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Proof step 5 from proof modex (previous step)

i % j = 0 ⊦ j = 0, 0 < j ∧ 0 ≤ i, ∃ j0. i = j0 * j

User applied rule insert elim lemma

And obtained the following new goals

Proof step 6:
⊦ j < 0 ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j))
Proof step 7:
i % j = 0 ⊦ j < 0 ∧ 0 ≤ i, j = 0, 0 < j ∧ 0 ≤ i, ∃ j0. i = j0 * j

Rule argument

spec: int-div inst: name: div-elim1 seq: ⊦ j < 0 ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j)) subst: [i, j, i0, j0] → [i, j, i0, j1] flags: rewrite, current

Additional information

Used simplifier rules: abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%neg: ⊦ 0 < ~ i ↔ i < 0; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-ge-01: ⊦ i < 0 → (0 ≤ i * j ↔ j ≤ 0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%eq-neg: ⊦ i < 0 → (i * j = i * j0 ↔ j = j0); from specification int-mult Used terms: i, j, i0, j1