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diff --git a/doc/sphinx/addendum/nsatz.rst b/doc/sphinx/addendum/nsatz.rst index 7ce400937c..9adeca46fc 100644 --- a/doc/sphinx/addendum/nsatz.rst +++ b/doc/sphinx/addendum/nsatz.rst @@ -1,37 +1,40 @@ .. include:: ../preamble.rst -.. _nsatz: +.. _nsatz_chapter: Nsatz: tactics for proving equalities in integral domains =========================================================== :Author: Loïc Pottier -The tactic `nsatz` proves goals of the form +.. tacn:: nsatz + :name: nsatz -: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}` + This tactic is for solving goals of the form -where :math:`P, Q, P_1, Q_1, \ldots, P_s, Q_s` are polynomials and :math:`A` is an integral -domain, i.e. a commutative ring with no zero divisors. 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:`tactics-enabled-on-user-provided-relations`) , not only Leibnitz equality. + :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}` -It also proves formulas + where :math:`P, Q, P_1, Q_1, \ldots, P_s, Q_s` are polynomials and :math:`A` is an integral + domain, i.e. a commutative ring with no zero divisors. 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:`tactics-enabled-on-user-provided-relations`) , not only Leibniz equality. -: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}` + It also proves formulas -doing automatic introductions. + :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}` -You can load the ``Nsatz`` module with the command ``Require Import Nsatz``. + doing automatic introductions. + + You can load the ``Nsatz`` module with the command ``Require Import Nsatz``. More about `nsatz` --------------------- @@ -57,34 +60,31 @@ 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}` +.. tacv:: nsatz with radicalmax:=@num%N strategy:=@num%Z parameters:=[{*, @ident}] variables:=[{*, @ident}] -where: + Most complete syntax for `nsatz`. -* `radicalmax` is a bound when searching for r such that - :math:`c (P−Q) r = \sum_{i=1..s} S_i (P i − Q i)` + * `radicalmax` is a bound when searching for r such that + :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` 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. + * 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. + * `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 the Buchberger algorithm. If `variables` = `(@nil R)`, - then `lvar` is replaced by all the variables which are not in - `parameters`. + * `variables` is the list of the variables in the decreasing order in + which they will be used in the 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. +See the file `Nsatz.v` for many examples, especially in geometry. |