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Proof step 10 from proof coef-le-2pot (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 11:
2 * 2 ^ (2 * j) < 2 * binom_coef(2 * j + 1, j + 1), binom_coef(2 * j + 1, j) = binom_coef(2 * j + 1, j + 1), 2 ^ (2 * j) * 2 = Σ countFun(0, 2 * j + 1, λ j0. binom_coef(2 * j + 1, j0)), 2 ^ (2 * j) < binom_coef(2 * j + 1, j + 1), 0 < j, 0 ≤ j ⊦

Rule argument

fma: 2 * 2 ^ (2 * j) < 2 * binom_coef(2 * j + 1, j + 1)

Additional information

Used simplifier rules: ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); from specification int-mult%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: 2 * 2 ^ (2 * j) < 2 * binom_coef(2 * j + 1, j + 1)