feat: BitVec.toNat BitVec.signExtend #6155
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This PR adds toInt theorems for BitVec.signExtend.
If the current width `w` is larger than the extended width `v`,
then the value when interpreted as an integer is truncated,
and we compute a modulo by `2^v`.
```lean
theorem toInt_signExtend_of_le (x : BitVec w) (hv : v ≤ w) :
(x.signExtend v).toInt = Int.bmod (x.toNat) (2^v)
```
Co-authored-by: Siddharth Bhat <[email protected]>
Co-authored-by: Harun Khan <[email protected]>
Stacked on top of #6155
---------
Co-authored-by: Harun Khan <[email protected]>
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This PR adds toNat theorems for BitVec.signExtend.
Sign extending to a larger bitwidth depends on the msb.
If the msb is false, then the result equals the original value.
If the msb is true, then we add a value of `(2^v - 2^w)`, which arises from the sign extension.
```lean
theorem toNat_signExtend (x : BitVec w) {v : Nat} :
(x.signExtend v).toNat = (x.setWidth v).toNat + if x.msb then 2^v - 2^w else 0
```
Co-authored-by: Siddharth Bhat <[email protected]>
Co-authored-by: Harun Khan <[email protected]>
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This PR adds
toNattheorems forBitVec.signExtend.Sign extending to a larger bitwidth depends on the msb. If the msb is false, then the result equals the original value. If the msb is true, then we add a value of
(2^v - 2^w), which arises from the sign extension.