1 % |
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2 \begin{isabellebody}% |
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3 \def\isabellecontext{Semigroups}% |
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4 \isamarkuptrue% |
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5 % |
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6 \isamarkupheader{Semigroups% |
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7 } |
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8 % |
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9 \isadelimtheory |
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10 % |
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11 \endisadelimtheory |
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12 % |
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13 \isatagtheory |
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14 \isamarkupfalse% |
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15 \isacommand{theory}\ Semigroups\ \isakeyword{imports}\ Main\ \isakeyword{begin}% |
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16 \endisatagtheory |
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17 {\isafoldtheory}% |
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18 % |
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19 \isadelimtheory |
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20 % |
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21 \endisadelimtheory |
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22 \isamarkuptrue% |
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23 % |
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24 \begin{isamarkuptext}% |
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25 \medskip\noindent An axiomatic type class is simply a class of types |
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26 that all meet certain properties, which are also called \emph{class |
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27 axioms}. Thus, type classes may be also understood as type |
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28 predicates --- i.e.\ abstractions over a single type argument \isa{{\isacharprime}a}. Class axioms typically contain polymorphic constants that |
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29 depend on this type \isa{{\isacharprime}a}. These \emph{characteristic |
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30 constants} behave like operations associated with the ``carrier'' |
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31 type \isa{{\isacharprime}a}. |
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32 |
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33 We illustrate these basic concepts by the following formulation of |
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34 semigroups.% |
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35 \end{isamarkuptext}% |
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36 \isamarkupfalse% |
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37 \isacommand{consts}\isanewline |
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38 \ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline |
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39 \isamarkupfalse% |
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40 \isacommand{axclass}\ semigroup\ {\isasymsubseteq}\ type\isanewline |
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41 \ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isamarkuptrue% |
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42 % |
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43 \begin{isamarkuptext}% |
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44 \noindent Above we have first declared a polymorphic constant \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} and then defined the class \isa{semigroup} of |
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45 all types \isa{{\isasymtau}} such that \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is indeed an |
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46 associative operator. The \isa{assoc} axiom contains exactly one |
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47 type variable, which is invisible in the above presentation, though. |
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48 Also note that free term variables (like \isa{x}, \isa{y}, |
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49 \isa{z}) are allowed for user convenience --- conceptually all of |
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50 these are bound by outermost universal quantifiers. |
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51 |
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52 \medskip In general, type classes may be used to describe |
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53 \emph{structures} with exactly one carrier \isa{{\isacharprime}a} and a fixed |
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54 \emph{signature}. Different signatures require different classes. |
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55 Below, class \isa{plus{\isacharunderscore}semigroup} represents semigroups \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymoplus}\isactrlsup {\isasymtau}{\isacharparenright}}, while the original \isa{semigroup} would |
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56 correspond to semigroups of the form \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymodot}\isactrlsup {\isasymtau}{\isacharparenright}}.% |
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57 \end{isamarkuptext}% |
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58 \isamarkupfalse% |
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59 \isacommand{consts}\isanewline |
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60 \ \ plus\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymoplus}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline |
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61 \isamarkupfalse% |
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62 \isacommand{axclass}\ plus{\isacharunderscore}semigroup\ {\isasymsubseteq}\ type\isanewline |
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63 \ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isasymoplus}\ z\ {\isacharequal}\ x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isachardoublequote}\isamarkuptrue% |
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64 % |
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65 \begin{isamarkuptext}% |
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66 \noindent Even if classes \isa{plus{\isacharunderscore}semigroup} and \isa{semigroup} both represent semigroups in a sense, they are certainly |
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67 not quite the same.% |
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68 \end{isamarkuptext}% |
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69 % |
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70 \isadelimtheory |
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71 % |
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72 \endisadelimtheory |
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73 % |
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74 \isatagtheory |
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75 \isamarkupfalse% |
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76 \isacommand{end}% |
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77 \endisatagtheory |
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78 {\isafoldtheory}% |
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79 % |
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80 \isadelimtheory |
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81 % |
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82 \endisadelimtheory |
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83 \isanewline |
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84 \end{isabellebody}% |
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85 %%% Local Variables: |
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88 %%% End: |
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