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Proof step 22 from proof max_prime_divisor-nls-02 (previous step)

0 < j, 1 < j0, 0 < j0, j0 < i0, j < j0, 0 < i0, prime(j), prime(j0), 2 ≤ j * i0, 2 ≤ i0, 0 ≤ i0, 0 ≤ j, ∀ j1. j0 < j1 ∧ j1 < i0 ∧ prime(j1) → ¬ j1 | i0, ∀ j0. j < j0 ∧ j0 < j * i0 ∧ prime(j0) → ¬ j0 | j * i0, ∃ i1. j0 * i1 = i0 ⊦ prime(i0), prime(j * i0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 23:
j < j0, 0 < j, 1 < j0, 0 < j0, 1 < i, 0 < i, prime(j), prime(j0), 0 ≤ i, 2 ≤ j * j0 * i, 2 ≤ j0 * i, 0 ≤ j, ∀ j1. j0 < j1 ∧ j1 < j0 * i ∧ prime(j1) → ¬ j1 | j0 * i, ∀ j1. j < j1 ∧ j1 < j * j0 * i ∧ prime(j1) → ¬ j1 | j * j0 * i ⊦ prime(j0 * i), prime(j * j0 * i)

Rule argument

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le: ⊦ i ≤ j ∧ 0 < i → 0 < j; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1% ⊦ 0 < i → (i < i * j ↔ 1 < j); (s)% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)