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| -rw-r--r-- | Makefile.doc | 2 | ||||
| -rw-r--r-- | doc/refman/Omega.tex | 249 | ||||
| -rw-r--r-- | doc/refman/Reference-Manual.tex | 1 | ||||
| -rw-r--r-- | doc/sphinx/addendum/omega.rst | 184 | ||||
| -rw-r--r-- | doc/sphinx/biblio.bib | 2 | ||||
| -rw-r--r-- | doc/sphinx/index.rst | 1 |
6 files changed, 187 insertions, 252 deletions
diff --git a/Makefile.doc b/Makefile.doc index d35f1b9d9..2f4727ef9 100644 --- a/Makefile.doc +++ b/Makefile.doc @@ -61,7 +61,7 @@ REFMANCOQTEXFILES:=$(addprefix doc/refman/, \ RefMan-oth.v.tex RefMan-ltac.v.tex \ RefMan-pro.v.tex \ Coercion.v.tex Extraction.v.tex \ - Program.v.tex Omega.v.tex Micromega.v.tex Polynom.v.tex Nsatz.v.tex \ + Program.v.tex Micromega.v.tex Polynom.v.tex Nsatz.v.tex \ Setoid.v.tex Classes.v.tex Universes.v.tex \ Misc.v.tex) diff --git a/doc/refman/Omega.tex b/doc/refman/Omega.tex deleted file mode 100644 index 82765da6e..000000000 --- a/doc/refman/Omega.tex +++ /dev/null @@ -1,249 +0,0 @@ -\achapter{Omega: a solver of quantifier-free problems in -Presburger Arithmetic} -%HEVEA\cutname{omega.html} -\aauthor{Pierre Crégut} -\label{OmegaChapter} - -\asection{Description of {\tt omega}} -\tacindex{omega} -\label{description} - -{\tt omega} solves a goal in Presburger arithmetic, i.e. a universally -quantified formula made of equations and inequations. Equations may -be specified either on the type \verb=nat= of natural numbers or on -the type \verb=Z= of binary-encoded integer numbers. Formulas on -\verb=nat= are automatically injected into \verb=Z=. The procedure -may use any hypothesis of the current proof session to solve the goal. - -Multiplication is handled by {\tt omega} but only goals where at -least one of the two multiplicands of products is a constant are -solvable. This is the restriction meant by ``Presburger arithmetic''. - -If the tactic cannot solve the goal, it fails with an error message. -In any case, the computation eventually stops. - -\asubsection{Arithmetical goals recognized by {\tt omega}} - -{\tt omega} applied only to quantifier-free formulas built from the -connectors - -\begin{quote} -\verb=/\, \/, ~, ->= -\end{quote} - -on atomic formulas. Atomic formulas are built from the predicates - -\begin{quote} -\verb!=, le, lt, gt, ge! -\end{quote} - - on \verb=nat= or from the predicates - -\begin{quote} -\verb!=, <, <=, >, >=! -\end{quote} - - on \verb=Z=. In expressions of type \verb=nat=, {\tt omega} recognizes - -\begin{quote} -\verb!plus, minus, mult, pred, S, O! -\end{quote} - -and in expressions of type \verb=Z=, {\tt omega} recognizes - -\begin{quote} -\verb!+, -, *, Z.succ!, and constants. -\end{quote} - -All expressions of type \verb=nat= or \verb=Z= not built on these -operators are considered abstractly as if they -were arbitrary variables of type \verb=nat= or \verb=Z=. - -\asubsection{Messages from {\tt omega}} -\label{errors} - -When {\tt omega} does not solve the goal, one of the following errors -is generated: - -\begin{ErrMsgs} - -\item \errindex{omega can't solve this system} - - This may happen if your goal is not quantifier-free (if it is - universally quantified, try {\tt intros} first; if it contains - existentials quantifiers too, {\tt omega} is not strong enough to solve your - goal). This may happen also if your goal contains arithmetical - operators unknown from {\tt omega}. Finally, your goal may be really - wrong! - -\item \errindex{omega: Not a quantifier-free goal} - - If your goal is universally quantified, you should first apply {\tt - intro} as many time as needed. - -\item \errindex{omega: Unrecognized predicate or connective: {\sl ident}} - -\item \errindex{omega: Unrecognized atomic proposition: {\sl prop}} - -\item \errindex{omega: Can't solve a goal with proposition variables} - -\item \errindex{omega: Unrecognized proposition} - -\item \errindex{omega: