def factorial_mul_desc

bend-collections-laws-crypto@1.0.0.0 · proofs/math/natural/fact.bend:102

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

102# (n - k)! * n.desc(k) == n!   (Mathlib Nat.factorial_mul_descFactorial)
103def factorial_mul_desc(+n: Nat, +k: Nat, +h: {Nat.is_le(k, n) == True{} : Bool}) -> {Nat.mul(S.factorial(Nat.sub(n, k)), S.desc(n, k)) == S.factorial(n) : Nat}:
104  match k:
105    case 0n:
106      %Equal.sym(Nat, Nat.sub(n, 0n), n, N.sub_zero(n)) : {Nat.mul(S.factorial(_), 1n) == S.factorial(n) : Nat}
107      A.mul_one(S.factorial(n))
108    case 1n+ +j:
109      +t = Nat.sub(n, 1n+j)
110      +hj = N.succ_le_lt(j, n, h)
111      +ih = factorial_mul_desc(n, j, N.lt_le(j, n, hj))
112      +es = sub_lt_succ(n, j, hj)
113      # t! * ((n - j) * n.desc(j)) == ((n - j) * t!) * n.desc(j) == (n - j)! * n.desc(j)
114      %Equal.sym(Nat, Nat.mul(S.factorial(t), Nat.mul(Nat.sub(n, j), S.desc(n, j))), Nat.mul(Nat.mul(Nat.sub(n, j), S.factorial(t)), S.desc(n, j)), Equal.sym(Nat, Nat.mul(Nat.mul(Nat.sub(n, j), S.factorial(t)), S.desc(n, j)), Nat.mul(S.factorial(t), Nat.mul(Nat.sub(n, j), S.desc(n, j))), mul_left_comm(Nat.sub(n, j), S.factorial(t), S.desc(n, j)))) : {_ == S.factorial(n) : Nat}
115      %Equal.sym(Nat, Nat.sub(n, j), 1n+t, es) : {Nat.mul(Nat.mul(_, S.factorial(t)), S.desc(n, j)) == S.factorial(n) : Nat}
116      %Equal.sym(Nat, Nat.mul(1n+t, S.factorial(t)), S.factorial(Nat.sub(n, j)), Equal.trans(Nat, Nat.mul(1n+t, S.factorial(t)), S.factorial(1n+t), S.factorial(Nat.sub(n, j)), {==}, Equal.cong(Nat, Nat, z => S.factorial(z), 1n+t, Nat.sub(n, j), Equal.sym(Nat, Nat.sub(n, j), 1n+t, es)))) : {Nat.mul(_, S.desc(n, j)) == S.factorial(n) : Nat}
117      ih