vstd/seq_lib.rs
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#[allow(unused_imports)]
use super::calc_macro::*;
#[allow(unused_imports)]
use super::multiset::Multiset;
#[allow(unused_imports)]
use super::pervasive::*;
#[allow(unused_imports)]
use super::prelude::*;
#[allow(unused_imports)]
use super::relations::*;
#[allow(unused_imports)]
use super::seq::*;
#[allow(unused_imports)]
use super::set::Set;
verus! {
broadcast use group_seq_axioms;
impl<A> Seq<A> {
/// Applies the function `f` to each element of the sequence, and returns
/// the resulting sequence.
/// The `int` parameter of `f` is the index of the element being mapped.
// TODO(verus): rename to map_entries, for consistency with Map::map
pub open spec fn map<B>(self, f: spec_fn(int, A) -> B) -> Seq<B> {
Seq::new(self.len(), |i: int| f(i, self[i]))
}
/// Applies the function `f` to each element of the sequence, and returns
/// the resulting sequence.
// TODO(verus): rename to map, because this is what everybody wants.
pub open spec fn map_values<B>(self, f: spec_fn(A) -> B) -> Seq<B> {
Seq::new(self.len(), |i: int| f(self[i]))
}
/// Applies the function `f` to each element of the sequence,
/// producing a sequence of sequences, and then concatenates (flattens)
/// those into a single flat sequence of `B`.
///
/// ## Example
///
/// ```rust
/// fn example() {
/// let s = seq![1, 2, 3];
/// let result = s.flat_map(|x| seq![x, x]);
/// assert_eq!(result, seq![1, 1, 2, 2, 3, 3]);
/// }
/// ``
pub open spec fn flat_map<B>(self, f: spec_fn(A) -> Seq<B>) -> Seq<B> {
self.map_values(f).flatten()
}
/// Is true if the calling sequence is a prefix of the given sequence 'other'.
///
/// ## Example
///
/// ```rust
/// proof fn prefix_test() {
/// let pre: Seq<int> = seq![1, 2, 3];
/// let whole: Seq<int> = seq![1, 2, 3, 4, 5];
/// assert(pre.is_prefix_of(whole));
/// }
/// ```
pub open spec fn is_prefix_of(self, other: Self) -> bool {
self.len() <= other.len() && self =~= other.subrange(0, self.len() as int)
}
/// Is true if the calling sequence is a suffix of the given sequence 'other'.
///
/// ## Example
///
/// ```rust
/// proof fn suffix_test() {
/// let end: Seq<int> = seq![3, 4, 5];
/// let whole: Seq<int> = seq![1, 2, 3, 4, 5];
/// assert(end.is_suffix_of(whole));
/// }
/// ```
pub open spec fn is_suffix_of(self, other: Self) -> bool {
self.len() <= other.len() && self =~= other.subrange(
(other.len() - self.len()) as int,
other.len() as int,
)
}
/// Sorts the sequence according to the given leq function
///
/// ## Example
///
/// ```rust
/// {{#include ../../../../examples/multiset.rs:sorted_by_leq}}
/// ```
pub closed spec fn sort_by(self, leq: spec_fn(A, A) -> bool) -> Seq<A>
recommends
total_ordering(leq),
decreases self.len(),
{
if self.len() <= 1 {
self
} else {
let split_index = self.len() / 2;
let left = self.subrange(0, split_index as int);
let right = self.subrange(split_index as int, self.len() as int);
let left_sorted = left.sort_by(leq);
let right_sorted = right.sort_by(leq);
merge_sorted_with(left_sorted, right_sorted, leq)
}
}
/// Tests if all elements in the sequence satisfy the predicate.
///
/// ## Example
///
/// ```rust
/// fn example() {
/// let s = seq![2, 4, 6, 8];
/// assert!(s.all(|x| x % 2 == 0));
/// }
/// ```
pub open spec fn all(self, pred: spec_fn(A) -> bool) -> bool {
forall|i: int| 0 <= i < self.len() ==> #[trigger] pred(self[i])
}
/// Tests if any element in the sequence satisfies the predicate.
///
/// ## Example
///
/// ```rust
/// fn example() {
/// let s = seq![1, 2, 3, 4];
/// assert!(s.any(|x| x > 3));
/// }
/// ```
pub open spec fn any(self, pred: spec_fn(A) -> bool) -> bool {
exists|i: int| 0 <= i < self.len() && #[trigger] pred(self[i])
}
/// Checks that exactly one element in the sequence satisfies the given predicate.
/// ## Example
///
/// ```rust
/// fn example() {
/// let s = seq![1, 2, 3];
/// assert!(s.exactly_one(|x| x == 2));
/// }
/// ```
pub open spec fn exactly_one(self, pred: spec_fn(A) -> bool) -> bool {
self.filter(pred).len() == 1
}
pub proof fn lemma_sort_by_ensures(self, leq: spec_fn(A, A) -> bool)
requires
total_ordering(leq),
ensures
self.to_multiset() =~= self.sort_by(leq).to_multiset(),
sorted_by(self.sort_by(leq), leq),
forall|x: A| !self.contains(x) ==> !(#[trigger] self.sort_by(leq).contains(x)),
decreases self.len(),
{
if self.len() <= 1 {
} else {
let split_index = self.len() / 2;
let left = self.subrange(0, split_index as int);
let right = self.subrange(split_index as int, self.len() as int);
assert(self =~= left + right);
let left_sorted = left.sort_by(leq);
left.lemma_sort_by_ensures(leq);
let right_sorted = right.sort_by(leq);
right.lemma_sort_by_ensures(leq);
lemma_merge_sorted_with_ensures(left_sorted, right_sorted, leq);
lemma_multiset_commutative(left, right);
lemma_multiset_commutative(left_sorted, right_sorted);
assert forall|x: A| !self.contains(x) implies !(#[trigger] self.sort_by(leq).contains(
x,
)) by {
broadcast use group_to_multiset_ensures;
assert(!self.contains(x) ==> self.to_multiset().count(x) == 0);
}
}
}
/// Returns the sequence containing only the elements of the original sequence
/// such that pred(element) is true.
///
/// ## Example
///
/// ```rust
/// proof fn filter_test() {
/// let seq: Seq<int> = seq![1, 2, 3, 4, 5];
/// let even: Seq<int> = seq.filter(|x| x % 2 == 0);
/// reveal_with_fuel(Seq::<int>::filter, 6); //Needed for Verus to unfold the recursive definition of filter
/// assert(even =~= seq![2, 4]);
/// }
/// ```
#[verifier::opaque]
pub open spec fn filter(self, pred: spec_fn(A) -> bool) -> Self
decreases self.len(),
{
if self.len() == 0 {
self
} else {
let subseq = self.drop_last().filter(pred);
if pred(self.last()) {
subseq.push(self.last())
} else {
subseq
}
}
}
pub broadcast proof fn lemma_filter_len(self, pred: spec_fn(A) -> bool)
ensures
// the filtered list can't grow
#[trigger] self.filter(pred).len() <= self.len(),
decreases self.len(),
{
reveal(Seq::filter);
let out = self.filter(pred);
if 0 < self.len() {
self.drop_last().lemma_filter_len(pred);
}
}
pub broadcast proof fn lemma_filter_pred(self, pred: spec_fn(A) -> bool, i: int)
requires
0 <= i < self.filter(pred).len(),
ensures
pred(#[trigger] self.filter(pred)[i]),
{
// TODO: remove this after proved filter_lemma is proved
#[allow(deprecated)]
self.filter_lemma(pred);
}
pub broadcast proof fn lemma_filter_contains(self, pred: spec_fn(A) -> bool, i: int)
requires
0 <= i < self.len() && pred(self[i]),
ensures
#[trigger] self.filter(pred).contains(self[i]),
{
// TODO: remove this after proved filter_lemma is proved
#[allow(deprecated)]
self.filter_lemma(pred);
}
// deprecated since the triggers inside of 2 of the conjuncts are blocked
#[deprecated = "Use `broadcast use group_filter_ensures` instead" ]
pub proof fn filter_lemma(self, pred: spec_fn(A) -> bool)
ensures
// we don't keep anything bad
// TODO(andrea): recommends didn't catch this error, where i isn't known to be in
// self.filter(pred).len()
//forall |i: int| 0 <= i < self.len() ==> pred(#[trigger] self.filter(pred)[i]),
forall|i: int|
0 <= i < self.filter(pred).len() ==> pred(#[trigger] self.filter(pred)[i]),
// we keep everything we should
forall|i: int|
0 <= i < self.len() && pred(self[i]) ==> #[trigger] self.filter(pred).contains(
self[i],
),
// the filtered list can't grow
#[trigger] self.filter(pred).len() <= self.len(),
decreases self.len(),
{
reveal(Seq::filter);
let out = self.filter(pred);
if 0 < self.len() {
self.drop_last().filter_lemma(pred);
assert forall|i: int| 0 <= i < out.len() implies pred(out[i]) by {
if i < out.len() - 1 {
assert(self.drop_last().filter(pred)[i] == out.drop_last()[i]); // trigger drop_last
assert(pred(out[i])); // TODO(andrea): why is this line required? It's the conclusion of the assert-forall.
}
}
assert forall|i: int|
0 <= i < self.len() && pred(self[i]) implies #[trigger] out.contains(self[i]) by {
if i == self.len() - 1 {
assert(self[i] == out[out.len() - 1]); // witness to contains
} else {
let subseq = self.drop_last().filter(pred);
assert(subseq.contains(self.drop_last()[i])); // trigger recursive invocation
let j = choose|j| 0 <= j < subseq.len() && subseq[j] == self[i];
assert(out[j] == self[i]); // TODO(andrea): same, seems needless
}
}
}
}
pub broadcast proof fn filter_distributes_over_add(a: Self, b: Self, pred: spec_fn(A) -> bool)
ensures
#[trigger] (a + b).filter(pred) == a.filter(pred) + b.filter(pred),
decreases b.len(),
{
reveal(Seq::filter);
if 0 < b.len() {
Self::drop_last_distributes_over_add(a, b);
Self::filter_distributes_over_add(a, b.drop_last(), pred);
if pred(b.last()) {
Self::push_distributes_over_add(
a.filter(pred),
b.drop_last().filter(pred),
b.last(),
);
}
} else {
Self::add_empty_right(a, b);
Self::add_empty_right(a.filter(pred), b.filter(pred));
}
}
pub broadcast proof fn add_empty_left(a: Self, b: Self)
requires
a.len() == 0,
ensures
#[trigger] (a + b) == b,
{
assert(a + b =~= b);
}
pub broadcast proof fn add_empty_right(a: Self, b: Self)
requires
b.len() == 0,
ensures
#[trigger] (a + b) == a,
{
assert(a + b =~= a);
}
pub broadcast proof fn push_distributes_over_add(a: Self, b: Self, elt: A)
ensures
#[trigger] (a + b).push(elt) == a + b.push(elt),
{
assert((a + b).push(elt) =~= a + b.push(elt));
}
/// Returns the maximum value in a non-empty sequence, given sorting function leq
pub open spec fn max_via(self, leq: spec_fn(A, A) -> bool) -> A
recommends
self.len() > 0,
decreases self.len(),
{
if self.len() > 1 {
if leq(self[0], self.subrange(1, self.len() as int).max_via(leq)) {
self.subrange(1, self.len() as int).max_via(leq)
} else {
self[0]
}
} else {
self[0]
}
}
/// Returns the minimum value in a non-empty sequence, given sorting function leq
pub open spec fn min_via(self, leq: spec_fn(A, A) -> bool) -> A
recommends
self.len() > 0,
decreases self.len(),
{
if self.len() > 1 {
let subseq = self.subrange(1, self.len() as int);
let elt = subseq.min_via(leq);
if leq(elt, self[0]) {
elt
} else {
self[0]
}
} else {
self[0]
}
}
// TODO is_sorted -- extract from summer_school e22
pub open spec fn contains(self, needle: A) -> bool {
exists|i: int| 0 <= i < self.len() && self[i] == needle
}
/// Returns an index where `needle` appears in the sequence.
/// Returns an arbitrary value if the sequence does not contain the `needle`.
pub open spec fn index_of(self, needle: A) -> int {
choose|i: int| 0 <= i < self.len() && self[i] == needle
}
/// For an element that occurs at least once in a sequence, if its first occurence
/// is at index i, Some(i) is returned. Otherwise, None is returned
pub closed spec fn index_of_first(self, needle: A) -> (result: Option<int>) {
if self.contains(needle) {
Some(self.first_index_helper(needle))
} else {
None
}
}
// Recursive helper function for index_of_first
spec fn first_index_helper(self, needle: A) -> int
recommends
self.contains(needle),
decreases self.len(),
{
if self.len() <= 0 {
-1 //arbitrary, will never get to this case
} else if self[0] == needle {
0
} else {
1 + self.subrange(1, self.len() as int).first_index_helper(needle)
}
}
pub proof fn index_of_first_ensures(self, needle: A)
ensures
match self.index_of_first(needle) {
Some(index) => {
&&& self.contains(needle)
&&& 0 <= index < self.len()
&&& self[index] == needle
&&& forall|j: int| 0 <= j < index < self.len() ==> self[j] != needle
},
None => { !self.contains(needle) },
},
decreases self.len(),
{
if self.contains(needle) {
let index = self.index_of_first(needle).unwrap();
if self.len() <= 0 {
} else if self[0] == needle {
} else {
assert(Seq::empty().push(self.first()).add(self.drop_first()) =~= self);
self.drop_first().index_of_first_ensures(needle);
}
}
}
/// For an element that occurs at least once in a sequence, if its last occurence
/// is at index i, Some(i) is returned. Otherwise, None is returned
pub closed spec fn index_of_last(self, needle: A) -> Option<int> {
if self.contains(needle) {
Some(self.last_index_helper(needle))
} else {
None
}
}
// Recursive helper function for last_index_of
spec fn last_index_helper(self, needle: A) -> int
recommends
self.contains(needle),
decreases self.len(),
{
if self.len() <= 0 {
-1 //arbitrary, will never get to this case
} else if self.last() == needle {
self.len() - 1
} else {
self.drop_last().last_index_helper(needle)
}
}
pub proof fn index_of_last_ensures(self, needle: A)
ensures
match self.index_of_last(needle) {
Some(index) => {
&&& self.contains(needle)
&&& 0 <= index < self.len()
&&& self[index] == needle
&&& forall|j: int| 0 <= index < j < self.len() ==> self[j] != needle
},
None => { !self.contains(needle) },
},
decreases self.len(),
{
if self.contains(needle) {
let index = self.index_of_last(needle).unwrap();
if self.len() <= 0 {
} else if self.last() == needle {
} else {
assert(self.drop_last().push(self.last()) =~= self);
self.drop_last().index_of_last_ensures(needle);
}
}
}
/// Drops the last element of a sequence and returns a sequence whose length is
/// thereby 1 smaller.
