[Sphinx] Add chapter 11

Thanks to Enrico Tassi, Assia Mahboubi, Laurence Rideau and Yves Bertot
for porting this chapter.

7 files changed, 5502 insertions, 4936 deletions

diff --git a/Makefile.doc b/Makefile.doc
index 7c68e5d8e..df1e2d3c4 100644
--- a/Makefile.doc
+++ b/Makefile.doc
@@ -68,7 +68,7 @@ REFMANCOQTEXFILES:=$(addprefix doc/refman/, \
 REFMANTEXFILES:=$(addprefix doc/refman/, \
   headers.sty Reference-Manual.tex \
   RefMan-uti.tex \
-  AsyncProofs.tex RefMan-ssr.tex) \
+  AsyncProofs.tex) \
   $(REFMANCOQTEXFILES) \
 
 REFMANEPSFILES:=doc/refman/coqide.eps doc/refman/coqide-queries.eps
diff --git a/doc/refman/Coercion.tex b/doc/refman/Coercion.tex
index 53b6b7827..9862a9b30 100644
--- a/doc/refman/Coercion.tex
+++ b/doc/refman/Coercion.tex
@@ -33,7 +33,8 @@ classes:
 
 \begin{itemize}
 \item {\tt Sortclass}, the class of sorts; 
-  its objects are the terms whose type is a sort (e.g., \ssrC{Prop} or \ssrC{Type}).
+  its objects are the terms whose type is a sort (e.g., \texttt{Prop}
+  or \texttt{Type}).
 \item {\tt Funclass}, the class of functions; 
   its objects are all the terms with a functional 
   type, i.e. of form $forall~ x:A, B$.
diff --git a/doc/refman/RefMan-ssr.tex b/doc/refman/RefMan-ssr.tex
deleted file mode 100644
index 31dabcdd4..000000000
--- a/doc/refman/RefMan-ssr.tex
+++ /dev/null
@@ -1,4933 +0,0 @@
-\achapter{The SSReflect proof language}
-%HEVEA\cutname{ssreflect.html}
-\aauthor{Georges Gonthier, Assia Mahboubi, Enrico Tassi}
-
-\newcommand{\ssr}{{\sc SSReflect}}
-
-% listing
-\ifhevea\newcommand{\ssrC}[1]{\texttt{#1}}\else\newcommand{\ssrC}[1]{\text{\lstinline!#1!}}\fi
-\ifhevea\renewenvironment{center}{\@open{div}{class="center"}\@open{div}{class="centered"}}{\@close{div}\@close{div}}\fi
-% non-terminal
-%\newcommand\ssrN[2][]{\ensuremath{\langle\mbox{\itshape\rmfamily\small #2}\rangle_{#1}}}
-\newcommand\ssrN[2][]{{\textsl {#2}}\ensuremath{_{#1}}}
-\ifhevea\newcommand{\underbar}[1]{\underline{#1}}\fi
-
-% TODO: only use \ssrC
-\let\ssrL=\lstinline
-
-\newcommand{\iitem}{{\it i-item}}
-\newcommand{\ditem}{{\it d-item}}
-\newcommand{\optional}[1]{{\it[}#1{\it]}}
-\newcommand{\optsep}{{\it|}}
-\newcommand{\idx}[1]{\tacindex{#1 (ssreflect)}}
-\newcommand{\idxC}[1]{\comindex{#1 (ssreflect)}}
-
-\newenvironment{new}%
-  {\begin{Sbox}\begin{minipage}{0.97\textwidth}%
-    \begin{flushright}\textcolor{red}{\fbox{Version 1.3}}%
-      \end{flushright}\noindent}%
-  {\end{minipage}\end{Sbox}\noindent\doublebox{\TheSbox}}
-\section{Introduction}\label{sec:intro}
-
-This chapter describes a set of tactics known as \ssr{}
-originally designed to provide support for the so-called \emph{small scale
-reflection} proof methodology.  Despite the original purpose this set of tactic
-is of general interest and is available in Coq starting from version 8.7.
