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Proof step 7 from proof mod-add-zero (previous step)

i % j + j * i1 = 0, 0 < j, 0 < i1, 0 ≤ i ⊦ i1 = 0

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 8:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 9:
j0 + j * i1 = 0, i = i0 * j + j0, 0 < i1, 0 < j, j0 < j, 0 ≤ i, 0 ≤ j0, 0 ≤ i0 ⊦ i1 = 0

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, j0] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: i, j, i0, j0