Nsatz: tactics for proving equalities in integral domains — Coq 8.8.1 documentation

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# Nsatz: tactics for proving equalities in integral domains

Author: | Loïc Pottier

The tactic 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 \(P, Q, P₁,Q₁,\ldots,Pₛ, Qₛ\) are polynomials and \(A\) is an integral domain, i.e. a commutative ring with no zero divisor. For example, \(A\) can be \(\mathbb{R}\), \(\mathbb{Z}\), or \(\mathbb{Q}\). Note that the equality \(=\) used in these goals can be any setoid equality (see Tactics enabled on user provided relations) , 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.

## Using the basic tactic nsatz

Load the Nsatz module:

Require Import Nsatz.
  [Loading ML file z_syntax_plugin.cmxs ... done] [Loading ML file quote_plugin.cmxs ... done] [Loading ML file newring_plugin.cmxs ... done] [Loading ML file omega_plugin.cmxs ... done] [Loading ML file r_syntax_plugin.cmxs ... done] [Loading ML file fourier_plugin.cmxs ... done] [Loading ML file micromega_plugin.cmxs ... done] [Loading ML file nsatz_plugin.cmxs ... done]

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 \(A\) 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 \(P_1,\ldots,P_s\) 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 ring.

This is achieved by the computation of a Gröbner 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 reification, which is performed using Type Classes.

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:= `var*` variables:= `var*`

where:

- radicalmax is a bound when for searching r s.t. \(c (P−Q) r = \sum_{i=1..s} S_i (P i − Q i)\)

- strategy gives the order on variables \(X_1,\ldots,X_n\) and the strategy used in Buchberger algorithm (see [GMN+91] 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 \(X_{i_1},\ldots,X_{i_k}\) among \(X_1,\ldots,X_n\) which are considered as parameters: computation will be performed with rational fractions in these variables, i.e. polynomials are considered with coefficients in \(R(X_{i_1},\ldots,X_{i_k})\). 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.

- 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.