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Proof step 8 from proof prime-divides-coef (previous step)

prime(j0), j0 ≤ j * 2 + 1, j + 2 ≤ j0, 0 < j, fac(j + 1) * fac(j) | fac(j * 2 + 1), 0 ≤ j ⊦ j0 | fac(2 * j + 1) / (fac(j + 1) * fac(j))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 10:
prime(j0), j0 ≤ j * 2 + 1, j + 2 ≤ j0, 0 < j, ∃ i0. (fac(j) * fac(j + 1)) * i0 = fac(j * 2 + 1), 0 ≤ j ⊦ j0 | fac(2 * j + 1) / (fac(j + 1) * fac(j))

Rule argument

spec: int-gcd inst: name: divides seq: ⊦ i | j ↔ (∃ i0. i * i0 = j) subst: [j, i] → [fac(j * 2 + 1), fac(j) * fac(j + 1)] flags: rewrite

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: fac(j * 2 + 1), fac(j) * fac(j + 1)