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feat: BitVec.toNat_{add,sub,mul_of_lt} for BitVector non-overflow reasoning #5411

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merged 2 commits into from
Sep 24, 2024
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feat: BitVec.toNat_{add,sub,mul_of_lt} for BitVector non-overflow reasoning #5411

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feat: BitVec.toNat_{add,sub,mul_of_lt} for BitVector non-overflow rea…
bollu Sep 20, 2024
d180789
Merge branch 'master' into bitvec-non-overflow-lemmas
kim-em Sep 24, 2024
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22 changes: 22 additions & 0 deletions src/Init/Data/BitVec/Lemmas.lean
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Expand Up @@ -2186,6 +2186,11 @@ theorem getLsbD_intMax (w : Nat) : (intMax w).getLsbD i = decide (i + 1 < w) :=

/-! ### Non-overflow theorems -/

/-- If `x.toNat * y.toNat < 2^w`, then the multiplication `(x * y)` does not overflow. -/
theorem toNat_add_of_lt {w} {x y : BitVec w} (h : x.toNat + y.toNat < 2^w) :
(x + y).toNat = x.toNat + y.toNat := by
rw [BitVec.toNat_add, Nat.mod_eq_of_lt h]

/--
If `y ≤ x`, then the subtraction `(x - y)` does not overflow.
Thus, `(x - y).toNat = x.toNat - y.toNat`
Expand All @@ -2199,6 +2204,23 @@ theorem toNat_sub_of_le {x y : BitVec n} (h : y ≤ x) :
· have : 2 ^ n - y.toNat + x.toNat = 2 ^ n + (x.toNat - y.toNat) := by omega
rw [this, Nat.add_mod_left, Nat.mod_eq_of_lt (by omega)]

/--
If `y > x`, then the subtraction `(x - y)` *does* overflow, and the result is the wraparound.
Thus, `(x - y).toNat = 2^w - (y.toNat - x.toNat)`.
-/
theorem toNat_sub_of_lt {x y : BitVec w} (h : x < y) :
(x - y).toNat = 2^w - (y.toNat - x.toNat) := by
simp only [toNat_sub]
rw [Nat.mod_eq_of_lt (by bv_omega)]
bv_omega

/-- If `x.toNat * y.toNat < 2^w`, then the multiplication `(x * y)` does not overflow.
Thus, `(x * y).toNat = x.toNat * y.toNat`.
-/
theorem toNat_mul_of_lt {w} {x y : BitVec w} (h : x.toNat * y.toNat < 2^w) :
(x * y).toNat = x.toNat * y.toNat := by
rw [BitVec.toNat_mul, Nat.mod_eq_of_lt h]

/-! ### Decidable quantifiers -/

theorem forall_zero_iff {P : BitVec 0 → Prop} :
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