Fix typos in the Micromega part of the reference manual.

1 files changed, 78 insertions, 57 deletions

diff --git a/doc/refman/Micromega.tex b/doc/refman/Micromega.tex
index 551f6c205..1efed6ef7 100644
--- a/doc/refman/Micromega.tex
+++ b/doc/refman/Micromega.tex
@@ -1,4 +1,4 @@
-\achapter{Micromega : tactics for solving arithmetic goals over ordered rings}
+\achapter{Micromega: tactics for solving arithmetic goals over ordered rings}
 \aauthor{Frédéric Besson and Evgeny Makarov}
 \newtheorem{theorem}{Theorem}
 
@@ -6,33 +6,40 @@
 \asection{Short description of the tactics}
 \tacindex{psatz}  \tacindex{lra} 
 \label{sec:psatz-hurry}
-The {\tt Psatz} module ({\tt Require Import Psatz.}) gives access to several tactics for solving arithmetic goals over
- {\tt Z}\footnote{Support for {\tt nat} and {\tt N} is obtained by pre-processing the goal with the {\tt zify} tactic.}, {\tt Q} and {\tt R}:
+The {\tt Psatz} module ({\tt Require Import Psatz.}) gives access to
+several tactics for solving arithmetic goals over {\tt Z}, {\tt Q}, and
+{\tt R}:\footnote{Support for {\tt nat} and {\tt N} is obtained by
+  pre-processing the goal with the {\tt zify} tactic.}
 \begin{itemize}
 \item {\tt lia} is a decision procedure for linear integer arithmetic (see Section~\ref{sec:lia});
 \item {\tt nia} is an incomplete proof procedure for integer non-linear arithmetic (see Section~\ref{sec:nia});
 \item {\tt lra} is a decision procedure for linear (real or rational) arithmetic goals (see Section~\ref{sec:lra});
-\item {\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 is
-is an incomplete proof procedure for non-linear arithmetic. It is based on 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} which allows rerunning scripts even without {\tt csdp} (see Section~\ref{sec:psatz}).
+\item {\tt psatz D n} where {\tt D} is {\tt Z} or {\tt Q} or {\tt R}, and
+  {\tt n} is an optional integer limiting the proof search depth is is an
+  incomplete proof procedure for non-linear arithmetic. It is based on
+  John Harrison's HOL Light driver to the external prover {\tt
+    csdp}\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} which makes it possible to
+  rerun scripts even without {\tt csdp} (see Section~\ref{sec:psatz}).
 \end{itemize}
 
-The tactics solve propositional formulas parameterised by atomic arithmetics expressions
+The tactics solve propositional formulas parameterized by atomic arithmetic expressions
 interpreted over a domain $D \in \{\mathbb{Z}, \mathbb{Q}, \mathbb{R} \}$.
 The syntax of the formulas is the following:
 \[
 \begin{array}{lcl}
- F &::=&  A \mid P \mid \mathit{True} \mid \mathit{False} \mid F_1 \land F_2 \mid F_1 \lor F_2 \mid F_1 \leftrightarrow F_2 \mid F_1 \to F_2 \mid \sim F\\
+ F &::=&  A \mid P \mid \mathit{True} \mid \mathit{False} \mid F_1 \land F_2 \mid F_1 \lor F_2 \mid F_1 \leftrightarrow F_2 \mid F_1 \to F_2 \mid \neg F\\
  A &::=& p_1 = p_2 \mid  p_1 > p_2 \mid p_1 < p_2 \mid p_1 \ge p_2 \mid p_1 \le p_2 \\
  p &::=& c \mid x \mid {-}p \mid p_1 - p_2 \mid p_1 + p_2 \mid p_1 \times p_2 \mid p \verb!^! n
- \end{array}
- \]
- 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.
+\end{array}
+\]
+where $c$ is a numeric constant, $x\in D$ is a numeric variable, the
+operators $-$, $+$, $\times$ are respectively subtraction, addition,
+product, $p \verb!^!n $ is exponentiation by a constant $n$, $P$ is an
+arbitrary proposition.
  %
- For {\tt Q}, equality is not leibnitz equality {\tt =} but the equality of rationals {\tt ==}.
+ For {\tt Q}, equality is not Leibniz equality {\tt =} but the equality of rationals {\tt ==}.
 
