Gcd.dvd_of_mod: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/gcd.bend as Gcd
Lines 25-32 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/gcd.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/gcd.bend&start_line=25&end_line=32

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/arith.bend as R

# d | n and d | a mod n give d | a: a == (q kn + kr) d
def dvd_of_mod(+bp: Nat, +a: Nat, +d: Nat, +kn: Nat, +kr: Nat, +en: {1n+bp == Nat.mul(kn, d) : Nat}, +er: {Nat.mod(a, 1n+bp) == Nat.mul(kr, d) : Nat}) -> {a == Nat.mul(Nat.add(Nat.mul(Nat.div(a, 1n+bp), kn), kr), d) : Nat}:
  +q = Nat.div(a, 1n+bp)
  %Equal.sym(Nat, Nat.mul(Nat.add(Nat.mul(q, kn), kr), d), Nat.add(Nat.mul(Nat.mul(q, kn), d), Nat.mul(kr, d)), A.mul_add_right(Nat.mul(q, kn), kr, d)) : {a == _ : Nat}
  %Equal.sym(Nat, Nat.mul(Nat.mul(q, kn), d), Nat.mul(q, Nat.mul(kn, d)), A.mul_assoc(q, kn, d)) : {a == Nat.add(_, Nat.mul(kr, d)) : Nat}
  %Equal.sym(Nat, Nat.mul(kn, d), 1n+bp, Equal.sym(Nat, 1n+bp, Nat.mul(kn, d), en)) : {a == Nat.add(Nat.mul(q, _), Nat.mul(kr, d)) : Nat}
  %Equal.sym(Nat, Nat.mul(kr, d), Nat.mod(a, 1n+bp), Equal.sym(Nat, Nat.mod(a, 1n+bp), Nat.mul(kr, d), er)) : {a == Nat.add(Nat.mul(q, 1n+bp), _) : Nat}
  R.dm_eq(bp, a)

Uses (9):
  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.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.div
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mod
  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_add_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/arith.bend&name=dm_eq