diff -r 0c86acc069ad -r 5deda0549f97 doc-src/TutorialI/CodeGen/document/CodeGen.tex --- a/doc-src/TutorialI/CodeGen/document/CodeGen.tex Thu Jul 26 16:54:44 2012 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,237 +0,0 @@ -% -\begin{isabellebody}% -\def\isabellecontext{CodeGen}% -% -\isadelimtheory -% -\endisadelimtheory -% -\isatagtheory -% -\endisatagtheory -{\isafoldtheory}% -% -\isadelimtheory -% -\endisadelimtheory -% -\isamarkupsection{Case Study: Compiling Expressions% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -\label{sec:ExprCompiler} -\index{compiling expressions example|(}% -The task is to develop a compiler from a generic type of expressions (built -from variables, constants and binary operations) to a stack machine. This -generic type of expressions is a generalization of the boolean expressions in -\S\ref{sec:boolex}. This time we do not commit ourselves to a particular -type of variables or values but make them type parameters. Neither is there -a fixed set of binary operations: instead the expression contains the -appropriate function itself.% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse% -\ {\isaliteral{27}{\isacharprime}}v\ binop\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}v\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}v\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}v{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isacommand{datatype}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}expr\ {\isaliteral{3D}{\isacharequal}}\ Cex\ {\isaliteral{27}{\isacharprime}}v\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\ Vex\ {\isaliteral{27}{\isacharprime}}a\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\ Bex\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}v\ binop{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}expr{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}expr{\isaliteral{22}{\isachardoublequoteclose}}% -\begin{isamarkuptext}% -\noindent -The three constructors represent constants, variables and the application of -a binary operation to two subexpressions. - -The value of an expression with respect to an environment that maps variables to -values is easily defined:% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{primrec}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}value{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}expr\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}v{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline -{\isaliteral{22}{\isachardoublequoteopen}}value\ {\isaliteral{28}{\isacharparenleft}}Cex\ v{\isaliteral{29}{\isacharparenright}}\ env\ {\isaliteral{3D}{\isacharequal}}\ v{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline -{\isaliteral{22}{\isachardoublequoteopen}}value\ {\isaliteral{28}{\isacharparenleft}}Vex\ a{\isaliteral{29}{\isacharparenright}}\ env\ {\isaliteral{3D}{\isacharequal}}\ env\ a{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline -{\isaliteral{22}{\isachardoublequoteopen}}value\ {\isaliteral{28}{\isacharparenleft}}Bex\ f\ e{\isadigit{1}}\ e{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ env\ {\isaliteral{3D}{\isacharequal}}\ f\ {\isaliteral{28}{\isacharparenleft}}value\ e{\isadigit{1}}\ env{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}value\ e{\isadigit{2}}\ env{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% -\begin{isamarkuptext}% -The stack machine has three instructions: load a constant value onto the -stack, load the contents of an address onto the stack, and apply a -binary operation to the two topmost elements of the stack, replacing them by -the result. As for \isa{expr}, addresses and values are type parameters:% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{datatype}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}\ instr\ {\isaliteral{3D}{\isacharequal}}\ Const\ {\isaliteral{27}{\isacharprime}}v\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\ Load\ {\isaliteral{27}{\isacharprime}}a\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\ Apply\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}v\ binop{\isaliteral{22}{\isachardoublequoteclose}}% -\begin{isamarkuptext}% -The execution of the stack machine is modelled by a function -\isa{exec} that takes a list of instructions, a store (modelled as a -function from addresses to values, just like the environment for -evaluating expressions), and a stack (modelled as a list) of values, -and returns the stack at the end of the execution --- the store remains -unchanged:% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{primrec}\isamarkupfalse% -\ exec\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}instr\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}{\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}v\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}v\ list{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isakeyword{where}\isanewline -{\isaliteral{22}{\isachardoublequoteopen}}exec\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ s\ vs\ {\isaliteral{3D}{\isacharequal}}\ vs{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline -{\isaliteral{22}{\isachardoublequoteopen}}exec\ {\isaliteral{28}{\isacharparenleft}}i{\isaliteral{23}{\isacharhash}}is{\isaliteral{29}{\isacharparenright}}\ s\ vs\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}case\ i\ of\isanewline -\ \ \ \ Const\ v\ \ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ exec\ is\ s\ {\isaliteral{28}{\isacharparenleft}}v{\isaliteral{23}{\isacharhash}}vs{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ {\isaliteral{7C}{\isacharbar}}\ Load\ a\ \ \ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ exec\ is\ s\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}s\ a{\isaliteral{29}{\isacharparenright}}{\isaliteral{23}{\isacharhash}}vs{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ {\isaliteral{7C}{\isacharbar}}\ Apply\ f\ \ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ exec\ is\ s\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}f\ {\isaliteral{28}{\isacharparenleft}}hd\ vs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}hd{\isaliteral{28}{\isacharparenleft}}tl\ vs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{23}{\isacharhash}}{\isaliteral{28}{\isacharparenleft}}tl{\isaliteral{28}{\isacharparenleft}}tl\ vs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% -\begin{isamarkuptext}% -\noindent -Recall that \isa{hd} and \isa{tl} -return the first element and the remainder of a list. -Because all functions are total, \cdx{hd} is defined even for the empty -list, although we do not know what the result is. Thus our model of the -machine always terminates properly, although the definition above does not -tell us much about the result in situations where \isa{Apply} was executed -with fewer than two elements on the stack. - -The compiler is a function from expressions to a list of instructions. Its -definition is obvious:% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{primrec}\isamarkupfalse% -\ compile\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}expr\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}v{\isaliteral{29}{\isacharparenright}}instr\ list{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline -{\isaliteral{22}{\isachardoublequoteopen}}compile\ {\isaliteral{28}{\isacharparenleft}}Cex\ v{\isaliteral{29}{\isacharparenright}}\ \ \ \ \ \ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}Const\ v{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline -{\isaliteral{22}{\isachardoublequoteopen}}compile\ {\isaliteral{28}{\isacharparenleft}}Vex\ a{\isaliteral{29}{\isacharparenright}}\ \ \ \ \ \ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}Load\ a{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline -{\isaliteral{22}{\isachardoublequoteopen}}compile\ {\isaliteral{28}{\isacharparenleft}}Bex\ f\ e{\isadigit{1}}\ e{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}compile\ e{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{28}{\isacharparenleft}}compile\ e{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}Apply\ f{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}% -\begin{isamarkuptext}% -Now we have to prove the correctness of the compiler, i.e.\ that the -execution of a compiled expression results in the value of the expression:% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{theorem}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}exec\ {\isaliteral{28}{\isacharparenleft}}compile\ e{\isaliteral{29}{\isacharparenright}}\ s\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}value\ e\ s{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\begin{isamarkuptext}% -\noindent -This theorem needs to be generalized:% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{theorem}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}vs{\isaliteral{2E}{\isachardot}}\ exec\ {\isaliteral{28}{\isacharparenleft}}compile\ e{\isaliteral{29}{\isacharparenright}}\ s\ vs\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}value\ e\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{23}{\isacharhash}}\ vs{\isaliteral{22}{\isachardoublequoteclose}}% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -% -\begin{isamarkuptxt}% -\noindent -It will be proved by induction on \isa{e} followed by simplification. -First, we must prove a lemma about executing the concatenation of two -instruction sequences:% -\end{isamarkuptxt}% -\isamarkuptrue% -% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -\isacommand{lemma}\isamarkupfalse% -\ exec{\isaliteral{5F}{\isacharunderscore}}app{\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline -\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}vs{\isaliteral{2E}{\isachardot}}\ exec\ {\isaliteral{28}{\isacharparenleft}}xs{\isaliteral{40}{\isacharat}}ys{\isaliteral{29}{\isacharparenright}}\ s\ vs\ {\isaliteral{3D}{\isacharequal}}\ exec\ ys\ s\ {\isaliteral{28}{\isacharparenleft}}exec\ xs\ s\ vs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -% -\begin{isamarkuptxt}% -\noindent -This requires induction on \isa{xs} and ordinary simplification for the -base cases. In the induction step, simplification leaves us with a formula -that contains two \isa{case}-expressions over instructions. Thus we add -automatic case splitting, which finishes the proof:% -\end{isamarkuptxt}% -\isamarkuptrue% -\isacommand{apply}\isamarkupfalse% -{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{2C}{\isacharcomma}}\ simp{\isaliteral{2C}{\isacharcomma}}\ simp\ split{\isaliteral{3A}{\isacharcolon}}\ instr{\isaliteral{2E}{\isachardot}}split{\isaliteral{29}{\isacharparenright}}% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\begin{isamarkuptext}% -\noindent -Note that because both \methdx{simp_all} and \methdx{auto} perform simplification, they can -be modified in the same way as \isa{simp}. Thus the proof can be -rewritten as% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -\isacommand{apply}\isamarkupfalse% -{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{2C}{\isacharcomma}}\ simp{\isaliteral{5F}{\isacharunderscore}}all\ split{\isaliteral{3A}{\isacharcolon}}\ instr{\isaliteral{2E}{\isachardot}}split{\isaliteral{29}{\isacharparenright}}% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\begin{isamarkuptext}% -\noindent -Although this is more compact, it is less clear for the reader of the proof. - -We could now go back and prove \isa{exec\ {\isaliteral{28}{\isacharparenleft}}compile\ e{\isaliteral{29}{\isacharparenright}}\ s\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}value\ e\ s{\isaliteral{5D}{\isacharbrackright}}} -merely by simplification with the generalized version we just proved. -However, this is unnecessary because the generalized version fully subsumes -its instance.% -\index{compiling expressions example|)}% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\isadelimtheory -% -\endisadelimtheory -% -\isatagtheory -% -\endisatagtheory -{\isafoldtheory}% -% -\isadelimtheory -% -\endisadelimtheory -\end{isabellebody}% -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "root" -%%% End: