8749
|
1 |
\begin{isabelle}%
|
|
2 |
%
|
|
3 |
\begin{isamarkuptext}%
|
|
4 |
\subsubsection{How can we model boolean expressions?}
|
|
5 |
|
|
6 |
We want to represent boolean expressions built up from variables and
|
|
7 |
constants by negation and conjunction. The following datatype serves exactly
|
|
8 |
that purpose:%
|
|
9 |
\end{isamarkuptext}%
|
|
10 |
\isacommand{datatype}~boolex~=~Const~bool~|~Var~nat~|~Neg~boolex\isanewline
|
|
11 |
~~~~~~~~~~~~~~~~|~And~boolex~boolex%
|
|
12 |
\begin{isamarkuptext}%
|
|
13 |
\noindent
|
|
14 |
The two constants are represented by \isa{Const~True} and
|
|
15 |
\isa{Const~False}. Variables are represented by terms of the form
|
|
16 |
\isa{Var~$n$}, where $n$ is a natural number (type \isa{nat}).
|
|
17 |
For example, the formula $P@0 \land \neg P@1$ is represented by the term
|
|
18 |
\isa{And~(Var~0)~(Neg(Var~1))}.
|
|
19 |
|
|
20 |
\subsubsection{What is the value of a boolean expression?}
|
|
21 |
|
|
22 |
The value of a boolean expression depends on the value of its variables.
|
|
23 |
Hence the function \isa{value} takes an additional parameter, an {\em
|
|
24 |
environment} of type \isa{nat \isasymFun\ bool}, which maps variables to
|
|
25 |
their values:%
|
|
26 |
\end{isamarkuptext}%
|
|
27 |
\isacommand{consts}~value~::~{"}boolex~{\isasymRightarrow}~(nat~{\isasymRightarrow}~bool)~{\isasymRightarrow}~bool{"}\isanewline
|
|
28 |
\isacommand{primrec}\isanewline
|
|
29 |
{"}value~(Const~b)~env~=~b{"}\isanewline
|
|
30 |
{"}value~(Var~x)~~~env~=~env~x{"}\isanewline
|
|
31 |
{"}value~(Neg~b)~~~env~=~({\isasymnot}~value~b~env){"}\isanewline
|
|
32 |
{"}value~(And~b~c)~env~=~(value~b~env~{\isasymand}~value~c~env){"}%
|
|
33 |
\begin{isamarkuptext}%
|
|
34 |
\noindent
|
|
35 |
\subsubsection{If-expressions}
|
|
36 |
|
|
37 |
An alternative and often more efficient (because in a certain sense
|
|
38 |
canonical) representation are so-called \emph{If-expressions} built up
|
|
39 |
from constants (\isa{CIF}), variables (\isa{VIF}) and conditionals
|
|
40 |
(\isa{IF}):%
|
|
41 |
\end{isamarkuptext}%
|
|
42 |
\isacommand{datatype}~ifex~=~CIF~bool~|~VIF~nat~|~IF~ifex~ifex~ifex%
|
|
43 |
\begin{isamarkuptext}%
|
|
44 |
\noindent
|
|
45 |
The evaluation if If-expressions proceeds as for \isa{boolex}:%
|
|
46 |
\end{isamarkuptext}%
|
|
47 |
\isacommand{consts}~valif~::~{"}ifex~{\isasymRightarrow}~(nat~{\isasymRightarrow}~bool)~{\isasymRightarrow}~bool{"}\isanewline
|
|
48 |
\isacommand{primrec}\isanewline
|
|
49 |
{"}valif~(CIF~b)~~~~env~=~b{"}\isanewline
|
|
50 |
{"}valif~(VIF~x)~~~~env~=~env~x{"}\isanewline
|
|
51 |
{"}valif~(IF~b~t~e)~env~=~(if~valif~b~env~then~valif~t~env\isanewline
|
|
52 |
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~else~valif~e~env){"}%
|
|
53 |
\begin{isamarkuptext}%
|
|
54 |
\subsubsection{Transformation into and of If-expressions}
|
|
55 |
|
|
56 |
The type \isa{boolex} is close to the customary representation of logical
|
8771
|
57 |
formulae, whereas \isa{ifex} is designed for efficiency. It is easy to
|
8749
|
58 |
translate from \isa{boolex} into \isa{ifex}:%
|
|
59 |
\end{isamarkuptext}%
|
|
60 |
\isacommand{consts}~bool2if~::~{"}boolex~{\isasymRightarrow}~ifex{"}\isanewline
|
|
61 |
\isacommand{primrec}\isanewline
|
|
62 |
{"}bool2if~(Const~b)~=~CIF~b{"}\isanewline
|
|
63 |
{"}bool2if~(Var~x)~~~=~VIF~x{"}\isanewline
|
|
64 |
{"}bool2if~(Neg~b)~~~=~IF~(bool2if~b)~(CIF~False)~(CIF~True){"}\isanewline
|
|
65 |
{"}bool2if~(And~b~c)~=~IF~(bool2if~b)~(bool2if~c)~(CIF~False){"}%
|
|
66 |
\begin{isamarkuptext}%
|
|
67 |
\noindent
|
|
68 |
At last, we have something we can verify: that \isa{bool2if} preserves the
|
|
69 |
value of its argument:%
|
|
70 |
\end{isamarkuptext}%
|
|
71 |
\isacommand{lemma}~{"}valif~(bool2if~b)~env~=~value~b~env{"}%
|
|
72 |
\begin{isamarkuptxt}%
|
|
73 |
\noindent
|
|
74 |
The proof is canonical:%
|
|
75 |
\end{isamarkuptxt}%
|
|
76 |
\isacommand{apply}(induct\_tac~b)\isanewline
|
9458
|
77 |
\isacommand{by}(auto)%
|
8749
|
78 |
\begin{isamarkuptext}%
|
|
79 |
\noindent
|
|
80 |
In fact, all proofs in this case study look exactly like this. Hence we do
|
|
81 |
not show them below.
