we keep running into examples where working with well-founded recursion
is slow because defeq checks (which are all over the place, including
failing ones that are back-tracked) unfold well-founded definitions.
The definition of a function defined by well-founded recursion should be
an implementation detail that should only be peeked inside by the
equation generator and the functional induction generator.
We now mark the mutual recursive function as irreducible (if the user
did not
set a flag explicitly), and use `withAtLeastTransparency .all` when
producing
the equations.
Proofs can be fixed by using rewriting, or – a bit blunt, but nice for
adjusting
existing proofs – using `unseal` (a.k.a. `attribute [local
semireducible]`).
Mathlib performance does not change a whole lot:
http://speed.lean-fro.org/mathlib4/compare/08b82265-75db-4a28-b12b-08751b9ad04a/to/16f46d5e-28b1-41c4-a107-a6f6594841f8
Build instructions -0.126 %, four modules with significant instructions
decrease.
To reduce impact, these definitions were changed:
* `Nat.mod`, to make `1 % n` reduce definitionally, so that `1` as a
`Fin 2` literal
works nicely. Theorems with larger `Fin` literals tend to need a `unseal
Nat.modCore`
https://github.com/leanprover/lean4/pull/4098
* `List.ofFn` rewritten to be structurally recursive and not go via
`Array.ofFn`:
https://github.com/leanprover-community/batteries/pull/784
Alternative designs explored were
* Making `WellFounded.fix` irreducible.
One benefit is that recursive functions with equal definitions (possibly
after
instantiating fixed parameters) are defeq; this is used in mathlib to
relate
[`OrdinalApprox.gfpApprox`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/FixedPointApproximants.html#OrdinalApprox.gfpApprox)
with `.lfpApprox`.
But the downside is that one cannot use `unseal` in a
targeted way, being explicit in which recursive function needs to be
reducible here.
And in cases where Lean does unwanted unfolding, we’d still unfold the
recursive
definition once to expose `WellFounded.fix`, leading to large terms for
often no good
reason.
* Defining `WellFounded.fix` to unroll defintionally once before hitting
a irreducible
`WellFounded.fixF`. This was explored in #4002. It shares most of the
ups and downs
with the previous variant, with the additional neat benefit that
function calls that
do not lead to recursive cases (e.g. a `[]` base case) reduce nicely.
This means that
the majority of existing `rfl` proofs continue to work.
Issue #4051, which demonstrates how badly things can go if wf recursive
functions can be
unrolled, showed that making the recursive function irreducible there
leads to noticeably
faster elaboration than making `WellFounded.fix` irreducible; this is
good evidence that
the present PR is the way to go.
This fixes https://github.com/leanprover/lean4/issues/3988
---------
Co-authored-by: Leonardo de Moura <leomoura@amazon.com>
39286862e3