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Proof step 7 from proof Extsimple (previous step)

rational(i1, j1).num = rational(i2, j2).num, abs(j1) / gcd(i1, j1) = abs(j2) / gcd(i2, j2) ⊦ j2 = 0, j1 = 0, i2 = 0, i1 = 0, abs(i1) / gcd(i1, j1) = abs(i2) / gcd(i2, j2) ∧ ( (¬ i1 < 0 ∧ ¬ j1 < 0 ∨ ¬ 0 < i1 ∧ ¬ 0 < j1) ∧ (¬ i2 < 0 ∧ ¬ j2 < 0 ∨ ¬ 0 < i2 ∧ ¬ 0 < j2) ∨ (¬ i1 < 0 ∧ ¬ 0 < j1 ∨ ¬ 0 < i1 ∧ ¬ j1 < 0) ∧ (¬ i2 < 0 ∧ ¬ 0 < j2 ∨ ¬ 0 < i2 ∧ ¬ j2 < 0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ i ≠ 0 ∧ 0 < j → rational(i, j).num = i / gcd(i, j)
Proof step 9:
¬(i1 ≠ 0 ∧ 0 < j1), rational(i1, j1).num = rational(i2, j2).num, abs(j1) / gcd(i1, j1) = abs(j2) / gcd(i2, j2) ⊦ j2 = 0, j1 = 0, i2 = 0, i1 = 0, abs(i1) / gcd(i1, j1) = abs(i2) / gcd(i2, j2) ∧ ( (¬ i1 < 0 ∧ ¬ j1 < 0 ∨ ¬ 0 < i1 ∧ ¬ 0 < j1) ∧ (¬ i2 < 0 ∧ ¬ j2 < 0 ∨ ¬ 0 < i2 ∧ ¬ 0 < j2) ∨ (¬ i1 < 0 ∧ ¬ 0 < j1 ∨ ¬ 0 < i1 ∧ ¬ j1 < 0) ∧ (¬ i2 < 0 ∧ ¬ 0 < j2 ∨ ¬ 0 < i2 ∧ ¬ j2 < 0))
Proof step 48:
i1 ≠ 0 ∧ 0 < j1, i1 / gcd(i1, j1) = rational(i2, j2).num, abs(j1) / gcd(i1, j1) = abs(j2) / gcd(i2, j2) ⊦ j2 = 0, j1 = 0, i2 = 0, i1 = 0, abs(i1) / gcd(i1, j1) = abs(i2) / gcd(i2, j2) ∧ ( (¬ i1 < 0 ∧ ¬ j1 < 0 ∨ ¬ 0 < i1 ∧ ¬ 0 < j1) ∧ (¬ i2 < 0 ∧ ¬ j2 < 0 ∨ ¬ 0 < i2 ∧ ¬ 0 < j2) ∨ (¬ i1 < 0 ∧ ¬ 0 < j1 ∨ ¬ 0 < i1 ∧ ¬ j1 < 0) ∧ (¬ i2 < 0 ∧ ¬ 0 < j2 ∨ ¬ 0 < i2 ∧ ¬ j2 < 0))

Rule argument

name: numerator-pos rotate: F subst: [j, i] → [j1, i1]

Additional information

Used terms: j1, i1