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feat: add Bitvec.(toInt, toFin)_twoPow #7225

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feat: add Bitvec.(toInt, toFin)_twoPow #7225

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4dfd4d2
chore: add theorems
luisacicolini Feb 25, 2025
2d3bd0d
chore: correct order"
luisacicolini Feb 25, 2025
bc3be0a
chore: update simp lemma sets
luisacicolini Feb 25, 2025
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24 changes: 24 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 @@ -3731,6 +3731,30 @@ theorem getLsbD_twoPow (i j : Nat) : (twoPow w i).getLsbD j = ((i < w) && (i = j
simp at hi
simp_all

theorem toInt_twoPow {w i : Nat} :
(BitVec.twoPow w i).toInt = (if w ≤ i then 0 else if i + 1 = w then -1 <<< i else 1 <<< i : Int) := by
simp only [BitVec.twoPow, BitVec.toInt]
rcases w with _|w
· simp
· by_cases h : w + 1 ≤ i
· simp [h]; norm_cast; omega
· simp only [toNat_shiftLeft, toNat_ofNat, Nat.zero_lt_succ, Nat.one_mod_two_pow,
Int.ofNat_emod, h, ↓reduceIte, Nat.add_right_cancel_iff]
have hy : (2 ^ i % 2 ^ (w + 1)) = 2 ^ i := by rw [Nat.mod_eq_of_lt (by rw [Nat.pow_lt_pow_iff_right (by omega)]; omega)]
have hj : 2 * 2 ^ i = 2 ^ (i + 1) := by rw [Nat.pow_add, Nat.mul_comm]
Comment on lines +3743 to +3744
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Suggested change
have hy : (2 ^ i % 2 ^ (w + 1)) = 2 ^ i := by rw [Nat.mod_eq_of_lt (by rw [Nat.pow_lt_pow_iff_right (by omega)]; omega)]
have hj : 2 * 2 ^ i = 2 ^ (i + 1) := by rw [Nat.pow_add, Nat.mul_comm]
have hy : 2 ^ i % 2 ^ (w + 1) = 2 ^ i := by rw [Nat.mod_eq_of_lt (by rw [Nat.pow_lt_pow_iff_right (by omega)]; omega)]
have hj : 2 * 2 ^ i = 2 ^ (i + 1) := by rw [Nat.pow_add, Nat.mul_comm]

I wonder if these two lemma above should not be named simp lemmas.

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In fact, the entire second half of the proof would be a lot nicer, if we can make avoid the casts in the middle and just make this all simp away. I am wondering if there is a set of simp lemmas which could help here?

norm_cast
simp only [Nat.shiftLeft_eq, Nat.one_mul, hy, hj]
by_cases i + 1 = (w + 1)
· simp [show i = w by omega]; omega
Comment on lines +3747 to +3748
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@tobiasgrosser tobiasgrosser Feb 25, 2025 •
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Suggested change
by_cases i + 1 = (w + 1)
· simp [show i = w by omega]; omega
by_cases i + 1 = w + 1
· simp [show i = w by omega]; omega

· simp [show 2 ^ (i + 1) < 2 ^ (w + 1) by rw [Nat.pow_lt_pow_iff_right (by omega)]; omega]
omega

theorem toFin_twoPow {w i : Nat} :
(BitVec.twoPow w i).toFin = Fin.ofNat' (2 ^ w) (2 ^ i) := by
rcases w with rfl | w
· simp [BitVec.twoPow, BitVec.toFin, toFin_shiftLeft, Fin.fin_one_eq_zero]
· simp [BitVec.twoPow, BitVec.toFin, toFin_shiftLeft, Nat.shiftLeft_eq]

@[simp]
theorem getElem_twoPow {i j : Nat} (h : j < w) : (twoPow w i)[j] = decide (j = i) := by
rw [←getLsbD_eq_getElem, getLsbD_twoPow]
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