///
/// If the input sequence is empty, the result is meaningless and arbitrary.
pub open spec fn drop_last(self) -> Seq<A>
recommends
self.len() >= 1,
{
self.subrange(0, self.len() as int - 1)
}
/// Dropping the last element of a concatenation of `a` and `b` is equivalent
/// to skipping the last element of `b` and then concatenating `a` and `b`
pub proof fn drop_last_distributes_over_add(a: Self, b: Self)
requires
0 < b.len(),
ensures
(a + b).drop_last() == a + b.drop_last(),
{
assert_seqs_equal!((a+b).drop_last(), a+b.drop_last());
}
pub open spec fn drop_first(self) -> Seq<A>
recommends
self.len() >= 1,
{
self.subrange(1, self.len() as int)
}
/// returns `true` if the sequence has no duplicate elements
pub open spec fn no_duplicates(self) -> bool {
forall|i, j| (0 <= i < self.len() && 0 <= j < self.len() && i != j) ==> self[i] != self[j]
}
/// Returns `true` if two sequences are disjoint
pub open spec fn disjoint(self, other: Self) -> bool {
forall|i: int, j: int| 0 <= i < self.len() && 0 <= j < other.len() ==> self[i] != other[j]
}
/// Converts a sequence into a set
pub open spec fn to_set(self) -> Set<A> {
Set::new(|a: A| self.contains(a))
}
/// Converts a sequence into a multiset
pub closed spec fn to_multiset(self) -> Multiset<A>
decreases self.len(),
{
if self.len() == 0 {
Multiset::<A>::empty()
} else {
Multiset::<A>::empty().insert(self.first()).add(self.drop_first().to_multiset())
}
}
// Parts of verified lemma used to be an axiom in the Dafny prelude
// Note: the inner triggers in this lemma are blocked by `to_multiset_len`
/// Proof of function to_multiset() correctness
pub broadcast proof fn to_multiset_ensures(self)
ensures
forall|a: A| #[trigger] (self.push(a).to_multiset()) =~= self.to_multiset().insert(a), // to_multiset_build
forall|i: int|
0 <= i < self.len() ==> #[trigger] (self.remove(i).to_multiset())
=~= self.to_multiset().remove(self[i]), // to_multiset_remove
self.len() == #[trigger] self.to_multiset().len(), // to_multiset_len
forall|a: A|
self.contains(a) <==> #[trigger] self.to_multiset().count(a)
> 0, // to_multiset_contains
{
broadcast use group_seq_properties;
}
/// Insert item a at index i, shifting remaining elements (if any) to the right
pub open spec fn insert(self, i: int, a: A) -> Seq<A>
recommends
0 <= i <= self.len(),
{
self.subrange(0, i).push(a) + self.subrange(i, self.len() as int)
}
/// Proof of correctness and expected properties of insert function
pub proof fn insert_ensures(self, pos: int, elt: A)
requires
0 <= pos <= self.len(),
ensures
self.insert(pos, elt).len() == self.len() + 1,
forall|i: int| 0 <= i < pos ==> #[trigger] self.insert(pos, elt)[i] == self[i],
forall|i: int| pos <= i < self.len() ==> self.insert(pos, elt)[i + 1] == self[i],
self.insert(pos, elt)[pos] == elt,
{
}
/// Remove item at index i, shifting remaining elements to the left
pub open spec fn remove(self, i: int) -> Seq<A>
recommends
0 <= i < self.len(),
{
self.subrange(0, i) + self.subrange(i + 1, self.len() as int)
}
/// Proof of function remove() correctness
pub proof fn remove_ensures(self, i: int)
requires
0 <= i < self.len(),
ensures
self.remove(i).len() == self.len() - 1,
forall|index: int| 0 <= index < i ==> #[trigger] self.remove(i)[index] == self[index],
forall|index: int|
i <= index < self.len() - 1 ==> #[trigger] self.remove(i)[index] == self[index + 1],
{
}
/// If a given element occurs at least once in a sequence, the sequence without
/// its first occurrence is returned. Otherwise the same sequence is returned.
pub open spec fn remove_value(self, val: A) -> Seq<A> {
let index = self.index_of_first(val);
match index {
Some(i) => self.remove(i),
None => self,
}
}
/// Returns the sequence that is in reverse order to a given sequence.
pub open spec fn reverse(self) -> Seq<A>
decreases self.len(),
{
if self.len() == 0 {
Seq::empty()
} else {
Seq::new(self.len(), |i: int| self[self.len() - 1 - i])
}
}
/// Zips two sequences of equal length into one sequence that consists of pairs.
/// If the two sequences are different lengths, returns an empty sequence
pub open spec fn zip_with<B>(self, other: Seq<B>) -> Seq<(A, B)>
recommends
self.len() == other.len(),
decreases self.len(),
{
if self.len() != other.len() {
Seq::empty()
} else if self.len() == 0 {
Seq::empty()
} else {
Seq::new(self.len(), |i: int| (self[i], other[i]))
}
}
/// Folds the sequence to the left, applying `f` to perform the fold.
///
/// Equivalent to `Iterator::fold` in Rust.
///
/// Given a sequence `s = [x0, x1, x2, ..., xn]`, applying this function `s.fold_left(b, f)`
/// returns `f(...f(f(b, x0), x1), ..., xn)`.
pub open spec fn fold_left<B>(self, b: B, f: spec_fn(B, A) -> B) -> (res: B)
decreases self.len(),
{
if self.len() == 0 {
b
} else {
f(self.drop_last().fold_left(b, f), self.last())
}
}
/// Equivalent to [`Self::fold_left`] but defined by breaking off the leftmost element when
/// recursing, rather than the rightmost. See [`Self::lemma_fold_left_alt`] that proves
/// equivalence.
pub open spec fn fold_left_alt<B>(self, b: B, f: spec_fn(B, A) -> B) -> (res: B)
decreases self.len(),
{
if self.len() == 0 {
b
} else {
self.subrange(1, self.len() as int).fold_left_alt(f(b, self[0]), f)
}
}
/// A lemma that proves how [`Self::fold_left`] distributes over splitting a sequence.
pub broadcast proof fn lemma_fold_left_split<B>(self, b: B, f: spec_fn(B, A) -> B, k: int)
requires
0 <= k <= self.len(),
ensures
self.subrange(k, self.len() as int).fold_left(
(#[trigger] self.subrange(0, k).fold_left(b, f)),
f,
) == self.fold_left(b, f),
decreases self.len(),
{
reveal_with_fuel(Seq::fold_left, 2);
if k == self.len() {
assert(self.subrange(0, self.len() as int) == self);
} else {
self.drop_last().lemma_fold_left_split(b, f, k);
assert_seqs_equal!(
self.drop_last().subrange(k, self.drop_last().len() as int) ==
self.subrange(k, self.len()-1)
);
assert_seqs_equal!(
self.drop_last().subrange(0, k) ==
self.subrange(0, k)
);
assert_seqs_equal!(
self.subrange(k, self.len() as int).drop_last() ==
self.subrange(k, self.len() - 1)
);
}
}
/// An auxiliary lemma for proving [`Self::lemma_fold_left_alt`].
proof fn aux_lemma_fold_left_alt<B>(self, b: B, f: spec_fn(B, A) -> B, k: int)
requires
0 < k <= self.len(),
ensures
self.subrange(k, self.len() as int).fold_left_alt(
self.subrange(0, k).fold_left_alt(b, f),
f,
) == self.fold_left_alt(b, f),
decreases k,
{
reveal_with_fuel(Seq::fold_left_alt, 2);
if k == 1 {
// trivial base case
} else {
self.subrange(1, self.len() as int).aux_lemma_fold_left_alt(f(b, self[0]), f, k - 1);
assert_seqs_equal!(
self.subrange(1, self.len() as int)
.subrange(k - 1, self.subrange(1, self.len() as int).len() as int) ==
self.subrange(k, self.len() as int)
);
assert_seqs_equal!(
self.subrange(1, self.len() as int).subrange(0, k - 1) ==
self.subrange(1, k)
);
assert_seqs_equal!(
self.subrange(0, k).subrange(1, self.subrange(0, k).len() as int) ==
self.subrange(1, k)
);
}
}
/// [`Self::fold_left`] and [`Self::fold_left_alt`] are equivalent.
pub proof fn lemma_fold_left_alt<B>(self, b: B, f: spec_fn(B, A) -> B)
ensures
self.fold_left(b, f) == self.fold_left_alt(b, f),
decreases self.len(),
{
reveal_with_fuel(Seq::fold_left, 2);
reveal_with_fuel(Seq::fold_left_alt, 2);
if self.len() <= 1 {
// trivial base cases
} else {
self.aux_lemma_fold_left_alt(b, f, self.len() - 1);
self.subrange(self.len() - 1, self.len() as int).lemma_fold_left_alt(
self.drop_last().fold_left_alt(b, f),
f,
);
self.subrange(0, self.len() - 1).lemma_fold_left_alt(b, f);
}
}
/// Folds the sequence to the right, applying `f` to perform the fold.
///
/// Equivalent to `DoubleEndedIterator::rfold` in Rust.
///
/// Given a sequence `s = [x0, x1, x2, ..., xn]`, applying this function `s.fold_right(b, f)`
/// returns `f(x0, f(x1, f(x2, ..., f(xn, b)...)))`.
pub open spec fn fold_right<B>(self, f: spec_fn(A, B) -> B, b: B) -> (res: B)
decreases self.len(),
{
if self.len() == 0 {
b
} else {
self.drop_last().fold_right(f, f(self.last(), b))
}
}
/// Equivalent to [`Self::fold_right`] but defined by breaking off the leftmost element when
/// recursing, rather than the rightmost. See [`Self::lemma_fold_right_alt`] that proves
/// equivalence.
pub open spec fn fold_right_alt<B>(self, f: spec_fn(A, B) -> B, b: B) -> (res: B)
decreases self.len(),
{
if self.len() == 0 {
b
} else {
f(self[0], self.subrange(1, self.len() as int).fold_right_alt(f, b))
}
}
/// A lemma that proves how [`Self::fold_right`] distributes over splitting a sequence.
pub broadcast proof fn lemma_fold_right_split<B>(self, f: spec_fn(A, B) -> B, b: B, k: int)
requires
0 <= k <= self.len(),
ensures
self.subrange(0, k).fold_right(
f,
(#[trigger] self.subrange(k, self.len() as int).fold_right(f, b)),
) == self.fold_right(f, b),
decreases self.len(),
{
reveal_with_fuel(Seq::fold_right, 2);
if k == self.len() {
assert(self.subrange(0, k) == self);
} else if k == self.len() - 1 {
// trivial base case
} else {
self.subrange(0, self.len() - 1).lemma_fold_right_split(f, f(self.last(), b), k);
assert_seqs_equal!(
self.subrange(0, self.len() - 1).subrange(0, k) ==
self.subrange(0, k)
);
assert_seqs_equal!(
self.subrange(0, self.len() - 1).subrange(k, self.subrange(0, self.len() - 1).len() as int) ==
self.subrange(k, self.len() - 1)
);
assert_seqs_equal!(
self.subrange(k, self.len() as int).drop_last() ==
self.subrange(k, self.len() - 1)
);
}
}
// Lemma that proves it's possible to commute a commutative operator across fold_right.
pub proof fn lemma_fold_right_commute_one<B>(self, a: A, f: spec_fn(A, B) -> B, v: B)
requires
commutative_foldr(f),
ensures
self.fold_right(f, f(a, v)) == f(a, self.fold_right(f, v)),
decreases self.len(),
{
if self.len() > 0 {
self.drop_last().lemma_fold_right_commute_one(a, f, f(self.last(), v));
}
}
/// [`Self::fold_right`] and [`Self::fold_right_alt`] are equivalent.
pub proof fn lemma_fold_right_alt<B>(self, f: spec_fn(A, B) -> B, b: B)
ensures
self.fold_right(f, b) == self.fold_right_alt(f, b),
decreases self.len(),
{
reveal_with_fuel(Seq::fold_right, 2);
reveal_with_fuel(Seq::fold_right_alt, 2);
if self.len() <= 1 {
// trivial base cases
} else {
self.subrange(1, self.len() as int).lemma_fold_right_alt(f, b);
self.lemma_fold_right_split(f, b, 1);
}
}
// Proven lemmas
/// Given a sequence with no duplicates, each element occurs only
/// once in its conversion to a multiset
pub proof fn lemma_multiset_has_no_duplicates(self)
requires
self.no_duplicates(),
ensures
forall|x: A| self.to_multiset().contains(x) ==> self.to_multiset().count(x) == 1,
decreases self.len(),
{
broadcast use super::multiset::group_multiset_axioms;
if self.len() == 0 {
assert(forall|x: A|
self.to_multiset().contains(x) ==> self.to_multiset().count(x) == 1);
} else {
broadcast use group_seq_properties;
assert(self.drop_last().push(self.last()) =~= self);
self.drop_last().lemma_multiset_has_no_duplicates();
}
}
/// If, in a sequence's conversion to a multiset, each element occurs only once,
/// the sequence has no duplicates.
pub proof fn lemma_multiset_has_no_duplicates_conv(self)
requires
forall|x: A| self.to_multiset().contains(x) ==> self.to_multiset().count(x) == 1,
ensures
self.no_duplicates(),
{
broadcast use super::multiset::group_multiset_axioms;
assert forall|i, j| (0 <= i < self.len() && 0 <= j < self.len() && i != j) implies self[i]
!= self[j] by {
let mut a = if (i < j) {
i
} else {
j
};
let mut b = if (i < j) {
j
} else {
i
};
if (self[a] == self[b]) {
let s0 = self.subrange(0, b);
let s1 = self.subrange(b, self.len() as int);
assert(self == s0 + s1);
broadcast use group_to_multiset_ensures;
lemma_multiset_commutative(s0, s1);
assert(self.to_multiset().count(self[a]) >= 2);
}
}
}
/// The concatenation of two subsequences derived from a non-empty sequence,
/// the first obtained from skipping the last element, the second consisting only
/// of the last element, is the original sequence.
pub proof fn lemma_add_last_back(self)
requires
0 < self.len(),
ensures
#[trigger] self.drop_last().push(self.last()) =~= self,
{
}
/// If a predicate is true at every index of a sequence,
/// it is true for every member of the sequence as a collection.
/// Useful for converting quantifiers between the two forms
/// to satisfy a precondition in the latter form.
pub proof fn lemma_indexing_implies_membership(self, f: spec_fn(A) -> bool)
requires
forall|i: int| 0 <= i < self.len() ==> #[trigger] f(#[trigger] self[i]),
ensures
forall|x: A| #[trigger] self.contains(x) ==> #[trigger] f(x),
{
assert(forall|i: int| 0 <= i < self.len() ==> #[trigger] self.contains(self[i]));
}
/// If a predicate is true for every member of a sequence as a collection,
/// it is true at every index of the sequence.
