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Proof step 25 from proof coef-le-2pot (previous step)

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)

User applied rule cut formula

And obtained the following new goals

Proof step 26:
2 * j - j = j, 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

fma: 2 * j - j = j

Additional information

Used simplifier rules: sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%add-mult-two: ⊦ i + i = 2 * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%eq-pos: ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: 2 * j - j = j