diff -r 0c86acc069ad -r 5deda0549f97 doc-src/TutorialI/Inductive/document/Even.tex --- a/doc-src/TutorialI/Inductive/document/Even.tex Thu Jul 26 16:54:44 2012 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,543 +0,0 @@ -% -\begin{isabellebody}% -\def\isabellecontext{Even}% -% -\isadelimtheory -% -\endisadelimtheory -% -\isatagtheory -% -\endisatagtheory -{\isafoldtheory}% -% -\isadelimtheory -% -\endisadelimtheory -% -\isadelimML -% -\endisadelimML -% -\isatagML -% -\endisatagML -{\isafoldML}% -% -\isadelimML -% -\endisadelimML -% -\isamarkupsection{The Set of Even Numbers% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -\index{even numbers!defining inductively|(}% -The set of even numbers can be inductively defined as the least set -containing 0 and closed under the operation $+2$. Obviously, -\emph{even} can also be expressed using the divides relation (\isa{dvd}). -We shall prove below that the two formulations coincide. On the way we -shall examine the primary means of reasoning about inductively defined -sets: rule induction.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsubsection{Making an Inductive Definition% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -Using \commdx{inductive\protect\_set}, we declare the constant \isa{even} to be -a set of natural numbers with the desired properties.% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}\isamarkupfalse% -\ even\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ set{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline -zero{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{0}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline -step{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{22}{\isachardoublequoteclose}}% -\begin{isamarkuptext}% -An inductive definition consists of introduction rules. The first one -above states that 0 is even; the second states that if $n$ is even, then so -is~$n+2$. Given this declaration, Isabelle generates a fixed point -definition for \isa{even} and proves theorems about it, -thus following the definitional approach (see {\S}\ref{sec:definitional}). -These theorems -include the introduction rules specified in the declaration, an elimination -rule for case analysis and an induction rule. We can refer to these -theorems by automatically-generated names. Here are two examples: -\begin{isabelle}% -{\isadigit{0}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\rulename{even{\isaliteral{2E}{\isachardot}}zero}\par\smallskip% -n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\rulename{even{\isaliteral{2E}{\isachardot}}step}% -\end{isabelle} - -The introduction rules can be given attributes. Here -both rules are specified as \isa{intro!},% -\index{intro"!@\isa {intro"!} (attribute)} -directing the classical reasoner to -apply them aggressively. Obviously, regarding 0 as even is safe. The -\isa{step} rule is also safe because $n+2$ is even if and only if $n$ is -even. We prove this equivalence later.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsubsection{Using Introduction Rules% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -Our first lemma states that numbers of the form $2\times k$ are even. -Introduction rules are used to show that specific values belong to the -inductive set. Such proofs typically involve -induction, perhaps over some other inductive set.% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{lemma}\isamarkupfalse% -\ two{\isaliteral{5F}{\isacharunderscore}}times{\isaliteral{5F}{\isacharunderscore}}even{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{2}}{\isaliteral{2A}{\isacharasterisk}}k\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -\isacommand{apply}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ k{\isaliteral{29}{\isacharparenright}}\isanewline -\ \isacommand{apply}\isamarkupfalse% -\ auto\isanewline -\isacommand{done}\isamarkupfalse% -% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -% -\begin{isamarkuptxt}% -\noindent -The first step is induction on the natural number \isa{k}, which leaves -two subgoals: -\begin{isabelle}% -\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isadigit{0}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\isanewline -\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ Suc\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even% -\end{isabelle} -Here \isa{auto} simplifies both subgoals so that they match the introduction -rules, which are then applied automatically. - -Our ultimate goal is to prove the equivalence between the traditional -definition of \isa{even} (using the divides relation) and our inductive -definition. One direction of this equivalence is immediate by the lemma -just proved, whose \isa{intro{\isaliteral{21}{\isacharbang}}} attribute ensures it is applied automatically.