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Proof step 3 from proof mod-mod-less (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j)

Rule argument

spec: <toplevel> name: modex seq: j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j) subst: [j, i] → [i, j] flags: <all paths>

Additional information

Used simplifier rules: modmod-mult-03: j ≠ 0 ⊦ 0 < j0 → (i % (j * j0)) % j = i % j;% ⊦ 0 < i → (i < i * j ↔ 1 < j); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s) Used terms: i, j