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-// BQueue.bpl
-// A queue program specified in the style of dynamic frames.
-// Rustan Leino, Michal Moskal, and Wolfram Schulte, 2007.
-
-// ---------------------------------------------------------------
-
-type ref;
-const null: ref;
-
-type Field x;
-
-// this variable represents the heap; read its type as \forall \alpha. ref * Field \alpha --> \alpha
-type HeapType = <x>[ref, Field x]x;
-var H: HeapType;
-
-// every object has an 'alloc' field, which says whether or not the object has been allocated
-const unique alloc: Field bool;
-
-// for simplicity, we say that every object has one field representing its abstract value and one
-// field representing its footprint (aka frame aka data group).
-
-const unique abstractValue: Field Seq;
-const unique footprint: Field [ref]bool;
-
-// ---------------------------------------------------------------
-
-type T; // the type of the elements of the queue
-const NullT: T; // some value of type T
-
-// ---------------------------------------------------------------
-
-// Queue:
-const unique head: Field ref;
-const unique tail: Field ref;
-const unique mynodes: Field [ref]bool;
-// Node:
-const unique data: Field T;
-const unique next: Field ref;
-
-function ValidQueue(HeapType, ref) returns (bool);
-axiom (forall h: HeapType, q: ref ::
- { ValidQueue(h, q) }
- q != null && h[q,alloc] ==>
- (ValidQueue(h, q) <==>
- h[q,head] != null && h[h[q,head],alloc] &&
- h[q,tail] != null && h[h[q,tail],alloc] &&
- h[h[q,tail], next] == null &&
- // The following line can be suppressed now that we have a ValidFootprint invariant
- (forall o: ref :: { h[q,footprint][o] } o != null && h[q,footprint][o] ==> h[o,alloc]) &&
- h[q,footprint][q] &&
- h[q,mynodes][h[q,head]] && h[q,mynodes][h[q,tail]] &&
- (forall n: ref :: { h[q,mynodes][n] }
- h[q,mynodes][n] ==>
- n != null && h[n,alloc] && ValidNode(h, n) &&
- SubSet(h[n,footprint], h[q,footprint]) &&
- !h[n,footprint][q] &&
- (h[n,next] == null ==> n == h[q,tail])
- ) &&
- (forall n: ref :: { h[n,next] }
- h[q,mynodes][n] ==>
- (h[n,next] != null ==> h[q,mynodes][h[n,next]])
- ) &&
- h[q,abstractValue] == h[h[q,head],abstractValue]
- ));
-
-// frame axiom for ValidQueue
-axiom (forall h0: HeapType, h1: HeapType, n: ref ::
- { ValidQueue(h0,n), ValidQueue(h1,n) }
- (forall<alpha> o: ref, f: Field alpha :: o != null && h0[o,alloc] && h0[n,footprint][o]
- ==> h0[o,f] == h1[o,f])
- &&
- (forall<alpha> o: ref, f: Field alpha :: o != null && h1[o,alloc] && h1[n,footprint][o]
- ==> h0[o,f] == h1[o,f])
- ==>
- ValidQueue(h0,n) == ValidQueue(h1,n));
-
-function ValidNode(HeapType, ref) returns (bool);
-axiom (forall h: HeapType, n: ref ::
- { ValidNode(h, n) }
- n != null && h[n,alloc] ==>
- (ValidNode(h, n) <==>
- // The following line can be suppressed now that we have a ValidFootprint invariant
- (forall o: ref :: { h[n,footprint][o] } o != null && h[n,footprint][o] ==> h[o,alloc]) &&
- h[n,footprint][n] &&
