Getting stuck in *Real World Haskell*'s chapter on pretty printing (chap 5), I decided to take a look at one of the suggested sources, i.e., John Hughes' *The Design of a Pretty-printing Library https://www.researchgate.net/publication/226525714_The_Design_of_a_Pretty-pr...*. Not that I fully got the first three section, still, let me skip ahead to section 4, *Designing a Sequence Type *where we're designing a supposedly generic sequence type, I'm guessing in the same universal, fundamental way that Peano's Axioms give a generic underpinning of the natural numbers. Good. I get it. This is why Haskell is unique. So he says we take the following *signature* as our starting point nil :: Seq a unit :: a -> Seq a cat :: Seq a -> Seq a -> Seq a list :: Seq a -> [a] where nil , unit , and cat give us ways to build sequences, and list is an *observation *[explained before]. The correspondence with the usual list operations is nil = [] unit x = [x] cat = (++) These operations are to satisfy the following laws: xs `cat` ( ys `cat` zs ) = ( xs `cat` ys ) `cat` zs nil `cat` xs = xs xs `cat` nil = xs list nil = [] list ( unit x `cat` xs ) = x : list xs My first question is why must we satisfy these laws? Where did these laws come from? Is this a universal truth thing? I've heard the word *monoid *and *semigroup* tossed around. Is this germane here? Next in *4.1 Term Representation* I'm immediately lost with the concept of *term. *On a hunch, I borrowed a book *Term Rewriting and All That *from Baader and Nipkow which I'm sure goes very deep into the concept of terms, but I'm not there yet. What is meant by *term*? As 4.1 continues, confusing is this passage The most direct way to represent values of [the?] sequence type is just as terms of the algebra [huh?], for example using data Seq a = Nil j Unit a j Seq a `Cat` Seq a But this trivial representation does not exploit the algebraic laws that we know to hold [huh?], and moreover the list observation will be a little tricky to de ne (ideally we would like to implement observations by very simple, non-recursive functions: the real work should be done in the implementations of the Seq operators themselves). Instead, we may choose a restricted subset of terms -- call them simplified forms -- into which every term can be put using the algebraic laws. Then we can represent sequences using a datatype that represents the syntax of simplified forms. In this case, there is an obvious candidate for simplified forms: terms of the form nil and unit x `cat` xs , where xs is also in simplified form. Simplified forms can be represented using the type data Seq a = Nil j a `UnitCat` Seq a with the interpretation Nil = nil x `UnitCat` xs = unit x `cat` xs All of this presupposes much math lore that I'm hereby asking someone on this mailing list to point me towards. I threw in a few [huh?] as specific points, but in general, I'm not there yet. Any explanations appreciated. ⨽ Lawrence Bottorff Grand Marais, MN, USA borgauf@gmail.com