This was upstreamed from Mathlib in #5478, but leaving off the `@[simp]`
attribute, thereby breaking Mathlib. (We could of course add the simp
attribute back in Mathlib, but wherever it lives it should have been in
place at the time we merged -- this way I have to add it temporarily in
Mathlib and then remove it again once it is redundant.)
This PR adds the theorems
```
@[simp]
theorem divRec_zero (qr : DivModState w) :
divRec w w 0 n d qr = qr
@[simp]
theorem divRec_succ' (wn : Nat) (qr : DivModState w) :
divRec w wr (wn + 1) n d qr =
let r' := shiftConcat qr.r (n.getLsbD wn)
let input : DivModState w :=
if r' < d then ⟨qr.q.shiftConcat false, r'⟩ else ⟨qr.q.shiftConcat true, r' - d⟩
divRec w (wr + 1) wn n d input
```
The final statements may need some masasging to interoperate with
`bv_decide`. We prove the recurrence for unsigned division by building a
shift-subtract circuit, and then showing that this circuit obeys the
division algorithm's invariant.
---
A `DivModState` is lawful if the remainder width `wr` plus the dividend
width `wn` equals `w`,
and the bitvectors `r` and `n` have values in the bounds given by
bitwidths `wr`, resp. `wn`.
This is a proof engineering choice: An alternative world could have
`r : BitVec wr` and `n : BitVec wn`, but this required much more
dependent typing coercions.
Instead, we choose to declare all involved bitvectors as length `w`, and
then prove that
the values are within their respective bounds.
---------
Co-authored-by: Tobias Grosser <github@grosser.es>
Co-authored-by: Alex Keizer <alex@keizer.dev>
Co-authored-by: Kim Morrison <scott@tqft.net>
Co-authored-by: Tobias Grosser <tobias@grosser.es>
Add iff version of `List.IsPrefix.getElem`, and `eq_of_length_le`
variants of `List.IsInfix.eq_of_length, List.IsPrefix.eq_of_length,
List.IsSuffix.eq_of_length`
We make sure that we can pull `List.toArray` out through all operations
(well, for now "most" rather than "all"). As we also push `Array.toList`
inwards, this hopefully has the effect of them cancelling as they meet,
and `simp` naturally rewriting Array operations into List operations
wherever possible.
This is not at all complete yet.
These theorems are useful when one wants to simplify the goal state,
under knowledge that the bitvector operations don't overflow. This can
produce much smaller goal states that eventually allows `bv_omega` to
quickly close the goal.
Note that the LHS of the theorem is *not* in `simp` normal form, since
e.g. `(x + y).toNat` is normalized to `(x.toNat + y.toNat) % 2^w`. It's
not immediately clear to me what should be done about this.
Co-authored-by: Kim Morrison <scott.morrison@gmail.com>
Previously, it was not possible to use `decide` with most Array
functions (including `==`).
Later, we may replace some of these functions with defeqs that go via
the `List` operations, and use `csimp` lemmas for fast runtime
behaviour. In the meantime, this allows using `decide`.
Given the derived `Repr` instance for types with parameters, the absence
of `Repr Empty` can cause `Repr` instance synthesis to fail. For
example, given
```lean
inductive Prim (special : Type) where
| plus
| other : special → Prim special
deriving Repr
```
this works:
```lean
#eval (Prim.plus : Prim Int)
```
but this fails:
```lean
#eval (Prim.plus : Prim Empty)
```
---------
Co-authored-by: Kyle Miller <kmill31415@gmail.com>
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
In LNSym we often use the pattern `ofBool (a.getLsbD i)` to pick out a
specific bit (`i`) from a bitvector (`a`).
By adding a rewrite to `extractLsb` to `bv_decide`s normalization set,
we can still automatically close goals that have this pattern. In the
process, I also added a simp-lemma about the value of a `Fin 1`.
