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Proof step 4 from proof est-4-big-02 (previous step)

0 ≤ j0, j ≤ 2 * j0, 2 < i0 * j + j0, j ≤ i0 * j + j0, (2 * (i0 * j + j0)) / 3 < j, j0 < j, 0 < i0 * j + j0, (2 * (i0 * j + j0)) / 3 < 2 * j0, 0 ≤ i0, 1 < j, 0 < j, 0 < i0, 0 < j0 ⊦ j = 3, j = 2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ 0 < j ∧ 0 ≤ i → (i / (* j) < j0 ↔ i < j * j0)
Proof step 6:
0 < 3 ∧ 0 ≤ 2 * (i0 * j + j0), 0 ≤ j0, j ≤ 2 * j0, 2 < i0 * j + j0, j ≤ i0 * j + j0, 2 * (i0 * j + j0) < 3 * j, j0 < j, 0 < i0 * j + j0, (2 * (i0 * j + j0)) / 3 < 2 * j0, 0 ≤ i0, 1 < j, 0 < j, 0 < i0, 0 < j0 ⊦ j = 3, j = 2

Rule argument

spec: int-div inst: name: num-ls seq: ⊦ 0 < j ∧ 0 ≤ i → (i / (* j) < j0 ↔ i < j * j0) subst: [j0, j, i] → [j, 3, 2 * (i0 * j + j0)] flags: rewrite

Additional information

Used simplifier rules: zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%zero-ge-add: ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%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%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: j, 3, 2 * (i0 * j + j0)