Can't solve a goal with non-linear products} - -\item \errindex{omega: Can't solve a goal with equality on {\sl type}} - -\end{ErrMsgs} - -%% This code is currently unplugged -%% -% \asubsection{Control over the output} -% There are some flags that can be set to get more information on the procedure - -% \begin{itemize} -% \item \verb=Time= to get the time used by the procedure -% \item \verb=System= to visualize the normalized systems. -% \item \verb=Action= to visualize the actions performed by the OMEGA -% procedure (see \ref{technical}). -% \end{itemize} - -% \comindex{Set omega Time} -% \comindex{UnSet omega Time} -% \comindex{Switch omega Time} -% \comindex{Set omega System} -% \comindex{UnSet omega System} -% \comindex{Switch omega System} -% \comindex{Set omega Action} -% \comindex{UnSet omega Action} -% \comindex{Switch omega Action} - -% Use {\tt Set omega {\rm\sl flag}} to set the flag -% {\rm\sl flag}. Use {\tt Unset omega {\rm\sl flag}} to unset it and -% {\tt Switch omega {\rm\sl flag}} to toggle it. - -\section{Using {\tt omega}} - -The {\tt omega} tactic does not belong to the core system. It should be -loaded by -\begin{coq_example*} -Require Import Omega. -Open Scope Z_scope. -\end{coq_example*} - -\example{} - -\begin{coq_example} -Goal forall m n:Z, 1 + 2 * m <> 2 * n. -intros; omega. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\example{} - -\begin{coq_example} -Goal forall z:Z, z > 0 -> 2 * z + 1 > z. -intro; omega. -\end{coq_example} - -% Other examples can be found in \verb+$COQLIB/theories/DEMOS/OMEGA+. - -\section{Options} - -\begin{quote} - \optindex{Stable Omega} - {\tt Unset Stable Omega} -\end{quote} -This deprecated option (on by default) is for compatibility with Coq -pre 8.5. It resets internal name counters to make executions of -{\tt omega} independent. - -\begin{quote} - \optindex{Omega UseLocalDefs} - {\tt Unset Omega UseLocalDefs} -\end{quote} -This option (on by default) allows {\tt omega} to use the bodies of -local variables. - -\begin{quote} - \optindex{Omega System} - {\tt Set Omega System} - \optindex{Omega Action} - {\tt Set Omega Action} -\end{quote} -These two options (off by default) activate the printing of debug -information. - -\asection{Technical data} -\label{technical} - -\asubsection{Overview of the tactic} -\begin{itemize} - -\item The goal is negated twice and the first negation is introduced as an - hypothesis. -\item Hypothesis are decomposed in simple equations or inequations. Multiple - goals may result from this phase. -\item Equations and inequations over \verb=nat= are translated over - \verb=Z=, multiple goals may result from the translation of - substraction. -\item Equations and inequations are normalized. -\item Goals are solved by the {\it OMEGA} decision procedure. -\item The script of the solution is replayed. - -\end{itemize} - -\asubsection{Overview of the {\it OMEGA} decision procedure} - -The {\it OMEGA} decision procedure involved in the {\tt omega} tactic uses -a small subset of the decision procedure presented in - -\begin{quote} - "The Omega Test: a fast and practical integer programming -algorithm for dependence analysis", William Pugh, Communication of the -ACM , 1992, p 102-114. -\end{quote} - -Here is an overview, look at the original paper for more information. - -\begin{itemize} - -\item Equations and inequations are normalized by division by the GCD of their - coefficients. -\item Equations are eliminated, using the Banerjee test to get a coefficient - equal to one. -\item Note that each inequation defines a half space in the space of real value - of the variables. - \item Inequations are solved by projecting on the hyperspace - defined by cancelling one of the variable. They are partitioned - according to the sign of the coefficient of the eliminated - variable. Pairs of inequations from different classes define a - new edge in the