author nipkow
Tue Oct 17 13:28:57 2000 +0200 (2000-10-17)
changeset 10236 7626cb4e1407
parent 10217 e61e7e1eacaf
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     1 %
     2 \begin{isabellebody}%
     3 \def\isabellecontext{AdvancedInd}%
     4 %
     5 \begin{isamarkuptext}%
     6 \noindent
     7 Now that we have learned about rules and logic, we take another look at the
     8 finer points of induction. The two questions we answer are: what to do if the
     9 proposition to be proved is not directly amenable to induction, and how to
    10 utilize and even derive new induction schemas.%
    11 \end{isamarkuptext}%
    12 %
    13 \isamarkupsubsection{Massaging the proposition}
    14 %
    15 \begin{isamarkuptext}%
    16 \label{sec:ind-var-in-prems}
    17 So far we have assumed that the theorem we want to prove is already in a form
    18 that is amenable to induction, but this is not always the case:%
    19 \end{isamarkuptext}%
    20 \isacommand{lemma}\ {\isachardoublequote}xs\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymLongrightarrow}\ hd{\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ last\ xs{\isachardoublequote}\isanewline
    21 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ xs{\isacharparenright}%
    22 \begin{isamarkuptxt}%
    23 \noindent
    24 (where \isa{hd} and \isa{last} return the first and last element of a
    25 non-empty list)
    26 produces the warning
    27 \begin{quote}\tt
    28 Induction variable occurs also among premises!
    29 \end{quote}
    30 and leads to the base case
    31 \begin{isabelle}
    32 \ 1.\ xs\ {\isasymnoteq}\ []\ {\isasymLongrightarrow}\ hd\ (rev\ [])\ =\ last\ []
    33 \end{isabelle}
    34 which, after simplification, becomes
    35 \begin{isabelle}
    36 \ 1.\ xs\ {\isasymnoteq}\ []\ {\isasymLongrightarrow}\ hd\ []\ =\ last\ []
    37 \end{isabelle}
    38 We cannot prove this equality because we do not know what \isa{hd} and
    39 \isa{last} return when applied to \isa{{\isacharbrackleft}{\isacharbrackright}}.
    41 The point is that we have violated the above warning. Because the induction
    42 formula is only the conclusion, the occurrence of \isa{xs} in the premises is
    43 not modified by induction. Thus the case that should have been trivial
    44 becomes unprovable. Fortunately, the solution is easy:
    45 \begin{quote}
    46 \emph{Pull all occurrences of the induction variable into the conclusion
    47 using \isa{{\isasymlongrightarrow}}.}
    48 \end{quote}
    49 This means we should prove%
    50 \end{isamarkuptxt}%
    51 \isacommand{lemma}\ hd{\isacharunderscore}rev{\isacharcolon}\ {\isachardoublequote}xs\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymlongrightarrow}\ hd{\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ last\ xs{\isachardoublequote}%
    52 \begin{isamarkuptext}%
    53 \noindent
    54 This time, induction leaves us with the following base case
    55 \begin{isabelle}
    56 \ 1.\ []\ {\isasymnoteq}\ []\ {\isasymlongrightarrow}\ hd\ (rev\ [])\ =\ last\ []
    57 \end{isabelle}
    58 which is trivial, and \isa{auto} finishes the whole proof.
    60 If \isa{hd{\isacharunderscore}rev} is meant to be a simplification rule, you are
    61 done. But if you really need the \isa{{\isasymLongrightarrow}}-version of
    62 \isa{hd{\isacharunderscore}rev}, for example because you want to apply it as an
    63 introduction rule, you need to derive it separately, by combining it with
    64 modus ponens:%
    65 \end{isamarkuptext}%
    66 \isacommand{lemmas}\ hd{\isacharunderscore}revI\ {\isacharequal}\ hd{\isacharunderscore}rev{\isacharbrackleft}THEN\ mp{\isacharbrackright}%
    67 \begin{isamarkuptext}%
    68 \noindent
    69 which yields the lemma we originally set out to prove.
    71 In case there are multiple premises $A@1$, \dots, $A@n$ containing the
    72 induction variable, you should turn the conclusion $C$ into
    73 \[ A@1 \longrightarrow \cdots A@n \longrightarrow C \]
    74 (see the remark?? in \S\ref{??}).
    75 Additionally, you may also have to universally quantify some other variables,
    76 which can yield a fairly complex conclusion.
