\section{Setoids}
A basic ingredient of constructive real numbers is the {\em apartness\/}
relation $\noto$. This is a constructive version of the (classical)
inequality on reals: two real numbers are apart if it can positively
be decided that they are distinct from each other. In constructive analysis, the apartness is more basic than equality. We therefore take
the notion of apartness as a basic ingredient of our structures
Usually this apartness is taken to be {\em tight}, saying that the
negation of
apartness is the equality. In \cite{Ruit82} and \cite{MRR88} also
apartness relations occur that are not necessarily tight, but in the
formalization
of reals one can restrict to a tight apartness. This also implies that,
in the formalization, we could have done without an equality
alltogether (and replace it with the negation of $\noto$). For reasons
of exposition and for relating to a more classical set-up we choose to
take equality as a primitive.

\begin{definition}
A binary relation $\noto$ on a set $S$ is an {\em apartness\/}
relation if
\begin{enumerate}
\item $\noto$ is {\em consistent}, i.e.\ $\neg a \noto a$ for all $a$.
\item $\noto$ is {\em symmetric}, i.e.\ $ a \noto b \implies b \noto a$
for
all $a, b$.
\item $\noto$ is cotransitive, i.e.\ $a\noto b \rightarrow\forall
z[a\noto z \vee z\noto b]$ for all $a, b$.
\end{enumerate}
An apartness relation is {\em tight\/} if its negation is the
equality, i.e.\ $\neg(a\noto b) \eqqe a = b$ for all $a, b$.
\end{definition}

\begin{fact} The negation of an apartness relation on $S$ is an
equivalence relation on $S$ which is stable, i.e.\ $\neg\neg\neg(a\noto b) 
\implies \neg(a\noto b)$.
\end{fact}

\begin{lemma}\label{lemaprespeq}
  A tight apartness relation respects the equality, i.e.
$$a\noto b \wedge b=b' \implies a\noto b' \mbox{ for all } a,b,b'.$$
\end{lemma}

\begin{proof}
  If $a\noto b$, then $a\noto b' \vee b\noto b'$. The latter is false,
  because $b=b'$. \qed
\end{proof}

\begin{definition}\label{defset} A {\em constructive setoid\/} is a triple 
 $\langle  S, =, \noto \rangle$, with  $S$ a set, $=$ an equivalence relation
 on $S$ and $\noto$ a tight apartness relation on $S$.
\end{definition}

In a structure, we want the operations and relations to respect
the equality and the apartness. For the equality this means
that we want to have the {\em replacement property\/} for all
predicates:
$$R(x_1,\ldots , x_n)\wedge x_1 = y_1 \wedge \ldots \wedge x_n = y_n
\implies R(y_1,\ldots , y_n).$$ 

\begin{fact}
The replacement property is closed under $\vee, \wedge, \neg,
\implies, \exists$ and $\forall$.
\end{fact}

So, we only have to require that the basic relations satisfy the
replacement property and that all basic operations are {\em
well-defined\/} with respect to the equality, i.e.\ for $f$ of arity
$n$ we have the following.
$$x_1 = y_1 \wedge \ldots \wedge x_n = y_n
\implies f(x_1,\ldots , x_n) = f(y_1,\ldots , y_n).$$ 

If we have a tight apartness relation, 
this immediately implies
$$\neg(x_1 \noto y_1) \wedge \ldots \wedge \neg(x_n \noto y_n)
\implies \neg(f(x_1,\ldots , x_n) \noto f(y_1,\ldots , y_n)),$$
but one would like to have a more positive
formulation saying
$$f(x_1,\ldots , x_n) \noto f(y_1,\ldots , y_n)\implies (x_1 \noto
y_1) \vee \ldots \vee (x_n \noto y_n).$$  

This property is called {\em strong extensionality\/} of $f$.

