Combnat.cmono: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/typed/combnat.bend as Combnat
Lines 69-73 of bend-collections-laws-math@1.0.0.0/proofs/math/typed/combnat.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/combnat.bend&start_line=69&end_line=73

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

# i <= k, 2 k <= n give C(n, i) <= C(n, k)
def 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}:
  +ek = N.sub_add(k, i, hi)
  +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)
  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))

Uses (11):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=sub_add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/combnat.bend&name=cmono_d
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=choose