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Proof step 54 from proof square-neq-any-pos-02 (previous step)

j * j / gcd(j0 * j1, j) = (j0 * j1) * (j0 * j1) / gcd(j0 * j1, j) * i, j / gcd(j, j0 * j1) * j / gcd(j0 * j1, j) = (j0 * j1) / gcd(j0 * j1, j) * (j0 * j1) / gcd(j0 * j1, j) * i, gcd(j, j0) = 1, 0 < j, 1 < j1 * j1 * i, 0 < j1, 0 < i, 0 < j0, 0 < j / gcd(j, j0 * j1), (j0 * j1) / gcd(j0 * j1, j) < j, (j0 * j1) / gcd(j0 * j1, j) < j1, (j0 * j1) / gcd(j0 * j1, j) < i, (j0 * j1) / gcd(j0 * j1, j) < j0, (j0 * j1) / gcd(j0 * j1, j) < j / gcd(j, j0 * j1), prime(j1), j1 * j1 ≤ j1 * j1 * i, (j0 * j1) / gcd(j0 * j1, j) ≤ 0, ∀ i1, j, j0. i→n(i1) < i→n(j1 * j1 * i) → j * j = i1 * j0 * j0 ∧ gcd(j, j0) = 1 ∧ 0 < j0 ∧ 0 < j ∧ 0 < i1 → (∃ i2. i1 = i2 * i2) ⊦ gcd(j0 * j1, j) = 1, j = 0, j0 = 0, j1 = 0, ∃ i1. j1 * j1 * i = i1 * i1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div name: div-lezero seq: ⊦ 0 < j → (i / j ≤ 0 ↔ i < j) subst: [j, i] → [gcd(j0 * j1, j), j0 * j1] flags: <all paths>

Additional information

Used simplifier rules: gcd-less-eq-zero: ⊦ gcd(i, j) ≤ 0 ↔ gcd(i, j) = 0;%gcd-zero-zero: ⊦ gcd(i, j) = 0 ↔ i = 0 ∧ j = 0;% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i * j = 0 ↔ i = 0 ∨ j = 0; (ws)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: gcd(j0 * j1, j), j0 * j1