- (h[n,next] != null ==>
- h[h[n,next],alloc] &&
- SubSet(h[h[n,next], footprint], h[n,footprint]) &&
- !h[h[n,next], footprint][n]) &&
- (h[n,next] == null ==> EqualSeq(h[n,abstractValue], EmptySeq)) &&
- (h[n,next] != null ==> EqualSeq(h[n,abstractValue],
- Append(Singleton(h[h[n,next],data]), h[h[n,next],abstractValue])))
- ));
-
-// frame axiom for ValidNode
-axiom (forall h0: HeapType, h1: HeapType, n: ref ::
- { ValidNode(h0,n), ValidNode(h1,n) }
- (forall<alpha> o: ref, f: Field alpha :: o != null && h0[o,alloc] && h0[n,footprint][o]
- ==> h0[o,f] == h1[o,f])
- &&
- (forall<alpha> o: ref, f: Field alpha :: o != null && h1[o,alloc] && h1[n,footprint][o]
- ==> h0[o,f] == h1[o,f])
- ==>
- ValidNode(h0,n) == ValidNode(h1,n));
-
-// ---------------------------------------------------------------
-
-procedure MakeQueue() returns (q: ref)
- requires ValidFootprints(H);
- modifies H;
- ensures ValidFootprints(H);
- ensures ModifiesOnlySet(old(H), H, EmptySet);
- ensures q != null && H[q,alloc];
- ensures AllNewSet(old(H), H[q,footprint]);
- ensures ValidQueue(H, q);
- ensures Length(H[q,abstractValue]) == 0;
-{
- var n: ref;
-
- assume Fresh(H,q);
- H[q,alloc] := true;
-
- call n := MakeNode(NullT);
- H[q,head] := n;
- H[q,tail] := n;
- H[q,mynodes] := SingletonSet(n);
- H[q,footprint] := UnionSet(SingletonSet(q), H[n,footprint]);
- H[q,abstractValue] := H[n,abstractValue];
-}
-
-procedure IsEmpty(q: ref) returns (isEmpty: bool)
- requires ValidFootprints(H);
- requires q != null && H[q,alloc] && ValidQueue(H, q);
- ensures isEmpty <==> Length(H[q,abstractValue]) == 0;
-{
- isEmpty := H[q,head] == H[q,tail];
-}
-
-procedure Enqueue(q: ref, t: T)
- requires ValidFootprints(H);
- requires q != null && H[q,alloc] && ValidQueue(H, q);
- modifies H;
- ensures ValidFootprints(H);
- ensures ModifiesOnlySet(old(H), H, old(H)[q,footprint]);
- ensures DifferenceIsNew(old(H), old(H)[q,footprint], H[q,footprint]);
- ensures ValidQueue(H, q);
- ensures EqualSeq(H[q,abstractValue], Append(old(H)[q,abstractValue], Singleton(t)));
-{
- var n: ref;
-
- call n := MakeNode(t);
-
- // foreach m in q.mynodes { m.footprint := m.footprint U n.footprint }
- call BulkUpdateFootprint(H[q,mynodes], H[n,footprint]);
- H[q,footprint] := UnionSet(H[q,footprint], H[n,footprint]);
-
- // foreach m in q.mynodes { m.abstractValue := Append(m.abstractValue, Singleton(t)) }
- call BulkUpdateAbstractValue(H[q,mynodes], t);
- H[q,abstractValue] := H[H[q,head],abstractValue];
-
- H[q,mynodes] := UnionSet(H[q,mynodes], SingletonSet(n));
-
- H[H[q,tail], next] := n;
- H[q,tail] := n;
-}
-
-procedure BulkUpdateFootprint(targetSet: [ref]bool, delta: [ref]bool);
- requires ValidFootprints(H);
- modifies H;
- ensures ValidFootprints(H);
- ensures ModifiesOnlySetField(old(H), H, targetSet, footprint);
- ensures (forall o: ref ::
- o != null && old(H)[o,alloc] && targetSet[o]
- ==> H[o,footprint] == UnionSet(old(H)[o,footprint], delta));
-
-procedure BulkUpdateAbstractValue(targetSet: [ref]bool, t: T);
