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Proof step 10 from proof coef-mult-rec-lower (previous step)

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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j))
Proof step 12:
0 ≤ j - 1 ∧ j - 1 ≤ i, 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 = fac(i) / (fac(i - (j - 1)) * fac(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 - 1, i] 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%le-add-03: ⊦ i ≤ i0 ∧ j0 ≥ 0 → i ≤ i0 + j0; from specification int-abs Used terms: j - 1, i