Nsatz: tactics for proving equalities in integral domains¶
| Author: | Loïc Pottier |
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nsatz¶ 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.
You can load the
Nsatzmodule with the commandRequire Import 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 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.
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Variant
nsatz with radicalmax:=num%N strategy:=num%Z parameters:=[ident*,] variables:=[ident*,] Most complete syntax for
nsatz.radicalmaxis a bound when searching for r such that \(c (P−Q) r = \sum_{i=1..s} S_i (P i − Q i)\)strategygives 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.
parametersis 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.variablesis the list of the variables in the decreasing order in which they will be used in the Buchberger algorithm. Ifvariables=(@nil R), thenlvaris replaced by all the variables which are not inparameters.
See the file Nsatz.v for many examples, especially in geometry.