% -\end{isamarkuptxt}% -\isamarkuptrue% -% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -\isacommand{lemma}\isamarkupfalse% -\ dvd{\isaliteral{5F}{\isacharunderscore}}imp{\isaliteral{5F}{\isacharunderscore}}even{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{2}}\ dvd\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -\isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}auto\ simp\ add{\isaliteral{3A}{\isacharcolon}}\ dvd{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\isamarkupsubsection{Rule Induction \label{sec:rule-induction}% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -\index{rule induction|(}% -From the definition of the set -\isa{even}, Isabelle has -generated an induction rule: -\begin{isabelle}% -{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{3B}{\isacharsemicolon}}\ P\ {\isadigit{0}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isaindent{\ }{\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{3B}{\isacharsemicolon}}\ P\ n{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline -{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ x\rulename{even{\isaliteral{2E}{\isachardot}}induct}% -\end{isabelle} -A property \isa{P} holds for every even number provided it -holds for~\isa{{\isadigit{0}}} and is closed under the operation -\isa{Suc(Suc \(\cdot\))}. Then \isa{P} is closed under the introduction -rules for \isa{even}, which is the least set closed under those rules. -This type of inductive argument is called \textbf{rule induction}. - -Apart from the double application of \isa{Suc}, the induction rule above -resembles the familiar mathematical induction, which indeed is an instance -of rule induction; the natural numbers can be defined inductively to be -the least set containing \isa{{\isadigit{0}}} and closed under~\isa{Suc}. - -Induction is the usual way of proving a property of the elements of an -inductively defined set. Let us prove that all members of the set -\isa{even} are multiples of two.% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{lemma}\isamarkupfalse% -\ even{\isaliteral{5F}{\isacharunderscore}}imp{\isaliteral{5F}{\isacharunderscore}}dvd{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isadigit{2}}\ dvd\ n{\isaliteral{22}{\isachardoublequoteclose}}% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -% -\begin{isamarkuptxt}% -We begin by applying induction. Note that \isa{even{\isaliteral{2E}{\isachardot}}induct} has the form -of an elimination rule, so we use the method \isa{erule}. We get two -subgoals:% -\end{isamarkuptxt}% -\isamarkuptrue% -\isacommand{apply}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}erule\ even{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}% -\begin{isamarkuptxt}% -\begin{isabelle}% -\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isadigit{2}}\ dvd\ {\isadigit{0}}\isanewline -\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{3B}{\isacharsemicolon}}\ {\isadigit{2}}\ dvd\ n{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isadigit{2}}\ dvd\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}% -\end{isabelle} -We unfold the definition of \isa{dvd} in both subgoals, proving the first -one and simplifying the second:% -\end{isamarkuptxt}% -\isamarkuptrue% -\isacommand{apply}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}simp{\isaliteral{5F}{\isacharunderscore}}all\ add{\isaliteral{3A}{\isacharcolon}}\ dvd{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}% -\begin{isamarkuptxt}% -\begin{isabelle}% -\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}k{\isaliteral{2E}{\isachardot}}\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ k{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}k{\isaliteral{2E}{\isachardot}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ k% -\end{isabelle} -The next command eliminates the existential quantifier from the assumption -and replaces \isa{n} by \isa{{\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ k}.% -\end{isamarkuptxt}% -\isamarkuptrue% -\isacommand{apply}\isamarkupfalse% -\ clarify% -\begin{isamarkuptxt}% -\begin{isabelle}% -\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n\ k{\isaliteral{2E}{\isachardot}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ k\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}ka{\isaliteral{2E}{\isachardot}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ k{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ ka% -\end{isabelle} -To conclude, we tell Isabelle that the desired value is -\isa{Suc\ k}. With this hint, the subgoal falls to \isa{simp}.% -\end{isamarkuptxt}% -\isamarkuptrue% -\isacommand{apply}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}rule{\isaliteral{5F}{\isacharunderscore}}tac\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}Suc\ k{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{in}\ exI{\isaliteral{2C}{\isacharcomma}}\ simp{\isaliteral{29}{\isacharparenright}}% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\begin{isamarkuptext}% -Combining the previous two results yields our objective, the -equivalence relating \isa{even} and \isa{dvd}. -% -%we don't want [iff]: discuss?% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{theorem}\isamarkupfalse% -\ even{\isaliteral{5F}{\isacharunderscore}}iff{\isaliteral{5F}{\isacharunderscore}}dvd{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}\ dvd\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -\isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}blast\ intro{\isaliteral{3A}{\isacharcolon}}\ dvd{\isaliteral{5F}{\isacharunderscore}}imp{\isaliteral{5F}{\isacharunderscore}}even\ even{\isaliteral{5F}{\isacharunderscore}}imp{\isaliteral{5F}{\isacharunderscore}}dvd{\isaliteral{29}{\isacharparenright}}% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\isamarkupsubsection{Generalization and Rule Induction \label{sec:gen-rule-induction}% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -\index{generalizing for induction}% -Before applying induction, we typically must generalize -the induction formula. With rule induction, the required generalization -can be hard to find and sometimes requires a complete reformulation of the -problem. In this example, our first attempt uses the obvious statement of -the result. It fails:% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{lemma}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -\isacommand{apply}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}erule\ even{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline -\isacommand{oops}\isamarkupfalse% -% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -% -\begin{isamarkuptxt}% -Rule induction finds no occurrences of \isa{Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}} in the -conclusion, which it therefore leaves unchanged. (Look at -\isa{even{\isaliteral{2E}{\isachardot}}induct} to see why this happens.) We have these subgoals: -\begin{isabelle}% -\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\isanewline -\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}na{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}na\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{3B}{\isacharsemicolon}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even% -\end{isabelle} -The first one is hopeless. Rule induction on -a non-variable term discards information, and usually fails. -How to deal with such situations -in general is described in {\S}\ref{sec:ind-var-in-prems} below. -In the current case the solution is easy because -we have the necessary inverse, subtraction:% -\end{isamarkuptxt}% -\isamarkuptrue% -% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -\isacommand{lemma}\isamarkupfalse% -\ even{\isaliteral{5F}{\isacharunderscore}}imp{\isaliteral{5F}{\isacharunderscore}}even{\isaliteral{5F}{\isacharunderscore}}minus{\isaliteral{5F}{\isacharunderscore}}{\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -\isacommand{apply}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}erule\ even{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline -\ \isacommand{apply}\isamarkupfalse% -\ auto\isanewline -\isacommand{done}\isamarkupfalse% -% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -% -\begin{isamarkuptxt}% -This lemma is trivially inductive. Here are the subgoals: -\begin{isabelle}% -\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isadigit{0}}\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\isanewline -\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{3B}{\isacharsemicolon}}\ n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even% -\end{isabelle} -The first is trivial because \isa{{\isadigit{0}}\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}} simplifies to \isa{{\isadigit{0}}}, which is -even. The second is trivial too: \isa{Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}} simplifies to -\isa{n}, matching the assumption.% -\index{rule induction|)} %the sequel isn't really about induction - -\medskip -Using our lemma, we can easily prove the result we originally wanted:% -\end{isamarkuptxt}% -\isamarkuptrue% -% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -\isacommand{lemma}\isamarkupfalse% -\ Suc{\isaliteral{5F}{\isacharunderscore}}Suc{\isaliteral{5F}{\isacharunderscore}}even{\isaliteral{5F}{\isacharunderscore}}imp{\isaliteral{5F}{\isacharunderscore}}even{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -\isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}drule\ even{\isaliteral{5F}{\isacharunderscore}}imp{\isaliteral{5F}{\isacharunderscore}}even{\isaliteral{5F}{\isacharunderscore}}minus{\isaliteral{5F}{\isacharunderscore}}{\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ simp{\isaliteral{29}{\isacharparenright}}% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\begin{isamarkuptext}% -We have just proved the converse of the introduction rule \isa{even{\isaliteral{2E}{\isachardot}}step}. -This suggests proving the following equivalence. We give it the -\attrdx{iff} attribute because of its obvious value for simplification.% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{lemma}\isamarkupfalse% -\ {\isaliteral{5B}{\isacharbrackleft}}iff{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -\isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}blast\ dest{\isaliteral{3A}{\isacharcolon}}\ Suc{\isaliteral{5F}{\isacharunderscore}}Suc{\isaliteral{5F}{\isacharunderscore}}even{\isaliteral{5F}{\isacharunderscore}}imp{\isaliteral{5F}{\isacharunderscore}}even{\isaliteral{29}{\isacharparenright}}% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\isamarkupsubsection{Rule Inversion \label{sec:rule-inversion}% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -\index{rule inversion|(}% -Case analysis on an inductive definition is called \textbf{rule -inversion}. It is frequently used in proofs about operational -semantics. It can be highly effective when it is applied -automatically. Let us look at how rule inversion is done in -Isabelle/HOL\@. - -Recall