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Proof step 38 from proof congmod-def-mod-02 (previous step)

j ≠ 0, i0 % j < 0, 0 ≤ i % j, i < 0, i0 < 0, 0 ≤ i % abs(j) ⊦ i % abs(j) = i0 % j + abs(j) ↔ i % j = i0 |%| j, j = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 39:
i0 % j < 0, 0 ≤ i % abs(j), i0 < 0, i < 0, 0 ≤ i % j ⊦ j = 0, i % abs(j) = i0 % j + abs(j) ↔ i % j = i0 |%| j

Rule argument

Additional information

Used simplifier rules: zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; 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