how to prove universal introduction in Dafny - dafny

Question

I am trying to find strategies to prove universally quantified assertions in Dafny. I see Dafny proves universal elimination
quite easily:
predicate P<X>(k:X)
lemma unElim<X>(x:X)
ensures (forall a:X :: P(a)) ==> P(x)
{ }
lemma elimHyp<H> ()
ensures forall k:H :: P(k)
lemma elimGoal<X> (x:X)
ensures P(x)
{ elimHyp<X>(); }
but I can not find how to prove the introduction rule:
//lemma unInto<X>(x:X)
// ensures P(x) ==> (forall a:X :: P(a))
// this definition is wrong
lemma introHyp<X> (x:X)
ensures P(x)
lemma introGoal<H> ()
ensures forall k:H :: P(k)
{ }
all ideas appreciated

Answer 1

Universal introduction is done using Dafny's forall statement.
lemma introHyp<X>(x: X)
ensures P(x)
lemma introGoal<H>()
ensures forall k: H :: P(k)
{
forall k: H
ensures P(k)
{
introHyp<H>(k);
}
}
In general, it looks like this:
forall x: X | R(x)
ensures P(x)
{
// for x of type X and satisfying R(x), prove P(x) here
// ...
}
So, inside the curly braces, you prove P(x) for one x. After the forall statement, you get to assume the universal quantifier
forall x: X :: R(x) ==> P(x)
If, like in my introGoal above, the body of the forall statement is exactly one lemma call and the postcondition of that lemma is what you what in the ensures clause of the forall statement, then you can omit the ensures clause of the forall statement and Dafny will infer it for you. Lemma introGoal then looks like this:
lemma introGoal<H>()
ensures forall k: H :: P(k)
{
forall k: H {
introHyp(k);
}
}
There's a Dafny Power User note on Automatic induction that may be helpful, or at least gives some additional examples.
PS. A natural next question would be how to do existential elimination. You do it using Dafny's "assign such that" statement. Here is an example:
type X
predicate P(x: X)
lemma ExistentialElimination() returns (y: X)
requires exists x :: P(x)
ensures P(y)
{
y :| P(y);
}
Some examples are found in this Dafny Power User note. Some advanced technical information about the :| operators are found in this paper.