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Proof step 2 from proof coef-rec-add-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j))
Proof step 4:
0 ≤ j ∧ j ≤ i, j ≤ i, 0 < j, 0 < i ⊦ fac(i) / (fac(i - j) * fac(j)) = binom_coef(i - 1, j - 1) + binom_coef(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: lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: j, i