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\begin{isabellebody}%
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\def\isabellecontext{WFrec}%
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\isamarkupfalse%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isatagtheory
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%
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\begin{isamarkuptext}%
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\noindent
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So far, all recursive definitions were shown to terminate via measure
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functions. Sometimes this can be inconvenient or
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impossible. Fortunately, \isacommand{recdef} supports much more
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general definitions. For example, termination of Ackermann's function
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can be shown by means of the \rmindex{lexicographic product} \isa{{\isacharless}{\isacharasterisk}lex{\isacharasterisk}{\isachargreater}}:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{consts}\isamarkupfalse%
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\ ack\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
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\isacommand{recdef}\isamarkupfalse%
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\ ack\ {\isachardoublequoteopen}measure{\isacharparenleft}{\isasymlambda}m{\isachardot}\ m{\isacharparenright}\ {\isacharless}{\isacharasterisk}lex{\isacharasterisk}{\isachargreater}\ measure{\isacharparenleft}{\isasymlambda}n{\isachardot}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ {\isachardoublequoteopen}ack{\isacharparenleft}{\isadigit{0}}{\isacharcomma}n{\isacharparenright}\ \ \ \ \ \ \ \ \ {\isacharequal}\ Suc\ n{\isachardoublequoteclose}\isanewline
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\ \ {\isachardoublequoteopen}ack{\isacharparenleft}Suc\ m{\isacharcomma}{\isadigit{0}}{\isacharparenright}\ \ \ \ \ {\isacharequal}\ ack{\isacharparenleft}m{\isacharcomma}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ {\isachardoublequoteopen}ack{\isacharparenleft}Suc\ m{\isacharcomma}Suc\ n{\isacharparenright}\ {\isacharequal}\ ack{\isacharparenleft}m{\isacharcomma}ack{\isacharparenleft}Suc\ m{\isacharcomma}n{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
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\noindent
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The lexicographic product decreases if either its first component
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decreases (as in the second equation and in the outer call in the
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third equation) or its first component stays the same and the second
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component decreases (as in the inner call in the third equation).
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In general, \isacommand{recdef} supports termination proofs based on
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arbitrary well-founded relations as introduced in \S\ref{sec:Well-founded}.
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This is called \textbf{well-founded
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recursion}\indexbold{recursion!well-founded}. A function definition
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is total if and only if the set of
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all pairs $(r,l)$, where $l$ is the argument on the
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left-hand side of an equation and $r$ the argument of some recursive call on
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the corresponding right-hand side, induces a well-founded relation. For a
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systematic account of termination proofs via well-founded relations see, for
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example, Baader and Nipkow~\cite{Baader-Nipkow}.
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Each \isacommand{recdef} definition should be accompanied (after the function's
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name) by a well-founded relation on the function's argument type.
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Isabelle/HOL formalizes some of the most important
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constructions of well-founded relations (see \S\ref{sec:Well-founded}). For
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example, \isa{measure\ f} is always well-founded. The lexicographic
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product of two well-founded relations is again well-founded, which we relied
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on when defining Ackermann's function above.
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Of course the lexicographic product can also be iterated:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{consts}\isamarkupfalse%
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\ contrived\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymtimes}\ nat\ {\isasymtimes}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
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\isacommand{recdef}\isamarkupfalse%
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\ contrived\isanewline
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\ \ {\isachardoublequoteopen}measure{\isacharparenleft}{\isasymlambda}i{\isachardot}\ i{\isacharparenright}\ {\isacharless}{\isacharasterisk}lex{\isacharasterisk}{\isachargreater}\ measure{\isacharparenleft}{\isasymlambda}j{\isachardot}\ j{\isacharparenright}\ {\isacharless}{\isacharasterisk}lex{\isacharasterisk}{\isachargreater}\ measure{\isacharparenleft}{\isasymlambda}k{\isachardot}\ k{\isacharparenright}{\isachardoublequoteclose}\isanewline
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{\isachardoublequoteopen}contrived{\isacharparenleft}i{\isacharcomma}j{\isacharcomma}Suc\ k{\isacharparenright}\ {\isacharequal}\ contrived{\isacharparenleft}i{\isacharcomma}j{\isacharcomma}k{\isacharparenright}{\isachardoublequoteclose}\isanewline
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{\isachardoublequoteopen}contrived{\isacharparenleft}i{\isacharcomma}Suc\ j{\isacharcomma}{\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ contrived{\isacharparenleft}i{\isacharcomma}j{\isacharcomma}j{\isacharparenright}{\isachardoublequoteclose}\isanewline
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{\isachardoublequoteopen}contrived{\isacharparenleft}Suc\ i{\isacharcomma}{\isadigit{0}}{\isacharcomma}{\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ contrived{\isacharparenleft}i{\isacharcomma}i{\isacharcomma}i{\isacharparenright}{\isachardoublequoteclose}\isanewline
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{\isachardoublequoteopen}contrived{\isacharparenleft}{\isadigit{0}}{\isacharcomma}{\isadigit{0}}{\isacharcomma}{\isadigit{0}}{\isacharparenright}\ \ \ \ \ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
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Lexicographic products of measure functions already go a long
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way. Furthermore, you may embed a type in an
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existing well-founded relation via the inverse image construction \isa{inv{\isacharunderscore}image}. All these constructions are known to \isacommand{recdef}. Thus you
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will never have to prove well-foundedness of any relation composed
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solely of these building blocks. But of course the proof of
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termination of your function definition --- that the arguments
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decrease with every recursive call --- may still require you to provide
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additional lemmas.
