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

binom_coef(2 * i, i) * (2 * i + 1) ≤ 4 ^ i, binom_coef(2 * i, i) * (2 * i + 1) ≤ binom_coef(2 * i, i) * 2 * i, 0 ≤ 4 ^ i, 4 ^ i ≤ binom_coef(2 * i, i) * 2 * i, 0 ≤ binom_coef(2 * i, i), 0 < i ⊦ binom_coef(2 * i, i) = 4 ^ i / (2 * i + 1), i = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: le-pos seq: ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0) subst: [i0, j, i] → [2 * i, 2 * i + 1, binom_coef(2 * i, i)] flags: rewrite

Additional information

Used simplifier rules: coef-le-zero: ⊦ binom_coef(i, j) ≤ 0 ↔ binom_coef(i, j) = 0;%numle-02: ⊦ (* i2) * i + j ≤ (* i1) * i ↔ (i2 - i1) * i + j ≤ 0; 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%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%pot-lez: ⊦ 0 < i → (i ^ j ≤ 0 ↔ i ^ j = 0); from specification int-pot%pot-zero-01: ⊦ 0 < i → (i ^ j = 0 ↔ j < 0 ∧ 1 < i); from specification int-pot%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: 2 * i, 2 * i + 1, binom_coef(2 * i, i)