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
This PR changes the order of steps tried when proving equational
theorems for structural recursion. In order to avoid goals that `split`
cannot handle, avoid unfolding the LHS of the equation to `.brecOn` and
`.rec` until after the RHS has been split into its final cases.
Fixes: #10195
This PR updates the formatting of, and adds explanations for, "unknown
identifier" errors as well as "failed to infer type" errors for binders
and definitions.
It attempts to ameliorate some of the confusion encountered in #1592 by
modifying the wording of the "header is elaborated before body is
processed" note and adding further discussion and examples of this
behavior in the corresponding error explanation.
This PR introduces an explicit `defeq` attribute to mark theorems that
can be used by `dsimp`. The benefit of an explicit attribute over the
prior logic of looking at the proof body is that we can reliably omit
theorem bodies across module boundaries. It also helps with intra-file
parallelism.
If a theorem is syntactically defined by `:= rfl`, then the attribute is
assumed and need not given explicitly. This is a purely syntactic check
and can be fooled, e.g. if in the current namespace, `rfl` is not
actually “the” `rfl` of `Eq`. In that case, some other syntax has be
used, such as `:= (rfl)`. This is also the way to go if a theorem can be
proved by `defeq`, but one does not actually want `dsimp` to use this
fact.
The `defeq` attribute will look at the *type* of the declaration, not
the body, to check if it really holds definitionally. Because of
different reduction settings, this can sometimes go wrong. Then one
should also write `:= (rfl)`, if one does not want this to be a defeq
theorem. (If one does then this is currently not possible, but it’s
probably a bad idea anyways).
The `set_option debug.tactic.simp.checkDefEqAttr true`, `dsimp` will
warn if could not apply a lemma due to a missing `defeq` attribute.
With `set_option backward.dsimp.useDefEqAttr.get false` one can revert
to the old behavior of inferring rfl-ness based on the theorem body.
Both options will go away eventually (too bad we can’t mark them as
deprecated right away, see #7969)
Meta programs that generate theorems (e.g. equational theorems) can use
`inferDefEqAttr` to set the attribute based on the theorem body of the
just created declaration.
This builds on #8501 to update Init to `@[expose]` a fair amount of
definitions that, if not exposed, would prevent some existing `:= rfl`
theorems from being `defeq` theorems. In the interest of starting
backwards compatible, I exposed these function. Hopefully many can be
un-exposed later again.
A mathlib adaption branch exists that includes both the meta programming
fixes and changes to the theorems (e.g. changing `:= by rfl` to `:=
rfl`).
With the module system there is now no special handling for `defeq`
theorem bodies, because we don’t look at the body anymore. The previous
hack is removed. The `defeq`-ness of the theorem needs to be checked in
the context of the theorem’s *type*; the error message contains a hint
if the defeq check fails because of the exported context.
This includes the examples from issues #2961, #3219 and #5667 in our
test suite, so that we know when (accidentially) fix them.
In fact this closes#3219, which (judging from the nightlies) was fixed
last week by #6901.
