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Library.dfy
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Library.dfy
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module Utils.Lib.ControlFlow {
function method Unreachable<A>() : A
requires false
{
match () {}
}
}
module Utils.Lib.Debug {
class {:compile false} Debugger {
static method {:extern "System.Diagnostics.Debugger", "Break"} Break() {}
}
function TODO<A>(a : A) : A {
a
} by method {
Debugger.Break();
return a;
}
}
module Utils.Lib.Datatypes {
datatype Option<+T> = | Some(value: T) | None
{
function method UnwrapOr(default: T) : T {
if this.Some? then value else default
}
function method ToSuccessOr<E>(f: E): Result<T, E> {
match this
case Some(v) => Success(v)
case None() => Failure(f)
}
function method All(P: T -> bool) : bool {
match this {
case Some(v) => P(v)
case None => true
}
}
function method Map<R>(f: T ~> R): Option<R>
reads f.reads
requires this.Some? ==> f.requires(this.value)
{
match this {
case None => None
case Some(v) => Some(f(v))
}
}
}
datatype Result<+T, +R> = | Success(value: T) | Failure(error: R) {
predicate method IsFailure() {
Failure?
}
function method PropagateFailure<U>(): Result<U, R>
requires Failure?
{
Failure(this.error)
}
function method Extract(): T
requires Success?
{
value
}
// FIXME: Port
function method MapFailure<R'>(f: R --> R') : Result<T, R'>
requires this.Failure? ==> f.requires(this.error)
{
match this
case Success(value) => Success(value)
case Failure(error) => Failure(f(error))
}
function method Then<K>(f: T --> Result<K, R>) : Result<K, R>
requires Success? ==> f.requires(value)
{
match this
case Success(value) => f(value)
case Failure(err) => Failure(err)
}
}
datatype Outcome<E> = Pass | Fail(error: E)
{
predicate method IsFailure() {
Fail?
}
// Note: PropagateFailure returns a Result, not an Outcome.
function method PropagateFailure<U>(): Result<U, E>
requires Fail?
{
Failure(this.error)
}
// Note: no Extract method
}
// A helper function to ensure a requirement is true at runtime
// :- Need(5 == |mySet|, "The set MUST have 5 elements.")
// FIXME: Move to library (added opaque)
function method {:opaque} Need<E>(condition: bool, error: E): (result: Outcome<E>)
ensures condition <==> result.Pass?
{
if condition then Pass else Fail(error)
}
}
module Utils.Lib.Seq {
// FIXME why not use a comprehension directly?
// FIXME move `f`
function method {:opaque} Map<T, Q>(f: T ~> Q, ts: seq<T>) : (qs: seq<Q>)
reads f.reads
requires forall t | t in ts :: f.requires(t)
ensures |qs| == |ts|
ensures forall i | 0 <= i < |ts| :: qs[i] == f(ts[i])
ensures forall q | q in qs :: exists t | t in ts :: q == f(t)
{
if ts == [] then [] else [f(ts[0])] + Map(f, ts[1..])
}
lemma Map_in<T, Q>(f: T ~> Q, ts: seq<T>)
requires forall t | t in ts :: f.requires(t)
ensures forall q :: q in Map(f, ts) <==> exists t | t in ts :: q == f(t)
{}
function method FoldL<TAcc(!new), T>(f: (TAcc, T) ~> TAcc, a0: TAcc, ts: seq<T>) : TAcc
reads f.reads
requires forall a, t | t in ts :: f.requires(a, t)
{
if ts == [] then a0 else FoldL(f, f(a0, ts[0]), ts[1..])
