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Proof step 34 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), j2 < i, 1 < i ^ i0, 0 ≤ i0, 0 < i ^ i0, i1 * i + j2 < i ^ i0, prime(i), 0 ≤ j2, 0 < i, 0 < i1 * i + j2, 0 < i1, 0 < 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 + max_exp(i, fac(i1)) = (i1 * i + j2) / i + Σ countFun(2, i0, λ j0. (i1 * i + j2) / i ^ j0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 35:
max_exp(i, fac(i1)) = Σ countFun(1, i0 - 1, λ j1. i1 / i ^ j1), (i1 * i + j2) / i = i1, 0 < i0, 0 < i1, 0 < i1 * i + j2, 0 < i, 0 ≤ j2, prime(i), i1 * i + j2 < i ^ i0, 0 < i ^ i0, 0 ≤ i0, 1 < i ^ i0, j2 < i, ∀ 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) ⊦ max_exp(i, fac(i1)) = Σ countFun(2, i0, λ j0. (i1 * i + j2) / i ^ j0)

Rule argument

Additional information

Used simplifier rules: add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2