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Proof step 7 from proof div-add1-negpos-neg-01 (previous step)

i0 = i2 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i2, i1 < j, i1 < 0, j + i1 * j ≤ 0, 0 ≤ i0, 1 + i1 * j < j, 1 + i1 * j + i0 < j, 0 < j, i1 * j + i0 < j, i1 * j + i0 < 0, i1 + 1 < j ⊦ (i1 * j + i0) / j = i1 + i2 + 1, j = 1, j0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
i1 * j < j0, i1 * j + i2 * j < j, i1 * j + i2 * j < j0, (i1 * j + i2 * j) + 1 < j0, 2 + i1 < j, i1 < j0, 0 ≤ i2, 0 < j, 1 < j, 1 + i1 * j + i2 * j + j0 < j, j + i1 * j ≤ 0, i1 < 0, i1 < j, j0 < j, 0 < j0, i1 * j + i2 * j < ~1, i1 * j + i2 * j + j0 < j, i1 * j + i2 * j + j0 < 0, 0 ≤ i2 * j + j0, j + i1 * j < j0, i1 + 1 < j0, i1 * j + i2 * j < 0, 2 + i1 * j + i2 * j < j ⊦ j0 = 0, j = 1, (i1 * j + i2 * j + j0) / j = i1 + i2 + 1

Rule argument

Additional information

Used simplifier rules: lsone: ⊦ i < 1 ↔ i ≤ 0; from specification int-abs%ls-same: ⊦ 0 < i → (i * j < i ↔ j < 1); from specification int-mult%modzero-02: ⊦ i < 0 ∧ j0 < j ∧ 0 < j → i * j + j0 < 0; 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-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%ls-add: ⊦ 0 ≤ j1 ∧ j1 < j → (i0 * j + j1 < j ↔ i0 ≤ 0); from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%lst-04: ⊦ i + j < i0 ∧ j0 < j → i + j0 < i0; from specification int-basic2%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%lst-03: ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%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-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (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%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs