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Proof step 5 from proof mod-add1-posneg-pos (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

Used simplifier rules: add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i < i + j ↔ 0 < j; (s)% ⊦ i + j < i + i0 ↔ j < i0; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (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-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-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)