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datatype List<A> = Nil | Cons(head: A,tail: List<A>);
datatype Tree<A> = Branch(val: A,trees: List<Tree<A>>);
function Set(xs : List) : set
ensures forall x :: x in Set(xs) ==> x < xs;
{
match xs
case Nil => {}
case Cons(x,xs) => {x} + Set(xs)
}
function TSet(t0 : Tree) : set
ensures forall t :: t in TSet(t0) ==> t < t0;
{
match t0 case Branch(x,ts) => {x} + set t, y | t in Set(ts) && y in TSet(t) :: y
}
// reads {}
function ReadNothingMap(f : A -> B, xs: List<A>): List<B>
requires forall x :: x in Set(xs) ==> f.requires(x);
requires forall x :: f.reads(x) == {};
decreases xs;
{
match xs
case Nil => Nil
case Cons(x,xs) => Cons(f(x),ReadNothingMap(f,xs))
}
function TReadNothingMap(f: A -> B, t0: Tree<A>): Tree<B>
requires forall x {:heapQuantifier} :: f.requires(x);
requires forall x {:heapQuantifier} :: f.reads(x) == {};
decreases t0;
{
var Branch(x,ts) := t0;
Branch(f(x), ReadNothingMap(t requires t in Set(ts) => TReadNothingMap(f,t), ts))
}
function TReadNothingMap2(f: A -> B, t0: Tree<A>): Tree<B>
requires forall x :: f.requires(x);
requires forall x :: f.reads(x) == {};
decreases t0;
{
var Branch(x,ts) := t0;
Branch(f(x), ReadNothingMap(t requires t in Set(ts) -> TReadNothingMap2(f,t), ts))
}
function TReadNothingMap3(f: A -> B, t0: Tree<A>): Tree<B>
requires forall x :: f.requires(x);
requires forall x :: f.reads(x) == {};
decreases t0;
{
var Branch(x,ts) := t0;
Branch(f(x), ReadNothingMap(
t requires t in Set(ts)
requires (forall x :: x in TSet(t) ==> f.requires(x))
=> TReadNothingMap(f,t), ts))
}
method TestReadNothingStar() {
assert TReadNothingMap(x => x + 1, Branch(1,Nil)).Branch?;
assert TReadNothingMap(x => x + 1, Branch(0,Nil)) == Branch(1,Nil);
assert TReadNothingMap(x => x + 1, Branch(1,Cons(Branch(0,Nil),Nil)))
== Branch(2,Cons(Branch(1,Nil),Nil));
assert TReadNothingMap2(x -> x + 1, Branch(1,Nil)).Branch?;
assert TReadNothingMap2(x -> x + 1, Branch(0,Nil)) == Branch(1,Nil);
assert TReadNothingMap2(x -> x + 1, Branch(1,Cons(Branch(0,Nil),Nil)))
== Branch(2,Cons(Branch(1,Nil),Nil));
}
/// reads *
function ReadStarMap(f : A -> B, xs: List<A>): List<B>
requires forall x :: x in Set(xs) ==> f.requires(x);
reads *;
decreases xs;
{
match xs
case Nil => Nil
case Cons(x,xs) => Cons(f(x),ReadStarMap(f,xs))
}
function TReadStarMap(f: A -> B, t0: Tree<A>): Tree<B>
requires forall x {:heapQuantifier} :: f.requires(x);
reads *;
decreases t0;
{
var Branch(x,ts) := t0;
Branch(f(x), ReadStarMap(t reads * requires t in Set(ts) => TReadStarMap(f,t), ts))
}
function TReadStarMap2(f: A -> B, t0: Tree<A>): Tree<B>
requires forall x :: f.requires(x);
reads *;
decreases t0;
{
var Branch(x,ts) := t0;
Branch(f(x), ReadStarMap(t reads * requires t in Set(ts) -> TReadStarMap2(f,t), ts))
}
lemma LitTReadStarMap2(f : A -> B, x : A, ts: List<Tree<A>>)
requires forall u :: f.requires(u);
ensures TReadStarMap2(f, Branch(x,ts)) ==
Branch(f(x), ReadStarMap(t reads * requires t in Set(ts) -> TReadStarMap2(f,t), ts));