}
lemma FoldL_induction'<TAcc(!new), T>(
f: (TAcc, T) ~> TAcc, a0: TAcc, ts: seq<T>,
prefix: seq<T>, P: (TAcc, seq<T>) -> bool
)
requires forall a, t | t in ts :: f.requires(a, t)
requires P(a0, prefix)
requires forall a, t, ts' | t in ts && P(a, ts') :: P(f(a, t), ts' + [t])
ensures P(FoldL(f, a0, ts), prefix + ts)
{
if ts == [] {
assert prefix + ts == prefix;
} else {
var t0, ts' := ts[0], ts[1..];
var a0' := f(a0, t0);
var prefix' := prefix + [t0];
FoldL_induction'(f, a0', ts[1..], prefix', P);
assert P(FoldL(f, a0', ts[1..]), prefix' + ts');
assert prefix' + ts' == prefix + ts;
}
}
lemma FoldL_induction<TAcc(!new), T>(
f: (TAcc, T) ~> TAcc, a0: TAcc, ts: seq<T>,
P: (TAcc, seq<T>) -> bool
)
requires forall a, t | t in ts :: f.requires(a, t)
requires P(a0, [])
requires forall a, t, ts' | t in ts && P(a, ts') :: P(f(a, t), ts' + [t])
ensures P(FoldL(f, a0, ts), ts)
{
assert [] + ts == ts;
FoldL_induction'(f, a0, ts, [], P);
}
function method FoldR<TAcc(!new), T>(f: (TAcc, T) ~> TAcc, a0: TAcc, ts: seq<T>) : TAcc
reads f.reads
requires forall a, t | t in ts :: f.requires(a, t)
{
if ts == [] then a0 else f(FoldL(f, a0, ts[1..]), ts[0])
}
function method Flatten<T>(tss: seq<seq<T>>): (ts: seq<T>)
ensures forall s <- ts :: exists ts <- tss :: s in ts
ensures forall ts0 <- tss, s <- ts0 :: s in ts
{
if tss == [] then []
else tss[0] + Flatten(tss[1..])
}
function method Interleave<T>(sep:T, ts: seq<T>): seq<T> {
if |ts| <= 1 then ts
else [ts[0], sep] + Interleave(sep, ts[1..])
}
// TODO: Why not use forall directly?
function method {:opaque} All<T>(P: T ~> bool, ts: seq<T>) : (b: bool)
reads P.reads
requires forall t | t in ts :: P.requires(t)
ensures b == forall t | t in ts :: P(t)
ensures b == forall i | 0 <= i < |ts| :: P(ts[i])
{
if ts == [] then true else P(ts[0]) && All(P, ts[1..])
}
lemma All_weaken<T>(P: T ~> bool, Q: T~> bool, ts: seq<T>)
requires forall t | t in ts :: P.requires(t)
requires forall t | t in ts :: Q.requires(t)
requires forall t | t in ts :: P(t) ==> Q(t)
ensures All(P, ts) ==> All(Q, ts)
{}
lemma All_weaken_auto<T>(ts: seq<T>)
ensures forall P: T ~> bool, Q: T ~> bool |
&& (forall t: T | t in ts :: P.requires(t))
&& (forall t: T | t in ts :: Q.requires(t))
&& (forall t: T | t in ts :: P(t) ==> Q(t)) ::
All(P, ts) ==> All(Q, ts)
{}
import Math
function method {:opaque} Max(s: seq<int>, default: int) : (m: int)
requires forall i | i in s :: default <= i
ensures if s == [] then m == default else m in s
ensures forall i | i in s :: i <= m
ensures default <= m
{
var P := (m, s) =>
&& (if s == [] then m == default else m in s)
&& (forall i | i in s :: i <= m);
FoldL_induction(Math.Max, default, s, P);
FoldL(Math.Max, default, s)
}
function method {:opaque} MaxF<T>(f: T ~> int, ts: seq<T>, default: int) : (m: int)
reads f.reads
requires forall t | t in ts :: f.requires(t)
requires forall t | t in ts :: default <= f(t)
ensures if ts == [] then m == default else exists t | t in ts :: f(t) == m
ensures forall t | t in ts :: f(t) <= m
ensures default <= m
{
var s := Map(f, ts);
var m := Max(s, default);
assert forall t | t in ts :: f(t) <= m by {
forall t | t in ts ensures f(t) <= m { assert f(t) in s; }
}
m
}
function method {:opaque} Zip<T, Q>(ts: seq<T>, qs: seq<Q>)
: (tqs: seq<(T, Q)>)
requires |ts| == |qs|
ensures |ts| == |tqs| == |qs|
ensures forall i | 0 <= i < |tqs| :: tqs[i] == (ts[i], qs[i])
ensures forall tq | tq in tqs :: tq.0 in ts && tq.1 in qs
{
if ts == [] then []
else [(ts[0], qs[0])] + Zip(ts[1..], qs[1..])
