ReductionPROOF.add_comm: def of bendtorch-core@0.1.0.0
Import: import bendtorch-core@0.1.0.0/src/tensor/reduction_PROOF.bend as ReductionPROOF
Lines 723-730 of bendtorch-core@0.1.0.0/src/tensor/reduction_PROOF.bend: https://jend.leeeet.dev/read.txt?path=bendtorch-core@0.1.0.0/src/tensor/reduction_PROOF.bend&start_line=723&end_line=730

import Base
import bendtorch-core@0.1.0.0/src/storage/arith.bend as Arith

def add_comm(+a: Nat, +b: Nat) -> {Nat.add(a, b) == Nat.add(b, a) : Nat}:
  match a:
    case 0n:
      Equal.sym(Nat, Nat.add(b, 0n), b, add_zero(b))
    case 1n+p:
      Equal.trans(Nat, 1n+Nat.add(p, b), 1n+Nat.add(b, p), Nat.add(b, 1n+p),
        Equal.cong(Nat, Nat, n => 1n+n, Nat.add(p, b), Nat.add(b, p), add_comm(p, b)),
        Equal.sym(Nat, Nat.add(b, 1n+p), 1n+Nat.add(b, p), Arith.add_succ(b, p)))

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.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=bendtorch-core@0.1.0.0/src/storage/arith.bend&name=add_succ
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/tensor/reduction_PROOF.bend&name=add_zero