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Proof step 40 from proof middle-coef-max_exp (previous step)

∃ j0. 1 ≤ j0 ∧ j0 ≤ i0 ∧ (2 * j) / i ^ j0 - 2 * j / i ^ j0 = i1, 0 < i, 1 < i, 0 < i ^ (i0 + 1), 1 < i ^ (i0 + 1), 0 < fac(2 * j), Σ countFun(1, i0, λ j0. 2 * j / i ^ j0) + i0 < Σ countFun(1, i0, λ j0. (2 * j) / i ^ j0), 0 ≤ i0, 0 < j, 2 * j < i ^ Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0), prime(i), i | fac(2 * j) / (fac(j) * fac(j)), 2 * j < i ^ (i0 + 1), i ^ i0 ≤ 2 * j, 0 ≤ j, fac(j) * fac(j) | fac(2 * j), j < i ^ (i0 + 1), 0 < fac(j), 0 ≤ Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0), 0 < Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0), i0 < Σ countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0), 1 < i1, 0 < i1 ⊦ Σ countFun(1, i0, λ j0. (2 * j) / i ^ j0) = Σ countFun(1, i0, λ j0. 2 * j / i ^ j0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-le: ⊦ i ≤ j ∧ 0 < i → 0 < j; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%ax-04: ⊦ i + 0 = i; from specification int-basic2%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs