doc-src/TutorialI/Inductive/document/Star.tex
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     1 %
     2 \begin{isabellebody}%
     3 \def\isabellecontext{Star}%
     4 %
     5 \isamarkupsection{The reflexive transitive closure%
     6 }
     7 %
     8 \begin{isamarkuptext}%
     9 \label{sec:rtc}
    10 Many inductive definitions define proper relations rather than merely set
    11 like \isa{even}. A perfect example is the
    12 reflexive transitive closure of a relation. This concept was already
    13 introduced in \S\ref{sec:Relations}, where the operator \isa{{\isacharcircum}{\isacharasterisk}} was
    14 defined as a least fixed point because inductive definitions were not yet
    15 available. But now they are:%
    16 \end{isamarkuptext}%
    17 \isacommand{consts}\ rtc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}set{\isachardoublequote}\ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharunderscore}{\isacharasterisk}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline
    18 \isacommand{inductive}\ {\isachardoublequote}r{\isacharasterisk}{\isachardoublequote}\isanewline
    19 \isakeyword{intros}\isanewline
    20 rtc{\isacharunderscore}refl{\isacharbrackleft}iff{\isacharbrackright}{\isacharcolon}\ \ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}x{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline
    21 rtc{\isacharunderscore}step{\isacharcolon}\ \ \ \ \ \ \ {\isachardoublequote}{\isasymlbrakk}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}%
    22 \begin{isamarkuptext}%
    23 \noindent
    24 The function \isa{rtc} is annotated with concrete syntax: instead of
    25 \isa{rtc\ r} we can read and write \isa{r{\isacharasterisk}}. The actual definition
    26 consists of two rules. Reflexivity is obvious and is immediately given the
    27 \isa{iff} attribute to increase automation. The
    28 second rule, \isa{rtc{\isacharunderscore}step}, says that we can always add one more
    29 \isa{r}-step to the left. Although we could make \isa{rtc{\isacharunderscore}step} an
    30 introduction rule, this is dangerous: the recursion in the second premise
    31 slows down and may even kill the automatic tactics.
    32 
    33 The above definition of the concept of reflexive transitive closure may
    34 be sufficiently intuitive but it is certainly not the only possible one:
    35 for a start, it does not even mention transitivity explicitly.
    36 The rest of this section is devoted to proving that it is equivalent to
    37 the ``standard'' definition. We start with a simple lemma:%
    38 \end{isamarkuptext}%
    39 \isacommand{lemma}\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isacharcolon}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline
    40 \isacommand{by}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isacharunderscore}step{\isacharparenright}%
    41 \begin{isamarkuptext}%
    42 \noindent
    43 Although the lemma itself is an unremarkable consequence of the basic rules,
    44 it has the advantage that it can be declared an introduction rule without the
    45 danger of killing the automatic tactics because \isa{r{\isacharasterisk}} occurs only in
    46 the conclusion and not in the premise. Thus some proofs that would otherwise
    47 need \isa{rtc{\isacharunderscore}step} can now be found automatically. The proof also
    48 shows that \isa{blast} is quite able to handle \isa{rtc{\isacharunderscore}step}. But
    49 some of the other automatic tactics are more sensitive, and even \isa{blast} can be lead astray in the presence of large numbers of rules.
    50 
    51 To prove transitivity, we need rule induction, i.e.\ theorem
    52 \isa{rtc{\isachardot}induct}:
    53 \begin{isabelle}%
    54 \ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}{\isacharquery}xb{\isacharcomma}\ {\isacharquery}xa{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ {\isacharquery}P\ x\ x{\isacharsemicolon}\isanewline
    55 \ \ \ \ \ \ \ \ {\isasymAnd}x\ y\ z{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}{\isacharsemicolon}\ {\isacharquery}P\ y\ z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ x\ z{\isasymrbrakk}\isanewline
    56 \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}xb\ {\isacharquery}xa%
    57 \end{isabelle}
    58 It says that \isa{{\isacharquery}P} holds for an arbitrary pair \isa{{\isacharparenleft}{\isacharquery}xb{\isacharcomma}{\isacharquery}xa{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}} if \isa{{\isacharquery}P} is preserved by all rules of the inductive definition,
    59 i.e.\ if \isa{{\isacharquery}P} holds for the conclusion provided it holds for the
    60 premises. In general, rule induction for an $n$-ary inductive relation $R$
    61 expects a premise of the form $(x@1,\dots,x@n) \in R$.
    62 
    63 Now we turn to the inductive proof of transitivity:%
    64 \end{isamarkuptext}%
    65 \isacommand{lemma}\ rtc{\isacharunderscore}trans{\isacharcolon}\ {\isachardoublequote}{\isasymlbrakk}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline
    66 \isacommand{apply}{\isacharparenleft}erule\ rtc{\isachardot}induct{\isacharparenright}%
    67 \begin{isamarkuptxt}%
    68 \noindent
    69 Unfortunately, even the resulting base case is a problem
    70 \begin{isabelle}%
    71 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%
    72 \end{isabelle}
    73 and maybe not what you had expected. We have to abandon this proof attempt.
