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Connecting: reading heads
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Adam Chlipala
parent
d537e28266
commit
daebf21dc0
1 changed files with 139 additions and 99 deletions
238
Connecting.v
238
Connecting.v
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@ -1060,7 +1060,7 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
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}.
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End fn.
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Definition heap := fmap (wrd) (wrd * wrd).
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Definition heap := fmap wrd wrd.
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Definition valuation := fmap var wrd.
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Inductive eval : heap -> valuation -> exp -> wrd -> Prop :=
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@ -1073,13 +1073,13 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
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eval H V e1 n1
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-> eval H V e2 n2
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-> eval H V (Add e1 e2) (n1 ^+ n2)
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| VHead : forall H V e1 p ph pt,
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| VHead : forall H V e1 p ph,
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eval H V e1 p
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-> H $? p = Some (ph, pt)
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-> H $? p = Some ph
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-> eval H V (Head e1) ph
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| VTail : forall H V e1 p ph pt,
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| VTail : forall H V e1 p pt,
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eval H V e1 p
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-> H $? p = Some (ph, pt)
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-> H $? (p ^+ ^1) = Some pt
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-> eval H V (Tail e1) pt
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| VNotNull : forall H V e1 p,
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eval H V e1 p
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@ -1089,16 +1089,16 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
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| StAssign : forall H V x e v,
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eval H V e v
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-> step (H, V, Assign x e) (H, V $+ (x, v), Skip)
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| StWriteHead : forall H V x e a v hv tv,
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| StWriteHead : forall H V x e a v hv,
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V $? x = Some a
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-> eval H V e v
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-> H $? a = Some (hv, tv)
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-> step (H, V, WriteHead x e) (H $+ (a, (v, tv)), V, Skip)
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| StWriteTail : forall H V x e a v hv tv,
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-> H $? a = Some hv
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-> step (H, V, WriteHead x e) (H $+ (a, v), V, Skip)
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| StWriteTail : forall H V x e a v tv,
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V $? x = Some a
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-> eval H V e v
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-> H $? a = Some (hv, tv)
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-> step (H, V, WriteTail x e) (H $+ (a, (hv, v)), V, Skip)
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-> H $? a = Some tv
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-> step (H, V, WriteTail x e) (H $+ (a, v), V, Skip)
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| StSeq1 : forall H V s2,
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step (H, V, Seq Skip s2) (H, V, s2)
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| StSeq2 : forall H V s1 s2 H' V' s1',
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@ -1167,17 +1167,18 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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| TrDone : forall V (v : RT) s,
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translate_return V v s
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-> translate translate_return V (Return v) s
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| TrAssign : forall V (B : Set) (v : wrd) (c : wrd -> cmd B) e x s1,
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| TrAssign : forall V B (v : wrd) (c : wrd -> cmd B) e x s1,
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translate_exp V v e
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-> (forall w, translate translate_return (V $+ (x, w)) (c w) s1)
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-> translate translate_return V (Bind (Return v) c) (Seq (Assign x e) s1)
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| TrAssigned : forall V (B : Set) (v : wrd) (c : wrd -> cmd B) x s1,
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| TrAssigned : forall V B (v : wrd) (c : wrd -> cmd B) x s1,
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V $? x = Some v
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-> translate translate_return (V $+ (x, v)) (c v) s1
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-> translate translate_return V (Bind (Return v) c) (Seq Skip s1).
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Example adder (a b c : wrd) :=
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Bind (Return (a ^+ b)) (fun ab => Return (ab ^+ c)).
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-> translate translate_return V (c v) s1
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-> translate translate_return V (Bind (Return v) c) (Seq Skip s1)
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| TrReadHead : forall V B (a : wrd) (c : wrd -> cmd B) e x s1,
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translate_exp V a e
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-> (forall w, translate translate_return (V $+ (x, w)) (c w) s1)
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-> translate translate_return V (Bind (Read a) c) (Seq (Assign x (Head e)) s1).
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Ltac freshFor vm k :=
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let rec keepTrying x :=
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@ -1199,8 +1200,14 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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| [ |- translate _ ?V (Bind (Return _) _) _ ] =>
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freshFor V ltac:(fun y =>
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eapply TrAssign with (x := y); [ | intro ])
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| [ |- translate _ ?V (Bind (Read _) _) _ ] =>
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freshFor V ltac:(fun y =>
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eapply TrReadHead with (x := y); [ | intro ])
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end.
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Example adder (a b c : wrd) :=
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Bind (Return (a ^+ b)) (fun ab => Return (ab ^+ c)).
