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Proof step 15 from proof middle-coef-largest-01 (previous step)

binom_coef(2 * i, j + 1) = (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) + binom_coef(2 * i, j), (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) < i, (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) < 0, binom_coef(2 * i, j + 1) < binom_coef(2 * i, j), binom_coef(2 * i, j + 1) ≤ binom_coef(2 * i, i), 0 < i, 0 ≤ j, j < i, binom_coef(2 * i, i) < binom_coef(2 * i, j), j < 2 * i, j + 1 < i, 2 * i + 1 < 2 * (j + 1), (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) < fac(2 * i - j) * fac(j + 1), 0 < fac(2 * i - j) * fac(j + 1), ∀ 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 ⊦ 2 * i + 1 < 2 * i

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-basic1 inst: name: trans-01 seq: ⊦ i < j ∧ j < j0 → i < j0 subst: [j, j0, i] → [2 * (j + 1), 2 * i, 2 * i + 1] flags: rewrite

Additional information

Used simplifier rules: numls-02: ⊦ (* i2) * i + j < (* i1) * i ↔ j < (i1 - i2) * i; from specification int-mult%Mult-zero: ⊦ i * 0 = 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%lst-03: ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0; from specification int-basic2%ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); from specification int-mult Used terms: 2 * (j + 1), 2 * i, 2 * i + 1