PROOFDiv.ud.cover_sum: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF_div.bend as PROOFDiv
Lines 160-165 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=160&end_line=165

import Base
import bend-osdev-u32@0.1.0.0/LAWS.bend as UL

# a + bp == (a + (bp - r)) + r, for r < 1 + bp
def ud.cover_sum(+bp: Nat, +a: Nat, +r: Nat, h: {Nat.is_lt(r, 1n+bp) == True{} : Bool}) -> {Nat.add(a, Nat.sub(bp, 0n)) == Nat.add(Nat.add(a, Nat.sub(bp, r)), r) : Nat}:
  %Equal.sym(Nat, Nat.sub(bp, 0n), bp, ud.sub_zero(bp)) : {Nat.add(a, _) == Nat.add(Nat.add(a, Nat.sub(bp, r)), r) : Nat}
  %Equal.sym(Nat, Nat.add(Nat.add(a, Nat.sub(bp, r)), r), Nat.add(a, Nat.add(Nat.sub(bp, r), r)), UL.nat_add_assoc(a, Nat.sub(bp, r), r)) : {Nat.add(a, bp) == _ : Nat}
  %ud.sub_add(bp, r, h) : {Nat.add(a, _) == Nat.add(a, Nat.add(Nat.sub(bp, r), r)) : Nat}
  {==}

Uses (9):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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.is_lt
  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/LAWS.bend&name=nat_add_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.sub_add
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.sub_zero