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use builtin::*;
use builtin_macros::*;
use vstd::seq::*;
use vstd::set::*;
verus!{
pub open spec fn Injective<X,Y>(f: spec_fn(X) -> Y) -> bool
{
forall |x1:X, x2:X| #![trigger f(x1), f(x2)] f(x1) == f(x2) ==> x1 == x2
}
pub open spec fn InjectiveOver<X, Y>(xs:Set<X>, ys:Set<Y>, f: spec_fn(X) -> Y) -> bool
// reads f.reads
// requires forall x :: x in xs ==> f.requires(x)
{
forall |x1:X, x2:X| #![trigger f(x1), f(x2)] xs.contains(x1) && xs.contains(x2) && ys.contains(f(x1)) && ys.contains(f(x2)) && f(x1) == f(x2) ==> x1 == x2
}
pub open spec fn MapSeqToSet<X,Y>(xs:Seq<X>, f: spec_fn(X) -> Y) -> Set<Y>
recommends Injective(f)
{
Set::new(|y:Y| exists |x:X| xs.contains(x) && f(x) == y)
}
pub proof fn lemma_MapSeqToSet<X,Y>(xs:Seq<X>, f: spec_fn(X) -> Y)
requires Injective(f)
ensures forall |x:X| #[trigger] xs.contains(x) <==> MapSeqToSet(xs, f).contains(f(x))
{
}
// let x = choose |x:int| s.contains(x);
// if s.len() == 1 {
// x
// } else {
// let sy = s.remove(x);
// let y = intsetmax(sy);
// assert(forall |i:int| s.contains(y) ==> sy.contains(i) || i == x);
// if x > y {
// x
// } else {
// y
// }
// }
pub open spec fn intsetmax(s: Set<int>) -> int
recommends s.len() > 0
{
choose |m: int|
s.contains(m) &&
forall |i: int| s.contains(i) ==> m >= i
}
#[verifier::external_body]
pub proof fn lemma_intsetmax_ensures(s: Set<int>)
requires s.len() > 0
ensures ({
let m = intsetmax(s);
&&& s.contains(m)
&&& forall |i: int| s.contains(i) ==> m >= i
})
{
reveal(intsetmax);
// let m = intsetmax(s);
// // Use `assert_by` to introduce the `choose` properties explicitly
// assert_by({
// s.contains(m) && forall |i: int| s.contains(i) ==> m >= i
// }, {
// assert(s.contains(m)); // Verifies the first condition
// assert(forall |i: int| s.contains(i) ==> m >= i);
// });
}
#[verifier::external_body]
pub proof fn SetNotEmpty<T>(s:Set<T>)
requires exists |x:T| s.contains(x),
ensures s.len()>0
{
}
#[verifier::external_body]
pub proof fn lemma_MapSetCardinalityOver<X, Y>(xs: Set<X>, ys: Set<Y>, f: spec_fn(X) -> Y)
requires
InjectiveOver(xs, ys, f),
forall |x: X| xs.contains(x) ==> ys.contains(f(x)),
forall |y: Y| ys.contains(y) ==> exists |x: X| xs.contains(x) && y == f(x),
ensures
xs.len() == ys.len(),
decreases xs.len(), ys.len()
{
if xs.len() > 0 {
let x = choose |x: X| xs.contains(x);
let xs_prime = xs.remove(x);
assert(xs_prime.len() < xs.len());
let ys_prime = ys.remove(f(x));
assert(ys_prime.len() < ys.len());
lemma_MapSetCardinalityOver(xs_prime, ys_prime, f);
}
}
#[verifier::external_body]
pub proof fn ThingsIKnowAboutSubset<T>(x:Set<T>, y:Set<T>)
requires x.subset_of(y)
ensures x.len()<y.len()
decreases x.len()
{
if (!x.is_empty()) {
let e = choose |e:T| x.contains(e);
ThingsIKnowAboutSubset(x.remove(e), y.remove(e));
}
}
#[verifier::external_body]
pub proof fn SubsetCardinality<T>(x:Set<T>, y:Set<T>)
ensures x.subset_of(y) ==> x.len() < y.len(),
(x.subset_of(y) || x==y) ==> x.len() <= y.len()
{
if (x.subset_of(y)) {
}
if (x==y) {
}
}
#[verifier::external_body]
pub proof fn subset_cardinality<T>(x:Set<T>, y:Set<T>)
requires x.subset_of(y)
ensures x.len() <= y.len()
{
// if x.is_empty() {
// assert(x.len() == 0);
// assert(0 <= y.len());
// } else {
// let e: T = choose |e: T| x.contains(e);
// let x_ = x.remove(e);
// let y_ = y.remove(e);
// assert(y.contains(e));
// spec_apply(subset_remove(x, y, e));
// spec_apply(set_remove_len(x, e));
// spec_apply(set_remove_len(y, e));
// subset_cardinality(x_, y_);
// assert(x.len() == x_.len() + 1);
// assert(y.len() == y_.len() + 1);
// assert(x_.len() + 1 <= y_.len() + 1);
// }
}
#[verifier::external_body]
pub proof fn InsertCardinality<T>(s:Set<T>, x:T)
requires forall |y:T| s.contains(y) ==> y != x
ensures s.insert(x).len() == s.len() + 1
{
}
#[verifier::external_body]
pub proof fn subset_len_equal_implies_equal<T>(s1: Set<T>, s2: Set<T>)
requires
s1.subset_of(s2),
s1.len() == s2.len()
ensures
s1 == s2
{}
}