# ExtractRunning Coq Programs in OCaml

*Logical Foundations*has a simple example of Coq's program extraction features, but it's not required reading. This chapter starts from scratch and goes deeper.

Fixpoint ins (i : nat) (l : list nat) :=

match l with

| [] ⇒ [i]

| h :: t ⇒ if i <=? h then i :: h :: t else h :: ins i t

end.

Fixpoint sort (l : list nat) : list nat :=

match l with

| [] ⇒ []

| h :: t ⇒ ins h (sort t)

end.

The Extraction command prints out a function as OCaml code.

Extraction sort.

You can see the translation of sort from Coq to OCaml in
your IDE. Examine it there, and notice the similarities and
differences. To get the whole program, we need Recursive
Extraction:

Recursive Extraction sort.

The first thing you see there is a redefinition of the bool type.
But OCaml already has a bool type whose inductive structure is
isomorphic. We want our extracted functions to be compatible
with, i.e. callable by, ordinary OCaml code. So we want to use
OCaml's standard definition of bool in place of Coq's inductive
definition, bool. You'll notice the same issue with lists. The
following directive causes Coq to use OCaml's definitions of bool
and list in the extracted code:

Extract Inductive bool ⇒ "bool" [ "true" "false" ].

Extract Inductive list ⇒ "list" [ "[]" "(::)" ].

Recursive Extraction sort.

But the program still uses a unary representation of natural
numbers: the number 7 is really (S (S (S (S (S (S (S O))))))),
which in OCaml will be a data structure that's seven pointers
deep. The leb function takes linear time, proportional to the
difference in value between n and m.
We could instead use Coq's Z, which is a binary representation
of integers. But that is logarithmic-time, not constant.

Require Import ZArith.

Open Scope Z_scope.

Fixpoint insertZ (i : Z) (l : list Z) :=

match l with

| [] ⇒ [i]

| h :: t ⇒ if i <=? h then i :: h :: t else h :: insertZ i t

end.

Fixpoint sortZ (l : list Z) : list Z :=

match l with

| [] ⇒ []

| h :: t ⇒ insertZ h (sortZ t)

end.

Recursive Extraction sortZ.

Of course, for that extraction to be meaningful, we would need
to prove that sortZ is a sorting algorithm.
Other alternatives include:

- Extract nat directly to OCaml int. But int is finite (2^63
in modern implementations), so there are theorems we could prove
in Coq that wouldn't hold in OCaml.
- Use Coq's Int63, which faithfully models 63-bit cyclic
arithmetic, and extract directly to OCaml int. But that's
painful.
- Define and axiomatize our own lightweight abstract type of naturals, but extract it to OCaml int. But, this is dangerous! If our axioms are inconsistent, we can prove anything at all. If they are not faithful to OCaml, our proofs will be meaningless.

# Lightweight Extraction to int

We'll abstract OCaml int to Coq Z. Every int does have a
representation as a Z, though the other direction cannot
hold.

Nothing else is known so far about int. Let's add a less-than
operators, which are extracted to OCaml's:

Parameter ltb: int → int → bool.

Extract Inlined Constant ltb ⇒ "(<)".

Axiom ltb_lt : ∀ (n m : int), ltb n m = true ↔ Abs n < Abs m.

Parameter leb: int → int → bool.

Extract Inlined Constant leb ⇒ "(<=)".

Axiom leb_le : ∀ (n m : int), leb n m = true ↔ Abs n ≤ Abs m.

Those axioms are sound: OCaml's < and ≤ are consistent with
Coq's on any int. Note that we do not give extraction directives
for Abs, ltb_lt, or leb_le. They will not appear in
programs, only in proofs --which are not meant to be extracted.
You could imagine doing the same thing we just did with (+), but
that would be wrong:

Parameter ocaml_plus : int → int → int.

Extract Inlined Constant ocaml_plus ⇒ "(+)".

Axiom ocaml_plus_plus: ∀ a b c: int,

ocaml_plus a b = c ↔ Abs a + Abs b = Abs c.
The first two lines are OK: there really is a + function in
OCaml, and its type really is int → int → int.
But ocaml_plus_plus is unsound. From it, you could prove,

Abs max_int + Abs max_int = Abs (ocaml_plus max_int max_int)
which is not true in OCaml because of overflow.
In Perm we proved several theorems showing that Boolean
operators were reflected in propositions. Below, we do that
for int and Z comparisons.

