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Proof step 44 from proof middle-coef-max_exp (previous step)

i0 < n→i(# 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) ≤ n→i(# 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), 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), 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, 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), 1 < i, 0 < i ⊦ Σ countFun(1, i0, λ j0. (2 * j) / i ^ j0) = Σ countFun(1, i0, λ j0. 2 * j / i ^ j0), countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ 0 ≤ i ∧ i ≤ j → # countFun(i, j, intfun) = i→n(1 + j - i)
Proof step 46:
¬(0 ≤ 1 ∧ 1 ≤ i0), i0 < n→i(# 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) ≤ n→i(# 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), 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), 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, 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), 1 < i, 0 < i ⊦ Σ countFun(1, i0, λ j0. (2 * j) / i ^ j0) = Σ countFun(1, i0, λ j0. 2 * j / i ^ j0), countFun(1, i0, λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0) = []

Rule argument

spec: intlist inst: name: countFun-length seq: ⊦ 0 ≤ i ∧ i ≤ j → # countFun(i, j, intfun) = i→n(1 + j - i) subst: [intfun, j, i] → [λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0, i0, 1] flags: rewrite

Additional information

Used simplifier rules: leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%numsub: ⊦ (* i) + i0 - (* j) = (i - j) + i0; from specification int-abs%ax-04: ⊦ i + 0 = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%i2i-pos: ⊦ 0 < i → n→i(i→n(i)) = i; from specification intnat%irreflexivity: ⊦ ¬ i < i; from specification oelem[intlista] Used terms: λ i0. (2 * j) / i ^ i0 - 2 * j / i ^ i0, i0, 1