Before this commit, the elaborator was solving constraints of the form
ctx |- (?m x) == (f x)
as
?m <- (fun x : A, f x) where A is the domain of f.
In our kernel, the terms f and (fun x, f x) are not definitionally equal.
So, the solution above is not the only one. Another possible solution is
?m <- f
Depending of the circumstances we want ?m <- (fun x : A, f x) OR ?m <- f.
For example, when Lean is elaborating the eta-theorem in kernel.lean, the first solution should be used:
?m <- (fun x : A, f x)
When we are elaborating the axiom_of_choice theorem, we need to use the second one:
?m <- f
Of course, we can always provide the parameters explicitly and bypass the elaborator.
However, this goes against the idea that the elaborator can do mechanical steps for us.
This commit addresses this issue by creating a case-split
?m <- (fun x : A, f x)
OR
?m <- f
Another solution is to implement eta-expanded normal forms in the Kernel.
With this change, we were able to cleanup the following "hacks" in kernel.lean:
@eps_ax A (nonempty_ex_intro H) P w Hw
@axiom_of_choice A B P H
where we had to explicitly provided the implicit arguments
This commit also improves the imitation step for Pi-terms that are actually arrows.
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit also adds several new theorems that are useful for implementing the simplifier.
TODO: perhaps we should remove the declarations at basic_thms.h?
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
It was not a good idea to use heterogeneous equality as the default equality in Lean.
It creates the following problems.
- Heterogeneous equality does not propagate constraints in the elaborator.
For example, suppose that l has type (List Int), then the expression
l = nil
will not propagate the type (List Int) to nil.
- It is easy to write false. For example, suppose x has type Real, and the user
writes x = 0. This is equivalent to false, since 0 has type Nat. The elaborator cannot introduce
the coercion since x = 0 is a type correct expression.
Homogeneous equality does not suffer from the problems above.
We keep heterogeneous equality because it is useful for generating proof terms.
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
