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SeparationLogic: soundness proof
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3 changed files with 1335 additions and 2 deletions
257
Map.v
257
Map.v
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@ -175,6 +175,101 @@ Module Type S.
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Axiom lookup_None_dom : forall K V (m : fmap K V) k,
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m $? k = None
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-> ~ k \in dom m.
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(* Bits meant for separation logic *)
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Section splitting.
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Variables K V : Type.
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Definition disjoint (h1 h2 : fmap K V) : Prop :=
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forall a, h1 $? a <> None
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-> h2 $? a <> None
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-> False.
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Definition split (h h1 h2 : fmap K V) : Prop :=
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h = h1 $++ h2.
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Axiom split_empty_fwd : forall h h1,
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split h h1 $0
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-> h = h1.
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Axiom split_empty_fwd' : forall h h1,
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split h $0 h1
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-> h = h1.
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Axiom split_empty_bwd : forall h,
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split h h $0.
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Axiom split_empty_bwd' : forall h,
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split h $0 h.
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Axiom disjoint_hemp : forall h,
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disjoint h $0.
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Axiom disjoint_hemp' : forall h,
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disjoint $0 h.
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Axiom disjoint_comm : forall h1 h2,
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disjoint h1 h2
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-> disjoint h2 h1.
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Axiom split_comm : forall h h1 h2,
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disjoint h1 h2
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-> split h h1 h2
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-> split h h2 h1.
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Axiom split_assoc1 : forall h h1 h' h2 h3,
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split h h1 h'
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-> split h' h2 h3
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-> split h (join h1 h2) h3.
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Axiom split_assoc2' : forall h h1 h' h2 h3,
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split h h1 h'
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-> split h' h2 h3
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-> disjoint h1 h'
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-> disjoint h2 h3
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-> split h h2 (join h3 h1).
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Axiom split_assoc2 : forall h h1 h' h2 h3,
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split h h' h1
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-> split h' h2 h3
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-> disjoint h' h1
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-> disjoint h2 h3
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-> split h h2 (join h3 h1).
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Axiom disjoint_assoc1 : forall h h1 h' h2 h3,
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split h h1 h'
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-> split h' h2 h3
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-> disjoint h1 h'
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-> disjoint h2 h3
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-> disjoint (join h1 h2) h3.
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Axiom disjoint_assoc2 : forall h h1 h' h2 h3,
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split h h' h1
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-> split h' h2 h3
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-> disjoint h' h1
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-> disjoint h2 h3
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-> disjoint h2 (join h3 h1).
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Axiom split_join : forall h1 h2,
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split (join h1 h2) h1 h2.
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Axiom split_disjoint : forall h h1 h2 h' h3,
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split h h1 h'
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-> split h' h2 h3
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-> disjoint h1 h'
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-> disjoint h2 h3
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-> disjoint h1 h2.
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Axiom disjoint_assoc3 : forall h h1 h2 h3,
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disjoint h h2
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-> split h h1 h3
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-> disjoint h1 h3
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-> disjoint h3 h2.
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End splitting.
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Hint Immediate disjoint_comm split_comm.
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Hint Immediate split_empty_bwd disjoint_hemp disjoint_hemp' split_assoc1 split_assoc2.
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Hint Immediate disjoint_assoc1 disjoint_assoc2 split_join split_disjoint disjoint_assoc3.
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End S.
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Module M : S.
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@ -479,6 +574,168 @@ Module M : S.
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Proof.
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unfold lookup, dom, In; congruence.
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Qed.
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Section splitting.
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Variables K V : Type.
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Notation "$0" := (@empty K V).
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Notation "m $+ ( k , v )" := (add m k v) (at level 50, left associativity).
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Infix "$-" := remove (at level 50, left associativity).
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Infix "$++" := join (at level 50, left associativity).
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Infix "$?" := lookup (at level 50, no associativity).
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Infix "$<=" := includes (at level 90).
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Definition disjoint (h1 h2 : fmap K V) : Prop :=
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forall a, h1 $? a <> None
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-> h2 $? a <> None
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-> False.
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Definition split (h h1 h2 : fmap K V) : Prop :=
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h = h1 $++ h2.
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Hint Extern 2 (_ <> _) => congruence.
