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Proof step 5 from proof auxexdivmod (previous step)

∀ i0, j, i. i0 * j ≤ i ∧ i < (i0 + i2) * j ∧ 0 ≤ i0 ∧ 0 < j ∧ 0 ≤ i ∧ 0 ≤ i2 → (∃ i0. i0 * j ≤ i ∧ i < (i0 + 1) * j ∧ 0 ≤ i0), i0 * j ≤ i, i < (i0 + i2 +1) * j, 0 ≤ i0, 0 < j, 0 ≤ i, 0 ≤ i2 +1 ⊦ i0 * j ≤ i ∧ i < (i0 + i2) * j ∧ 0 ≤ i0 ∧ 0 < j ∧ 0 ≤ i ∧ 0 ≤ i2, (i0 + i2) * j ≤ i ∧ i < ((i0 + i2) + 1) * j ∧ 0 ≤ i0 + i2, ∃ i0. i0 * j ≤ i ∧ i < (i0 + 1) * j ∧ 0 ≤ i0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
~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) ⊦ ∃ i1. i1 * j ≤ i ∧ i < i1 * j + j ∧ 0 ≤ i1, i0 * j + i2 * j ≤ i ∧ 0 ≤ i0 + i2, i < i0 * j + i2 * j ∧ 0 ≤ i2

Rule argument

Additional information

Used simplifier rules: dist-lsplit: ⊦ (i + i0) * j = i * j + i0 * j;%one: ⊦ 1 * i = i;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%succ-01: ⊦ i +1 = i + 1; from specification int-basic2