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Proof step 4 from proof mod-rec-01 (previous step)

~ i % ~ j = (~ i - ~ j) % ~ j, i ≤ j, i < 0, j < 0 ⊦ i % j = (i - j) % j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
i % j + (j - i) % j = 0, i ≤ j, i < 0, j < 0 ⊦ i % j = (i - j) % j

Rule argument

Additional information

Used simplifier rules: mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j);%mod-neg: j ≠ 0 ⊦ i % ~ j = i % j;%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%negeq: ⊦ ~ i = j ↔ i + j = 0; from specification int-abs%sub-minus: ⊦ ~ i - ~ j = j - i; from specification int-abs%sub-zero-01: ⊦ i - 0 = i; from specification int-abs%neg-10: ⊦ ~ i = 0 ↔ i = 0; from specification int-abs