- requires ValidFootprints(H);
- modifies H;
- ensures ValidFootprints(H);
- ensures ModifiesOnlySetField(old(H), H, targetSet, abstractValue);
- ensures (forall o: ref ::
- o != null && old(H)[o,alloc] && targetSet[o]
- ==> EqualSeq(H[o,abstractValue], Append(old(H)[o,abstractValue], Singleton(t))));
-
-procedure Front(q: ref) returns (t: T)
- requires ValidFootprints(H);
- requires q != null && H[q,alloc] && ValidQueue(H, q);
- requires 0 < Length(H[q,abstractValue]);
- ensures t == Index(H[q,abstractValue], 0);
-{
- t := H[H[H[q,head], next], data];
-}
-
-procedure Dequeue(q: ref)
- requires ValidFootprints(H);
- requires q != null && H[q,alloc] && ValidQueue(H, q);
- requires 0 < Length(H[q,abstractValue]);
- modifies H;
- ensures ValidFootprints(H);
- ensures ModifiesOnlySet(old(H), H, old(H)[q,footprint]);
- ensures DifferenceIsNew(old(H), old(H)[q,footprint], H[q,footprint]);
- ensures ValidQueue(H, q);
- ensures EqualSeq(H[q,abstractValue], Drop(old(H)[q,abstractValue], 1));
-{
- var n: ref;
-
- n := H[H[q,head], next];
- H[q,head] := n;
- // we could also remove old(H)[q,head] from H[q,mynodes], and similar for the footprints
- H[q,abstractValue] := H[n,abstractValue];
-}
-
-// --------------------------------------------------------------------------------
-
-procedure MakeNode(t: T) returns (n: ref)
- requires ValidFootprints(H);
- modifies H;
- ensures ValidFootprints(H);
- ensures ModifiesOnlySet(old(H), H, EmptySet);
- ensures n != null && H[n,alloc];
- ensures AllNewSet(old(H), H[n,footprint]);
- ensures ValidNode(H, n);
- ensures H[n,data] == t && H[n,next] == null;
-{
- assume Fresh(H,n);
- H[n,alloc] := true;
-
- H[n,next] := null;
- H[n,data] := t;
- H[n,footprint] := SingletonSet(n);
- H[n,abstractValue] := EmptySeq;
-}
-
-// --------------------------------------------------------------------------------
-
-procedure Main(t: T, u: T, v: T)
- requires ValidFootprints(H);
- modifies H;
- ensures ValidFootprints(H);
- ensures ModifiesOnlySet(old(H), H, EmptySet);
-{
- var q0, q1: ref;
- var w: T;
-
- call q0 := MakeQueue();
- call q1 := MakeQueue();
-
- call Enqueue(q0, t);
- call Enqueue(q0, u);
-
- call Enqueue(q1, v);
-
- assert Length(H[q0,abstractValue]) == 2;
-
- call w := Front(q0);
- assert w == t;
- call Dequeue(q0);
-
- call w := Front(q0);
- assert w == u;
-
- assert Length(H[q0,abstractValue]) == 1;
- assert Length(H[q1,abstractValue]) == 1;
-}
-
-// --------------------------------------------------------------------------------
-
-procedure Main2(t: T, u: T, v: T, q0: ref, q1: ref)
- requires q0 != null && H[q0,alloc] && ValidQueue(H, q0);
- requires q1 != null && H[q1,alloc] && ValidQueue(H, q1);
- requires DisjointSet(H[q0,footprint], H[q1,footprint]);
- requires Length(H[q0,abstractValue]) == 0;
-
- requires ValidFootprints(H);
- modifies H;
- ensures ValidFootprints(H);
- ensures ModifiesOnlySet(old(H), H, UnionSet(old(H)[q0,footprint], old(H)[q1,footprint]));
-{
- var w: T;
-
- call Enqueue(q0, t);
