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Proof step 8 from proof modmod-add-if (previous step)

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

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-add2-posneg-neg: ⊦ i + i0 % j < 0 ∧ 0 ≤ i ∧ i0 < 0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j;%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2