this improves support for structural recursion over inductive
*predicates* when there are reflexive arguments.
Consider
```lean
inductive F: Prop where
| base
| step (fn: Nat → F)
-- set_option trace.Meta.IndPredBelow.search true
set_option pp.proofs true
def F.asdf1 : (f : F) → True
| base => trivial
| step f => F.asdf1 (f 0)
termination_by structural f => f`
```
Previously the search for the right induction hypothesis would fail with
```
could not solve using backwards chaining x✝¹ : F
x✝ : x✝¹.below
f : Nat → F
a✝¹ : ∀ (a : Nat), (f a).below
a✝ : Nat → True
⊢ True
```
The backchaining process will try to use `a✝ : Nat → True`, but then has
no idea what to use for `Nat`.
There are three steps here to fix this.
1. We let-bind the function's type before the whole process. Now the
goal is
```
funType : F → Prop := fun x => True
x✝ : x✝¹.below
f : Nat → F
a✝¹ : ∀ (a : Nat), (f a).below
a✝ : ∀ (a : Nat), funType (f a)
⊢ funType (f 0)
```
2. Instead of using the general purpose backchaining proof search, which
is more
powerful than we need here (we need on recursive search and no
backtracking),
we have a custom search that looks for local assumptions that
provide evidence of `funType`, and extracts the arguments from that
“type” application to construct the recursive call.
Above, it will thus unify `f a =?= f 0`.
3. In order to make progress here, we also turn on use
`withoutProofIrrelevance`,
because else `isDefEq` is happy to say “they are equal” without actually
looking
at the terms and thus assigning `?a := 0`.
This idea of let-binding the function's motive may also be useful for
the other recursion compilers, as it may simplify the FunInd
construction. This is to be investigated.
fixes#4751
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.)
@Kha The structural recursion is working :)
It is so much more powerful than the one in Lean3.
I added examples with `let rec`, nested and multiple
`match`-expressions.
I will keep testing and adding missing features tomorrow, and
will hopefully start porting stdlib to new frontend before the end of
the week.
