def factorial_go

bend-collections-laws-crypto@1.0.0.0 · proofs/math/natural/fact.bend:25

import Base
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/src/math/natural.bend as M
import bend-collections-laws-crypto@1.0.0.0/spec/math/natural.bend as S

25def factorial_go(+n: Nat, +acc: Nat) -> {M.factorial_go(n, acc) == Nat.mul(acc, S.factorial(n)) : Nat}:
26  match n:
27    case 0n:
28      Equal.sym(Nat, Nat.mul(acc, 1n), acc, A.mul_one(acc))
29    case 1n+ +p:
30      Equal.trans(Nat, M.factorial_go(p, Nat.mul(1n+p, acc)), Nat.mul(Nat.mul(1n+p, acc), S.factorial(p)), Nat.mul(acc, Nat.mul(1n+p, S.factorial(p))), factorial_go(p, Nat.mul(1n+p, acc)), mul_left_comm(1n+p, acc, S.factorial(p)))