/// Useful for converting quantifiers between the two forms
/// to satisfy a precondition in the latter form.
pub proof fn lemma_membership_implies_indexing(self, f: spec_fn(A) -> bool)
requires
forall|x: A| #[trigger] self.contains(x) ==> #[trigger] f(x),
ensures
forall|i: int| 0 <= i < self.len() ==> #[trigger] f(self[i]),
{
assert forall|i: int| 0 <= i < self.len() implies #[trigger] f(self[i]) by {
assert(self.contains(self[i]));
}
}
/// A sequence that is sliced at the pos-th element, concatenated
/// with that same sequence sliced from the pos-th element, is equal to the
/// original unsliced sequence.
pub proof fn lemma_split_at(self, pos: int)
requires
0 <= pos <= self.len(),
ensures
self.subrange(0, pos) + self.subrange(pos, self.len() as int) =~= self,
{
}
/// Any element in a slice is included in the original sequence.
pub proof fn lemma_element_from_slice(self, new: Seq<A>, a: int, b: int, pos: int)
requires
0 <= a <= b <= self.len(),
new == self.subrange(a, b),
a <= pos < b,
ensures
pos - a < new.len(),
new[pos - a] == self[pos],
{
}
/// A slice (from s2..e2) of a slice (from s1..e1) of a sequence is equal to just a
/// slice (s1+s2..s1+e2) of the original sequence.
pub proof fn lemma_slice_of_slice(self, s1: int, e1: int, s2: int, e2: int)
requires
0 <= s1 <= e1 <= self.len(),
0 <= s2 <= e2 <= e1 - s1,
ensures
self.subrange(s1, e1).subrange(s2, e2) =~= self.subrange(s1 + s2, s1 + e2),
{
}
/// A sequence of unique items, when converted to a set, produces a set with matching length
pub proof fn unique_seq_to_set(self)
requires
self.no_duplicates(),
ensures
self.len() == self.to_set().len(),
decreases self.len(),
{
broadcast use super::set::group_set_axioms;
seq_to_set_equal_rec::<A>(self);
if self.len() == 0 {
} else {
let rest = self.drop_last();
rest.unique_seq_to_set();
seq_to_set_equal_rec::<A>(rest);
seq_to_set_rec_is_finite::<A>(rest);
assert(!seq_to_set_rec(rest).contains(self.last()));
assert(seq_to_set_rec(rest).insert(self.last()).len() == seq_to_set_rec(rest).len()
+ 1);
}
}
/// The cardinality of a set of elements is always less than or
/// equal to that of the full sequence of elements.
pub proof fn lemma_cardinality_of_set(self)
ensures
self.to_set().len() <= self.len(),
decreases self.len(),
{
broadcast use {super::set::group_set_axioms, seq_to_set_is_finite};
broadcast use group_seq_properties;
broadcast use super::set_lib::group_set_properties;
if self.len() == 0 {
} else {
assert(self.drop_last().to_set().insert(self.last()) =~= self.to_set());
self.drop_last().lemma_cardinality_of_set();
}
}
/// A sequence is of length 0 if and only if its conversion to
/// a set results in the empty set.
pub proof fn lemma_cardinality_of_empty_set_is_0(self)
ensures
self.to_set().len() == 0 <==> self.len() == 0,
{
broadcast use {super::set::group_set_axioms, seq_to_set_is_finite};
assert(self.len() == 0 ==> self.to_set().len() == 0) by { self.lemma_cardinality_of_set() }
assert(!(self.len() == 0) ==> !(self.to_set().len() == 0)) by {
if self.len() > 0 {
assert(self.to_set().contains(self[0]));
assert(self.to_set().remove(self[0]).len() <= self.to_set().len());
}
}
}
/// A sequence with cardinality equal to its set has no duplicates.
/// Inverse property of that shown in lemma unique_seq_to_set
pub proof fn lemma_no_dup_set_cardinality(self)
requires
self.to_set().len() == self.len(),
ensures
self.no_duplicates(),
decreases self.len(),
{
broadcast use {super::set::group_set_axioms, seq_to_set_is_finite};
if self.len() == 0 {
} else {
assert(self =~= Seq::empty().push(self.first()).add(self.drop_first()));
if self.drop_first().contains(self.first()) {
// If there is a duplicate, then we show that |s.to_set()| == |s| cannot hold.
assert(self.to_set() =~= self.drop_first().to_set());
assert(self.to_set().len() <= self.drop_first().len()) by {
self.drop_first().lemma_cardinality_of_set()
}
} else {
assert(self.to_set().len() == 1 + self.drop_first().to_set().len()) by {
assert(self.drop_first().to_set().insert(self.first()) =~= self.to_set());
}
self.drop_first().lemma_no_dup_set_cardinality();
}
}
}
/// Mapping a function over a sequence and converting to a set is the same
/// as mapping it over the sequence converted to a set.
pub broadcast proof fn lemma_to_set_map_commutes<B>(self, f: spec_fn(A) -> B)
ensures
#[trigger] self.to_set().map(f) =~= self.map_values(f).to_set(),
{
broadcast use crate::vstd::group_vstd_default;
assert forall|elem: B|
self.to_set().map(f).contains(elem) <==> self.map_values(f).to_set().contains(elem) by {
if self.to_set().map(f).contains(elem) {
let x = choose|x: A| self.to_set().contains(x) && f(x) == elem;
let i = choose|i: int| 0 <= i < self.len() && self[i] == x;
assert(self.map_values(f)[i] == elem);
}
if self.map_values(f).to_set().contains(elem) {
let i = choose|i: int|
0 <= i < self.map_values(f).len() && self.map_values(f)[i] == elem;
let x = self[i];
assert(self.to_set().contains(x));
}
};
}
/// Appending an element to a sequence and converting to set, is equal
/// to converting to set and inserting it.
pub broadcast proof fn lemma_to_set_insert_commutes(sq: Seq<A>, elt: A)
requires
ensures
#[trigger] (sq + seq![elt]).to_set() =~= sq.to_set().insert(elt),
{
broadcast use crate::vstd::group_vstd_default;
broadcast use lemma_seq_concat_contains_all_elements;
broadcast use lemma_seq_empty_contains_nothing;
broadcast use lemma_seq_contains_after_push;
broadcast use super::seq::group_seq_axioms;
broadcast use super::set_lib::group_set_properties;
}
/// Update a subrange of a sequence starting at `off` to values `vs`.
/// Expects that the updated subrange `off` up to `off+vs.len()` fits
/// in the existing sequence.
pub open spec fn update_subrange_with(self, off: int, vs: Self) -> Self
recommends
0 <= off,
off + vs.len() <= self.len(),
{
Seq::new(
self.len(),
|i: int|
if off <= i < off + vs.len() {
vs[i - off]
} else {
self[i]
},
)
}
/// Skipping `i` elements and then 1 more is equivalent to skipping `i + 1` elements.
///
/// ## Example
///
/// ```rust
/// proof fn example() {
/// let s = seq![1, 2, 3, 4];
/// s.lemma_seq_skip_skip(2);
/// assert(s.skip(2).skip(1) =~= s.skip(3));
/// }
/// ```
pub broadcast proof fn lemma_seq_skip_skip(self, i: int)
ensures
0 <= i < self.len() ==> (self.skip(i)).skip(1) =~= #[trigger] self.skip(i + 1),
{
broadcast use group_seq_properties;
}
/// If an element is contained in a sequence, then there exists an index where that element appears.
///
/// ## Example
///
/// ```rust
/// proof fn example() {
/// let s = seq![10, 20, 30];
/// assert(s.contains(20));
/// let idx = s.lemma_contains_to_index(20);
/// assert(s[idx] == 20);
/// }
/// ```
pub proof fn lemma_contains_to_index(self, elem: A) -> (idx: int)
requires
self.contains(elem),
ensures
0 <= idx < self.len() && self[idx] == elem,
decreases self.len(),
{
broadcast use group_seq_properties;
if self[0] == elem {
0
} else {
let i = self.skip(1).lemma_contains_to_index(elem);
i + 1
}
}
/// If a predicate holds for the first element and for all elements in the tail,
/// then it holds for the entire sequence.
///
/// ## Example
///
/// ```rust
/// proof fn example() {
/// let s = seq![2, 4, 6, 8];
/// let is_even = |x| x % 2 == 0;
/// assert(is_even(s[0]));
/// assert(s.skip(1).all(is_even));
/// s.lemma_all_from_head_tail(is_even);
/// assert(s.all(is_even));
/// }
/// ```
pub proof fn lemma_all_from_head_tail(self, pred: spec_fn(A) -> bool)
requires
self.len() > 0,
pred(self[0]) && self.skip(1).all(|x| pred(x)),
ensures
self.all(|x| pred(x)),
{
broadcast use group_seq_properties;
assert(seq![self[0]] + self.skip(1) == self);
}
/// If a predicate holds for any element in the sequence and does not hold for the first element,
/// then the predicate must hold for some element in the tail.
///
/// ## Example
///
/// ```rust
/// proof fn example() {
/// let s = seq![1, 4, 6, 8];
/// let is_even = |x| x % 2 == 0;
/// assert(s.any(is_even));
/// assert(!is_even(s[0]));
/// s.lemma_any_tail(is_even);
/// assert(s.skip(1).any(is_even));
/// }
/// ```
pub proof fn lemma_any_tail(self, pred: spec_fn(A) -> bool)
requires
self.any(|x| pred(x)),
ensures
!pred(self[0]) ==> self.skip(1).any(|x| pred(x)),
{
broadcast use group_seq_properties;
}
/// Removes duplicate elements from a sequence, maintaining the order of first appearance.
/// Takes a `seen` sequence parameter to track previously encountered elements.
///
/// ## Example
///
/// ```rust
/// fn example() {
/// let s = seq![1, 2, 1, 3, 2, 4];
/// let seen = seq![];
/// let result = s.remove_duplicates(seen);
/// assert_eq!(result, seq![1, 2, 3, 4]);
///
/// let seen2 = seq![2, 3];
/// let result2 = s.remove_duplicates(seen2);
/// assert_eq!(result2, seq![1, 4]);
/// }
/// ```
pub open spec fn remove_duplicates(self, seen: Seq<A>) -> Seq<A>
decreases self.len(),
{
if self.len() == 0 {
seen
} else if seen.contains(self[0]) {
self.skip(1).remove_duplicates(seen)
} else {
self.skip(1).remove_duplicates(seen + seq![self[0]])
}
}
/// Properties of remove_duplicates:
/// - The output contains x if and only if x was in the input sequence or seen set
/// - The output length is at most the sum of input and seen lengths
///
/// ## Example
///
/// ```rust
/// proof fn example() {
/// let s = seq![1, 2, 1, 3];
/// let seen = seq![2];
/// s.lemma_remove_duplicates_properties(seen);
/// assert(s.remove_duplicates(seen).contains(1));
/// assert(s.remove_duplicates(seen).contains(3));
/// assert(!s.remove_duplicates(seen).contains(2));
/// assert(s.remove_duplicates(seen).len() <= s.len() + seen.len());
/// }
/// ```
pub broadcast proof fn lemma_remove_duplicates_properties(self, seen: Seq<A>)
ensures
forall|x|
(self + seen).contains(x) <==> #[trigger] self.remove_duplicates(seen).contains(x),
#[trigger] self.remove_duplicates(seen).len() <= self.len() + seen.len(),
decreases self.len(),
{
broadcast use group_seq_properties;
if self.len() == 0 {
} else if seen.contains(self[0]) {
let rest = self.skip(1);
rest.lemma_remove_duplicates_properties(seen);
} else {
let rest = self.skip(1);
rest.lemma_remove_duplicates_properties(seen + seq![self[0]]);
}
}
/// Shows that removing duplicates from a sequence is equivalent to:
/// 1. First removing duplicates from the prefix up to index i (with the given seen set)
/// 2. Using that result as the new seen set for removing duplicates from the suffix after i
///
/// ## Example
///
/// ```rust
/// proof fn example() {
/// let s = seq![1, 2, 1, 3, 2, 4];
/// let seen = seq![];
/// s.lemma_remove_duplicates_append_index(seen, 2);
/// assert(s.remove_duplicates(seen)
/// =~= seq![1, 3, 2, 4].remove_duplicates(seq![1, 2].remove_duplicates(seen)));
/// }
/// ```
pub proof fn lemma_remove_duplicates_append_index(self, i: int, seen: Seq<A>)
requires
0 <= i < self.len(),
ensures
self.remove_duplicates(seen) == self.skip(i).remove_duplicates(
self.take(i).remove_duplicates(seen),
),
decreases self.len(),
{
#[allow(deprecated)]
lemma_seq_properties::<A>(); // new broadcast group not working here
broadcast use Seq::lemma_remove_duplicates_properties;
if i == 0 {
} else if i == self.len() {
assert(self.take(i) == self);
} else {
assert(self.skip(1).take(i - 1) == self.subrange(1, i));
assert(self.take(i).skip(1) == self.subrange(1, i));
assert(self.skip(1).take(i - 1) == self.take(i).skip(1));
if seen.contains(self[0]) {
self.skip(1).lemma_remove_duplicates_append_index(i - 1, seen);
} else {
self.skip(1).lemma_remove_duplicates_append_index(i - 1, seen + seq![self[0]]);
}
}
}
/// For two sequences, skipping one element after concatenation equals concatenating
/// the result of skipping one element of the first sequence (which must be non-empty)
/// with the second sequence.
///
/// ## Example
/// ```rust
/// proof fn example() {
/// let s1 = seq![1, 2];
/// let s2 = seq![3, 4, 5];
///
/// lemma_skip1_concat(s1, s2);
/// assert((s1 + s2).skip(1) =~= seq![2, 3, 4, 5]);
/// }
/// ```
proof fn lemma_skip1_concat(xs: Seq<A>, ys: Seq<A>)
requires
xs.len() > 0,
ensures
(xs + ys).skip(1) == xs.skip(1) + ys,
{
broadcast use group_seq_properties;
assert((xs + ys).skip(1) == xs.skip(1) + ys);
}
/// When appending an element `x` to a sequence:
/// - If `x` is in `self + seen`, removing duplicates equals removing duplicates from self
/// - If x is not in (self + seen), removing duplicates equals removing duplicates from self,
/// concatenated with [x]
///
/// ## Example
/// ```rust
/// proof fn example() {
/// let s1 = seq![1, 2];
/// let seen = seq![];
/// assert!(!s1.contains(3));
/// lemma_remove_duplicates_append(s1, 3, seen);
/// assert((s1 + seq![3]).remove_duplicates(seen) =~= s1.remove_duplicates(seen) + seq![3]);
/// }
/// ```
pub proof fn lemma_remove_duplicates_append(self, x: A, seen: Seq<A>)
ensures
(self + seen).contains(x) ==> (self + seq![x]).remove_duplicates(seen)
== self.remove_duplicates(seen),
!(self + seen).contains(x) ==> (self + seq![x]).remove_duplicates(seen)
== self.remove_duplicates(seen) + seq![x],
decreases self.len(),
{
broadcast use group_seq_properties;
reveal_with_fuel(Seq::remove_duplicates, 2);
if self.len() != 0 {
let head = self[0];
let tail = self.skip(1);
let seen2 = if seen.contains(head) {
seen
} else {
seen + seq![head]
};
tail.lemma_remove_duplicates_append(x, seen2);
assert((self + seq![x]).skip(1) == tail + seq![x]) by {
Seq::lemma_skip1_concat(self, seq![x]);
};
}
}
/// If all elements in a sequence fail the predicate,
/// filtering by that predicate yields an empty sequence
///
/// ## Example
/// ```rust
/// proof fn example() {
/// let s = seq![1, 2, 3];
/// let pred = |x| x > 5;
/// lemma_all_neg_filter_empty(s, pred);
/// assert(s.filter(pred).len() == 0);
/// }
/// ```
pub proof fn lemma_all_neg_filter_empty(self, pred: spec_fn(A) -> bool)
requires
self.all(|x: A| !pred(x)),
ensures
self.filter(pred).len() == 0,
decreases self.len(),
{
broadcast use group_seq_properties;
reveal(Seq::filter);
if self.len() != 0 {
let rest = self.drop_last();
rest.lemma_all_neg_filter_empty(pred);
rest.lemma_filter_len_push(pred, self.last());
let neg_pred = |x| !pred(x);
assert(neg_pred(self.last()));
}
}
/// Applies an Option-returning function to each element, keeping only successful (Some) results
///
/// ## Example
/// ```rust
/// let s = seq![1, 2, 3];
/// let f = |x| if x % 2 == 0 { Some(x * 2) } else { None };
/// assert(s.filter_map(f) =~= seq![4]);
/// ```
pub open spec fn filter_map<B>(self, f: spec_fn(A) -> Option<B>) -> Seq<B>
decreases self.len(),
{
// We're defining this by starting at the end of the list since it makes it
// easier to reason about in the common case of looping over a vector in the
// implementation.
