This PR lets recursive functions defined by well-founded recursion use a
different `fix` function when the termination measure is of type `Nat`.
This fix-point operator use structural recursion on “fuel”, initialized
by the given measure, and is thus reasonable to reduce, e.g. in `by
decide` proofs.
Extra provisions are in place that the fixpoint operator only starts
reducing when the fuel is fully known, to prevent “accidential” defeqs
when the remaining fuel for the recursive calls match the initial fuel
for that recursive argument.
To opt-out, the idiom `termination_by (n,0)` can be used.
We still use `@[irreducible]` as the default for such recursive
definitions, to avoid unexpected `defeq` lemmas. Making these functions
`@[semireducible]` by default showed performance regressions in lean.
When the measure is of type `Nat`, the system will accept an explicit
`@[semireducible]` without the usual warning.
Fixes#5234. Fixes: #11181.
This PR enables transforming nondependent `let`s into `have`s in a
number of contexts: the bodies of nonrecursive definitions, equation
lemmas, smart unfolding definitions, and types of theorems. A motivation
for this change is that when zeta reduction is disabled, `simp` can only
effectively rewrite `have` expressions (e.g. `split` uses `simp` with
zeta reduction disabled), and so we cache the nondependence calculations
by transforming `let`s to `have`s. The transformation can be disabled
using `set_option cleanup.letToHave false`.
Uses `Meta.letToHave`, introduced in #8954.
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
