def sub_add

bend-collections-laws-math@1.0.0.0 · proofs/lib/nat.bend:272

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

272def sub_add(+a: Nat, +b: Nat, e: {Nat.is_le(b, a) == True{} : Bool}) -> {Nat.add(b, Nat.sub(a, b)) == a : Nat}:
273  match a b:
274    case 0n 0n:
275      {==}
276    case 1n+p 0n:
277      {==}
278    case 0n 1n+q:
279      Empty.absurd({Nat.add(1n+q, Nat.sub(0n, 1n+q)) == 0n : Nat}, L.false_true(e))
280    case 1n+p 1n+q:
281      succ_cong(Nat.add(q, Nat.sub(p, q)), p, sub_add(p, q, e))