Natfuel.fstep: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend as Natfuel
Lines 347-350 of bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend&start_line=347&end_line=350

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/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as S

def fstep(+y: Nat) -> {Nat.is_le(S.factorial(y), S.factorial(1n+y)) == True{} : Bool}:
  +f = S.factorial(y)
  +e = Equal.trans(Nat, Nat.mul(1n+y, f), Nat.mul(f, 1n+y), Nat.add(f, Nat.mul(f, y)), NA.mul_comm(1n+y, f), NA.mul_succ(f, y))
  L.subst(Nat, z => {Nat.is_le(f, z) == True{} : Bool}, Nat.add(f, Nat.mul(f, y)), Nat.mul(1n+y, f), Equal.sym(Nat, Nat.mul(1n+y, f), Nat.add(f, Nat.mul(f, y)), e), N.le_add_right(f, Nat.mul(f, y)))

Uses (12):
  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.add
  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.mul
  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/lib/lemmas/proofs/nat_algebra.bend&name=mul_succ
  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=le_add_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=factorial