if self.len() == 0 {
Seq::empty()
} else {
let rest = self.drop_last();
match f(self.last()) {
Option::Some(s) => rest.filter_map(f) + seq![s],
Option::None => rest.filter_map(f),
}
}
}
/// If an element exists in the filtered sequence,
/// it must exist in the original sequence
/// ```
pub broadcast proof fn lemma_filter_contains_rev(self, p: spec_fn(A) -> bool, elem: A)
requires
#[trigger] self.filter(p).contains(elem),
ensures
self.contains(elem),
decreases self.len(),
{
broadcast use group_seq_properties;
reveal(Seq::filter);
if self.len() == 0 {
} else {
let rest = self.drop_last();
let last = self.last();
if !p(last) || last != elem {
rest.lemma_filter_contains_rev(p, elem);
}
}
}
/// If an element exists in filter_map's output,
/// there must be an input element that mapped to it
/// ```
pub broadcast proof fn lemma_filter_map_contains<B>(self, f: spec_fn(A) -> Option<B>, elt: B)
requires
#[trigger] self.filter_map(f).contains(elt),
ensures
exists|t: A| #[trigger] self.contains(t) && f(t) == Some(elt),
decreases self.len(),
{
broadcast use group_seq_properties;
if self.len() == 0 {
} else {
let last = self.last();
let rest = self.drop_last();
if f(last) == Some(elt) {
assert(self.contains(last));
} else {
rest.lemma_filter_map_contains(f, elt);
let t = choose|t: A| #[trigger] rest.contains(t) && f(t) == Some(elt);
assert(self.contains(t));
}
}
}
/// Taking k+1 elements is the same as taking k elements plus the kth element
///
/// ## Example
/// ```rust
/// let s = seq![1, 2, 3];
/// lemma_take_plus_one(s, 1);
/// seq![1, 2] == seq![1] + seq![2]
/// ```
pub proof fn lemma_take_succ(xs: Seq<A>, k: int)
requires
0 <= k < xs.len(),
ensures
xs.take(k + 1) =~= xs.take(k) + seq![xs[k]],
{
broadcast use group_seq_properties;
}
/// filter_map on a single element sequence
/// either produces a new single element sequence (if f returns Some)
/// or an empty sequence (if f returns None)
pub proof fn lemma_filter_map_singleton<B>(a: A, f: spec_fn(A) -> Option<B>)
ensures
seq![a].filter_map(f) =~= match f(a) {
Option::Some(b) => seq![b],
Option::None => Seq::empty(),
},
{
reveal_with_fuel(Seq::filter_map, 2);
}
/// filter_map of take(i+1) equals
/// filter_map of take(i) plus maybe the mapped i'th element
///
/// ## Example
/// ```rust
/// let s = seq![1, 2, 3];
/// let f = |x| if x % 2 == 0 { Some(x * 2) } else { None };
/// s.lemma_filter_map_take_succ(s, f, 1);
/// assert(s.take(2).filter_map(f) == s.take(1).filter_map(f) + seq![f(s[1]).unwrap()]);
/// assert(s.take(2).filter_map(f) == seq![] + seq![4]);
/// ```
pub broadcast proof fn lemma_filter_map_take_succ<B>(self, f: spec_fn(A) -> Option<B>, i: int)
requires
0 <= i < self.len(),
ensures
#[trigger] self.take(i + 1).filter_map(f) =~= self.take(i).filter_map(f) + (match f(
self[i],
) {
Option::Some(s) => seq![s],
Option::None => Seq::empty(),
}),
decreases self.len(),
{
broadcast use group_seq_properties;
if i != 0 {
self.drop_last().lemma_filter_map_take_succ(f, i - 1);
assert(self.take(i + 1).drop_last() == self.take(i));
}
}
/// An alternative implementation of filter that processes the sequence recursively from
/// left to right, in contrast to the standard filter which processes from right to left.
pub open spec fn filter_alt(self, p: spec_fn(A) -> bool) -> Seq<A> {
if self.len() == 0 {
Seq::empty()
} else {
let rest = self.drop_first().filter(p);
let first = self.first();
if p(first) {
seq![first] + rest
} else {
rest
}
}
}
/// When filtering (x + sequence), if x satisfies the predicate, x is prepended to
/// the filtered sequence. Otherwise, only the filtered sequence remains.
///
/// ## Example
/// ```rust
/// proof fn filter_prepend_test() {
/// let s = seq![2, 3, 4];
/// let is_even = |x: int| x % 2 == 0;
/// let with_five = seq![5] + s;
/// assert(with_five.filter(is_even) =~= seq![2, 4]); // 5 filtered out
/// let with_six = seq![6] + s;
/// assert(with_six.filter(is_even) =~= seq![6, 2, 4]); // 6 included
/// }
/// ```
pub broadcast proof fn lemma_filter_prepend(self, x: A, p: spec_fn(A) -> bool)
ensures
#[trigger] (seq![x] + self).filter(p) == (if p(x) {
seq![x]
} else {
Seq::empty()
}) + self.filter(p),
decreases self.len(),
{
broadcast use group_seq_properties;
reveal(Seq::filter);
let lhs = (seq![x] + self).filter(p);
let rhs = (if p(x) {
seq![x]
} else {
Seq::empty()
}) + self.filter(p);
if self.len() == 0 {
assert(lhs =~= rhs);
} else {
let tail_seq = if p(self.last()) {
seq![self.last()]
} else {
Seq::empty()
};
assert(((seq![x] + self).drop_last()) =~= seq![x] + self.drop_last());
let sub = (seq![x] + self.drop_last()).filter(p);
assert(lhs =~= sub + tail_seq);
assert(rhs =~= (if p(x) {
seq![x]
} else {
Seq::empty()
}) + self.drop_last().filter(p) + tail_seq);
self.drop_last().lemma_filter_prepend(x, p);
}
}
/// The filter() and filter_alt() methods produce equivalent results for any sequence
pub proof fn lemma_filter_eq_filter_alt(self, p: spec_fn(A) -> bool)
ensures
self.filter(p) =~= self.filter_alt(p),
decreases self.len(),
{
broadcast use group_seq_properties;
broadcast use Seq::lemma_filter_prepend;
reveal(Seq::filter);
if self.len() == 0 {
} else {
let first = self.first();
let but_first = self.drop_first();
assert(self =~= seq![first] + but_first);
self.drop_first().lemma_filter_eq_filter_alt(p);
}
}
/// Filtering preserves the prefix relationship between sequences.
///
/// ## Example
/// ```rust
/// proof fn filter_monotone_test() {
/// let s = seq![1, 2, 3];
/// let ys = seq![1, 2, 3, 4, 5];
/// let is_even = |x: int| x % 2 == 0;
/// assert(s.is_prefix_of(ys));
/// assert(s.filter(is_even).is_prefix_of(ys.filter(is_even)));
/// assert(s.filter(is_even) =~= seq![2]);
/// assert(ys.filter(is_even) =~= seq![2, 4]);
/// }
/// ```
pub proof fn lemma_filter_monotone(self, ys: Seq<A>, p: spec_fn(A) -> bool)
requires
self.is_prefix_of(ys),
ensures
self.filter(p).is_prefix_of(ys.filter(p)),
decreases self.len(),
{
broadcast use group_seq_properties;
self.lemma_filter_eq_filter_alt(p);
ys.lemma_filter_eq_filter_alt(p);
if self.len() == 0 {
} else {
self.drop_first().lemma_filter_monotone(ys.drop_first(), p);
}
}
/// The length of filter(take(i)) is never greater than the length of filter(entire_sequence).
///
/// ## Example
/// ```rust
/// proof fn filter_take_len_test() {
/// let s = seq![1, 2, 3, 4, 5];
/// let is_even = |x: int| x % 2 == 0;
/// let i = 3;
/// assert(s.take(i) =~= seq![1, 2, 3]);
/// assert(s.take(i).filter(is_even) =~= seq![2]);
/// assert(s.filter(is_even) =~= seq![2, 4]);
/// assert(s.filter(is_even).len() >= s.take(i).filter(is_even).len());
/// }
/// ```
pub proof fn lemma_filter_take_len(self, p: spec_fn(A) -> bool, i: int)
requires
0 <= i <= self.len(),
ensures
self.filter(p).len() >= self.take(i).filter(p).len(),
decreases i,
{
broadcast use group_seq_properties;
broadcast use Seq::lemma_filter_len_push;
broadcast use Seq::lemma_filter_push;
self.take(i).lemma_filter_monotone(self, p);
}
/// Filtering a prefix of a sequence produces the same number or fewer elements
/// as filtering the entire sequence.
///
/// ## Example
/// ```rust
/// proof fn filter_take_len_test() {
/// let s = seq![1, 2, 3, 4, 5];
/// let is_even = |x: int| x % 2 == 0;
/// assert(s.filter(is_even).len() >= s.take(3).filter(is_even).len());
/// }
/// ```
pub broadcast proof fn lemma_filter_len_push(self, p: spec_fn(A) -> bool, elem: A)
ensures
#[trigger] self.push(elem).filter(p).len() == self.filter(p).len() + (if p(elem) {
1int
} else {
0int
}),
{
broadcast use group_seq_properties;
broadcast use Seq::lemma_filter_push;
}
/// If an index i is valid for a sequence (0 ≤ i < len), then the element at that index
/// is contained in the sequence.
pub broadcast proof fn lemma_index_contains(self, i: int)
requires
0 <= i < self.len(),
ensures
self.contains(#[trigger] self[i]),
{
}
/// Taking i+1 elements from a sequence is equivalent to taking i elements
/// and then pushing the element at index i.
pub broadcast proof fn lemma_take_succ_push(self, i: int)
requires
0 <= i < self.len(),
ensures
#[trigger] self.take(i + 1) =~= self.take(i).push(self[i]),
{
broadcast use group_seq_properties;
}
/// Taking the full length of a sequence returns the sequence itself.
pub broadcast proof fn lemma_take_len(self)
ensures
#[trigger] self.take(self.len() as int) == self,
{
broadcast use group_seq_properties;
}
/// Taking i+1 elements and checking if any element satisfies predicate p is equivalent to:
/// either taking i elements and checking if any satisfies p, OR checking if the i-th element satisfies p.
///
/// ## Example
/// ```rust
/// proof fn take_any_succ_test() {
/// let s = seq![1, 2, 3];
/// let is_even = |x| x % 2 == 0;
/// let i = 1;
/// assert(s.take(i + 1).any(is_even) == (s.take(i).any(is_even) || is_even(s[i])));
/// }
/// ```
pub broadcast proof fn lemma_take_any_succ(self, p: spec_fn(A) -> bool, i: int)
requires
0 <= i < self.len(),
ensures
#[trigger] self.take(i + 1).any(p) <==> self.take(i).any(p) || p(self[i]),
{
broadcast use group_seq_properties;
self.lemma_take_succ_push(i);
if self.take(i + 1).any(p) {
let x = choose|x: A| self.take(i + 1).contains(x) && #[trigger] p(x);
assert(self.take(i).contains(x) || x == self[i]);
}
if self.take(i).any(p) {
let x = choose|x: A| self.take(i).contains(x) && #[trigger] p(x);
assert(self.take(i + 1).contains(x));
}
if p(self[i]) {
assert(self.take(i + 1).contains(self[i]));
}
}
/// A sequence has no duplicates iff mapping an injective function over it
/// produces a sequence with no duplicates.
///
/// ## Example
/// ```rust
/// proof fn no_duplicates_injective_test() {
/// let s = seq![1, 2];
/// let f = |x| x + 1; // injective function
/// assert(s.no_duplicates() == s.map_values(f).no_duplicates());
/// }
/// ```
pub proof fn lemma_no_duplicates_injective<B>(self, f: spec_fn(A) -> B)
requires
injective(f),
ensures
self.no_duplicates() <==> self.map_values(f).no_duplicates(),
{
broadcast use group_seq_properties;
broadcast use super::set_lib::group_set_properties;
let mapped = self.map_values(f);
assert(mapped.len() == self.len());
if mapped.no_duplicates() {
assert forall|i: int, j: int| 0 <= i < j < mapped.len() implies self[i] != self[j] by {
assert(mapped[i] == f(self[i]));
assert(mapped[j] == f(self[j]));
}
}
}
/// Pushing an element and then mapping a function over a sequence is equivalent to
/// mapping the function over the sequence and then pushing the function applied to that element.
///
/// ## Example
/// ```rust
/// proof fn push_map_test() {
/// let s = seq![1, 2];
/// let f = |x| x + 1;
/// assert(s.push(3).map_values(f) =~= s.map_values(f).push(f(3)));
/// }
/// ```
pub broadcast proof fn lemma_push_map_commute<B>(self, f: spec_fn(A) -> B, x: A)
ensures
self.map_values(f).push(f(x)) =~= #[trigger] self.push(x).map_values(f),
decreases self.len(),
{
broadcast use group_seq_properties;
}
/// Converting a sequence to a set after pushing an element is equivalent to
/// converting to a set first and then inserting that element.
///
/// ## Example
/// ```rust
/// proof fn push_to_set_test() {
/// let s = seq![1, 2];
/// assert(s.push(3).to_set() =~= s.to_set().insert(3));
/// }
/// ```
pub broadcast proof fn lemma_push_to_set_commute(self, elem: A)
ensures
#[trigger] self.push(elem).to_set() =~= self.to_set().insert(elem),
{
broadcast use group_seq_properties;
broadcast use super::set::group_set_axioms;
let lhs = self.push(elem).to_set();
let rhs = self.to_set().insert(elem);
assert(lhs.subset_of(rhs));
assert forall|x: A| rhs.contains(x) implies lhs.contains(x) by {
lemma_seq_contains_after_push(self, elem, x);
if x == elem {
} else {
lemma_seq_contains_after_push(self, elem, x);
}
}
}
/// Filtering a sequence after pushing an element is equivalent to:
/// if the element satisfies the predicate, filter the sequence and push the element
/// otherwise, just filter the sequence without the element.
///
/// ## Example
/// ```rust
/// proof fn filter_push_test() {
/// let s = seq![1, 2];
/// let is_even = |x| x % 2 == 0;
/// assert(s.push(4).filter(is_even) == s.filter(is_even).push(4));
/// assert(s.push(3).filter(is_even) == s.filter(is_even));
/// }
/// ```
pub broadcast proof fn lemma_filter_push(self, elem: A, pred: spec_fn(A) -> bool)
ensures
#[trigger] self.push(elem).filter(pred) == if pred(elem) {
self.filter(pred).push(elem)
} else {
self.filter(pred)
},
{
broadcast use group_seq_properties;
reveal(Seq::filter);
assert(self.push(elem).drop_last() =~= self);
}
/// If two sequences have the same length and `i` is a valid index,
/// then the pair (a[i], b[i]) is contained in their zip.
