Fround.sub_cancel_r: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend as Fround
Lines 260-265 of bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend&start_line=260&end_line=265

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N

def sub_cancel_r(+a: Nat, +b: Nat, +c: Nat) -> {Nat.sub(Nat.add(a, c), Nat.add(b, c)) == Nat.sub(a, b) : Nat}:
  match c:
    case 0n:
      Equal.trans(Nat, Nat.sub(Nat.add(a, 0n), Nat.add(b, 0n)), Nat.sub(a, Nat.add(b, 0n)), Nat.sub(a, b), Equal.cong(Nat, Nat, z => Nat.sub(z, Nat.add(b, 0n)), Nat.add(a, 0n), a, N.add_zero(a)), Equal.cong(Nat, Nat, z => Nat.sub(a, z), Nat.add(b, 0n), b, N.add_zero(b)))
    case 1n+ +cp:
      Equal.trans(Nat, Nat.sub(Nat.add(a, 1n+cp), Nat.add(b, 1n+cp)), Nat.sub(1n+Nat.add(a, cp), Nat.add(b, 1n+cp)), Nat.sub(a, b), Equal.cong(Nat, Nat, z => Nat.sub(z, Nat.add(b, 1n+cp)), Nat.add(a, 1n+cp), 1n+Nat.add(a, cp), N.add_succ(a, cp)), Equal.trans(Nat, Nat.sub(1n+Nat.add(a, cp), Nat.add(b, 1n+cp)), Nat.sub(1n+Nat.add(a, cp), 1n+Nat.add(b, cp)), Nat.sub(a, b), Equal.cong(Nat, Nat, z => Nat.sub(1n+Nat.add(a, cp), z), Nat.add(b, 1n+cp), 1n+Nat.add(b, cp), N.add_succ(b, cp)), sub_cancel_r(a, b, cp)))

Uses (7):
  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.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_zero