--- a/doc-src/AxClass/Group/Semigroups.thy Wed Mar 04 11:05:02 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,54 +0,0 @@
-
-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