    74 To understand what is going on, let us look again at \isa{rtc{\isachardot}induct}.
    75 In the above application of \isa{erule}, the first premise of
    76 \isa{rtc{\isachardot}induct} is unified with the first suitable assumption, which
    77 is \isa{{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}} rather than \isa{{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}}. Although that
    78 is what we want, it is merely due to the order in which the assumptions occur
    79 in the subgoal, which it is not good practice to rely on. As a result,
    80 \isa{{\isacharquery}xb} becomes \isa{x}, \isa{{\isacharquery}xa} becomes
    81 \isa{y} and \isa{{\isacharquery}P} becomes \isa{{\isasymlambda}u\ v{\isachardot}\ {\isacharparenleft}u{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}}, thus
    82 yielding the above subgoal. So what went wrong?
    83 
    84 When looking at the instantiation of \isa{{\isacharquery}P} we see that it does not
    85 depend on its second parameter at all. The reason is that in our original
    86 goal, of the pair \isa{{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}} only \isa{x} appears also in the
    87 conclusion, but not \isa{y}. Thus our induction statement is too
    88 weak. Fortunately, it can easily be strengthened:
    89 transfer the additional premise \isa{{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}} into the conclusion:%
    90 \end{isamarkuptxt}%
    91 \isacommand{lemma}\ rtc{\isacharunderscore}trans{\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}{\isacharcolon}\isanewline
    92 \ \ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}y{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}%
    93 \begin{isamarkuptxt}%
    94 \noindent
    95 This is not an obscure trick but a generally applicable heuristic:
    96 \begin{quote}\em
    97 Whe proving a statement by rule induction on $(x@1,\dots,x@n) \in R$,
    98 pull all other premises containing any of the $x@i$ into the conclusion
    99 using $\longrightarrow$.
   100 \end{quote}
   101 A similar heuristic for other kinds of inductions is formulated in
   102 \S\ref{sec:ind-var-in-prems}. The \isa{rule{\isacharunderscore}format} directive turns
   103 \isa{{\isasymlongrightarrow}} back into \isa{{\isasymLongrightarrow}}. Thus in the end we obtain the original
   104 statement of our lemma.%
   105 \end{isamarkuptxt}%
   106 \isacommand{apply}{\isacharparenleft}erule\ rtc{\isachardot}induct{\isacharparenright}%
   107 \begin{isamarkuptxt}%
   108 \noindent
   109 Now induction produces two subgoals which are both proved automatically:
   110 \begin{isabelle}%
   111 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\isanewline
   112 \ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y\ za{\isachardot}\isanewline
   113 \ \ \ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ za{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isacharsemicolon}\ {\isacharparenleft}za{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isasymrbrakk}\isanewline
   114 \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}za{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%
   115 \end{isabelle}%
   116 \end{isamarkuptxt}%
   117 \ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
   118 \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isacharunderscore}step{\isacharparenright}\isanewline
   119 \isacommand{done}%
   120 \begin{isamarkuptext}%
   121 Let us now prove that \isa{r{\isacharasterisk}} is really the reflexive transitive closure
   122 of \isa{r}, i.e.\ the least reflexive and transitive
   123 relation containing \isa{r}. The latter is easily formalized%
   124 \end{isamarkuptext}%
   125 \isacommand{consts}\ rtc{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}set{\isachardoublequote}\isanewline
   126 \isacommand{inductive}\ {\isachardoublequote}rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline
   127 \isakeyword{intros}\isanewline
   128 {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline
   129 {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}x{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline
   130 {\isachardoublequote}{\isasymlbrakk}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}%
   131 \begin{isamarkuptext}%
   132 \noindent
   133 and the equivalence of the two definitions is easily shown by the obvious rule
   134 inductions:%
   135 \end{isamarkuptext}%
   136 \isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline
   137 \isacommand{apply}{\isacharparenleft}erule\ rtc{\isadigit{2}}{\isachardot}induct{\isacharparenright}\isanewline
   138 \ \ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
   139 \ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
   140 \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isacharunderscore}trans{\isacharparenright}\isanewline
   141 \isacommand{done}\isanewline
   142 \isanewline
   143 \isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline
   144 \isacommand{apply}{\isacharparenleft}erule\ rtc{\isachardot}induct{\isacharparenright}\isanewline
   145 \ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isadigit{2}}{\isachardot}intros{\isacharparenright}\isanewline
   146 \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isadigit{2}}{\isachardot}intros{\isacharparenright}\isanewline
   147 \isacommand{done}%
   148 \begin{isamarkuptext}%
   149 So why did we start with the first definition? Because it is simpler. It
   150 contains only two rules, and the single step rule is simpler than
   151 transitivity.  As a consequence, \isa{rtc{\isachardot}induct} is simpler than
   152 \isa{rtc{\isadigit{2}}{\isachardot}induct}. Since inductive proofs are hard enough, we should
   153 certainly pick the simplest induction schema available for any concept.
   154 Hence \isa{rtc} is the definition of choice.
   155 
   156 \begin{exercise}\label{ex:converse-rtc-step}
   157 Show that the converse of \isa{rtc{\isacharunderscore}step} also holds:
   158 \begin{isabelle}%
   159 \ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%
   160 \end{isabelle}
   161 \end{exercise}
   162 \begin{exercise}
   163 Repeat the development of this section, but starting with a definition of
   164 \isa{rtc} where \isa{rtc{\isacharunderscore}step} is replaced by its converse as shown
   165 in exercise~\ref{ex:converse-rtc-step}.
   166 \end{exercise}%
   167 \end{isamarkuptext}%
   168 \end{isabellebody}%
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