PROOF.un.add_comm: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF.bend as PROOF
Lines 76-81 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=76&end_line=81

import Base

def un.add_comm(x: Nat, +y: Nat) -> {Nat.add(x, y) == Nat.add(y, x) : Nat}:
  match x:
    case 0n:
      Equal.sym(Nat, Nat.add(y, 0n), y, un.add_zero(y))
    case 1n+p:
      Equal.trans(Nat, 1n+Nat.add(p, y), 1n+Nat.add(y, p), Nat.add(y, 1n+p), un.succ_cong(Nat.add(p, y), Nat.add(y, p), un.add_comm(p, y)), Equal.sym(Nat, Nat.add(y, 1n+p), 1n+Nat.add(y, p), un.add_succ(y, p)))

Uses (7):
  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=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.add_succ
  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.succ_cong