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Proof step 2 from proof div-elim-negpos (previous step)

0 < j, i < 0 ⊦ i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
i % j < 0, 0 < j, i < 0, i < j, j + i % j ≤ 0 ⊦ i % j = 0

Rule argument

Additional information

Used simplifier rules: divmod-rew: j ≠ 0 ⊦ i / j * j + i % j = i;%div-lezero: ⊦ 0 < j → (i / j ≤ 0 ↔ i < j);%div-mult-neg: ⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) / j = i;%mod-mult-neg: ⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) % j = i0;%mod-grz: ⊦ 0 < j ∧ i < 0 → ¬ 0 < i % j;%mod-less-01: ⊦ 0 < j → i % j < j;% ⊦ i + j < j ↔ i < 0; (s)% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ 0 < abs(i) ↔ i ≠ 0; (s)%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%forward-le: ⊦ i ≤ j ∧ 0 < i → 0 < j; (forward) from specification int-abs% ⊦ i < 0 ∧ 0 < j → ¬ j ≤ i; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ ¬ abs(i) < 0; (s)