This PR changes how Lean proves the equational theorems for structural
recursion. The core idea is to let-bind the `f` argument to `brecOn` and
rewriting `.brecOn` with an unfolding theorem. This means no extra case
split for the `.rec` in `.brecOn` is needed, and `simp` doesn't change
the `f` argument which can break the definitional equality with the
defined function. With this, we can prove the unfolding theorem first,
and derive the equational theorems from that, like for all other ways of
defining recursive functions.
Backs out the changes from #10415, the old strategy works well with the
new goals.
Fixes#5667Fixes#10431Fixes#10195Fixes#2962
Many of our tests in `tests/lean/run/` produce output from `#eval` (or
`#check`) statements, that is then ignored.
This PR tries to capture all the useful output using `#guard_msgs`. I've
only done a cursory check that the output is still sane --- there is a
chance that some "unchecked" tests have already accumulated regressions
and this just cements them!
In the other direction, I did identify two rotten tests:
* a minor one in `setStructInstNotation.lean`, where a comment says `Set
Nat`, but `#check` actually prints `?_`. Weird?
* `CompilerProbe.lean` is generating empty output, apparently indicating
that something is broken, but I don't know the signficance of this file.
In any case, I'll ask about these elsewhere.
(This started by noticing that a recent `grind` test file had an
untested `trace_state`, and then got carried away.)
Given a definition `foo`, they were previously called `foo._unfold`
until 4.7.0. We tried to rename them to `foo.def`, but it created too
many issues in the Mathlib repo. We decided to rename it again to
`foo.eq_def`. The new name is also consistent with the `eq_<idx>`
theorems generated for different "cases". That is, `foo.eq_def` is the
equality theorem for the whole definition, and `foo.eq_<idx>` is the
equality theorem for case `<idx>`.
cc @semorrison
Reason: avoid unnecessary case-analysis explosion
It is also unnecessary since we only refine the `below` and `F`
arguments over `match`-expressions.
This fixes the last case at issue #998
```
attribute [simp] BinaryHeap.heapifyDown
```
closes#998
