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Proof step 30 from proof coef-rec-add-01 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: fac-zero-02: ⊦ fac(i) ≠ 0;%mod-less-01: ⊦ 0 < j → i % j < j; from specification int-div%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%ax-04: ⊦ i + 0 = i; from specification int-basic2%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%zero-ge-02: ⊦ 0 ≤ i → (0 ≤ i * j ↔ ¬ ¬(i = 0 ∨ 0 ≤ j)); from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%Sub-def: ⊦ i + (j - i) = j; from specification int-abs