    77 Here is a simple example (which is proved by \isa{blast}):%
    78 \end{isamarkuptext}%
    79 \isacommand{lemma}\ simple{\isacharcolon}\ {\isachardoublequote}{\isasymforall}y{\isachardot}\ A\ y\ {\isasymlongrightarrow}\ B\ y\ {\isasymlongrightarrow}\ B\ y\ {\isacharampersand}\ A\ y{\isachardoublequote}%
    80 \begin{isamarkuptext}%
    81 \noindent
    82 You can get the desired lemma by explicit
    83 application of modus ponens and \isa{spec}:%
    84 \end{isamarkuptext}%
    85 \isacommand{lemmas}\ myrule\ {\isacharequal}\ simple{\isacharbrackleft}THEN\ spec{\isacharcomma}\ THEN\ mp{\isacharcomma}\ THEN\ mp{\isacharbrackright}%
    86 \begin{isamarkuptext}%
    87 \noindent
    88 or the wholesale stripping of \isa{{\isasymforall}} and
    89 \isa{{\isasymlongrightarrow}} in the conclusion via \isa{rule{\isacharunderscore}format}%
    90 \end{isamarkuptext}%
    91 \isacommand{lemmas}\ myrule\ {\isacharequal}\ simple{\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}%
    92 \begin{isamarkuptext}%
    93 \noindent
    94 yielding \isa{{\isasymlbrakk}A\ y{\isacharsemicolon}\ B\ y{\isasymrbrakk}\ {\isasymLongrightarrow}\ B\ y\ {\isasymand}\ A\ y}.
    95 You can go one step further and include these derivations already in the
    96 statement of your original lemma, thus avoiding the intermediate step:%
    97 \end{isamarkuptext}%
    98 \isacommand{lemma}\ myrule{\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}{\isacharcolon}\ \ {\isachardoublequote}{\isasymforall}y{\isachardot}\ A\ y\ {\isasymlongrightarrow}\ B\ y\ {\isasymlongrightarrow}\ B\ y\ {\isacharampersand}\ A\ y{\isachardoublequote}%
    99 \begin{isamarkuptext}%
   100 \bigskip
   102 A second reason why your proposition may not be amenable to induction is that
   103 you want to induct on a whole term, rather than an individual variable. In
   104 general, when inducting on some term $t$ you must rephrase the conclusion $C$
   105 as
   106 \[ \forall y@1 \dots y@n.~ x = t \longrightarrow C \]
   107 where $y@1 \dots y@n$ are the free variables in $t$ and $x$ is new, and
   108 perform induction on $x$ afterwards. An example appears in
   109 \S\ref{sec:complete-ind} below.
   111 The very same problem may occur in connection with rule induction. Remember
   112 that it requires a premise of the form $(x@1,\dots,x@k) \in R$, where $R$ is
   113 some inductively defined set and the $x@i$ are variables.  If instead we have
   114 a premise $t \in R$, where $t$ is not just an $n$-tuple of variables, we
   115 replace it with $(x@1,\dots,x@k) \in R$, and rephrase the conclusion $C$ as
   116 \[ \forall y@1 \dots y@n.~ (x@1,\dots,x@k) = t \longrightarrow C \]
   117 For an example see \S\ref{sec:CTL-revisited} below.%
   118 \end{isamarkuptext}%
   119 %
   120 \isamarkupsubsection{Beyond structural and recursion induction}
   121 %
   122 \begin{isamarkuptext}%
   123 \label{sec:complete-ind}
   124 So far, inductive proofs where by structural induction for
   125 primitive recursive functions and recursion induction for total recursive
   126 functions. But sometimes structural induction is awkward and there is no
   127 recursive function in sight either that could furnish a more appropriate
   128 induction schema. In such cases some existing standard induction schema can
   129 be helpful. We show how to apply such induction schemas by an example.
