PROOFDiv.ud.cover: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF_div.bend as PROOFDiv
Lines 167-172 of bend-osdev-u32@0.1.0.0/PROOF_div.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&start_line=167&end_line=172

import Base

def ud.cover(+bp: Nat, +a: Nat) -> {Nat.mul(Nat.div(Nat.add(a, Nat.sub(1n+bp, 1n)), 1n+bp), 1n+bp) == Nat.add(a, Nat.sub(Nat.sub(1n+bp, 1n), Nat.mod(Nat.add(a, Nat.sub(1n+bp, 1n)), 1n+bp))) : Nat}:
  +t = Nat.add(a, Nat.sub(bp, 0n))
  +q = Nat.div(t, 1n+bp)
  +r = Nat.mod(t, 1n+bp)
  %Equal.sym(Nat, Nat.sub(bp, 0n), bp, ud.sub_zero(bp)) : {Nat.mul(q, 1n+bp) == Nat.add(a, Nat.sub(_, r)) : Nat}
  ud.add_cancel(r, Nat.mul(q, 1n+bp), Nat.add(a, Nat.sub(bp, r)), Equal.trans(Nat, Nat.add(Nat.mul(q, 1n+bp), r), t, Nat.add(Nat.add(a, Nat.sub(bp, r)), r), ud.div_mod(t, bp), ud.cover_sum(bp, a, r, ud.mod_lt(t, bp))))

Uses (13):
  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=Base/base.bend&name=Nat.sub
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.add_cancel
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.cover_sum
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.div_mod
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.mod_lt
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.sub_zero