Due to nested recursion, we do two passes of `getRecArgInfo`: One on
each argument in isolation, to see which inductive types are around
(e.g. `Tree` and `List`), and
then we later refine/replace this result with the data for the nested
type former (the implicit `ListTree`).
If we have nested recursion through a non-recursive data type like
`Array` or `Prod` then arguemnts of these types should survive the first
phase, so that we can still use them when looking for, say, `Array
Tree`.
This was helpfully reported by @arthur-adjedj.
This refactoring PR changes the structure of the `FunInd` module, with
the main purpose to make it easier to support mutual structural
recursion.
In particular the recursive calls are now longer recognized by their
terms (simple for well-founded recursion, `.app oldIH [arg, proof]`, but
tedious for structural recursion and even more so for mutual structural
recursion), but the type after replacing `oldIH` with `newIH`, where the
type will be simply and plainly `mkAppN motive args`).
We also no longer try to guess whether we deal with well-founded or
structural recursion but instead rely on the `EqnInfo` environment
extensions. The previous code tried to handle both variants, but they
differ too much, so having separate top-level functions is easier.
This also fuses the `foldCalls` and `collectIHs` traversals and
introduces a suitable monad for collecting the inductive hypotheses.
This now works:
```lean
inductive Tree where | node : List Tree → Tree
mutual
def Tree.size : Tree → Nat
| node ts => list_size ts
def Tree.list_size : List Tree → Nat
| [] => 0
| t::ts => t.size + list_size ts
end
```
It is still out of scope to expect to be able to use nested recursion
(e.g. through `List.map` or `List.foldl`) here.
Depends on #4718.
---------
Co-authored-by: Tobias Grosser <tobias@grosser.es>
the support for mutual structural recursion (new since #4575) is
extended so that Lean tries to infer it even without annotations.
* The error message when termination checking fails looks quite
different now. Maybe a bit better, maybe with more room for
improvements.
* If there are too many combinations (with an arbitrary cut-off) for a
given argument type, it will just give up and ask the user to use
`termination_by structural`.
* It is now legal to specify `termination_by structural` on not
necessarily all functions of a clique; this simply restricts the
combinations of arguments that Lean considers.
---------
Co-authored-by: Tobias Grosser <tobias@grosser.es>
We now get `.below` and `.brecOn` definitions for nested inductives.
No surprises in the implementation: the kernel already gives us suitable
`.rec_1` etc. recursors, and our construction follows the structure of
this recursor.
---------
Co-authored-by: Tobias Grosser <tobias@grosser.es>
if will fail otherwise, but with a worse error message, and it's helpful
in later transformation to know that the parameters are the same for the
whole group.
This adds support for mutual structural recursive functions.
For now this is opt-in: The functions must have a `termination_by
structural …` annotation (new since #4542) for this to work:
```lean
mutual
inductive A
| self : A → A
| other : B → A
| empty
inductive B
| self : B → B
| other : A → B
| empty
end
mutual
def A.size : A → Nat
| .self a => a.size + 1
| .other b => b.size + 1
| .empty => 0
termination_by structural x => x
def B.size : B → Nat
| .self b => b.size + 1
| .other a => a.size + 1
| .empty => 0
termination_by structural x => x
end
```
The recursive functions don’t have to be in a one-to-one relation to a
set of mutually recursive inductive data types. It is possible to ignore
some of the types:
```lean
def A.self_size : A → Nat
| .self a => a.self_size + 1
| .other _ => 0
| .empty => 0
termination_by structural x => x
```
or have more than one function per argument type:
```lean
def isEven : Nat → Prop
| 0 => True
| n+1 => ¬ isOdd n
termination_by structural x => x
def isOdd : Nat → Prop
| 0 => False
| n+1 => ¬ isEven n
termination_by structural x => x
```
This does not include
* Support for nested inductive data types or nested recursion
* Inferring mutual structural recursion in the absence of
`termination_by`.
* Functional induction principles for these.
* Mutually recursive functions that live in different universes. This
may be possible,
maybe after beefing up the `.below` and `.brecOn` functions; we can look
into this some
other time, maybe when there are concrete use cases.
---------
Co-authored-by: Richard Kiss <him@richardkiss.com>
Co-authored-by: Tobias Grosser <tobias@grosser.es>
