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\begin{document}

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\section*{coinduction}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{induction}{}\paragraph*{{Induction}}\label{induction}

[[!include induction - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Coinduction} is a method of [[proof]] which relies on the fact that any two states of the [[terminal coalgebra]] [[algebra for an endofunctor|for an endofunctor]] $H$ must be equal if they are indistinguishable under repeated operations of $H$. That is, there are no proper coalgebra [[quotient objects]]. Generally, we show the existence of a [[bisimulation]] between states of terminal coalgebra, that is a relation between states, such that when the coalgebra function is applied, the respective outputs are still related. Since any bisimulation must be contained within the identity relation, we can then conclude that the states are equal.

Coinduction is [[duality|dual]] to [[induction]]. It generalises to [[corecursion]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
Take $add\colon \bar{\mathbb{N}} \times \bar{\mathbb{N}} \to 1 + \bar{\mathbb{N}} \times \bar{\mathbb{N}}$ as defined at [[corecursion]], which defines an addition $+$ on the extended natural numbers. We can then establish a bisimulation between the terms $(n + m)$ and $(m + n)$, from which we can conclude that this addition is commutative. (See p. 52 of Rutten \href{http://homepages.cwi.nl/~janr/papers/files-of-papers/universal_coalgebra.pdf}{Universal coalgebra: a theory of systems}.)

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Many theorems of calculus involving formal [[power series]] $f(x) = \sum_{n \geq 0} \frac{a_n x^n}{n!}$ are naturally viewed as proofs by coinduction. The collection of power series $k[ [x] ]$ over a [[field]] $k$ (say of [[characteristic]] $0$) may be viewed as a \emph{stream coalgebra} $k^\mathbb{N}$, i.e., the [[terminal coalgebra of the endofunctor]] $k \times -:\; Set \to Set$, with the coalgebra structure

\begin{displaymath}
\itexarray{
\langle eval_0, D = \frac{d}{d x} \rangle: k[ [x] ] & \to & k \times k[ [x] ] \\ 
\sum_{n \geq 0} \frac{a_n x^n}{n!} & \mapsto & (a_0, \sum_{n \geq 0} \frac{a_{n+1} x^n}{n!})
}
\end{displaymath}
Thus if we set up a bisimulation $\sim$ between two power series $f, g$, establishing $f(0) = g(0)$ and $(D f) \sim (D g)$, we may conclude by coinduction that $f = g$ (else, $k[ [x] ]/\sim$ would be a proper quotient).

As an illustration: consider a field $k$ of characteristic $0$ and, for $r \in k$, define $(1 + x)^r$ to be the power series expansion of $\exp(r \cdot \log(1 + x))$, where $\exp(x) = \sum_{n \geq 0} \frac{x^n}{n!}$ and $\log(1 + x) = \sum_{n \geq 1} (-1)^{n+1}\; \frac{x^n}{n}$ is the inverse of $\exp(x)-1$. To prove the generalized [[binomial theorem]]

\begin{displaymath}
(1 + x)^r = f(r) \coloneqq \sum_{k \geq 0} \frac{r^\underline{k} x^k}{k!}
\end{displaymath}
where $r^\underline{k}$ is the falling power $r(r-1)\ldots (r-k+1)$, we introduce a bisimilarity

\begin{displaymath}
\sum_{k \geq 0} \frac{r^\underline{k} x^k}{k!}\; \sim\; \exp(r \cdot \log(1 + x))
\end{displaymath}
and observe first that the [[constant coefficients]] of $f(x)$ and $(1+x)^r$ are both $1$, and that

\begin{displaymath}
D\left(\sum_{k \geq 0} \frac{r^\underline{k} x^k}{k!}\right) = r \sum_{k \geq 0} \frac{(r-1)^\underline{k} x^k}{k!}
\end{displaymath}
and similarly

\begin{displaymath}
D\left(\exp(r \cdot \log(1 + x))\right) = r \cdot \exp((r-1) \cdot \log(1 + x))
\end{displaymath}
by the [[chain rule]], whence $D(f/r) \sim D((1+x)^r/r)$. Hence $f/r$ is bisimilar to $(1+x)^r/r$, and we conclude in the terminal coalgebra that $f(x) = (1+x)^r$.

\end{example}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\begin{itemize}%
\item [[Bart Jacobs]], [[Jan Rutten]], \emph{A tutorial on (Co)Algebras and (Co)Induction} (\href{http://www.cs.ru.nl/~bart/PAPERS/JR.pdf}{pdf})


\item Davide Sangiorgi, \emph{Introduction to Bisimulation and Coinduction}, Cambridge Universtity Press (2012) (\href{http://www.cs.unibo.it/~sangio/IntroBook.html}{web})


\item Davide Sangiorgi, Jan Rutten (eds.), \emph{Advanced Topics in Bisimulation and Coinduction}, Cambridge Universtity Press (2012) (\href{http://www.cs.unibo.it/~sangio/AdvancedBook.html}{web})



\end{itemize}
Discussion of [[differential calculus]] in terms of coinduction is in

\begin{itemize}%
\item Martin Escardo, Du\v{s}ko Pavlovi, \emph{Calculus in coinductive form} (1998) (\href{http://www.isg.rhul.ac.uk/dusko/papers/1998-lapl-LICS.pdf}{pdf})

\end{itemize}
[[!redirects coinductions]]

[[!redirects coinductive definition]]



\end{document}