PROOFDiv.ud.sub_add: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF_div.bend as PROOFDiv
Lines 57-67 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=57&end_line=67

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

# r < bp + 1 gives (bp - r) + r == bp
def ud.sub_add(+bp: Nat, +r: Nat, h: {Nat.is_lt(r, 1n+bp) == True{} : Bool}) -> {Nat.add(Nat.sub(bp, r), r) == bp : Nat}:
  match bp r:
    case 0n 0n:
      {==}
    case 0n 1n+q:
      ud.ex_falso({Nat.add(Nat.sub(0n, 1n+q), 1n+q) == 0n : Nat}, ud.no_lt_one(q, h))
    case 1n+b 0n:
      UL.nat_add_zero(1n+b)
    case 1n+b 1n+q:
      Equal.trans(Nat, Nat.add(Nat.sub(b, q), 1n+q), 1n+Nat.add(Nat.sub(b, q), q), 1n+b, UL.nat_add_succ(Nat.sub(b, q), q), Equal.cong(Nat, Nat, z => 1n+z, Nat.add(Nat.sub(b, q), q), b, ud.sub_add(b, q, h)))

Uses (11):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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.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.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_succ
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.ex_falso
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.no_lt_one