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Proof step 4 from proof modmod-add2-negpos-neg (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
i + i1 * j < j, i + i1 * j < 0, i + i1 * j + j0 < j, j0 < j, 0 < j, i + 1 < j, i < j, i < 0, (i + i1 * j + j0) + 1 < j, 0 ≤ j0, 0 ≤ i1, i + i1 * j + j0 < 0, (i + i1 * j) + 1 < j ⊦ (i + j0) % j = (i + i1 * j + j0) % j

Rule argument

Additional information

Used simplifier rules: ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%lst-03: ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0; from specification int-basic2%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; 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%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-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%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-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-ge-add: ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); from specification int-mult