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Ltac splt := unfold disjoint, split, join, lookup in *; intros; subst;
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try match goal with
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| [ |- @eq (fmap K V) _ _ ] => let a := fresh "a" in extensionality a; simpl
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end;
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repeat match goal with
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| [ a : K, H : forall a : K, _ |- _ ] => specialize (H a)
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end;
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repeat match goal with
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| [ H : _ |- _ ] => rewrite H
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| [ |- context[match ?E with Some _ => _ | None => _ end] ] => destruct E
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| [ _ : context[match ?E with Some _ => _ | None => _ end] |- _ ] => destruct E
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end; eauto; try solve [ exfalso; eauto ].
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Lemma split_empty_fwd : forall h h1,
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split h h1 $0
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-> h = h1.
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Proof.
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splt.
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Qed.
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Lemma split_empty_fwd' : forall h h1,
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split h $0 h1
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-> h = h1.
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Proof.
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splt.
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Qed.
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Lemma split_empty_bwd : forall h,
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split h h $0.
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Proof.
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splt.
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Qed.
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Lemma split_empty_bwd' : forall h,
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split h $0 h.
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Proof.
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splt.
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Qed.
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Lemma disjoint_hemp : forall h,
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disjoint h $0.
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Proof.
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splt.
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Qed.
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Lemma disjoint_hemp' : forall h,
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disjoint $0 h.
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Proof.
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splt.
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Qed.
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Lemma disjoint_comm : forall h1 h2,
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disjoint h1 h2
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-> disjoint h2 h1.
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Proof.
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splt.
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Qed.
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Lemma split_comm : forall h h1 h2,
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disjoint h1 h2
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-> split h h1 h2
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-> split h h2 h1.
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Proof.
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splt.
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Qed.
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Hint Immediate disjoint_comm split_comm.
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Lemma split_assoc1 : forall h h1 h' h2 h3,
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split h h1 h'
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-> split h' h2 h3
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-> split h (join h1 h2) h3.
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Proof.
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splt.
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Qed.
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Lemma split_assoc2' : forall h h1 h' h2 h3,
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split h h1 h'
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-> split h' h2 h3
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-> disjoint h1 h'
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-> disjoint h2 h3
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-> split h h2 (join h3 h1).
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Proof.
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splt.
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Qed.
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Lemma split_assoc2 : forall h h1 h' h2 h3,
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split h h' h1
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-> split h' h2 h3
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-> disjoint h' h1
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-> disjoint h2 h3
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-> split h h2 (join h3 h1).
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Proof.
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intros; eapply split_assoc2'; eauto.
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Qed.
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Lemma disjoint_assoc1 : forall h h1 h' h2 h3,
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split h h1 h'
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-> split h' h2 h3
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-> disjoint h1 h'
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-> disjoint h2 h3
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-> disjoint (join h1 h2) h3.
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Proof.
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splt.
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Qed.
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Lemma disjoint_assoc2 : forall h h1 h' h2 h3,
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split h h' h1
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-> split h' h2 h3
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-> disjoint h' h1
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-> disjoint h2 h3
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-> disjoint h2 (join h3 h1).
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Proof.
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splt.
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Qed.
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Lemma split_join : forall h1 h2,
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split (join h1 h2) h1 h2.
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Proof.
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splt.
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Qed.
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Lemma split_disjoint : forall h h1 h2 h' h3,
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split h h1 h'
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-> split h' h2 h3
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-> disjoint h1 h'
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-> disjoint h2 h3
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-> disjoint h1 h2.
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Proof.
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splt.
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Qed.
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Lemma disjoint_assoc3 : forall h h1 h2 h3,
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disjoint h h2
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-> split h h1 h3
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-> disjoint h1 h3
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-> disjoint h3 h2.
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Proof.
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splt.
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Qed.
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End splitting.
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End M.
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Export M.
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1079
SeparationLogic.v
1079
SeparationLogic.v
File diff suppressed because it is too large
Load diff
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@ -24,3 +24,4 @@ LambdaCalculusAndTypeSoundness.v
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TypesAndMutation.v
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DeepAndShallowEmbeddings_template.v
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DeepAndShallowEmbeddings.v
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SeparationLogic.v
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