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

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

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9 changes: 9 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 @@ -1664,6 +1664,15 @@ 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]
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Can we make these binders explicit for consistency with the respective Nat theorems?

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It's not totally clear to me, because right above, there's theorem BitVec.mul_add {x y z : BitVec w} : .... I made these implicit following the above, because x, y can be determined by the equality.

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This is generally treated inconsistently in the BitVec file and I would really like this to become consistent. @hargoniX, is there official guidance how to choose here -- at best with an explanation why a certain style is preferred?

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]
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