear (filter PurelyFunctional langs)) 0.
Proof.
simplify.
rewrite filterK_ok with (f_base := PurelyFunctional); trivial.
rewrite mapK_ok with (f_base := AppearedInYear); trivial.
apply fold_leftK_ok with (f_base := max); trivial.
Qed.
(** * Tree traversals *)
(* Let's see how the way of continuations can guide us toward defining a tree
* traversal as a "loop" rather than a general recursive function. *)
(* Recall this type from last week. *)
Inductive tree {A} :=
| Leaf
| Node (l : tree) (d : A) (r : tree).
Arguments tree : clear implicits.
(* And here's an in-order traversal that we also already worked with. *)
Fixpoint flatten {A} (t : tree A) : list A :=
match t with
| Leaf => []
| Node l d r => flatten l ++ d :: flatten r
end.
(* This flattening does some wasteful extra copying-around of list elements,
* with all those [++] operations. We can surface the quadratic running time
* with large enough test cases. *)
Fixpoint big (n : nat) : tree nat :=
match n with
| O => Leaf
| S n' => Node (big n') n Leaf
end.
Compute big 3.
Time Compute length (flatten (big 5000)).
(* That one takes long enough to notice (and larger trees lead to stack
* overflows!). *)
(* Let's write a version that avoids repeated list concatenation, by maintaining
* an *accumulator*, where we "accumulate" the answer is reverse order. *)
Fixpoint flattenAcc {A} (t : tree A) (acc : list A) : list A :=
match t with
| Leaf => acc
| Node l d r => flattenAcc l (d :: flattenAcc r acc)
end.
(* It gives the same answer as the original. *)
Theorem flattenAcc_ok : forall {A} (t : tree A) acc,
flattenAcc t acc = flatten t ++ acc.
Proof.
induct t; simplify; try equality.
rewrite IHt1, IHt2.
rewrite <- app_assoc.
simplify.
equality.
Qed.
Time Compute length (flattenAcc (big 5000) []).
(* Much faster! *)
(* There is a generic transformation of any function into CPS. We won't spell
* the transformation out formally, but here's what it does for [flattenAcc]. *)
Fixpoint flattenK {A R} (t : tree A) (acc : list A) (k : list A -> R) : R :=
match t with
| Leaf => k acc
| Node l d r => flattenK r acc (fun acc' =>
flattenK l (d :: acc') k)
end.
(* Note how the first recursive call takes as an argument a continuation that
* makes a further recursive call. We have made all recursive calls into tail
* calls, which wasn't true in the original function. *)
(* This version is still correct. *)
Theorem flattenK_ok : forall {A R} (t : tree A) acc (k : list A -> R),
flattenK t acc k = k (flattenAcc t acc).
Proof.
induct t; simplify; try equality.
rewrite IHt2, IHt1.
equality.
Qed.
(* Continuations can feel something like magic. Let's concretize them by
* replacing them with a datatype that doesn't appeal to first-class functions.
* This kind of transformation is called *defunctionalization*, and it can also
* be done quite mechanically. *)
Inductive flatten_continuation {A} :=
| KDone
(* This is a base-case identity function, which we might use to kick off the
* recursion for [flattenK]. *)
| KMore (l : tree A) (d : A) (k : flatten_continuation)
(* For given arguments [l d k], this one corresponds to:
* [fun acc' => flattenK l (d :: acc') k] *).
Arguments flatten_continuation : clear implicits.
(* This function explains how to apply one of our defunctionalized continuations
* to an accumulator. We also need to take the new flattening function as an
* argument. *)
Definition apply_continuation {A} (acc : list A) (k : flatten_continuation A)
(flattenKD : tree A -> list A -> flatten_continuation A -> list A)
: list A :=
match k with
| KDone => acc
| KMore l d k' => flattenKD l (d :: acc) k'
(* Note how this case just copies back in the code that
* inspired our inclusion of the constructor [KMore]. *)
end.
(* Here's the overall function. Note a pesky element: we add an extra [nat]
* parameter of *fuel*, a count that goes down across recursive calls, just to
* convince Coq that our function terminates. Otherwise, the recursion
* structure is too intricate for Coq to make sense of. *)
Fixpoint flattenKD {A} (fuel : nat) (t : tree A) (acc : list A)
(k : flatten_continuation A) : list A :=
match fuel with
| O => []
| S fuel' =>
match t with
| Leaf => apply_continuation acc k (flattenKD fuel')
(* Note the partial
* application of [flattenKD]. --^ *)
| Node l d r => flattenKD fuel' r acc (KMore l d k)
end
end.
