DeepAndShallowEmbeddings: Deep

.gitignore

@@ -16,3 +16,4 @@ Makefile.coq
 frap.tgz
 .coq-native
 Deep.ml*
+Deeper.ml*

DeepAndShallowEmbeddings.v

@@ -5,6 +5,8 @@
 
 Require Import Frap.
 
+Set Implicit Arguments.
+
 (** * Shared notations and definitions *)
 
 Notation "m $! k" := (match m $? k with Some n => n | None => O end) (at level 30).
@@ -146,8 +148,8 @@ End Shallow.
 (** * A basic deep embedding *)
 
 Module Deep.
-  Inductive cmd : Type -> Type :=
-  | Return {result} (r : result) : cmd result
+  Inductive cmd : Set -> Type :=
+  | Return {result : Set} (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.
@@ -197,7 +199,7 @@ Module Deep.
   Qed.
 
   Inductive hoare_triple : assertion -> forall {result}, cmd result -> (result -> assertion) -> Prop :=
-  | HtReturn : forall P {result} (v : result),
+  | HtReturn : forall P {result : Set} (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
@@ -213,6 +215,111 @@ Module Deep.
       -> (forall r h, Q r h -> Q' r h)
       -> hoare_triple P' c Q'.
 
+  Lemma HtStrengthen : forall {result} (c : cmd result) P Q (Q' : _ -> assertion),
+      hoare_triple P c Q
+      -> (forall r h, Q r h -> Q' r h)
+      -> hoare_triple P c Q'.
+  Proof.
+    simplify.
+    eapply HtConsequence; eauto.
+  Qed.
+
+  Notation "{{ h ~> P }} c {{ r & h' ~> Q }}" :=
+    (hoare_triple (fun h => P) c (fun r h' => Q)) (at level 90, c at next level).
+
+  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; simplify.
+
+    eapply HtStrengthen.
+    econstructor.
+    simplify.
+    propositional.
+    subst.
+    auto.
+
+    econstructor.
+    constructor.
+    simplify.
+    eapply HtConsequence.
+    apply IHi.
+    simplify; propositional.
+    subst.
+    cases (j ==n i); subst; auto.
+    assert (h $! j <= acc) by auto.
+    linear_arithmetic.
+
+    simplify; auto.
+  Qed.
+
+  Theorem array_max_ok : forall len,
+    {{ _ ~> True }}
+      array_max len 0
+    {{ r&h ~> forall i, i < len -> h $! i <= r }}.
+  Proof.
+    simplify.
+    eapply HtConsequence.
+    apply array_max_ok' with (len := len).
+
+    simplify.
+    linear_arithmetic.
+
+    auto.
+  Qed.
+
+  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; simplify; propositional.
+
+    eapply HtStrengthen.
+    econstructor.
+    simplify.
+    propositional.
+    auto.
+
+    econstructor.
+    econstructor.
+    simplify.
+    econstructor.
+    econstructor.
+    simplify.
+    eapply HtConsequence.
+    apply IHi.
+    simplify.
+    invert H; propositional; subst.
+    simplify.
+    auto.
+
+    cases (j ==n i); subst; auto.
+    simplify; 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 HtConsequence.
+    apply increment_all_ok' with (len := len).
+
+    simplify; subst; propositional.
+    linear_arithmetic.
+
+    simplify.
+    auto.
+  Qed.
+
   Theorem hoare_triple_sound : forall P {result} (c : cmd result) Q,
       hoare_triple P c Q
       -> forall h, P h
@@ -232,3 +339,319 @@ Module Deep.
 
