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Proof step 14 from proof middle-coef-larger-strongest (previous step)

0 ≤ 2 * i, ∀ i0. i0 ∈ binom_coef(2 * i, 0) ' + countFun(0 + 1, 2 * i, λ j. binom_coef(2 * i, j)) → i0 ≤ binom_coef(2 * i, i), binom_coef(2 * i, i) ∈ binom_coef(2 * i, 0) ' + countFun(0 + 1, 2 * i, λ j. binom_coef(2 * i, j)), 4 ^ i = Σ(binom_coef(2 * i, 0) ' + countFun(0 + 1, 2 * i, λ j. binom_coef(2 * i, j))), binom_coef(2 * i, i) * 2 * i < 4 ^ i, 0 < i, 0 ≤ i ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: coef-zero: ⊦ 0 ≤ i → binom_coef(i, 0) = 1;%in-17: ⊦ i ∈ i0 ' ↔ i = i0; from specification list[intlista]%in-append: ⊦ i ∈ ints + ints0 ↔ ¬ ¬(i ∈ ints ∨ i ∈ ints0); from specification list[intlista]%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%sum-rec: ⊦ Σ(i ' + ints) = i + Σ ints; from specification intlist%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2