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Proof step 5 from proof modmod-add (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 6:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j)
Proof step 7:
j0 + i0 = i2 * j + j1 ∧ 0 ≤ j1 ∧ j1 < j ∧ 0 ≤ j0 + i0, 0 ≤ i0, 0 ≤ j0, j0 < j, 0 ≤ i1, 0 < j ⊦ j1 = ((i1 * j + j0) + i0) % j

Rule argument

spec: int-div inst: name: div-elim-pos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [j0 + i0, j, i, j1] flags: rewrite, current

Additional information

Used simplifier rules: lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%lels-04: ⊦ i ≤ j → (¬ ¬ j < i ↔ false); from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: j0 + i0, j, i, j1