}
// TODO: Merge. Removed unnecessary trigger and strengthened postcondition
function method {:opaque} Filter<T>(s: seq<T>, f: (T ~> bool)): (result: seq<T>)
requires forall i | 0 <= i < |s| :: f.requires(s[i])
ensures |result| <= |s| // DISCUSS: This allows duplication / unifiqation
ensures forall x | x in result :: f.requires(x) && x in s && f(x)
ensures forall x | x in s && f(x) :: x in result
reads f.reads
{
if |s| == 0 then []
else (if f(s[0]) then [s[0]] else []) + Filter(s[1..], f)
}
import Datatypes
function method {:opaque} MapResult<T, Q, E>(ts: seq<T>, f: T ~> Datatypes.Result<Q, E>)
: (qs: Datatypes.Result<seq<Q>, E>)
reads f.reads
requires forall t | t in ts :: f.requires(t)
decreases ts
ensures qs.Success? ==> |qs.value| == |ts|
ensures qs.Success? ==> forall i | 0 <= i < |ts| :: f(ts[i]).Success? && qs.value[i] == f(ts[i]).value
ensures qs.Failure? ==> exists i | 0 <= i < |ts| :: f(ts[i]).Failure? && qs.error == f(ts[i]).error
{
if ts == [] then
Datatypes.Success([])
else
var hd :- f(ts[0]);
var tl :- MapResult(ts[1..], f);
Datatypes.Success([hd] + tl)
}
function method FoldLResult<TAcc(!new), T, TErr>(f: (TAcc, T) ~> Datatypes.Result<TAcc, TErr>, a0: TAcc, ts: seq<T>)
: (acc: Datatypes.Result<TAcc, TErr>)
reads f.reads
requires forall a, t | t in ts :: f.requires(a, t)
{
if ts == [] then Datatypes.Success(a0)
else
var a0 :- f(a0, ts[0]);
FoldLResult(f, a0, ts[1..])
}
function method {:opaque} MapFilter<T, Q>(s: seq<T>, f: (T ~> Datatypes.Option<Q>)): (result: seq<Q>)
requires forall i | 0 <= i < |s| :: f.requires(s[i])
ensures |result| <= |s|
ensures forall y | y in result :: exists x | x in s :: f.requires(x) && f(x) == Datatypes.Some(y)
ensures forall x | x in s && f(x).Some? :: f(x).value in result
reads f.reads
{
if |s| == 0 then []
else (match f(s[0])
case Some(y) => [y]
case None => []) + MapFilter(s[1..], f)
}
function method Break<T>(P: T -> bool, s: seq<T>): (result: (seq<T>, seq<T>))
// Break a sequence into a prefix that does not satisfy a predicate and
// a suffix beginning with an element that does (or an empty suffix if
// no elements satisfy the predicate).
ensures |result.1| <= |s|
ensures forall c <- result.0 :: !P(c)
ensures |result.1| > 0 ==> P(result.1[0])
ensures result.0 + result.1 == s
{
if |s| == 0 then ([], [])
else if P(s[0]) then ([], s)
else
var (h, t) := Break(P, s[1..]);
([s[0]] + h, t)
}
function method Split<T(==)>(x: T, s: seq<T>): (result: seq<seq<T>>)
// Split a sequence into the subsequences separated by the given delimiter.
decreases |s|
ensures forall s <- result :: x !in s
// The following should be true, but doesn't go through automatically.
// ensures Flatten(result) == Filter(s, y => y != x)
{
var (h, t) := Break(y => y == x, s);
assert !(x in h);
[h] + if |t| == 0 then [] else Split(x, t[1..])
}
}
module Utils.Lib.Outcome.OfSeq { // FIXME rename to Seq
import Seq
import opened Datatypes
function method Combine<E>(so: seq<Outcome<E>>): (os: Outcome<seq<E>>)
ensures os.Pass? <==> forall o | o in so :: o.Pass?
{
var fails := Seq.Filter(so, (x: Outcome<E>) => x.Fail?);
if |fails| == 0 then
Pass
else
assert fails[0] in so;
Fail(Seq.Map((x: Outcome<E>) requires x.Fail? => x.error, fails))
}
function method CombineSeq<E>(so: seq<Outcome<seq<E>>>): (os: Outcome<seq<E>>)
ensures os.Pass? <==> forall o | o in so :: o.Pass?
