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Proof step 27 from proof middle-coef-max_exp (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ i0 * Σ countFun(i, j, intfun) = Σ countFun(i, j, λ j0. i0 * intfun(j0))
Proof step 29:
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 j0. (2 * j) / i ^ j0) - Σ countFun(1, i0, λ j0. 2 * j / i ^ j0)), 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), i ^ i0 < i ^ (Σ countFun(1, i0, λ j0. (2 * j) / i ^ j0) - Σ countFun(1, i0, λ j0. 2 * j / i ^ j0)) ⊦ Σ countFun(1, i0, λ j0. (2 * j) / i ^ j0) = Σ countFun(1, i0, λ j0. 2 * j / i ^ j0)

Rule argument

spec: intlist inst: name: countFun-mult seq: ⊦ i0 * Σ countFun(i, j, intfun) = Σ countFun(i, j, λ j0. i0 * intfun(j0)) subst: [intfun, j, i, i0] → [λ j0. j / i ^ j0, i0, 1, 2] flags: rewrite

Additional information

Used terms: λ j0. j / i ^ j0, i0, 1, 2