def cmono

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/combnat.bend:69

import Base
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

69# i <= k, 2 k <= n give C(n, i) <= C(n, k)
70def cmono(+n: Nat, +i: Nat, +k: Nat, +hi: {Nat.is_le(i, k) == True{} : Bool}, +hk: {Nat.is_le(Nat.double(k), n) == True{} : Bool}) -> {Nat.is_le(S.choose(n, i), S.choose(n, k)) == True{} : Bool}:
71  +ek = N.sub_add(k, i, hi)
72  +h1 = L.subst(Nat, z => {Nat.is_le(Nat.double(z), n) == True{} : Bool}, k, Nat.add(i, Nat.sub(k, i)), Equal.sym(Nat, Nat.add(i, Nat.sub(k, i)), k, ek), hk)
73  L.subst(Nat, z => {Nat.is_le(S.choose(n, i), S.choose(n, z)) == True{} : Bool}, Nat.add(i, Nat.sub(k, i)), k, ek, cmono_d(n, i, Nat.sub(k, i), h1))