Lcm.gcd_pos: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/lcm.bend as Lcm
Lines 104-110 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/lcm.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/lcm.bend&start_line=104&end_line=110

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/lemmas/proofs/nat_algebra.bend as A

# a == la g with a > 0 makes g positive
def gcd_pos(+ap: Nat, +l: Nat, +g: Nat, +e: {1n+ap == Nat.mul(l, g) : Nat}) -> {Nat.is_lt(0n, g) == True{} : Bool}:
  match g:
    case 0n:
      Empty.absurd({Nat.is_lt(0n, 0n) == True{} : Bool}, N.succ_zero(ap, Equal.trans(Nat, 1n+ap, Nat.mul(l, 0n), 0n, e, A.mul_zero(l))))
    case 1n+gp:
      {==}

Uses (8):
  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.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.is_lt
  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_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=succ_zero