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Proof step 21 from proof coef-rec-add-01 (previous step)

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

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: 1 < j

Additional information

Used simplifier rules: fac-zero: ⊦ i ≤ 0 → fac(i) = 1;%fac-zero-02: ⊦ fac(i) ≠ 0;%fac-one: ⊦ fac(1) = 1;%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%bet: ⊦ 0 < i → (i ≤ 1 ↔ i = 1); 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%one-01: ⊦ i = 1 → i * j = j; from specification int-mult%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%irreflexivity: ⊦ ¬ i < i; from specification oelem[intlista]%div-same: i ≠ 0 ⊦ i / i = 1; from specification int-div%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%div-mult-03: j ≠ 0 ⊦ (i * j) / j = i; from specification int-div%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%mod-same: i ≠ 0 ⊦ i % i = 0; from specification int-div Used formulas: 1 < j