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Proof step 4 from proof coef-le-2pot (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
2 ^ (2 * j + 1) = Σ countFun(0, 2 * j + 1, λ j0. binom_coef(2 * j + 1, j0)), 2 ^ (2 * j) < binom_coef(2 * j + 1, j), 0 < j, 0 ≤ j ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%one: ⊦ 1 * i = i; from specification int-mult%one-01: ⊦ 1 ^ j = 1; from specification int-pot%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-ge-add: ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); from specification int-mult