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Proof step 11 from proof modmod-add-zero (previous step)

(i + i0 % j) % j = 0, 0 < j, i0 < 0 ⊦ (i + i0) % j = 0, 0 = j, 0 ≤ i0, i ≤ 0 ∧ i0 ≤ 0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 12:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0)
Proof step 13:
(i + j0) % j = 0, i0 = i1 * j + j0, i0 < 0, 0 < j, abs(j0) < j, j0 ≤ 0, i1 ≤ 0 ⊦ (i + i0) % j = 0, 0 = j, 0 ≤ i0, i ≤ 0 ∧ i0 ≤ 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] → [i0, j, i1, j0] flags: <rotated> <all paths>

Additional information

Used terms: i0, j, i1, j0