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Proof step 9 from proof modmod-add-law-pos (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%numsucc: ⊦ (* i0) * i + i = (i0 + 1) * i; from specification int-mult%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%one: ⊦ 1 * i = i; from specification int-mult%numeq-02: ⊦ (* i2) * i + j = i + j0 ↔ (i2 - 1) * i + j = j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-neq-01: ⊦ 0 < i → (i * j = 0 ↔ j = 0); from specification int-mult%eq: ⊦ i = i + j ↔ j = 0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-mult-two: ⊦ i + i = 2 * i; from specification int-mult