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Connecting: proved DeeplyEmbedded.hoare_triple_sound

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Adam Chlipala 2018-04-28 21:23:41 -04:00
parent 625458d80e
commit 26abb7b8a0
1 changed files with 254 additions and 5 deletions
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@ -503,12 +503,12 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
-> hoare_triple Q s2 R
-> hoare_triple P (Seq s1 s2) R
| HtIfThenElse : forall P e s1 s2 Q R,
hoare_triple_exp P e Q
hoare_triple_exp P e (fun r V => Q r V * exists n, [| r = VScalar n |])%sep
-> hoare_triple (fun V => exists n, Q (VScalar n) V * [| n <> ^0 |])%sep s1 R
-> hoare_triple (fun V => Q (VScalar (^0)) V)%sep s2 R
-> hoare_triple P (IfThenElse e s1 s2) R
| HtWhileLoop : forall I e s1 Q,
hoare_triple_exp I e Q
hoare_triple_exp I e (fun r V => Q r V * exists n, [| r = VScalar n |])%sep
-> hoare_triple (fun V => exists n, Q (VScalar n) V * [| n <> ^0 |])%sep s1 I
-> hoare_triple I (WhileLoop e s1) (Q (VScalar (^0)))
| HtConsequence : forall P s Q P' Q',
@ -863,7 +863,7 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
Lemma invert_IfThenElse : forall P e s1 s2 Q,
hoare_triple P (IfThenElse e s1 s2) Q
-> exists R, hoare_triple_exp P e R
-> exists R, hoare_triple_exp P e (fun r V => R r V * exists n, [| r = VScalar n |])%sep
/\ hoare_triple (fun V => exists n, R (VScalar n) V * [| n <> ^0 |])%sep s1 Q
/\ hoare_triple (fun V => R (VScalar (^0)) V)%sep s2 Q.
Proof.
@ -873,6 +873,7 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
eapply HtExpConsequence.
eassumption.
assumption.
simplify.
reflexivity.
eapply HtStrengthen.
eassumption.
@ -882,6 +883,13 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
assumption.
exists (fun r V => x r V * R V)%sep; propositional.
eapply HtExpConsequence.
eapply HtExpFrame.
eauto.
simplify.
reflexivity.
simplify.
cancel.
eauto.
eapply HtWeaken.
eapply HtFrame.
@ -903,7 +911,7 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
Lemma invert_WhileLoop : forall P e s1 Q,
hoare_triple P (WhileLoop e s1) Q
-> exists I R, (forall V, P V ===> I V)
/\ hoare_triple_exp I e R
/\ hoare_triple_exp I e (fun r V => R r V * exists n, [| r = VScalar n |])%sep
/\ hoare_triple (fun V => exists n, R (VScalar n) V * [| n <> ^0 |])%sep s1 I
/\ (forall V, R (VScalar (^0)) V ===> Q V).
Proof.
@ -917,6 +925,13 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
exists (fun V => x V * R V)%sep, (fun r V => x0 r V * R V)%sep; propositional.
rewrite H1; reflexivity.
eapply HtExpConsequence.
eapply HtExpFrame.
eauto.
simplify.
reflexivity.
simplify.
cancel.
eauto.
eapply HtWeaken.
eauto.
@ -1055,6 +1070,11 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
unfold himp; propositional; subst.
eapply preservation_exp in H0; eauto.
exists n, H', $0; propositional; eauto.
invert H0; simp.
invert H7.
invert H6.
apply split_empty_fwd in H4; subst.
auto.
constructor; auto.
simplify; auto.
@ -1063,6 +1083,19 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
simplify.
unfold himp; propositional; subst.
eapply preservation_exp in H0; eauto.
eapply HtExpConsequence.
eauto.
simplify.
instantiate (1 := fun V H'0 => H'0 = H' /\ V = V').
simplify.
reflexivity.
simplify.
unfold himp; simplify.
invert H3; simp.
invert H7.
invert H6.
apply split_empty_fwd in H4; subst.
auto.
simplify; auto.
apply invert_WhileLoop in H; first_order.
@ -1072,6 +1105,11 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
unfold himp; propositional; subst.
eapply preservation_exp in H0; eauto.
exists n, H', $0; propositional; eauto.
invert H0; simp.
invert H8.
invert H7.
apply split_empty_fwd in H5; subst.
auto.
constructor; auto.
apply H; simplify; auto.
eapply HtStrengthen.
@ -1084,6 +1122,12 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
unfold himp; propositional; subst.
eapply preservation_exp in H0; eauto.
apply H3; auto.
simplify.
invert H0; simp.
invert H7.
invert H6.
apply split_empty_fwd in H4; subst.
auto.
apply H; simplify; auto.
Qed.
@ -1112,6 +1156,200 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
eapply preservation; eauto.
Qed.
Lemma progress_exp : forall P e Q,
hoare_triple_exp P e Q
-> forall V H H1 H2, split H H1 H2
-> disjoint H1 H2
-> P V H1
-> exists v, eval H V e v.
