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Proof step 9 from proof add-def-02 (previous step)

gcd(i1, j1) = 1, 0 < j1, 0 < j2, ∀ j2, j1, i, i0. i→n(i0) + i→n(i) < i→n(i1) + i→n(i2) → 0 < j1 ∧ 0 < j2 → rational(i0, j1) + rational(i, j2) = rational(i0 * j2 + j1 * i, j1 * j2) ⊦ rational(i1, j1) + rational(i2, j2) = rational(i1 * j2 + j1 * i2, j1 * j2), ∀ i1. ¬(i1 > 1 ∧ i1 | i2 ∧ i1 | j2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
gcd(i1, j1) = 1, 0 < j1, 0 < i, 1 < i, 0 < i3, ∀ j2, j1, i4, i3. i→n(i3) + i→n(i4) < i→n(i1) + i→n(i * i0) → 0 < j1 ∧ 0 < j2 → rational(i3, j1) + rational(i4, j2) = rational(i3 * j2 + j1 * i4, j1 * j2) ⊦ rational(i0, i3) + rational(i1, j1) = rational(i * (j1 * i0 + i1 * i3), j1 * i * i3)

Rule argument

Additional information

Used simplifier rules: factor-rat-03: ⊦ 0 < j ∧ 0 < i1 → rational(i1 * i, i1 * j) = rational(i, j);% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j + i * j0 = i * (j + j0);% ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward)% ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward)% ⊦ i | j ↔ (∃ i0. i * i0 = j);% ⊦ i > j ↔ j < i;