-
-\ssr{} was developed independently of the tactics described in
-Chapter~\ref{Tactics}.  Indeed the scope of the tactics part of
-\ssr{} largely overlaps with the standard set of tactics.  Eventually
-the overlap will be reduced in future releases of Coq.
-
-Proofs written in \ssr{} typically look quite different from the
-ones written using only tactics as per Chapter~\ref{Tactics}.
-We try to summarise here the most ``visible'' ones in order to
-help the reader already accustomed to the tactics described in
-Chapter~\ref{Tactics} to read this chapter.
-
-The first difference between the tactics described in this
-chapter and the tactics described in Chapter~\ref{Tactics} is the way
-hypotheses are managed (we call this \emph{bookkeeping}).
-In Chapter~\ref{Tactics} the most common
-approach is to avoid moving explicitly hypotheses back and forth
-between the context and the conclusion of the goal. On the contrary
-in \ssr{}
-all bookkeeping is performed on the conclusion of the goal, using for
-that purpose a couple of syntactic constructions behaving similar to
-tacticals (and often named as such in this chapter).
-The \ssrC{:} tactical  moves hypotheses from the context to the
-conclusion, while \ssrC{=>} moves hypotheses from the
-conclusion to the context, and \ssrC{in} moves back
-and forth an hypothesis from the context to the conclusion for the
-time of applying an action to it.
-
-While naming hypotheses is commonly done by means of an \ssrC{as}
-clause in the basic model of Chapter~\ref{Tactics}, it is here to
-\ssrC{=>} that this task is devoted. As tactics leave
-new assumptions in the conclusion, and are often followed by
-\ssrC{=>} to explicitly name them.
-While generalizing the goal is normally
-not explicitly needed in Chapter~\ref{Tactics}, it is an explicit
-operation performed by \ssrC{:}.
-
-Beside the difference of bookkeeping model, this chapter includes
-specific tactics which have no explicit counterpart in
-Chapter~\ref{Tactics} such as tactics to mix forward steps and
-generalizations as \ssrC{generally have} or \ssrC{without loss}.
-
-\ssr{} adopts the point of view that rewriting, definition
-expansion and partial evaluation participate all to a same concept of
-rewriting a goal in a larger sense. As such, all these functionalities are
-provided by the \ssrC{rewrite} tactic.
-
-\ssr{} includes a little language of patterns to select subterms in tactics
-or tacticals where it matters.  Its most notable application
-is in the \ssrC{rewrite} tactic, where patterns are used to specify
-where the rewriting step has to take place.
-
-Finally, \ssr{} supports so-called reflection steps, typically
-allowing to switch back and forth between the computational view and
-logical view of a concept.
-
-To conclude it is worth mentioning that \ssr{} tactics
-can be mixed with non \ssr{} tactics in the same proof,
-or in the same Ltac expression.  The few exceptions
-to this statement are described in section~\ref{sec:compat}.
-
-\iffalse
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection*{How to read this documentation}
-
-The syntax of the tactics is presented as follows:
-\begin{itemize}
-\item \ssrC{terminals} are in typewriter font and \ssrN{non terminals} are
-  between angle brackets.
-\item Optional parts of the grammar are surrounded by \optional{ }
-  brackets. These should not be confused with verbatim brackets
-  \ssrC{[ ]}, which are delimiters in the \ssr{} syntax.
-\item A vertical rule {\optsep} indicates an alternative in the syntax, and
-  should not be confused with a
-  verbatim vertical rule between verbatim brackets \ssrC{[ | ]}.
-\item A non empty list of non terminals (at least one item should be
-  present) is represented by \ssrN{non terminals}$^+$. A possibly empty
-  one is represented by \ssrN{non terminals}$^*$.
-\item In a non empty list of non terminals, items are separated by blanks.
-\end{itemize}
-\fi
-
-% Hevea has no colors
-\ifhevea \else
-\noindent We follow the default color scheme of the \ssr{} mode for
-ProofGeneral provided in the distribution:
-
-\centerline{
-\textcolor{dkblue}{\texttt{tactic}} or \textcolor{dkviolet}{\tt
-  Command} or \textcolor{dkgreen}{\tt keyword} or
-\textcolor{dkpink}{\tt tactical}}
-
-\noindent Closing tactics/tacticals like \ssrC{exact} or \ssrC{by} (see section
-\ref{ssec:termin}) are in red.
