I want nice operators for complex arithmetic to make my code more readable. Ocaml has a Complex module, so I just want to add operators that call those functions.

The most intuitive way for me is to make a new complex operator from all of the usual operators by appending '&' to the operator symbol. Thus +& and *& will be complex addition and multiplication. I would also like ~& to be complex conjugation.

If I'm going to use these operators, I want them to associate the same way that normal arithmetic associates. Based on the following sessions, they are automatically behaving the way I want, but I would like to understand why, so that I don't get horrible bugs when I introduce more operators.

My current guess is that their precedence is done by lexically sorting the operator symbols according to an ordering that is consistent with normal arithmetic precedence. But I cannot confirm this.

Session one:

# open Complex;;
# let (+&) a b = add a b;;
val ( +& ) : Complex.t -> Complex.t -> Complex.t = <fun>
# let ( *&) a b = mul a b;;
val ( *& ) : Complex.t -> Complex.t -> Complex.t = <fun>
# one +& zero *& one +& zero *& one;;
- : Complex.t = {re = 1.; im = 0.}
# zero +& one *& zero +& one *& zero;;
- : Complex.t = {re = 0.; im = 0.}
# i +& i *& i +& i *& i *& i;;
- : Complex.t = {re = -1.; im = 0.}

Session two:

# open Complex;;
# let ( *&) a b = mul a b;;
val ( *& ) : Complex.t -> Complex.t -> Complex.t = <fun>
# let (+&) a b = add a b;;
val ( +& ) : Complex.t -> Complex.t -> Complex.t = <fun>
# one +& zero *& one +& zero *& one;;
- : Complex.t = {re = 1.; im = 0.}
# zero +& one *& zero +& one *& zero;;
- : Complex.t = {re = 0.; im = 0.}
# i +& i *& i +& i *& i *& i;;
- : Complex.t = {re = -1.; im = 0.}
# let (~&) a = conj a;;
val ( ~& ) : Complex.t -> Complex.t = <fun>
# (one +& i) *& ~& (one +& i);;
- : Complex.t = {re = 2.; im = 0.}