projection. - \item Redundant inequations are eliminated or merged in new - equations that can be eliminated by the Banerjee test. -\item The last two steps are iterated until a contradiction is reached - (success) or there is no more variable to eliminate (failure). - -\end{itemize} - -It may happen that there is a real solution and no integer one. The last -steps of the Omega procedure (dark shadow) are not implemented, so the -decision procedure is only partial. - -\asection{Bugs} - -\begin{itemize} -\item The simplification procedure is very dumb and this results in - many redundant cases to explore. - -\item Much too slow. - -\item Certainly other bugs! You can report them to \url{https://coq.inria.fr/bugs/}. - -\end{itemize} - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Reference-Manual.tex b/doc/refman/Reference-Manual.tex index d65749336..a49637bb2 100644 --- a/doc/refman/Reference-Manual.tex +++ b/doc/refman/Reference-Manual.tex @@ -119,7 +119,6 @@ Options A and B of the licence are {\em not} elected.} \include{AddRefMan-pre}% \include{Coercion.v}% \include{Classes.v}% -\include{Omega.v}% \include{Micromega.v} \include{Extraction.v}% \include{Program.v}% diff --git a/doc/sphinx/addendum/omega.rst b/doc/sphinx/addendum/omega.rst new file mode 100644 index 000000000..20e40c550 --- /dev/null +++ b/doc/sphinx/addendum/omega.rst @@ -0,0 +1,184 @@ +.. _omega: + +Omega: a solver for quantifier-free problems in Presburger Arithmetic +===================================================================== + +:Author: Pierre Crégut + +Description of ``omega`` +------------------------ + +This tactic does not need any parameter: + +.. tacn:: omega + +``omega`` solves a goal in Presburger arithmetic, i.e. a universally +quantified formula made of equations and inequations. Equations may +be specified either on the type ``nat`` of natural numbers or on +the type ``Z`` of binary-encoded integer numbers. Formulas on +``nat`` are automatically injected into ``Z``. The procedure +may use any hypothesis of the current proof session to solve the goal. + +Multiplication is handled by ``omega`` but only goals where at +least one of the two multiplicands of products is a constant are +solvable. This is the restriction meant by "Presburger arithmetic". + +If the tactic cannot solve the goal, it fails with an error message. +In any case, the computation eventually stops. + +Arithmetical goals recognized by ``omega`` +------------------------------------------ + +``omega`` applied only to quantifier-free formulas built from the +connectors:: + + /\ \/ ~ -> + +on atomic formulas. Atomic formulas are built from the predicates:: + + = < <= > >= + +on ``nat`` or ``Z``. In expressions of type ``nat``, ``omega`` recognizes:: + + + - * S O pred + +and in expressions of type ``Z``, ``omega`` recognizes numeral constants and:: + + + - * Z.succ Z.pred + +All expressions of type ``nat`` or ``Z`` not built on these +operators are considered abstractly as if they +were arbitrary variables of type ``nat`` or ``Z``. + +Messages from ``omega`` +----------------------- + +When ``omega`` does not solve the goal, one of the following errors +is generated: + +.. exn:: omega can't solve this system + + This may happen if your goal is not quantifier-free (if it is + universally quantified, try ``intros`` first; if it contains + existentials quantifiers too, ``omega`` is not strong enough to solve your + goal). This may happen also if your goal contains arithmetical + operators unknown from ``omega``. Finally, your goal may be really + wrong! + +.. exn:: omega: Not a quantifier-free goal + + If your goal is universally quantified, you should first apply + ``intro`` as many time as needed. + +.. exn:: omega: Unrecognized predicate or connective: @ident + +.. exn:: omega: Unrecognized atomic proposition: ... + +.. exn:: omega: Can't solve a goal with proposition