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