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Proof step 48 from proof max_exp-fac-gen (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: intnat inst: name: num-i2n-ls-02 seq: 0 ≤ i ⊦ (* m) < i→n(i) ↔ n→i(m) < i subst: [i, m] → [i0, n + 1] flags: rewrite

Additional information

Used simplifier rules: grz: ⊦ 0 < i → (0 < i ^ j ↔ i = 1 ∨ 0 ≤ j); from specification int-pot%one-01: ⊦ i = 1 → i * j = j; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: i0, n + 1