[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 22 from proof coef-le-2pot (previous step)

2 ^ (2 * j) * 2 = Σ countFun(0, 2 * j + 1, λ j0. binom_coef(2 * j + 1, j0)), 0 < j, 0 ≤ j, 2 ^ (2 * j) < fac(2 * j + 1) / (fac(j + 1) * fac(j)) ⊦ fac(2 * j + 1) / (fac(j + 1) * fac(j)) = binom_coef(2 * j + 1, j + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j))
Proof step 24:
0 ≤ j + 1 ∧ j + 1 ≤ 2 * j + 1, 2 ^ (2 * j) * 2 = Σ countFun(0, 2 * j + 1, λ j0. binom_coef(2 * j + 1, j0)), 0 < j, 0 ≤ j, 2 ^ (2 * j) < fac(2 * j + 1) / (fac(j + 1) * fac(j)) ⊦ fac(2 * j + 1) / (fac(j + 1) * fac(j)) = fac(2 * j + 1) / (fac(2 * j + 1 - j + 1) * fac(j + 1))

Rule argument

spec: inst: name: binom-def seq: ⊦ 0 ≤ j ∧ j ≤ i → binom_coef(i, j) = fac(i) / (fac(i - j) * fac(j)) subst: [j, i] → [j + 1, 2 * j + 1] flags: rewrite, current

Additional information

Used simplifier rules: numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* 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%le-add: ⊦ i + j ≤ i0 + j ↔ i ≤ i0; from specification int-abs%le-same-01: ⊦ 0 < i → (i ≤ i * j ↔ 1 ≤ j); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j + 1, 2 * j + 1