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Proof step 5 from proof modmod-add2-posneg-neg-01 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
i1 < 0, i1 * j < i, abs(j0) < j, i + i1 * j + j0 < 0, 0 < j, i1 < j, j0 < j, j + i1 * j ≤ i, j + i1 * j + j0 ≤ 0, j0 ≤ 0, 0 ≤ i, 0 ≤ i + j0 ⊦ (i + j0) % j = 0, (i + j0) % j = (i + i1 * j + j0) % j + j, j = 1

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i + j ≤ i0 + j ↔ i ≤ i0; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i < 0 ∧ j0 < j ∧ 0 < j → i * j + j0 < 0; (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% ⊦ 0 < i → (i * j < 0 ↔ j < 0); (s)% ⊦ i + j < j ↔ i < 0; (s)% ⊦ i + j < i + i0 ↔ j < i0; (s)%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (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%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