-\fi
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection*{Acknowledgments}
-The authors would like to thank Frédéric Blanqui, François Pottier
-and Laurence Rideau for their comments and suggestions.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\newpage\section{Usage}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Getting started}\label{sec:files}
-To be available, the tactics presented in this manual need the
-following minimal set of libraries to loaded: {\tt ssreflect.v}, {\tt
-ssrfun.v} and {\tt ssrbool.v}. Moreover, these tactics come with a
-methodology specific to the authors of Ssreflect and which requires a
-few options to be set in a different way than in their default
-way. All in all, this corresponds to working in the following context:
-
-\begin{lstlisting}
-  From Coq Require Import ssreflect ssrfun ssrbool.
-  Set Implicit Arguments.
-  Unset Strict Implicit.
-  Unset Printing Implicit Defensive.
-\end{lstlisting}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Compatibility issues}\label{sec:compat}
-Requiring the above modules creates an environment which
-is mostly compatible with the rest of \Coq{}, up to a few discrepancies:
-\begin{itemize}
-\item New keywords (\ssrC{is}) might clash with variable, constant,
-tactic or tactical names, or with quasi-keywords in tactic or
-vernacular notations.
-\item New tactic(al)s names (\ssrC{last}, \ssrC{done}, \ssrC{have},
-  \ssrC{suffices}, \ssrC{suff},
-  \ssrC{without loss}, \ssrC{wlog}, \ssrC{congr}, \ssrC{unlock}) might clash
-  with user tactic names.
-\item Identifiers with both leading and trailing \ssrC{_}, such as \ssrC{_x_},
-are reserved by \ssr{} and cannot appear in scripts.
-\item The extensions to the \ssrC{rewrite} tactic are partly
-incompatible with those available in current versions of \Coq{};
-in particular:
-\ssrC{rewrite .. in (type of k)} or \\ \ssrC{rewrite .. in *} or any other
-variant of \ssrC{rewrite} will not work, and the \ssr{} syntax and semantics for occurrence selection and
-rule chaining is different.
-
-Use an explicit rewrite direction (\ssrC{rewrite <-} $\dots$ or \ssrC{rewrite ->} $\dots$)
-to access the \Coq{} \ssrC{rewrite} tactic.
-\item New symbols (\ssrC{//, /=, //=}) might clash with adjacent existing
-  symbols (e.g., '\ssrC{//}') instead of '\ssrC{/}''\ssrC{/}'). This can be avoided
-  by inserting white spaces.
-\item New constant and theorem names might clash with the user
-theory. This can be avoided by not importing all of \ssr{}:
-\begin{lstlisting}
-  From Coq Require ssreflect.
-  Import ssreflect.SsrSyntax.
-\end{lstlisting}
-Note that the full syntax of \ssr{}'s {\tt rewrite} and reserved identifiers are
-enabled only if the \ssrC{ssreflect} module has been required and if
-\ssrC{SsrSyntax} has been imported. Thus a file that requires (without importing)
- \ssrC{ssreflect} and imports \ssrC{SsrSyntax}, can be
-required and imported without automatically enabling \ssr{}'s
-extended rewrite syntax and reserved identifiers.
-\item Some user notations (in particular, defining an infix ';') might
-interfere with the "open term", parenthesis free, syntax of tactics
-such as \ssrC{have}, \ssrC{set} and \ssrC{pose}.
-\item The generalization of \ssrC{if} statements to non-Boolean
-conditions is turned off by \ssr{}, because it is mostly subsumed by
-\ssrC{Coercion} to \ssrC{bool} of the \ssrC{sum}XXX types (declared in
-\ssrC{ssrfun.v})
-and the \ssrC{if} {\term} \ssrC{is} \ssrN{pattern} \ssrC{then} {\term} \ssrC{else} {\term} construct (see
-\ref{ssec:patcond}). To use the generalized form, turn off the \ssr{}
-Boolean \ssrC{if} notation using the command:
-\begin{lstlisting}
-  Close Scope boolean_if_scope.
