[Sphinx] Add chapter 26

Thanks to Paul Steckler for porting this chapter.

1 files changed, 97 insertions, 98 deletions

diff --git a/doc/sphinx/addendum/nsatz.rst b/doc/sphinx/addendum/nsatz.rst
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-\achapter{Nsatz: tactics for proving equalities in integral domains}
-%HEVEA\cutname{nsatz.html}
-\aauthor{Loïc 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.
+.. include:: ../preamble.rst
+
+.. _nsatz:
+
+Nsatz: tactics for proving equalities in integral domains
+===========================================================
+
+:Author: Loïc Pottier
+
+The tactic `nsatz` proves goals of the form
+
+:math:`\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 :math:`P, Q, P₁,Q₁,\ldots,Pₛ, Qₛ` are polynomials and :math:`A` is an integral
+domain, i.e. a commutative ring with no zero divisor. For example, :math:`A`
+can be :math:`\mathbb{R}`, :math:`\mathbb{Z}`, or :math:`\mathbb{Q}`.
+Note that the equality :math:`=` used in these goals can be
+any setoid equality (see :ref:`TODO-27.2.2`) , 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 tactic
-\texttt{nsatz}, which can be used without arguments: \\
-\vspace*{3mm}
-\texttt{nsatz}\\
-or with the syntax: \\
-\vspace*{3mm}
-\texttt{nsatz with radicalmax:={\em number}\%N strategy:={\em number}\%Z parameters:={\em list of variables} variables:={\em list of variables}}\\
+
+:math:`\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.
+
+
+Using the basic tactic `nsatz`
+------------------------------
+
+
+Load the Nsatz module:
+
+.. coqtop:: all
+
+  Require Import Nsatz.
+
+and use the tactic `nsatz`.
+
+More about `nsatz`
+---------------------
+
+Hilbert’s Nullstellensatz theorem shows how to reduce proofs of
+equalities on polynomials on a commutative ring :math:`A` with no zero divisor
+to algebraic computations: it is easy to see that if a polynomial :math:`P` in
+:math:`A[X_1,\ldots,X_n]` verifies :math:`c P^r = \sum_{i=1}^{s} S_i P_i`, with
+:math:`c \in A`, :math:`c \not = 0`,
+:math:`r` a positive integer, and the :math:`S_i` s in :math:`A[X_1,\ldots,X_n ]`,
+then :math:`P` is zero whenever polynomials :math:`P_1,\ldots,P_s` are zero
+(the converse is also true when :math:`A` is an algebraic closed field: the method is
+complete).
+
+So, proving our initial problem can reduce into finding :math:`S_1,\ldots,S_s`,
+:math:`c` and :math:`r` such that :math:`c (P-Q)^r = \sum_{i} S_i (P_i-Q_i)`,
+which will be proved by the tactic ring.
+
+This is achieved by the computation of a Gröbner basis of the ideal
+generated by :math:`P_1-Q_1,...,P_s-Q_s`, with an adapted version of the
+Buchberger algorithm.
+
+This computation is done after a step of *reification*, which is
+performed using :ref:`typeclasses`.
+
+The ``Nsatz`` module defines the tactic `nsatz`, which can be used without
+arguments, or with the syntax:
+
+| nsatz with radicalmax:=num%N strategy:=num%Z parameters:= :n:`{* var}` variables:= :n:`{* var}`
+
 where:
 
-\begin{itemize}
-    \item \texttt{radicalmax} 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{parameters} 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{variables} is the list of the variables
-in the decreasing order in which they will be used in Buchberger algorithm. If \texttt{variables} = {(@nil
-R)}, then \texttt{lvar} is replaced by all the variables which are not in
-parameters.
-
-\end{itemize}
-
-See file \texttt{Nsatz.v} for many examples, specially in geometry.
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End: 
+* `radicalmax` is a bound when for searching r s.t.
+  :math:`c (P−Q) r = \sum_{i=1..s} S_i (P i − Q i)`
+
+* `strategy` gives the order on variables :math:`X_1,\ldots,X_n` and the strategy
+  used in Buchberger algorithm (see :cite:`sugar` for details):
+
+    * strategy = 0: reverse lexicographic order and newest s-polynomial.
+    * strategy = 1: reverse lexicographic order and sugar strategy.
+    * strategy = 2: pure lexicographic order and newest s-polynomial.
+    * strategy = 3: pure lexicographic order and sugar strategy.
+
+* `parameters` is the list of variables :math:`X_{i_1},\ldots,X_{i_k}` among
+  :math:`X_1,\ldots,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 :math:`R(X_{i_1},\ldots,X_{i_k})`. In this case, the coefficient
+  :math:`c` can be a non constant polynomial in :math:`X_{i_1},\ldots,X_{i_k}`, and the tactic
+  produces a goal which states that :math:`c` is not zero.
+
+* `variables` is the list of the variables in the decreasing order in
+  which they will be used in Buchberger algorithm. If `variables` = `(@nil R)`,
+  then `lvar` is replaced by all the variables which are not in
+  `parameters`.
+
+See file `Nsatz.v` for many examples, especially in geometry.