--- a/doc-src/AxClass/Nat/document/NatClass.tex Fri Aug 19 22:25:14 2005 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,96 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{NatClass}%
-\isamarkuptrue%
-%
-\isamarkupheader{Defining natural numbers in FOL \label{sec:ex-natclass}%
-}
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isamarkupfalse%
-\isacommand{theory}\ NatClass\ \isakeyword{imports}\ FOL\ \isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\medskip\noindent Axiomatic type classes abstract over exactly one
- type argument. Thus, any \emph{axiomatic} theory extension where each
- axiom refers to at most one type variable, may be trivially turned
- into a \emph{definitional} one.
-
- We illustrate this with the natural numbers in
- Isabelle/FOL.\footnote{See also
- \url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}}%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{consts}\isanewline
-\ \ zero\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymzero}{\isachardoublequote}{\isacharparenright}\isanewline
-\ \ Suc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline
-\ \ rec\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline
-\isanewline
-\isamarkupfalse%
-\isacommand{axclass}\ nat\ {\isasymsubseteq}\ {\isachardoublequote}term{\isachardoublequote}\isanewline
-\ \ induct{\isacharcolon}\ {\isachardoublequote}P{\isacharparenleft}{\isasymzero}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ P{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}Suc{\isacharparenleft}x{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}n{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ Suc{\isacharunderscore}inject{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ Suc{\isacharparenleft}n{\isacharparenright}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n{\isachardoublequote}\isanewline
-\ \ Suc{\isacharunderscore}neq{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ {\isasymzero}\ {\isasymLongrightarrow}\ R{\isachardoublequote}\isanewline
-\ \ rec{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequote}rec{\isacharparenleft}{\isasymzero}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ a{\isachardoublequote}\isanewline
-\ \ rec{\isacharunderscore}Suc{\isacharcolon}\ {\isachardoublequote}rec{\isacharparenleft}Suc{\isacharparenleft}m{\isacharparenright}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ f{\isacharparenleft}m{\isacharcomma}\ rec{\isacharparenleft}m{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
-\isanewline
-\isamarkupfalse%
-\isacommand{constdefs}\isanewline
-\ \ add\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isacharcolon}{\isacharcolon}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isacharplus}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ {\isachardoublequote}m\ {\isacharplus}\ n\ {\isasymequiv}\ rec{\isacharparenleft}m{\isacharcomma}\ n{\isacharcomma}\ {\isasymlambda}x\ y{\isachardot}\ Suc{\isacharparenleft}y{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-This is an abstract version of the plain \isa{Nat} theory in
- FOL.\footnote{See
- \url{http://isabelle.in.tum.de/library/FOL/ex/Nat.html}} Basically,
- we have just replaced all occurrences of type \isa{nat} by \isa{{\isacharprime}a} and used the natural number axioms to define class \isa{nat}.
- There is only a minor snag, that the original recursion operator
- \isa{rec} had to be made monomorphic.
-
- Thus class \isa{nat} contains exactly those types \isa{{\isasymtau}} that
- are isomorphic to ``the'' natural numbers (with signature \isa{{\isasymzero}}, \isa{Suc}, \isa{rec}).
-
- \medskip What we have done here can be also viewed as \emph{type
- specification}. Of course, it still remains open if there is some
- type at all that meets the class axioms. Now a very nice property of
- axiomatic type classes is that abstract reasoning is always possible
- --- independent of satisfiability. The meta-logic won't break, even
- if some classes (or general sorts) turns out to be empty later ---
- ``inconsistent'' class definitions may be useless, but do not cause
- any harm.
-
- Theorems of the abstract natural numbers may be derived in the same
- way as for the concrete version. The original proof scripts may be
- re-used with some trivial changes only (mostly adding some type
- constraints).%
-\end{isamarkuptext}%
-%
-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isamarkupfalse%
-\isacommand{end}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
-\end{isabellebody}%
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