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Connecting: added a heap relation, but at the moment it could be anything, because no heap-accessing commands are supported
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1 changed files with 44 additions and 25 deletions
69
Connecting.v
69
Connecting.v
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@ -738,8 +738,8 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
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Lemma hoare_triple_sound' : forall P {result} (c : cmd result) Q,
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hoare_triple P c Q
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-> P $0
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-> invariantFor (trsys_of $0 c)
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-> forall h, P h
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-> invariantFor (trsys_of h c)
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(fun p =>
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hoare_triple (fun h' => h' = fst p)
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(snd p)
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@ -761,8 +761,8 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
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Theorem hoare_triple_sound : forall P {result} (c : cmd result) Q,
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hoare_triple P c Q
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-> P $0
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-> invariantFor (trsys_of $0 c)
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-> forall h, P h
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-> invariantFor (trsys_of h c)
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(fun p => (exists r, snd p = Return r)
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\/ (exists p', step p p')).
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Proof.
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@ -1205,9 +1205,17 @@ 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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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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translate "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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Lemma eval_translate : forall H V e v,
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@ -1243,19 +1251,22 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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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 out V c s
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-> exists c', ME.step^* (h, c) (h, c')
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/\ translate 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 out V' c' s'
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/\ heaps_compat H' h'.
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Proof.
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induct 1; invert 1; simplify.
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apply inj_pair2 in H1; subst.
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eapply eval_translate in H5; eauto; subst.
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eexists; propositional.
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do 2 eexists; propositional.
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eauto.
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apply TrReturned; simplify; auto.
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assumption.
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apply inj_pair2 in H0; subst.
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eexists; propositional.
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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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@ -1263,10 +1274,11 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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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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apply inj_pair2 in H2; subst.
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invert H0.
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eexists; propositional.
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do 2 eexists; propositional.
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eauto.
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eapply TrAssigned with (x := x).
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eapply eval_translate in H7; eauto.
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@ -1277,12 +1289,14 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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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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eauto.
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assumption.
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invert H0.
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Qed.
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Theorem translated_simulates : forall H V c h s,
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translate "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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Proof.
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constructor; simplify.
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@ -1292,16 +1306,18 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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econstructor.
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eassumption.
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eauto.
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auto.
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invert H1.
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apply inj_pair2 in H4; subst.
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invert H2.
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apply inj_pair2 in H5; subst.
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cases st1'.
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cases p.
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eapply step_translate in H2; eauto.
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first_order.
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eapply step_translate in H6; 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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@ -1339,12 +1355,13 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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econstructor.
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Qed.
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Theorem hoare_triple_sound : forall P (c : cmd wrd) Q V s,
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Theorem hoare_triple_sound : forall P (c : cmd wrd) Q V s h H,
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hoare_triple P c Q
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-> P $0
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-> P h
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-> heaps_compat H h
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-> translate "result" V c s
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-> V $? "result" = None
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-> invariantFor (DE.trsys_of $0 V s)
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-> invariantFor (DE.trsys_of H V s)
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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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@ -1353,23 +1370,24 @@ 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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apply invariant_multistepify with (sys := trsys_of $0 c).
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eassumption.
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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; first_order.
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invert 1; simp.
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cases st2; simplify; subst.
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invert H4; simplify.
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apply inj_pair2 in H9; subst.
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invert H8.
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apply inj_pair2 in H4; 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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right; eexists.
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econstructor; eauto.
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auto.
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invert H4.
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apply inj_pair2 in H6; subst; simplify.
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invert H6; simp.
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apply inj_pair2 in H8; subst; simplify.
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cases x.
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eapply not_stuck in H3; eauto.
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eapply not_stuck in H7; eauto.
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Qed.
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Theorem adder_ok : forall a b c,
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@ -1398,6 +1416,7 @@ 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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