{
var fails := Seq.Filter(so, (x: Outcome<seq<E>>) => x.Fail?);
if |fails| == 0 then
Pass
else
assert fails[0] in so;
Fail(Seq.Flatten(Seq.Map((x: Outcome<seq<E>>) requires x.Fail? => x.error, fails)))
}
}
module Utils.Lib.Str {
module Private {
function method digits(n: int, base: int): (digits: seq<int>)
requires base > 1
requires n >= 0
decreases n
ensures forall d | d in digits :: 0 <= d < base
{
if n == 0 then
[]
else
assert n > 0;
assert base > 1;
assert n < base * n;
assert n / base < n;
digits(n / base, base) + [n % base]
}
function method string_of_digits(digits: seq<int>, chars: seq<char>) : string
requires forall d | d in digits :: 0 <= d < |chars|
{
if digits == [] then ""
else
assert digits[0] in digits;
assert forall d | d in digits[1..] :: d in digits;
[chars[digits[0]]] + string_of_digits(digits[1..], chars)
}
}
function method of_int_any(n: int, chars: seq<char>) : string
requires |chars| > 1
{
var base := |chars|;
if n == 0 then
"0"
else if n > 0 then
Private.string_of_digits(Private.digits(n, base), chars)
else
"-" + Private.string_of_digits(Private.digits(-n, base), chars)
}
const HEX_DIGITS := [
'0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
'A', 'B', 'C', 'D', 'E', 'F']
// FIXME rename
function method of_int(n: int, base: int := 10) : string
requires 2 <= base <= 16
{
of_int_any(n, HEX_DIGITS[..base])
}
method Test() { // FIXME {:test}?
expect of_int(0, 10) == "0";
expect of_int(3, 10) == "3";
expect of_int(302, 10) == "302";
expect of_int(-3, 10) == "-3";
expect of_int(-302, 10) == "-302";
}
function method of_bool(b: bool) : string {
if b then "true" else "false"
}
function method of_char(c: char) : string {
[c]
}
function method Join(sep: string, strs: seq<string>) : string {
if |strs| == 0 then ""
else if |strs| == 1 then strs[0]
else strs[0] + sep + Join(sep, strs[1..])
}
function method Concat(strs: seq<string>) : string {
Join("", strs)
}
}
module Utils.Lib.Set {
function method Map<T, T'>(ts: set<T>, f: T ~> T'): set<T'>
reads set t <- ts, o <- f.reads(t) :: o
requires forall t <- ts :: f.requires(t)
{
set t <- ts :: f(t)
}
function method OfSeq<T(==)>(sq: seq<T>): set<T> {
set x <- sq
}
function method OfSlice<T(==)>(arr: array<T>, lo: int, hi: int): set<T>
requires 0 <= lo <= hi <= arr.Length
reads arr
{
OfSeq(arr[lo..hi])
}
function method OfArray<T(==)>(arr: array<T>): set<T>
reads arr
{
OfSlice(arr, 0, arr.Length)
}
}
module Utils.Lib.Multiset {
function method OfSlice<T(==)>(arr: array<T>, lo: int, hi: int): multiset<T>
requires 0 <= lo <= hi <= arr.Length
reads arr
{
multiset(arr[lo..hi])
}
function method OfArray<T(==)>(arr: array<T>): multiset<T>
reads arr
{
OfSlice(arr, 0, arr.Length)
}
}
module Utils.Lib.Array {
import Set
import Multiset
method OfSet<T>(st: set<T>) returns (arr: array<T>)
ensures fresh(arr)
ensures Set.OfArray(arr) == st
ensures Multiset.OfArray(arr) == multiset(st)
{
if |st| == 0 {
arr := new T[|st|];
} else {
var s := st;
var t0 :| t0 in s;
arr := new T[|st|](_ => t0);
for i := 0 to |s|
invariant |s| == |st| - i
invariant Set.OfSlice(arr, 0, i) + s == st
invariant Multiset.OfSlice(arr, 0, i) + multiset(s) == multiset(st)
{
var t :| t in s;
arr[i] := t;
s := s - {t};
assert Set.OfSlice(arr, 0, i + 1) == Set.OfSlice(arr, 0, i) + {t};
}
}
}
}
module Utils.Lib.Int.Comparison {
import C = Sort.Comparison
function method Cmp(i0: int, i1: int): C.Cmp {
if i0 < i1 then C.Lt
else if i0 > i1 then C.Gt
else C.Eq
}
const Comparison := C.Comparison(Cmp);
lemma {:induction ints} Total(ints: set<int>)
ensures Comparison.Total?(ints)
{
reveal Comparison.Total?();
}
}
module Utils.Lib.Char.Comparison {
import C = Sort.Comparison
function method Cmp(i0: char, i1: char): C.Cmp {
if i0 < i1 then C.Lt
else if i0 > i1 then C.Gt
else C.Eq
}
const Comparison := C.Comparison(Cmp);
lemma {:induction chars} Total(chars: set<char>)
ensures Comparison.Total?(chars)
{
reveal Comparison.Total?();
}
}
module Utils.Lib.Seq.Comparison {
import Set
import C = Sort.Comparison
function Flatten<T>(tss: set<seq<T>>): set<T> {
set ts <- tss, t <- ts :: t
}
datatype SeqComparison<!T> = SeqComparison(cmp: C.Comparison<T>) {
function method Cmp(s0: seq<T>, s1: seq<T>): C.Cmp {
if s0 == [] && s1 == [] then C.Eq
else if s0 == [] then C.Lt
else if s1 == [] then C.Gt
else
match cmp.Compare(s0[0], s1[0])
case Eq => Cmp(s0[1..], s1[1..])
case other => other
}
const Comparison := C.Comparison(Cmp);
lemma {:induction false} Complete1(s0: seq<T>, s1: seq<T>)
requires cmp.Total?(Set.OfSeq(s0) + Set.OfSeq(s1))
ensures Comparison.Complete??(s0, s1)
{
reveal cmp.Total?();
if s0 != [] && s1 != [] {
assert s0[0] in s0 && s1[0] in s1;
assert s0[0] in Set.OfSeq(s0) && s1[0] in Set.OfSeq(s1);
Complete1(s0[1..], s1[1..]);
}
}
lemma {:induction false} Antisymmetric1(s0: seq<T>, s1: seq<T>)
requires cmp.Total?(Set.OfSeq(s0) + Set.OfSeq(s1))
ensures Comparison.Antisymmetric??(s0, s1)
{
reveal cmp.Total?();
if s0 != [] && s1 != [] {
assert s0[0] in s0 && s1[0] in s1;
assert s0[0] in Set.OfSeq(s0) && s1[0] in Set.OfSeq(s1);
assert s0 == [s0[0]] + s0[1..] && s1 == [s1[0]] + s1[1..];
Antisymmetric1(s0[1..], s1[1..]);
}
}
lemma {:induction false} Transitive1(s0: seq<T>, s1: seq<T>, s2: seq<T>)
requires cmp.Total?(Set.OfSeq(s0) + Set.OfSeq(s1) + Set.OfSeq(s2))
ensures Comparison.Transitive??(s0, s1, s2)
{
reveal cmp.Total?();
if s0 != [] && s1 != [] && s1 != [] && Comparison.Compare(s0, s1).Le? && Comparison.Compare(s1, s2).Le? {
assert s0[0] in s0 && s1[0] in s1 && s2[0] in s2;
assert s0[0] in Set.OfSeq(s0) && s1[0] in Set.OfSeq(s1) && s2[0] in Set.OfSeq(s2);
