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|
// RUN: %dafny /rprint:"%t.rprint" /autoTriggers:1 "%s" > "%t"
// RUN: %diff "%s.expect" "%t"
// Rustan Leino, Nov 2015
// Module M0 gives the high-level specification of the UnionFind data structure
abstract module M0 {
class Element { }
class {:autocontracts} UnionFind {
ghost var M: map<Element, Element>
protected predicate Valid()
reads this, Repr
constructor ()
ensures M == map[]
{
Repr, M:= {this}, map[];
}
method New() returns (e: Element)
ensures old(e !in M && forall e' :: e' in M ==> M[e'] != e)
ensures M == old(M)[e := e]
method Find(e: Element) returns (r: Element)
requires e in M
ensures M == old(M) && M[e] == r
method Union(e0: Element, e1: Element) returns (ghost r: Element)
requires e0 in M && e1 in M
ensures r in old(M) && old(M[r] == M[e0] || M[r] == M[e1])
ensures M == map e | e in old(M) :: old(if M[e] == M[e0] || M[e] == M[e1] then r else M[e])
}
}
// Module M1 gives the concrete data structure used in UnionFind, along with the invariant of
// that data structure. It also implements the New method and, by declaring a new method Join,
// the Union method.
abstract module M1 refines M0 {
datatype Contents = Root(depth: nat) | Link(next: Element)
class Element {
var c: Contents
}
type CMap = map<Element, Contents>
predicate GoodCMap(C: CMap)
{
null !in C && forall f :: f in C && C[f].Link? ==> C[f].next in C
}
class UnionFind {
protected predicate Valid...
{
(forall e :: e in M ==> e in Repr && M[e] in M && M[M[e]] == M[e]) &&
(forall e :: e in M && e.c.Link? ==> e.c.next in M) &&
(forall e :: e in M ==> M[e].c.Root? && Reaches(M[e].c.depth, e, M[e], Collect()))
}
// This function returns a snapshot of the .c fields of the objects in the domain of M
function Collect(): CMap
requires null !in M && forall f :: f in M && f.c.Link? ==> f.c.next in M
reads this, set a | a in M
ensures GoodCMap(Collect())
{
map e | e in M :: e.c
}
predicate Reaches(d: nat, e: Element, r: Element, C: CMap)
requires GoodCMap(C)
requires e in C
{
match C[e]
case Root(_) => e == r
case Link(next) => d != 0 && Reaches(d-1, next, r, C)
}
lemma {:autocontracts false} Reaches_Monotonic(d: nat, e: Element, r: Element, C: CMap, C': CMap)
requires GoodCMap(C) && GoodCMap(C')
requires e in C
requires Reaches(d, e, r, C) && forall f :: f in C ==> f in C' && C[f] == C'[f]
ensures Reaches(d, e, r, C')
{
}
method New...
{
e := new Element;
e.c := Root(0);
Repr := Repr + {e};
M := M[e := e];
assert Reaches(0, e, e, Collect());
forall f | f in M
ensures M[f].c.Root? && Reaches(M[f].c.depth, f, M[f], Collect())
{
if f != e {
Reaches_Monotonic(M[f].c.depth, f, M[f], old(Collect()), Collect());
}
}
}
method Union...
{
var r0 := Find(e0);
var r1 := Find(e1);
r := Join(r0, r1);
}
method Join(r0: Element, r1: Element) returns (ghost r: Element)
requires r0 in M && r1 in M && M[r0] == r0 && M[r1] == r1
ensures r == r0 || r == r1
ensures M == map e | e in old(M) :: old(if M[e] == r0 || M[e] == r1 then r else M[e])
}
// stupid help lemma to get around boxing
lemma MapsEqual<D>(m: map<D, D>, n: map<D, D>)
requires forall d :: d in m <==> d in n;
requires forall d :: d in m ==> m[d] == n[d]
ensures m == n
{
}
}
// Module M2 adds the implementation of Find, together with the proofs needed for the verification.
abstract module M2 refines M1 {
class UnionFind {
method Find...
