Arith.mul_swap4: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as Arith
Lines 178-185 of bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&start_line=178&end_line=185

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A

# (a b)(c d) == (a c)(b d)
def mul_swap4(+a: Nat, +b: Nat, +c: Nat, +d: Nat) -> {Nat.mul(Nat.mul(a, b), Nat.mul(c, d)) == Nat.mul(Nat.mul(a, c), Nat.mul(b, d)) : Nat}:
  %Equal.sym(Nat, Nat.mul(Nat.mul(a, b), Nat.mul(c, d)), Nat.mul(a, Nat.mul(b, Nat.mul(c, d))), A.mul_assoc(a, b, Nat.mul(c, d))) : {_ == Nat.mul(Nat.mul(a, c), Nat.mul(b, d)) : Nat}
  %Equal.sym(Nat, Nat.mul(Nat.mul(a, c), Nat.mul(b, d)), Nat.mul(a, Nat.mul(c, Nat.mul(b, d))), A.mul_assoc(a, c, Nat.mul(b, d))) : {Nat.mul(a, Nat.mul(b, Nat.mul(c, d))) == _ : Nat}
  %Equal.sym(Nat, Nat.mul(b, Nat.mul(c, d)), Nat.mul(Nat.mul(b, c), d), Equal.sym(Nat, Nat.mul(Nat.mul(b, c), d), Nat.mul(b, Nat.mul(c, d)), A.mul_assoc(b, c, d))) : {Nat.mul(a, _) == Nat.mul(a, Nat.mul(c, Nat.mul(b, d))) : Nat}
  %Equal.sym(Nat, Nat.mul(c, Nat.mul(b, d)), Nat.mul(Nat.mul(c, b), d), Equal.sym(Nat, Nat.mul(Nat.mul(c, b), d), Nat.mul(c, Nat.mul(b, d)), A.mul_assoc(c, b, d))) : {Nat.mul(a, Nat.mul(Nat.mul(b, c), d)) == Nat.mul(a, _) : Nat}
  %Equal.sym(Nat, Nat.mul(b, c), Nat.mul(c, b), A.mul_comm(b, c)) : {Nat.mul(a, Nat.mul(_, d)) == Nat.mul(a, Nat.mul(Nat.mul(c, b), d)) : Nat}
  {==}

Uses (5):
  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.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_comm