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Proof step 7 from proof modmod-add-law-pos (previous step)

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

User applied rule case distinction

And obtained the following new goals

Proof step 8:
(i + i0 % j) % j = (i + i0) % j + j, (i0 % j + i % j) % j = (i0 % j + i) % j + j, 0 < j ⊦ (i0 % j + i) % j + j = (i + i0) % j ∨ (i0 % j + i) % j + j = (i + i0) % j + j ∨ (i0 % j + i) % j + j = (i + i0) % j - j

Rule argument

pos: left 2

Additional information

Used simplifier rules: sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2