Natfuel.fmono: 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 357-366 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=357&end_line=366

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/arith.bend as AR2
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as S

def fmono(+a: Nat, +b: Nat, +h: {Nat.is_le(a, b) == True{} : Bool}) -> {Nat.is_le(S.factorial(a), S.factorial(b)) == True{} : Bool}:
  match a b:
    case 0n 0n:
      {==}
    case 0n 1n+ +y:
      N.le_trans(S.factorial(0n), S.factorial(y), S.factorial(1n+y), fmono(0n, y, N.zero_le(y)), fstep(y))
    case 1n+x 0n:
      Empty.absurd({Nat.is_le(S.factorial(1n+x), S.factorial(0n)) == True{} : Bool}, true_ne_false(Equal.sym(Bool, False{}, True{}, h)))
    case 1n+ +x 1n+ +y:
      N.le_trans(Nat.mul(1n+x, S.factorial(x)), Nat.mul(1n+y, S.factorial(x)), Nat.mul(1n+y, S.factorial(y)), AR2.mul_le(1n+x, 1n+y, S.factorial(x), h), mul_le_r(1n+y, S.factorial(x), S.factorial(y), fmono(x, y, h)))

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  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_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/arith.bend&name=mul_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=zero_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend&name=fstep
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend&name=mul_le_r
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend&name=true_ne_false
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=factorial