{
r := FindAux(M[e].c.depth, e);
}
lemma NextReachesSame(e: Element)
requires e in M && e.c.Link?
ensures M[e] == M[e.c.next]
{
var next := e.c.next;
var d0, d1 := M[e].c.depth, M[next].c.depth;
assert Reaches(d0 - 1, next, M[e], Collect());
assert Reaches(d1, next, M[next], Collect());
Reaches_SinkIsFunctionOfStart(d0 - 1, d1, next, M[e], M[next], Collect());
}
lemma {:autocontracts false} Reaches_SinkIsFunctionOfStart(d0: nat, d1: nat, e: Element, r0: Element, r1: Element, C: CMap)
requires GoodCMap(C)
requires e in C
requires Reaches(d0, e, r0, C) && Reaches(d1, e, r1, C)
ensures r0 == r1
{
}
method FindAux(ghost d: nat, e: Element) returns (r: Element)
requires e in M && Reaches(d, e, M[e], Collect())
ensures M == old(M) && M[e] == r
ensures forall d: nat, e, r :: e in old(Collect()) && Reaches(d, e, r, old(Collect())) ==> Reaches(d, e, r, Collect())
{
match e.c
case Root(_) =>
r := e;
case Link(next) =>
NextReachesSame(e);
r := FindAux(d-1, next);
ghost var C := Collect(); // take a snapshot of all the .c fields
e.c := Link(r);
UpdateMaintainsReaches(d, e, r, C, Collect());
}
lemma {:autocontracts false} UpdateMaintainsReaches(td: nat, tt: Element, tm: Element, C: CMap, C': CMap)
requires GoodCMap(C)
requires tt in C && Reaches(td, tt, tm, C)
requires C' == C[tt := Link(tm)] && C'[tt].Link? && tm in C' && C'[tm].Root?
requires null !in C' && forall f :: f in C && C'[f].Link? ==> C'[f].next in C
ensures forall d: nat, e, r :: e in C && Reaches(d, e, r, C) ==> Reaches(d, e, r, C')
{
forall d: nat, e, r | e in C && Reaches(d, e, r, C)
ensures Reaches(d, e, r, C')
{
ConstructReach(d, e, r, C, td, tt, tm, C');
}
}
lemma {:autocontracts false} ConstructReach(d: nat, e: Element, r: Element, C: CMap, td: nat, tt: Element, tm: Element, C': CMap)
requires GoodCMap(C)
requires e in C
requires Reaches(d, e, r, C)
requires tt in C && Reaches(td, tt, tm, C);
requires C' == C[tt := Link(tm)] && C'[tt].Link? && tm in C' && C'[tm].Root?
requires GoodCMap(C')
ensures Reaches(d, e, r, C')
{
if e == tt {
Reaches_SinkIsFunctionOfStart(d, td, e, r, tm, C);
} else {
match C[e]
case Root(_) =>
case Link(next) =>
ConstructReach(d-1, next, r, C, td, tt, tm, C');
}
}
}
}
// Finally, module M3 adds the implementation of Join, along with what's required to
// verify its correctness.
module M3 refines M2 {
class UnionFind {
method Join...
{
if r0 == r1 {
r := r0;
MapsEqual(M, map e | e in M :: if M[e] == r0 || M[e] == r1 then r else M[e]);
return;
} else if r0.c.depth < r1.c.depth {
r0.c := Link(r1);
r := r1;
M := map e | e in M :: if M[e] == r0 || M[e] == r1 then r else M[e];
forall e | e in Collect()
ensures M[e].c.Root? && Reaches(M[e].c.depth, e, M[e], Collect())
{
JoinMaintainsReaches0(r0, r1, old(Collect()), Collect());
assert Reaches(old(M[e].c).depth, e, old(M)[e], old(Collect()));
}
} else if r1.c.depth < r0.c.depth {
r1.c := Link(r0);
r := r0;
M := map e | e in M :: if M[e] == r0 || M[e] == r1 then r else M[e];
forall e | e in Collect()
ensures M[e].c.Root? && Reaches(M[e].c.depth, e, M[e], Collect())
{
JoinMaintainsReaches0(r1, r0, old(Collect()), Collect());
assert Reaches(old(M[e].c).depth, e, old(M)[e], old(Collect()));
}
} else {
r0.c := Link(r1);
r1.c := Root(r1.c.depth + 1);
r := r1;
M := map e | e in M :: if M[e] == r0 || M[e] == r1 then r else M[e];
forall e | e in Collect()
ensures M[e].c.Root? && Reaches(M[e].c.depth, e, M[e], Collect())
{
JoinMaintainsReaches1(r0, r1, old(Collect()), Collect());
assert Reaches(old(M[e].c).depth, e, old(M)[e], old(Collect()));
}
}
}
lemma {:autocontracts false} JoinMaintainsReaches1(r0: Element, r1: Element, C: CMap, C': CMap)