\begin{definition}\label{defstrext}
Let $S$ be a set with an apartness relation $\noto$ defined on it. For
$f$ a $n$-ary function on $S$, we say that $f$ is {\em strongly
extensional\/} if
$$\forall x_1, \ldots, x_n, y_1, \ldots, y_n [f(\vec{x}) \noto
f(\vec{y}) \implies (x_1\noto y_1 \vee \ldots \vee x_n \noto y_n)].$$
For $R$ a $n$-ary relation on $S$, we say that $R$ is {\em strongly
extensional\/} if
$$\forall x_1, \ldots, x_n, y_1, \ldots, y_n [R(\vec{x}) \implies 
(R(\vec{y})
\vee x_1\noto y_1 \vee \ldots \vee x_n \noto y_n)].$$
\end{definition}

\begin{fact}
Strong extensionality of functions is closed under composition. Strong
extensionality of relations is closed under $\vee$, $\wedge$,
$\exists$ and the substitution of strongly extensional terms.
\end{fact}

\begin{lemma}
Strong extensionality implies well-definedness for functions.
\end{lemma}

\begin{proof}
Suppose $f(\vec{x}) \noto f(\vec{y}) \implies (x_1\noto y_1 \vee
\ldots \vee x_n \noto y_n)$ for all $x_1, \ldots, x_n, y_1, \ldots,
y_n$. Suppose $x_1 = y_1\wedge \ldots \wedge x_n = y_n$ and
$f(\vec{x}) \noto f(\vec{y})$. Then $x_1\noto y_1 \vee \ldots \vee x_n
\noto y_n$ by strong extensionality of $f$. Contradiction, so
$\neg(f(\vec{x}) \noto f(\vec{y}))$, i.e.\ $f(\vec{x}) = f(\vec{y})$.
\qed
\end{proof}

 
\begin{remark}
Strong extensionality (for functions) says that a function can only
distinguish elements that can already be distinguished.  We will
require all basic functions in constructive structures to be strongly
extensional.  As a consequence, all composed functions will be
strongly extensional.
%
%Strong extensionality for relations says -- roughly --
%that a relation can only distinguish elements that can already be
%distinguished. This intuition can be made more precise if we define
%an apartness relation $\noto'$ on formulas by
%$$ \phi \noto'\psi := (\phi\wedge \neg \psi)\vee(\neg \phi \vee
%\psi),$$
%and we define a relation to be {\em modified strongly extensional\/}
%if
%$$ \forall x_1, \ldots, x_n, y_1, \ldots, y_n [R(\vec{x}) \noto'
%R(\vec{y}) \implies (x_1\noto y_1 \vee \ldots \vee x_n \noto y_n)].$$
%Now, strong extensionality implies modified strong extensionality, but
%not the other way around (QUESTION: Give CEX!). Obviously, both
%notions are classically equivalent.

We do not want all relations to be strongly extensional. For example,
equality is not strongly extensional: if it were, then $x = y \implies
p=q\vee x\noto p\vee y\noto q$ for all $x,y,p,q$, which implies the
decidability of equality (take $x$ for $y$ and $p$).
\end{remark}

\begin{lemma}\label{lemstrextarg}
If a binary function $f$ is strongly extensional in both arguments,
i.e.\
\begin{eqnarray*}
\forall x_1, x_2, y [f(x_1,y) \noto f(x_2, y) \implies (x_1\noto
x_2)],\\
\forall x, y_1,y_2 [f(x,y_1) \noto f(x, y_2) \implies (y_1\noto y_2)],
\end{eqnarray*}
then it is strongly extensional. Similarly for functions of higher
arity.
\end{lemma}

\begin{proof}
Suppose the binary function $f$ is strongly extensional in both
arguments and suppose $f(x_1,y_1) \noto f(x_2, y_2)$. Then $f(x_1,y_1)
\noto f(x_1, y_2) \vee f(x_1,y_2) \noto f(x_2, y_2)$ by
cotransitivity.  Hence $ y_1 \noto y_2 \vee x_1 \noto x_2$. \qed
\end{proof}

