This PR upstreams many of the results from `Mathlib/Data/Int/Init.lean`.
Notably, we upstream the `simp` tag on `Int.natCast_pow`. While this is
desirable as a `simp` lemma, it is non-confluent with other good `simp`
lemmas like `Int.emod_bmod_congr`, and this will need to be addressed in
the future.
This PR adds lemmas about `Int.bmod` to achieve parity between
`Int.bmod` and `Int.emod`/`Int.fmod`/`Int.tmod`. Furthermore, it adds
missing lemmas for `emod`/`fmod`/`tmod` and performs cleanup on names
and statements for all four operations, also with a view towards
increasing consistency with the corresponding `Nat.mod` lemmas.
This PR shuffles some results about integers around to make sure that
all material that currently exists about `Int.bmod` is located in
`DivMod/Lemmas.lean` and not downstream of that.
This PR adds `BitVec.[toNat|toFin|toInt]_[sshiftRight|sshiftRight']`
plus variants with `of_msb_*`. While at it, we also add
`toInt_zero_length` and `toInt_of_zero_length`. In support of our main
theorem we add `toInt_shiftRight_lt` and `le_toInt_shiftRight`, which
make the main theorem automatically derivable via omega.
We also add four shift lemmas for `Int`: `le_shiftRight_of_nonpos`,
`shiftRight_le_of_nonneg`, `le_shiftRight_of_nonneg`,
`shiftRight_le_of_nonpos`, as well as `emod_eq_add_self_emod`,
`ediv_nonpos_of_nonpos_of_neg `, and`bmod_eq_emod_of_lt `. For `Nat` we
add `shiftRight_le`.
Beyond the lemmas directly needed in the proof, we added a couple more
to ensure the API is complete.
We also fix the casing of `toFin_ushiftRight` and rename `lt_toInt` to
`two_mul_lt_toInt` to avoid `'`-ed lemmas.
This PR fills further gaps in the integer division API, and mostly
achieves parity between the three variants of integer division. There
are still some inequality lemmas about `tdiv` and `fdiv` that are
missing, but as they would have quite awkward statements I'm hoping that
for now no one is going to miss them.
This PR continues alignment of lemmas about `Int.ediv/fdiv/tdiv`,
including adding notes about "missing" lemmas that do not apply in one
case. Also lemmas about `emod/fmod/tmod`. There's still more to do.
This PR adds theorems comparing `Int.ediv` with `tdiv` and `fdiv`, for
all signs of arguments. (Previously we just had the statements about the
cases in which they agree.)
This PR splits `Int.DivModLemmas` into a `Bootstrap` and `Lemmas` file,
where it is possible to use `omega` in `Lemmas`.
I'm going to add more theory, particularly about `fdiv` and `tdiv` to
the `Lemmas` file, and would prefer to have access to `omega`.
