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Proof step 11 from proof div-add1-posneg-pos-01 (previous step)

0 < j, j0 < 0, i1 * j + j0 < 0, 0 < j + j0, i1 < j + j0, i1 < j, i1 < i0 * j, i1 < i0, 0 < i0, 0 ≤ i0, 0 ≤ j * (i1 + i0) + j0, i1 ≤ 0 ⊦ i0 + i1 = 1 + (j * (i1 + i0) + j0) / j, j0 = 0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 12:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 13:
j * (i1 + i0) + j0 = i2 * j + j1, 0 < i0, i1 < i0, i1 < i0 * j, i1 < j, i1 < j + j0, 0 < j + j0, i1 * j + j0 < 0, j0 < 0, 0 < j, j1 < j, i1 ≤ 0, 0 ≤ j * (i1 + i0) + j0, 0 ≤ i0, 0 ≤ j1, 0 ≤ i2 ⊦ i0 + i1 = 1 + i2, j0 = 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] → [j * (i1 + i0) + j0, j, i, j1]

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: j * (i1 + i0) + j0, j, i, j1