///
/// ## Example
/// ```rust
/// proof fn zip_contains_test() {
/// let a = seq![1, 2];
/// let b = seq!["a", "b"];
/// assert(a.zip_with(b).contains((a[0], b[0])));
/// assert(a.zip_with(b).contains((a[1], b[1])));
/// }
/// ```
pub proof fn lemma_zip_with_contains_index<B>(self, b: Seq<B>, i: int)
requires
0 <= i < self.len(),
self.len() == b.len(),
ensures
self.zip_with(b).contains((self[i], b[i])),
{
assert(self.zip_with(b)[i] == (self[i], b[i]));
}
/// Proves equivalence between checking a predicate over zipped sequences and checking
/// corresponding elements by index. Requires sequences of equal length.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let xs = seq![1, 2];
/// let ys = seq![2, 3];
/// let f = |x, y| x < y;
/// assert(xs.zip_with(ys).all(|(x, y)| f(x, y)) <==>
/// forall|i| 0 <= i < xs.len() ==> f(xs[i], ys[i]));
/// // We can now prove specific index relationships
/// assert(xs[0] < ys[0]); // 1 < 2
/// assert(xs[1] < ys[1]); // 2 < 3
/// }
/// ```
pub proof fn lemma_zip_with_uncurry_all<B>(self, b: Seq<B>, f: spec_fn(A, B) -> bool)
requires
self.len() == b.len(),
ensures
self.zip_with(b).all(|p: (A, B)| f(p.0, p.1)) <==> forall|i: int|
0 <= i < self.len() ==> f(self[i], b[i]),
{
broadcast use group_seq_properties;
let zipped = self.zip_with(b);
let f_uncurr = |p: (A, B)| f(p.0, p.1);
let lhs = zipped.all(f_uncurr);
let rhs = (forall|i: int| 0 <= i < self.len() ==> f(self[i], b[i]));
if lhs {
assert forall|i: int| 0 <= i < self.len() implies f(self[i], b[i]) by {
self.lemma_zip_with_contains_index(b, i);
assert(forall|j| 0 <= j < zipped.len() ==> f_uncurr(zipped[j]));
}
}
}
/// flat_mapping after pushing an element is the same as
/// flat_mapping first and then appending f of that element.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let xs = seq![1, 2];
/// let f = |x| seq![x, x + 1];
/// assert(xs.push(3).flat_map(f) =~= xs.flat_map(f) + f(3));
/// // xs.push(3).flat_map(f) = [1,2,2,3,3,4]
/// // xs.flat_map(f) + f(3) = [1,2,2,3] + [3,4]
/// }
/// ```
pub proof fn lemma_flat_map_push<B>(self, f: spec_fn(A) -> Seq<B>, elem: A)
ensures
self.push(elem).flat_map(f) =~= self.flat_map(f) + f(elem),
decreases self.len(),
{
broadcast use group_seq_properties;
broadcast use Seq::lemma_flatten_push;
broadcast use Seq::lemma_push_map_commute;
}
/// flat_mapping a sequence up to index i+1 is equivalent to
/// flat_mapping up to index i and appending f of the element at index i.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let xs = seq![1, 2, 3];
/// let f = |x| seq![x, x + 1];
///
/// assert(xs.take(2).flat_map(f) =~= xs.take(1).flat_map(f) + f(xs[1]));
/// // xs.take(2).flat_map(f) = [1,2,2,3]
/// // xs.take(1).flat_map(f) + f(2) = [1,2] + [2,3]
/// }
/// ```
pub broadcast proof fn lemma_flat_map_take_append<B>(self, f: spec_fn(A) -> Seq<B>, i: int)
requires
0 <= i < self.len(),
ensures
#[trigger] self.take(i + 1).flat_map(f) =~= self.take(i).flat_map(f) + f(self[i]),
decreases i,
{
broadcast use group_seq_properties;
self.lemma_take_succ_push(i);
self.take(i).lemma_flat_map_push(f, self[i]);
}
/// flat_mapping a sequence with a single element
/// is equivalent to applying the function f to that element.
pub broadcast proof fn lemma_flat_map_singleton<B>(self, f: spec_fn(A) -> Seq<B>)
requires
#[trigger] self.len() == 1,
ensures
#[trigger] self.flat_map(f) == f(self[0]),
{
broadcast use Seq::lemma_flatten_singleton;
}
/// Mapping a sequence's first i+1 elements equals
/// mapping its first i elements plus f of the i-th element.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let xs = seq![1, 2, 3];
/// let f = |x| x * 2;
///
/// assert(xs.take(2).map_values(f) =~= xs.take(1).map_values(f).push(f(xs[1])));
/// // Left: [1,2].map(f) = [2,4]
/// // Right: [1].map(f).push(f(2)) = [2].push(4)
/// }
/// ```
pub broadcast proof fn lemma_map_take_succ<B>(self, f: spec_fn(A) -> B, i: int)
requires
0 <= i < self.len(),
ensures
#[trigger] self.take(i + 1).map_values(f) =~= self.take(i).map_values(f).push(
f(self[i]),
),
{
broadcast use group_seq_properties;
self.lemma_take_succ_push(i);
}
/// If a sequence is a prefix of another sequence,
/// their elements match at all indices within the prefix length.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let xs = seq![1, 2, 3];
/// let prefix = seq![1, 2];
/// assert(prefix.is_prefix_of(xs));
/// assert(prefix[0] == xs[0] && prefix[1] == xs[1]);
/// }
/// ```
pub broadcast proof fn lemma_prefix_index_eq(self, prefix: Seq<A>)
requires
#[trigger] prefix.is_prefix_of(self),
ensures
forall|i: int| 0 <= i < prefix.len() ==> prefix[i] == self[i],
{
}
/// If a concatenated sequence (prefix1 + prefix2) is a prefix of another sequence,
/// then prefix1 by itself is also a prefix of that sequence.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let xs = seq![1, 2, 3, 4];
/// let prefix1 = seq![1, 2];
/// let prefix2 = seq![3];
/// assert((prefix1 + prefix2).is_prefix_of(xs));
/// assert(prefix1.is_prefix_of(xs));
/// }
/// ```
pub broadcast proof fn lemma_prefix_concat(self, prefix1: Seq<A>, prefix2: Seq<A>)
requires
#[trigger] (prefix1 + prefix2).is_prefix_of(self),
ensures
prefix1.is_prefix_of(self),
{
broadcast use Seq::lemma_prefix_index_eq;
}
/// If `prefix1 + [t]` is a prefix of a sequence,
/// `prefix1` is a prefix of `prefix2`,
/// `prefix2` is a prefix of the sequence,
/// `prefix1` and `prefix2` are different, and
/// `prefix1` doesn't contain `t`,
/// then `prefix2` must contain t.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let xs = seq![1, 2, 3, 4];
/// let prefix1 = seq![1];
/// let prefix2 = seq![1, 2];
/// let t = 2;
/// assert((prefix1 + seq![t]).is_prefix_of(xs));
/// assert(prefix1.is_prefix_of(prefix2));
/// assert(prefix2.is_prefix_of(xs));
/// assert(prefix2.contains(t));
/// }
/// ```
pub broadcast proof fn lemma_prefix_chain_contains(self, prefix1: Seq<A>, prefix2: Seq<A>, t: A)
requires
#[trigger] (prefix1 + seq![t]).is_prefix_of(self),
#[trigger] prefix1.is_prefix_of(prefix2),
prefix2.is_prefix_of(self),
prefix1 != prefix2,
!prefix1.contains(t),
ensures
prefix2.contains(t),
{
broadcast use Seq::lemma_prefix_concat;
broadcast use Seq::lemma_prefix_index_eq;
assert(prefix2[prefix1.len() as int] == t);
}
/// If `prefix1 + [t]` and `prefix2 + [t]` are both prefixes of a sequence,
/// and neither `prefix1` nor `prefix2` contains `t`,
/// then `prefix1` equals `prefix2`.
pub broadcast proof fn lemma_prefix_append_unique(self, prefix1: Seq<A>, prefix2: Seq<A>, t: A)
requires
#[trigger] (prefix1 + seq![t]).is_prefix_of(self),
#[trigger] (prefix2 + seq![t]).is_prefix_of(self),
!prefix1.contains(t),
!prefix2.contains(t),
ensures
prefix1 == prefix2,
{
broadcast use Seq::lemma_prefix_concat;
broadcast use Seq::lemma_prefix_index_eq;
broadcast use Seq::lemma_prefix_chain_contains;
if prefix1 != prefix2 {
assert(prefix1.is_prefix_of(prefix2) || prefix2.is_prefix_of(prefix1));
}
}
/// If a predicate `p` is true for all elements in a sequence,
/// and `p` is true for an element `e`, then `p` remains true for all elements
/// after pushing `e` to the sequence.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let xs = seq![2, 4, 6];
/// let is_even = |x| x % 2 == 0;
/// assert(xs.all(is_even));
/// assert(is_even(8));
/// assert(xs.push(8).all(is_even));
/// }
/// ```
pub broadcast proof fn lemma_all_push(self, p: spec_fn(A) -> bool, elem: A)
requires
self.all(p),
p(elem),
ensures
#[trigger] self.push(elem).all(p),
{
broadcast use group_seq_properties;
assert forall|x: A| self.push(elem).contains(x) implies p(x) by {
lemma_seq_contains_after_push(self, elem, x);
}
}
/// Two sequences are equal when concatenated with the same prefix
/// iff those two sequences are equal.
pub proof fn lemma_concat_injective(self, s1: Seq<A>, s2: Seq<A>)
ensures
(self + s1 == self + s2) <==> (s1 == s2),
{
broadcast use group_seq_properties;
assert((self + s1).skip(self.len() as int) == s1);
}
pub broadcast group group_seq_extra {
Seq::<_>::lemma_seq_skip_skip,
Seq::<_>::lemma_remove_duplicates_properties,
Seq::<_>::lemma_filter_contains_rev,
Seq::<_>::lemma_filter_map_take_succ,
Seq::<_>::lemma_filter_prepend,
Seq::<_>::lemma_filter_len_push,
Seq::<_>::lemma_take_len,
Seq::<_>::lemma_take_any_succ,
Seq::<_>::lemma_push_map_commute,
Seq::<_>::lemma_push_to_set_commute,
Seq::<_>::lemma_filter_push,
Seq::<_>::lemma_flat_map_take_append,
Seq::<_>::lemma_flat_map_singleton,
Seq::<_>::lemma_map_take_succ,
Seq::<_>::lemma_prefix_index_eq,
Seq::<_>::lemma_prefix_concat,
Seq::<_>::lemma_prefix_chain_contains,
Seq::<_>::lemma_prefix_append_unique,
Seq::<_>::lemma_all_push,
}
}
/// Filtering a sequence and then viewing its elements produces the same result as
/// viewing the elements first and then filtering with the corresponding predicate.
/// The predicates p and sp must be equivalent under view.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let s = seq!["hello".to_string(), "world".to_string()];
/// let p = |x: String| x.len() > 4;
/// let sp = |x: Seq<char>| x.len() > 4;
///
/// let way1 = s.filter(p).map_values(|x| x.view());
/// let way2 = s.map_values(|x| x.view()).filter(sp);
/// assert(way1 == way2);
/// }
/// ```
pub proof fn lemma_filter_view_commute<S: View>(
s: Seq<S>,
p: spec_fn(S) -> bool,
sp: spec_fn(S::V) -> bool,
)
requires
forall|s: S| p(s) <==> sp(s.view()),
ensures
s.filter(p).map_values(|x: S| x.view()) == s.map_values(|x: S| x.view()).filter(sp),
decreases s.len(),
{
broadcast use group_seq_properties;
broadcast use Seq::lemma_push_map_commute;
broadcast use Seq::lemma_filter_push;
reveal(Seq::filter);
let view = |x: S| x.view();
if s.len() > 0 {
let rest = s.drop_last();
let last = s.last();
assert(s =~= rest.push(last));
assert(s.map_values(view).last() == view(last));
lemma_filter_view_commute(rest, p, sp);
}
}
/// A sequence has exactly one element satisfying a predicate iff
/// viewing all elements and filtering with the corresponding predicate
/// produces a sequence with exactly one element.
///
/// # Example
/// ```rust
/// proof fn example() {
/// let s = seq!["hello".to_string(), "world".to_string()];
/// let p = |x: String| x.len() == 5;
/// let sp = |x: Seq<char>| x.len() == 5;
///
/// assert(s.exactly_one(p) <==> s.map_values(|x| x.view()).exactly_one(sp));
/// }
/// ```
pub proof fn lemma_exactly_one_view<S: View>(
s: Seq<S>,
p: spec_fn(S) -> bool,
sp: spec_fn(S::V) -> bool,
)
requires
forall|s: S| p(s) <==> sp(s.view()),
injective(|x: S| x.view()),
ensures
s.exactly_one(p) <==> s.map_values(|x: S| x.view()).exactly_one(sp),
{
lemma_filter_view_commute(s, p, sp);
}
impl<A, B> Seq<(A, B)> {
/// Unzips a sequence that contains pairs into two separate sequences.
pub closed spec fn unzip(self) -> (Seq<A>, Seq<B>) {
(Seq::new(self.len(), |i: int| self[i].0), Seq::new(self.len(), |i: int| self[i].1))
}
/// Proof of correctness and expected properties of unzip function
pub proof fn unzip_ensures(self)
ensures
self.unzip().0.len() == self.unzip().1.len(),
self.unzip().0.len() == self.len(),
self.unzip().1.len() == self.len(),
forall|i: int|
0 <= i < self.len() ==> (#[trigger] self.unzip().0[i], #[trigger] self.unzip().1[i])
== self[i],
decreases self.len(),
{
if self.len() > 0 {
self.drop_last().unzip_ensures();
}
}
/// Unzipping a sequence of sequences and then zipping the resulting two sequences
/// back together results in the original sequence of sequences.
pub proof fn lemma_zip_of_unzip(self)
ensures
self.unzip().0.zip_with(self.unzip().1) =~= self,
{
}
}
impl<A> Seq<Seq<A>> {
/// Flattens a sequence of sequences into a single sequence by concatenating
/// subsequences, starting from the first element.
///
/// ## Example
///
/// ```rust
/// proof fn flatten_test() {
/// let seq: Seq<Seq<int>> = seq![seq![1, 2, 3], seq![4, 5, 6], seq![7, 8, 9]];
/// let flat: Seq<int> = seq.flatten();
/// reveal_with_fuel(Seq::<Seq<int>>::flatten, 5); //Needed for Verus to unfold the recursive definition of flatten
/// assert(flat =~= seq![1, 2, 3, 4, 5, 6, 7, 8, 9]);
/// }
/// ```
pub open spec fn flatten(self) -> Seq<A>
decreases self.len(),
{
if self.len() == 0 {
Seq::empty()
} else {
self.first().add(self.drop_first().flatten())
}
}
/// Flattens a sequence of sequences into a single sequence by concatenating
/// subsequences in reverse order, i.e. starting from the last element.
/// This is equivalent to a call to `flatten`, but with concatenation operation
/// applied along the oppositive associativity for the sake of proof reasoning in that direction.
pub open spec fn flatten_alt(self) -> Seq<A>
decreases self.len(),
{
if self.len() == 0 {
Seq::empty()
} else {
self.drop_last().flatten_alt().add(self.last())
}
}
/// Flattening a sequence of a sequence x, where x has length 1,
/// results in a sequence equivalent to the single element of x
pub proof fn lemma_flatten_one_element(self)
ensures
self.len() == 1 ==> self.flatten() == self.first(),
{
broadcast use Seq::add_empty_right;
if self.len() == 1 {
assert(self.flatten() =~= self.first().add(self.drop_first().flatten()));
}
}
/// The length of a flattened sequence of sequences x is greater than or
/// equal to any of the lengths of the elements of x.
pub proof fn lemma_flatten_length_ge_single_element_length(self, i: int)
requires
0 <= i < self.len(),
ensures
self.flatten_alt().len() >= self[i].len(),
decreases self.len(),
{
if self.len() == 1 {
self.lemma_flatten_one_element();
self.lemma_flatten_and_flatten_alt_are_equivalent();
} else if i < self.len() - 1 {
self.drop_last().lemma_flatten_length_ge_single_element_length(i);
} else {
assert(self.flatten_alt() == self.drop_last().flatten_alt().add(self.last()));
}
}
/// The length of a flattened sequence of sequences x is less than or equal
/// to the length of x multiplied by a number greater than or equal to the
/// length of the longest sequence in x.
pub proof fn lemma_flatten_length_le_mul(self, j: int)
requires
forall|i: int| 0 <= i < self.len() ==> (#[trigger] self[i]).len() <= j,
ensures
self.flatten_alt().len() <= self.len() * j,
decreases self.len(),
{
broadcast use group_seq_properties;
if self.len() == 0 {
} else {
self.drop_last().lemma_flatten_length_le_mul(j);
assert((self.len() - 1) * j == (self.len() * j) - (1 * j)) by (nonlinear_arith); //TODO: use math library after imported
}
}
/// Flattening sequences of sequences in order (starting from the beginning)
/// and in reverse order (starting from the end) results in the same sequence.
