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Proof step 4 from proof coef-mult-rec (previous step)

0 ≤ j ∧ j ≤ i, j ≤ i, 0 < j, 0 < i ⊦ i * binom_coef(i - 1, j - 1) = j * fac(i) / (fac(i - j) * fac(j))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j))
Proof step 6:
0 ≤ j - 1 ∧ j - 1 ≤ i - 1, 0 ≤ j ∧ j ≤ i, j ≤ i, 0 < j, 0 < i ⊦ i * fac(i - 1) / (fac(i - 1 - (j - 1)) * fac(j - 1)) = j * fac(i) / (fac(i - j) * fac(j))

Rule argument

spec: int-fac inst: name: binom-def seq: ⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j)) subst: [j, i] → [j - 1, i - 1] flags: rewrite, current

Additional information

Used simplifier rules: sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%sub-le: ⊦ i - j ≤ i0 ↔ i ≤ j + i0; from specification int-abs%Sub-def: ⊦ i + (j - i) = j; from specification int-abs Used terms: j - 1, i - 1