   131 Structural induction on \isa{nat} is
   132 usually known as ``mathematical induction''. There is also ``complete
   133 induction'', where you must prove $P(n)$ under the assumption that $P(m)$
   134 holds for all $m<n$. In Isabelle, this is the theorem \isa{nat{\isacharunderscore}less{\isacharunderscore}induct}:
   135 \begin{isabelle}%
   136 \ \ \ \ \ {\isacharparenleft}{\isasymAnd}n{\isachardot}\ {\isasymforall}m{\isachardot}\ m\ {\isacharless}\ n\ {\isasymlongrightarrow}\ P\ m\ {\isasymLongrightarrow}\ P\ n{\isacharparenright}\ {\isasymLongrightarrow}\ P\ n%
   137 \end{isabelle}
   138 Here is an example of its application.%
   139 \end{isamarkuptext}%
   140 \isacommand{consts}\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat\ {\isacharequal}{\isachargreater}\ nat{\isachardoublequote}\isanewline
   141 \isacommand{axioms}\ f{\isacharunderscore}ax{\isacharcolon}\ {\isachardoublequote}f{\isacharparenleft}f{\isacharparenleft}n{\isacharparenright}{\isacharparenright}\ {\isacharless}\ f{\isacharparenleft}Suc{\isacharparenleft}n{\isacharparenright}{\isacharparenright}{\isachardoublequote}%
   142 \begin{isamarkuptext}%
   143 \noindent
   144 From the above axiom\footnote{In general, the use of axioms is strongly
   145 discouraged, because of the danger of inconsistencies. The above axiom does
   146 not introduce an inconsistency because, for example, the identity function
   147 satisfies it.}
   148 for \isa{f} it follows that \isa{n\ {\isasymle}\ f\ n}, which can
   149 be proved by induction on \isa{f\ n}. Following the recipy outlined
   150 above, we have to phrase the proposition as follows to allow induction:%
   151 \end{isamarkuptext}%
   152 \isacommand{lemma}\ f{\isacharunderscore}incr{\isacharunderscore}lem{\isacharcolon}\ {\isachardoublequote}{\isasymforall}i{\isachardot}\ k\ {\isacharequal}\ f\ i\ {\isasymlongrightarrow}\ i\ {\isasymle}\ f\ i{\isachardoublequote}%
   153 \begin{isamarkuptxt}%
   154 \noindent
   155 To perform induction on \isa{k} using \isa{nat{\isacharunderscore}less{\isacharunderscore}induct}, we use the same
   156 general induction method as for recursion induction (see
   157 \S\ref{sec:recdef-induction}):%
   158 \end{isamarkuptxt}%
   159 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ k\ rule{\isacharcolon}\ nat{\isacharunderscore}less{\isacharunderscore}induct{\isacharparenright}%
   160 \begin{isamarkuptxt}%
   161 \noindent
   162 which leaves us with the following proof state:
   163 \begin{isabelle}
   164 \ 1.\ {\isasymAnd}\mbox{n}.\ {\isasymforall}\mbox{m}.\ \mbox{m}\ <\ \mbox{n}\ {\isasymlongrightarrow}\ ({\isasymforall}\mbox{i}.\ \mbox{m}\ =\ f\ \mbox{i}\ {\isasymlongrightarrow}\ \mbox{i}\ {\isasymle}\ f\ \mbox{i})\isanewline
   165 \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymforall}\mbox{i}.\ \mbox{n}\ =\ f\ \mbox{i}\ {\isasymlongrightarrow}\ \mbox{i}\ {\isasymle}\ f\ \mbox{i}
   166 \end{isabelle}
   167 After stripping the \isa{{\isasymforall}i}, the proof continues with a case
   168 distinction on \isa{i}. The case \isa{i\ {\isacharequal}\ {\isadigit{0}}} is trivial and we focus on
   169 the other case:
   170 \begin{isabelle}
   171 \ 1.\ {\isasymAnd}\mbox{n}\ \mbox{i}\ \mbox{nat}.\isanewline
   172 \ \ \ \ \ \ \ {\isasymlbrakk}{\isasymforall}\mbox{m}.\ \mbox{m}\ <\ \mbox{n}\ {\isasymlongrightarrow}\ ({\isasymforall}\mbox{i}.\ \mbox{m}\ =\ f\ \mbox{i}\ {\isasymlongrightarrow}\ \mbox{i}\ {\isasymle}\ f\ \mbox{i});\ \mbox{i}\ =\ Suc\ \mbox{nat}{\isasymrbrakk}\isanewline
   173 \ \ \ \ \ \ \ {\isasymLongrightarrow}\ \mbox{n}\ =\ f\ \mbox{i}\ {\isasymlongrightarrow}\ \mbox{i}\ {\isasymle}\ f\ \mbox{i}
   174 \end{isabelle}%
   175 \end{isamarkuptxt}%
   176 \isacommand{by}{\isacharparenleft}blast\ intro{\isacharbang}{\isacharcolon}\ f{\isacharunderscore}ax\ Suc{\isacharunderscore}leI\ intro{\isacharcolon}\ le{\isacharunderscore}less{\isacharunderscore}trans{\isacharparenright}%
   177 \begin{isamarkuptext}%
   178 \noindent
   179 It is not surprising if you find the last step puzzling.
   180 The proof goes like this (writing \isa{j} instead of \isa{nat}).
   181 Since \isa{i\ {\isacharequal}\ Suc\ j} it suffices to show
   182 \isa{j\ {\isacharless}\ f\ {\isacharparenleft}Suc\ j{\isacharparenright}} (by \isa{Suc{\isacharunderscore}leI}: \isa{m\ {\isacharless}\ n\ {\isasymLongrightarrow}\ Suc\ m\ {\isasymle}\ n}). This is
   183 proved as follows. From \isa{f{\isacharunderscore}ax} we have \isa{f\ {\isacharparenleft}f\ j{\isacharparenright}\ {\isacharless}\ f\ {\isacharparenleft}Suc\ j{\isacharparenright}}
   184 (1) which implies \isa{f\ j\ {\isasymle}\ f\ {\isacharparenleft}f\ j{\isacharparenright}} (by the induction hypothesis).
