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Proof step 8 from proof divmod2-2pot-neg (previous step)

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) + i0 * 2 ^ i) / 2) % 2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ i0 % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j
Proof step 10:
(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) / 2) % 2

Rule argument

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

Additional information

Used simplifier rules: mod-2pot: ⊦ 0 < i → (j * 2 ^ i) % 2 = 0;%pot2-pos: ⊦ 0 < 2 ^ i ↔ 0 ≤ i;%sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%sub-ls: ⊦ i - j < i0 ↔ i < j + i0; from specification int-abs%lsone: ⊦ i < 1 ↔ i ≤ 0; from specification int-abs%zero-ls-03: ⊦ i ≤ 0 → (i * j < 0 ↔ ¬ ¬(0 < j ∧ i ≠ 0)); from specification int-mult%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%add-zero: ⊦ i + 0 = i; from specification int-basic2 Used terms: 2, i0 * 2 ^ i, j0 - 1