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Proof step 9 from proof mult-split-mod (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: div-add-pos seq: ⊦ i0 % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [j, (i1 * j) * i0, j0 * i0] flags: rewrite, current

Additional information

Used simplifier rules: mod-mult-04: j ≠ 0 ⊦ (i * j) % j = 0;%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%commutativity: ⊦ i * j = j * i; from specification int-mult%zero-ge-01: ⊦ i < 0 → (0 ≤ i * j ↔ j ≤ 0); from specification int-mult%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%eq: ⊦ i = i + j ↔ j = 0; from specification int-basic2%zero-eq: ⊦ i * j = 0 ↔ i = 0 ∨ j = 0; from specification int-mult%zero-ge-02: ⊦ 0 ≤ i → (0 ≤ i * j ↔ ¬ ¬(i = 0 ∨ 0 ≤ j)); from specification int-mult%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult Used terms: j, (i1 * j) * i0, j0 * i0