   185 Using (1) once more we obtain \isa{f\ j\ {\isacharless}\ f\ {\isacharparenleft}Suc\ j{\isacharparenright}} (2) by transitivity
   186 (\isa{le{\isacharunderscore}less{\isacharunderscore}trans}: \isa{{\isasymlbrakk}i\ {\isasymle}\ j{\isacharsemicolon}\ j\ {\isacharless}\ k{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharless}\ k}).
   187 Using the induction hypothesis once more we obtain \isa{j\ {\isasymle}\ f\ j}
   188 which, together with (2) yields \isa{j\ {\isacharless}\ f\ {\isacharparenleft}Suc\ j{\isacharparenright}} (again by
   189 \isa{le{\isacharunderscore}less{\isacharunderscore}trans}).
   191 This last step shows both the power and the danger of automatic proofs: they
   192 will usually not tell you how the proof goes, because it can be very hard to
   193 translate the internal proof into a human-readable format. Therefore
   194 \S\ref{sec:part2?} introduces a language for writing readable yet concise
   195 proofs.
   197 We can now derive the desired \isa{i\ {\isasymle}\ f\ i} from \isa{f{\isacharunderscore}incr}:%
   198 \end{isamarkuptext}%
   199 \isacommand{lemmas}\ f{\isacharunderscore}incr\ {\isacharequal}\ f{\isacharunderscore}incr{\isacharunderscore}lem{\isacharbrackleft}rule{\isacharunderscore}format{\isacharcomma}\ OF\ refl{\isacharbrackright}%
   200 \begin{isamarkuptext}%
   201 \noindent
   202 The final \isa{refl} gets rid of the premise \isa{{\isacharquery}k\ {\isacharequal}\ f\ {\isacharquery}i}. Again,
   203 we could have included this derivation in the original statement of the lemma:%
   204 \end{isamarkuptext}%
   205 \isacommand{lemma}\ f{\isacharunderscore}incr{\isacharbrackleft}rule{\isacharunderscore}format{\isacharcomma}\ OF\ refl{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isasymforall}i{\isachardot}\ k\ {\isacharequal}\ f\ i\ {\isasymlongrightarrow}\ i\ {\isasymle}\ f\ i{\isachardoublequote}%
   206 \begin{isamarkuptext}%
   207 \begin{exercise}
   208 From the above axiom and lemma for \isa{f} show that \isa{f} is the
   209 identity.
   210 \end{exercise}
   212 In general, \isa{induct{\isacharunderscore}tac} can be applied with any rule $r$
   213 whose conclusion is of the form ${?}P~?x@1 \dots ?x@n$, in which case the
   214 format is
   215 \begin{quote}
   216 \isacommand{apply}\isa{{\isacharparenleft}induct{\isacharunderscore}tac} $y@1 \dots y@n$ \isa{rule{\isacharcolon}} $r$\isa{{\isacharparenright}}
   217 \end{quote}\index{*induct_tac}%
   218 where $y@1, \dots, y@n$ are variables in the first subgoal.
