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Proof step 8 from proof mod-add-01 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
i * (i0 + 1) + j0 = 0, 0 < i, i0 * i + j0 < 0, i0 < i, j0 < i, i0 < i + j0, j0 < 0, 0 < i + j0, i0 ≤ 0 ⊦ 0 = j0

Rule argument

Additional information

Used simplifier rules: mod-zero-04: j ≠ 0 ⊦ 0 % j = 0;%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i + 0 = i; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ i * 0 = 0; (s)%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)