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Proof step 7 from proof auxexdivmod (previous step)

~1 ≤ i2, i < i0 * j + i2 * j + j, i0 * j ≤ i, 0 < i0 * j + i2 * j + j, 0 ≤ i, 0 < j, 0 ≤ i0, 0 < i2 * j + j, ∀ i1, j0, i3. i1 * j0 ≤ i3 ∧ i3 < i1 * j0 + i2 * j0 ∧ 0 ≤ i1 ∧ 0 < j0 ∧ 0 ≤ i3 ∧ 0 ≤ i2 → (∃ i4. i4 * j0 ≤ i3 ∧ i3 < i4 * j0 + j0 ∧ 0 ≤ i4) ⊦ i < i0 * j + i2 * j, ∃ i1. i1 * j ≤ i ∧ i < i1 * j + j ∧ 0 ≤ i1, i0 * j + i2 * j ≤ i ∧ 0 ≤ i0 + i2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
i0 * j + i2 * j ≤ i, 0 < i2 * j + j, 0 ≤ i0, 0 ≤ i, 0 < i0 * j + i2 * j + j, i0 * j ≤ i, i < i0 * j + i2 * j + j, ~1 ≤ i2, 0 < j, ∀ i1, j0, i3. i1 * j0 ≤ i3 ∧ i3 < i1 * j0 + i2 * j0 ∧ 0 ≤ i1 ∧ 0 < j0 ∧ 0 ≤ i3 ∧ 0 ≤ i2 → (∃ i4. i4 * j0 ≤ i3 ∧ i3 < i4 * j0 + j0 ∧ 0 ≤ i4) ⊦ 0 ≤ i0 + i2, ∃ i1. i1 * j ≤ i ∧ i < i1 * j + j ∧ 0 ≤ i1

Rule argument

Additional information

Used simplifier rules: add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs