Nat.sub_add: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as Nat
Lines 272-281 of bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&start_line=272&end_line=281

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as L

def sub_add(+a: Nat, +b: Nat, e: {Nat.is_le(b, a) == True{} : Bool}) -> {Nat.add(b, Nat.sub(a, b)) == a : Nat}:
  match a b:
    case 0n 0n:
      {==}
    case 1n+p 0n:
      {==}
    case 0n 1n+q:
      Empty.absurd({Nat.add(1n+q, Nat.sub(0n, 1n+q)) == 0n : Nat}, L.false_true(e))
    case 1n+p 1n+q:
      succ_cong(Nat.add(q, Nat.sub(p, q)), p, sub_add(p, q, e))

Uses (8):
  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_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=false_true
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=succ_cong