requires GoodCMap(C)
requires r0 in C && r1 in C && C[r0].Root? && C[r1].Root? && C[r0].depth == C[r1].depth && r0 != r1
requires C' == C[r0 := Link(r1)][r1 := Root(C[r1].depth + 1)]
requires GoodCMap(C')
ensures forall d: nat, e, r :: e in C && Reaches(d, e, r, C) && r != r0 && r != r1 ==> Reaches(d, e, r, C') // proved automatically by induction
ensures forall e :: e in C && Reaches(C[r0].depth, e, r0, C) ==> Reaches(C'[r1].depth, e, r1, C')
ensures forall e :: e in C && Reaches(C[r1].depth, e, r1, C) ==> Reaches(C'[r1].depth, e, r1, C')
{
forall e | e in C && Reaches(C[r0].depth, e, r0, C)
ensures Reaches(C'[r1].depth, e, r1, C')
{
ExtendedReach'(e, C, C[r0].depth, C'[r1].depth, r0, r1, C');
}
forall e | e in C && Reaches(C[r1].depth, e, r1, C)
ensures Reaches(C'[r1].depth, e, r1, C')
{
ReachUnaffectedByChangeFromRoot'(C[r1].depth, e, r1, C, C[r0].depth, r0, r1, C');
}
}
lemma {:autocontracts false} ReachUnaffectedByChangeFromRoot'(d: nat, e: Element, r: Element, C: CMap, td: nat, r0: Element, r1: Element, C': CMap)
requires GoodCMap(C)
requires e in C && Reaches(d, e, r, C)
requires r0 in C && r1 in C && C[r0].Root? && C[r1].Root? && C[r0].depth == C[r1].depth && r0 != r1
requires C[r0].Root? && C' == C[r0 := Link(r1)][r1 := Root(C[r1].depth + 1)]
requires Reaches(td, r0, r0, C) && r0 != r
requires GoodCMap(C')
ensures Reaches(d+1, e, r, C')
{
}
lemma {:autocontracts false} ExtendedReach'(e: Element, C: CMap, d0: nat, d1: nat, r0: Element, r1: Element, C': CMap)
requires GoodCMap(C) && GoodCMap(C')
requires r0 in C && r1 in C && C[r0].Root? && C[r1].Root? && C[r0].depth == C[r1].depth && r0 != r1
requires C' == C[r0 := Link(r1)][r1 := Root(C[r1].depth + 1)]
requires C[r0].Root? && d0 <= C[r0].depth && C[r1].Root? && d1 <= C'[r1].depth && d0 < d1
requires e in C && Reaches(d0, e, r0, C)
ensures Reaches(d1, e, r1, C')
{
match C[e]
case Root(_) =>
case Link(next) =>
ExtendedReach'(next, C, d0-1, d1-1, r0, r1, C');
}
lemma {:autocontracts false} JoinMaintainsReaches0(r0: Element, r1: Element, C: CMap, C': CMap)
requires GoodCMap(C)
requires r0 in C && r1 in C && C[r0].Root? && C[r1].Root? && C[r0].depth < C[r1].depth
requires C' == C[r0 := Link(r1)]
requires GoodCMap(C')
ensures forall d: nat, e, r :: e in C && Reaches(d, e, r, C) && r != r0 ==> Reaches(d, e, r, C')
ensures forall e :: e in C && Reaches(C[r0].depth, e, r0, C) ==> Reaches(C[r1].depth, e, r1, C')
{
forall d: nat, e, r | e in C && Reaches(d, e, r, C) && r != r0
ensures Reaches(d, e, r, C')
{
ReachUnaffectedByChangeFromRoot(d, e, r, C, C[r0].depth, r0, r1, C');
}
forall e | e in C && Reaches(C[r0].depth, e, r0, C)
ensures Reaches(C[r1].depth, e, r1, C')
{
ExtendedReach(e, C, C[r0].depth, C[r1].depth, r0, r1, C');
}
}
lemma {:autocontracts false} ReachUnaffectedByChangeFromRoot(d: nat, e: Element, r: Element, C: CMap, td: nat, tt: Element, tm: Element, C': CMap)
requires GoodCMap(C)
requires e in C && Reaches(d, e, r, C)
requires tt in C && Reaches(td, tt, tt, C) && tt != r
requires C[tt].Root? && C' == C[tt := Link(tm)]
requires GoodCMap(C')
ensures Reaches(d, e, r, C')
{
}
lemma {:autocontracts false} ExtendedReach(e: Element, C: CMap, d0: nat, d1: nat, r0: Element, r1: Element, C': CMap)
requires GoodCMap(C) && GoodCMap(C')
requires r0 in C && r1 in C && r0 != r1
requires C' == C[r0 := Link(r1)]
requires C[r0].Root? && d0 <= C[r0].depth && C[r1].Root? && d1 <= C[r1].depth && d0 < d1
requires e in C && Reaches(d0, e, r0, C)
ensures Reaches(d1, e, r1, C')
{
match C[e]
case Root(_) =>
case Link(next) =>
ExtendedReach(next, C, d0-1, d1-1, r0, r1, C');
}
}
}
|
