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Proof step 35 from proof coef-rec-add-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: inst: name: fac-rec seq: ⊦ 0 < i → fac(i) = i * fac(i - 1) subst: [i] → [j] flags: rewrite, current

Additional information

Used simplifier rules: fac-not-zero: ⊦ fac(i) * fac(j) ≠ 0;%fac-zero: ⊦ i ≤ 0 → fac(i) = 1;%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%sub-zero-01: ⊦ i - 0 = i; from specification int-abs%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%one: ⊦ 1 * i = i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%div-mult-01: j ≠ 0, i0 ≠ 0 ⊦ (i * i0) / (j * i0) = i / j; from specification int-div Used terms: j