This PR implements `BitVec.toInt_abs`.
The absolute value of `x : BitVec w` is naively a case split on the sign
of `x`.
However, recall that when `x = intMin w`, `-x = x`.
Thus, the full value of `abs x` is computed by the case split:
- If `x : BitVec w` is `intMin`, then its absolute value is also `intMin
w`, and
thus `toInt` will equal `intMin.toInt`.
- Otherwise, if `x` is negative, then `x.abs.toInt = (-x).toInt`.
- Finally, when `x` is nonnegative, then `x.abs.toInt = x.toInt`.
```lean
theorem toInt_abs {x : BitVec w} :
x.abs.toInt =
if x = intMin w then (intMin w).toInt
else if x.msb then -x.toInt
else x.toInt
```
We also provide a variant of `toInt_abs` that
hides the case split for `x` being positive or negative by using
`natAbs`.
```lean
theorem toInt_abs_eq_natAbs {x : BitVec w} : x.abs.toInt =
if x = intMin w then (intMin w).toInt else x.toInt.natAbs
```
Supercedes https://github.com/leanprover/lean4/pull/5787
---------
Co-authored-by: Tobias Grosser <github@grosser.es>
af4a3f2251