src/Doc/Tutorial/Inductive/Star.thy
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     1 (*<*)theory Star imports Main begin(*>*)

     2

     3 section{*The Reflexive Transitive Closure*}

     4

     5 text{*\label{sec:rtc}

     6 \index{reflexive transitive closure!defining inductively|(}%

     7 An inductive definition may accept parameters, so it can express

     8 functions that yield sets.

     9 Relations too can be defined inductively, since they are just sets of pairs.

    10 A perfect example is the function that maps a relation to its

    11 reflexive transitive closure.  This concept was already

    12 introduced in \S\ref{sec:Relations}, where the operator @{text"\<^sup>*"} was

    13 defined as a least fixed point because inductive definitions were not yet

    14 available. But now they are:

    15 *}

    16

    17 inductive_set

    18   rtc :: "('a \<times> 'a)set \<Rightarrow> ('a \<times> 'a)set"   ("_*"  999)

    19   for r :: "('a \<times> 'a)set"

    20 where

    21   rtc_refl[iff]:  "(x,x) \<in> r*"

    22 | rtc_step:       "\<lbrakk> (x,y) \<in> r; (y,z) \<in> r* \<rbrakk> \<Longrightarrow> (x,z) \<in> r*"

    23

    24 text{*\noindent

    25 The function @{term rtc} is annotated with concrete syntax: instead of

    26 @{text"rtc r"} we can write @{term"r*"}. The actual definition

    27 consists of two rules. Reflexivity is obvious and is immediately given the

    28 @{text iff} attribute to increase automation. The

    29 second rule, @{thm[source]rtc_step}, says that we can always add one more

    30 @{term r}-step to the left. Although we could make @{thm[source]rtc_step} an

    31 introduction rule, this is dangerous: the recursion in the second premise

    32 slows down and may even kill the automatic tactics.

    33

    34 The above definition of the concept of reflexive transitive closure may

    35 be sufficiently intuitive but it is certainly not the only possible one:

    36 for a start, it does not even mention transitivity.

    37 The rest of this section is devoted to proving that it is equivalent to

    38 the standard definition. We start with a simple lemma:

    39 *}

    40

    41 lemma [intro]: "(x,y) \<in> r \<Longrightarrow> (x,y) \<in> r*"

    42 by(blast intro: rtc_step);

    43

    44 text{*\noindent

    45 Although the lemma itself is an unremarkable consequence of the basic rules,

    46 it has the advantage that it can be declared an introduction rule without the

    47 danger of killing the automatic tactics because @{term"r*"} occurs only in

    48 the conclusion and not in the premise. Thus some proofs that would otherwise

    49 need @{thm[source]rtc_step} can now be found automatically. The proof also

    50 shows that @{text blast} is able to handle @{thm[source]rtc_step}. But

    51 some of the other automatic tactics are more sensitive, and even @{text

    52 blast} can be lead astray in the presence of large numbers of rules.

    53

    54 To prove transitivity, we need rule induction, i.e.\ theorem

    55 @{thm[source]rtc.induct}:

    56 @{thm[display]rtc.induct}

    57 It says that @{text"?P"} holds for an arbitrary pair @{thm (prem 1) rtc.induct}

    58 if @{text"?P"} is preserved by all rules of the inductive definition,

    59 i.e.\ if @{text"?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 *}

    65

    66 lemma rtc_trans: "\<lbrakk> (x,y) \<in> r*; (y,z) \<in> r* \<rbrakk> \<Longrightarrow> (x,z) \<in> r*"

    67 apply(erule rtc.induct)

    68

    69 txt{*\noindent

    70 Unfortunately, even the base case is a problem:

    71 @{subgoals[display,indent=0,goals_limit=1]}

    72 We have to abandon this proof attempt.

    73 To understand what is going on, let us look again at @{thm[source]rtc.induct}.

    74 In the above application of @{text erule}, the first premise of

    75 @{thm[source]rtc.induct} is unified with the first suitable assumption, which

    76 is @{term"(x,y) \<in> r*"} rather than @{term"(y,z) \<in> r*"}. Although that

    77 is what we want, it is merely due to the order in which the assumptions occur

    78 in the subgoal, which it is not good practice to rely on. As a result,

    79 @{text"?xb"} becomes @{term x}, @{text"?xa"} becomes

    80 @{term y} and @{text"?P"} becomes @{term"%u v. (u,z) : r*"}, thus

    81 yielding the above subgoal. So what went wrong?

