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Proof step 10 from proof num-div-denom (previous step)

0 < i ∧ i1 < 0, i1 < 0 ⊦ i = 0, i1 = 0, ~(i / gcd(i, i1)) / i1 / gcd(i, i1) + i / i1 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ j ≠ 0 ∧ i ≠ 0 ∧ i % j = 0 → i1 / i / j = (i1 * j) / i

Rule argument

seq: ⊦ j ≠ 0 ∧ i ≠ 0 ∧ i % j = 0 → i1 / i / j = (i1 * j) / i subst: [j, i, i1] → [gcd(i, i1), i1, ~(i / gcd(i, i1))] rotate: F spec: int-div inst: name: div-div-switch

Additional information

Used simplifier rules: ⊦ gcd(i, j) = 0 ↔ i = 0 ∧ j = 0;% ⊦ i + 0 = i;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i + ~ j = i - j;% ⊦ i0 < 0 → i0 / gcd(i, i0) ≠ 0;% ⊦ i ≠ 0 → i % gcd(i, j) = 0;% ⊦ gcd(i, j) = gcd(j, i);% ⊦ ~ i * j = ~(i * j);% ⊦ i ≠ 0 ∧ j ≠ 0 → i / gcd(i, j) * gcd(i, j) = i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;%j ≠ 0 ⊦ ~ i / j = ~(i / j);% ⊦ ~ i + i = 0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i; Used terms: gcd(i, i1), i1, ~(i / gcd(i, i1))