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Proof step 11 from proof coef-fac-divides-h (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
fac(j) * fac(i - 1 + j) | fac(i - 1), j < i, fac(j) * (i - j) * fac(i - j + 1) | j * fac(i - 1), fac(j - 1) * (i - j) * fac(i - j + 1) | fac(i - 1), 0 < j, 0 ≤ j, 1 < i, 0 < i ⊦ fac(j) * (i - j) * fac(i - j + 1) | (i - j) * fac(i - 1), fac(j) * (i - j) * fac(i - j + 1) | fac(i)

Rule argument

Additional information

Used simplifier rules: forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%sub-sub: ⊦ 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%ax-04: ⊦ i + 0 = i; from specification int-basic2%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs