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Proof step 10 from proof Extsimple (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ 0 < i ∧ j < 0 → rational(i, j).num = ~(i / gcd(i, j))
Proof step 12:
¬(0 < i1 ∧ j1 < 0), rational(i1, j1).num = rational(i2, j2).num, j1 / gcd(i1, j1) + abs(j2) / gcd(i2, j2) = 0, j1 < 0 ⊦ j2 = 0, j1 = 0, i2 = 0, i1 = 0, abs(i1) / gcd(i1, j1) = abs(i2) / gcd(i2, j2) ∧ (¬ 0 < i1 ∧ (¬ i2 < 0 ∧ ¬ j2 < 0 ∨ ¬ 0 < i2 ∧ ¬ 0 < j2) ∨ ¬ i1 < 0 ∧ (¬ i2 < 0 ∧ ¬ 0 < j2 ∨ ¬ 0 < i2 ∧ ¬ j2 < 0))
Proof step 31:
0 < i1 ∧ j1 < 0, ~(i1 / gcd(i1, j1)) = rational(i2, j2).num, j1 / gcd(i1, j1) + abs(j2) / gcd(i2, j2) = 0, j1 < 0 ⊦ j2 = 0, j1 = 0, i2 = 0, i1 = 0, abs(i1) / gcd(i1, j1) = abs(i2) / gcd(i2, j2) ∧ (¬ 0 < i1 ∧ (¬ i2 < 0 ∧ ¬ j2 < 0 ∨ ¬ 0 < i2 ∧ ¬ 0 < j2) ∨ ¬ i1 < 0 ∧ (¬ i2 < 0 ∧ ¬ 0 < j2 ∨ ¬ 0 < i2 ∧ ¬ j2 < 0))

Rule argument

name: numerator-neg1 rotate: F subst: [j, i] → [j1, i1]

Additional information

Used terms: j1, i1