def cbig

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/combnat.bend:110

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as S
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L

110# K <= i, 2 i <= n give 2^K <= C(n, i)
111def cbig(+K: Nat, +n: Nat, +i: Nat, +hK: {Nat.is_le(K, i) == True{} : Bool}, +h2: {Nat.is_le(Nat.double(i), n) == True{} : Bool}) -> {Nat.is_le(C.pow2(K), S.choose(n, i)) == True{} : Bool}:
112  +d = Nat.sub(n, Nat.double(i))
113  +en = Equal.trans(Nat, Nat.add(d, Nat.double(i)), Nat.add(Nat.double(i), d), n, N.add_comm(d, Nat.double(i)), N.sub_add(n, Nat.double(i), h2))
114  +h3 = L.subst(Nat, z => {Nat.is_le(S.choose(Nat.double(i), i), S.choose(z, i)) == True{} : Bool}, Nat.add(d, Nat.double(i)), n, en, cn_mono_d(Nat.double(i), d, i))
115  N.le_trans(C.pow2(K), C.pow2(i), S.choose(n, i), N.pow2_mono(K, i, hK), N.le_trans(C.pow2(i), S.choose(Nat.double(i), i), S.choose(n, i), central(i), h3))