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feat: setup simp lemmas: 'msb -> getLsb -> decide ...' (#3436)

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This is a follow up to 'https://github.com/leanprover/std4/pull/645',
where the simp lemmas were requested:
https://github.com/leanprover/std4/pull/645#issuecomment-1944862251

---

Note that @semorrison asked to use `(Fin.last _)` to index. Now that we
use a `Nat` to index `msb` , the pattern `(Fin.last _)` would not have
the width be automatically inferred. Therefore, I've changed the
definitions to use `Nat` for indexing.

---------

Co-authored-by: Siddharth Bhat Mala <sb2743@cl.cam.ac.uk>
Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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Siddharth 2024-02-22 00:07:14 +00:00 • committed by GitHub
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commit b6ed97bb3d
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1 changed files with 14 additions and 3 deletions
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17
src/Init/Data/BitVec/Lemmas.lean
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@ -117,9 +117,16 @@ private theorem lt_two_pow_of_le {x m n : Nat} (lt : x < 2 ^ m) (le : m ≤ n) :
/-! ### msb -/
theorem msb_eq_decide (x : BitVec (Nat.succ w)) : BitVec.msb x = decide (2 ^ w ≤ x.toNat) := by
simp only [BitVec.msb, getMsb, Nat.zero_lt_succ,
decide_True, getLsb, Nat.testBit, Nat.succ_sub_succ_eq_sub,
theorem msb_eq_getLsb_last (x : BitVec w) :
x.msb = x.getLsb (w - 1) := by
simp [BitVec.msb, getMsb, getLsb]
rcases w with rfl | w
· simp [BitVec.eq_nil x]
· simp
@[simp] theorem getLsb_last (x : BitVec (w + 1)) :
x.getLsb w = decide (2 ^ w ≤ x.toNat) := by
simp only [Nat.zero_lt_succ, decide_True, getLsb, Nat.testBit, Nat.succ_sub_succ_eq_sub,
Nat.sub_zero, Nat.and_one_is_mod, Bool.true_and, Nat.shiftRight_eq_div_pow]
rcases (Nat.lt_or_ge (BitVec.toNat x) (2 ^ w)) with h | h
· simp [Nat.div_eq_of_lt h, h]
@ -129,6 +136,10 @@ theorem msb_eq_decide (x : BitVec (Nat.succ w)) : BitVec.msb x = decide (2 ^ w
· have : BitVec.toNat x < 2^w + 2^w := by simpa [Nat.pow_succ, Nat.mul_two] using x.isLt
omega
@[simp]
theorem msb_eq_decide (x : BitVec (w + 1)) : BitVec.msb x = decide (2 ^ w ≤ x.toNat) := by
simp [msb_eq_getLsb_last]
/-! ### cast -/
@[simp] theorem toNat_cast (h : w = v) (x : BitVec w) : (cast h x).toNat = x.toNat := rfl