that \isa{even} is the minimal set closed under these two rules: -\begin{isabelle}% -{\isadigit{0}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\isasep\isanewline% -n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even% -\end{isabelle} -Minimality means that \isa{even} contains only the elements that these -rules force it to contain. If we are told that \isa{a} -belongs to -\isa{even} then there are only two possibilities. Either \isa{a} is \isa{{\isadigit{0}}} -or else \isa{a} has the form \isa{Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}}, for some suitable \isa{n} -that belongs to -\isa{even}. That is the gist of the \isa{cases} rule, which Isabelle proves -for us when it accepts an inductive definition: -\begin{isabelle}% -{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}a\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{3B}{\isacharsemicolon}}\ a\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isaindent{\ }{\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}a\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline -{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\rulename{even{\isaliteral{2E}{\isachardot}}cases}% -\end{isabelle} -This general rule is less useful than instances of it for -specific patterns. For example, if \isa{a} has the form -\isa{Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}} then the first case becomes irrelevant, while the second -case tells us that \isa{n} belongs to \isa{even}. Isabelle will generate -this instance for us:% -\end{isamarkuptext}% -\isamarkuptrue% -\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}\isamarkupfalse% -\ Suc{\isaliteral{5F}{\isacharunderscore}}Suc{\isaliteral{5F}{\isacharunderscore}}cases\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{22}{\isachardoublequoteclose}}% -\begin{isamarkuptext}% -The \commdx{inductive\protect\_cases} command generates an instance of -the \isa{cases} rule for the supplied pattern and gives it the supplied name: -\begin{isabelle}% -{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{3B}{\isacharsemicolon}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\rulename{Suc{\isaliteral{5F}{\isacharunderscore}}Suc{\isaliteral{5F}{\isacharunderscore}}cases}% -\end{isabelle} -Applying this as an elimination rule yields one case where \isa{even{\isaliteral{2E}{\isachardot}}cases} -would yield two. Rule inversion works well when the conclusions of the -introduction rules involve datatype constructors like \isa{Suc} and \isa{{\isaliteral{23}{\isacharhash}}} -(list ``cons''); freeness reasoning discards all but one or two cases. - -In the \isacommand{inductive\_cases} command we supplied an -attribute, \isa{elim{\isaliteral{21}{\isacharbang}}}, -\index{elim"!@\isa {elim"!} (attribute)}% -indicating that this elimination rule can be -applied aggressively. The original -\isa{cases} rule would loop if used in that manner because the -pattern~\isa{a} matches everything. - -The rule \isa{Suc{\isaliteral{5F}{\isacharunderscore}}Suc{\isaliteral{5F}{\isacharunderscore}}cases} is equivalent to the following implication: -\begin{isabelle}% -Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even% -\end{isabelle} -Just above we devoted some effort to reaching precisely -this result. Yet we could have obtained it by a one-line declaration, -dispensing with the lemma \isa{even{\isaliteral{5F}{\isacharunderscore}}imp{\isaliteral{5F}{\isacharunderscore}}even{\isaliteral{5F}{\isacharunderscore}}minus{\isaliteral{5F}{\isacharunderscore}}{\isadigit{2}}}. -This example also justifies the terminology -\textbf{rule inversion}: the new rule inverts the introduction rule -\isa{even{\isaliteral{2E}{\isachardot}}step}. In general, a rule can be inverted when the set of elements -it introduces is disjoint from those of the other introduction rules. - -For one-off applications of rule inversion, use the \methdx{ind_cases} method. -Here is an example:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimproof -% -\endisadelimproof -% -\isatagproof -\isacommand{apply}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}ind{\isaliteral{5F}{\isacharunderscore}}cases\ {\isaliteral{22}{\isachardoublequoteopen}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}% -\endisatagproof -{\isafoldproof}% -% -\isadelimproof -% -\endisadelimproof -% -\begin{isamarkuptext}% -The specified instance of the \isa{cases} rule is generated, then applied -as an elimination rule. - -To summarize, every inductive definition produces a \isa{cases} rule. The -\commdx{inductive\protect\_cases} command stores an instance of the -\isa{cases} rule for a given pattern. Within a proof, the -\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases} method applies an instance of the \isa{cases} -rule. - -The even numbers example has shown how inductive definitions can be -used. Later examples will show that they are actually worth using.% -\index{rule inversion|)}% -\index{even numbers!defining inductively|)}% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimtheory -% -\endisadelimtheory -% -\isatagtheory -% -\endisatagtheory -{\isafoldtheory}% -% -\isadelimtheory -% -\endisadelimtheory -\end{isabellebody}% -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "root" -%%% End: