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Proof step 3 from proof modmod-add2-negpos-neg (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 5:
i0 = i1 * j + j0, 0 < j, i < 0, i + i0 < 0, j0 < j, 0 ≤ i0, 0 ≤ j0, 0 ≤ i1 ⊦ (i + j0) % j = (i + i0) % j

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] → [i0, j, i1, j0] flags: <rotated> <all paths>

Additional information

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