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Proof step 12 from proof divmod2-2pot-neg (previous step)

0 < i, (i0 * 2 ^ i) % 2 = 0 ∧ j0 - 1 < 0 ∧ i0 * 2 ^ i < 0 ∧ 0 < 2, 0 ≤ i, i0 < i, i0 ≤ 0, j0 < i, j0 ≤ 0, i0 * 2 ^ i + j0 < 0, abs(j0) < 2 ^ i, 1 < i, 0 < i ⊦ ((j0 - 1) / 2) % 2 = ((j0 - 1) / 2 + i0 * 2 ^ (i - 1)) % 2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ i0 % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) % j = i % j
Proof step 14:
¬((i0 * 2 ^ (i - 1)) % 2 = 0 ∧ (j0 - 1) / 2 < 0 ∧ i0 * 2 ^ (i - 1) < 0 ∧ 0 < 2), 0 < i, (i0 * 2 ^ i) % 2 = 0 ∧ j0 - 1 < 0 ∧ i0 * 2 ^ i < 0 ∧ 0 < 2, 0 ≤ i, i0 < i, i0 ≤ 0, j0 < i, j0 ≤ 0, i0 * 2 ^ i + j0 < 0, abs(j0) < 2 ^ i, 1 < i, 0 < i ⊦ ((j0 - 1) / 2) % 2 = ((j0 - 1) / 2 + i0 * 2 ^ (i - 1)) % 2

Rule argument

spec: int-div inst: name: mod-add-neg seq: ⊦ i0 % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) % j = i % j subst: [j, i0, i] → [2, i0 * 2 ^ (i - 1), (j0 - 1) / 2] flags: rewrite

Additional information

Used terms: 2, i0 * 2 ^ (i - 1), (j0 - 1) / 2