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\begin{isabellebody}%
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\begin{isamarkuptext}%
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\noindent
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The task is to develop a compiler from a generic type of expressions (built
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up from variables, constants and binary operations) to a stack machine. This
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generic type of expressions is a generalization of the boolean expressions in
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\S\ref{sec:boolex}. This time we do not commit ourselves to a particular
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type of variables or values but make them type parameters. Neither is there
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a fixed set of binary operations: instead the expression contains the
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appropriate function itself.%
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\end{isamarkuptext}%
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\isacommand{types}\ {\isacharprime}v\ binop\ {\isacharequal}\ {\isachardoublequote}{\isacharprime}v\ {\isasymRightarrow}\ {\isacharprime}v\ {\isasymRightarrow}\ {\isacharprime}v{\isachardoublequote}\isanewline
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\isacommand{datatype}\ {\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}v{\isacharparenright}expr\ {\isacharequal}\ Cex\ {\isacharprime}v\isanewline
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Vex\ {\isacharprime}a\isanewline
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Bex\ {\isachardoublequote}{\isacharprime}v\ binop{\isachardoublequote}\ \ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}v{\isacharparenright}expr{\isachardoublequote}\ \ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}v{\isacharparenright}expr{\isachardoublequote}%
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\begin{isamarkuptext}%
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\noindent
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The three constructors represent constants, variables and the application of
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a binary operation to two subexpressions.
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The value of an expression w.r.t.\ an environment that maps variables to
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values is easily defined:%
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\end{isamarkuptext}%
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\isacommand{consts}\ value\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}v{\isacharparenright}expr\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}v{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}v{\isachardoublequote}\isanewline
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\isacommand{primrec}\isanewline
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{\isachardoublequote}value\ {\isacharparenleft}Cex\ v{\isacharparenright}\ env\ {\isacharequal}\ v{\isachardoublequote}\isanewline
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{\isachardoublequote}value\ {\isacharparenleft}Vex\ a{\isacharparenright}\ env\ {\isacharequal}\ env\ a{\isachardoublequote}\isanewline
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{\isachardoublequote}value\ {\isacharparenleft}Bex\ f\ e\isadigit{1}\ e\isadigit{2}{\isacharparenright}\ env\ {\isacharequal}\ f\ {\isacharparenleft}value\ e\isadigit{1}\ env{\isacharparenright}\ {\isacharparenleft}value\ e\isadigit{2}\ env{\isacharparenright}{\isachardoublequote}%
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\begin{isamarkuptext}%
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The stack machine has three instructions: load a constant value onto the
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stack, load the contents of a certain address onto the stack, and apply a
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binary operation to the two topmost elements of the stack, replacing them by
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the result. As for \isa{expr}, addresses and values are type parameters:%
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\end{isamarkuptext}%
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\isacommand{datatype}\ {\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}v{\isacharparenright}\ instr\ {\isacharequal}\ Const\ {\isacharprime}v\isanewline
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Load\ {\isacharprime}a\isanewline
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Apply\ {\isachardoublequote}{\isacharprime}v\ binop{\isachardoublequote}%
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\begin{isamarkuptext}%
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The execution of the stack machine is modelled by a function
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\isa{exec} that takes a list of instructions, a store (modelled as a
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function from addresses to values, just like the environment for
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evaluating expressions), and a stack (modelled as a list) of values,
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and returns the stack at the end of the execution---the store remains
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unchanged:%
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\end{isamarkuptext}%
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\isacommand{consts}\ exec\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}v{\isacharparenright}instr\ list\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a{\isasymRightarrow}{\isacharprime}v{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}v\ list\ {\isasymRightarrow}\ {\isacharprime}v\ list{\isachardoublequote}\isanewline
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\isacommand{primrec}\isanewline
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{\isachardoublequote}exec\ {\isacharbrackleft}{\isacharbrackright}\ s\ vs\ {\isacharequal}\ vs{\isachardoublequote}\isanewline
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{\isachardoublequote}exec\ {\isacharparenleft}i{\isacharhash}is{\isacharparenright}\ s\ vs\ {\isacharequal}\ {\isacharparenleft}case\ i\ of\isanewline
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\ \ \ \ Const\ v\ \ {\isasymRightarrow}\ exec\ is\ s\ {\isacharparenleft}v{\isacharhash}vs{\isacharparenright}\isanewline
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\ \ {\isacharbar}\ Load\ a\ \ \ {\isasymRightarrow}\ exec\ is\ s\ {\isacharparenleft}{\isacharparenleft}s\ a{\isacharparenright}{\isacharhash}vs{\isacharparenright}\isanewline
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\ \ {\isacharbar}\ Apply\ f\ \ {\isasymRightarrow}\ exec\ is\ s\ {\isacharparenleft}{\isacharparenleft}f\ {\isacharparenleft}hd\ vs{\isacharparenright}\ {\isacharparenleft}hd{\isacharparenleft}tl\ vs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isacharhash}{\isacharparenleft}tl{\isacharparenleft}tl\ vs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequote}%
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\begin{isamarkuptext}%
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\noindent
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Recall that \isa{hd} and \isa{tl}
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return the first element and the remainder of a list.
