DeepAndShallowEmbeddings: ran some code in OCaml

This commit is contained in:
Adam Chlipala 2016-04-10 13:48:58 -04:00
parent d5c82fa62e
commit 01d550e4b0
3 changed files with 259 additions and 0 deletions

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.gitignore vendored
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@ -15,3 +15,4 @@ Makefile.coq
*.vo *.vo
frap.tgz frap.tgz
.coq-native .coq-native
Deep.ml*

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DeepAndShallowEmbeddings.v Normal file
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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 11: Deep and Shallow Embeddings
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Require Import Frap.
(** * Shared notations and definitions *)
Notation "m $! k" := (match m $? k with Some n => n | None => O end) (at level 30).
Definition heap := fmap nat nat.
Definition assertion := heap -> Prop.
Hint Extern 1 (_ <= _) => linear_arithmetic.
Hint Extern 1 (@eq nat _ _) => linear_arithmetic.
Example h0 : heap := $0 $+ (0, 2) $+ (1, 1) $+ (2, 8) $+ (3, 6).
Hint Rewrite max_l max_r using linear_arithmetic.
(** * Shallow embedding of a language very similar to the one we used last chapter *)
Module Shallow.
Definition cmd result := heap -> heap * result.
Definition hoare_triple (P : assertion) {result} (c : cmd result) (Q : result -> assertion) :=
forall h, P h
-> let (h', r) := c h in
Q r h'.
Notation "{{ h ~> P }} c {{ r & h' ~> Q }}" :=
(hoare_triple (fun h => P) c (fun r h' => Q)) (at level 90, c at next level).
Theorem consequence : forall P {result} (c : cmd result) Q
(P' : assertion) (Q' : _ -> assertion),
hoare_triple P c Q
-> (forall h, P' h -> P h)
-> (forall r h, Q r h -> Q' r h)
-> hoare_triple P' c Q'.
Proof.
unfold hoare_triple; simplify.
specialize (H h).
specialize (H0 h).
cases (c h).
auto.
Qed.
Fixpoint array_max (i acc : nat) : cmd nat :=
fun h =>
match i with
| O => (h, acc)
| S i' =>
let h_i' := h $! i' in
array_max i' (max h_i' acc) h
end.
Lemma array_max_ok' : forall len i acc,
{{ h ~> forall j, i <= j < len -> h $! j <= acc }}
array_max i acc
{{ r&h ~> forall j, j < len -> h $! j <= r }}.
Proof.
induct i; unfold hoare_triple in *; simplify; propositional; auto.
specialize (IHi (max (h $! i) acc) h); propositional.
cases (array_max i (max (h $! i) acc)); simplify; propositional; subst.
apply IHi; auto.
simplify.
cases (j0 ==n i); subst; auto.
assert (h $! j0 <= acc) by auto.
linear_arithmetic.
Qed.
Theorem array_max_ok : forall len,
{{ _ ~> True }}
array_max len 0
{{ r&h ~> forall i, i < len -> h $! i <= r }}.
Proof.
simplify.
eapply consequence.
apply array_max_ok' with (len := len).
simplify.
linear_arithmetic.
auto.
Qed.
Example run_array_max0 : array_max 4 0 h0 = (h0, 8).
Proof.
unfold h0.
simplify.
reflexivity.
Qed.
Fixpoint increment_all (i : nat) : cmd unit :=
fun h =>
match i with
| O => (h, tt)
| S i' => increment_all i' (h $+ (i', S (h $! i')))
end.
Lemma increment_all_ok' : forall len h0 i,
{{ h ~> (forall j, j < i -> h $! j = h0 $! j)
/\ (forall j, i <= j < len -> h $! j = S (h0 $! j)) }}
increment_all i
{{ _&h ~> forall j, j < len -> h $! j = S (h0 $! j) }}.
Proof.
induct i; unfold hoare_triple in *; simplify; propositional; auto.
specialize (IHi (h $+ (i, S (h $! i)))); propositional.
cases (increment_all i (h $+ (i, S (h $! i)))); simplify; propositional; subst.
