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Proof step 10 from proof modmod-add-if3 (previous step)

i + i0 < j, i + i0 < 0, i0 + 1 < j, 0 < j, 0 ≤ i, i0 < 0, i0 < j, (i + i0) + 1 < j ⊦ (i + i0 % j < 0 ⊃ (i + i0) % j;((i + i0 % j) % j ≠ 0 ⊃ (i + i0) % j + j;0)) = (0 ≤ i + i0 % j ∧ (i + i0 % j) % j ≠ 0 ⊃ (i + i0) % j + j;(i + i0) % j)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Used simplifier rules: modmod-add-zero: 0 < j ⊦ (i + i0 % j) % j = 0 → (i + i0) % j = 0;%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs