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-\achapter{Nsatz: tactics for proving equalities in integral domains}
-\aauthor{Loīc Pottier}
-
-The tactic \texttt{nsatz} proves goals of the form
-
-\[ \begin{array}{l}
-  \forall X_1,\ldots,X_n \in A,\\
-   P_1(X_1,\ldots,X_n) = Q_1(X_1,\ldots,X_n) , \ldots ,  P_s(X_1,\ldots,X_n) =Q_s(X_1,\ldots,X_n)\\ 
- \vdash P(X_1,\ldots,X_n) = Q(X_1,\ldots,X_n)\\
-  \end{array}
-\]
-where $P,Q, P_1,Q_1,\ldots,P_s,Q_s$ are polynomials and A is an integral
-domain, i.e. a commutative ring with no zero divisor. For example, A can be
-$\mathbb{R}$, $\mathbb{Z}$, of $\mathbb{Q}$. Note that the equality $=$ used in these
-goals can be any setoid equality 
-(see \ref{setoidtactics})
-, not only Leibnitz equality.
-
-It also proves formulas
-\[ \begin{array}{l}
-  \forall X_1,\ldots,X_n \in A,\\
-   P_1(X_1,\ldots,X_n) = Q_1(X_1,\ldots,X_n) \wedge \ldots \wedge  P_s(X_1,\ldots,X_n) =Q_s(X_1,\ldots,X_n)\\ 
- \rightarrow P(X_1,\ldots,X_n) = Q(X_1,\ldots,X_n)\\
-  \end{array}
-\] doing automatic introductions.
- 
-\asection{Using the basic tactic \texttt{nsatz}}
-\tacindex{nsatz}
-
-Load the
-\texttt{Nsatz} module: \texttt{Require Import Nsatz}.\\
- and use the tactic \texttt{nsatz}.
-
-\asection{More about \texttt{nsatz}}
-
-Hilbert's Nullstellensatz theorem shows how to reduce proofs of equalities on
-polynomials on a commutative ring A with no zero divisor to algebraic computations: it is easy to see that if a polynomial
-$P$ in $A[X_1,\ldots,X_n]$ verifies $c P^r = \sum_{i=1}^{s} S_i P_i$, with $c
-\in A$, $c \not = 0$, $r$ a positive integer, and the $S_i$s in
-$A[X_1,\ldots,X_n]$, then $P$ is zero whenever polynomials $P_1,...,P_s$  are
-zero (the converse is also true when A is an algebraic closed field:
-the method is complete). 
-
-So, proving our initial problem can reduce into finding $S_1,\ldots,S_s$, $c$
-and $r$ such that $c (P-Q)^r = \sum_{i} S_i (P_i-Q_i)$, which will be proved by the
-tactic \texttt{ring}.
-
-This is achieved by the computation of a Groebner basis of the
-ideal generated by $P_1-Q_1,...,P_s-Q_s$, with an adapted version of the Buchberger
-algorithm.
-
-This computation is done after a step of {\em reification}, which is
-performed using {\em Type Classes} 
-(see \ref{typeclasses})
-.
-
-The \texttt{Nsatz} module defines the generic tactic
-\texttt{nsatz}, which uses the low-level tactic \texttt{nsatz\_domainpv}: \\
-\vspace*{3mm}
-\texttt{nsatz\_domainpv pretac rmax strategy lparam lvar simpltac domain}
-
-where:
-
-\begin{itemize}
-    \item \texttt{pretac} is a tactic depending on the ring A; its goal is to
-make apparent the generic operations of a domain (ring\_eq, ring\_plus, etc),
-both in the goal and the hypotheses; it is executed first. By default it is \texttt{ltac:idtac}.
-
-    \item \texttt{rmax} is a bound when for searching r s.t.$c (P-Q)^r =
-\sum_{i=1..s} S_i (P_i - Q_i)$
-	
-    \item \texttt{strategy} gives the order on variables $X_1,...X_n$ and
-the strategy used in Buchberger algorithm (see
-\cite{sugar} for details): 
-
-     	\begin{itemize}
-		\item  strategy = 0: reverse lexicographic order and newest s-polynomial.
-		\item   strategy = 1: reverse lexicographic order and sugar strategy.
-	        \item  strategy = 2: pure lexicographic order and newest s-polynomial.
-	        \item   strategy = 3: pure lexicographic order and sugar strategy.
-	\end{itemize}
-
-	\item \texttt{lparam} is the list of variables
-$X_{i_1},\ldots,X_{i_k}$  among $X_1,...,X_n$ which are considered as
-   parameters: computation will be performed with rational fractions in these
-   variables, i.e. polynomials are considered with coefficients in
-$R(X_{i_1},\ldots,X_{i_k})$. In this case, the coefficient $c$ can be a non
-constant polynomial in $X_{i_1},\ldots,X_{i_k}$, and the tactic produces a goal
-which states that $c$ is not zero.
-
-	\item \texttt{lvar} is the list of the variables
-in the decreasing order in which they will be used in Buchberger algorithm. If \texttt{lvar} = {(@nil
-R)}, then \texttt{lvar} is replaced by all the variables which are not in
-lparam.
-
-    \item \texttt{simpltac} is a tactic depending on the ring A; its goal is to
-simplify goals and make apparent the generic operations of a domain after
-simplifications. By default it is \texttt{ltac:simpl}.
-
- \item \texttt{domain} is the object of type Domain representing A, its
-operations and properties of integral domain.
-
-\end{itemize}
-
-See file \texttt{Nsatz.v} for examples.
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End: