doc-src/AxClass/Group/Semigroups.thy

structure Display: less pervasive operations;

header {* Semigroups *}

theory Semigroups imports Main begin

text {*
  \medskip\noindent An axiomatic type class is simply a class of types
  that all meet certain properties, which are also called \emph{class
  axioms}. Thus, type classes may be also understood as type
  predicates --- i.e.\ abstractions over a single type argument @{typ
  'a}.  Class axioms typically contain polymorphic constants that
  depend on this type @{typ 'a}.  These \emph{characteristic
  constants} behave like operations associated with the ``carrier''
  type @{typ 'a}.

  We illustrate these basic concepts by the following formulation of
  semigroups.
*}

consts
  times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "\<odot>" 70)
axclass semigroup \<subseteq> type
  assoc: "(x \<odot> y) \<odot> z = x \<odot> (y \<odot> z)"

text {*
  \noindent Above we have first declared a polymorphic constant @{text
  "\<odot> \<Colon> 'a \<Rightarrow> 'a \<Rightarrow> 'a"} and then defined the class @{text semigroup} of
  all types @{text \<tau>} such that @{text "\<odot> \<Colon> \<tau> \<Rightarrow> \<tau> \<Rightarrow> \<tau>"} is indeed an
  associative operator.  The @{text assoc} axiom contains exactly one
  type variable, which is invisible in the above presentation, though.
  Also note that free term variables (like @{term x}, @{term y},
  @{term z}) are allowed for user convenience --- conceptually all of
  these are bound by outermost universal quantifiers.

  \medskip In general, type classes may be used to describe
  \emph{structures} with exactly one carrier @{typ 'a} and a fixed
  \emph{signature}.  Different signatures require different classes.
  Below, class @{text plus_semigroup} represents semigroups @{text
  "(\<tau>, \<oplus>\<^sup>\<tau>)"}, while the original @{text semigroup} would
  correspond to semigroups of the form @{text "(\<tau>, \<odot>\<^sup>\<tau>)"}.
*}

consts
  plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "\<oplus>" 70)
axclass plus_semigroup \<subseteq> type
  assoc: "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"

text {*
  \noindent Even if classes @{text plus_semigroup} and @{text
  semigroup} both represent semigroups in a sense, they are certainly
  not quite the same.
*}

end