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Proof step 33 from proof middle-coef-max_exp (previous step)

1 < i ∧ 0 ≤ i0, 0 ≤ 1, 0 < fac(j), j < i ^ (i0 + 1), fac(j) * fac(j) | fac(2 * j), 0 ≤ j, i ^ i0 ≤ 2 * j, 2 * j < i ^ (i0 + 1), i | fac(2 * j) / (fac(j) * fac(j)), prime(i), 2 * j < i ^ Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0), 0 < j, 0 ≤ i0, i ^ i0 < i ^ (i0 + 1), Σ countFun(1, i0, λ j0. 2 * j / i ^ j0) + i0 < Σ countFun(1, i0, λ j0. (2 * j) / i ^ j0), 0 < fac(2 * j), 1 < i ^ (i0 + 1), 0 < i ^ (i0 + 1), i0 < Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) ⊦ Σ countFun(1, i0, λ j0. (2 * j) / i ^ j0) = Σ countFun(1, i0, λ j0. 2 * j / i ^ j0)

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: 0 ≤ Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) ∧ Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) ≤ i0 * 1

Additional information

Used simplifier rules: 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%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs Used formulas: 0 ≤ Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) ∧ Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) ≤ i0 * 1