NatProofs.mul_comm: def of bendtorch-core@0.1.0.0
Import: import bendtorch-core@0.1.0.0/src/shape/nat_proofs.bend as NatProofs
Lines 75-81 of bendtorch-core@0.1.0.0/src/shape/nat_proofs.bend: https://jend.leeeet.dev/read.txt?path=bendtorch-core@0.1.0.0/src/shape/nat_proofs.bend&start_line=75&end_line=81

import Base

def mul_comm(+a: Nat, +b: Nat) -> {Nat.mul(a, b) == Nat.mul(b, a) : Nat}:
  match a:
    case 0n:
      Equal.sym(Nat, Nat.mul(b, 0n), 0n, mul_zero_right(b))
    case 1n+p:
      %Equal.sym(Nat, Nat.mul(b, 1n+p), Nat.add(b, Nat.mul(b, p)), mul_succ_right(b, p)) : {Nat.add(b, Nat.mul(p, b)) == _ : Nat}
      Equal.cong(Nat, Nat, value => Nat.add(b, value), Nat.mul(p, b), Nat.mul(b, p), mul_comm(p, b))

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=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.mul
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/shape/nat_proofs.bend&name=mul_succ_right
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/shape/nat_proofs.bend&name=mul_zero_right