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Proof step 19 from proof modmod-add-case (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
i0 < 0, 0 < j, 0 ≤ i ⊦ (i + i0) % j = (i + i0 % j) % j + j, (i + i0 % j) % j = (i + i0) % j, (i + i0 % j) % j = (i + i0) % j + j

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)