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Proof step 25 from proof middle-coef-larger-strongest (previous step)

Σ countFun(1, 2 * i - 1, λ j. binom_coef(2 * i, j)) ≤ binom_coef(2 * i, i) * n→i(# countFun(1, 2 * i - 1, λ j. binom_coef(2 * i, j))), 4 ^ i = 2 + Σ countFun(1, 2 * i - 1, λ j. binom_coef(2 * i, j)), binom_coef(2 * i, i) * 2 * i < 4 ^ i, 0 ≤ i, 0 < i, ∀ i0. i0 = binom_coef(2 * i, 0) ∨ i0 ∈ countFun(1, 2 * i - 1, λ j. binom_coef(2 * i, j)) ∨ i0 = 1 → i0 ≤ binom_coef(2 * i, i) ⊦ binom_coef(2 * i, i) ≠ binom_coef(2 * i, 0) ∧ ¬ binom_coef(2 * i, i) ∈ countFun(1, 2 * i - 1, λ j. binom_coef(2 * i, j))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: intlist inst: name: countFun-length seq: ⊦ 0 ≤ i ∧ i ≤ j → # countFun(i, j, intfun) = i→n(1 + j - i) subst: [intfun, j, i] → [λ j. binom_coef(2 * i, j), 2 * i - 1, 1] flags: rewrite

Additional information

Used simplifier rules: leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%nummult-ls-02: ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0); from specification int-div%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: λ j. binom_coef(2 * i, j), 2 * i - 1, 1