\begin{lemma}\label{lemstrextneq} If $f$ is strongly extensional, then
$$ f(\vec{x}) \neq f(\vec{y}) \implies \neg(x_1 = y_1 \wedge \ldots
\wedge x_n = y_n).$$ 
\end{lemma}

\begin{proof}
Suppose $ f(\vec{x}) \neq f(\vec{y})$, i.e. $\neg\neg(f(\vec{x}) \neq
f(\vec{y}))$. Suppose also that $x_1 = y_1 \wedge \ldots
\wedge x_n = y_n$. Now, if $f(\vec{x}) \noto f(\vec{y})$, then $x_1
\noto y_1 \vee \ldots \vee x_n \noto y_n$, contradicting $x_1 = y_1
\wedge \ldots \wedge x_n = y_n$. So $\neg(f(\vec{x}) \noto
f(\vec{y}))$, contradicting $ f(\vec{x}) \neq f(\vec{y})$. So we
conclude that $\neg(x_1 = y_1 \wedge \ldots \wedge x_n = y_n)$. \qed
\end{proof}

If a function $f$ has an inverse, we want it to {\em respect\/} the
apartness. Note that, if $f$ has no inverse we do not want that in
general (e.g.\ consider multiplication in $\ZZ_4$).  That $f$ respects
$\noto$ comes as a consequence of strong extensionality and the
existence of an inverse.

\begin{lemma}\label{lemstrextinv}
Suppose that the unary function $f$ has an inverse $g$ which is
strongly extensional. Then $f$ respects the apartness, i.e.\
$$x\noto y \implies f(x) \noto f(y).$$
\end{lemma}

\begin{proof}
We know that $g(x) \noto g(y) \implies x\noto y$
and that $g(f(x))=x$. Now suppose $x\noto y$, i.e.\ $g(f(x))\noto
g(f(y))$. Then $f(x) \noto f(y)$ by strong extensionality of $g$. \qed
\end{proof}

\begin{lemma}\label{respneq}
If $f$ respects the apartness, then $f$ respects the inequality
\end{lemma}

\begin{proof}
Let $f$ respect the apartness(i.e.\ $(x_1\noto y_1 \vee \ldots \vee
x_n \noto y_n) \implies 
f(\vec{x}) \noto f(\vec{y})$). Suppose  $x_1\neq y_1 \vee \ldots \vee
x_n \neq y_n$ and suppose $f(\vec{x} = f(\vec{y})$. Now, if $x_i \noto
y_i$ for some $i$, then $f(\vec{x} \noto f(\vec{y})$, contradiction,
so $x_i = y_i$ for all $i$. This is again a contradiction, so
$f(\vec{x}) \neq f(\vec{y})$. \qed
\end{proof}

\begin{lemma}
If a relation $R$ is strongly extensional in each of its arguments, it
is strongly extensional.
\end{lemma}

\begin{proof}
We give the proof for a binary relation $R$. Suppose $R$ is strongly
extensional in both arguments, i.e.
\begin{eqnarray*}
R(x,y) &\implies & R(x,q)\vee y\noto q,\\
R(x,y) &\implies & R(p,y)\vee p\noto x.
\end{eqnarray*}
for all $x,y,p,q$.
Now, if $R(x,y)$, then $R(x,q)\vee y\noto q$. If $R(x,q)$, then
$R(p,q)\vee p\noto x$, so $R(p,q)\vee p\noto x\vee y\noto q$. \qed
\end{proof}


\begin{lemma}\label{tapstrext}
Apartness is strongly extensional.
\end{lemma}

\begin{proof}
Suppose $x\noto y$. Then $x\noto p \vee y\noto p$ by cotransitivity
and hence $x\noto p \vee y\noto q \vee p\noto q$ by again
cotransitivity.
\qed
\end{proof}

