Chapter 26  Nsatz: tactics for proving equalities in integral domains

Loïc Pottier

The tactic nsatz proves goals of the form

 ∀ X1,…,Xn ∈ A, P1(X1,…,Xn) = Q1(X1,…,Xn) , … ,  Ps(X1,…,Xn) =Qs(X1,…,Xn) ⊢ P(X1,…,Xn) = Q(X1,…,Xn)

where P,Q, P1,Q1,…,Ps,Qs are polynomials and A is an integral domain, i.e. a commutative ring with no zero divisor. For example, A can be ℝ, ℤ, of ℚ. Note that the equality = used in these goals can be any setoid equality (see 27.2.2) , not only Leibnitz equality.

It also proves formulas

 ∀ X1,…,Xn ∈ A, P1(X1,…,Xn) = Q1(X1,…,Xn) ∧ … ∧  Ps(X1,…,Xn) =Qs(X1,…,Xn) → P(X1,…,Xn) = Q(X1,…,Xn)

doing automatic introductions.

26.1  Using the basic tactic nsatz

Load the Nsatz module: Require Import Nsatz.
and use the tactic 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[X1,…,Xn] verifies c Pr = ∑i=1s Si Pi, with cA, c ≠ 0, r a positive integer, and the Sis in A[X1,…,Xn], then P is zero whenever polynomials P1,...,Ps 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 S1,…,Ss, c and r such that c (PQ)r = ∑i Si (PiQi), which will be proved by the tactic ring.

This is achieved by the computation of a Groebner basis of the ideal generated by P1Q1,...,PsQs, with an adapted version of the Buchberger algorithm.

This computation is done after a step of reification, which is performed using Type Classes (see 20) .

The Nsatz module defines the tactic nsatz, which can be used without arguments:

nsatz
or with the syntax:

nsatz with radicalmax:=number%N strategy:=number%Z parameters:=list of variables variables:=list of variables
where:

• radicalmax is a bound when for searching r s.t.c (PQ)r = ∑i=1..s Si (PiQi)
• strategy gives the order on variables X1,...Xn and the strategy used in Buchberger algorithm (see [72] 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 Xi1,…,Xik among X1,...,Xn which are considered as parameters: computation will be performed with rational fractions in these variables, i.e. polynomials are considered with coefficients in R(Xi1,…,Xik). In this case, the coefficient c can be a non constant polynomial in Xi1,…,Xik, 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, specially in geometry.