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Lemma translate_adder : sig (fun s =>
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forall a b c, translate (translate_result "result") ($0 $+ ("a", a) $+ ("b", b) $+ ("c", c)) (adder a b c) s).
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Proof.
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@ -1211,18 +1218,23 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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Definition adder_compiled := Eval simpl in proj1_sig translate_adder.
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Record heaps_compat (H : DE.heap) (h : ME.heap) : Prop := {
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DeepToMixedHd : forall a v1 v2, H $? a = Some (v1, v2) -> h $? a = Some v1;
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DeepToMixedTl : forall a v1 v2, H $? a = Some (v1, v2) -> h $? (a ^+ ^1) = Some v2;
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MixedToDeep : forall a v1 v2, h $? a = Some v1 -> h $? (a ^+ ^1) = Some v2
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-> H $? a = Some (v1, v2)
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}.
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Example reader (p1 p2 : wrd) :=
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Bind (Read p1) (fun v1 => Bind (Read p2) (fun v2 => Return (v1 ^+ v2))).
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Lemma translate_reader : sig (fun s =>
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forall p1 p2, translate (translate_result "result") ($0 $+ ("p1", p1) $+ ("p2", p2)) (reader p1 p2) s).
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Proof.
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eexists; simplify.
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unfold reader.
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repeat translate.
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Defined.
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Definition reader_compiled := Eval simpl in proj1_sig translate_reader.
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Inductive translated : forall {A}, DE.heap * valuation * stmt -> ME.heap * cmd A -> Prop :=
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| Translated : forall A H h V s (c : cmd A),
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| Translated : forall A H V s (c : cmd A),
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translate (translate_result "result") V c s
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-> heaps_compat H h
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-> translated (H, V, s) (h, c).
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-> translated (H, V, s) (H, c).
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Lemma eval_translate : forall H V e v,
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eval H V e v
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@ -1253,75 +1265,57 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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eauto.
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Qed.
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Lemma step_translate : forall H V s H' V' s',
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DE.step (H, V, s) (H', V', s')
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-> forall h (c : cmd wrd) out,
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translate (translate_result out) V c s
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-> heaps_compat H h
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-> exists c' h', ME.step^* (h, c) (h', c')
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/\ translate (translate_result out) V' c' s'
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/\ heaps_compat H' h'.
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Lemma step_translate : forall out V (c : cmd wrd) s,
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translate (translate_result out) V c s
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-> forall H H' V' s',
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DE.step (H, V, s) (H', V', s')
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-> exists c', ME.step^* (H, c) (H', c')
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/\ translate (translate_result out) V' c' s'.
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Proof.
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induct 1; invert 1; simplify.
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induct 1.
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invert H3.
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apply inj_pair2 in H1; subst.
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eapply eval_translate in H4; eauto; subst.
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invert H; invert 1; simplify.
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eapply eval_translate in H0; eauto; subst.
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do 2 eexists; propositional.
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eauto.
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apply TrDone; apply TrReturned; simplify; auto.
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assumption.
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invert 1; simplify.
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invert H6.
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eapply eval_translate in H5; eauto; subst.
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do 2 eexists; propositional.
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eauto.
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econstructor.
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instantiate (1 := x); simplify.
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reflexivity.
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eauto.
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invert H6.
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invert H2.
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apply inj_pair2 in H0; subst.
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invert 1; simplify.
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do 2 eexists; propositional.
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eapply TrcFront.
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eauto.
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eauto.
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replace V' with (V' $+ (x, v)).
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assumption.
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maps_equal.
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symmetry; assumption.
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assumption.
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invert H5.
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invert H4.
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apply inj_pair2 in H2; subst.
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invert H0.
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do 2 eexists; propositional.
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eauto.
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eapply TrAssigned.
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instantiate (1 := x).
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invert 1.
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invert H6.
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invert H5.
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eapply eval_translate in H4; eauto; subst.
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simplify.
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eapply eval_translate in H7; eauto.
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subst.
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reflexivity.
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match goal with
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| [ |- translate _ ?V' _ _ ] => replace V' with (V $+ (x, v)) by (maps_equal; equality)
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end.
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do 2 eexists; propositional.
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eapply TrcFront.
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eauto.
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eauto.
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econstructor.
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instantiate (1 := x); simplify; reflexivity.
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eauto.
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assumption.
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invert H0.
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invert H4.
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invert H3.
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invert H4.
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invert H3.
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Qed.
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Theorem translated_simulates : forall H V c h s,
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Theorem translated_simulates : forall H V c s,
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translate (translate_result "result") V c s
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-> heaps_compat H h
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-> simulates (translated (A := wrd)) (DE.trsys_of H V s) (ME.multistep_trsys_of h c).
