@FooAC.mk : {α : Type} →
  [toFooComm : FooComm α] →
    [toMul : Mul.{0} α] →
      (∀ (a b c : α),
          @Eq.{1} α
            (@HAdd.hAdd.{0, 0, 0} α α α (@instHAdd.{0} α (@Foo.toAdd α (@FooComm.toFoo α toFooComm)))
              (@HAdd.hAdd.{0, 0, 0} α α α (@instHAdd.{0} α (@Foo.toAdd α (@FooComm.toFoo α toFooComm))) a b) c)
            (@HAdd.hAdd.{0, 0, 0} α α α (@instHAdd.{0} α (@Foo.toAdd α (@FooComm.toFoo α toFooComm))) a
              (@HAdd.hAdd.{0, 0, 0} α α α (@instHAdd.{0} α (@Foo.toAdd α (@FooComm.toFoo α toFooComm))) b c))) →
        (∀ (a b c : α),
            @Eq.{1} α
              (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul)
                (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) a b) c)
              (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) a
                (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) b c))) →
          (∀ (a b : α),
              @Eq.{1} α (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) a b)
                (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) b a)) →
            FooAC α
def FooAC.toFooAssoc : {α : Type} → [self : FooAC α] → FooAssoc α :=
fun (α : Type) [self : FooAC α] =>
  @FooAssoc.mk α (@FooComm.toFoo α (@FooAC.toFooComm α self)) (@FooAC.toMul α self) (@FooAC.add_assoc α self)
    (@FooAC.mul_assoc α self)
@FooAC'.mk : {α : Type} →
  [toFooComm : FooComm α] →
    [toMul : Mul.{0} α] →
      (∀ (a b c : α),
          @Eq.{1} α
            (@HAdd.hAdd.{0, 0, 0} α α α (@instHAdd.{0} α (@Foo.toAdd α (@FooComm.toFoo α toFooComm)))
              (@HAdd.hAdd.{0, 0, 0} α α α (@instHAdd.{0} α (@Foo.toAdd α (@FooComm.toFoo α toFooComm))) a b) c)
            (@HAdd.hAdd.{0, 0, 0} α α α (@instHAdd.{0} α (@Foo.toAdd α (@FooComm.toFoo α toFooComm))) a
              (@HAdd.hAdd.{0, 0, 0} α α α (@instHAdd.{0} α (@Foo.toAdd α (@FooComm.toFoo α toFooComm))) b c))) →
        (∀ (a b c : α),
            @Eq.{1} α
              (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul)
                (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) a b) c)
              (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) a
                (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) b c))) →
          α →
            (∀ (a b : α),
                @Eq.{1} α (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) a b)
                  (@HMul.hMul.{0, 0, 0} α α α (@instHMul.{0} α toMul) b a)) →
              FooAC' α
def FooAC'.toFooAssoc' : {α : Type} → [self : FooAC' α] → FooAssoc' α :=
fun (α : Type) [self : FooAC' α] =>
  @FooAssoc'.mk α
    (@FooAssoc.mk α (@FooComm.toFoo α (@FooAC'.toFooComm α self)) (@FooAC'.toMul α self) (@FooAC'.add_assoc α self)
      (@FooAC'.mul_assoc α self))
    (@FooAC'.one α self)