assert s0 == [s0[0]] + s0[1..] && s1 == [s1[0]] + s1[1..] && s2 == [s2[0]] + s2[1..];
assert cmp.Compare(s0[0], s1[0]).Le?;
assert cmp.Compare(s1[0], s2[0]).Le?;
assert cmp.Transitive??(s0[0], s1[0], s2[0]);
assert cmp.Compare(s0[0], s2[0]).Le?;
Transitive1(s0[1..], s1[1..], s2[1..]);
}
}
lemma {:induction false} Total(tss: set<seq<T>>)
requires cmp.Total?(Flatten(tss))
ensures Comparison.Total?(tss)
{
reveal cmp.Total?();
reveal Comparison.Total?();
forall s0, s1 | s0 in tss && s1 in tss ensures Comparison.Complete??(s0, s1) {
Complete1(s0, s1);
}
forall s0, s1 | s0 in tss && s1 in tss ensures Comparison.Antisymmetric??(s0, s1) {
Antisymmetric1(s0, s1);
}
forall s0, s1, s2 | s0 in tss && s1 in tss && s2 in tss ensures Comparison.Transitive??(s0, s1, s2) {
Transitive1(s0, s1, s2);
}
}
}
}
module Utils.Lib.Str.Comparison {
import C = Sort.Comparison
import SeqCmp = Seq.Comparison
import CharCmp = Char.Comparison
const StrComparison := SeqCmp.SeqComparison(CharCmp.Comparison);
const Comparison := StrComparison.Comparison;
lemma {:induction false} Total(strs: set<string>)
ensures Comparison.Total?(strs)
{
CharCmp.Total(SeqCmp.Flatten(strs));
StrComparison.Total(strs);
}
}
module Utils.Lib.Math {
function method {:opaque} Max(x: int, y: int) : (m: int)
ensures x <= m
ensures y <= m
ensures x == m || y == m
{
if (x <= y) then y else x
}
function method {:opaque} IntPow(x: int, n: nat) : int {
if n == 0 then 1
else x * IntPow(x, n - 1)
}
}
module Utils.Lib.Sort.Comparison {
import opened Datatypes
datatype Cmp = Lt | Eq | Gt {
function method Flip(): Cmp {
match this
case Lt => Gt
case Eq => Eq
case Gt => Lt
}
const Le? := this != Gt
const Ge? := this != Lt
static function method ComputeTransitivity(c0: Cmp, c1: Cmp): Option<Cmp> {
match (c0, c1)
case (Lt, Lt) => Some(Lt)
case (Lt, Eq) => Some(Lt)
case (Lt, Gt) => None
case (Eq, Lt) => Some(Lt)
case (Eq, Eq) => Some(Eq)
case (Eq, Gt) => Some(Gt)
case (Gt, Lt) => None
case (Gt, Eq) => Some(Gt)
case (Gt, Gt) => Some(Gt)
}
}
datatype Comparison<!T> = Comparison(cmp: (T, T) -> Cmp) {
function method Compare(t0: T, t1: T): Cmp {
cmp(t0, t1)
}
predicate Complete??(t0: T, t1: T) {
cmp(t0, t1) == cmp(t1, t0).Flip()
}
predicate Antisymmetric??(t0: T, t1: T) {
cmp(t0, t1) == Eq ==> t0 == t1
}
predicate Transitive??(t0: T, t1: T, t2: T) {
cmp(t0, t1).Le? && cmp(t1, t2).Le? ==> cmp(t0, t2).Le?
}
predicate Reflexive??(t0: T) {
cmp(t0, t0) == Eq
}
lemma AlwaysReflexive(t0: T)
requires Complete??(t0, t0)
ensures Reflexive??(t0)
{}
lemma PreciselyTransitive'(t0: T, t1: T, t2: T)
requires Complete??(t0, t1) && Complete??(t0, t2) && Complete??(t1, t2)
requires Antisymmetric??(t0, t1) && Antisymmetric??(t0, t2) && Antisymmetric??(t1, t2)
requires Transitive??(t0, t1, t2) && Transitive??(t1, t2, t0)
requires cmp(t0, t1).Le? && cmp(t1, t2).Le?