 For {\tt Z} (resp. {\tt Q} ), $c$ ranges over integer constants (resp. rational constants).
 %% The following table details for each domain $D \in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\}$ the range of constants $c$ and exponent $n$.
@@ -47,79 +54,89 @@ For {\tt Z} (resp. {\tt Q} ), $c$ ranges over integer constants (resp. rational
 %%   \hline
 %% \end{array}
 %% \]
-For {\tt R}, the tactic recognises as real constants the following expressions:
+For {\tt R}, the tactic recognizes as real constants the following expressions:
 \begin{verbatim}
-c ::= R0 | R1 | Rmul(c,c) | Rplus(c,c) | Rminus(c,c) | IZR z | IQR q | Rdiv(c,c) | Rinv c
+c ::= R0 | R1 | Rmul(c,c) | Rplus(c,c) | Rminus(c,c) | IZR z | IQR q
+    | Rdiv(c,c) | Rinv c
 \end{verbatim}
-where ${\tt z}$ is a constant in {\tt Z} and {\tt q} is a constant in {\tt Q}.
+where {\tt z} is a constant in {\tt Z} and {\tt q} is a constant in {\tt Q}.
 This includes integer constants written using the decimal notation \emph{i.e.,} {\tt c\%R}.
 
 \asection{\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}.
+generalizes 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:
+It relies on the notion of $\mathit{Cone}$. Given a (finite) set of
+polynomials $S$, $\mathit{Cone}(S)$ is inductively defined as the
+smallest set of polynomials closed under the following rules:
 \[
 \begin{array}{l}
-\dfrac{p \in S}{p \in Cone(S)} \quad 
-\dfrac{}{p^2 \in Cone(S)} \quad
-\dfrac{p_1 \in Cone(S) \quad p_2 \in Cone(S) \quad \Join \in \{+,*\}} {p_1 \Join p_2 \in Cone(S)}\\
+\dfrac{p \in S}{p \in \mathit{Cone}(S)} \quad
+\dfrac{}{p^2 \in \mathit{Cone}(S)} \quad
+\dfrac{p_1 \in \mathit{Cone}(S) \quad p_2 \in \mathit{Cone}(S) \quad
+\Join \in \{+,*\}} {p_1 \Join p_2 \in \mathit{Cone}(S)}\\
 \end{array}
 \]
-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.}:
+The following theorem provides a proof principle for checking that a set
+of polynomial inequalities does 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.
+  If ${-}1$ belongs to $\mathit{Cone}(S)$ 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
+The tactics work as follows. Formulas are normalized into conjunctive 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
+Upon success, the oracle returns a \emph{cone expression} that is normalized by the {\tt ring} tactic (see chapter~\ref{ring}) and checked to be
 $-1$.
 
 
-\asection{{\tt lra} : a decision procedure for linear real and rational arithmetic}
+\asection{{\tt lra}: a decision procedure for linear real and rational arithmetic}
 \label{sec:lra}
 The {\tt lra} tactic is searching for \emph{linear} refutations using
-Fourier elimination\footnote{More efficient linear programming techniques could equally be employed}.  As a
-result, this tactic explores a subset of the $Cone$ defined as:
+Fourier elimination.\footnote{More efficient linear programming
+  techniques could equally be employed.} As a result, this tactic
+explores a subset of the $\mathit{Cone}$ defined as
 \[
-LinCone(S) =\left\{ \left. \sum_{p \in S} \alpha_p \times p\ \right|\ \alpha_p \mbox{ are positive constants} \right\}
+\mathit{LinCone}(S) =\left\{ \left. \sum_{p \in S} \alpha_p \times p~\right|
+~\alpha_p \mbox{ are positive constants} \right\}.
 \]
 The deductive power of {\tt lra} is the combined deductive power of {\tt ring\_simplify} and {\tt fourier}.
 %
 There is also an overlap with the {\tt field} tactic {\emph e.g.}, {\tt x = 10 * x / 10} is solved by {\tt lra}.
 
-\asection{ {\tt psatz} : a proof procedure for non-linear arithmetic}
+\asection{{\tt psatz}: a proof procedure for non-linear arithmetic}
 \label{sec:psatz}
-The {\tt psatz} tactic explores the $Cone$ by increasing degrees -- hence the depth parameter $n$.
+The {\tt psatz} tactic explores the $\mathit{Cone}$ 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
+Unfortunately, the external oracle is using numeric (approximate) optimization techniques that might miss a
 refutation. 
 
