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Proof step 8 from proof coef-rec-add-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: inst: name: binom-def seq: ⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j)) subst: [j, i] → [j, i - 1] flags: rewrite, current

Additional information

Used simplifier rules: fac-zero: ⊦ i ≤ 0 → fac(i) = 1;%fac-zero-02: ⊦ fac(i) ≠ 0;%coef-eq-zero: ⊦ binom_coef(i, j) = 0 ↔ j < 0 ∨ i < j;%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%succ-01: ⊦ i + 1 ≤ j ↔ i < j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%sub-eq-01: ⊦ i = j → i - j = 0; from specification int-abs%one: ⊦ 1 * i = i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%div-same: i ≠ 0 ⊦ i / i = 1; from specification int-div%numadd-01: ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0; from specification int-abs%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%sub-ls: ⊦ i - j < i0 ↔ i < j + i0; from specification int-abs%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs Used terms: j, i - 1