--- 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
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-\begin{isabellebody}%
-\def\isabellecontext{CodeGen}%
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\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
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-\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
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-\endisadelimtheory
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-\isatagtheory
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
-\end{isabellebody}%
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