From d652155ae013f36a1ee17653a8e458baad2d9c2c Mon Sep 17 00:00:00 2001 From: Checkmate50 Date: Mon, 6 Jun 2016 23:14:18 -0600 Subject: Merging complete. Everything looks good *crosses fingers* --- Test/textbook/BQueue.bpl | 864 +++++++++++++++++++++++------------------------ 1 file changed, 432 insertions(+), 432 deletions(-) (limited to 'Test/textbook/BQueue.bpl') diff --git a/Test/textbook/BQueue.bpl b/Test/textbook/BQueue.bpl index f224334c..3fdc407c 100644 --- a/Test/textbook/BQueue.bpl +++ b/Test/textbook/BQueue.bpl @@ -1,432 +1,432 @@ -// RUN: %boogie "%s" > "%t" -// RUN: %diff "%s.expect" "%t" -// 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 = [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 o: ref, f: Field alpha :: o != null && h0[o,alloc] && h0[n,footprint][o] - ==> h0[o,f] == h1[o,f]) - && - (forall 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 o: ref, f: Field alpha :: o != null && h0[o,alloc] && h0[n,footprint][o] - ==> h0[o,f] == h1[o,f]) - && - (forall 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 o: ref, f: Field alpha :: { newHeap[o,f] } - o != null && oldHeap[o,alloc] ==> - oldHeap[o,f] == newHeap[o,f] || set[o])); - -function ModifiesOnlySetField(oldHeap: HeapType, newHeap: HeapType, - set: [ref]bool, field: Field alpha) returns (bool); -axiom (forall oldHeap: HeapType, newHeap: HeapType, set: [ref]bool, field: Field alpha :: - { ModifiesOnlySetField(oldHeap, newHeap, set, field) } - ModifiesOnlySetField(oldHeap, newHeap, set, field) <==> - NoDeallocs(oldHeap, newHeap) && - (forall 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)); - -// --------------------------------------------------------------- +// RUN: %boogie "%s" > "%t" +// RUN: %diff "%s.expect" "%t" +// 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 = [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 o: ref, f: Field alpha :: o != null && h0[o,alloc] && h0[n,footprint][o] + ==> h0[o,f] == h1[o,f]) + && + (forall 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 o: ref, f: Field alpha :: o != null && h0[o,alloc] && h0[n,footprint][o] + ==> h0[o,f] == h1[o,f]) + && + (forall 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 o: ref, f: Field alpha :: { newHeap[o,f] } + o != null && oldHeap[o,alloc] ==> + oldHeap[o,f] == newHeap[o,f] || set[o])); + +function ModifiesOnlySetField(oldHeap: HeapType, newHeap: HeapType, + set: [ref]bool, field: Field alpha) returns (bool); +axiom (forall oldHeap: HeapType, newHeap: HeapType, set: [ref]bool, field: Field alpha :: + { ModifiesOnlySetField(oldHeap, newHeap, set, field) } + ModifiesOnlySetField(oldHeap, newHeap, set, field) <==> + NoDeallocs(oldHeap, newHeap) && + (forall 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)); + +// --------------------------------------------------------------- -- cgit v1.2.3