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Proof step 24 from proof coef-le-2pot (previous step)

0 ≤ j + 1 ∧ j + 1 ≤ 2 * j + 1, 2 ^ (2 * j) * 2 = Σ countFun(0, 2 * j + 1, λ j0. binom_coef(2 * j + 1, j0)), 0 < j, 0 ≤ j, 2 ^ (2 * j) < fac(2 * j + 1) / (fac(j + 1) * fac(j)) ⊦ fac(2 * j + 1) / (fac(j + 1) * fac(j)) = fac(2 * j + 1) / (fac(2 * j + 1 - j + 1) * fac(j + 1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 25:
2 ^ (2 * j) * 2 = Σ countFun(0, 2 * j + 1, λ j0. binom_coef(2 * j + 1, j0)), 0 < j, 0 ≤ j, 2 ^ (2 * j) < fac(2 * j + 1) / (fac(j + 1) * fac(j)) ⊦ fac(2 * j + 1) / (fac(j + 1) * fac(j)) = fac(2 * j + 1) / (fac(2 * j - j) * fac(j + 1)), fac(j + 1) * fac(j) = fac(2 * j - j) * fac(j + 1)

Rule argument

Additional information

Used simplifier rules: fac-zero-02: ⊦ fac(i) ≠ 0;%neq-not-zero: ⊦ i * j ≠ i * j0 → i ≠ 0; (forward) from specification int-mult%neqf: ⊦ i / j ≠ i / i0 → j ≠ i0; (forward) from specification int-div%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%le-same-01: ⊦ 0 < i → (i ≤ i * j ↔ 1 ≤ j); from specification int-mult%le-add: ⊦ i + j ≤ i0 + j ↔ i ≤ i0; from specification int-abs%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%sub-add-06: ⊦ i + j - i0 + j = i - i0; from specification int-abs