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Proof step 2 from proof mod-mod-less (previous step)

j % i = 0, i < j, 0 < i, 1 < j, 0 < j, 0 ≤ i1 ⊦ (i1 % j) % i = i1 % i

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j)

Rule argument

spec: int-div inst: name: modex seq: j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j) subst: [j, i] → [i, j] flags: rewrite, current

Additional information

Used simplifier rules: modmod-mult-04: ⊦ 0 < j ∧ 0 ≤ i ∧ 0 < j0 → (i % (j * j0)) % j = i % j;%ls-same-01: ⊦ 0 < i → (i < i * j ↔ 1 < j); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult Used terms: i, j