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Proof step 21 from proof max_exp-fac-gen (previous step)

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, ∀ 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)) = Σ countFun(1, i0, λ j0. (i1 * i + j2) / i ^ j0)

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [j0, i2, i0] → [i1, i, i0 - 1]

Additional information

Used simplifier rules: labs: ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j); from specification intnat%abs-16: ⊦ 0 < j → abs(j) = j; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%mod-nle: ⊦ 0 ≤ i0 ∧ i0 < i ∧ 0 < j ∧ 0 < i → (j * i + i0 ≤ j ↔ i = 1 ∧ i0 = 0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%one-01: ⊦ i = 1 → i * j = j; from specification int-mult%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%ax-04: ⊦ i + 0 = i; from specification int-basic2%one-01: ⊦ 1 ^ j = 1; from specification int-pot%one: ⊦ i / 1 = i; from specification int-div Used terms: i1, i, i0 - 1