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Proof step 8 from proof coef-mult-rec-lower (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j))
Proof step 10:
0 ≤ j ∧ j ≤ i, j ≤ i, 0 < j, 0 < i, fac(i - j) * fac(j) | fac(i), 0 ≤ j, 0 ≤ i, fac(i + 1 - j) * fac(j - 1) | fac(i) ⊦ fac(i) / (fac(i - j) * fac(j)) * j = binom_coef(i, j - 1) * (i + 1 - j)

Rule argument

spec: inst: name: binom-def seq: ⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j)) subst: [j, i] → [j, i] flags: rewrite, current

Additional information

Used simplifier rules: zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs Used terms: j, i