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feat: sdiv reduction to udiv #23

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feat: sdiv reduction to udiv #23

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35 changes: 35 additions & 0 deletions src/Init/Data/BitVec/Lemmas.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -1312,6 +1312,41 @@ theorem umod_eq {x y : BitVec n} :
theorem toNat_umod {x y : BitVec n} :
(x.umod y).toNat = x.toNat % y.toNat := rfl

/-! ### sdiv -/

/-- Equation theorem for `sdiv` in terms of `udiv`. -/
theorem sdiv_eq (x y : BitVec w) : x.sdiv y =
match x.msb, y.msb with
| false, false => udiv x y
| false, true => - (x.udiv (- y))
| true, false => - ((- x).udiv y)
| true, true => (- x).udiv (- y) := by
rw [BitVec.sdiv]
rcases x.msb <;> rcases y.msb <;> simp

theorem sdiv_eq_and (x y : BitVec 1) : x.sdiv y = x &&& y := by
have hx : x = 0#1 ∨ x = 1#1 := by bv_omega
have hy : y = 0#1 ∨ y = 1#1 := by bv_omega
rcases hx with rfl | rfl <;>
rcases hy with rfl | rfl <;>
rfl

/-! ### smod -/

/-- Equation theorem for `smod` in terms of `umod`. -/
theorem smod_eq (x y : BitVec w) : x.smod y =
match x.msb, y.msb with
| false, false => x.umod y
| false, true =>
let u := x.umod (- y)
(if u = 0#w then u else u + y)
| true, false =>
let u := umod (- x) y
(if u = 0#w then u else y - u)
| true, true => - ((- x).umod (- y)) := by
rw [BitVec.smod]
rcases x.msb <;> rcases y.msb <;> simp

/-! ### signExtend -/

/-- Equation theorem for `Int.sub` when both arguments are `Int.ofNat` -/
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