def commutative

stelliferous@0.0.2.0 · proofs/nat_order.bend:272

import Base
import stelliferous@0.0.2.0/proofs/natural_addition.bend as Addition

272def commutative(+a: Nat,+b: Nat) -> {Nat.add(a,b) == Nat.add(b,a) : Nat}:
273  match a:
274    case 0n:
275      Equal.sym(Nat,Nat.add(b,0n),b,Addition.zero_right(b))
276    case 1n+ +p:
277      %Equal.sym(Nat,Nat.add(b,1n+p),1n+Nat.add(b,p),Addition.successor_right(b,p)) : {1n+Nat.add(p,b) == _ : Nat}
278      %commutative(p,b) : {1n+Nat.add(p,b) == 1n+_ : Nat}
279      {==}