author nipkow Wed, 19 Apr 2000 12:54:56 +0200 changeset 8746 ccbb5e0dccdf parent 8745 13b32661dde4 child 8747 22580c8bc62f
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/CodeGen/document/CodeGen.tex	Wed Apr 19 12:54:56 2000 +0200
@@ -0,0 +1,113 @@
+\begin{isabelle}%
+%
+\begin{isamarkuptext}%
+\noindent
+The task is to develop a compiler from a generic type of expressions (built
+up 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}%
+\isacommand{types}~'v~binop~=~{"}'v~{\isasymRightarrow}~'v~{\isasymRightarrow}~'v{"}\isanewline
+\isacommand{datatype}~('a,'v)expr~=~Cex~'v\isanewline
+~~~~~~~~~~~~~~~~~~~~~|~Vex~'a\isanewline
+~~~~~~~~~~~~~~~~~~~~~|~Bex~{"}'v~binop{"}~~{"}('a,'v)expr{"}~~{"}('a,'v)expr{"}%
+\begin{isamarkuptext}%
+\noindent
+The three constructors represent constants, variables and the combination of
+two subexpressions with a binary operation.
+
+The value of an expression w.r.t.\ an environment that maps variables to
+values is easily defined:%
+\end{isamarkuptext}%
+\isacommand{consts}~value~::~{"}('a~{\isasymRightarrow}~'v)~{\isasymRightarrow}~('a,'v)expr~{\isasymRightarrow}~'v{"}\isanewline
+\isacommand{primrec}\isanewline
+{"}value~env~(Cex~v)~=~v{"}\isanewline
+{"}value~env~(Vex~a)~=~env~a{"}\isanewline
+{"}value~env~(Bex~f~e1~e2)~=~f~(value~env~e1)~(value~env~e2){"}%
+\begin{isamarkuptext}%
+The stack machine has three instructions: load a constant value onto the
+stack, load the contents of a certain 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}%
+\isacommand{datatype}~('a,'v)~instr~=~Const~'v\isanewline
+~~~~~~~~~~~~~~~~~~~~~~~|~Apply~{"}'v~binop{"}%
+\begin{isamarkuptext}%
+The execution of the stack machine is modelled by a function \isa{exec}
+that takes a store (modelled as a function from addresses to values, just
+like the environment for evaluating expressions), a stack (modelled as a
+list) of values, and a list of instructions, and returns the stack at the end
+of the execution---the store remains unchanged:%
+\end{isamarkuptext}%
+\isacommand{consts}~exec~::~{"}('a{\isasymRightarrow}'v)~{\isasymRightarrow}~'v~list~{\isasymRightarrow}~('a,'v)instr~list~{\isasymRightarrow}~'v~list{"}\isanewline
+\isacommand{primrec}\isanewline
+{"}exec~s~vs~[]~=~vs{"}\isanewline
+{"}exec~s~vs~(i\#is)~=~(case~i~of\isanewline
+~~~~Const~v~~{\isasymRightarrow}~exec~s~(v\#vs)~is\isanewline
+~~|~Apply~f~~{\isasymRightarrow}~exec~s~(~(f~(hd~vs)~(hd(tl~vs)))\#(tl(tl~vs))~)~is){"}%
+\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, \isa{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 above definition 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 pretty much obvious:%
+\end{isamarkuptext}%
+\isacommand{consts}~comp~::~{"}('a,'v)expr~{\isasymRightarrow}~('a,'v)instr~list{"}\isanewline
+\isacommand{primrec}\isanewline
+{"}comp~(Cex~v)~~~~~~~=~[Const~v]{"}\isanewline
+{"}comp~(Bex~f~e1~e2)~=~(comp~e2)~@~(comp~e1)~@~[Apply~f]{"}%
+\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}%
+\isacommand{theorem}~{"}exec~s~[]~(comp~e)~=~[value~s~e]{"}%
+\begin{isamarkuptext}%
+\noindent
+This theorem needs to be generalized to%
+\end{isamarkuptext}%
+\isacommand{theorem}~{"}{\isasymforall}vs.~exec~s~vs~(comp~e)~=~(value~s~e)~\#~vs{"}%
+\begin{isamarkuptxt}%
+\noindent
+which is proved by induction on \isa{e} followed by simplification, once
+we have the following lemma about executing the concatenation of two
+instruction sequences:%
+\end{isamarkuptxt}%
+\isacommand{lemma}~exec\_app[simp]:\isanewline
+~~{"}{\isasymforall}vs.~exec~s~vs~(xs@ys)~=~exec~s~(exec~s~vs~xs)~ys{"}%
+\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 as well, which finishes the proof:%
+\end{isamarkuptxt}%
+\isacommand{apply}(induct\_tac~xs,~simp,~simp~split:~instr.split)\isacommand{.}%
+\begin{isamarkuptext}%
+\noindent
+Note that because \isaindex{auto} performs simplification, it can
+also be modified in the same way \isa{simp} can. Thus the proof can be
+rewritten as%
+\end{isamarkuptext}%
+\isacommand{apply}(induct\_tac~xs,~auto~split:~instr.split)\isacommand{.}%
+\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 s [] (comp e) = [value s e]}
+merely by simplification with the generalized version we just proved.
+However, this is unnecessary because the generalized version fully subsumes
+its instance.%
+\end{isamarkuptext}%
+\end{isabelle}%
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/CodeGen/document/session.tex	Wed Apr 19 12:54:56 2000 +0200
@@ -0,0 +1,1 @@
+\input{CodeGen.tex}