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Proof step 11 from proof coef-le-2pot (previous step)

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 ⊦

User applied rule cut formula

And obtained the following new goals

Proof step 12:
2 * binom_coef(2 * j + 1, j + 1) = binom_coef(2 * j + 1, j) + binom_coef(2 * j + 1, j + 1), 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 * binom_coef(2 * j + 1, j + 1) = binom_coef(2 * j + 1, j) + binom_coef(2 * j + 1, j + 1)

Additional information

Used simplifier rules: numeq-03: ⊦ (* i2) * i = i + j0 ↔ (i2 - 1) * i = j0; from specification int-mult%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%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used formulas: 2 * binom_coef(2 * j + 1, j + 1) = binom_coef(2 * j + 1, j) + binom_coef(2 * j + 1, j + 1)