Fact.comb_small: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend as Fact
Lines 268-275 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&start_line=268&end_line=275

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/src/math/natural.bend as M
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as S

def comb_small(+n: Nat, +k: Nat, +b: Bool, +hb: {Nat.is_lt(n, k) == b : Bool}) -> {M.comb_small(n, k, b) == S.choose(n, k) : Nat}:
  match b:
    case True{}:
      Equal.sym(Nat, S.choose(n, k), 0n, choose_eq_zero(n, k, hb))
    case False{}:
      +m = Nat.min(k, Nat.sub(n, k))
      %Equal.sym(Nat, 1n, S.choose(n, 0n), Equal.sym(Nat, S.choose(n, 0n), 1n, choose_zero_right(n))) : {M.comb_go(m, n, 0n, _) == S.choose(n, k) : Nat}
      Equal.trans(Nat, M.comb_go(m, n, 0n, S.choose(n, 0n)), S.choose(n, m), S.choose(n, k), comb_go(m, n, 0n), comb_min(n, k, N.not_lt_le(n, k, hb), Nat.is_lt(k, Nat.sub(n, k)), {==}))

Uses (15):
  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=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.min
  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/nat.bend&name=not_lt_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&name=choose_eq_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&name=choose_zero_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&name=comb_go
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&name=comb_min
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=choose
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/natural.bend&name=comb_go
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/natural.bend&name=comb_small