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feat: add BitVec.signExtend_extractLsb_setWidth theorem (#11943)

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This PR introduces the theorem
`BitVec.sshiftRight_eq_setWidth_extractLsb_signExtend` theorem, proving
`x.sshiftRight n` is equivalent to first sign-extending `x`, extracting
the appropriate least significant bits, and then setting the width back
to `w`.

---------

Co-authored-by: Tobias Grosser <github@grosser.es>
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Osman Yasar 2026-02-16 22:50:10 +00:00 • committed by GitHub
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commit 6833b6dba8
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src/Init/Data/BitVec/Lemmas.lean
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@ -2581,6 +2581,19 @@ theorem msb_signExtend {x : BitVec w} :
· simp [h, BitVec.msb, getMsbD_signExtend, show v - w = 0 by omega]
· simp [h, BitVec.msb, getMsbD_signExtend, show ¬ (v - w = 0) by omega]
/-- Sign-extending to `w + n` bits, extracting bits `[w - 1 + n..n]`, and setting width
back to `w` is equivalent to arithmetic right shift by `n`, since both sides discard the `n`
least significant bits and replicate the sign bit into the upper bits. -/
@[simp]
theorem signExtend_extractLsb_setWidth {x : BitVec w} {n : Nat} :
((x.signExtend (w + n)).extractLsb (w - 1 + n) n).setWidth w = x.sshiftRight n := by
ext i hi
simp only [getElem_sshiftRight, getElem_setWidth, getLsbD_extract,
Nat.add_sub_cancel, show i ≤ w - 1 by omega, decide_true, getLsbD_signExtend,
Bool.true_and]
by_cases hni : n + i < w
<;> (simp [hni]; omega)
/-- Sign extending to a width smaller than the starting width is a truncation. -/
theorem signExtend_eq_setWidth_of_le (x : BitVec w) {v : Nat} (hv : v ≤ w) :
x.signExtend v = x.setWidth v := by