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SeparationLogic: soundness proof

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Adam Chlipala 2016-04-17 16:55:52 -04:00
parent 9dc96733d4
commit ef310e2b1e
3 changed files with 1335 additions and 2 deletions
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257
Map.v
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@ -175,6 +175,101 @@ Module Type S.
Axiom lookup_None_dom : forall K V (m : fmap K V) k,
m $? k = None
-> ~ k \in dom m.
(* Bits meant for separation logic *)
Section splitting.
Variables K V : Type.
Definition disjoint (h1 h2 : fmap K V) : Prop :=
forall a, h1 $? a <> None
-> h2 $? a <> None
-> False.
Definition split (h h1 h2 : fmap K V) : Prop :=
h = h1 $++ h2.
Axiom split_empty_fwd : forall h h1,
split h h1 $0
-> h = h1.
Axiom split_empty_fwd' : forall h h1,
split h $0 h1
-> h = h1.
Axiom split_empty_bwd : forall h,
split h h $0.
Axiom split_empty_bwd' : forall h,
split h $0 h.
Axiom disjoint_hemp : forall h,
disjoint h $0.
Axiom disjoint_hemp' : forall h,
disjoint $0 h.
Axiom disjoint_comm : forall h1 h2,
disjoint h1 h2
-> disjoint h2 h1.
Axiom split_comm : forall h h1 h2,
disjoint h1 h2
-> split h h1 h2
-> split h h2 h1.
Axiom split_assoc1 : forall h h1 h' h2 h3,
split h h1 h'
-> split h' h2 h3
-> split h (join h1 h2) h3.
Axiom split_assoc2' : forall h h1 h' h2 h3,
split h h1 h'
-> split h' h2 h3
-> disjoint h1 h'
-> disjoint h2 h3
-> split h h2 (join h3 h1).
Axiom split_assoc2 : forall h h1 h' h2 h3,
split h h' h1
-> split h' h2 h3
-> disjoint h' h1
-> disjoint h2 h3
-> split h h2 (join h3 h1).
Axiom disjoint_assoc1 : forall h h1 h' h2 h3,
split h h1 h'
-> split h' h2 h3
-> disjoint h1 h'
-> disjoint h2 h3
-> disjoint (join h1 h2) h3.
Axiom disjoint_assoc2 : forall h h1 h' h2 h3,
split h h' h1
-> split h' h2 h3
-> disjoint h' h1
-> disjoint h2 h3
-> disjoint h2 (join h3 h1).
Axiom split_join : forall h1 h2,
split (join h1 h2) h1 h2.
Axiom split_disjoint : forall h h1 h2 h' h3,
split h h1 h'
-> split h' h2 h3
-> disjoint h1 h'
-> disjoint h2 h3
-> disjoint h1 h2.
Axiom disjoint_assoc3 : forall h h1 h2 h3,
disjoint h h2
-> split h h1 h3
-> disjoint h1 h3
-> disjoint h3 h2.
End splitting.
Hint Immediate disjoint_comm split_comm.
Hint Immediate split_empty_bwd disjoint_hemp disjoint_hemp' split_assoc1 split_assoc2.
Hint Immediate disjoint_assoc1 disjoint_assoc2 split_join split_disjoint disjoint_assoc3.
End S.
Module M : S.
@ -479,6 +574,168 @@ Module M : S.
Proof.
unfold lookup, dom, In; congruence.
Qed.
Section splitting.
Variables K V : Type.
Notation "$0" := (@empty K V).
Notation "m $+ ( k , v )" := (add m k v) (at level 50, left associativity).
Infix "$-" := remove (at level 50, left associativity).
Infix "$++" := join (at level 50, left associativity).
Infix "$?" := lookup (at level 50, no associativity).
Infix "$<=" := includes (at level 90).
Definition disjoint (h1 h2 : fmap K V) : Prop :=
forall a, h1 $? a <> None
-> h2 $? a <> None
-> False.
