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Proof step 14 from proof mod-neg (previous step)

0 ≤ ~(i0 * j + j0) % j, 0 < j + j0, i0 * j + j0 < j, i0 * j + j0 < 0, j0 < j, j0 < 0, i0 < j + j0, i0 < j, i0 ≤ 0, 0 < j ⊦ j0 + ~(i0 * j + j0) % j = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: mod-less-01: ⊦ 0 < j → i % j < j; from specification int-div%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%negle-07: ⊦ i ≤ ~ j ↔ i + j ≤ 0; from specification int-abs%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%neg-15: ⊦ i + ~ j = i - j; from specification int-abs%mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j); from specification int-div%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%ax-04: ⊦ i + 0 = i; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%Mult-zero: ⊦ i * 0 = 0; from specification int-mult