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Proof step 12 from proof coef-fac-divides-h (previous step)

fac(j) * fac(i - 1 + j) | fac(i - 1), j < i, fac(j) * (i - j) * fac(i - j + 1) | j * fac(i - 1), fac(j - 1) * (i - j) * fac(i - j + 1) | fac(i - 1), 0 < j, 0 ≤ j, 1 < i, 0 < i ⊦ fac(j) * (i - j) * fac(i - j + 1) | (i - j) * fac(i - 1), fac(j) * (i - j) * fac(i - j + 1) | fac(i)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ i * j = j * i
Proof step 14:
fac(j) * fac(i - 1 + j) | fac(i - 1), j < i, ((i - j) * fac(i - j + 1)) * fac(j) | j * fac(i - 1), fac(j - 1) * (i - j) * fac(i - j + 1) | fac(i - 1), 0 < j, 0 ≤ j, 1 < i, 0 < i ⊦ ((i - j) * fac(i - j + 1)) * fac(j) | (i - j) * fac(i - 1), ((i - j) * fac(i - j + 1)) * fac(j) | fac(i)

Rule argument

spec: int-mult inst: name: commutativity seq: ⊦ i * j = j * i subst: [j, i] → [(i - j) * fac(i - j + 1), fac(j)] flags: rewrite

Additional information

Used terms: (i - j) * fac(i - j + 1), fac(j)