PROOF.un.borrow: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF.bend as PROOF
Lines 624-636 of bend-osdev-u32@0.1.0.0/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&start_line=624&end_line=636

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

# s + B = A + 2^n * C with s < 2^n: C is whether A < B
def un.borrow(c: Bool, +n: Nat, +s: Nat, +a: Nat, +b: Nat, e: {Nat.add(s, b) == Nat.add(a, U.U32Nat.dbl(n, U.U32Nat.bit(c))) : Nat}, hs: {Nat.is_lt(s, U.U32Nat.dbl(n, 1n)) == True{} : Bool}) -> {Nat.is_lt(a, b) == c : Bool}:
  match c:
    case True{}:
      +k = U.U32Nat.dbl(n, 1n)
      +h1 = Equal.trans(Bool, Nat.is_lt(Nat.add(s, b), Nat.add(k, b)), Nat.is_lt(s, k), True{}, un.lt_add_r(s, k, b), hs)
      +h2 = un.lt_rw(Nat.add(s, b), Nat.add(a, k), Nat.add(k, b), e, h1)
      +h3 = un.lt_rw2(Nat.add(a, k), Nat.add(k, b), Nat.add(b, k), un.add_comm(k, b), h2)
      Equal.trans(Bool, Nat.is_lt(a, b), Nat.is_lt(Nat.add(a, k), Nat.add(b, k)), True{}, Equal.sym(Bool, Nat.is_lt(Nat.add(a, k), Nat.add(b, k)), Nat.is_lt(a, b), un.lt_add_r(a, b, k)), h3)
    case False{}:
      +ea = Equal.trans(Nat, Nat.add(s, b), Nat.add(a, U.U32Nat.dbl(n, 0n)), a, e, Equal.trans(Nat, Nat.add(a, U.U32Nat.dbl(n, 0n)), Nat.add(a, 0n), a, Equal.cong(Nat, Nat, v => Nat.add(a, v), U.U32Nat.dbl(n, 0n), 0n, Laws.dbl_zero(n)), un.add_zero(a)))
      %ea : {Nat.is_lt(_, b) == False{} : Bool}
      un.not_lt_add_l(s, b)

Uses (16):
  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.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.is_lt
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=dbl_zero
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.add_comm
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.lt_add_r
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.lt_rw
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.lt_rw2
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.not_lt_add_l
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/u32.bend&name=U32Nat.bit
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/u32.bend&name=U32Nat.dbl