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Proof step 9 from proof intshiftrsign-2pot-neg-02 (previous step)

0 ≤ j, i0 < 2 ^ j + j0, i0 < j, i0 ≤ 0, 0 < 2 ^ j + j0, j0 < 0, i0 * 2 ^ j + j0 < 0, 0 < j, j < 32 ⊦ i0 = 1 + (2 ^ j * (i0 - 1) + j0 + 1) / 2 ^ j, j0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) / j = i

Rule argument

seq: ⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) / j = i subst: [i0, i, j] → [j0 + 1, i0 - 1, 2 ^ j] rotate: F spec: int-div inst: name: div-mult-neg

Additional information

Used simplifier rules: ⊦ i + (j - i) = j;% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;% ⊦ i - j ≤ i0 ↔ i ≤ j + i0;%j0 ≠ 0, j ≤ j0 ⊦ i ≤ (* j) → i ≤ (* j0);% ⊦ i + (* j) ≤ (* j0) ↔ i ≤ j0 - j;% ⊦ i ≤ ~1 ↔ i < 0;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i < 0 → abs(i + 1) = abs(i) - 1;% ⊦ i < 0 → abs(i) = ~ i;% ⊦ ~ i - j = ~(i + j);% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ 0 < i ∧ 0 < j → 0 < i + j; Used terms: j0 + 1, i0 - 1, 2 ^ j