Fact.choose_mul_factorial: 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 181-192 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=181&end_line=192

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as S

# C(n, k) k! == n.desc(k)   (Mathlib Nat.descFactorial_eq_factorial_mul_choose)
def choose_mul_factorial(+n: Nat, +k: Nat) -> {Nat.mul(S.choose(n, k), S.factorial(k)) == S.desc(n, k) : Nat}:
  match k:
    case 0n:
      %Equal.sym(Nat, S.choose(n, 0n), 1n, choose_zero_right(n)) : {Nat.mul(_, 1n) == 1n : Nat}
      {==}
    case 1n+ +j:
      # C(n, 1+j) ((1+j) j!) == (C(n, 1+j) (1+j)) j! == (C(n, j) (n - j)) j! == (n - j) (C(n, j) j!)
      %Equal.sym(Nat, Nat.mul(S.choose(n, 1n+j), Nat.mul(1n+j, S.factorial(j))), Nat.mul(Nat.mul(S.choose(n, 1n+j), 1n+j), S.factorial(j)), Equal.sym(Nat, Nat.mul(Nat.mul(S.choose(n, 1n+j), 1n+j), S.factorial(j)), Nat.mul(S.choose(n, 1n+j), Nat.mul(1n+j, S.factorial(j))), A.mul_assoc(S.choose(n, 1n+j), 1n+j, S.factorial(j)))) : {_ == S.desc(n, 1n+j) : Nat}
      %Equal.sym(Nat, Nat.mul(S.choose(n, 1n+j), 1n+j), Nat.mul(S.choose(n, j), Nat.sub(n, j)), choose_succ_right(n, j)) : {Nat.mul(_, S.factorial(j)) == S.desc(n, 1n+j) : Nat}
      %Equal.sym(Nat, Nat.mul(Nat.mul(S.choose(n, j), Nat.sub(n, j)), S.factorial(j)), Nat.mul(Nat.sub(n, j), Nat.mul(S.choose(n, j), S.factorial(j))), Equal.trans(Nat, Nat.mul(Nat.mul(S.choose(n, j), Nat.sub(n, j)), S.factorial(j)), Nat.mul(Nat.mul(Nat.sub(n, j), S.choose(n, j)), S.factorial(j)), Nat.mul(Nat.sub(n, j), Nat.mul(S.choose(n, j), S.factorial(j))), Equal.cong(Nat, Nat, z => Nat.mul(z, S.factorial(j)), Nat.mul(S.choose(n, j), Nat.sub(n, j)), Nat.mul(Nat.sub(n, j), S.choose(n, j)), A.mul_comm(S.choose(n, j), Nat.sub(n, j))), A.mul_assoc(Nat.sub(n, j), S.choose(n, j), S.factorial(j)))) : {_ == S.desc(n, 1n+j) : Nat}
      Equal.cong(Nat, Nat, z => Nat.mul(Nat.sub(n, j), z), Nat.mul(S.choose(n, j), S.factorial(j)), S.desc(n, j), choose_mul_factorial(n, j))

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  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.mul
  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/lemmas/proofs/nat_algebra.bend&name=mul_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&name=choose_succ_right
  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/spec/math/natural.bend&name=choose
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=desc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=factorial