Shuffle.swap_ends: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/shuffle.bend as Shuffle
Lines 40-46 of bend-collections-laws-math@1.0.0.0/proofs/math/random/shuffle.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/shuffle.bend&start_line=40&end_line=46

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

# (a + c) + b == (b + c) + a
def swap_ends(+a: Nat, +b: Nat, +c: Nat) -> {Nat.add(Nat.add(a, c), b) == Nat.add(Nat.add(b, c), a) : Nat}:
  Equal.trans(Nat, Nat.add(Nat.add(a, c), b), Nat.add(a, Nat.add(c, b)), Nat.add(Nat.add(b, c), a),
    N.add_assoc(a, c, b),
    Equal.trans(Nat, Nat.add(a, Nat.add(c, b)), Nat.add(a, Nat.add(b, c)), Nat.add(Nat.add(b, c), a),
      Equal.cong(Nat, Nat, t => Nat.add(a, t), Nat.add(c, b), Nat.add(b, c), N.add_comm(c, b)),
      N.add_comm(a, Nat.add(b, c))))

Uses (6):
  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=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_comm