|
1 % |
|
2 \begin{isabellebody}% |
|
3 \def\isabellecontext{Product}% |
|
4 \isamarkuptrue% |
|
5 % |
|
6 \isamarkupheader{Syntactic classes% |
|
7 } |
|
8 % |
|
9 \isadelimtheory |
|
10 % |
|
11 \endisadelimtheory |
|
12 % |
|
13 \isatagtheory |
|
14 \isamarkupfalse% |
|
15 \isacommand{theory}\ Product\ \isakeyword{imports}\ Main\ \isakeyword{begin}% |
|
16 \endisatagtheory |
|
17 {\isafoldtheory}% |
|
18 % |
|
19 \isadelimtheory |
|
20 % |
|
21 \endisadelimtheory |
|
22 \isamarkuptrue% |
|
23 % |
|
24 \begin{isamarkuptext}% |
|
25 \medskip\noindent There is still a feature of Isabelle's type system |
|
26 left that we have not yet discussed. When declaring polymorphic |
|
27 constants \isa{c\ {\isasymColon}\ {\isasymsigma}}, the type variables occurring in \isa{{\isasymsigma}} |
|
28 may be constrained by type classes (or even general sorts) in an |
|
29 arbitrary way. Note that by default, in Isabelle/HOL the |
|
30 declaration \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} is actually an abbreviation |
|
31 for \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}type\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} Since class \isa{type} is the |
|
32 universal class of HOL, this is not really a constraint at all. |
|
33 |
|
34 The \isa{product} class below provides a less degenerate example of |
|
35 syntactic type classes.% |
|
36 \end{isamarkuptext}% |
|
37 \isamarkupfalse% |
|
38 \isacommand{axclass}\isanewline |
|
39 \ \ product\ {\isasymsubseteq}\ type\isanewline |
|
40 \isamarkupfalse% |
|
41 \isacommand{consts}\isanewline |
|
42 \ \ product\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isamarkuptrue% |
|
43 % |
|
44 \begin{isamarkuptext}% |
|
45 Here class \isa{product} is defined as subclass of \isa{type} |
|
46 without any additional axioms. This effects in logical equivalence |
|
47 of \isa{product} and \isa{type}, as is reflected by the trivial |
|
48 introduction rule generated for this definition. |
|
49 |
|
50 \medskip So what is the difference of declaring \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} vs.\ declaring \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}type\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} anyway? In this particular case where \isa{product\ {\isasymequiv}\ type}, it should be obvious that both declarations are the same |
|
51 from the logic's point of view. It even makes the most sense to |
|
52 remove sort constraints from constant declarations, as far as the |
|
53 purely logical meaning is concerned \cite{Wenzel:1997:TPHOL}. |
|
54 |
|
55 On the other hand there are syntactic differences, of course. |
|
56 Constants \isa{{\isasymodot}} on some type \isa{{\isasymtau}} are rejected by the |
|
57 type-checker, unless the arity \isa{{\isasymtau}\ {\isasymColon}\ product} is part of the |
|
58 type signature. In our example, this arity may be always added when |
|
59 required by means of an $\INSTANCE$ with the default proof $\DDOT$. |
|
60 |
|
61 \medskip Thus, we may observe the following discipline of using |
|
62 syntactic classes. Overloaded polymorphic constants have their type |
|
63 arguments restricted to an associated (logically trivial) class |
|
64 \isa{c}. Only immediately before \emph{specifying} these |
|
65 constants on a certain type \isa{{\isasymtau}} do we instantiate \isa{{\isasymtau}\ {\isasymColon}\ c}. |
|
66 |
|
67 This is done for class \isa{product} and type \isa{bool} as |
|
68 follows.% |
|
69 \end{isamarkuptext}% |
|
70 \isamarkupfalse% |
|
71 \isacommand{instance}\ bool\ {\isacharcolon}{\isacharcolon}\ product% |
|
72 \isadelimproof |
|
73 \ % |
|
74 \endisadelimproof |
|
75 % |
|
76 \isatagproof |
|
77 \isamarkupfalse% |
|
78 \isacommand{{\isachardot}{\isachardot}}% |
|
79 \endisatagproof |
|
80 {\isafoldproof}% |
|
81 % |
|
82 \isadelimproof |
|
83 % |
|
84 \endisadelimproof |
|
85 \isanewline |
|
86 \isamarkupfalse% |
|
87 \isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline |
|
88 \ \ product{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymand}\ y{\isachardoublequote}\isamarkuptrue% |
|
89 % |
|
90 \begin{isamarkuptext}% |
|
91 The definition \isa{prod{\isacharunderscore}bool{\isacharunderscore}def} becomes syntactically |
|
92 well-formed only after the arity \isa{bool\ {\isasymColon}\ product} is made |
|
93 known to the type checker. |
|
94 |
|
95 \medskip It is very important to see that above $\DEFS$ are not |
|
96 directly connected with $\INSTANCE$ at all! We were just following |
|
97 our convention to specify \isa{{\isasymodot}} on \isa{bool} after having |
|
98 instantiated \isa{bool\ {\isasymColon}\ product}. Isabelle does not require |
|
99 these definitions, which is in contrast to programming languages like |
|
100 Haskell \cite{haskell-report}. |
|
101 |
|
102 \medskip While Isabelle type classes and those of Haskell are almost |
|
103 the same as far as type-checking and type inference are concerned, |
|
104 there are important semantic differences. Haskell classes require |
|
105 their instances to \emph{provide operations} of certain \emph{names}. |
|
106 Therefore, its \texttt{instance} has a \texttt{where} part that tells |
|
107 the system what these ``member functions'' should be. |
|
108 |
|
109 This style of \texttt{instance} would not make much sense in |
|
110 Isabelle's meta-logic, because there is no internal notion of |
|
111 ``providing operations'' or even ``names of functions''.% |
|
112 \end{isamarkuptext}% |
|
113 % |
|
114 \isadelimtheory |
|
115 % |
|
116 \endisadelimtheory |
|
117 % |
|
118 \isatagtheory |
|
119 \isamarkupfalse% |
|
120 \isacommand{end}% |
|
121 \endisatagtheory |
|
122 {\isafoldtheory}% |
|
123 % |
|
124 \isadelimtheory |
|
125 % |
|
126 \endisadelimtheory |
|
127 \isanewline |
|
128 \end{isabellebody}% |
|
129 %%% Local Variables: |
|
130 %%% mode: latex |
|
131 %%% TeX-master: "root" |
|
132 %%% End: |