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Proof step 6 from proof mod-add-01 (previous step)

j0 < i, i0 < i, i0 * i + j0 < 0, 0 < i, abs(j0) < i, i * (i0 + 1) + j0 ≤ 0, j0 ≤ 0, i0 ≤ 0 ⊦ (i * (i0 + 1) + j0) % i = j0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 7:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0)
Proof step 8:
abs(j0) < i, 0 < i, i0 * i + j0 < 0, i0 < i, j0 < i, i0 ≤ 0, j0 ≤ 0, i * (i0 + 1) + j0 ≤ 0 ⊦ (i * (i0 + 1) + j0) % i = j0, 0 < i ∧ i * (i0 + 1) + j0 < 0
Proof step 10:
i * (i0 + 1) + j0 = i1 * i + j1, abs(j0) < i, 0 < i, i0 * i + j0 < 0, i0 < i, j0 < i, abs(j1) < i, i * (i0 + 1) + j0 < 0, i0 ≤ 0, j0 ≤ 0, i * (i0 + 1) + j0 ≤ 0, j1 ≤ 0, i1 ≤ 0 ⊦ j1 = j0

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] → [i * (i0 + 1) + j0, i, i1, j] flags: <rotated> <all paths>

Additional information

Used terms: i * (i0 + 1) + j0, i, i1, j