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// The following are Dafny versions from Section 8 of
// "Mechanizing Coinduction and Corecursion in Higher-order Logic"
// by Lawrence C. Paulson, 1996.
codatatype LList<T> = Nil | Cons(head: T, tail: LList);
// Simulate function as arguments
datatype FunctionHandle<T> = FH(T);
function Apply<T>(f: FunctionHandle<T>, argument: T): T
// Function composition
function After(f: FunctionHandle, g: FunctionHandle): FunctionHandle
ghost method Definition_After<T>(f: FunctionHandle<T>, g: FunctionHandle<T>, arg: T)
ensures Apply(After(f, g), arg) == Apply(f, Apply(g, arg));
function Lmap(f: FunctionHandle, a: LList): LList
{
match a
case Nil => Nil
case Cons(x, xs) => Cons(Apply(f, x), Lmap(f, xs))
}
function Lappend(a: LList, b: LList): LList
{
match a
case Nil => b
case Cons(x, xs) => Cons(x, Lappend(xs, b))
}
// ---------- Section 8.3 ----------
comethod Example6(f: FunctionHandle, g: FunctionHandle, M: LList)
ensures Lmap(After(f, g), M) == Lmap(f, Lmap(g, M));
{
match M {
case Nil =>
case Cons(x, xs) =>
calc {
Lmap(After(f, g), M);
Lmap(After(f, g), Cons(x, xs));
// def. Lmap
Cons(Apply(After(f, g), x), Lmap(After(f, g), xs));
{ Definition_After(f, g, x); }
Cons(Apply(f, Apply(g, x)), Lmap(After(f, g), xs));
==#[_k] // use co-induction hypothesis
Cons(Apply(f, Apply(g, x)), Lmap(f, Lmap(g, xs)));
// def. Lmap
Lmap(f, Cons(Apply(g, x), Lmap(g, xs)));
// def. Lmap
Lmap(f, Lmap(g, Cons(x, xs)));
Lmap(f, Lmap(g, M));
}
}
}
// ---------- Section 8.4 ----------
// Iterates(f, M) == [M, f M, f^2 M, ..., f^n M, ...]
function Iterates<A>(f: FunctionHandle<LList<A>>, M: LList<A>): LList<LList<A>>
{
Cons(M, Iterates(f, Apply(f, M)))
}
comethod Eq24<A>(f: FunctionHandle<LList<A>>, M: LList<A>)
ensures Lmap(f, Iterates(f, M)) == Iterates(f, Apply(f, M));
{
calc {
Lmap(f, Iterates(f, M));
Lmap(f, Cons(M, Iterates(f, Apply(f, M))));
Cons(Apply(f, M), Lmap(f, Iterates(f, Apply(f, M))));
==#[_k]
calc {
Lmap(f, Iterates(f, Apply(f, M)));
==#[_k-1] { Eq24(f, Apply(f, M)); } // co-induction hypothesis
Iterates(f, Apply(f, Apply(f, M)));
}
Cons(Apply(f, M), Iterates(f, Apply(f, Apply(f, M))));
// def. Iterates
Iterates(f, Apply(f, M));
}
}
ghost method CorollaryEq24<A>(f: FunctionHandle<LList<A>>, M: LList<A>)
ensures Iterates(f, M) == Cons(M, Lmap(f, Iterates(f, M)));
{
Eq24(f, M);
}
// Now prove that equation in CorollaryEq24 uniques characterizes Iterates.
// Paulson says "The bisimulation for this proof is unusually complex".
// Let h be any function
function h<A>(f: FunctionHandle<LList<A>>, M: LList<A>): LList<LList<A>>
// ... that satisfies the property in CorollaryEq24:
ghost method Definition_h<A>(f: FunctionHandle<LList<A>>, M: LList<A>)
ensures h(f, M) == Cons(M, Lmap(f, h(f, M)));
// Functions to support the proof:
function Iter<A>(n: nat, f: FunctionHandle<A>, arg: A): A
{
if n == 0 then arg else Apply(f, Iter(n-1, f, arg))
}
function LmapIter(n: nat, f: FunctionHandle, arg: LList): LList
{
if n == 0 then arg else Lmap(f, LmapIter(n-1, f, arg))
}
ghost method Lemma25<A>(n: nat, f: FunctionHandle<A>, b: A, M: LList<A>)
ensures LmapIter(n, f, Cons(b, M)) == Cons(Iter(n, f, b), LmapIter(n, f, M));
{
// proof is by (automatic) induction
}
ghost method Lemma26<A>(n: nat, f: FunctionHandle, x: LList) // (26) in the paper, but with f := LMap f
ensures LmapIter(n, f, Lmap(f, x)) == LmapIter(n+1, f, x);
{
// proof is by (automatic) induction
}
comethod BisimulationLemma<A>(n: nat, f: FunctionHandle<LList<A>>, u: LList<A>)
ensures LmapIter(n, f, h(f, u)) == LmapIter(n, f, Iterates(f, u));
{
calc {
LmapIter(n, f, h(f, u));
{ Definition_h(f, u); }
LmapIter(n, f, Cons(u, Lmap(f, h(f, u))));
{ Lemma25(n, f, u, Lmap(f, h(f, u))); }
Cons(Iter(n, f, u), LmapIter(n, f, Lmap(f, h(f, u))));
{ Lemma26(n, f, h(f, u)); }
Cons(Iter(n, f, u), LmapIter(n+1, f, h(f, u)));
==#[_k]
calc {
LmapIter(n+1, f, h(f, u));
==#[_k-1]
LmapIter(n+1, f, Iterates(f, u));
}
Cons(Iter(n, f, u), LmapIter(n+1, f, Iterates(f, u)));
{ Lemma26(n, f, Iterates(f, u)); }
Cons(Iter(n, f, u), LmapIter(n, f, Lmap(f, Iterates(f, u))));
{ Lemma25(n, f, u, Lmap(f, Iterates(f, u))); }
LmapIter(n, f, Cons(u, Lmap(f, Iterates(f, u))));
{ CorollaryEq24(f, u); }
LmapIter(n, f, Iterates(f, u));
}
}
ghost method Example7<A>(f: FunctionHandle<LList<A>>)
// Given the definition of h, prove h(f, _) == Iterates(f, _):
ensures forall M :: h(f, M) == Iterates(f, M);
{
forall M {
BisimulationLemma(0, f, M);
}
}
// ---------- Section 8.5 ----------
comethod Example8<A>(f: FunctionHandle<A>, M: LList<A>, N: LList<A>)
ensures Lmap(f, Lappend(M, N)) == Lappend(Lmap(f, M), Lmap(f, N));
{
// A proof doesn't get any simpler than this
}
comethod Associativity(a: LList, b: LList, c: LList)
ensures Lappend(Lappend(a, b), c) == Lappend(a, Lappend(b, c));
{
// Again, Dafny does this proof completely automatically
}
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