   219 A further example of a useful induction rule is \isa{length{\isacharunderscore}induct},
   220 induction on the length of a list:\indexbold{*length_induct}
   221 \begin{isabelle}%
   222 \ \ \ \ \ {\isacharparenleft}{\isasymAnd}xs{\isachardot}\ {\isasymforall}ys{\isachardot}\ length\ ys\ {\isacharless}\ length\ xs\ {\isasymlongrightarrow}\ P\ ys\ {\isasymLongrightarrow}\ P\ xs{\isacharparenright}\ {\isasymLongrightarrow}\ P\ xs%
   223 \end{isabelle}
   225 In fact, \isa{induct{\isacharunderscore}tac} even allows the conclusion of
   226 $r$ to be an (iterated) conjunction of formulae of the above form, in
   227 which case the application is
   228 \begin{quote}
   229 \isacommand{apply}\isa{{\isacharparenleft}induct{\isacharunderscore}tac} $y@1 \dots y@n$ \isa{and} \dots\ \isa{and} $z@1 \dots z@m$ \isa{rule{\isacharcolon}} $r$\isa{{\isacharparenright}}
   230 \end{quote}%
   231 \end{isamarkuptext}%
   232 %
   233 \isamarkupsubsection{Derivation of new induction schemas}
   234 %
   235 \begin{isamarkuptext}%
   236 \label{sec:derive-ind}
   237 Induction schemas are ordinary theorems and you can derive new ones
   238 whenever you wish.  This section shows you how to, using the example
   239 of \isa{nat{\isacharunderscore}less{\isacharunderscore}induct}. Assume we only have structural induction
   240 available for \isa{nat} and want to derive complete induction. This
   241 requires us to generalize the statement first:%
   242 \end{isamarkuptext}%
   243 \isacommand{lemma}\ induct{\isacharunderscore}lem{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isasymAnd}n{\isacharcolon}{\isacharcolon}nat{\isachardot}\ {\isasymforall}m{\isacharless}n{\isachardot}\ P\ m\ {\isasymLongrightarrow}\ P\ n{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymforall}m{\isacharless}n{\isachardot}\ P\ m{\isachardoublequote}\isanewline
   244 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ n{\isacharparenright}%
   245 \begin{isamarkuptxt}%
   246 \noindent
   247 The base case is trivially true. For the induction step (\isa{m\ {\isacharless}\ Suc\ n}) we distinguish two cases: case \isa{m\ {\isacharless}\ n} is true by induction
   248 hypothesis and case \isa{m\ {\isacharequal}\ n} follows from the assumption, again using
   249 the induction hypothesis:%
   250 \end{isamarkuptxt}%
   251 \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
   252 \isacommand{by}{\isacharparenleft}blast\ elim{\isacharcolon}less{\isacharunderscore}SucE{\isacharparenright}%
   253 \begin{isamarkuptext}%
   254 \noindent
   255 The elimination rule \isa{less{\isacharunderscore}SucE} expresses the case distinction:
   256 \begin{isabelle}%
   257 \ \ \ \ \ {\isasymlbrakk}m\ {\isacharless}\ Suc\ n{\isacharsemicolon}\ m\ {\isacharless}\ n\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ m\ {\isacharequal}\ n\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P%
   258 \end{isabelle}
   260 Now it is straightforward to derive the original version of
   261 \isa{nat{\isacharunderscore}less{\isacharunderscore}induct} by manipulting the conclusion of the above lemma:
   262 instantiate \isa{n} by \isa{Suc\ n} and \isa{m} by \isa{n} and
   263 remove the trivial condition \isa{n\ {\isacharless}\ Sc\ n}. Fortunately, this
   264 happens automatically when we add the lemma as a new premise to the
   265 desired goal:%
   266 \end{isamarkuptext}%
   267 \isacommand{theorem}\ nat{\isacharunderscore}less{\isacharunderscore}induct{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isasymAnd}n{\isacharcolon}{\isacharcolon}nat{\isachardot}\ {\isasymforall}m{\isacharless}n{\isachardot}\ P\ m\ {\isasymLongrightarrow}\ P\ n{\isacharparenright}\ {\isasymLongrightarrow}\ P\ n{\isachardoublequote}\isanewline
   268 \isacommand{by}{\isacharparenleft}insert\ induct{\isacharunderscore}lem{\isacharcomma}\ blast{\isacharparenright}%
   269 \begin{isamarkuptext}%
   270 Finally we should mention that HOL already provides the mother of all
   271 inductions, \textbf{wellfounded
   272 induction}\indexbold{induction!wellfounded}\index{wellfounded
   273 induction|see{induction, wellfounded}} (\isa{wf{\isacharunderscore}induct}):
   274 \begin{isabelle}%
   275 \ \ \ \ \ {\isasymlbrakk}wf\ r{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ {\isasymforall}y{\isachardot}\ {\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ P\ y\ {\isasymLongrightarrow}\ P\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ a%
   276 \end{isabelle}
   277 where \isa{wf\ r} means that the relation \isa{r} is wellfounded
   278 (see \S\ref{sec:wellfounded}).
   279 For example, theorem \isa{nat{\isacharunderscore}less{\isacharunderscore}induct} can be viewed (and
   280 derived) as a special case of \isa{wf{\isacharunderscore}induct} where 
   281 \isa{r} is \isa{{\isacharless}} on \isa{nat}. The details can be found in the HOL library.
   282 For a mathematical account of wellfounded induction see, for example, \cite{Baader-Nipkow}.%
   283 \end{isamarkuptext}%
   284 \end{isabellebody}%
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