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Proof step 60 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) ⊦ max_exp(i, fac(i1)) = Σ countFun(2, i0, λ j0. (i1 * i + j2) / i ^ j0), i0 = 0, i1 / i ^ (1 + n→i(n)) = (i1 * i + j2) / (i ^ (1 + n→i(n)) * i)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: mult-02 seq: ⊦ 0 ≤ i0 ∧ i0 < j ∧ 0 ≤ i ∧ 0 < j0 → (i * j + i0) / (j0 * j) = i / j0 subst: [j0, i0, j, i] → [i ^ (1 + n→i(n)), j2, i, i1] flags: rewrite

Additional information

Used simplifier rules: zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%grz: ⊦ 0 < i → (0 < i ^ j ↔ i = 1 ∨ 0 ≤ j); from specification int-pot%numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs%num-n2i-ls-07: i < 0 ⊦ (* i) ≤ n→i(n); from specification intnat%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: i ^ (1 + n→i(n)), j2, i, i1