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Proof step 8 from proof mod-pred-neg-01 (previous step)

i0 * j + j0 = i1 + 1, j0 < j, i0 < j, 0 < j, abs(j0) < j, i1 < ~1, i1 < 0, j0 ≤ 0, i0 ≤ 0 ⊦ i1 % j = 0, j0 = i1 % j + 1

User applied rule insert elim lemma

And obtained the following new goals

Proof step 9:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0)
Proof step 10:
i0 * j + j0 = i1 + 1, i1 = i2 * j + j1, i1 < 0, i1 < ~1, abs(j0) < j, 0 < j, i0 < j, j0 < j, abs(j1) < j, i0 ≤ 0, j0 ≤ 0, j1 ≤ 0, i2 ≤ 0 ⊦ j0 = j1 + 1, j1 = 0

Rule argument

spec: <toplevel> name: div-elim-negpos seq: ⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0) 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