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Proof step 6 from proof modmod-add-law-pos (previous step)

0 < j, (i0 % j + i % j) % j = (i0 % j + i) % j ∨ (i0 % j + i % j) % j = (i0 % j + i) % j + j ∨ (i0 % j + i % j) % j = (i0 % j + i) % j - j, (i + i0 % j) % j = (i + i0) % j ∨ (i + i0 % j) % j = (i + i0) % j + j ∨ (i + i0 % j) % j = (i + i0) % j - j ⊦ (i % j + i0 % j) % j = (i + i0) % j, (i % j + i0 % j) % j = (i + i0) % j + j, (i % j + i0 % j) % j = (i + i0) % j - j

User applied rule case distinction

And obtained the following new goals

Proof step 7:
(i0 % j + i % j) % j = (i0 % j + i) % j + j, 0 < j, (i + i0 % j) % j = (i + i0) % j ∨ (i + i0 % j) % j = (i + i0) % j + j ∨ (i + i0 % j) % j = (i + i0) % j - j ⊦ (i % j + i0 % j) % j = (i + i0) % j - j, (i % j + i0 % j) % j = (i + i0) % j + j, (i % j + i0 % j) % j = (i + i0) % j
Proof step 15:
(i0 % j + i % j) % j = (i0 % j + i) % j - j, 0 < j, (i + i0 % j) % j = (i + i0) % j ∨ (i + i0 % j) % j = (i + i0) % j + j ∨ (i + i0 % j) % j = (i + i0) % j - j ⊦ (i % j + i0 % j) % j = (i + i0) % j - j, (i % j + i0 % j) % j = (i + i0) % j + j, (i % j + i0 % j) % j = (i + i0) % j

Rule argument

pos: left 2

Additional information

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