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Proof step 14 from proof coef-le-2pot (previous step)

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 ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
∃ i1, j1. i0 ≤ i1 ∧ i1 ≤ j0 ∧ intfun(i1) = i ∧ i0 ≤ j1 ∧ j1 ≤ j0 ∧ intfun(j1) = j ∧ i1 ≠ j1, ∀ i1. i0 ≤ i1 ∧ i1 ≤ j0 → 0 ≤ intfun(i1), 0 ≤ i0 ⊦ ¬ Σ countFun(i0, j0, intfun) < i + j
Proof step 16:
¬ ( (∃ 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 ⊦

Rule argument

spec: intlist inst: name: sum-countfun-not-ls seq: ∃ i1, j1. i0 ≤ i1 ∧ i1 ≤ j0 ∧ intfun(i1) = i ∧ i0 ≤ j1 ∧ j1 ≤ j0 ∧ intfun(j1) = j ∧ i1 ≠ j1, ∀ i1. i0 ≤ i1 ∧ i1 ≤ j0 → 0 ≤ intfun(i1), 0 ≤ i0 ⊦ ¬ Σ countFun(i0, j0, intfun) < i + j subst: [j, i, intfun, j0, i0] → [binom_coef(2 * j + 1, j + 1), binom_coef(2 * j + 1, j), λ j0. binom_coef(2 * j + 1, j0), 2 * j + 1, 0] flags: rewrite

Additional information

Used terms: binom_coef(2 * j + 1, j + 1), binom_coef(2 * j + 1, j), λ j0. binom_coef(2 * j + 1, j0), 2 * j + 1, 0