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// RUN: %dafny /compile:0 /dprint:"%t.dprint" "%s" > "%t"
// RUN: %diff "%s.expect" "%t"
// Dafny transcription of the Coq development of insertion sort, as found in Coq'Art.
datatype List<T> = Nil | Cons(head: T, tail: List)
predicate sorted(l: List<int>)
{
match l
case Nil => true
case Cons(x, rest) =>
match rest
case Nil => true
case Cons(y, _) => x <= y && sorted(rest)
}
lemma example_sort_2357()
ensures sorted(Cons(2, Cons(3, Cons(5, Cons(7, Nil)))));
{
}
lemma sorted_inv(z: int, l: List<int>)
requires sorted(Cons(z, l));
ensures sorted(l);
{
match l {
case Nil =>
case Cons(_, _) =>
}
}
// Number of occurrences of z in l
function nb_occ(z: int, l: List<int>): nat
{
match l
case Nil => 0
case Cons(z', l') =>
(if z == z' then 1 else 0) + nb_occ(z, l')
}
lemma example_nb_occ_0()
ensures nb_occ(3, Cons(3, Cons(7, Cons(3, Nil)))) == 2;
{
}
lemma example_nb_occ_1()
ensures nb_occ(36725, Cons(3, Cons(7, Cons(3, Nil)))) == 0;
{
}
// list l' is a permutation of list l
predicate equiv(l: List<int>, l': List<int>)
{
forall z :: nb_occ(z, l) == nb_occ(z, l')
}
// equiv is an equivalence
lemma equiv_refl(l: List<int>)
ensures equiv(l, l);
{
}
lemma equiv_sym(l: List<int>, l': List<int>)
requires equiv(l, l');
ensures equiv(l', l);
{
}
lemma equiv_trans(l: List<int>, l': List<int>, l'': List<int>)
requires equiv(l, l') && equiv(l', l'');
ensures equiv(l, l'');
{
}
lemma equiv_cons(z: int, l: List<int>, l': List<int>)
requires equiv(l, l');
ensures equiv(Cons(z, l), Cons(z, l'));
{
}
lemma equiv_perm(a: int, b: int, l: List<int>, l': List<int>)
requires equiv(l, l');
ensures equiv(Cons(a, Cons(b, l)), Cons(b, Cons(a, l')));
{
var L, L' := Cons(a, Cons(b, l)), Cons(b, Cons(a, l'));
forall z {
calc {
nb_occ(z, L);
(if z == a && z == b then 2 else if z == a || z == b then 1 else 0) + nb_occ(z, l);
(if z == a && z == b then 2 else if z == a || z == b then 1 else 0) + nb_occ(z, l');
nb_occ(z, L');
}
}
}
// insertion of z into l at the right place (assuming l is sorted)
function method aux(z: int, l: List<int>): List<int>
{
match l
case Nil => Cons(z, Nil)
case Cons(a, l') =>
if z <= a then Cons(z, l) else Cons(a, aux(z, l'))
}
lemma example_aux_0()
ensures aux(4, Cons(2, Cons(5, Nil))) == Cons(2, Cons(4, Cons(5, Nil)));
{
}
lemma example_aux_1()
ensures aux(4, Cons(24, Cons(50, Nil))) == Cons(4, Cons(24, Cons(50, Nil)));
{
}
// the aux function seems to be a good tool for sorting...
lemma aux_equiv(l: List<int>, x: int)
ensures equiv(Cons(x, l), aux(x, l));
{
match l {
case Nil =>
case Cons(_, _) =>
}
}
lemma aux_sorted(l: List<int>, x: int)
requires sorted(l);
ensures sorted(aux(x, l));
{
match l {
case Nil =>
case Cons(_, l') =>
match l' {
case Nil =>
case Cons(_, _) =>
}
}
}
// the sorting function
function sort(l: List<int>): List<int>
ensures var l' := sort(l); equiv(l, l') && sorted(l');
{
existence_proof(l);
var l' :| equiv(l, l') && sorted(l'); l'
}
lemma existence_proof(l: List<int>)
ensures exists l' :: equiv(l, l') && sorted(l');
{
match l {
case Nil =>
case Cons(x, m) =>
existence_proof(m);
var m' :| equiv(m, m') && sorted(m');
calc ==> {
equiv(m, m') && sorted(m');
equiv(l, Cons(x, m')) && sorted(m');
{ aux_equiv(m', x); }
equiv(l, aux(x, m')) && sorted(m');
{ aux_sorted(m', x); }
equiv(l, aux(x, m')) && sorted(aux(x, m'));
}
}
}
// to get a compilable function in Dafny
function method Sort(l: List<int>): List<int>
ensures equiv(l, Sort(l)) && sorted(Sort(l));
{
match l
case Nil => l
case Cons(x, m) =>
var m' := Sort(m);
assert equiv(l, Cons(x, m'));
aux_equiv(m', x);
aux_sorted(m', x);
aux(x, m')
}
predicate p_aux_equiv(l: List<int>, x: int)
ensures equiv(Cons(x, l), aux(x, l));
{
aux_equiv(l, x);
true
}
predicate p_aux_sorted(l: List<int>, x: int)
requires sorted(l);
ensures sorted(aux(x, l));
{
aux_sorted(l, x);
true
}
|
