\[ \newcommand{\alors}{\textsf{then}} \newcommand{\alter}{\textsf{alter}} \newcommand{\as}{\kw{as}} \newcommand{\Assum}[3]{\kw{Assum}(#1)(#2:#3)} \newcommand{\bool}{\textsf{bool}} \newcommand{\case}{\kw{case}} \newcommand{\conc}{\textsf{conc}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\conshl}{\textsf{cons\_hl}} \newcommand{\Def}[4]{\kw{Def}(#1)(#2:=#3:#4)} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\endkw}{\kw{end}} \newcommand{\EqSt}{\textsf{EqSt}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even_O}} \newcommand{\evenS}{\textsf{even_S}} \newcommand{\false}{\textsf{false}} \newcommand{\filter}{\textsf{filter}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\from}{\textsf{from}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\haslength}{\textsf{has\_length}} \newcommand{\hd}{\textsf{hd}} \newcommand{\ident}{\textsf{ident}} \newcommand{\In}{\kw{in}} \newcommand{\Ind}[4]{\kw{Ind}[#2](#3:=#4)} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[5]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)} \newcommand{\Indpstr}[6]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)/{#6}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\lb}{\lambda} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\Nat}{\mathbb{N}} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd_S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\Prod}{\textsf{prod}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\si}{\textsf{if}} \newcommand{\sinon}{\textsf{else}} \newcommand{\Sort}{\cal S} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\true}{\textsf{true}} \newcommand{\Type}{\textsf{Type}} \newcommand{\unfold}{\textsf{unfold}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\cal W\!F}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\cal W\!F}(#2)} \newcommand{\WFTWOLINES}[2]{{\cal W\!F}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \newcommand{\zeros}{\textsf{zeros}} \]

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.