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It is also possible to use your own well-founded relations with
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\isacommand{recdef}. For example, the greater-than relation can be made
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well-founded by cutting it off at a certain point. Here is an example
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of a recursive function that calls itself with increasing values up to ten:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{consts}\isamarkupfalse%
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\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
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\isacommand{recdef}\isamarkupfalse%
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\ f\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharparenleft}i{\isacharcomma}j{\isacharparenright}{\isachardot}\ j{\isacharless}i\ {\isasymand}\ i\ {\isasymle}\ {\isacharparenleft}{\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
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{\isachardoublequoteopen}f\ i\ {\isacharequal}\ {\isacharparenleft}if\ {\isadigit{1}}{\isadigit{0}}\ {\isasymle}\ i\ then\ {\isadigit{0}}\ else\ i\ {\isacharasterisk}\ f{\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
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\noindent
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Since \isacommand{recdef} is not prepared for the relation supplied above,
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Isabelle rejects the definition. We should first have proved that
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our relation was well-founded:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{lemma}\isamarkupfalse%
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\ wf{\isacharunderscore}greater{\isacharcolon}\ {\isachardoublequoteopen}wf\ {\isacharbraceleft}{\isacharparenleft}i{\isacharcomma}j{\isacharparenright}{\isachardot}\ j{\isacharless}i\ {\isasymand}\ i\ {\isasymle}\ {\isacharparenleft}N{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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%
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\begin{isamarkuptxt}%
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\noindent
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The proof is by showing that our relation is a subset of another well-founded
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relation: one given by a measure function.\index{*wf_subset (theorem)}%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\isacommand{apply}\isamarkupfalse%
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\ {\isacharparenleft}rule\ wf{\isacharunderscore}subset\ {\isacharbrackleft}of\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}k{\isacharcolon}{\isacharcolon}nat{\isachardot}\ N{\isacharminus}k{\isacharparenright}{\isachardoublequoteclose}{\isacharbrackright}{\isacharcomma}\ blast{\isacharparenright}%
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\begin{isamarkuptxt}%
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\begin{isabelle}%
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\ {\isadigit{1}}{\isachardot}\ {\isacharbraceleft}{\isacharparenleft}i{\isacharcomma}\ j{\isacharparenright}{\isachardot}\ j\ {\isacharless}\ i\ {\isasymand}\ i\ {\isasymle}\ N{\isacharbraceright}\ {\isasymsubseteq}\ measure\ {\isacharparenleft}op\ {\isacharminus}\ N{\isacharparenright}%
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\end{isabelle}
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\noindent
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The inclusion remains to be proved. After unfolding some definitions,
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we are left with simple arithmetic:%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\isacommand{apply}\isamarkupfalse%
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\ {\isacharparenleft}clarify{\isacharcomma}\ simp\ add{\isacharcolon}\ measure{\isacharunderscore}def\ inv{\isacharunderscore}image{\isacharunderscore}def{\isacharparenright}%
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\begin{isamarkuptxt}%
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\begin{isabelle}%
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\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}a\ b{\isachardot}\ {\isasymlbrakk}b\ {\isacharless}\ a{\isacharsemicolon}\ a\ {\isasymle}\ N{\isasymrbrakk}\ {\isasymLongrightarrow}\ N\ {\isacharminus}\ a\ {\isacharless}\ N\ {\isacharminus}\ b%
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\end{isabelle}
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\noindent
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And that is dispatched automatically:%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\isacommand{by}\isamarkupfalse%
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\ arith%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\begin{isamarkuptext}%
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\noindent
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Armed with this lemma, we use the \attrdx{recdef_wf} attribute to attach a
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crucial hint\cmmdx{hints} to our definition:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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{\isacharparenleft}\isakeyword{hints}\ recdef{\isacharunderscore}wf{\isacharcolon}\ wf{\isacharunderscore}greater{\isacharparenright}%
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\begin{isamarkuptext}%
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\noindent
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Alternatively, we could have given \isa{measure\ {\isacharparenleft}{\isasymlambda}k{\isacharcolon}{\isacharcolon}nat{\isachardot}\ {\isadigit{1}}{\isadigit{0}}{\isacharminus}k{\isacharparenright}} for the
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well-founded relation in our \isacommand{recdef}. However, the arithmetic
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goal in the lemma above would have arisen instead in the \isacommand{recdef}
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termination proof, where we have less control. A tailor-made termination
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relation makes even more sense when it can be used in several function
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declarations.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isadelimtheory
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\endisadelimtheory
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%
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\isatagtheory
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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\end{isabellebody}%
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "root"
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%%% End:
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