(* Now, again, all function calls are tail calls, but we also don't rely on
* first-class functions. *)
(* Next, to prove correctness, we will need good notions of sizes of things, to
* tell us how much fuel is needed. *)
(* A somewhat peculiar notion of size for trees. Why that 2 instead of 1?
* Because it lets the proof below work out! *)
Fixpoint tree_size {A} (t : tree A) : nat :=
match t with
| Leaf => 0
| Node l _ r => 2 + tree_size l + tree_size r
end.
Fixpoint continuation_size {A} (k : flatten_continuation A) : nat :=
match k with
| KDone => 0
| KMore l d k' => 1 + tree_size l + continuation_size k'
end.
(* A continuation encodes a flattening call, waiting to be run.
* We can go ahead and run all of it, using the original, simple [flatten]. *)
Fixpoint flatten_cont {A} (k : flatten_continuation A) : list A :=
match k with
| KDone => []
| KMore l d k' => flatten_cont k' ++ flatten l ++ [d]
end.
(* That operation turns out to be just what we need to state correctness.
* We also have to fiddle with fuel, effectively building in a kind of
* *strong induction* via the parameter [fuel], which bounds the actual fuel
* amount [fuel']. *)
Lemma flattenKD_ok' : forall {A} fuel fuel' (t : tree A) acc k,
tree_size t + continuation_size k < fuel' < fuel
-> flattenKD fuel' t acc k
= flatten_cont k ++ flatten t ++ acc.
Proof.
induct fuel; simplify; cases fuel'; simplify; try linear_arithmetic.
cases t; simplify; trivial.
cases k; simplify; trivial.
rewrite IHfuel; try linear_arithmetic.
repeat rewrite <- app_assoc.
simplify.
equality.
rewrite IHfuel.
simplify.
repeat rewrite <- app_assoc.
simplify.
equality.
simplify.
linear_arithmetic.
Qed.
(* A nice, simple final theorem can be stated, when we initialize fuel in the
* right way. *)
Theorem flattenKD_ok : forall {A} (t : tree A),
flattenKD (tree_size t + 1) t [] KDone = flatten t.
Proof.
simplify.
rewrite flattenKD_ok' with (fuel := tree_size t + 2).
simplify.
apply app_nil_r.
simplify.
linear_arithmetic.
Qed.
(* The author was once asked a programming interview question, of how to perform
* some tree traversal with a loop but not recursion. Our last step shows how
* to do that for flattening, just relying on explicit lists that effectively
* represent call stacks! Actually, such data have been implicit in our
* defunctionalized continuations. *)
(* Specifically, one of our continuations is really just a list of arguments to
* pending calls to [flatten]. *)
Definition call_stack A := list (tree A * A).
(* This analogue to [apply_continuation] explains how to "pop the current stack
* frame" and return to the most recent suspended call. *)
Definition pop_call_stack {A} (acc : list A) (st : call_stack A)
(flattenS : tree A -> list A -> call_stack A -> list A)
: list A :=
match st with
| [] => acc
| (l, d) :: st' => flattenS l (d :: acc) st'
end.
(* And here's the rewritten main function. *)
Fixpoint flattenS {A} (fuel : nat) (t : tree A) (acc : list A)
(st : call_stack A) : list A :=
match fuel with
| O => []
| S fuel' =>
match t with
| Leaf => pop_call_stack acc st (flattenS fuel')
| Node l d r => flattenS fuel' r acc ((l, d) :: st)
end
end.
(* To prove correctness, we will want a translation from the new kind of
* continuation to the old. *)
Fixpoint call_stack_to_continuation {A} (st : call_stack A) : flatten_continuation A :=
match st with
| [] => KDone
| (l, d) :: st' => KMore l d (call_stack_to_continuation st')
end.
Lemma flattenS_flattenKD : forall {A} fuel (t : tree A) acc st,
flattenS fuel t acc st = flattenKD fuel t acc (call_stack_to_continuation st).
Proof.
induct fuel; simplify; trivial.
cases t.
cases st; simplify; trivial.
cases p; simplify.
apply IHfuel.
apply IHfuel.
