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// RUN: %dafny /compile:0 /rprint:"%t.rprint" /autoTriggers:1 "%s" > "%t"
// RUN: %diff "%s.expect" "%t"
// Monadic Laws
// Niki Vazou and Rustan Leino
// 28 March 2016
datatype List<T> = Nil | Cons(head: T, tail: List)
predicate Total<T,U>(p: T -> U)
reads p.reads
{
forall x :: p.reads(x) == {} && p.requires(x)
}
function append(xs: List, ys: List): List
{
match xs
case Nil => ys
case Cons(x, xs') => Cons(x, append(xs', ys))
}
lemma AppendNil(xs: List)
ensures append(xs, Nil) == xs
{
}
lemma AppendAssoc(xs: List, ys: List, zs: List)
ensures append(append(xs, ys), zs) == append(xs, append(ys, zs));
{
}
function Return<T>(a: T): List
{
Cons(a, Nil)
}
function Bind<T,U>(xs: List<T>, f: T -> List<U>): List<U>
requires Total(f)
{
match xs
case Nil => Nil
case Cons(x, xs') => append(f(x), Bind(xs', f))
}
lemma LeftIdentity<T>(a: T, f: T -> List)
requires Total(f)
ensures Bind(Return(a), f) == f(a)
{
AppendNil(f(a));
}
lemma RightIdentity<T>(m: List)
ensures Bind(m, Return) == m
{
match m
case Nil =>
assert Bind<T,T>(Nil, Return) == Nil;
case Cons(x, m') =>
calc {
Bind(Cons(x, m'), Return);
append(Return(x), Bind(m', Return));
Cons(x, Bind(m', Return));
}
}
lemma Associativity<T>(m: List, f: T -> List, g: T -> List)
requires Total(f) && Total(g)
ensures Bind(Bind(m, f), g) == Bind(m, x => Bind(f(x), g))
{
match m
case Nil =>
assert Bind(m, x => Bind(f(x), g)) == Nil;
case Cons(x, xs) =>
match f(x)
case Nil =>
calc {
Bind(xs, y => Bind(f(y), g));
Bind(Cons(x, xs), y => Bind(f(y), g));
}
case Cons(y, ys) =>
calc {
append(g(y), Bind(append(ys, Bind(xs, f)), g));
{ BindOverAppend(ys, Bind(xs, f), g); }
append(g(y), append(Bind(ys, g), Bind(Bind(xs, f), g)));
{ AppendAssoc(g(y), Bind(ys, g), Bind(Bind(xs, f), g)); }
append(append(g(y), Bind(ys, g)), Bind(Bind(xs, f), g));
Bind(Cons(x, xs), z => Bind(f(z), g));
}
}
lemma BindOverAppend<T>(xs: List, ys: List, g: T -> List)
requires Total(g)
ensures Bind(append(xs, ys), g) == append(Bind(xs, g), Bind(ys, g))
{
match xs
case Nil =>
case Cons(x, xs') =>
AppendAssoc(g(x), Bind(xs', g), Bind(ys, g));
}
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