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Proof step 18 from proof div-addthree-rew (previous step)

j2 < j, j1 < j, j0 < j, 0 < j, 0 ≤ j0, 0 ≤ i2, 0 ≤ j1, 0 ≤ i3, 0 ≤ j2, 0 ≤ i0 ⊦ ((i2 + i3 + i0) * j + j0 + j1 + j2) / j = i2 + i3 + i0 + (j0 + j1 + j2) / j, ((i2 * j + j0) + (i3 * j + j1) + i0 * j + j2) / j = i2 + i3 + i0 + (j0 + j1 + j2) / j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: inst: name: div-add1-pospos seq: ⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [j, j0 + j1 + j2, j * (i0 + i3 + i2)] flags: rewrite, current

Additional information

Used simplifier rules: div-mult-pos-01: ⊦ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0 * j) / j = i / j + i0;%div-mult-03: j ≠ 0 ⊦ (i * j) / j = i;%zero-left: ⊦ 0 * i = 0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%eq: ⊦ i = i + j ↔ j = 0; from specification int-basic2%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%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%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%zero-gs: ⊦ 0 ≤ i ∧ 0 ≤ j → ¬ i + j < 0; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; 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%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j, j0 + j1 + j2, j * (i0 + i3 + i2)