Lemire.div_add_mul: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend as Lemire
Lines 262-269 of bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend&start_line=262&end_line=269

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as AR
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA

# (q n + r) / n = q + r / n
def div_add_mul(+q: Nat, +m: Nat, +r: Nat) -> {Nat.div(Nat.add(Nat.mul(q, 1n+m), r), 1n+m) == Nat.add(q, Nat.div(r, 1n+m)) : Nat}:
  +rq = Nat.div(r, 1n+m)
  +rr = Nat.mod(r, 1n+m)
  %Equal.sym(Nat, r, Nat.add(Nat.mul(rq, 1n+m), rr), AR.dm_eq(m, r)) : {Nat.div(Nat.add(Nat.mul(q, 1n+m), _), 1n+m) == Nat.add(q, rq) : Nat}
  %N.add_assoc(Nat.mul(q, 1n+m), Nat.mul(rq, 1n+m), rr) : {Nat.div(_, 1n+m) == Nat.add(q, rq) : Nat}
  %NA.mul_add_right(q, rq, 1n+m) : {Nat.div(Nat.add(_, rr), 1n+m) == Nat.add(q, rq) : Nat}
  AR.div_of(Nat.add(q, rq), m, rr, AR.dm_lt(m, r))

Uses (11):
  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-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/lib/nat.bend&name=add_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=div_of
  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