   Extraction "Deep.ml" array_max increment_all.
 End Deep.
+
+
+(** * A slightly fancier deep embedding, adding unbounded loops *)
+
+Module Deeper.
+  Inductive loop_outcome acc :=
+  | Done (a : acc)
+  | Again (a : acc).
+
+  Inductive cmd : Set -> Type :=
+  | Return {result : Set} (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
+  | Loop {acc : Set} (init : acc) (body : acc -> cmd (loop_outcome acc)) : cmd acc.
+
+  Notation "x <- c1 ; c2" := (Bind c1 (fun x => c2)) (right associativity, at level 80).
+  Notation "'for' x := i 'loop' c1 'done'" := (Loop i (fun x => c1)) (right associativity, at level 80).
+
+  Definition index_of (needle : nat) : cmd nat :=
+    for i := 0 loop
+      h_i <- Read i;
+      if h_i ==n needle then
+        Return (Done i)
+      else
+        Return (Again (S i))
+    done.
+
+  Inductive stepResult (result : Set) :=
+  | Answer (r : result)
+  | Stepped (h : heap) (c : cmd result). 
+
+  Fixpoint step {result} (c : cmd result) (h : heap) : stepResult result :=
+    match c with
+    | Return r => Answer r
+    | Bind c1 c2 =>
+      match step c1 h with
+      | Answer r => Stepped h (c2 r)
+      | Stepped h' c1' => Stepped h' (Bind c1' c2)
+      end
+    | Read a => Answer (h $! a)
+    | Write a v => Stepped (h $+ (a, v)) (Return tt)
+    | Loop init body =>
+      Stepped h (r <- body init;
+                 match r with
+                 | Done r' => Return r'
+                 | Again r' => Loop r' body
+                 end)
+    end.
+
+  Fixpoint multiStep {result} (c : cmd result) (h : heap) (n : nat) : stepResult result :=
+    match n with
+    | O => Stepped h c
+    | S n' => match step c h with
+              | Answer r => Answer r
+              | Stepped h' c' => multiStep c' h' n'
+              end
+    end.
+
+  Example run_index_of : multiStep (index_of 6) h0 20 = Answer 3.
+  Proof.
+    unfold h0.
+    simplify.
+    reflexivity.
+  Qed.
+
+  Inductive hoare_triple : assertion -> forall {result}, cmd result -> (result -> assertion) -> Prop :=
+  | HtReturn : forall P {result : Set} (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'
+
+  | HtLoop : forall {acc : Set} (init : acc) (body : acc -> cmd (loop_outcome acc)) I,
+      (forall acc, hoare_triple (I (Again acc)) (body acc) I)
+      -> hoare_triple (I (Again init)) (Loop init body) (fun r h => I (Done 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).
+
+  Lemma HtStrengthen : forall {result} (c : cmd result) P Q (Q' : _ -> assertion),
+      hoare_triple P c Q
+      -> (forall r h, Q r h -> Q' r h)
+      -> hoare_triple P c Q'.
+  Proof.
+    simplify.
+    eapply HtConsequence; eauto.
+  Qed.
+
+  Lemma HtWeaken : forall {result} (c : cmd result) P Q (P' : assertion),
+      hoare_triple P c Q
+      -> (forall h, P' h -> P h)
+      -> hoare_triple P' c Q.
+  Proof.
+    simplify.
+    eapply HtConsequence; eauto.
+  Qed.
+
+  Theorem index_of_ok : forall hinit needle,
+    {{ h ~> h = hinit }}
+      index_of needle
+    {{ r&h ~> h = hinit
+         /\ hinit $! r = needle
+         /\ forall i, i < r -> hinit $! i <> needle }}.
+  Proof.
+    unfold index_of.
+    simplify.
+    eapply HtConsequence.
+    apply HtLoop with (I := fun r h => h = hinit
+                                       /\ match r with
+                                          | Done r' => hinit $! r' = needle
+                                                       /\ forall i, i < r' -> hinit $! i <> needle
+                                          | Again r' => forall i, i < r' -> hinit $! i <> needle
+                                          end); simplify.
+
+    econstructor.
+    econstructor.
+
+    simplify.
+    cases (r ==n needle); subst.
+    eapply HtStrengthen.
+    econstructor.
+    simplify; propositional; subst.
+    auto.
+
+    eapply HtStrengthen.
+    econstructor.
+    simplify.
+    propositional; subst.
+    simplify.
+    cases (i ==n acc); subst; auto.
+    apply H3 with (i0 := i); auto.
+
+    simplify.
+    propositional.
+    linear_arithmetic.
+
+    simplify.
+    propositional.
+  Qed.
+
+  Definition trsys_of {result} (c : cmd result) (h : heap) := {|
+    Initial := {(c, h)};
+    Step := fun p1 p2 => step (fst p1) (snd p1) = Stepped (snd p2) (fst p2)
+  |}.
+
+  Lemma invert_Return : forall {result : Set} (r : result) P Q,
+    hoare_triple P (Return r) Q
+    -> forall h, P h -> Q r h.
+  Proof.
+    induct 1; propositional; eauto.