Qed.
Theorem flattenS_ok : forall {A} (t : tree A),
flattenS (tree_size t + 1) t [] [] = flatten t.
Proof.
simplify.
rewrite flattenS_flattenKD.
apply flattenKD_ok.
Qed.
(** * Proof of our motivating example *)
(* This theorem is quite intricate to get right. At this point in the class, it
* is not important to follow anything about this proof, really, but it's kinda
* cool, once digested. *)
Theorem allSublistsK_ok : forall {A B} (ls : list A) (failed : unit -> B) found,
(* First, we describe what makes for a legit [found] continuation. *)
(forall sol,
(* For any solution we might ask it about,
* either [found] is going to accept that solution,
* returning the same answer no matter which failure continuation we
* pass: *)
(exists ans, (forall failed', found sol failed' = ans)
/\ ans <> failed tt)
(* ...and, by the way, this answer is never the same as the failure
* value (or we could get confused in case analysis). *)
(* OR [found] is going to reject this solution, invoking its failure
* continuation: *)
\/ (forall failed', found sol failed' = failed' tt))
(* Then we conclude a rather similar property for [allSublistsK]. *)
->
(* Option 1: there is a correct answer [sol], for which [found] returns
* [ans]. *)
(exists sol ans, In sol (allSublists ls)
/\ (forall failed', found sol failed' = ans)
/\ allSublistsK ls failed found = ans
/\ ans <> failed tt)
(* Option 2: there is no correct answer. *)
\/ ((forall sol, In sol (allSublists ls)
-> forall failed', found sol failed' = failed' tt)
/\ allSublistsK ls failed found = failed tt).
Proof.
induct ls; simplify.
specialize (H []).
first_order.
right.
propositional.
subst.
trivial.
trivial.
assert (let found := (fun (sol : list A) (failed' : unit -> B) =>
found sol (fun _ : unit => found (a :: sol) failed')) in
(exists (sol : list A) (ans : B),
In sol (allSublists ls) /\
(forall failed' : unit -> B, found sol failed' = ans) /\
allSublistsK ls failed found = ans /\ ans <> failed tt) \/
(forall sol : list A,
In sol (allSublists ls) -> forall failed' : unit -> B, found sol failed' = failed' tt) /\
allSublistsK ls failed found = failed tt).
apply IHls.
first_order.
generalize (H sol).
first_order.
specialize (H (a :: sol)).
first_order.
left.
exists x; propositional.
rewrite H0.
trivial.
right.
simplify.
rewrite H0.
trivial.
clear IHls.
simplify.
first_order.
generalize (H x); first_order.
left; exists x, x1; propositional.
apply in_or_app; propositional.
specialize (H1 failed).
specialize (H4 (fun _ => found (a :: x) failed)).
equality.
left; exists (a :: x), x0; propositional.
apply in_or_app; right; apply in_map_iff.
first_order.
specialize (H1 failed').
rewrite H4 in H1.
trivial.
right; propositional.
apply in_app_or in H2; propositional.
generalize (H sol); first_order.
apply H0 with (failed' := failed') in H3.
rewrite H2 in H3.
equality.
apply in_map_iff in H3.
first_order.
subst.
generalize (H x); first_order.
apply H0 with (failed' := failed) in H3.
equality.
apply H0 with (failed' := failed') in H3.
rewrite H2 in H3; trivial.
Qed.
(* At least we can wrap it all up in a simple correctness theorem! *)
Theorem sublistSummingToK_ok : forall ns target,
match sublistSummingToK ns target with
| None => forall sol, In sol (allSublists ns) -> sum sol <> target
| Some sol => In sol (allSublists ns) /\ sum sol = target
end.
Proof.
simplify.
unfold sublistSummingToK.
pose proof (allSublistsK_ok ns (fun _ => None)
(fun sol failed => if sum sol ==n target then Some sol else failed tt)).
cases H.
simplify.
cases (sum sol ==n target).
left; exists (Some sol); equality.
propositional.
first_order.
specialize (H0 (fun _ => None)).
cases (sum x ==n target); try equality.
subst.
rewrite H1.
propositional.
first_order.
rewrite H0.
simplify.
apply H with (failed' := fun _ => None) in H1.
cases (sum sol ==n target); equality.
Qed.
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