pub proof fn lemma_flatten_and_flatten_alt_are_equivalent(self)
ensures
self.flatten() =~= self.flatten_alt(),
decreases self.len(),
{
broadcast use {Seq::add_empty_right, Seq::push_distributes_over_add};
if self.len() != 0 {
self.drop_last().lemma_flatten_and_flatten_alt_are_equivalent();
// let s = self.drop_last().flatten();
// let s2 = self.drop_last().flatten_alt();
// assert(s == s2);
seq![self.last()].lemma_flatten_one_element();
assert(seq![self.last()].flatten() == self.last());
lemma_flatten_concat(self.drop_last(), seq![self.last()]);
assert((self.drop_last() + seq![self.last()]).flatten() == self.drop_last().flatten()
+ self.last());
assert(self.drop_last() + seq![self.last()] =~= self);
assert(self.flatten_alt() == self.drop_last().flatten_alt() + self.last());
}
}
/// Flattening a sequence of sequences after pushing a new sequence is equivalent to
/// concatenating that sequence to the original flattened result.
pub broadcast proof fn lemma_flatten_push(self, elem: Seq<A>)
ensures
#[trigger] self.push(elem).flatten() =~= self.flatten() + elem,
decreases self.len(),
{
broadcast use group_seq_properties;
assert(self.push(elem).last() == elem);
assert(self.push(elem).drop_last() =~= self);
calc! {
(==)
self.push(elem).flatten(); {
self.push(elem).lemma_flatten_and_flatten_alt_are_equivalent();
}
self.push(elem).flatten_alt(); {}
self.flatten_alt() + elem; {
self.lemma_flatten_and_flatten_alt_are_equivalent();
}
self.flatten() + elem;
}
}
/// Flattening a sequence containing a single sequence yields that inner sequence.
pub broadcast proof fn lemma_flatten_singleton(self)
requires
#[trigger] self.len() == 1,
ensures
#[trigger] self.flatten() == self[0],
{
assert(self.flatten() == self[0] + self.drop_first().flatten());
assert(self.flatten() == self[0]);
}
pub broadcast group group_seq_flatten {
Seq::<_>::lemma_flatten_push,
Seq::<_>::lemma_flatten_singleton,
}
}
/********************************* Extrema in Sequences *********************************/
impl Seq<int> {
/// Returns the maximum integer value in a non-empty sequence of integers.
pub open spec fn max(self) -> int
recommends
0 < self.len(),
decreases self.len(),
{
if self.len() == 1 {
self[0]
} else if self.len() == 0 {
0
} else {
let later_max = self.drop_first().max();
if self[0] >= later_max {
self[0]
} else {
later_max
}
}
}
/// Proof of correctness and expected properties for max function
pub proof fn max_ensures(self)
ensures
forall|x: int| self.contains(x) ==> x <= self.max(),
forall|i: int| 0 <= i < self.len() ==> self[i] <= self.max(),
self.len() == 0 || self.contains(self.max()),
decreases self.len(),
{
if self.len() <= 1 {
} else {
let elt = self.drop_first().max();
assert(self.drop_first().contains(elt)) by { self.drop_first().max_ensures() }
assert forall|i: int| 0 <= i < self.len() implies self[i] <= self.max() by {
assert(i == 0 || self[i] == self.drop_first()[i - 1]);
assert(forall|j: int|
0 <= j < self.drop_first().len() ==> self.drop_first()[j]
<= self.drop_first().max()) by { self.drop_first().max_ensures() }
}
}
}
/// Returns the minimum integer value in a non-empty sequence of integers.
pub open spec fn min(self) -> int
recommends
0 < self.len(),
decreases self.len(),
{
if self.len() == 1 {
self[0]
} else if self.len() == 0 {
0
} else {
let later_min = self.drop_first().min();
if self[0] <= later_min {
self[0]
} else {
later_min
}
}
}
/// Proof of correctness and expected properties for min function
pub proof fn min_ensures(self)
ensures
forall|x: int| self.contains(x) ==> self.min() <= x,
forall|i: int| 0 <= i < self.len() ==> self.min() <= self[i],
self.len() == 0 || self.contains(self.min()),
decreases self.len(),
{
if self.len() <= 1 {
} else {
let elt = self.drop_first().min();
assert(self.subrange(1, self.len() as int).contains(elt)) by {
self.drop_first().min_ensures()
}
assert forall|i: int| 0 <= i < self.len() implies self.min() <= self[i] by {
assert(i == 0 || self[i] == self.drop_first()[i - 1]);
assert(forall|j: int|
0 <= j < self.drop_first().len() ==> self.drop_first().min()
<= self.drop_first()[j]) by { self.drop_first().min_ensures() }
}
}
}
pub closed spec fn sort(self) -> Self {
self.sort_by(|x: int, y: int| x <= y)
}
pub proof fn lemma_sort_ensures(self)
ensures
self.to_multiset() =~= self.sort().to_multiset(),
sorted_by(self.sort(), |x: int, y: int| x <= y),
{
self.lemma_sort_by_ensures(|x: int, y: int| x <= y);
}
/// The maximum element in a non-empty sequence is greater than or equal to
/// the maxima of its non-empty subsequences.
pub proof fn lemma_subrange_max(self, from: int, to: int)
requires
0 <= from < to <= self.len(),
ensures
self.subrange(from, to).max() <= self.max(),
{
self.max_ensures();
self.subrange(from, to).max_ensures();
}
/// The minimum element in a non-empty sequence is less than or equal to
/// the minima of its non-empty subsequences.
pub proof fn lemma_subrange_min(self, from: int, to: int)
requires
0 <= from < to <= self.len(),
ensures
self.subrange(from, to).min() >= self.min(),
{
self.min_ensures();
self.subrange(from, to).min_ensures();
}
}
// Helper function to aid with merge sort
spec fn merge_sorted_with<A>(left: Seq<A>, right: Seq<A>, leq: spec_fn(A, A) -> bool) -> Seq<A>
recommends
sorted_by(left, leq),
sorted_by(right, leq),
total_ordering(leq),
decreases left.len(), right.len(),
{
if left.len() == 0 {
right
} else if right.len() == 0 {
left
} else if leq(left.first(), right.first()) {
Seq::<A>::empty().push(left.first()) + merge_sorted_with(left.drop_first(), right, leq)
} else {
Seq::<A>::empty().push(right.first()) + merge_sorted_with(left, right.drop_first(), leq)
}
}
proof fn lemma_merge_sorted_with_ensures<A>(left: Seq<A>, right: Seq<A>, leq: spec_fn(A, A) -> bool)
requires
sorted_by(left, leq),
sorted_by(right, leq),
total_ordering(leq),
ensures
(left + right).to_multiset() =~= merge_sorted_with(left, right, leq).to_multiset(),
sorted_by(merge_sorted_with(left, right, leq), leq),
decreases left.len(), right.len(),
{
// TODO: lemma_seq_skip_of_skip and lemma_seq_skip_index2 cause a lot of QIs
broadcast use group_seq_properties;
if left.len() == 0 {
assert(left + right =~= right);
} else if right.len() == 0 {
assert(left + right =~= left);
} else if leq(left.first(), right.first()) {
let result = Seq::<A>::empty().push(left.first()) + merge_sorted_with(
left.drop_first(),
right,
leq,
);
lemma_merge_sorted_with_ensures(left.drop_first(), right, leq);
let rest = merge_sorted_with(left.drop_first(), right, leq);
assert(rest.len() == 0 || rest.first() == left.drop_first().first() || rest.first()
== right.first()) by {
if left.drop_first().len() == 0 {
} else if leq(left.drop_first().first(), right.first()) {
assert(rest =~= Seq::<A>::empty().push(left.drop_first().first())
+ merge_sorted_with(left.drop_first().drop_first(), right, leq));
} else {
assert(rest =~= Seq::<A>::empty().push(right.first()) + merge_sorted_with(
left.drop_first(),
right.drop_first(),
leq,
));
}
}
lemma_new_first_element_still_sorted_by(left.first(), rest, leq);
assert((left.drop_first() + right) =~= (left + right).drop_first());
} else {
let result = Seq::<A>::empty().push(right.first()) + merge_sorted_with(
left,
right.drop_first(),
leq,
);
lemma_merge_sorted_with_ensures(left, right.drop_first(), leq);
let rest = merge_sorted_with(left, right.drop_first(), leq);
assert(rest.len() == 0 || rest.first() == left.first() || rest.first()
== right.drop_first().first()) by {
assert(left.len() > 0);
if right.drop_first().len() == 0 { /*assert(rest =~= left);*/
} else if leq(left.first(), right.drop_first().first()) { //right might be length 1
assert(rest =~= Seq::<A>::empty().push(left.first()) + merge_sorted_with(
left.drop_first(),
right.drop_first(),
leq,
));
} else {
assert(rest =~= Seq::<A>::empty().push(right.drop_first().first())
+ merge_sorted_with(left, right.drop_first().drop_first(), leq));
}
}
lemma_new_first_element_still_sorted_by(
right.first(),
merge_sorted_with(left, right.drop_first(), leq),
leq,
);
lemma_seq_union_to_multiset_commutative(left, right);
assert((right.drop_first() + left) =~= (right + left).drop_first());
lemma_seq_union_to_multiset_commutative(right.drop_first(), left);
}
}
/// The maximum of the concatenation of two non-empty sequences is greater than or
/// equal to the maxima of its two non-empty subsequences.
pub proof fn lemma_max_of_concat(x: Seq<int>, y: Seq<int>)
requires
0 < x.len() && 0 < y.len(),
ensures
x.max() <= (x + y).max(),
y.max() <= (x + y).max(),
forall|elt: int| (x + y).contains(elt) ==> elt <= (x + y).max(),
decreases x.len(),
{
broadcast use group_seq_properties;
x.max_ensures();
y.max_ensures();
(x + y).max_ensures();
assert(x.drop_first().len() == x.len() - 1);
if x.len() == 1 {
assert(y.max() <= (x + y).max()) by {
assert((x + y).contains(y.max()));
}
} else {
assert(x.max() <= (x + y).max()) by {
assert(x.contains(x.max()));
assert((x + y).contains(x.max()));
}
assert(x.drop_first() + y =~= (x + y).drop_first());
lemma_max_of_concat(x.drop_first(), y);
}
}
/// The minimum of the concatenation of two non-empty sequences is less than or
/// equal to the minimum of its two non-empty subsequences.
pub proof fn lemma_min_of_concat(x: Seq<int>, y: Seq<int>)
requires
0 < x.len() && 0 < y.len(),
ensures
(x + y).min() <= x.min(),
(x + y).min() <= y.min(),
forall|elt: int| (x + y).contains(elt) ==> (x + y).min() <= elt,
decreases x.len(),
{
x.min_ensures();
y.min_ensures();
(x + y).min_ensures();
broadcast use group_seq_properties;
if x.len() == 1 {
assert((x + y).min() <= y.min()) by {
assert((x + y).contains(y.min()));
}
} else {
assert((x + y).min() <= x.min()) by {
assert((x + y).contains(x.min()));
}
assert((x + y).min() <= y.min()) by {
assert((x + y).contains(y.min()));
}
assert(x.drop_first() + y =~= (x + y).drop_first());
lemma_max_of_concat(x.drop_first(), y)
}
}
/************************* Sequence to Multiset Conversion **************************/
/// push(a) o to_multiset = to_multiset o insert(a)
pub broadcast proof fn to_multiset_build<A>(s: Seq<A>, a: A)
ensures
#![trigger s.push(a).to_multiset()]
s.push(a).to_multiset() =~= s.to_multiset().insert(a),
decreases s.len(),
{
broadcast use super::multiset::group_multiset_axioms;
if s.len() == 0 {
assert(s.to_multiset() =~= Multiset::<A>::empty());
assert(s.push(a).drop_first() =~= Seq::<A>::empty());
assert(s.push(a).to_multiset() =~= Multiset::<A>::empty().insert(a).add(
Seq::<A>::empty().to_multiset(),
));
} else {
to_multiset_build(s.drop_first(), a);
assert(s.drop_first().push(a).to_multiset() =~= s.drop_first().to_multiset().insert(a));
assert(s.push(a).drop_first() =~= s.drop_first().push(a));
}
}
pub broadcast proof fn to_multiset_remove<A>(s: Seq<A>, i: int)
requires
0 <= i < s.len(),
ensures
#![trigger s.remove(i).to_multiset()]
s.remove(i).to_multiset() =~= s.to_multiset().remove(s[i]),
{
broadcast use super::multiset::group_multiset_axioms;
let s0 = s.subrange(0, i);
let s1 = s.subrange(i, s.len() as int);
let s2 = s.subrange(i + 1, s.len() as int);
lemma_seq_union_to_multiset_commutative(s0, s2);
lemma_seq_union_to_multiset_commutative(s0, s1);
assert(s == s0 + s1);
assert(s2 + s0 == (s1 + s0).drop_first());
}
/// to_multiset() preserves length
pub broadcast proof fn to_multiset_len<A>(s: Seq<A>)
ensures
s.len() == #[trigger] s.to_multiset().len(),
decreases s.len(),
{
broadcast use super::multiset::group_multiset_axioms;
if s.len() == 0 {
assert(s.to_multiset() =~= Multiset::<A>::empty());
assert(s.len() == 0);
} else {
to_multiset_len(s.drop_first());
assert(s.len() == s.drop_first().len() + 1);
assert(s.to_multiset().len() == s.drop_first().to_multiset().len() + 1);
}
}
/// to_multiset() contains only the elements of the sequence
pub broadcast proof fn to_multiset_contains<A>(s: Seq<A>, a: A)
ensures
#![trigger s.to_multiset().count(a)]
s.contains(a) <==> s.to_multiset().count(a) > 0,
decreases s.len(),
{
broadcast use super::multiset::group_multiset_axioms;
if s.len() != 0 {
// ==>
if s.contains(a) {
if s.first() == a {
to_multiset_build(s, a);
assert(s.to_multiset() =~= Multiset::<A>::empty().insert(s.first()).add(
s.drop_first().to_multiset(),
));
assert(Multiset::<A>::empty().insert(s.first()).contains(s.first()));
} else {
to_multiset_contains(s.drop_first(), a);
assert(s.skip(1) =~= s.drop_first());
lemma_seq_skip_contains(s, 1, a);
assert(s.to_multiset().count(a) == s.drop_first().to_multiset().count(a));
assert(s.contains(a) <==> s.to_multiset().count(a) > 0);
}
}
// <==
if s.to_multiset().count(a) > 0 {
to_multiset_contains(s.drop_first(), a);
assert(s.contains(a) <==> s.to_multiset().count(a) > 0);
} else {
assert(s.contains(a) <==> s.to_multiset().count(a) > 0);
}
}
}
/// The last element of two concatenated sequences, the second one being non-empty, will be the
/// last element of the latter sequence.
