U32div.mod_identify: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/u32div.bend as U32div
Lines 202-207 of bend-collections-laws-math@1.0.0.0/proofs/lib/u32div.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32div.bend&start_line=202&end_line=207

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/natural_division.bend as ND

def mod_identify(+q: Nat, +d: Nat, +r: Nat, +bound: {Nat.is_lt(r, d) == True{} : Bool}) -> {Nat.mod(Nat.add(Nat.mul(q, d), r), d) == r : Nat}:
  match d:
    case 0n:
      Empty.absurd({Nat.mod(Nat.add(Nat.mul(q, 0n), r), 0n) == r : Nat}, N.lt_zero_absurd(r, bound))
    case 1n+p:
      ND.remainder(q, p, r, N.lt_succ_le(r, p, bound))

Uses (10):
  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=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.is_lt
  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/natural_division.bend&name=remainder
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=lt_succ_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=lt_zero_absurd