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Proof step 7 from proof div-add1-posneg-pos-01 (previous step)

i = i0 * j + j1 ∧ 0 ≤ j1 ∧ j1 < j ∧ 0 ≤ i0, j1 = 0, 0 < j, 1 < j, i1 < i, i1 * j + j0 < j, j0 < 0, j0 < j, j0 + 1 < j, 0 < j + j0, (i1 * j + j0) + 1 < j, 0 ≤ i, i1 ≤ 0, i1 < j, i1 < j + j0, 0 ≤ i + i1 * j + j0, i1 * j + j0 < 0, 0 < i, (i1 * j + j0) + 1 < i, j0 + 1 < i ⊦ j0 = 0, i0 + i1 = (i + i1 * j + j0) / j + 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
0 ≤ i0, j0 + 1 < i0 * j, (i1 * j + j0) + 1 < i0 * j, i1 * j + j0 < 0, i1 < j + j0, i1 < j, i1 ≤ 0, (i1 * j + j0) + 1 < j, 0 < j + j0, j0 + 1 < j, j0 < j, j0 < 0, i1 * j + j0 < j, i1 < i0 * j, 1 < j, 0 < i0, (i1 * j + j0) + 1 < i0, 0 < j, 0 ≤ i0 * j + i1 * j + j0, i1 < i0, j0 + 1 < i0 ⊦ i0 + i1 = (i0 * j + i1 * j + j0) / j + 1, j0 = 0

Rule argument

Additional information

Used simplifier rules: ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%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%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2