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Proof step 8 from proof middle-coef-max_exp (previous step)

i0 < max_exp(i, binom_coef(2 * j, j)), i ^ i0 < i ^ max_exp(i, binom_coef(2 * j, j)), i ^ i0 < i ^ (i0 + 1), 0 ≤ i0, 0 < j, 2 * j < i ^ max_exp(i, binom_coef(2 * j, j)), prime(i), i | binom_coef(2 * j, j), 2 * j < i ^ (i0 + 1), i ^ i0 ≤ 2 * j ⊦ max_exp(i, binom_coef(2 * j, j)) = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j))
Proof step 10:
0 ≤ j ∧ j ≤ 2 * j, i0 < max_exp(i, fac(2 * j) / (fac(2 * j - j) * fac(j))), i ^ i0 < i ^ max_exp(i, fac(2 * j) / (fac(2 * j - j) * fac(j))), i ^ i0 < i ^ (i0 + 1), 0 ≤ i0, 0 < j, 2 * j < i ^ max_exp(i, fac(2 * j) / (fac(2 * j - j) * fac(j))), prime(i), i | fac(2 * j) / (fac(2 * j - j) * fac(j)), 2 * j < i ^ (i0 + 1), i ^ i0 ≤ 2 * j ⊦ max_exp(i, fac(2 * j) / (fac(2 * j - j) * fac(j))) = 0

Rule argument

spec: inst: name: binom-def seq: ⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j)) subst: [j, i] → [j, 2 * j] flags: rewrite, current

Additional information

Used simplifier rules: le-same-01: ⊦ 0 < i → (i ≤ i * j ↔ 1 ≤ 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: j, 2 * j