apply H; simplify; auto.
cases (j0 ==n i); subst; auto.
simplify; auto.
simplify; auto.
Qed.
Theorem increment_all_ok : forall len h0,
{{ h ~> h = h0 }}
increment_all len
{{ _&h ~> forall j, j < len -> h $! j = S (h0 $! j) }}.
Proof.
simplify.
eapply consequence.
apply increment_all_ok' with (len := len).
simplify; subst; propositional.
linear_arithmetic.
simplify.
auto.
Qed.
Example run_increment_all0 : increment_all 4 h0 = ($0 $+ (0, 3) $+ (1, 2) $+ (2, 9) $+ (3, 7), tt).
Proof.
unfold h0.
simplify.
f_equal.
maps_equal.
Qed.
End Shallow.
(** * A basic deep embedding *)
Module Deep.
Inductive cmd : Type -> Type :=
| Return {result} (r : result) : cmd result
| Bind {result result'} (c1 : cmd result') (c2 : result' -> cmd result) : cmd result
| Read (a : nat) : cmd nat
| Write (a v : nat) : cmd unit.
Notation "x <- c1 ; c2" := (Bind c1 (fun x => c2)) (right associativity, at level 80).
Fixpoint array_max (i acc : nat) : cmd nat :=
match i with
| O => Return acc
| S i' =>
h_i' <- Read i';
array_max i' (max h_i' acc)
end.
Fixpoint increment_all (i : nat) : cmd unit :=
match i with
| O => Return tt
| S i' =>
v <- Read i';
_ <- Write i' (S v);
increment_all i'
end.
Fixpoint interp {result} (c : cmd result) (h : heap) : heap * result :=
match c with
| Return r => (h, r)
| Bind c1 c2 =>
let (h', r) := interp c1 h in
interp (c2 r) h'
| Read a => (h, h $! a)
| Write a v => (h $+ (a, v), tt)
end.
Example run_array_max0 : interp (array_max 4 0) h0 = (h0, 8).
Proof.
unfold h0.
simplify.
reflexivity.
Qed.
Example run_increment_all0 : interp (increment_all 4) h0 = ($0 $+ (0, 3) $+ (1, 2) $+ (2, 9) $+ (3, 7), tt).
Proof.
unfold h0.
simplify.
f_equal.
maps_equal.
Qed.
Inductive hoare_triple : assertion -> forall {result}, cmd result -> (result -> assertion) -> Prop :=
| HtReturn : forall P {result} (v : result),
hoare_triple P (Return v) (fun r h => P h /\ r = v)
| HtBind : forall P {result' result} (c1 : cmd result') (c2 : result' -> cmd result) Q R,
hoare_triple P c1 Q
-> (forall r, hoare_triple (Q r) (c2 r) R)
-> hoare_triple P (Bind c1 c2) R
| HtRead : forall P a,
hoare_triple P (Read a) (fun r h => P h /\ r = h $! a)
| HtWrite : forall P a v,
hoare_triple P (Write a v) (fun _ h => exists h', P h' /\ h = h' $+ (a, v))
| HtConsequence : forall {result} (c : cmd result) P Q (P' : assertion) (Q' : _ -> assertion),
hoare_triple P c Q
-> (forall h, P' h -> P h)
-> (forall r h, Q r h -> Q' r h)
-> hoare_triple P' c Q'.
Theorem hoare_triple_sound : forall P {result} (c : cmd result) Q,
hoare_triple P c Q
-> forall h, P h
-> let (h', r) := interp c h in
Q r h'.
Proof.
induct 1; simplify; propositional; eauto.
specialize (IHhoare_triple h).
cases (interp c1 h).
apply H1; eauto.
specialize (IHhoare_triple h).
cases (interp c h).
eauto.
Qed.
Extraction "Deep.ml" array_max increment_all.
End Deep.

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let rec i2n n =
match n with
| 0 -> O
| _ -> S (i2n (n - 1))
let interp c =
let h : (nat, nat) Hashtbl.t = Hashtbl.create 0 in
Hashtbl.add h (i2n 0) (i2n 2);
Hashtbl.add h (i2n 1) (i2n 1);
Hashtbl.add h (i2n 2) (i2n 8);
Hashtbl.add h (i2n 3) (i2n 6);
let rec interp' (c : 'a cmd) : 'a =
match c with
| Return v -> v
| Bind (c1, c2) -> interp' (c2 (interp' c1))
| Read a ->
Obj.magic (try
Hashtbl.find h a
with Not_found -> O)
| Write (a, v) -> Obj.magic (Hashtbl.replace h a v)
in let v = interp' c in
h, v