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Proof step 6 from proof middle-coef-largest-01 (previous step)

0 < i, 0 ≤ j, j < i, binom_coef(2 * i, i) < binom_coef(2 * i, j), ∀ i0, j0. i→n(i0 - j0) < i→n(i - j) → binom_coef(2 * i0, i0) < binom_coef(2 * i0, j0) ∧ j0 < i0 ∧ 0 ≤ j0 ∧ 0 < i0 → false ⊦ binom_coef(2 * i, i) < binom_coef(2 * i, j + 1) ∧ j + 1 < i

User applied rule insert lemma

And obtained the following new goals

Proof step 7:
⊦ 1 ≤ j ∧ j ≤ i → binom_coef(i, j) - binom_coef(i, j - 1) = (fac(i) * (i + 1 - 2 * j)) / (fac(j) * fac(i + 1 - j))
Proof step 8:
1 ≤ j + 1 ∧ j + 1 ≤ 2 * i → binom_coef(2 * i, j + 1) - binom_coef(2 * i, j + 1 - 1) = (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i + 1 - j + 1)), 0 < i, 0 ≤ j, j < i, binom_coef(2 * i, i) < binom_coef(2 * i, j), ∀ i0, j0. i→n(i0 - j0) < i→n(i - j) → binom_coef(2 * i0, i0) < binom_coef(2 * i0, j0) ∧ j0 < i0 ∧ 0 ≤ j0 ∧ 0 < i0 → false ⊦ binom_coef(2 * i, i) < binom_coef(2 * i, j + 1) ∧ j + 1 < i

Rule argument

spec: inst: name: coef-sub seq: ⊦ 1 ≤ j ∧ j ≤ i → binom_coef(i, j) - binom_coef(i, j - 1) = (fac(i) * (i + 1 - 2 * j)) / (fac(j) * fac(i + 1 - j)) subst: [j, i] → [j + 1, 2 * i] flags: current, []

Additional information

Used terms: j + 1, 2 * i