+  Qed.
+
+  Lemma invert_Bind : forall {result' result} (c1 : cmd result') (c2 : result' -> cmd result) P Q,
+    hoare_triple P (Bind c1 c2) Q
+    -> exists R, hoare_triple P c1 R
+                 /\ forall r, hoare_triple (R r) (c2 r) Q.
+  Proof.
+    induct 1; propositional; eauto.
+
+    invert IHhoare_triple; propositional.
+    eexists; propositional.
+    eapply HtWeaken.
+    eassumption.
+    auto.
+    eapply HtStrengthen.
+    apply H4.
+    auto.
+  Qed.
+
+  Lemma unit_not_nat : unit = nat -> False.
+  Proof.
+    simplify.
+    assert (exists x : unit, forall y : unit, x = y).
+    exists tt; simplify.
+    cases y; reflexivity.
+    rewrite H in H0.
+    invert H0.
+    specialize (H1 (S x)).
+    linear_arithmetic.
+  Qed.
+
+  Lemma invert_Read : forall a P Q,
+    hoare_triple P (Read a) Q
+    -> forall h, P h -> Q (h $! a) h.
+  Proof.
+    induct 1; propositional; eauto.
+    apply unit_not_nat in x0.
+    propositional.
+  Qed.
+
+  Lemma invert_Write : forall a v P Q,
+    hoare_triple P (Write a v) Q
+    -> forall h, P h -> Q tt (h $+ (a, v)).
+  Proof.
+    induct 1; propositional; eauto.
+    symmetry in x0.
+    apply unit_not_nat in x0.
+    propositional.
+  Qed.
+
+  Lemma invert_Loop : forall {acc : Set} (init : acc) (body : acc -> cmd (loop_outcome acc)) P Q,
+      hoare_triple P (Loop init body) Q
+      -> exists I, (forall acc, hoare_triple (I (Again acc)) (body acc) I)
+                   /\ (forall h, P h -> I (Again init) h)
+                   /\ (forall r h, I (Done r) h -> Q r h).
+  Proof.
+    induct 1; propositional; eauto.
+
+    invert IHhoare_triple; propositional.
+    exists x; propositional; eauto.
+  Qed.
+
+  Lemma step_sound : forall {result} (c : cmd result) h Q,
+      hoare_triple (fun h' => h' = h) c Q
+      -> match step c h with
+         | Answer r => Q r h
+         | Stepped h' c' => hoare_triple (fun h'' => h'' = h') c' Q
+         end.
+  Proof.
+    induct c; simplify; propositional.
+
+    eapply invert_Return.
+    eauto.
+    simplify; auto.
+
+    apply invert_Bind in H0.
+    invert H0; propositional.
+    apply IHc in H0.
+    cases (step c h); auto.
+    econstructor.
+    apply H2.
+    equality.
+    auto.
+    econstructor; eauto.
+
+    eapply invert_Read; eauto.
+    simplify; auto.
+
+    eapply HtStrengthen.
+    econstructor.
+    simplify; propositional; subst.
+    eapply invert_Write; eauto.
+    simplify; auto.
+
+    apply invert_Loop in H0.
+    invert H0; propositional.
+    econstructor.
+    eapply HtWeaken.
+    apply H0.
+    equality.
+    simplify.
+    cases r.
+    eapply HtStrengthen.
+    econstructor.
+    simplify.
+    propositional; subst; eauto.
+    eapply HtStrengthen.
+    eapply HtLoop.
+    auto.
+    simplify.
+    eauto.
+  Qed.
+
+  Lemma hoare_triple_sound' : forall P {result} (c : cmd result) Q,
+      hoare_triple P c Q
+      -> forall h, P h
+                   -> invariantFor (trsys_of c h)
+                                   (fun p => hoare_triple (fun h => h = snd p)
+                                                          (fst p)
+                                                          Q).
+  Proof.
+    simplify.
+
+    apply invariant_induction; simplify.
+
+    propositional; subst; simplify.
+    eapply HtConsequence.
+    eassumption.
+    equality.
+    auto.
+
+    eapply step_sound in H1.
+    rewrite H2 in H1.
+    auto.
+  Qed.
+
+  Theorem hoare_triple_sound : forall P {result} (c : cmd result) Q,
+      hoare_triple P c Q
+      -> forall h, P h
+                   -> invariantFor (trsys_of c h)
+                                   (fun p => forall r, fst p = Return r
+                                                       -> Q r (snd p)).
+  Proof.
+    simplify.
+
+    eapply invariant_weaken.
+    eapply hoare_triple_sound'; eauto.
+    simplify.
+    rewrite H2 in H1.
+    eapply invert_Return; eauto.
+    simplify; auto.
+  Qed.
+
+  Extraction "Deeper.ml" index_of.
+End Deeper.

DeepInterp.ml

@@ -20,5 +20,4 @@ let interp c =
         with Not_found -> O)
     | Write (a, v) -> Obj.magic (Hashtbl.replace h a v)
 
-  in let v = interp' c in
-     h, v
+  in h, interp' c

DeeperInterp.ml

@@ -0,0 +1,27 @@
+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)
+    | Loop (i, b) ->
+      match Obj.magic (interp' (Obj.magic (b i))) with
+      | Done r -> r
+      | Again r -> interp' (Loop (r, b))
+
+  in h, interp' c