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Proof step 86 from proof Extsimple (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 87:
⊦ 0 < i ∧ j < 0 → rational(i, j).num = ~(i / gcd(i, j))

Rule argument

name: numerator-neg1 rotate: F subst: [j, i] → [j2, i2]

Additional information

Used simplifier rules: numerator-pos: ⊦ i ≠ 0 ∧ 0 < j → rational(i, j).num = i / gcd(i, j);% ⊦ ~ i = j ↔ i + j = 0;% ⊦ i < 0 → abs(i) = ~ i;% ⊦ gcd(i, j) = 0 ↔ i = 0 ∧ j = 0;%j ≠ 0 ⊦ ~ i / j = ~(i / j);% ⊦ i + ~ j = i - j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i + 0 = i;% ⊦ 0 < j → abs(j) = j;% ⊦ ~ i + i = 0; Used terms: j2, i2