- call Enqueue(q0, u);
-
- call Enqueue(q1, v);
-
- assert Length(H[q0,abstractValue]) == 2;
-
- call w := Front(q0);
- assert w == t;
- call Dequeue(q0);
-
- call w := Front(q0);
- assert w == u;
-
- assert Length(H[q0,abstractValue]) == 1;
- assert Length(H[q1,abstractValue]) == old(Length(H[q1,abstractValue])) + 1;
-}
-
-// ---------------------------------------------------------------
-
-// Helpful predicates used in specs
-
-function ModifiesOnlySet(oldHeap: HeapType, newHeap: HeapType, set: [ref]bool) returns (bool);
-axiom (forall oldHeap: HeapType, newHeap: HeapType, set: [ref]bool ::
- { ModifiesOnlySet(oldHeap, newHeap, set) }
- ModifiesOnlySet(oldHeap, newHeap, set) <==>
- NoDeallocs(oldHeap, newHeap) &&
- (forall<alpha> o: ref, f: Field alpha :: { newHeap[o,f] }
- o != null && oldHeap[o,alloc] ==>
- oldHeap[o,f] == newHeap[o,f] || set[o]));
-
-function ModifiesOnlySetField<alpha>(oldHeap: HeapType, newHeap: HeapType,
- set: [ref]bool, field: Field alpha) returns (bool);
-axiom (forall<alpha> oldHeap: HeapType, newHeap: HeapType, set: [ref]bool, field: Field alpha ::
- { ModifiesOnlySetField(oldHeap, newHeap, set, field) }
- ModifiesOnlySetField(oldHeap, newHeap, set, field) <==>
- NoDeallocs(oldHeap, newHeap) &&
- (forall<beta> o: ref, f: Field beta :: { newHeap[o,f] }
- o != null && oldHeap[o,alloc] ==>
- oldHeap[o,f] == newHeap[o,f] || (set[o] && f == field)));
-
-function NoDeallocs(oldHeap: HeapType, newHeap: HeapType) returns (bool);
-axiom (forall oldHeap: HeapType, newHeap: HeapType ::
- { NoDeallocs(oldHeap, newHeap) }
- NoDeallocs(oldHeap, newHeap) <==>
- (forall o: ref :: { newHeap[o,alloc] }
- o != null && oldHeap[o,alloc] ==> newHeap[o,alloc]));
-
-function AllNewSet(oldHeap: HeapType, set: [ref]bool) returns (bool);
-axiom (forall oldHeap: HeapType, set: [ref]bool ::
- { AllNewSet(oldHeap, set) }
- AllNewSet(oldHeap, set) <==>
- (forall o: ref :: { oldHeap[o,alloc] }
- o != null && set[o] ==> !oldHeap[o,alloc]));
-
-function DifferenceIsNew(oldHeap: HeapType, oldSet: [ref]bool, newSet: [ref]bool) returns (bool);
-axiom (forall oldHeap: HeapType, oldSet: [ref]bool, newSet: [ref]bool ::
- { DifferenceIsNew(oldHeap, oldSet, newSet) }
- DifferenceIsNew(oldHeap, oldSet, newSet) <==>
- (forall o: ref :: { oldHeap[o,alloc] }
- o != null && !oldSet[o] && newSet[o] ==> !oldHeap[o,alloc]));
-
-function ValidFootprints(h: HeapType) returns (bool);
-axiom (forall h: HeapType ::
- { ValidFootprints(h) }
- ValidFootprints(h) <==>
- (forall o: ref, r: ref :: { h[o,footprint][r] }
- o != null && h[o,alloc] && r != null && h[o,footprint][r] ==> h[r,alloc]));
-
-function Fresh(h: HeapType, o: ref) returns (bool);
-axiom (forall h: HeapType, o: ref ::
- { Fresh(h,o) }
- Fresh(h,o) <==>
- o != null && !h[o,alloc] && h[o,footprint] == SingletonSet(o));
-
-// ---------------------------------------------------------------
-
-const EmptySet: [ref]bool;
-axiom (forall o: ref :: { EmptySet[o] } !EmptySet[o]);
-
-function SingletonSet(ref) returns ([ref]bool);
