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feat: add BitVec.and_add_xor_eq_or #304

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May 15, 2024
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feat: add BitVec.and_add_xor_eq_or #304

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feat: add BitVec.and_add_xor_eq_or
tobiasgrosser May 14, 2024
34ff84d
Update SSA/Projects/InstCombine/ForLean.lean
tobiasgrosser May 15, 2024
f35dab0
chore: extract lemma
tobiasgrosser May 15, 2024
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18 changes: 18 additions & 0 deletions SSA/Projects/InstCombine/ForLean.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -810,6 +810,24 @@ lemma toInt_zero_iff (w : Nat) (x : BitVec w) : BitVec.toInt x = 0 ↔ x = 0 :=
lemma toInt_nonzero_iff (w : Nat) (x : BitVec w) : BitVec.toInt x ≠ 0 ↔ x ≠ 0 := by
simp [toInt_ne]

@[simp]
lemma carry_and_xor_false : carry i (a &&& b) (a ^^^ b) false = false := by
induction i
case zero =>
simp [carry, Nat.mod_one]
case succ v v_h =>
simp only [carry_succ, v_h, Bool.atLeastTwo_false_right, getLsb_and,
getLsb_xor, Bool.and_eq_false_imp, Bool.and_eq_true, bne_eq_false_iff_eq,
and_imp]
intros; simp [*]

@[simp]
theorem and_add_xor_eq_or {a b : BitVec w} : (a &&& b) + (a ^^^ b) = a ||| b := by
ext i
rw [getLsb_add (by omega), getLsb_and, getLsb_xor, getLsb_or]
simp only [Bool.bne_assoc]
cases a.getLsb ↑i <;> simp [carry_and_xor_false]

end BitVec

-- Given (a, b) that are less than a modulus m, to show (a + b) % m < k, it suffices to consider two cases.
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