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Proof step 7 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), gcd(i2, j2) = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ gcd(i, j) = 1 ↔ ¬(∃ i1. i1 > 1 ∧ i1 | i ∧ i1 | j)
Proof step 9:
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)

Rule argument

seq: ⊦ gcd(i, j) = 1 ↔ ¬(∃ i1. i1 > 1 ∧ i1 | i ∧ i1 | j) subst: [j, i] → [j2, i2] rotate: F spec: int-gcd inst: name: gcd-one

Additional information

Used terms: j2, i2