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Proof step 12 from proof middle-coef-larger-strongest (previous step)

∀ i0. i0 ∈ countFun(0, 2 * i, λ j. binom_coef(2 * i, j)) → i0 ≤ binom_coef(2 * i, i), binom_coef(2 * i, i) ∈ countFun(0, 2 * i, λ j. binom_coef(2 * i, j)), 4 ^ i = Σ countFun(0, 2 * i, λ j. binom_coef(2 * i, j)), binom_coef(2 * i, i) * 2 * i < 4 ^ i, 0 < i, 0 ≤ i ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%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, 0