pub proof fn lemma_append_last<A>(s1: Seq<A>, s2: Seq<A>)
requires
0 < s2.len(),
ensures
(s1 + s2).last() == s2.last(),
{
}
/// The concatenation of sequences is associative
pub proof fn lemma_concat_associative<A>(s1: Seq<A>, s2: Seq<A>, s3: Seq<A>)
ensures
s1.add(s2.add(s3)) =~= s1.add(s2).add(s3),
{
}
/// Recursive definition of seq to set conversion
spec fn seq_to_set_rec<A>(seq: Seq<A>) -> Set<A>
decreases seq.len(),
{
if seq.len() == 0 {
Set::empty()
} else {
seq_to_set_rec(seq.drop_last()).insert(seq.last())
}
}
// Helper function showing that the recursive definition of set_to_seq produces a finite set
proof fn seq_to_set_rec_is_finite<A>(seq: Seq<A>)
ensures
seq_to_set_rec(seq).finite(),
decreases seq.len(),
{
broadcast use super::set::group_set_axioms;
if seq.len() > 0 {
let sub_seq = seq.drop_last();
assert(seq_to_set_rec(sub_seq).finite()) by {
seq_to_set_rec_is_finite(sub_seq);
}
}
}
// Helper function showing that the resulting set contains all elements of the sequence
proof fn seq_to_set_rec_contains<A>(seq: Seq<A>)
ensures
forall|a| #[trigger] seq.contains(a) <==> seq_to_set_rec(seq).contains(a),
decreases seq.len(),
{
broadcast use super::set::group_set_axioms;
if seq.len() > 0 {
assert(forall|a| #[trigger]
seq.drop_last().contains(a) <==> seq_to_set_rec(seq.drop_last()).contains(a)) by {
seq_to_set_rec_contains(seq.drop_last());
}
assert(seq =~= seq.drop_last().push(seq.last()));
assert forall|a| #[trigger] seq.contains(a) <==> seq_to_set_rec(seq).contains(a) by {
if !seq.drop_last().contains(a) {
if a == seq.last() {
assert(seq.contains(a));
assert(seq_to_set_rec(seq).contains(a));
} else {
assert(!seq_to_set_rec(seq).contains(a));
}
}
}
}
}
// Helper function showing that the recursive definition matches the set comprehension one
proof fn seq_to_set_equal_rec<A>(seq: Seq<A>)
ensures
seq.to_set() == seq_to_set_rec(seq),
{
broadcast use super::set::group_set_axioms;
assert(forall|n| #[trigger] seq.contains(n) <==> seq_to_set_rec(seq).contains(n)) by {
seq_to_set_rec_contains(seq);
}
assert(forall|n| #[trigger] seq.contains(n) <==> seq.to_set().contains(n));
assert(seq.to_set() =~= seq_to_set_rec(seq));
}
/// The set obtained from a sequence is finite
pub broadcast proof fn seq_to_set_is_finite<A>(seq: Seq<A>)
ensures
#[trigger] seq.to_set().finite(),
{
broadcast use super::set::group_set_axioms;
assert(seq.to_set().finite()) by {
seq_to_set_equal_rec(seq);
seq_to_set_rec_is_finite(seq);
}
}
pub proof fn seq_to_set_distributes_over_add<T>(s1: Seq<T>, s2: Seq<T>)
ensures
s1.to_set() + s2.to_set() =~= (s1 + s2).to_set(),
{
broadcast use super::group_vstd_default;
broadcast use super::set_lib::group_set_properties;
broadcast use group_seq_properties;
}
/// If sequences a and b don't have duplicates, and there are no
/// elements in common between them, then the concatenated sequence
/// a + b will not contain duplicates either.
pub proof fn lemma_no_dup_in_concat<A>(a: Seq<A>, b: Seq<A>)
requires
a.no_duplicates(),
b.no_duplicates(),
forall|i: int, j: int| 0 <= i < a.len() && 0 <= j < b.len() ==> a[i] != b[j],
ensures
#[trigger] (a + b).no_duplicates(),
{
}
/// Flattening sequences of sequences is distributive over concatenation. That is, concatenating
/// the flattening of two sequences of sequences is the same as flattening the
/// concatenation of two sequences of sequences.
pub proof fn lemma_flatten_concat<A>(x: Seq<Seq<A>>, y: Seq<Seq<A>>)
ensures
(x + y).flatten() =~= x.flatten() + y.flatten(),
decreases x.len(),
{
if x.len() == 0 {
assert(x + y =~= y);
} else {
assert((x + y).drop_first() =~= x.drop_first() + y);
assert(x.first() + (x.drop_first() + y).flatten() =~= x.first() + x.drop_first().flatten()
+ y.flatten()) by {
lemma_flatten_concat(x.drop_first(), y);
}
}
}
/// Flattening sequences of sequences in reverse order is distributive over concatentation.
/// That is, concatenating the flattening of two sequences of sequences in reverse
/// order is the same as flattening the concatenation of two sequences of sequences
/// in reverse order.
pub proof fn lemma_flatten_alt_concat<A>(x: Seq<Seq<A>>, y: Seq<Seq<A>>)
ensures
(x + y).flatten_alt() =~= x.flatten_alt() + y.flatten_alt(),
decreases y.len(),
{
if y.len() == 0 {
assert(x + y =~= x);
} else {
assert((x + y).drop_last() =~= x + y.drop_last());
assert((x + y.drop_last()).flatten_alt() + y.last() =~= x.flatten_alt()
+ y.drop_last().flatten_alt() + y.last()) by {
lemma_flatten_alt_concat(x, y.drop_last());
}
}
}
/// The multiset of a concatenated sequence `a + b` is equivalent to the multiset of the
/// concatenated sequence `b + a`.
pub proof fn lemma_seq_union_to_multiset_commutative<A>(a: Seq<A>, b: Seq<A>)
ensures
(a + b).to_multiset() =~= (b + a).to_multiset(),
{
broadcast use super::multiset::group_multiset_axioms;
lemma_multiset_commutative(a, b);
lemma_multiset_commutative(b, a);
}
/// The multiset of a concatenated sequence `a + b` is equivalent to the multiset of just
/// sequence `a` added to the multiset of just sequence `b`.
pub proof fn lemma_multiset_commutative<A>(a: Seq<A>, b: Seq<A>)
ensures
(a + b).to_multiset() =~= a.to_multiset().add(b.to_multiset()),
decreases a.len(),
{
broadcast use super::multiset::group_multiset_axioms;
if a.len() == 0 {
assert(a + b =~= b);
} else {
lemma_multiset_commutative(a.drop_first(), b);
assert(a.drop_first() + b =~= (a + b).drop_first());
}
}
/// Any two sequences that are sorted by a total order and that have the same elements are equal.
pub proof fn lemma_sorted_unique<A>(x: Seq<A>, y: Seq<A>, leq: spec_fn(A, A) -> bool)
requires
sorted_by(x, leq),
sorted_by(y, leq),
total_ordering(leq),
x.to_multiset() == y.to_multiset(),
ensures
x =~= y,
decreases x.len(), y.len(),
{
broadcast use super::multiset::group_multiset_axioms;
broadcast use group_to_multiset_ensures;
if x.len() == 0 || y.len() == 0 {
} else {
assert(x.to_multiset().contains(x[0]));
assert(x.to_multiset().contains(y[0]));
let i = choose|i: int| #![trigger x.spec_index(i) ] 0 <= i < x.len() && x[i] == y[0];
assert(leq(x[i], x[0]));
assert(leq(x[0], x[i]));
assert(x.drop_first().to_multiset() =~= x.to_multiset().remove(x[0]));
assert(y.drop_first().to_multiset() =~= y.to_multiset().remove(y[0]));
lemma_sorted_unique(x.drop_first(), y.drop_first(), leq);
assert(x.drop_first() =~= y.drop_first());
assert(x.first() == y.first());
assert(x =~= Seq::<A>::empty().push(x.first()).add(x.drop_first()));
assert(x =~= y);
}
}
// This verified lemma used to be an axiom in the Dafny prelude
pub broadcast proof fn lemma_seq_contains<A>(s: Seq<A>, x: A)
ensures
#[trigger] s.contains(x) <==> exists|i: int| 0 <= i < s.len() && #[trigger] s[i] == x,
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// The empty sequence contains nothing
pub broadcast proof fn lemma_seq_empty_contains_nothing<A>(x: A)
ensures
!(#[trigger] Seq::<A>::empty().contains(x)),
{
}
// This verified lemma used to be an axiom in the Dafny prelude
// Note: Dafny only does one way implication, but theoretically it could go both ways
/// A sequence with length 0 is equivalent to the empty sequence
pub broadcast proof fn lemma_seq_empty_equality<A>(s: Seq<A>)
ensures
#[trigger] s.len() == 0 ==> s =~= Seq::<A>::empty(),
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// The concatenation of two sequences contains only the elements
/// of the two sequences
pub broadcast proof fn lemma_seq_concat_contains_all_elements<A>(x: Seq<A>, y: Seq<A>, elt: A)
ensures
#[trigger] (x + y).contains(elt) <==> x.contains(elt) || y.contains(elt),
decreases x.len(),
{
if x.len() == 0 && y.len() > 0 {
assert((x + y) =~= y);
} else {
assert forall|elt: A| #[trigger] x.contains(elt) implies #[trigger] (x + y).contains(
elt,
) by {
let index = choose|i: int| 0 <= i < x.len() && x[i] == elt;
assert((x + y)[index] == elt);
}
assert forall|elt: A| #[trigger] y.contains(elt) implies #[trigger] (x + y).contains(
elt,
) by {
let index = choose|i: int| 0 <= i < y.len() && y[i] == elt;
assert((x + y)[index + x.len()] == elt);
}
}
}
// This verified lemma used to be an axiom in the Dafny prelude
/// After pushing an element onto a sequence, the sequence contains that element
pub broadcast proof fn lemma_seq_contains_after_push<A>(s: Seq<A>, v: A, x: A)
ensures
#[trigger] s.push(v).contains(x) <==> v == x || s.contains(x),
{
assert forall|elt: A| #[trigger] s.contains(elt) implies #[trigger] s.push(v).contains(elt) by {
let index = choose|i: int| 0 <= i < s.len() && s[i] == elt;
assert(s.push(v)[index] == elt);
}
assert(s.push(v)[s.len() as int] == v);
}
// This verified lemma used to be an axiom in the Dafny prelude
/// The subrange of a sequence contains only the elements within the indices `start` and `stop`
/// of the original sequence.
pub broadcast proof fn lemma_seq_subrange_elements<A>(s: Seq<A>, start: int, stop: int, x: A)
requires
0 <= start <= stop <= s.len(),
ensures
#[trigger] s.subrange(start, stop).contains(x) <==> (exists|i: int|
0 <= start <= i < stop <= s.len() && #[trigger] s[i] == x),
{
assert((exists|i: int| 0 <= start <= i < stop <= s.len() && s[i] == x) ==> s.subrange(
start,
stop,
).contains(x)) by {
if exists|i: int| 0 <= start <= i < stop <= s.len() && s[i] == x {
let index = choose|i: int| 0 <= start <= i < stop <= s.len() && s[i] == x;
assert(s.subrange(start, stop)[index - start] == s[index]);
}
}
}
// Definition of a commutative fold_right operator.
pub open spec fn commutative_foldr<A, B>(f: spec_fn(A, B) -> B) -> bool {
forall|x: A, y: A, v: B| #[trigger] f(x, f(y, v)) == f(y, f(x, v))
}
// For a commutative fold_right operator, any folding order
// (i.e., any permutation) produces the same result.
pub proof fn lemma_fold_right_permutation<A, B>(l1: Seq<A>, l2: Seq<A>, f: spec_fn(A, B) -> B, v: B)
requires
commutative_foldr(f),
l1.to_multiset() == l2.to_multiset(),
ensures
l1.fold_right(f, v) == l2.fold_right(f, v),
decreases l1.len(),
{
broadcast use group_to_multiset_ensures;
if l1.len() > 0 {
let a = l1.last();
let i = l2.index_of(a);
let l2r = l2.subrange(i + 1, l2.len() as int).fold_right(f, v);
assert(l1.to_multiset().count(a) > 0);
l1.drop_last().lemma_fold_right_commute_one(a, f, v);
l2.subrange(0, i).lemma_fold_right_commute_one(a, f, l2r);
l2.lemma_fold_right_split(f, v, i + 1);
l2.remove(i).lemma_fold_right_split(f, v, i);
assert(l2.subrange(0, i + 1).drop_last() == l2.subrange(0, i));
assert(l1.drop_last() == l1.remove(l1.len() - 1));
assert(l2.remove(i).subrange(0, i) == l2.subrange(0, i));
assert(l2.remove(i).subrange(i, l2.remove(i).len() as int) == l2.subrange(
i + 1,
l2.len() as int,
));
lemma_fold_right_permutation(l1.drop_last(), l2.remove(i), f, v);
} else {
assert(l2.to_multiset().len() == 0);
}
}
/************************** Lemmas about Take/Skip ***************************/
// This verified lemma used to be an axiom in the Dafny prelude
/// Taking the first `n` elements of a sequence results in a sequence of length `n`,
/// as long as `n` is within the bounds of the original sequence.
pub broadcast proof fn lemma_seq_take_len<A>(s: Seq<A>, n: int)
ensures
0 <= n <= s.len() ==> #[trigger] s.take(n).len() == n,
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// The resulting sequence after taking the first `n` elements from sequence `s` contains
/// element `x` if and only if `x` is contained in the first `n` elements of `s`.
pub broadcast proof fn lemma_seq_take_contains<A>(s: Seq<A>, n: int, x: A)
requires
0 <= n <= s.len(),
ensures
#[trigger] s.take(n).contains(x) <==> (exists|i: int|
0 <= i < n <= s.len() && #[trigger] s[i] == x),
{
assert((exists|i: int| 0 <= i < n <= s.len() && #[trigger] s[i] == x) ==> s.take(n).contains(x))
by {
if exists|i: int| 0 <= i < n <= s.len() && #[trigger] s[i] == x {
let index = choose|i: int| 0 <= i < n <= s.len() && #[trigger] s[i] == x;
assert(s.take(n)[index] == s[index]);
}
}
}
// This verified lemma used to be an axiom in the Dafny prelude
/// If `j` is a valid index less than `n`, then the `j`th element of the sequence `s`
/// is the same as `j`th element of the sequence after taking the first `n` elements of `s`.
pub broadcast proof fn lemma_seq_take_index<A>(s: Seq<A>, n: int, j: int)
ensures
0 <= j < n <= s.len() ==> #[trigger] s.take(n)[j] == s[j],
{
}
pub proof fn subrange_of_matching_take<T>(a: Seq<T>, b: Seq<T>, s: int, e: int, l: int)
requires
a.take(l) == b.take(l),
l <= a.len(),
l <= b.len(),
0 <= s <= e <= l,
ensures
a.subrange(s, e) == b.subrange(s, e),
{
assert forall|i| 0 <= i < e - s implies a.subrange(s, e)[i] == b.subrange(s, e)[i] by {
assert(a.subrange(s, e)[i] == a.take(l)[i + s]);
// assert( b.subrange(s, e)[i] == b.take(l)[i + s] ); // either trigger will do
}
// trigger extn equality (verus issue #1257)
assert(a.subrange(s, e) == b.subrange(s, e));
}
// This verified lemma used to be an axiom in the Dafny prelude
/// Skipping the first `n` elements of a sequence gives a sequence of length `n` less than
/// the original sequence's length.
pub broadcast proof fn lemma_seq_skip_len<A>(s: Seq<A>, n: int)
ensures
0 <= n <= s.len() ==> #[trigger] s.skip(n).len() == s.len() - n,
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// The resulting sequence after skipping the first `n` elements from sequence `s` contains
/// element `x` if and only if `x` is contained in `s` before index `n`.
pub broadcast proof fn lemma_seq_skip_contains<A>(s: Seq<A>, n: int, x: A)
requires
0 <= n <= s.len(),
ensures
#[trigger] s.skip(n).contains(x) <==> (exists|i: int|
0 <= n <= i < s.len() && #[trigger] s[i] == x),
{
assert((exists|i: int| 0 <= n <= i < s.len() && #[trigger] s[i] == x) ==> s.skip(n).contains(x))
by {
let index = choose|i: int| 0 <= n <= i < s.len() && #[trigger] s[i] == x;
lemma_seq_skip_index(s, n, index - n);
}
}
// This verified lemma used to be an axiom in the Dafny prelude
/// If `j` is a valid index less than `s.len() - n`, then the `j`th element of the sequence
/// `s.skip(n)` is the same as the `j+n`th element of the sequence `s`.
