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Proof step 2 from proof modmod-add2-negpos-pos (previous step)

i < j, i < 0, 0 ≤ i + i0 % j, 0 < j, 0 ≤ i0, i + 1 < j ⊦ (i + i0 % j) % j = (i + i0) % j

User applied rule insert elim lemma

And obtained the following new goals

Proof step 3:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)

Rule argument

spec: int-div inst: 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: rewrite, current

Additional information

Used simplifier rules: mod-mult-pos-02: ⊦ 0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 → (i * j + i0) % j = i0 % j;%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs Used terms: i0, j, i1, j0