-To illustrate the working of the tactic, consider we wish to prove the following Coq goal.\\
+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.
+Require Import ZArith Psatz.
+Open Scope Z_scope.
 \end{coq_eval}
 \begin{coq_example*}
-  Goal forall x, -x^2 >= 0 -> x - 1 >= 0 -> False.
+Goal forall x, -x^2 >= 0 -> x - 1 >= 0 -> False.
 \end{coq_example*}
 \begin{coq_eval}
 intro x; psatz Z 2.
 \end{coq_eval}
-Such a goal is solved by {\tt intro x; psatz Z 2}. 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
+Such a goal is solved by {\tt intro x; psatz Z 2}. 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 $\mathit{Cone}(\{-x^2,
+x -1\})$. Moreover, by running {\tt ring} we obtain $-1$. By
 Theorem~\ref{thm:psatz}, the goal is valid.
 %
 
@@ -128,14 +145,14 @@ Theorem~\ref{thm:psatz}, the goal is valid.
 %% This amounts to searching for $p$ in the cone without generators \emph{i.e.}, $Cone(\{\})$.
 %
 
-\asection{ {\tt lia} : a tactic for linear integer arithmetic }
+\asection{{\tt lia}: a tactic for linear integer arithmetic}
 \tacindex{lia}
 \label{sec:lia}
 
-The tactic {\tt lia} ({\tt Require Lia.}) offers an alternative to the {\tt omega} and {\tt romega} tactic (see
-Chapter~\ref{OmegaChapter}). 
+The tactic {\tt lia} offers an alternative to the {\tt omega} and {\tt
+  romega} tactic (see Chapter~\ref{OmegaChapter}).
 %
-Rougthly speaking, the deductive power of {\tt lia} is the combined deductive power of {\tt ring\_simplify} and {\tt omega}.
+Roughly speaking, the deductive power of {\tt lia} is the combined deductive power of {\tt ring\_simplify} and {\tt omega}.
 %
 However, it solves linear goals that {\tt omega} and {\tt romega} do not solve, such as the
 following so-called \emph{omega nightmare}~\cite{TheOmegaPaper}.
@@ -147,18 +164,20 @@ Goal forall x y,
 \begin{coq_eval}
 intros x y; lia.
 \end{coq_eval}
-The estimation of the relative efficiency of lia \emph{vs} {\tt omega}
+The estimation of the relative efficiency of {\tt 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.}.
+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.
+Actually, \emph{positivstellensatz} refutations are not even sufficient
+to decide linear \emph{integer} arithmetic.
 %
-The canonical exemple is {\tt 2 * x = 1 -> False} which is a theorem of $\mathbb{Z}$ but not a theorem of $\mathbb{R}$.
+The canonical example 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:
 %
@@ -177,17 +196,18 @@ To remedy this weakness, the {\tt lia} tactic is using recursively a combination
   p \ge c \Rightarrow p \ge \lceil c \rceil
   \]
 \end{theorem}
-For instance, from $2 * x = 1$ we can deduce 
+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 ($-1 \equiv \mathbf{x-1} + \mathbf{-x}  \in Cone(\{x-1,x\})$).
+By combining these two facts (in normal form) $x - 1 \ge 0$ and $-x \ge
+0$, we conclude by exhibiting a \emph{positivstellensatz} refutation: $-1
+\equiv \mathbf{x-1} + \mathbf{-x} \in \mathit{Cone}(\{x-1,x\})$.
 
 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. 
+\paragraph{Case split} enumerates over the possible values of an expression.
 \begin{theorem}
   Let $p$ be an integer and $c_1$ and $c_2$  integer constants.
   \[
@@ -199,7 +219,7 @@ 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.
 
-\asection{ {\tt nia} : a proof procedure for non-linear integer arithmetic}
+\asection{{\tt nia}: a proof procedure for non-linear integer arithmetic}
 \tacindex{nia}
 \label{sec:nia}
 The {\tt nia} tactic is an {\emph experimental} proof procedure for non-linear  integer arithmetic.
@@ -212,7 +232,8 @@ This pre-processing does the following:
   monomial, the context is enriched with $x^2\ge 0$;
 \item For all pairs of hypotheses $e_1\ge 0$, $e_2 \ge 0$, the context is enriched with $e_1 \times e_2 \ge 0$.
 \end{itemize}
-After pre-processing, the linear prover of {\tt lia} is searching for a proof by abstracting monomials by variables.
+After pre-processing, the linear prover of {\tt lia} searches for a proof
+by abstracting monomials by variables.