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Proof step 28 from proof coef-mult-rec-lower (previous step)

0 < j, (fac(i - j) * j * fac(j - 1)) * i0 = fac(i), fac(i) = (fac(j - 1) * fac(i + 1 - j)) * i1, 0 < j, 0 ≤ i, 0 ≤ j, 0 < i, j ≤ i ⊦ i0 * j = i1 * (i + 1 - j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ 0 < i → fac(i) = i * fac(i - 1)
Proof step 30:
0 < i + 1 - j, 0 < j, (fac(i - j) * j * fac(j - 1)) * i0 = fac(i), fac(i) = (fac(j - 1) * (i + 1 - j) * fac(i + 1 - j - 1)) * i1, 0 < j, 0 ≤ i, 0 ≤ j, 0 < i, j ≤ i ⊦ i0 * j = i1 * (i + 1 - j)

Rule argument

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

Additional information

Used simplifier rules: sub-ls-01: ⊦ 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%add-c: ⊦ i + j = j + i; from specification int-basic2%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs Used terms: i + 1 - j