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Proof step 8 from proof mod-mult-neg-02 (previous step)

i0 = i1 * j + j0, 0 < j, i < j, i0 < j, i0 < 0, i + i0 < 0, abs(j0) < j, j + i0 ≤ 0, i ≤ 0, j0 ≤ 0, i1 ≤ 0 ⊦ (j * i + i0) % j = j0, i0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
j0 < j, i1 < j, i1 * j + j0 < 0, i + i1 * j + j0 < 0, 0 < j, i < j, i1 * j + j0 < j, abs(j0) < j, j * (i1 + 1) + j0 ≤ 0, i ≤ 0, j0 ≤ 0, i1 ≤ 0 ⊦ (j * (i1 + i) + j0) % j = j0, i1 * j + j0 = 0

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i * (j + 1) + j1 < i ↔ i * j + j1 < 0; (s)%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs% ⊦ i * j + i = i * (j + 1); (s)%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% ⊦ i * j + i * j0 = i * (j + j0); (s)