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Proof step 7 from proof mod-rec (previous step)

i0 * j + j0 = i1 + j, 0 < i1 + j, j0 < i1 + j, 0 < i0, j0 < j, 0 < j, 0 ≤ j0, 0 ≤ i1 ⊦ j0 = i1 % j

User 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)
Proof step 9:
i0 * j + j0 = i1 + j, i1 = i2 * j + j1, 0 < j, j0 < j, 0 < i0, j0 < i1 + j, 0 < i1 + j, j1 < j, 0 ≤ i1, 0 ≤ j0, 0 ≤ j1 ⊦ j0 = j1

Rule argument

spec: <toplevel> name: div-elim-pos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [i1, j, i, j1]

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: i1, j, i, j1