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Proof step 3 from proof middle-coef-max_exp-le (previous step)

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

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 4:
∀ i. i ∈ ints → i ≤ i0 ⊦ Σ ints ≤ i0 * n→i(# ints)
Proof step 5:
i ^ i0 ≤ 2 * j ∧ 2 * j < i ^ (i0 + 1) ∧ 0 ≤ i0 ∧ prime(i) ∧ 0 < j, i ^ i0 ≤ 2 * j, 2 * j < i ^ (i0 + 1), 0 ≤ i0, prime(i), 0 < j ⊦ ∀ i1. i1 ∈ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) → i1 ≤ 1, Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) ≤ i0
Proof step 14:
Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) ≤ 1 * n→i(# countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0)), i ^ i0 ≤ 2 * j ∧ 2 * j < i ^ (i0 + 1) ∧ 0 ≤ i0 ∧ prime(i) ∧ 0 < j, i ^ i0 ≤ 2 * j, 2 * j < i ^ (i0 + 1), 0 ≤ i0, prime(i), 0 < j ⊦ Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) ≤ i0

Rule argument

spec: intlist inst: name: sum-all-elems-le seq: ∀ i. i ∈ ints → i ≤ i0 ⊦ Σ ints ≤ i0 * n→i(# ints) subst: [ints, i0] → [countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0), 1] flags: []

Additional information

Used terms: countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0), 1