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Proof step 54 from proof max_exp-fac-gen (previous step)

(i1 * i + j2) / i = i1, max_exp(i, fac(i1)) = Σ countFun(1, i0 - 1, λ j1. i1 / i ^ j1), 0 < i ^ i0, i1 * i + j2 < i ^ i0, prime(i), 0 ≤ j2, 0 < i, 0 < i1 * i + j2, 0 < i1, 2 ≤ i0, n + 1 < i→n(i0), j2 < i, 1 < i ^ i0, 0 < i0, n→i(n) + 2 ≤ i0, ∀ j0, i2, i0. i→n(j0) < i→n(i1 * i + j2) → prime(i2) ∧ j0 < i2 ^ i0 ∧ 0 < j0 ∧ 0 < i2 ^ i0 ∧ 0 ≤ i0 ∧ 1 < i2 ^ i0 → max_exp(i2, fac(j0)) = Σ countFun(1, i0, λ j1. j0 / i2 ^ j1) ⊦ i1 / i ^ (1 + n→i(n)) = countFun(2, i0, λ j0. (i1 * i + j2) / i ^ j0)[n], max_exp(i, fac(i1)) = Σ countFun(2, i0, λ j0. (i1 * i + j2) / i ^ j0), i0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: intlist inst: name: countFun-at seq: ⊦ n→i(n) ≤ j - i ∧ 0 ≤ i ∧ i ≤ j → countFun(i, j, intfun)[n] = intfun(i + n→i(n)) subst: [n, intfun, j, i] → [n, λ j0. (i1 * i + j2) / i ^ j0, i0, 2] flags: rewrite

Additional information

Used simplifier rules: sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs Used terms: n, λ j0. (i1 * i + j2) / i ^ j0, i0, 2