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Proof step 13 from proof modmod-add-case (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
0 < i0, 0 < j, i < 0, 0 ≤ i + i0, 0 ≤ i0 ⊦ (i + i0) % j = (i + i0 % j) % j + j, (i + i0 % j) % j = (i + i0) % j, (i + i0 % j) % j = (i + i0) % j + j, 0 = j

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i < j → ¬ j < i; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i + 0 = i; (s)% ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; (s)