ensures Cmp.ComputeTransitivity(cmp(t0, t1), cmp(t1, t2)) == Some(cmp(t0, t2))
{}
lemma PreciselyTransitive(t0: T, t1: T, t2: T)
requires Reflexive??(t0) && Reflexive??(t1) && Reflexive??(t2)
requires Complete??(t0, t1) && Complete??(t0, t2) && Complete??(t1, t2)
requires Antisymmetric??(t0, t1) && Antisymmetric??(t0, t2) && Antisymmetric??(t1, t2)
requires Transitive??(t0, t1, t2) && Transitive??(t1, t2, t0)
requires Transitive??(t2, t1, t0) && Transitive??(t1, t0, t2)
ensures match Cmp.ComputeTransitivity(cmp(t0, t1), cmp(t1, t2))
case Some(c12) => c12 == cmp(t0, t2)
case None => true
{
match (cmp(t0, t1), cmp(t1, t2))
case (Lt, Lt) | (Lt, Eq) | (Eq, Lt) | (Eq, Eq) =>
PreciselyTransitive'(t0, t1, t2);
case (Eq, Gt) | (Gt, Eq) | (Gt, Gt) =>
PreciselyTransitive'(t2, t1, t0);
case (Lt, Gt) | (Gt, Lt) =>
}
predicate Complete?(ts: set<T>) {
forall t0, t1 | t0 in ts && t1 in ts :: Complete??(t0, t1)
}
predicate Antisymmetric?(ts: set<T>) {
forall t0, t1 | t0 in ts && t1 in ts :: Antisymmetric??(t0, t1)
}
predicate Transitive?(ts: set<T>) {
forall t0, t1, t2 | t0 in ts && t1 in ts && t2 in ts :: Transitive??(t0, t1, t2)
}
predicate {:opaque} Valid?(ts: set<T>) {
Complete?(ts) && /* Antisymmetric?(ts) && */ Transitive?(ts)
}
predicate {:opaque} Total?(ts: set<T>) {
Complete?(ts) && Antisymmetric?(ts) && Transitive?(ts)
}
lemma TotalValid(ts: set<T>)
ensures Total?(ts) ==> Valid?(ts)
{
reveal Total?();
reveal Valid?();
}
predicate Sorted(sq: seq<T>) {
forall i, j | 0 <= i < j < |sq| :: cmp(sq[i], sq[j]).Le?
}
lemma SortedConcat(sq0: seq<T>, sq1: seq<T>)
requires Sorted(sq0)
requires Sorted(sq1)
requires forall i, j | 0 <= i < |sq0| && 0 <= j < |sq1| :: cmp(sq0[i], sq1[j]).Le?
ensures Sorted(sq0 + sq1)
{}
// lemma SortedUnique(sq0: seq<T>, sq1: seq<T>)
// requires Sorted(sq0)
// requires Sorted(sq1)
// requires multiset(sq0) == multiset(sq1)
// requires Antisymmetric?(set x <- sq0)
// ensures sq0 == sq1
// {
// calc { set x <- sq0; set x <- multiset(sq0); set x <- multiset(sq1); set x <- sq1; }
// }
predicate {:opaque} Striped(sq: seq<T>, pivot: T, lo: int, left: int, mid: int, right: int, hi: int)
requires 0 <= lo <= left <= mid <= right <= hi <= |sq|
{
&& (forall i | lo <= i < left :: cmp(sq[i], pivot).Lt?)
&& (forall i | left <= i < mid :: cmp(sq[i], pivot).Eq?)
&& (forall i | right <= i < hi :: cmp(sq[i], pivot).Gt?)
}
}
}
module Utils.Lib.Sort.DerivedComparison {
import C = Comparison
import Set
datatype DerivedComparison<!T, !T'> = DerivedComparison(cmp: C.Comparison<T'>, fn: T -> T') {
const Comparison := C.Comparison((t0, t1) => cmp.Compare(fn(t0), fn(t1)));
lemma Valid(ts: set<T>)
requires cmp.Valid?(Set.Map(ts, fn))
ensures Comparison.Valid?(ts)
{
reveal cmp.Valid?();
reveal Comparison.Valid?();
forall s0, s1 | s0 in ts && s1 in ts ensures Comparison.Complete??(s0, s1) {
assert cmp.Complete??(fn(s0), fn(s1));
}
forall s0, s1, s2 | s0 in ts && s1 in ts && s2 in ts ensures Comparison.Transitive??(s0, s1, s2) {
assert cmp.Transitive??(fn(s0), fn(s1), fn(s2));
}
}
lemma Total(ts: set<T>)
requires cmp.Total?(Set.Map(ts, fn))
requires forall i, j | fn(i) == fn(j) :: i == j
ensures Comparison.Total?(ts)
{
reveal cmp.Valid?();
reveal cmp.Total?();
reveal Comparison.Valid?();
reveal Comparison.Total?();
Valid(ts);
forall s0, s1 | s0 in ts && s1 in ts ensures Comparison.Antisymmetric??(s0, s1) {
calc ==> {
Comparison.cmp(s0, s1).Eq?;
cmp.cmp(fn(s0), fn(s1)).Eq?;
{ assert cmp.Antisymmetric??(fn(s0), fn(s1)); }
fn(s0) == fn(s1);
s0 == s1;
}
}
}
}
}