|
|
82 |
|
|
83 |
More interesting is the transformation of If-expressions into a normal form
|
|
84 |
where the first argument of \isa{IF} cannot be another \isa{IF} but
|
|
85 |
must be a constant or variable. Such a normal form can be computed by
|
|
86 |
repeatedly replacing a subterm of the form \isa{IF~(IF~b~x~y)~z~u} by
|
|
87 |
\isa{IF b (IF x z u) (IF y z u)}, which has the same value. The following
|
|
88 |
primitive recursive functions perform this task:%
|
|
89 |
\end{isamarkuptext}%
|
|
90 |
\isacommand{consts}~normif~::~{"}ifex~{\isasymRightarrow}~ifex~{\isasymRightarrow}~ifex~{\isasymRightarrow}~ifex{"}\isanewline
|
|
91 |
\isacommand{primrec}\isanewline
|
|
92 |
{"}normif~(CIF~b)~~~~t~e~=~IF~(CIF~b)~t~e{"}\isanewline
|
|
93 |
{"}normif~(VIF~x)~~~~t~e~=~IF~(VIF~x)~t~e{"}\isanewline
|
|
94 |
{"}normif~(IF~b~t~e)~u~f~=~normif~b~(normif~t~u~f)~(normif~e~u~f){"}\isanewline
|
|
95 |
\isanewline
|
|
96 |
\isacommand{consts}~norm~::~{"}ifex~{\isasymRightarrow}~ifex{"}\isanewline
|
|
97 |
\isacommand{primrec}\isanewline
|
|
98 |
{"}norm~(CIF~b)~~~~=~CIF~b{"}\isanewline
|
|
99 |
{"}norm~(VIF~x)~~~~=~VIF~x{"}\isanewline
|
|
100 |
{"}norm~(IF~b~t~e)~=~normif~b~(norm~t)~(norm~e){"}%
|
|
101 |
\begin{isamarkuptext}%
|
|
102 |
\noindent
|
|
103 |
Their interplay is a bit tricky, and we leave it to the reader to develop an
|
|
104 |
intuitive understanding. Fortunately, Isabelle can help us to verify that the
|
|
105 |
transformation preserves the value of the expression:%
|
|
106 |
\end{isamarkuptext}%
|
|
107 |
\isacommand{theorem}~{"}valif~(norm~b)~env~=~valif~b~env{"}%
|
|
108 |
\begin{isamarkuptext}%
|
|
109 |
\noindent
|
|
110 |
The proof is canonical, provided we first show the following simplification
|
|
111 |
lemma (which also helps to understand what \isa{normif} does):%
|
|
112 |
\end{isamarkuptext}%
|
|
113 |
\isacommand{lemma}~[simp]:\isanewline
|
|
114 |
~~{"}{\isasymforall}t~e.~valif~(normif~b~t~e)~env~=~valif~(IF~b~t~e)~env{"}%
|
|
115 |
\begin{isamarkuptext}%
|
|
116 |
\noindent
|
|
117 |
Note that the lemma does not have a name, but is implicitly used in the proof
|
|
118 |
of the theorem shown above because of the \isa{[simp]} attribute.
|
|
119 |
|
|
120 |
But how can we be sure that \isa{norm} really produces a normal form in
|
|
121 |
the above sense? We define a function that tests If-expressions for normality%
|
|
122 |
\end{isamarkuptext}%
|
|
123 |
\isacommand{consts}~normal~::~{"}ifex~{\isasymRightarrow}~bool{"}\isanewline
|
|
124 |
\isacommand{primrec}\isanewline
|
|
125 |
{"}normal(CIF~b)~=~True{"}\isanewline
|
|
126 |
{"}normal(VIF~x)~=~True{"}\isanewline
|
|
127 |
{"}normal(IF~b~t~e)~=~(normal~t~{\isasymand}~normal~e~{\isasymand}\isanewline
|
|
128 |
~~~~~(case~b~of~CIF~b~{\isasymRightarrow}~True~|~VIF~x~{\isasymRightarrow}~True~|~IF~x~y~z~{\isasymRightarrow}~False)){"}%
|
|
129 |
\begin{isamarkuptext}%
|
|
130 |
\noindent
|
|
131 |
and prove \isa{normal(norm b)}. Of course, this requires a lemma about
|
|
132 |
normality of \isa{normif}:%
|
|
133 |
\end{isamarkuptext}%
|
9458
|
134 |
\isacommand{lemma}[simp]:~{"}{\isasymforall}t~e.~normal(normif~b~t~e)~=~(normal~t~{\isasymand}~normal~e){"}\end{isabelle}%
|
9145
|
135 |
%%% Local Variables:
|
|
136 |
%%% mode: latex
|
|
137 |
%%% TeX-master: "root"
|
|
138 |
%%% End:
|