Proof.
induct 1; simplify; eauto.
invert H4.
invert H5.
eauto.
invert H4.
invert H5.
eexists.
econstructor.
eauto.
invert H4.
invert H5; simp.
invert H6.
apply split_empty_fwd' in H4; subst.
invert H8.
invert H1; simp.
invert H6.
cases x1.
eexists.
econstructor.
eauto.
unfold heap1, split in *; subst.
rewrite lookup_join1.
rewrite lookup_join1.
simplify.
eauto.
eapply lookup_Some_dom; simplify; sets.
eapply lookup_Some_dom; simplify.
rewrite lookup_join1.
simplify; eauto.
eapply lookup_Some_dom; simplify; sets.
invert H4.
invert H5; simp.
invert H6.
apply split_empty_fwd' in H4; subst.
invert H8.
invert H1; simp.
invert H6.
cases x1.
eexists.
econstructor.
eauto.
unfold heap1, split in *; subst.
rewrite lookup_join1.
rewrite lookup_join1.
simplify.
eauto.
eapply lookup_Some_dom; simplify; sets.
eapply lookup_Some_dom; simplify.
rewrite lookup_join1.
simplify; eauto.
eapply lookup_Some_dom; simplify; sets.
invert H4.
invert H5.
eauto.
eapply IHhoare_triple_exp; eauto.
apply H0; auto.
invert H5; simp.
eapply IHhoare_triple_exp in H7; eauto.
Qed.
Lemma progress : forall P s Q,
hoare_triple P s Q
-> forall V H H1 H2, split H H1 H2
-> disjoint H1 H2
-> P V H1
-> s = Skip \/ (exists H' V' s', step (H, V, s) (H', V', s')).
Proof.
induct 1; simplify; eauto.
eapply (progress_exp _ _ _ H) in H5; eauto.
simp.
right; do 3 eexists.
econstructor.
eauto.
pose proof (progress_exp _ _ _ H _ _ _ _ H3 H4 H5) as Hprog.
invert Hprog.
assert (Hdropped : eval H1 V e x0).
eapply drop_cells; eauto.
unfold split in H3; subst.
simplify.
rewrite lookup_join1; auto.
eapply lookup_Some_dom; eauto.
pose proof (preservation_exp _ _ _ H _ _ _ H5 Hdropped) as Hpres.
simplify.
invert Hpres.
invert H7; simp.
invert H9.
apply split_empty_fwd' in H8; subst.
invert H11.
invert H1.
invert H8; simp.
invert H9.
right; do 3 eexists.
econstructor.
eauto.
eauto.
unfold heap1, split in *; subst.
rewrite lookup_join1.
rewrite lookup_join1.
simplify.
eauto.
eapply lookup_Some_dom; simplify; sets.
eapply lookup_Some_dom; simplify.
rewrite lookup_join1.
simplify; eauto.
eapply lookup_Some_dom; simplify; sets.
pose proof (progress_exp _ _ _ H _ _ _ _ H3 H4 H5) as Hprog.
invert Hprog.
assert (Hdropped : eval H1 V e x0).
eapply drop_cells; eauto.
unfold split in H3; subst.
simplify.
rewrite lookup_join1; auto.
eapply lookup_Some_dom; eauto.
pose proof (preservation_exp _ _ _ H _ _ _ H5 Hdropped) as Hpres.
simplify.
invert Hpres.
invert H7; simp.
invert H9.
apply split_empty_fwd' in H8; subst.
invert H11.
invert H1.
invert H8; simp.
invert H9.
right; do 3 eexists.
econstructor.
eauto.
eauto.
unfold heap1, split in *; subst.
rewrite lookup_join1.
rewrite lookup_join1.
simplify.
eauto.
eapply lookup_Some_dom; simplify; sets.
eapply lookup_Some_dom; simplify.
rewrite lookup_join1.
simplify; eauto.
eapply lookup_Some_dom; simplify; sets.
specialize (IHhoare_triple1 _ _ _ _ H4 H5 H6); simp; eauto 6.
pose proof (progress_exp _ _ _ H _ _ _ _ H5 H6 H7) as Hprog.
invert Hprog.
assert (Hdropped : eval H3 V e x).
eapply drop_cells; eauto.
unfold split in H5; subst.
simplify.
rewrite lookup_join1; auto.
eapply lookup_Some_dom; eauto.
pose proof (preservation_exp _ _ _ H _ _ _ H7 Hdropped) as Hpres.
simplify.
invert Hpres; simp.
invert H13.
invert H12.
cases (weq x2 (^0)); subst; eauto 10.
pose proof (progress_exp _ _ _ H _ _ _ _ H4 H5 H6) as Hprog.
invert Hprog.
assert (Hdropped : eval H2 V e x).
eapply drop_cells; eauto.
unfold split in H4; subst.
simplify.
rewrite lookup_join1; auto.
eapply lookup_Some_dom; eauto.
pose proof (preservation_exp _ _ _ H _ _ _ H6 Hdropped) as Hpres.
simplify.
invert Hpres; simp.
invert H12.
invert H11.
cases (weq x2 (^0)); subst; eauto 10.
apply H0 in H7.
eapply IHhoare_triple in H7; eauto.
invert H6; simp.
eapply IHhoare_triple in H8; eauto.
Qed.
Theorem hoare_triple_sound : forall P s Q,
hoare_triple P s Q
-> forall H V, P V H
@ -1123,6 +1361,17 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
eapply invariant_weaken.
eapply hoare_triple_sound'; eauto.
simplify.
Admitted.
cases s0.
cases p.
simplify.
pose proof (progress _ _ _ H2 v h h $0) as Hprog; simplify.
cases Hprog; eauto.
subst.
eapply invert_Skip in H2.
left; propositional.
apply H2; auto.
first_order.
Qed.
End DeeplyEmbedded.