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Because all functions are total, \isa{hd} is defined even for the empty
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list, although we do not know what the result is. Thus our model of the
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machine always terminates properly, although the above definition does not
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tell us much about the result in situations where \isa{Apply} was executed
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with fewer than two elements on the stack.
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The compiler is a function from expressions to a list of instructions. Its
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definition is pretty much obvious:%
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\end{isamarkuptext}%
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\isacommand{consts}\ comp\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}v{\isacharparenright}expr\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}v{\isacharparenright}instr\ list{\isachardoublequote}\isanewline
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\isacommand{primrec}\isanewline
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{\isachardoublequote}comp\ {\isacharparenleft}Cex\ v{\isacharparenright}\ \ \ \ \ \ \ {\isacharequal}\ {\isacharbrackleft}Const\ v{\isacharbrackright}{\isachardoublequote}\isanewline
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{\isachardoublequote}comp\ {\isacharparenleft}Vex\ a{\isacharparenright}\ \ \ \ \ \ \ {\isacharequal}\ {\isacharbrackleft}Load\ a{\isacharbrackright}{\isachardoublequote}\isanewline
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{\isachardoublequote}comp\ {\isacharparenleft}Bex\ f\ e\isadigit{1}\ e\isadigit{2}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}comp\ e\isadigit{2}{\isacharparenright}\ {\isacharat}\ {\isacharparenleft}comp\ e\isadigit{1}{\isacharparenright}\ {\isacharat}\ {\isacharbrackleft}Apply\ f{\isacharbrackright}{\isachardoublequote}%
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\begin{isamarkuptext}%
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Now we have to prove the correctness of the compiler, i.e.\ that the
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execution of a compiled expression results in the value of the expression:%
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\end{isamarkuptext}%
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\isacommand{theorem}\ {\isachardoublequote}exec\ {\isacharparenleft}comp\ e{\isacharparenright}\ s\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}value\ e\ s{\isacharbrackright}{\isachardoublequote}%
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\begin{isamarkuptext}%
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\noindent
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This theorem needs to be generalized to%
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\end{isamarkuptext}%
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\isacommand{theorem}\ {\isachardoublequote}{\isasymforall}vs{\isachardot}\ exec\ {\isacharparenleft}comp\ e{\isacharparenright}\ s\ vs\ {\isacharequal}\ {\isacharparenleft}value\ e\ s{\isacharparenright}\ {\isacharhash}\ vs{\isachardoublequote}%
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\begin{isamarkuptxt}%
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\noindent
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which is proved by induction on \isa{e} followed by simplification, once
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we have the following lemma about executing the concatenation of two
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instruction sequences:%
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\end{isamarkuptxt}%
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\isacommand{lemma}\ exec{\isacharunderscore}app{\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\isanewline
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\ \ {\isachardoublequote}{\isasymforall}vs{\isachardot}\ exec\ {\isacharparenleft}xs{\isacharat}ys{\isacharparenright}\ s\ vs\ {\isacharequal}\ exec\ ys\ s\ {\isacharparenleft}exec\ xs\ s\ vs{\isacharparenright}{\isachardoublequote}%
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\begin{isamarkuptxt}%
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\noindent
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This requires induction on \isa{xs} and ordinary simplification for the
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base cases. In the induction step, simplification leaves us with a formula
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that contains two \isa{case}-expressions over instructions. Thus we add
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automatic case splitting as well, which finishes the proof:%
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\end{isamarkuptxt}%
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\isacommand{by}{\isacharparenleft}induct{\isacharunderscore}tac\ xs{\isacharcomma}\ simp{\isacharcomma}\ simp\ split{\isacharcolon}\ instr{\isachardot}split{\isacharparenright}%
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\begin{isamarkuptext}%
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\noindent
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Note that because \isaindex{auto} performs simplification, it can
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also be modified in the same way \isa{simp} can. Thus the proof can be
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rewritten as%
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\end{isamarkuptext}%
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\isacommand{by}{\isacharparenleft}induct{\isacharunderscore}tac\ xs{\isacharcomma}\ auto\ split{\isacharcolon}\ instr{\isachardot}split{\isacharparenright}%
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\begin{isamarkuptext}%
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\noindent
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Although this is more compact, it is less clear for the reader of the proof.
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We could now go back and prove \isa{exec (comp e) s [] = [value e s]}
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merely by simplification with the generalized version we just proved.
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However, this is unnecessary because the generalized version fully subsumes
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its instance.%
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\end{isamarkuptext}%
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\end{isabellebody}%
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "root"
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%%% End:
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