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

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 5:
i = i0 * j + j1, 0 < j, 0 < j0, j1 < j, 0 ≤ i, 0 ≤ j1, 0 ≤ i0 ⊦ i % (j * j0) = j1 + i0 % j0 * j

Rule argument

spec: <toplevel> name: div-elim-pospos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0) subst: [i, j, i0, j0] → [i, j, i0, j1] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: i, j, i0, j1