pub broadcast proof fn lemma_seq_skip_index<A>(s: Seq<A>, n: int, j: int)
ensures
0 <= n && 0 <= j < (s.len() - n) ==> #[trigger] s.skip(n)[j] == s[j + n],
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// If `k` is a valid index between `n` (inclusive) and the length of sequence `s` (exclusive),
/// then the `k-n`th element of the sequence `s.skip(n)` is the same as the `k`th element of the
/// original sequence `s`.
pub broadcast proof fn lemma_seq_skip_index2<A>(s: Seq<A>, n: int, k: int)
ensures
0 <= n <= k < s.len() ==> (#[trigger] s.skip(n))[k - n] == #[trigger] s[k],
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// If `n` is the length of sequence `a`, then taking the first `n` elements of the concatenation
/// `a + b` is equivalent to the sequence `a` and skipping the first `n` elements of the concatenation
/// `a + b` is equivalent to the sequence `b`.
pub broadcast proof fn lemma_seq_append_take_skip<A>(a: Seq<A>, b: Seq<A>, n: int)
ensures
#![trigger (a + b).take(n)]
#![trigger (a + b).skip(n)]
n == a.len() ==> ((a + b).take(n) =~= a && (a + b).skip(n) =~= b),
{
}
/************* Lemmas about the Commutability of Take and Skip with Update ************/
// This verified lemma used to be an axiom in the Dafny prelude
/// If `i` is in the first `n` indices of sequence `s`, updating sequence `s` at index `i` with
/// value `v` and then taking the first `n` elements is equivalent to first taking the first `n`
/// elements of `s` and then updating index `i` to value `v`.
pub broadcast proof fn lemma_seq_take_update_commut1<A>(s: Seq<A>, i: int, v: A, n: int)
ensures
#![trigger s.update(i, v).take(n)]
0 <= i < n <= s.len() ==> #[trigger] s.update(i, v).take(n) =~= s.take(n).update(i, v),
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// If `i` is a valid index after the first `n` indices of sequence `s`, updating sequence `s` at
/// index `i` with value `v` and then taking the first `n` elements is equivalent to just taking the first `n`
/// elements of `s` without the update.
pub broadcast proof fn lemma_seq_take_update_commut2<A>(s: Seq<A>, i: int, v: A, n: int)
ensures
0 <= n <= i < s.len() ==> #[trigger] s.update(i, v).take(n) =~= s.take(n),
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// If `i` is a valid index after the first `n` indices of sequence `s`, updating sequence `s` at
/// index `i` with value `v` and then skipping the first `n` elements is equivalent to skipping the first `n`
/// elements of `s` and then updating index `i-n` to value `v`.
pub broadcast proof fn lemma_seq_skip_update_commut1<A>(s: Seq<A>, i: int, v: A, n: int)
ensures
0 <= n <= i < s.len() ==> #[trigger] s.update(i, v).skip(n) =~= s.skip(n).update(i - n, v),
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// If `i` is a valid index in the first `n` indices of sequence `s`, updating sequence `s` at
/// index `i` with value `v` and then skipping the first `n` elements is equivalent to just skipping
/// the first `n` elements without the update.
pub broadcast proof fn lemma_seq_skip_update_commut2<A>(s: Seq<A>, i: int, v: A, n: int)
ensures
0 <= i < n <= s.len() ==> #[trigger] s.update(i, v).skip(n) =~= s.skip(n),
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// Pushing element `v` onto the end of sequence `s` and then skipping the first `n` elements is
/// equivalent to skipping the first `n` elements of `s` and then pushing `v` onto the end.
pub broadcast proof fn lemma_seq_skip_build_commut<A>(s: Seq<A>, v: A, n: int)
ensures
#![trigger s.push(v).skip(n)]
0 <= n <= s.len() ==> s.push(v).skip(n) =~= s.skip(n).push(v),
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// `s.skip(0)` is equivalent to `s`.
pub broadcast proof fn lemma_seq_skip_nothing<A>(s: Seq<A>, n: int)
ensures
n == 0 ==> #[trigger] s.skip(n) =~= s,
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// `s.take(0)` is equivalent to the empty sequence.
pub broadcast proof fn lemma_seq_take_nothing<A>(s: Seq<A>, n: int)
ensures
n == 0 ==> #[trigger] s.take(n) =~= Seq::<A>::empty(),
{
}
// This verified lemma used to be an axiom in the Dafny prelude
/// If `m + n` is less than or equal to the length of sequence `s`, then skipping the first `m` elements
/// and then skipping the first `n` elements of the resulting sequence is equivalent to just skipping
/// the first `m + n` elements.
pub broadcast proof fn lemma_seq_skip_of_skip<A>(s: Seq<A>, m: int, n: int)
ensures
(0 <= m && 0 <= n && m + n <= s.len()) ==> #[trigger] s.skip(m).skip(n) =~= s.skip(m + n),
{
}
/// Properties of sequences from the Dafny prelude (which were axioms in Dafny, but proven here in Verus)
// TODO: seems like this warning doesn't come up?
#[deprecated = "Use `broadcast use group_seq_properties` instead"]
pub proof fn lemma_seq_properties<A>()
ensures
forall|s: Seq<A>, x: A|
s.contains(x) <==> exists|i: int| 0 <= i < s.len() && #[trigger] s[i] == x, //from lemma_seq_contains(s, x),
forall|x: A| !(#[trigger] Seq::<A>::empty().contains(x)), //from lemma_seq_empty_contains_nothing(x),
forall|s: Seq<A>| #[trigger] s.len() == 0 ==> s =~= Seq::<A>::empty(), //from lemma_seq_empty_equality(s),
forall|x: Seq<A>, y: Seq<A>, elt: A| #[trigger]
(x + y).contains(elt) <==> x.contains(elt) || y.contains(elt), //from lemma_seq_concat_contains_all_elements(x, y, elt),
forall|s: Seq<A>, v: A, x: A| #[trigger] s.push(v).contains(x) <==> v == x || s.contains(x), //from lemma_seq_contains_after_push(s, v, x)
forall|s: Seq<A>, start: int, stop: int, x: A|
(0 <= start <= stop <= s.len() && #[trigger] s.subrange(start, stop).contains(x)) <==> (
exists|i: int| 0 <= start <= i < stop <= s.len() && #[trigger] s[i] == x), //from lemma_seq_subrange_elements(s, start, stop, x),
forall|s: Seq<A>, n: int| 0 <= n <= s.len() ==> #[trigger] s.take(n).len() == n, //from lemma_seq_take_len(s, n)
forall|s: Seq<A>, n: int, x: A|
(#[trigger] s.take(n).contains(x) && 0 <= n <= s.len()) <==> (exists|i: int|
0 <= i < n <= s.len() && #[trigger] s[i] == x), //from lemma_seq_take_contains(s, n, x),
forall|s: Seq<A>, n: int, j: int| 0 <= j < n <= s.len() ==> #[trigger] s.take(n)[j] == s[j], //from lemma_seq_take_index(s, n, j),
forall|s: Seq<A>, n: int| 0 <= n <= s.len() ==> #[trigger] s.skip(n).len() == s.len() - n, //from lemma_seq_skip_len(s, n),
forall|s: Seq<A>, n: int, x: A|
(#[trigger] s.skip(n).contains(x) && 0 <= n <= s.len()) <==> (exists|i: int|
0 <= n <= i < s.len() && #[trigger] s[i] == x), //from lemma_seq_skip_contains(s, n, x),
forall|s: Seq<A>, n: int, j: int|
0 <= n && 0 <= j < (s.len() - n) ==> #[trigger] s.skip(n)[j] == s[j + n], //from lemma_seq_skip_index(s, n, j),
forall|a: Seq<A>, b: Seq<A>, n: int|
#![trigger (a+b).take(n)]
#![trigger (a+b).skip(n)]
n == a.len() ==> ((a + b).take(n) =~= a && (a + b).skip(n) =~= b), //from lemma_seq_append_take_skip(a, b, n),
forall|s: Seq<A>, i: int, v: A, n: int|
0 <= i < n <= s.len() ==> #[trigger] s.update(i, v).take(n) == s.take(n).update(i, v), //from lemma_seq_take_update_commut1(s, i, v, n),
forall|s: Seq<A>, i: int, v: A, n: int|
0 <= n <= i < s.len() ==> #[trigger] s.update(i, v).take(n) == s.take(n), //from lemma_seq_take_update_commut2(s, i, v, n),
forall|s: Seq<A>, i: int, v: A, n: int|
0 <= n <= i < s.len() ==> #[trigger] s.update(i, v).skip(n) == s.skip(n).update(
i - n,
v,
), //from lemma_seq_skip_update_commut1(s, i, v, n),
forall|s: Seq<A>, i: int, v: A, n: int|
0 <= i < n <= s.len() ==> #[trigger] s.update(i, v).skip(n) == s.skip(n), //from lemma_seq_skip_update_commut2(s, i, v, n),
forall|s: Seq<A>, v: A, n: int|
0 <= n <= s.len() ==> #[trigger] s.push(v).skip(n) == s.skip(n).push(v), //from lemma_seq_skip_build_commut(s, v, n),
forall|s: Seq<A>, n: int| n == 0 ==> #[trigger] s.skip(n) == s, //from lemma_seq_skip_nothing(s, n),
forall|s: Seq<A>, n: int| n == 0 ==> #[trigger] s.take(n) == Seq::<A>::empty(), //from lemma_seq_take_nothing(s, n),
forall|s: Seq<A>, m: int, n: int|
(0 <= m && 0 <= n && m + n <= s.len()) ==> #[trigger] s.skip(m).skip(n) == s.skip(
m + n,
), //from lemma_seq_skip_of_skip(s, m, n),
forall|s: Seq<A>, a: A| #[trigger] (s.push(a).to_multiset()) =~= s.to_multiset().insert(a), //from o_multiset_properties
forall|s: Seq<A>| s.len() == #[trigger] s.to_multiset().len(), //from to_multiset_ensures
forall|s: Seq<A>, a: A|
s.contains(a) <==> #[trigger] s.to_multiset().count(a)
> 0, //from to_multiset_ensures
{
broadcast use {group_seq_properties, lemma_seq_skip_of_skip};
// TODO: for some reason this still needs to be explicitly stated
assert forall|s: Seq<A>, v: A, x: A| v == x || s.contains(x) implies #[trigger] s.push(
v,
).contains(x) by {
lemma_seq_contains_after_push(s, v, x);
}
}
#[doc(hidden)]
#[verifier::inline]
pub open spec fn check_argument_is_seq<A>(s: Seq<A>) -> Seq<A> {
s
}
/// Prove two sequences `s1` and `s2` are equal by proving that their elements are equal at each index.
///
/// More precisely, `assert_seqs_equal!` requires:
/// * `s1` and `s2` have the same length (`s1.len() == s2.len()`), and
/// * for all `i` in the range `0 <= i < s1.len()`, we have `s1[i] == s2[i]`.
///
/// The property that equality follows from these facts is often called _extensionality_.
///
/// `assert_seqs_equal!` can handle many trivial-looking
/// identities without any additional help:
///
/// ```rust
/// proof fn subrange_concat(s: Seq<u64>, i: int) {
/// requires([
/// 0 <= i && i <= s.len(),
/// ]);
///
/// let t1 = s.subrange(0, i);
/// let t2 = s.subrange(i, s.len());
/// let t = t1.add(t2);
///
/// assert_seqs_equal!(s == t);
///
/// assert(s == t);
/// }
/// ```
///
/// In more complex cases, a proof may be required for the equality of each element pair.
/// For example,
///
/// ```rust
/// proof fn bitvector_seqs() {
/// let s = Seq::<u64>::new(5, |i| i as u64);
/// let t = Seq::<u64>::new(5, |i| i as u64 | 0);
///
/// assert_seqs_equal!(s == t, i => {
/// // Need to show that s[i] == t[i]
/// // Prove that the elements are equal by appealing to a bitvector solver:
/// let j = i as u64;
/// assert_bit_vector(j | 0 == j);
/// assert(s[i] == t[i]);
/// });
/// }
/// ```
#[macro_export]
macro_rules! assert_seqs_equal {
[$($tail:tt)*] => {
::builtin_macros::verus_proof_macro_exprs!($crate::vstd::seq_lib::assert_seqs_equal_internal!($($tail)*))
};
}
#[macro_export]
#[doc(hidden)]
macro_rules! assert_seqs_equal_internal {
(::builtin::spec_eq($s1:expr, $s2:expr)) => {
$crate::vstd::seq_lib::assert_seqs_equal_internal!($s1, $s2)
};
(::builtin::spec_eq($s1:expr, $s2:expr), $idx:ident => $bblock:block) => {
$crate::vstd::seq_lib::assert_seqs_equal_internal!($s1, $s2, $idx => $bblock)
};
(crate::builtin::spec_eq($s1:expr, $s2:expr)) => {
$crate::vstd::seq_lib::assert_seqs_equal_internal!($s1, $s2)
};
(crate::builtin::spec_eq($s1:expr, $s2:expr), $idx:ident => $bblock:block) => {
$crate::vstd::seq_lib::assert_seqs_equal_internal!($s1, $s2, $idx => $bblock)
};
($s1:expr, $s2:expr $(,)?) => {
$crate::vstd::seq_lib::assert_seqs_equal_internal!($s1, $s2, idx => { })
};
($s1:expr, $s2:expr, $idx:ident => $bblock:block) => {
#[verifier::spec] let s1 = $crate::vstd::seq_lib::check_argument_is_seq($s1);
#[verifier::spec] let s2 = $crate::vstd::seq_lib::check_argument_is_seq($s2);
$crate::vstd::prelude::assert_by($crate::vstd::prelude::equal(s1, s2), {
$crate::vstd::prelude::assert_(s1.len() == s2.len());
$crate::vstd::prelude::assert_forall_by(|$idx : $crate::vstd::prelude::int| {
$crate::vstd::prelude::requires(::builtin_macros::verus_proof_expr!(0 <= $idx && $idx < s1.len()));
$crate::vstd::prelude::ensures($crate::vstd::prelude::equal(s1.index($idx), s2.index($idx)));
{ $bblock }
});
$crate::vstd::prelude::assert_($crate::vstd::prelude::ext_equal(s1, s2));
});
}
}
pub broadcast group group_filter_ensures {
Seq::lemma_filter_len,
Seq::lemma_filter_pred,
Seq::lemma_filter_contains,
}
pub broadcast group group_seq_lib_default {
group_filter_ensures,
Seq::add_empty_left,
Seq::add_empty_right,
Seq::push_distributes_over_add,
Seq::filter_distributes_over_add,
seq_to_set_is_finite,
Seq::lemma_fold_right_split,
Seq::lemma_fold_left_split,
}
pub broadcast group group_to_multiset_ensures {
to_multiset_build,
to_multiset_remove,
to_multiset_len,
to_multiset_contains,
}
// include all the Dafny prelude lemmas
pub broadcast group group_seq_properties {
lemma_seq_contains,
lemma_seq_empty_contains_nothing,
lemma_seq_empty_equality,
lemma_seq_concat_contains_all_elements,
lemma_seq_contains_after_push,
lemma_seq_subrange_elements,
lemma_seq_take_len,
lemma_seq_take_contains,
lemma_seq_take_index,
lemma_seq_skip_len,
lemma_seq_skip_contains,
lemma_seq_skip_index,
lemma_seq_skip_index2,
lemma_seq_append_take_skip,
lemma_seq_take_update_commut1,
lemma_seq_take_update_commut2,
lemma_seq_skip_update_commut1,
lemma_seq_skip_update_commut2,
lemma_seq_skip_build_commut,
lemma_seq_skip_nothing,
lemma_seq_take_nothing,
// Removed the following from group due to bad verification performance
// for `lemma_merge_sorted_with_ensures`
// lemma_seq_skip_of_skip,
group_to_multiset_ensures,
}
#[doc(hidden)]
pub use assert_seqs_equal_internal;
pub use assert_seqs_equal;
} // verus!