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feat: support all the SMTLIB BitVec divison/remainder operations in bv_decide #5869

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Oct 28, 2024
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feat: support all the SMTLIB BitVec divison/remainder operations in bv_decide #5869

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4ab1b28
feat: add _eq lemmas for SMTLIB BitVec ops
hargoniX Oct 28, 2024
f891cf7
chore: integrate with bv_decide
hargoniX Oct 28, 2024
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27 changes: 27 additions & 0 deletions src/Init/Data/BitVec/Lemmas.lean
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Expand Up @@ -2383,6 +2383,11 @@ theorem umod_eq_and {x y : BitVec 1} : x % y = x &&& (~~~y) := by
rcases hy with rfl | rfl <;>
rfl

/-! ### smtUDiv -/

theorem smtUDiv_eq (x y : BitVec w) : smtUDiv x y = if y = 0#w then allOnes w else x / y := by
simp [smtUDiv]

/-! ### sdiv -/

/-- Equation theorem for `sdiv` in terms of `udiv`. -/
Expand Down Expand Up @@ -2439,6 +2444,28 @@ theorem sdiv_self {x : BitVec w} :
rcases x.msb with msb | msb <;> simp
· rcases x.msb with msb | msb <;> simp [h]

/-! ### smtSDiv -/

theorem smtSDiv_eq (x y : BitVec w) : smtSDiv x y =
match x.msb, y.msb with
| false, false => smtUDiv x y
| false, true => -(smtUDiv x (-y))
| true, false => -(smtUDiv (-x) y)
| true, true => smtUDiv (-x) (-y) := by
rw [BitVec.smtSDiv]
rcases x.msb <;> rcases y.msb <;> simp

/-! ### srem -/

theorem srem_eq (x y : BitVec w) : srem x y =
match x.msb, y.msb with
| false, false => x % y
| false, true => x % (-y)
| true, false => - ((-x) % y)
| true, true => -((-x) % (-y)) := by
rw [BitVec.srem]
rcases x.msb <;> rcases y.msb <;> simp

/-! ### smod -/

/-- Equation theorem for `smod` in terms of `umod`. -/
Expand Down
34 changes: 34 additions & 0 deletions src/Std/Tactic/BVDecide/Normalize/BitVec.lean
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Expand Up @@ -84,6 +84,40 @@ theorem BitVec.smod_umod (x y : BitVec w) :
rw [BitVec.smod_eq]
cases x.msb <;> cases y.msb <;> simp

attribute [bv_normalize] BitVec.smtUDiv_eq

@[bv_normalize]
theorem BitVec.smtSDiv_smtUDiv (x y : BitVec w) :
x.smtSDiv y =
if x.msb then
if y.msb then
(-x).smtUDiv (-y)
else
-((-x).smtUDiv y)
else
if y.msb then
-(x.smtUDiv (-y))
else
x.smtUDiv y := by
rw [BitVec.smtSDiv_eq]
cases x.msb <;> cases y.msb <;> simp

@[bv_normalize]
theorem BitVec.srem_umod (x y : BitVec w) :
x.srem y =
if x.msb then
if y.msb then
-((-x) % (-y))
else
-((-x) % y)
else
if y.msb then
x % (-y)
else
x % y := by
rw [BitVec.srem_eq]
cases x.msb <;> cases y.msb <;> simp

attribute [bv_normalize] Bool.cond_eq_if
attribute [bv_normalize] BitVec.abs_eq

Expand Down
12 changes: 12 additions & 0 deletions tests/lean/run/bv_arith.lean
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Expand Up @@ -60,3 +60,15 @@ theorem arith_unit_14 (x y : BitVec 8) (hx : x.msb = true) (hy : y.msb = true) :

theorem arith_unit_15 (x : BitVec 32) : BitVec.sle x (BitVec.abs x) := by
bv_decide

theorem arith_unit_16 (x y : BitVec 8) (hy : y ≠ 0) : x.smtUDiv y = x / y := by
bv_decide

theorem arith_unit_17 (x y : BitVec 8) (hy : y = 0) : x.smtUDiv y = -1#8 := by
bv_decide

theorem arith_unit_18 (x y : BitVec 8) (hx : x.msb = true) (h : y.msb = true) : x.smtSDiv y = (-x).smtUDiv (-y) := by
bv_decide

theorem arith_unit_19 (x y : BitVec 8) (hx : x.msb = true) (h : y.msb = true) : x.srem y = -((-x) % (-y)) := by
bv_decide
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