Arith.absorb: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as Arith
Lines 119-129 of bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&start_line=119&end_line=129

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A

# (q n + r) mod n == r mod n   (Mathlib Nat.mul_add_mod)
def absorb(+bp: Nat, +q: Nat, +r: Nat) -> {Nat.mod(Nat.add(Nat.mul(q, 1n+bp), r), 1n+bp) == Nat.mod(r, 1n+bp) : Nat}:
  +n = {1n+bp : Nat}
  +qr = Nat.div(r, n)
  +m = Nat.mod(r, n)
  +e = Equal.trans(Nat, Nat.add(Nat.mul(q, n), r), Nat.add(Nat.mul(q, n), Nat.add(Nat.mul(qr, n), m)), Nat.add(Nat.mul(Nat.add(q, qr), n), m),
    Equal.cong(Nat, Nat, z => Nat.add(Nat.mul(q, n), z), r, Nat.add(Nat.mul(qr, n), m), dm_eq(bp, r)),
    Equal.trans(Nat, Nat.add(Nat.mul(q, n), Nat.add(Nat.mul(qr, n), m)), Nat.add(Nat.add(Nat.mul(q, n), Nat.mul(qr, n)), m), Nat.add(Nat.mul(Nat.add(q, qr), n), m),
      Equal.sym(Nat, Nat.add(Nat.add(Nat.mul(q, n), Nat.mul(qr, n)), m), Nat.add(Nat.mul(q, n), Nat.add(Nat.mul(qr, n), m)), A.add_assoc(Nat.mul(q, n), Nat.mul(qr, n), m)),
      Equal.cong(Nat, Nat, z => Nat.add(z, m), Nat.add(Nat.mul(q, n), Nat.mul(qr, n)), Nat.mul(Nat.add(q, qr), n), Equal.sym(Nat, Nat.mul(Nat.add(q, qr), n), Nat.add(Nat.mul(q, n), Nat.mul(qr, n)), A.mul_add_right(q, qr, n)))))
  Equal.trans(Nat, Nat.mod(Nat.add(Nat.mul(q, n), r), n), Nat.mod(Nat.add(Nat.mul(Nat.add(q, qr), n), m), n), m, Equal.cong(Nat, Nat, z => Nat.mod(z, n), Nat.add(Nat.mul(q, n), r), Nat.add(Nat.mul(Nat.add(q, qr), n), m), e), mod_of(Nat.add(q, qr), bp, m, dm_lt(bp, r)))

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  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.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-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@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-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=dm_eq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=dm_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=mod_of