We want to make sure string users do not depend on the string
implementation. This is the first step.
We need this refactoring *now* to make sure it will not be
super painful to address issue #1175
This commit fixes issue #1631. However, it is not a perfect solution.
This commit improves the predicate that checks whether a definition is
noncomputable or not. This predicate was implemented before we had
a code generator.
We should refactor the code and use the code generator to check
whether a definition is noncomputable or not. Otherwise, we will
keep finding mismatches between the predicate at noncomputable.cpp
and what the code generator implements.
See Section "Other goodies" at
https://github.com/leanprover/lean/wiki/Refactoring-structures
This commit also improves the support for projections in the
unifier/matcher.
Now, we consider the extra case-split for projections.
Given a projection `proj`, and the constraint `proj s =?= proj t`, we need to try first `s =?= t` and if it fails, then try to reduce.
This is needed in the standard library because we now have constraints such as:
```
@has_le.le ?A ?s ?a ?b =?= @has_le.le nat nat.has_add x y
```
If we reduce the right hand side, we get the unsolvable constraint
```
@has_le.le ?A ?s ?a ?b =?= nat.le x y
```
Before this change, the constraint was `@le ?A ?s ?a ?b =?= @le nat nat.has_add x y`, and we already perform a case-split in this case.
Moreover, projections were eagerly reduced whenever possible.
The extra case-split generates a performance problem in several tests. For example `fib 8 = 34` was timing out.
I worked around this issue by performing the case-split only when the constraint contains meta-variables.
There are also minor issues. Example. `<` is notation for `has_lt.lt`, but `>` is for `gt`.
This is a hard coded extra case. It is not an instance of has_coe.
Even if we change has_coe to accomodate this case, it will not be a
satisfactory solution because this coercion depends on the element and
not the type, and the element usually contains metavariables.
We should eventually write a tactic for synthesizing coercions.