-\end{lstlisting}
-\item The following two options can be unset to disable the
-      incompatible \ssrC{rewrite} syntax and allow
-      reserved identifiers to appear in scripts.
-\begin{lstlisting}
-  Unset SsrRewrite.
-  Unset SsrIdents.
-\end{lstlisting}
-\end{itemize}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Gallina extensions}
-
-Small-scale reflection makes an extensive use of the programming
-subset of Gallina, \Coq{}'s logical specification language. This subset
-is quite suited to the description of functions on representations,
-because it closely follows the well-established design of the ML
-programming language. The \ssr{} extension provides three additions
-to Gallina, for pattern assignment, pattern testing, and polymorphism;
-these mitigate minor but annoying discrepancies between Gallina and ML.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Pattern assignment}\label{ssec:patass}
-The \ssr{} extension provides the following construct for
-irrefutable pattern matching, that is, destructuring assignment:
-
-\ssrC{let: } \ssrN{pattern}  \ssrC{:=} \ssrN[1]{term} \ssrC{in} \ssrN[2]{term}
-
-Note the colon `\ssrC{:}' after the \ssrC{let} keyword, which avoids any
-ambiguity with a function
-definition or \Coq{}'s basic destructuring \ssrC{let}. The \ssrC{let:}
-construct differs from the latter in that
-\begin{itemize}
-\item The pattern can be nested (deep pattern matching), in
-  particular, this allows expression of the form:
-\begin{lstlisting}
- let: exist (x, y) p_xy := Hp in ...
-\end{lstlisting}
-\item The destructured constructor is explicitly given in the
-  pattern, and is used for type inference, e.g.,
-\begin{lstlisting}
-  Let f u := let: (m, n) := u in m + n.
-\end{lstlisting}
-using a colon \ssrC{let:}, infers \ssrC{f : nat * nat -> nat}, whereas
-\begin{lstlisting}
-  Let f u := let (m, n) := u in m + n.
-\end{lstlisting}
-with a usual \ssrC{let}, requires an extra type annotation.
-\end{itemize}
-The \ssrC{let:} construct is just (more legible) notation for the primitive Gallina expression
-
-\begin{center}
-\ssrC{match} \ssrN[1]{term} \ssrC{with} \ssrN{pattern} \ssrC{=>} \ssrN[2]{term} \ssrC{end}
-\end{center}
-
-The \ssr{} destructuring assignment supports all the dependent match
-annotations; the  full syntax is
-
-\begin{center}
-\ssrC{let:} \ssrN[1]{pattern} \ssrC{as} \ssrN{ident} \ssrC{in} \ssrN[2]{pattern} \ssrC{:=} \ssrN[1]{term} \ssrC{return} \ssrN[2]{term} \ssrC{in} \ssrN[3]{term}
-\end{center}
-
-where \ssrN[2]{pattern} is a \emph{type} pattern and \ssrN[1]{term} and
-\ssrN[2]{term} are types.
-
-When the \ssrC{as} and \ssrC{return} are both present, then \ssrN{ident} is bound
-in both the type \ssrN[2]{term} and the expression \ssrN[3]{term};
-variables in the optional type pattern \ssrN[2]{pattern} are
-bound only in the type \ssrN[2]{term}, and other variables in \ssrN[1]{pattern} are
-bound only in the expression \ssrN[3]{term}, however.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Pattern conditional}\label{ssec:patcond}
-The following construct can be used for a refutable pattern matching,
-that is, pattern testing:
-
-\begin{center}
-\ssrC{if}\ \ssrN[1]{term} \ssrC{is} \ssrN[1]{pattern} \ssrC{then} \ssrN[2]{term} \ssrC{else} \ssrN[3]{term}
-\end{center}
-
-Although this construct is not strictly ML (it does exits in variants
-such as the pattern calculus or the $\rho$-calculus), it turns out to be
-very convenient for writing functions on representations,
-because most such functions manipulate simple datatypes such as Peano
-integers, options,
-lists, or binary trees, and the pattern conditional above is almost
-always the right construct
-for analyzing such simple types. For example, the \ssrC{null} and
-\ssrC{all} list function(al)s can be defined as follows:
-\begin{lstlisting}
-  Variable d: Set.