Definition split (h h1 h2 : fmap K V) : Prop :=
h = h1 $++ h2.
Hint Extern 2 (_ <> _) => congruence.
Ltac splt := unfold disjoint, split, join, lookup in *; intros; subst;
try match goal with
| [ |- @eq (fmap K V) _ _ ] => let a := fresh "a" in extensionality a; simpl
end;
repeat match goal with
| [ a : K, H : forall a : K, _ |- _ ] => specialize (H a)
end;
repeat match goal with
| [ H : _ |- _ ] => rewrite H
| [ |- context[match ?E with Some _ => _ | None => _ end] ] => destruct E
| [ _ : context[match ?E with Some _ => _ | None => _ end] |- _ ] => destruct E
end; eauto; try solve [ exfalso; eauto ].
Lemma split_empty_fwd : forall h h1,
split h h1 $0
-> h = h1.
Proof.
splt.
Qed.
Lemma split_empty_fwd' : forall h h1,
split h $0 h1
-> h = h1.
Proof.
splt.
Qed.
Lemma split_empty_bwd : forall h,
split h h $0.
Proof.
splt.
Qed.
Lemma split_empty_bwd' : forall h,
split h $0 h.
Proof.
splt.
Qed.
Lemma disjoint_hemp : forall h,
disjoint h $0.
Proof.
splt.
Qed.
Lemma disjoint_hemp' : forall h,
disjoint $0 h.
Proof.
splt.
Qed.
Lemma disjoint_comm : forall h1 h2,
disjoint h1 h2
-> disjoint h2 h1.
Proof.
splt.
Qed.
Lemma split_comm : forall h h1 h2,
disjoint h1 h2
-> split h h1 h2
-> split h h2 h1.
Proof.
splt.
Qed.
Hint Immediate disjoint_comm split_comm.
Lemma split_assoc1 : forall h h1 h' h2 h3,
split h h1 h'
-> split h' h2 h3
-> split h (join h1 h2) h3.
Proof.
splt.
Qed.
Lemma split_assoc2' : forall h h1 h' h2 h3,
split h h1 h'
-> split h' h2 h3
-> disjoint h1 h'
-> disjoint h2 h3
-> split h h2 (join h3 h1).
Proof.
splt.
Qed.
Lemma split_assoc2 : forall h h1 h' h2 h3,
split h h' h1
-> split h' h2 h3
-> disjoint h' h1
-> disjoint h2 h3
-> split h h2 (join h3 h1).
Proof.
intros; eapply split_assoc2'; eauto.
Qed.
Lemma disjoint_assoc1 : forall h h1 h' h2 h3,
split h h1 h'
-> split h' h2 h3
-> disjoint h1 h'
-> disjoint h2 h3
-> disjoint (join h1 h2) h3.
Proof.
splt.
Qed.
Lemma disjoint_assoc2 : forall h h1 h' h2 h3,
split h h' h1
-> split h' h2 h3
-> disjoint h' h1
-> disjoint h2 h3
-> disjoint h2 (join h3 h1).
Proof.
splt.
Qed.
Lemma split_join : forall h1 h2,
split (join h1 h2) h1 h2.
Proof.
splt.
Qed.
Lemma split_disjoint : forall h h1 h2 h' h3,
split h h1 h'
-> split h' h2 h3
-> disjoint h1 h'
-> disjoint h2 h3
-> disjoint h1 h2.
Proof.
splt.
Qed.
Lemma disjoint_assoc3 : forall h h1 h2 h3,
disjoint h h2
-> split h h1 h3
-> disjoint h1 h3
-> disjoint h3 h2.
Proof.
splt.
Qed.
End splitting.
End M.
Export M.

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SeparationLogic.v
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_CoqProject
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@ -24,3 +24,4 @@ LambdaCalculusAndTypeSoundness.v
TypesAndMutation.v
DeepAndShallowEmbeddings_template.v
DeepAndShallowEmbeddings.v
SeparationLogic.v