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\chapter{Advanced Simplification, Recursion and Induction}
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Although we have already learned a lot about simplification, recursion and
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induction, there are some advanced proof techniques that we have not covered
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yet and which are worth learning. The three sections of this chapter are almost
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independent of each other and can be read in any order. Only the notion of
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\emph{congruence rules}, introduced in the section on simplification, is
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required for parts of the section on recursion.
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\input{Advanced/document/simp.tex}
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\section{Advanced Forms of Recursion}
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\index{recdef@\isacommand {recdef} (command)|(}
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This section introduces advanced forms of
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\isacommand{recdef}: how to establish termination by means other than measure
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functions, how to define recursive functions over nested recursive datatypes
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and how to deal with partial functions.
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If, after reading this section, you feel that the definition of recursive
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functions is overly complicated by the requirement of
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totality, you should ponder the alternatives. In a logic of partial functions,
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recursive definitions are always accepted. But there are many
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such logics, and no clear winner has emerged. And in all of these logics you
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are (more or less frequently) required to reason about the definedness of
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terms explicitly. Thus one shifts definedness arguments from definition time to
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proof time. In HOL you may have to work hard to define a function, but proofs
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can then proceed unencumbered by worries about undefinedness.
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\subsection{Beyond Measure}
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\label{sec:beyond-measure}
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\input{Advanced/document/WFrec.tex}
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\subsection{Recursion Over Nested Datatypes}
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\label{sec:nested-recdef}
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\input{Recdef/document/Nested0.tex}
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\input{Recdef/document/Nested1.tex}
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\input{Recdef/document/Nested2.tex}
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\subsection{Partial Functions}
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\index{functions!partial}
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\input{Advanced/document/Partial.tex}
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\index{recdef@\isacommand {recdef} (command)|)}
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\section{Advanced Induction Techniques}
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\label{sec:advanced-ind}
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\index{induction|(}
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\input{Misc/document/AdvancedInd.tex}
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\input{CTL/document/CTLind.tex}
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\index{induction|)}
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