-  Fixpoint |*null*| (s : list d) := if s is nil then true else false.
-  Variable a : d -> bool.
-  Fixpoint |*all*| (s : list d) : bool :=
-     if s is cons x s' then a x && all s' else true.
-\end{lstlisting}
-
-The pattern conditional also provides a notation for destructuring
-assignment with a refutable pattern, adapted to the pure functional
-setting of Gallina, which lacks a \\\texttt{Match\_Failure} exception.
-
-Like \ssrC{let:} above, the \ssrC{if}$\dots$\ssrC{is} construct is just (more legible)
-notation for the primitive Gallina expression:
-
-\begin{center}
-\ssrC{match} \ssrN[1]{term} \ssrC{with} \ssrN{pattern} \ssrC{=>} \ssrN[2]{term} \ssrC{| _ =>} \ssrN[2]{term} \ssrC{end}
-\end{center}
-
-Similarly, it will always be displayed as the expansion of this form
-in terms of primitive \ssrC{match} expressions (where the default
-expression $\ssrN[3]{term}$ may be replicated).
-
-
-Explicit pattern testing also largely subsumes the generalization of
-the \ssrC{if} construct to all binary datatypes; compare:
-
-\begin{center}
-\ssrC{if} {\term} \ssrC{is inl _ then} \ssrN[l]{term} \ssrC{else} \ssrN[r]{term}
-\end{center}
-
-and:
-
-\begin{center}
-\ssrC{if} {\term} \ssrC{then} \ssrN[l]{term} \ssrC{else} \ssrN[r]{term}
-\end{center}
-
-The latter appears to be marginally shorter, but it is quite
-ambiguous, and indeed often
-requires an explicit annotation term : \ssrC{\{_\}+\{_\}} to type-check,
-which evens the character count.
-
-Therefore, \ssr{} restricts by default the condition of a plain \ssrC{if}
-construct to the standard \ssrC{bool} type; this avoids spurious type
-annotations, e.g., in:
-\begin{lstlisting}
-  Definition |*orb*| b1 b2 := if b1 then true else b2.
-\end{lstlisting}
-As pointed out in section~\ref{sec:compat}, this restriction can be removed with
-the command:
-\begin{lstlisting}
-  Close Scope boolean_if_scope.
-\end{lstlisting}
-Like \ssrC{let:} above, the \ssrC{if} {\term} \ssrC{is} \ssrN{pattern}
-\ssrC{else} {\term} construct
-supports the dependent \ssrC{match} annotations:
-
-\begin{center}
-\ssrC{if} \ssrN[1]{term} \ssrC{is} \ssrN[1]{pattern} \ssrC{as} \ssrN{ident} \ssrC{in} \ssrN[2]{pattern} \ssrC{return} \ssrN[2]{term} \ssrC{then} \ssrN[3]{term} \ssrC{else} \ssrN[4]{term}
-\end{center}
-
-As in \ssrC{let:} the variable \ssrN{ident} (and those in
-the type pattern \ssrN[2]{pattern}) are bound in \ssrN[2]{term}; \ssrN{ident} is
-also bound in \ssrN[3]{term} (but not in \ssrN[4]{term}), while the
-variables in \ssrN[1]{pattern} are bound only in \ssrN[3]{term}.
-
-\noindent
-Another variant allows to treat the else case first:
-
-\begin{center}
-\ssrC{if} \ssrN[1]{term} \ssrC{isn't} \ssrN[1]{pattern} \ssrC{then} \ssrN[2]{term} \ssrC{else} \ssrN[3]{term}
-\end{center}
-
-Note that \ssrN[1]{pattern} eventually binds variables in \ssrN[3]{term}
-and not \ssrN[2]{term}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Parametric polymorphism}\label{ssec:parampoly}
-
-Unlike ML, polymorphism in core Gallina is explicit: the type
-parameters of polymorphic  functions must be declared explicitly, and
-supplied at each point of use. However, \Coq{} provides two features
-to suppress redundant parameters:
-\begin{itemize}
-\item Sections are used to provide (possibly implicit) parameters for
-  a set of definitions.
