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Proof step 3 from proof modeq-pos (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
j ≠ 0 ⊦ i % j = i - i / j * j
Proof step 5:
0 < j, j0 < j, 0 ≤ i0, 0 ≤ j0 ⊦ i0 * j + j0 - (i0 * j + j0) / j * j = j0, j = 0

Rule argument

spec: <toplevel> name: mod-rew seq: j ≠ 0 ⊦ i % j = i - i / j * j subst: [j, i] → [j, i0 * j + j0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ i * 0 = 0; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i + 0 = i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i ≤ j → ¬ j < i; (s) Used terms: j, i0 * j + j0