    82

    83 When looking at the instantiation of @{text"?P"} we see that it does not

    84 depend on its second parameter at all. The reason is that in our original

    85 goal, of the pair @{term"(x,y)"} only @{term x} appears also in the

    86 conclusion, but not @{term y}. Thus our induction statement is too

    87 general. Fortunately, it can easily be specialized:

    88 transfer the additional premise @{prop"(y,z):r*"} into the conclusion:*}

    89 (*<*)oops(*>*)

    90 lemma rtc_trans[rule_format]:

    91   "(x,y) \<in> r* \<Longrightarrow> (y,z) \<in> r* \<longrightarrow> (x,z) \<in> r*"

    92

    93 txt{*\noindent

    94 This is not an obscure trick but a generally applicable heuristic:

    95 \begin{quote}\em

    96 When proving a statement by rule induction on $(x@1,\dots,x@n) \in R$,

    97 pull all other premises containing any of the $x@i$ into the conclusion

    98 using $\longrightarrow$.

    99 \end{quote}

   100 A similar heuristic for other kinds of inductions is formulated in

   101 \S\ref{sec:ind-var-in-prems}. The @{text rule_format} directive turns

   102 @{text"\<longrightarrow>"} back into @{text"\<Longrightarrow>"}: in the end we obtain the original

   103 statement of our lemma.

   104 *}

   105

   106 apply(erule rtc.induct)

   107

   108 txt{*\noindent

   109 Now induction produces two subgoals which are both proved automatically:

   110 @{subgoals[display,indent=0]}

   111 *}

   112

   113  apply(blast);

   114 apply(blast intro: rtc_step);

   115 done

   116

   117 text{*

   118 Let us now prove that @{term"r*"} is really the reflexive transitive closure

   119 of @{term r}, i.e.\ the least reflexive and transitive

   120 relation containing @{term r}. The latter is easily formalized

   121 *}

   122

   123 inductive_set

   124   rtc2 :: "('a \<times> 'a)set \<Rightarrow> ('a \<times> 'a)set"

   125   for r :: "('a \<times> 'a)set"

   126 where

   127   "(x,y) \<in> r \<Longrightarrow> (x,y) \<in> rtc2 r"

   128 | "(x,x) \<in> rtc2 r"

   129 | "\<lbrakk> (x,y) \<in> rtc2 r; (y,z) \<in> rtc2 r \<rbrakk> \<Longrightarrow> (x,z) \<in> rtc2 r"

   130

   131 text{*\noindent

   132 and the equivalence of the two definitions is easily shown by the obvious rule

   133 inductions:

   134 *}

   135

   136 lemma "(x,y) \<in> rtc2 r \<Longrightarrow> (x,y) \<in> r*"

   137 apply(erule rtc2.induct);

   138   apply(blast);

   139  apply(blast);

   140 apply(blast intro: rtc_trans);

   141 done

   142

   143 lemma "(x,y) \<in> r* \<Longrightarrow> (x,y) \<in> rtc2 r"

   144 apply(erule rtc.induct);

   145  apply(blast intro: rtc2.intros);

   146 apply(blast intro: rtc2.intros);

   147 done

   148

   149 text{*

   150 So why did we start with the first definition? Because it is simpler. It

   151 contains only two rules, and the single step rule is simpler than

   152 transitivity.  As a consequence, @{thm[source]rtc.induct} is simpler than

   153 @{thm[source]rtc2.induct}. Since inductive proofs are hard enough

   154 anyway, we should always pick the simplest induction schema available.

   155 Hence @{term rtc} is the definition of choice.

   156 \index{reflexive transitive closure!defining inductively|)}

   157

   158 \begin{exercise}\label{ex:converse-rtc-step}

   159 Show that the converse of @{thm[source]rtc_step} also holds:

   160 @{prop[display]"[| (x,y) : r*; (y,z) : r |] ==> (x,z) : r*"}

   161 \end{exercise}

   162 \begin{exercise}

   163 Repeat the development of this section, but starting with a definition of

   164 @{term rtc} where @{thm[source]rtc_step} is replaced by its converse as shown

   165 in exercise~\ref{ex:converse-rtc-step}.

   166 \end{exercise}

   167 *}

   168 (*<*)

   169 lemma rtc_step2[rule_format]: "(x,y) : r* \<Longrightarrow> (y,z) : r --> (x,z) : r*"

   170 apply(erule rtc.induct);

   171  apply blast;

   172 apply(blast intro: rtc_step)

   173 done

   174

   175 end

   176 (*>*)