variables + +.. exn:: omega: Unrecognized proposition + +.. exn:: omega: Can't solve a goal with non-linear products + +.. exn:: omega: Can't solve a goal with equality on type ... + + +Using ``omega`` +--------------- + +The ``omega`` tactic does not belong to the core system. It should be +loaded by + +.. coqtop:: in + + Require Import Omega. + +.. example:: + + .. coqtop:: all + + Require Import Omega. + + Open Scope Z_scope. + + Goal forall m n:Z, 1 + 2 * m <> 2 * n. + intros; omega. + Abort. + + Goal forall z:Z, z > 0 -> 2 * z + 1 > z. + intro; omega. + Abort. + + +Options +------- + +.. opt:: Stable Omega + +This deprecated option (on by default) is for compatibility with Coq pre 8.5. It +resets internal name counters to make executions of ``omega`` independent. + +.. opt:: Omega UseLocalDefs + +This option (on by default) allows ``omega`` to use the bodies of local +variables. + +.. opt:: Omega System + +This option (off by default) activate the printing of debug information + +.. opt:: Omega Action + +This option (off by default) activate the printing of debug information + +Technical data +-------------- + +Overview of the tactic +~~~~~~~~~~~~~~~~~~~~~~ + + * The goal is negated twice and the first negation is introduced as an hypothesis. + * Hypothesis are decomposed in simple equations or inequations. Multiple + goals may result from this phase. + * Equations and inequations over ``nat`` are translated over + ``Z``, multiple goals may result from the translation of substraction. + * Equations and inequations are normalized. + * Goals are solved by the OMEGA decision procedure. + * The script of the solution is replayed. + +Overview of the OMEGA decision procedure +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The OMEGA decision procedure involved in the ``omega`` tactic uses +a small subset of the decision procedure presented in :cite:`TheOmegaPaper` +Here is an overview, look at the original paper for more information. + + * Equations and inequations are normalized by division by the GCD of their + coefficients. + * Equations are eliminated, using the Banerjee test to get a coefficient + equal to one. + * Note that each inequation defines a half space in the space of real value + of the variables. + * Inequations are solved by projecting on the hyperspace + defined by cancelling one of the variable. They are partitioned + according to the sign of the coefficient of the eliminated + variable. Pairs of inequations from different classes define a + new edge in the projection. + * Redundant inequations are eliminated or merged in new + equations that can be eliminated by the Banerjee test. + * The last two steps are iterated until a contradiction is reached + (success) or there is no more variable to eliminate (failure). + +It may happen that there is a real solution and no integer one. The last +steps of the Omega procedure (dark shadow) are not implemented, so the +decision procedure is only partial. + +Bugs +---- + + * The simplification procedure is very dumb and this results in + many redundant cases to explore. + + * Much too slow. + + * Certainly other bugs! You can report them to https://coq.inria.fr/bugs/. diff --git a/doc/sphinx/biblio.bib b/doc/sphinx/biblio.bib index 4a9bd6c1a..247f32103 100644 --- a/doc/sphinx/biblio.bib +++ b/doc/sphinx/biblio.bib @@ -1312,7 +1312,7 @@ Languages}, year = {1994} } -@article{ TheOmegaPaper, +@article{TheOmegaPaper, author = "W. Pugh", title = "The Omega test: a fast and practical integer programming algorithm for dependence analysis", journal = "Communication of the ACM", diff --git a/doc/sphinx/index.rst b/doc/sphinx/index.rst index af93060fd..d42e73217 100644 --- a/doc/sphinx/index.rst +++ b/doc/sphinx/index.rst @@ -45,6 +45,7 @@ Table of contents addendum/extended-pattern-matching addendum/canonical-structures + addendum/omega .. toctree:: :caption: Reference |