\achapter{Micromega : tactics for solving arithmetics goals over ordered rings}
\aauthor{Frédéric Besson and Evgeny Makarov}
\newtheorem{theorem}{Theorem}

For using the tactics out-of-the-box, read Section~\ref{sec:psatz-hurry}.
%
Section~\ref{sec:psatz-back} presents some background explaining the proof principle for solving polynomials goals.
%
Section~\ref{sec:lia} explains how to get a complete procedure for linear integer arithmetic.

\section{The {\tt psatz} tactic in a hurry}
\label{sec:psatz-hurry}
Load the {\tt Psatz} module ({\tt Require Psatz}.).  This module defines the tactics:
{\tt lia}, {\tt psatzl D}, %{\tt sos D} 
and {\tt psatz D n} where {\tt D} is {\tt Z}, {\tt Q} or {\tt R} and {\tt n} is an optional integer limiting the proof search depth.
  % 
  \begin{itemize}
  \item The {\tt psatzl} tactic solves linear goals using an embedded (naive) linear programming prover \emph{i.e.},
    fourier elimination.
  \item The {\tt psatz} tactic solves polynomial goals using John Harrison's Hol light driver to the external prover {\tt cspd}\footnote{Sources and binaries can be found at \url{https://projects.coin-or.org/Csdp}}. Note that the {\tt csdp} driver is generating 
    a \emph{proof cache} thus allowing to rerun scripts even without {\tt csdp}. 
  \item The {\tt lia} (linear integer arithmetic) tactic is specialised to solve linear goals over $\mathbb{Z}$.
    It extends {\tt psatzl Z} and exploits the discreetness of $\mathbb{Z}$.
%%   \item The {\tt sos} tactic is another Hol light driver to the {\tt csdp} prover.  In theory, it is less general than
%%     {\tt psatz}. In practice, even when {\tt psatz} fails, it can be worth a try -- see
%%     Section~\ref{sec:psatz-back} for details.
  \end{itemize}

These tactics solve propositional formulas parameterised by atomic arithmetics expressions
interpreted over a domain $D \in \{\mathbb{Z}, \mathbb{Q}, \mathbb{R} \}$.
The syntax of the formulas is the following:
\[
\begin{array}{lcl}
 F &::=&  A \mid P \mid \mathit{True} \mid \mathit{False} \mid F_1 \land F_2 \mid F_1 \lor F_2 \mid F_1 \leftrightarrow F_2 \mid F_1 \to F_2 \mid \sim F\\
 A &::=& p_1 = p_2 \mid  p_1 > p_2 \mid p_1 < p_2 \mid p_1 \ge p_2 \mid p_1 \le p_2 \\
 p &::=& c \mid x \mid {-}p \mid p_1 - p_2 \mid p_1 + p_2 \mid p_1 \times p_2 \mid p \verb!^! n
 \end{array}
 \]
 where $c$ is a numeric constant, $x\in D$ is a numeric variable and the operators $-$, $+$, $\times$, are
 respectively subtraction, addition, product, $p \verb!^!n $ is exponentiation by a constant $n$, $P$ is an
 arbitrary proposition. %that is mostly ignored.
%%
%% Over $\mathbb{Z}$, $c$ is an integer ($c \in \mathtt{Z}$), over $\mathbb{Q}$, $c$ is 
The following table details for each domain $D \in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\}$ the range of constants $c$ and exponent $n$.
\[
\begin{array}{|c|c|c|c|}
  \hline
  &\mathbb{Z} & \mathbb{Q} & \mathbb{R} \\
  \hline
  c &\mathtt{Z} & \mathtt{Q} & \{R1, R0\} \\
  \hline
  n &\mathtt{Z} & \mathtt{Z} & \mathtt{nat}\\
  \hline
\end{array}
\]

\section{\emph{Positivstellensatz} refutations}
\label{sec:psatz-back}

The name {\tt psatz} is an abbreviation for \emph{positivstellensatz} -- literally positivity theorem -- which
generalises Hilbert's \emph{nullstellensatz}.
%
It relies on the notion of $\mathit{Cone}$. Given  a (finite) set of polynomials $S$, $Cone(S)$ is
inductively defined as the smallest set of polynomials closed under the following rules:
\[
\begin{array}{l}
\dfrac{p \in S}{p \in Cone(S)} \quad 
\dfrac{}{p^2 \in Cone(S)} \quad
\dfrac{p_1 \in Cone(S) \quad p_2 \in Cone(S) \quad \Join \in \{+,*\}} {p_1 \Join p_2 \in Cone(S)}\\
\end{array}
\]
The following theorem provides a proof principle for checking that a set of polynomial inequalities do not have solutions\footnote{Variants deal with equalities and strict inequalities.}:
\begin{theorem}
  \label{thm:psatz}
  Let $S$ be a set of polynomials.\\
  If ${-}1$ belongs to $Cone(S)$ then the conjunction $\bigwedge_{p \in S} p\ge 0$ is unsatisfiable.
\end{theorem}
A proof based on this theorem is called a \emph{positivstellensatz} refutation.
%
The tactics work as follows. Formulas are normalised into conjonctive normal form $\bigwedge_i C_i$ where
$C_i$ has the general form $(\bigwedge_{j\in S_i} p_j \Join 0) \to \mathit{False})$ and $\Join \in \{>,\ge,=\}$ for $D\in
\{\mathbb{Q},\mathbb{R}\}$ and $\Join \in \{\ge, =\}$ for $\mathbb{Z}$.
%
For each conjunct $C_i$, the tactic calls a oracle which searches for $-1$ within the cone.
%
Upon success, the oracle returns a \emph{cone expression} that is normalised by the {\tt ring} tactic (see chapter~\ref{ring}) and checked to be
$-1$.

