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Proof step 1 from proof modeq-pos-03 (conclusion)

0 ≤ i0 * j + j0, 0 < j, j0 < 0, j0 % j ≠ 0 ⊦ (i0 * j + j0) % j = j0 % j + j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 2:
j0 < i0, 0 < i0, j0 < j, j0 + 1 < j, 0 ≤ i0 * j + j0, 0 < j, j0 < 0, j0 + 1 < i0 ⊦ (i0 * j + j0) % j = j0 % j + j, j0 % j = 0

Rule argument

Additional information

Used simplifier rules: add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult%ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1