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Proof step 15 from proof div-est-num (previous step)

0 < i / j, 0 < j0, i1 < i / j, 0 < j, i1 < j0, j1 < j, j0 ≤ i / j, i ≤ i1 * j + j1, 0 ≤ j1, 0 ≤ i1 ⊦

User applied rule insert elim lemma

And obtained the following new goals

Proof step 16:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 17:
i = i0 * j + j2, j1 < j, i1 < j0, 0 < j, i1 < i0, 0 < j0, 0 < i0, j2 < j, 0 ≤ i1, 0 ≤ j1, i ≤ i1 * j + j1, j0 ≤ i0, 0 ≤ j2, 0 ≤ i0, 0 ≤ i ⊦

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, j2] flags: <rotated> <all paths>

Additional information

Used simplifier rules: div-zero-ngr: ⊦ i < 0 ∧ 0 < j → ¬ 0 < i / j;% ⊦ i ≤ j ↔ ¬ j < i; (ss) Used terms: i, j, i0, j2