To illustrate the working of the tactic, consider we wish to prove the following Coq goal.\\
\begin{coq_eval}
  Require Import ZArith Psatz.
  Open Scope Z_scope.
\end{coq_eval}
\begin{coq_example*}
  Goal forall x, -x^2 >= 0 -> x - 1 >= 0 -> False.
\end{coq_example*}
\begin{coq_eval}
intro x; psatz Z.
\end{coq_eval}
Such a goal is solved by {\tt intro x; psatz Z}. The oracle returns the cone expression $2 \times
(\mathbf{x-1}) + \mathbf{x-1}\times\mathbf{x-1} + \mathbf{-x^2}$ (polynomial hypotheses are printed in bold). By construction, this
expression belongs to $Cone(\{-x^2, x -1\})$.  Moreover, by running {\tt ring} we obtain $-1$. By
Theorem~\ref{thm:psatz}, the goal is valid.
%

\paragraph{The {\tt psatzl} tactic} is searching for \emph{linear} refutations using a fourier
elimination\footnote{More efficient linear programming techniques could equally be employed}.
As a result, this tactic explore a subset of the $Cone$  defined as:
\[
LinCone(S) =\left\{ \left. \sum_{p \in S} \alpha_p \times p\ \right|\ \alpha_p \mbox{ are positive constants} \right\}
\]
Basically, the deductive power of {\tt psatzl} is the combined deductive power of {\tt ring\_simplify} and {\tt fourier}.

\paragraph{The {\tt psatz} tactic} explores the $Cone$ by increasing degrees -- hence the depth parameter $n$.
In theory, such a proof search is complete -- if the goal is provable the search eventually stops.
Unfortunately, the external oracle is using numeric (approximate) optimisation techniques that might miss a
refutation. 

%% \paragraph{The {\tt sos} tactic} -- where {\tt sos} stands for \emph{sum of squares} -- tries to prove that a
%% single polynomial $p$ is positive by expressing it as a sum of squares \emph{i.e.,} $\sum_{i\in S} p_i^2$.
%% This amounts to searching for $p$ in the cone without generators \emph{i.e.}, $Cone(\{\})$.
%

\section{ {\tt lia} : the linear integer arithmetic tactic }
\label{sec:lia}

The tactic {\tt lia} offers an alternative to the {\tt omega} and {\tt romega} tactic (see
Chapter~\ref{OmegaChapter}). It solves goals that {\tt omega} and {\tt romega} do not solve, such as the
following so-called \emph{omega nightmare}~\cite{TheOmegaPaper}.
\begin{coq_example*}
  Goal forall x y, 
       27 <= 11 * x + 13 * y <= 45 -> 
       -10 <= 7 * x - 9 * y <= 4 ->   False.
\end{coq_example*}
\begin{coq_eval}
intro x; lia;
\end{coq_eval}
The estimation of the relative efficiency of lia \emph{vs} {\tt omega}
and {\tt romega} is under evaluation.

\paragraph{High level view of {\tt lia}.}
Over $\mathbb{R}$,  \emph{positivstellensatz} refutations are a complete proof principle\footnote{In practice, the oracle might fail to produce such a refutation.}.
%
However, this is not the case over $\mathbb{Z}$.
%
Actually, \emph{positivstellensatz} refutations are not even sufficient to decide linear \emph{integer} 
arithmetics.
%
The canonical exemple is {\tt 2 * x = 1 -> False} which is a theorem of $\mathbb{Z}$ but not a theorem of $\mathbb{R}$.
%
To remedy this weakness, the {\tt lia} tactic is using recursively a combination of:
%
\begin{itemize}
\item linear \emph{positivstellensatz} refutations \emph{i.e.}, {\tt psatzl Z};
\item cutting plane proofs;
\item case split.
\end{itemize}

\paragraph{Cutting plane proofs} are a way to take into account the discreetness of $\mathbb{Z}$ by rounding up
(rational) constants up-to the closest integer. 
%
\begin{theorem}
  Let $p$ be an integer and $c$ a rational constant.
  \[
  p \ge c \Rightarrow p \ge \lceil c \rceil
  \]
\end{theorem}
For instance, from $2 * x = 1$ we can deduce 
\begin{itemize}
\item $x \ge 1/2$ which cut plane is $ x \ge \lceil 1/2 \rceil = 1$;
\item $ x \le 1/2$ which cut plane is $ x \le \lfloor 1/2 \rfloor = 0$.
\end{itemize}
By combining these two facts (in normal form) $x - 1 \ge 0$ and $-x \ge 0$, we conclude by exhibiting a
\emph{positivstellensatz} refutation ($-1 \equiv \mathbf{x-1} + \mathbf{-x}  \in Cone(\{x-1,x\})$).

Cutting plane proofs and linear \emph{positivstellensatz} refutations are a complete proof principle for integer linear arithmetic.

\paragraph{Case split} allow to enumerate over the possible values of an expression. 
\begin{theorem}
  Let $p$ be an integer and $c_1$ and $c_2$  integer constants.
  \[
  c_1 \le p \le c_2 \Rightarrow \bigvee_{x \in [c_1,c_2]} p = x
  \]
\end{theorem}
Our current oracle tries to find an expression $e$ with a small range $[c_1,c_2]$.
%
We generate $c_2 - c_1$ subgoals which contexts are enriched with an equation $e = i$ for $i \in [c_1,c_2]$ and
recursively search for a proof.

% This technique is used to solve so-called \emph{Omega nightmare}


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