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Proof step 16 from proof coef-le-2pot (previous step)

¬ ( (∃ i1, j1. 0 ≤ i1 ∧ i1 ≤ 2 * j + 1 ∧ binom_coef(2 * j + 1, i1) = binom_coef(2 * j + 1, j) ∧ 0 ≤ j1 ∧ j1 ≤ 2 * j + 1 ∧ binom_coef(2 * j + 1, j1) = binom_coef(2 * j + 1, j + 1) ∧ i1 ≠ j1) ∧ (∀ i1. 0 ≤ i1 ∧ i1 ≤ 2 * j + 1 → 0 ≤ binom_coef(2 * j + 1, i1)) ∧ 0 ≤ 0), Σ countFun(0, 2 * j + 1, λ j0. binom_coef(2 * j + 1, j0)) < binom_coef(2 * j + 1, j) + 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 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 17:
2 ^ (2 * j) * 2 = Σ countFun(0, 2 * j + 1, λ j0. binom_coef(2 * j + 1, j0)), binom_coef(2 * j + 1, j) = binom_coef(2 * j + 1, j + 1), 0 ≤ j, 0 < j, 2 ^ (2 * j) < binom_coef(2 * j + 1, j + 1) ⊦ (∃ i1, j1. 0 ≤ i1 ∧ i1 ≤ 2 * j + 1 ∧ binom_coef(2 * j + 1, i1) = binom_coef(2 * j + 1, j + 1) ∧ 0 ≤ j1 ∧ j1 ≤ 2 * j + 1 ∧ binom_coef(2 * j + 1, j1) = binom_coef(2 * j + 1, j + 1) ∧ i1 ≠ j1) ∧ (0 ≤ i ∧ i ≤ 2 * j + 1 → 0 ≤ binom_coef(2 * j + 1, i))

Rule argument

Additional information

Used simplifier rules: ls-forward: ⊦ i * j < i * j0 → i ≠ 0; (forward) from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); from specification int-mult%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-mult-two: ⊦ i + i = 2 * i; from specification int-mult%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]