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Proof step 5 from proof mult-02 (previous step)

0 < j0, i0 < j, 0 ≤ i0, j1 < j0, 0 < j, 0 ≤ i1, 0 ≤ j1 ⊦ ((i1 * j0 + j1) * j + i0) / (j0 * j) = i1

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:
(i1 * j0 + j1) * j + i0 = i2 * j0 * j + j2 ∧ 0 ≤ j2 ∧ j2 < j0 * j ∧ 0 < j0 * j ∧ 0 ≤ (i1 * j0 + j1) * j + i0, 0 < j0, i0 < j, 0 ≤ i0, j1 < j0, 0 < j, 0 ≤ i1, 0 ≤ j1 ⊦ i2 = i1

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] → [(i1 * j0 + j1) * j + i0, j0 * j, i, j2] flags: rewrite, current

Additional information

Used simplifier rules: zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); 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-add: ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); 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 Used terms: (i1 * j0 + j1) * j + i0, j0 * j, i, j2