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feat: BitVec analogues of Nat.{mul_two, two_mul, mul_succ, succ_mul} (#5410)

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As requested by @hargoniX .
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Siddharth 2024-09-22 23:37:04 -05:00 • committed by GitHub
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src/Init/Data/BitVec/Lemmas.lean
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@ -1724,6 +1724,15 @@ theorem BitVec.mul_add {x y z : BitVec w} :
rw [Nat.mul_mod, Nat.mod_mod (y.toNat + z.toNat),
← Nat.mul_mod, Nat.mul_add]
theorem mul_succ {x y : BitVec w} : x * (y + 1#w) = x * y + x := by simp [BitVec.mul_add]
theorem succ_mul {x y : BitVec w} : (x + 1#w) * y = x * y + y := by simp [BitVec.mul_comm, BitVec.mul_add]
theorem mul_two {x : BitVec w} : x * 2#w = x + x := by
have : 2#w = 1#w + 1#w := by apply BitVec.eq_of_toNat_eq; simp
simp [this, mul_succ]
theorem two_mul {x : BitVec w} : 2#w * x = x + x := by rw [BitVec.mul_comm, mul_two]
@[simp, bv_toNat] theorem toInt_mul (x y : BitVec w) :
(x * y).toInt = (x.toInt * y.toInt).bmod (2^w) := by
simp [toInt_eq_toNat_bmod]