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Proof step 57 from proof isSquare-mult-aux (previous step)

i0 * i0 = i1 * i1 * j, max_prime_divisor(j0) * i0 = j0, j0 ≤ abs(i0), 0 < max_prime_divisor(j0) * i1, 0 < j0, 0 < j, prime(max_prime_divisor(j0)), ∀ j1, j, i. i→n(j1) < i→n(j0) → j1 * j1 = j * i * i ∧ 0 < j1 ∧ 0 < j ∧ 0 < i → i = 0 ∨ isSquare(j) ⊦ max_prime_divisor(j0) = 0, i1 = 0, i0 = 0, prime(j0), isSquare(j)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: Abs-pos seq: ⊦ ¬ j < 0 → abs(j) = j subst: [j] → [i0] flags: rewrite

Additional information

Used simplifier rules: prime-not-neg: ⊦ j < 0 → ¬ prime(j);%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%negle-07: ⊦ i ≤ ~ j ↔ i + j ≤ 0; from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%ls-same-03: ⊦ i < 0 → (i < i * j ↔ j < 1); from specification int-mult%lsone: ⊦ i < 1 ↔ i ≤ 0; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%zero-gr-01: ⊦ i < 0 → (0 < i * j ↔ j < 0); from specification int-mult Used terms: i0