\subsection{On the choice of the primitives}
In view of the fact that we require an apartness relation in  a setoid
to be tight, we could have chosen to define a setoid as a pair
$\langle S, \noto\rangle$ with $\noto$ an apartness relation and then
{\em define\/} equality by
$$x=y \;\;\;\; := \;\;\;\; \neg(x\noto y).$$ 
Then the following can be shown.
\begin{enumerate}
\item If an operation $f$ is strongly extensional, then it respects
$=$.
\item If a relation $R$ is strongly extensional, then it satisfies the
replacement property.
\item Hence all relations satisfy the replacement property.
\end{enumerate}

So, we could have done without an equality alltogether.  However, we
have not chosen this option, because equality is a natural
primitive. Furthermore one may at some point encounter structures in
which apartness is not tight.

\subsection{Subsetoids and Quotient Setoids}
\begin{definition}\label{defsubset}
Given a constructive setoid $\langle S, =, \noto \rangle$ and a
predicate $P$ on $S$, we define the {\em subsetoid of the $x\in S$
that satisfy $P$\/} as the setoid $\langle \{ x\in S \mid P (x) \},
=', \noto' \rangle$, where $='$ and $\noto'$ are the equality and
apartness inherited from $S$, i.e.\ for $q,t\in \{ x\in S \mid P (x) \}$,
\begin{eqnarray*}
 t='q &\iff & t =q,\\
 t\noto'q &\iff & t \noto q,\\ 
\end{eqnarray*}
We denote this subsetoid just by $\{ x\in S \mid P (x) \}$.
\end{definition}

For this definition to be correct, it has to be shown that $='$ is
indeed an equivalence relation and that $\noto'$ is a tight apartness
relation (w.r.t.\ $=$) on $\{ x\in S \mid P (x) \}$. This is trivially
the case. As the equivalence and apartness are directly inherited from
$S$, we never write them explicitly, but use the ones from $S$.

\begin{definition}\label{quotset}
Given a constructive setoid $\langle S, =, \noto \rangle$ and a
strongly extensional apartness relation $Q$ on $S$, we define the {\em
co-quotient setoid $S/R$\/} as the setoid $\langle S,\overline{R} , R
\rangle$, where $\overline{R}$ is the complement of $R$, i.e.\
$\overline{R} (x,y)$ iff $\neg R(x,y)$.
\end{definition}

For this definition to be correct, it has to be shown that
$\overline{R}$ is an equivalence relation and that $R$ is a tight
apartness relation (w.r.t.\ $\overline{R}$) on $\{ x\in S \mid P (x)
\}$. This follows trivially from the definition of $\overline{R}$ and
the fact that $R$ is an apartness.

If we do not require $R$ to be strongly extensional, $S/R$ as defined
above is still a constructive setoid. However, we only want to
consider the situation where the new apartness $R$ is a subset of the
old one, i.e.\ $R(x,y) \implies x\noto y$. This is a consequence of
strong extensionality of $R$: take $x$ for $p$ and for $q$ in $R(x,y)
\implies (R(p,q) \vee x\noto p \vee y \noto q)$.  As a consequence we
then find that $=$ is a subset of $\overline{R}$, so the new equality
is a refinement of the old one. So, the definition of co-quotient
setoid subsumes the ordinary definition of quotient set.

For a strongly extensional function $f$ on a setoid $\langle S,
=,\noto\rangle$ we find that, if $f$ is strongly extensional w.r.t.\
$R$, with $R$ a strongly extensional apartness relation on $S$, then
$f$ is also strongly extensional on the co-quotient setoid.

The real numbers form a primary example of a co-quotient setoid. They
can be seen as the setoid $(\NN \arr \QQ)/R$, where $\NN\arr \QQ$ is
the set of infinite sequences of rational numbers and $R$ is the
apartness relation between such sequences: for $r$ and $s$ two
sequences, $R(r,s)$ iff $\exists k, N\in\NN\forall m>N (|r_m - s_m| 
>\frac{1}{k})$.


