--- a/doc-src/TutorialI/Datatype/document/ABexpr.tex Thu May 29 13:27:13 2008 +0200
+++ b/doc-src/TutorialI/Datatype/document/ABexpr.tex Thu May 29 22:45:33 2008 +0200
@@ -49,59 +49,54 @@
The semantics is given by two evaluation functions:%
\end{isamarkuptext}%
\isamarkuptrue%
-\isacommand{consts}\isamarkupfalse%
-\ \ evala\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ aexp\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ evalb\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ bexp\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}%
+\isacommand{primrec}\isamarkupfalse%
+\ evala\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ aexp\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
+\ \ \ \ \ \ \ \ \ evalb\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ bexp\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+{\isachardoublequoteopen}evala\ {\isacharparenleft}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\isanewline
+\ \ \ {\isacharparenleft}if\ evalb\ b\ env\ then\ evala\ a{\isadigit{1}}\ env\ else\ evala\ a{\isadigit{2}}\ env{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}evala\ {\isacharparenleft}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ evala\ a{\isadigit{1}}\ env\ {\isacharplus}\ evala\ a{\isadigit{2}}\ env{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}evala\ {\isacharparenleft}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ evala\ a{\isadigit{1}}\ env\ {\isacharminus}\ evala\ a{\isadigit{2}}\ env{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}evala\ {\isacharparenleft}Var\ v{\isacharparenright}\ env\ {\isacharequal}\ env\ v{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}evala\ {\isacharparenleft}Num\ n{\isacharparenright}\ env\ {\isacharequal}\ n{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+\isanewline
+{\isachardoublequoteopen}evalb\ {\isacharparenleft}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}evala\ a{\isadigit{1}}\ env\ {\isacharless}\ evala\ a{\isadigit{2}}\ env{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}evalb\ {\isacharparenleft}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}evalb\ b{\isadigit{1}}\ env\ {\isasymand}\ evalb\ b{\isadigit{2}}\ env{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}evalb\ {\isacharparenleft}Neg\ b{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}{\isasymnot}\ evalb\ b\ env{\isacharparenright}{\isachardoublequoteclose}%
\begin{isamarkuptext}%
\noindent
-Both take an expression and an environment (a mapping from variables \isa{{\isacharprime}a} to values
-\isa{nat}) and return its arithmetic/boolean
-value. Since the datatypes are mutually recursive, so are functions that
-operate on them. Hence they need to be defined in a single \isacommand{primrec}
-section:%
+
+Both take an expression and an environment (a mapping from variables
+\isa{{\isacharprime}a} to values \isa{nat}) and return its arithmetic/boolean
+value. Since the datatypes are mutually recursive, so are functions
+that operate on them. Hence they need to be defined in a single
+\isacommand{primrec} section. Notice the \isakeyword{and} separating
+the declarations of \isa{evala} and \isa{evalb}. Their defining
+equations need not be split into two groups;
+the empty line is purely for readability.
+
+In the same fashion we also define two functions that perform substitution:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{primrec}\isamarkupfalse%
+\ substa\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ aexp\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
+\ \ \ \ \ \ \ \ \ substb\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ bexp\ {\isasymRightarrow}\ {\isacharprime}b\ bexp{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+{\isachardoublequoteopen}substa\ s\ {\isacharparenleft}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ IF\ {\isacharparenleft}substb\ s\ b{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}substa\ s\ {\isacharparenleft}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ Sum\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}substa\ s\ {\isacharparenleft}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ Diff\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}substa\ s\ {\isacharparenleft}Var\ v{\isacharparenright}\ {\isacharequal}\ s\ v{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}substa\ s\ {\isacharparenleft}Num\ n{\isacharparenright}\ {\isacharequal}\ Num\ n{\isachardoublequoteclose}\ {\isacharbar}\isanewline
\isanewline
-\ \ {\isachardoublequoteopen}evala\ {\isacharparenleft}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\isanewline
-\ \ \ \ \ {\isacharparenleft}if\ evalb\ b\ env\ then\ evala\ a{\isadigit{1}}\ env\ else\ evala\ a{\isadigit{2}}\ env{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}evala\ {\isacharparenleft}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ evala\ a{\isadigit{1}}\ env\ {\isacharplus}\ evala\ a{\isadigit{2}}\ env{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}evala\ {\isacharparenleft}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ evala\ a{\isadigit{1}}\ env\ {\isacharminus}\ evala\ a{\isadigit{2}}\ env{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}evala\ {\isacharparenleft}Var\ v{\isacharparenright}\ env\ {\isacharequal}\ env\ v{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}evala\ {\isacharparenleft}Num\ n{\isacharparenright}\ env\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ {\isachardoublequoteopen}evalb\ {\isacharparenleft}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}evala\ a{\isadigit{1}}\ env\ {\isacharless}\ evala\ a{\isadigit{2}}\ env{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}evalb\ {\isacharparenleft}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}evalb\ b{\isadigit{1}}\ env\ {\isasymand}\ evalb\ b{\isadigit{2}}\ env{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}evalb\ {\isacharparenleft}Neg\ b{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}{\isasymnot}\ evalb\ b\ env{\isacharparenright}{\isachardoublequoteclose}%
+{\isachardoublequoteopen}substb\ s\ {\isacharparenleft}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ Less\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}substb\ s\ {\isacharparenleft}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ And\ {\isacharparenleft}substb\ s\ b{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substb\ s\ b{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbar}\isanewline
+{\isachardoublequoteopen}substb\ s\ {\isacharparenleft}Neg\ b{\isacharparenright}\ {\isacharequal}\ Neg\ {\isacharparenleft}substb\ s\ b{\isacharparenright}{\isachardoublequoteclose}%
\begin{isamarkuptext}%
\noindent
-In the same fashion we also define two functions that perform substitution:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{consts}\isamarkupfalse%
-\ substa\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ aexp\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ substb\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ bexp\ {\isasymRightarrow}\ {\isacharprime}b\ bexp{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent
-The first argument is a function mapping variables to expressions, the
+Their first argument is a function mapping variables to expressions, the
substitution. It is applied to all variables in the second argument. As a
result, the type of variables in the expression may change from \isa{{\isacharprime}a}
-to \isa{{\isacharprime}b}. Note that there are only arithmetic and no boolean variables.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{primrec}\isamarkupfalse%
-\isanewline
-\ \ {\isachardoublequoteopen}substa\ s\ {\isacharparenleft}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ \ IF\ {\isacharparenleft}substb\ s\ b{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}substa\ s\ {\isacharparenleft}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ Sum\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}substa\ s\ {\isacharparenleft}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ Diff\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}substa\ s\ {\isacharparenleft}Var\ v{\isacharparenright}\ {\isacharequal}\ s\ v{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}substa\ s\ {\isacharparenleft}Num\ n{\isacharparenright}\ {\isacharequal}\ Num\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ {\isachardoublequoteopen}substb\ s\ {\isacharparenleft}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ Less\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}substb\ s\ {\isacharparenleft}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ And\ {\isacharparenleft}substb\ s\ b{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substb\ s\ b{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}substb\ s\ {\isacharparenleft}Neg\ b{\isacharparenright}\ {\isacharequal}\ Neg\ {\isacharparenleft}substb\ s\ b{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
+to \isa{{\isacharprime}b}. Note that there are only arithmetic and no boolean variables.
+
Now we can prove a fundamental theorem about the interaction between
evaluation and substitution: applying a substitution $s$ to an expression $a$
and evaluating the result in an environment $env$ yields the same result as