-axiom (forall r: ref :: { SingletonSet(r) } SingletonSet(r)[r]);
-axiom (forall r: ref, o: ref :: { SingletonSet(r)[o] } SingletonSet(r)[o] <==> r == o);
-
-function UnionSet([ref]bool, [ref]bool) returns ([ref]bool);
-axiom (forall a: [ref]bool, b: [ref]bool, o: ref :: { UnionSet(a,b)[o] }
- UnionSet(a,b)[o] <==> a[o] || b[o]);
-
-function SubSet([ref]bool, [ref]bool) returns (bool);
-axiom(forall a: [ref]bool, b: [ref]bool :: { SubSet(a,b) }
- SubSet(a,b) <==> (forall o: ref :: {a[o]} {b[o]} a[o] ==> b[o]));
-
-function EqualSet([ref]bool, [ref]bool) returns (bool);
-axiom(forall a: [ref]bool, b: [ref]bool :: { EqualSet(a,b) }
- EqualSet(a,b) <==> (forall o: ref :: {a[o]} {b[o]} a[o] <==> b[o]));
-
-function DisjointSet([ref]bool, [ref]bool) returns (bool);
-axiom (forall a: [ref]bool, b: [ref]bool :: { DisjointSet(a,b) }
- DisjointSet(a,b) <==> (forall o: ref :: {a[o]} {b[o]} !a[o] || !b[o]));
-
-// ---------------------------------------------------------------
-
-// Sequence of T
-type Seq;
-
-function Length(Seq) returns (int);
-axiom (forall s: Seq :: { Length(s) } 0 <= Length(s));
-
-const EmptySeq: Seq;
-axiom Length(EmptySeq) == 0;
-axiom (forall s: Seq :: { Length(s) } Length(s) == 0 ==> s == EmptySeq);
-
-function Singleton(T) returns (Seq);
-axiom (forall t: T :: { Length(Singleton(t)) } Length(Singleton(t)) == 1);
-
-function Append(Seq, Seq) returns (Seq);
-axiom (forall s0: Seq, s1: Seq :: { Length(Append(s0,s1)) }
- Length(Append(s0,s1)) == Length(s0) + Length(s1));
-
-function Index(Seq, int) returns (T);
-axiom (forall t: T :: { Index(Singleton(t), 0) } Index(Singleton(t), 0) == t);
-axiom (forall s0: Seq, s1: Seq, n: int :: { Index(Append(s0,s1), n) }
- (n < Length(s0) ==> Index(Append(s0,s1), n) == Index(s0, n)) &&
- (Length(s0) <= n ==> Index(Append(s0,s1), n) == Index(s1, n - Length(s0))));
-
-function EqualSeq(Seq, Seq) returns (bool);
-axiom (forall s0: Seq, s1: Seq :: { EqualSeq(s0,s1) }
- EqualSeq(s0,s1) <==>
- Length(s0) == Length(s1) &&
- (forall j: int :: { Index(s0,j) } { Index(s1,j) }
- 0 <= j && j < Length(s0) ==> Index(s0,j) == Index(s1,j)));
-
-function Take(s: Seq, howMany: int) returns (Seq);
-axiom (forall s: Seq, n: int :: { Length(Take(s,n)) }
- 0 <= n ==>
- (n <= Length(s) ==> Length(Take(s,n)) == n) &&
- (Length(s) < n ==> Length(Take(s,n)) == Length(s)));
-axiom (forall s: Seq, n: int, j: int :: { Index(Take(s,n), j) }
- 0 <= j && j < n && j < Length(s) ==>
- Index(Take(s,n), j) == Index(s, j));
-
-function Drop(s: Seq, howMany: int) returns (Seq);
-axiom (forall s: Seq, n: int :: { Length(Drop(s,n)) }
- 0 <= n ==>
- (n <= Length(s) ==> Length(Drop(s,n)) == Length(s) - n) &&
- (Length(s) < n ==> Length(Drop(s,n)) == 0));
-axiom (forall s: Seq, n: int, j: int :: { Index(Drop(s,n), j) }
- 0 <= n && 0 <= j && j < Length(s)-n ==>
- Index(Drop(s,n), j) == Index(s, j+n));
-
-// ---------------------------------------------------------------
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