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-> simulates (translated (A := wrd)) (DE.trsys_of H V s) (ME.multistep_trsys_of H c).
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Proof.
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constructor; simplify.
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@ -1332,16 +1326,15 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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eauto.
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auto.
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invert H2.
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apply inj_pair2 in H5; subst.
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invert H1.
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apply inj_pair2 in H4; subst.
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cases st1'.
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cases p.
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eapply step_translate in H6; eauto.
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eapply step_translate in H7; eauto.
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simp.
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eexists; split; [ | eassumption ].
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econstructor.
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assumption.
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eauto.
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Qed.
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Hint Constructors eval DE.step.
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@ -1355,10 +1348,11 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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Hint Resolve translate_exp_sound.
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Lemma not_stuck : forall A h (c : cmd A) h' c',
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step (h, c) (h', c')
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-> forall out V s, translate (translate_result out) V c s
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-> forall H, exists p', DE.step (H, V, s) p'.
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Lemma not_stuck : forall out V (c : cmd wrd) s,
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translate (translate_result out) V c s
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-> forall H H' c',
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step (H, c) (H', c')
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-> exists p', DE.step (H, V, s) p'.
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Proof.
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induct 1; invert 1; simplify.
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@ -1369,20 +1363,27 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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eexists.
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econstructor.
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econstructor.
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eauto.
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eexists.
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econstructor.
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econstructor.
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eauto.
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apply inj_pair2 in H8; subst.
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invert H6.
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eexists.
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econstructor.
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econstructor.
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econstructor.
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eauto.
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eauto.
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Qed.
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Theorem hoare_triple_sound : forall P (c : cmd wrd) Q V s h H,
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Theorem hoare_triple_sound : forall P (c : cmd wrd) Q V s H,
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hoare_triple P c Q
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-> P h
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-> heaps_compat H h
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-> P H
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-> translate (translate_result "result") V c s
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-> V $? "result" = None
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-> invariantFor (DE.trsys_of H V s)
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@ -1394,25 +1395,24 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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eapply invariant_simulates.
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apply translated_simulates.
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eassumption.
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eassumption.
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apply invariant_multistepify with (sys := trsys_of h c).
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apply invariant_multistepify with (sys := trsys_of H c).
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eauto using hoare_triple_sound.
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invert 1; simp.
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cases st2; simplify; subst.
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invert H6; simplify.
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apply inj_pair2 in H8; subst.
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invert H11.
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apply inj_pair2 in H6; subst.
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invert H8.
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invert H5; simplify.
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apply inj_pair2 in H10; subst.
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invert H9.
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apply inj_pair2 in H4; subst.
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invert H6.
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right; eexists.
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econstructor; eauto.
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auto.
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invert H6; simp.
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apply inj_pair2 in H8; subst; simplify.
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invert H5; simp.
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apply inj_pair2 in H7; subst; simplify.
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cases x.
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eapply not_stuck in H7; eauto.
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eapply not_stuck in H10; eauto.
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Qed.
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Theorem adder_ok : forall a b c,
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@ -1441,10 +1441,50 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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eapply hoare_triple_sound.
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apply adder_ok.
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constructor; auto.
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constructor; simplify; equality.
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apply (proj2_sig translate_adder).
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simplify; equality.
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Qed.
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Theorem reader_ok : forall p1 p2 v1 v2,
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{{p1 |-> v1 * p2 |-> v2}}
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reader p1 p2
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{{r ~> [| r = v1 ^+ v2 |] * p1 |-> v1 * p2 |-> v2}}.
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Proof.
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unfold reader.
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simplify.
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step.
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step.
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simp.
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step.
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step.
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simp.
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step.
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cancel.
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equality.
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Qed.
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Theorem reader_compiled_ok : forall p1 p2 v1 v2,
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p1 <> p2
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-> invariantFor (DE.trsys_of ($0 $+ (p1, v1) $+ (p2, v2)) ($0 $+ ("p1", p1) $+ ("p2", p2)) reader_compiled)
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(fun p => snd p = Skip
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\/ exists p', DE.step p p').
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Proof.
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simplify.
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eapply hoare_triple_sound.
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apply reader_ok.
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exists ($0 $+ (p1, v1)), ($0 $+ (p2, v2)); propositional.
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unfold split.
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maps_equal.
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rewrite lookup_join2; simplify; auto; sets.
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rewrite lookup_join1; simplify; auto; sets.
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rewrite lookup_join2; simplify; auto; sets.
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unfold disjoint; simplify.
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cases (weq a p1); simplify; propositional.
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constructor.
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constructor.
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apply (proj2_sig translate_reader).
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simplify; equality.
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Qed.
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End MixedToDeep.
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