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Proof step 14 from proof coef-fac-divides-h (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-gcd inst: name: div-mult-01 seq: ⊦ i ≠ 0 → (i * i0 | i * i1 ↔ i0 | i1) subst: [i1, i0, i] → [fac(i - 1), fac(i - j + 1) * fac(j), i - j] flags: rewrite

Additional information

Used simplifier rules: fac-zero-02: ⊦ fac(i) ≠ 0;%add-c: ⊦ i + j = j + i; from specification int-basic2%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%ax-04: ⊦ i + 0 = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%sub-eq-01: ⊦ i = j → i - j = 0; from specification int-abs%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%div-zero-zero: ⊦ 0 | i ↔ i = 0; from specification int-gcd%zero-neq-01: ⊦ 0 < i → (i * j = 0 ↔ j = 0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: fac(i - 1), fac(i - j + 1) * fac(j), i - j