Nsatz: a solver for equalities in integral domains — Coq 8.16.1 documentation

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# Nsatz: a solver for equalities in integral domains

Author:

  Loïc Pottier

To use the tactics described in this section, load the `Nsatz` module with the command `Require Import Nsatz`. Alternatively, if you prefer not to transitively depend on the files that declare the axioms used to define the real numbers, you can `Require Import NsatzTactic` instead; this will still allow `nsatz` to solve goals defined about \(\mathbb{Z}\), \(\mathbb{Q}\) and any user-registered rings.

Tactic nsatz with radicalmax := one_term strategy := one_term parameters := one_term variables := one_term?

  This tactic is for solving 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 divisors. 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 Leibniz 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.

  `radicalmax`

    bound when searching for r such that \(c (P−Q) r = \sum_{i=1..s} S_i (P i − Q i)\). This argument must be of type `N` (binary natural numbers).

  `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%Z`: reverse lexicographic order and newest s-polynomial.

    > - `strategy := 1%Z`: reverse lexicographic order and sugar strategy.

    > - `strategy := 2%Z`: pure lexicographic order and newest s-polynomial.

    > - `strategy := 3%Z`: pure lexicographic order and sugar strategy.

  `parameters`

    a list of parameters of type `R`, containing the variables \(X_{i_1},\ldots,X_{i_k}\) among \(X_1,\ldots,X_n\). Computation will be performed with rational fractions in these parameters, i.e. polynomials have coefficients in \(R(X_{i_1},\ldots,X_{i_k})\). In this case, the coefficient \(c\) can be a nonconstant polynomial in \(X_{i_1},\ldots,X_{i_k}\), and the tactic produces a goal which states that \(c\) is not zero.

  `variables`

    a list of variables of type `R` in the decreasing order in which they will be used in the Buchberger algorithm. If the list is empty, then `lvar` is replaced by all the variables which are not in `parameters`.

  See the file Nsatz.v for examples, especially in geometry.

## More about `nsatz`

Hilbert’s Nullstellensatz theorem shows how to reduce proofs of equalities on polynomials on a commutative ring \(A\) with no zero divisors 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 algebraically closed field: the method is complete).

So, solving our initial problem reduces to 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 Typeclasses.

Tactic nsatz_compute one_term

Other versions v: 8.16.1

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