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Proof step 11 from proof div-add-pos (previous step)

0 < j, j0 < j, 0 ≤ i2, 0 ≤ i1, 0 ≤ j0 ⊦ ((i2 + i1) * j + j0) / j = i1 + i2

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: diveq-pos seq: 0 ≤ i0 * j + j0, 0 ≤ j0, j0 < j ⊦ (i0 * j + j0) / j = i0 subst: [j0, j, i0] → [j0, j, i1 + i2] flags: <all paths>

Additional information

Used simplifier rules: ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); (s)%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% ⊦ i ≤ j ↔ ¬ j < i; (ss)%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% ⊦ i + j < j ↔ i < 0; (s)% ⊦ i ≤ j → (j < i ↔ false); (as)%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: j0, j, i1 + i2