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Proof step 22 from proof square-neq-any-pos-02 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: le-one seq: ⊦ 1 < j → (i * j ≤ 1 ↔ i ≤ 0) subst: [j, i] → [j1, j1] flags: rewrite

Additional information

Used simplifier rules: prime-one: ⊦ i ≤ 1 → ¬ prime(i);%prime-not-neg: ⊦ j < 0 → ¬ prime(j);%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%zero-gr-01: ⊦ i < 0 → (0 < i * j ↔ j < 0); from specification int-mult%zero-ls-01: ⊦ i < 0 → (i * j < 0 ↔ 0 < j); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%le-neg: ⊦ i < 0 → (i * j ≤ i * i0 ↔ i0 ≤ j); from specification int-mult%le-same-02: ⊦ i < 0 → (i * j ≤ i ↔ 1 ≤ j); from specification int-mult%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs Used terms: j1, j1