def fmono

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/natfuel.bend:357

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

357def fmono(+a: Nat, +b: Nat, +h: {Nat.is_le(a, b) == True{} : Bool}) -> {Nat.is_le(S.factorial(a), S.factorial(b)) == True{} : Bool}:
358  match a b:
359    case 0n 0n:
360      {==}
361    case 0n 1n+ +y:
362      N.le_trans(S.factorial(0n), S.factorial(y), S.factorial(1n+y), fmono(0n, y, N.zero_le(y)), fstep(y))
363    case 1n+x 0n:
364      Empty.absurd({Nat.is_le(S.factorial(1n+x), S.factorial(0n)) == True{} : Bool}, true_ne_false(Equal.sym(Bool, False{}, True{}, h)))
365    case 1n+ +x 1n+ +y:
366      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)))