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Proof step 5 from proof modmod-add2-negpos-pos-01 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

Used simplifier rules: ⊦ i + j < j ↔ i < 0; (s)% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s)% ⊦ i < i + j ↔ 0 < j; (s)% ⊦ i + j < i + i0 ↔ j < i0; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)%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%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (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% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); (s)