{
assert TReadStarMap2(f, Branch(x,ts)).val == f(x);
if (ts.Nil?) {
assert TReadStarMap2(f, Branch(x,ts)).trees ==
ReadStarMap(t reads * requires t in Set(ts) -> TReadStarMap2(f,t),ts);
} else {
assert TReadStarMap2(f, Branch(x,ts)).trees ==
ReadStarMap(t reads * requires t in Set(ts) -> TReadStarMap2(f,t),ts);
}
}
method TestReadStar() {
assert TReadStarMap(x => x + 1, Branch(1,Nil)).Branch?;
assert TReadStarMap(x => x + 1, Branch(0,Nil)) == Branch(1,Nil);
assert ReadStarMap(t reads * requires t in Set(Cons(Branch(0,Nil),Nil)) -> TReadStarMap(x => x + 1,t)
, Cons(Branch(0,Nil),Nil))
== Cons(Branch(1,Nil),Nil);
assert TReadStarMap(x => x + 1, Branch(1,Cons(Branch(0,Nil),Nil))).Branch?;
assert TReadStarMap(x => x + 1, Branch(1,Cons(Branch(0,Nil),Nil))).val == 2;
assert TReadStarMap(x => x + 1, Branch(1,Cons(Branch(0,Nil),Nil))).trees == Cons(Branch(1,Nil),Nil);
assert TReadStarMap(x => x + 1, Branch(1,Cons(Branch(0,Nil),Nil)))
== Branch(2,Cons(Branch(1,Nil),Nil));
assert TReadStarMap2(x -> x + 1, Branch(1,Nil)).Branch?;
assert TReadStarMap2(x -> x + 1, Branch(0,Nil)) == Branch(1,Nil);
assert TReadStarMap2(x -> x + 1, Branch(1,Cons(Branch(0,Nil),Nil)))
== Branch(2,Cons(Branch(1,Nil),Nil));
}
/// reads exact
function Map(f : A -> B, xs: List<A>): List<B>
requires forall x :: x in Set(xs) ==> f.requires(x);
reads set x, y | x in Set(xs) && y in f.reads(x) :: y;
decreases xs;
{
match xs
case Nil => Nil
case Cons(x,xs) => Cons(f(x),Map(f,xs))
}
function TMap(f : A -> B, t0: Tree<A>): Tree<B>
requires forall t :: t in TSet(t0) ==> f.requires(t);
reads set x, y | x in TSet(t0) && y in f.reads(x) :: y;
decreases t0;
{
var Branch(x,ts) := t0;
Branch(
f(x),
Map( t requires t in Set(ts)
reads set x, y | x in TSet(t) && y in f.reads(x) :: y
-> TMap(f,t)
, ts)
)
}
lemma LitTMap(f : A -> B,x : A, ts: List<Tree<A>>)
requires f.requires(x);
requires forall t, u :: t in Set(ts) && u in TSet(t) ==> f.requires(u);
ensures TMap(f, Branch(x,ts)) ==
Branch(f(x),
Map( t requires t in Set(ts)
reads set x, y | x in TSet(t) && y in f.reads(x) :: y
-> TMap(f,t),ts));
{
assert TMap(f, Branch(x,ts)).val == f(x);
assert TMap(f, Branch(x,ts)).trees ==
Map(t requires t in Set(ts)
reads set x, y | x in TSet(t) && y in f.reads(x) :: y
-> TMap(f,t),ts);
}
method Test() {
assert TMap(x -> x + 1, Branch(1,Nil)).Branch?;
assert TMap(x -> x + 1, Branch(0,Nil)) == Branch(1,Nil);
calc {
TMap(x -> x + 1, Branch(1,Cons(Branch(0,Nil),Nil)));
== { LitTMap(x -> x + 1,1, Cons(Branch(0,Nil),Nil)); }
Branch((x -> x + 1)(1),Map(t -> TMap(x -> x + 1,t),Cons(Branch(0,Nil),Nil)));
==
Branch(2,Map(t -> TMap(x -> x + 1,t),Cons(Branch(0,Nil),Nil)));
==
Branch(2,Map(t -> TMap(x -> x + 1,t),Cons(Branch(0,Nil),Nil)));
==
Branch(2,Cons(TMap(x -> x + 1,Branch(0,Nil)),Nil));
==
Branch(2,Cons(Branch((x -> x + 1)(0),Nil),Nil));
==
Branch(2,Cons(Branch(1,Nil),Nil));
}
assert TMap(x -> x + 1, Branch(1,Cons(Branch(0,Nil),Nil)))
== Branch(2,Cons(Branch(1,Nil),Nil));
}
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