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Proof step 20 from proof coef-rec-add-01 (previous step)

0 < j - 1 ∧ j - 1 ≤ i - 1 → fac(j - 1) * fac(i - 1 - (j - 1)) | fac(i - 1), fac(i - 1) / (fac(i - 1 + j) * fac(j)) = ((i - j) * fac(i - 1)) / (fac(i - j) * fac(j)), fac(i - 1) / (fac(i - j) * fac(j - 1)) = (j * fac(i - 1)) / (fac(i - j) * fac(j)), 0 ≤ j, 0 ≤ i, 0 < j, j ≤ i, j < i, 1 < i, 0 < i, 0 < fac(i - j) * fac(j), 0 ≤ fac(i - 1) ⊦ fac(i) / (fac(i - j) * fac(j)) = (j * fac(i - 1)) / (fac(i - j) * fac(j)) + ((i - j) * fac(i - 1)) / (fac(i - j) * fac(j)), (j * fac(i - 1)) % (fac(i - j) * fac(j)) = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
fac(i - 1) / (fac(i - 1 + j) * fac(j)) = ((i - j) * fac(i - 1)) / (fac(i - j) * fac(j)), fac(i - 1) / (fac(i - j) * fac(j - 1)) = (j * fac(i - 1)) / (fac(i - j) * fac(j)), 0 ≤ j, 0 ≤ i, 0 < j, j ≤ i, j < i, 1 < i, 0 < i, 0 < fac(i - j) * fac(j), 0 ≤ fac(i - 1), 1 < j → fac(j - 1) * fac(i - j) | fac(i - 1) ⊦ fac(i) / (fac(i - j) * fac(j)) = (j * fac(i - 1)) / (fac(i - j) * fac(j)) + ((i - j) * fac(i - 1)) / (fac(i - j) * fac(j)), (j * fac(i - 1)) % (fac(i - j) * fac(j)) = 0

Rule argument

Additional information

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