Parameter ocaml_plus : int → int → int.

Extract Inlined Constant ocaml_plus ⇒ "(+)".

Axiom ocaml_plus_plus: ∀ a b c: int,

ocaml_plus a b = c ↔ Abs a + Abs b = Abs c.

Abs max_int + Abs max_int = Abs (ocaml_plus max_int max_int)

Lemma int_ltb_reflect : ∀ x y, reflect (Abs x < Abs y) (ltb x y).

Proof.

intros x y.

apply iff_reflect. symmetry. apply ltb_lt.

Qed.

Lemma int_leb_reflect : ∀ x y, reflect (Abs x ≤ Abs y) (leb x y).

Proof.

intros x y.

apply iff_reflect. symmetry. apply leb_le.

Qed.

Lemma Z_eqb_reflect : ∀ x y, reflect (x = y) (Z.eqb x y).

Proof.

intros x y.

apply iff_reflect. symmetry. apply Z.eqb_eq.

Qed.

Lemma Z_ltb_reflect : ∀ x y, reflect (x < y) (Z.ltb x y).

Proof.

intros x y.

apply iff_reflect. symmetry. apply Z.ltb_lt.

Qed.

Lemma Z_leb_reflect : ∀ x y, reflect (x ≤ y) (Z.leb x y).

Proof.

intros x y.

apply iff_reflect. symmetry. apply Z.leb_le.

Qed.

Lemma Z_gtb_reflect : ∀ x y, reflect (x > y) (Z.gtb x y).

Proof.

intros x y.

apply iff_reflect. symmetry. rewrite Z.gtb_ltb. rewrite Z.gt_lt_iff. apply Z.ltb_lt.

Qed.

Lemma Z_geb_reflect : ∀ x y, reflect (x ≥ y) (Z.geb x y).

Proof.

intros x y.

apply iff_reflect. symmetry. rewrite Z.geb_leb. rewrite Z.ge_le_iff. apply Z.leb_le.

Qed.

Now we upgrade bdall to work with Z and int.

Hint Resolve

int_ltb_reflect int_leb_reflect

Z_eqb_reflect Z_ltb_reflect Z_leb_reflect Z_gtb_reflect Z_geb_reflect

: bdestruct.

Ltac bdestruct_guard:=

match goal with

| ⊢ context [ if Nat.eqb ?X ?Y then _ else _] ⇒ bdestruct (Nat.eqb X Y)

| ⊢ context [ if Nat.ltb ?X ?Y then _ else _] ⇒ bdestruct (Nat.ltb X Y)

| ⊢ context [ if Nat.leb ?X ?Y then _ else _] ⇒ bdestruct (Nat.leb X Y)

| ⊢ context [ if Z.eqb ?X ?Y then _ else _] ⇒ bdestruct (Z.eqb X Y)

| ⊢ context [ if Z.ltb ?X ?Y then _ else _] ⇒ bdestruct (Z.ltb X Y)

| ⊢ context [ if Z.leb ?X ?Y then _ else _] ⇒ bdestruct (Z.leb X Y)

| ⊢ context [ if Z.gtb ?X ?Y then _ else _] ⇒ bdestruct (Z.gtb X Y)

| ⊢ context [ if Z.geb ?X ?Y then _ else _] ⇒ bdestruct (Z.geb X Y)

| ⊢ context [ if ltb ?X ?Y then _ else _] ⇒ bdestruct (ltb X Y)

| ⊢ context [ if leb ?X ?Y then _ else _] ⇒ bdestruct (leb X Y)

end.

Ltac bdall :=

repeat (simpl; bdestruct_guard; try omega; auto).

Fixpoint ins_int (i : int) (l : list int) :=

match l with

| [] ⇒ [i]

| h :: t ⇒ if leb i h then i :: h :: t else h :: ins_int i t

end.

Fixpoint sort_int (l : list int) : list int :=

match l with

| [] ⇒ []

| h :: t ⇒ ins_int h (sort_int t)

end.

Recursive Extraction sort_int.

Again, for that extraction to be meaningful, we need to prove that
sort_int is a sorting algorithm. We can do that with the same
techniques we used in Sort. In particular, omega works
with Z, so we can enjoy automation without having to do any
unnecessary work axiomatizing and proving lemmas about int.

Inductive sorted : list int → Prop :=

| sorted_nil:

sorted []

| sorted_1: ∀ x,

sorted [x]

| sorted_cons: ∀ x y l,

Abs x ≤ Abs y → sorted (y :: l) → sorted (x :: y :: l).

Hint Constructors sorted.

#### Exercise: 3 stars, standard (sort_int_correct)

Theorem sort_int_correct : ∀ (al : list int),

Permutation al (sort_int al) ∧ sorted (sort_int al).

Proof.

(* FILL IN HERE *) Admitted.

☐

Definition key := int.

Inductive tree (V : Type) : Type :=

| E : tree V

| T : tree V → key → V → tree V → tree V.

Arguments E {V}.

Arguments T {V}.

Definition empty_tree {V : Type} : tree V := E.

Fixpoint lookup {V : Type} (default : V) (x : key) (t : tree V) : V :=

match t with

| E ⇒ default

| T l k v r ⇒ if ltb x k then lookup default x l

else if ltb k x then lookup default x r

else v

end.

Fixpoint insert {V : Type} (x : key) (v : V) (t : tree V) : tree V :=

match t with

| E ⇒ T E x v E

| T l y v' r ⇒ if ltb x y then T (insert x v l) y v' r

else if ltb y x then T l y v' (insert x v r)

else T l x v r

end.

Fixpoint elements_tr {V : Type}

(t : tree V) (acc : list (key × V)) : list (key × V) :=

match t with

| E ⇒ acc

| T l k v r ⇒ elements_tr l ((k, v) :: elements_tr r acc)

end.

Definition elements {V : Type} (t : tree V) : list (key × V) :=

elements_tr t [].

Theorem lookup_empty : ∀ (V : Type) (default : V) (k : key),

lookup default k empty_tree = default.

Proof. auto. Qed.

Theorem lookup_insert_eq :

∀ (V : Type) (default : V) (t : tree V) (k : key) (v : V),

lookup default k (insert k v t) = v.

Proof.

(* FILL IN HERE *) Admitted.

☐

∀ (V : Type) (default : V) (t : tree V) (k : key) (v : V),

lookup default k (insert k v t) = v.

Proof.

(* FILL IN HERE *) Admitted.

☐

Theorem lookup_insert_neq :

∀ (V : Type) (default : V) (t : tree V) (k k' : key) (v : V),

k ≠ k' → lookup default k' (insert k v t) = lookup default k' t.

Proof.

(* FILL IN HERE *) Admitted.

☐

∀ (V : Type) (default : V) (t : tree V) (k k' : key) (v : V),

k ≠ k' → lookup default k' (insert k v t) = lookup default k' t.

Proof.

(* FILL IN HERE *) Admitted.

☐

#### Exercise: 5 stars, standard, optional (int_elements)

Extract Inductive prod ⇒ "(*)" [ "(,)" ]. (* extract pairs natively *)

Recursive Extraction empty_tree insert lookup elements.

Extraction "searchtree.ml" empty_tree insert lookup elements.

Second, in the same directory as this file (Extract.v)
you will find the file test_searchtree.ml. You can
run it using the OCaml toplevel with these commands:
On a recent machine with a 2.9 GHz Intel Core i
That execution uses the bytecode interpreter. The native compiler
will have better performance:
On the same machine that prints,
Of course, the reason why the performance is so much worse with
consecutive integers is that BSTs exhibit worst-case performance
under that workload: linear time instead of logarithmic. We need
balanced search trees to achieve logarithmic. Redblack
will do that.

# #use "searchtree.ml";; # #use "test_searchtree.ml";;

_{9}that prints:Insert and lookup 1000000 random integers in .889566 seconds. Insert and lookup 20000 random integers in 0.009918 seconds. Insert and lookup 20000 consecutive integers in 2.777335 seconds.

$ ocamlopt -c searchtree.mli searchtree.ml $ ocamlopt searchtree.cmx -open Searchtree test_searchtree.ml -o test_searchtree $ ./test_searchtree

Insert and lookup 1000000 random integers in 0.488973 seconds. Insert and lookup 20000 random integers in 0.003237 seconds. Insert and lookup 20000 consecutive integers in 0.387535 seconds.

(* Mon May 11 15:03:16 EDT 2020 *)