-\item Implicit arguments declarations are used to tell \Coq{} to use
-  type inference to deduce some parameters from the context at each
-  point of call.
-\end{itemize}
-The combination of these features provides a fairly good emulation of ML-style
-polymorphism, but unfortunately this emulation breaks down for
-higher-order programming. Implicit arguments are indeed not inferred
-at all points of use, but only at
-points of call, leading to  expressions such as
-\begin{lstlisting}
-  Definition |*all_null*| (s : list T) := all (@null T) s.
-\end{lstlisting}
-Unfortunately, such higher-order expressions are quite frequent in
-representation functions, especially those which use \Coq{}'s
-\ssrC{Structure}s to emulate Haskell type classes.
-
-Therefore, \ssr{} provides a variant of \Coq{}'s implicit argument
-declaration, which  causes \Coq{} to fill in some implicit parameters
-at each point of use, e.g., the above definition can be written:
-\begin{lstlisting}
-  Definition |*all_null*| (s : list d) := all null s.
-\end{lstlisting}
-Better yet, it can be omitted entirely, since \ssrC{all_null s} isn't
-much of an improvement over \ssrC{all null s}.
-
-The syntax of the new declaration is
-
-\begin{center}
-\ssrC{Prenex Implicits} \ssrN{ident}$^+$.
-\end{center}
-
-Let us denote $_1 \dots c_n$ the list of identifiers given to a
-\ssrC{Prenex Implicits} command.
-The command checks that each $c_i$ is the name of a functional
-constant, whose implicit arguments are prenex, i.e., the first $n_i >
-0$ arguments of $c_i$ are implicit; then it assigns
-\ssrC{Maximal Implicit} status to these arguments.
-
-As these prenex implicit arguments are ubiquitous and have often large
-display strings, it is strongly recommended to change the default
-display settings of \Coq{} so that they are not printed (except after a
-\ssrC{Set Printing All} command).
-All \ssr{} library files thus start with the incantation
-\begin{lstlisting}
-  Set Implicit Arguments.
-  Unset Strict Implicit.
-  Unset Printing Implicit Defensive.
-\end{lstlisting}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Anonymous arguments}
-
-When in a definition, the type of a certain argument is mandatory, but
-not its name, one usually use ``arrow'' abstractions for prenex
-arguments, or the \ssrC{(_ : }{\term}\ssrC{)} syntax for inner arguments.
-In \ssr{}, the latter can be replaced by the open syntax `\ssrC{of\ }{\term}'
-or (equivalently)  `\ssrC{& }{\term}', which are both syntactically
-equivalent to a \ssrC{(_ : }{\term}\ssrC{)} expression.
-
-For instance, the usual two-contrsuctor polymorphic type \ssrC{list},
-i.e. the one of the
-standard {\tt List} library, can be defined by the following
-declaration:
-\begin{lstlisting}
-  Inductive list (A : Type) : Type := nil | cons of A & list A.
-\end{lstlisting}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Wildcards}\label{ssec:wild}
-
-The terms passed as arguments
-to \ssr{} tactics can contain \emph{holes}, materialized by wildcards
-\ssrC{_}.
-Since \ssr{} allows a more powerful form of type inference for these
-arguments, it enhances the possibilities of using such wildcards.
-These holes are in particular used as a convenient shorthand for
-abstractions, especially in local definitions or type expressions.
-
-Wildcards may be interpreted as abstractions (see for example sections
-\ref{ssec:pose} and \ref{ssec:struct}), or their content can be
-inferred from the whole
-context of the goal (see for example section \ref{ssec:set}).
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Definitions}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Definitions}\label{ssec:pose}
-\idx{pose \dots{} := \dots{}}
-\idx{pose fix \dots{} := \dots{}}
-\idx{pose cofix \dots{} := \dots{}}
-
-The \ssrC{pose} tactic allows to add a defined constant to a
-proof context. \ssr{} generalizes this tactic in several ways.
-In particular, the \ssr{} \ssrC{pose} tactic supports \emph{open syntax}:
-the body of
-the definition does not need surrounding parentheses. For instance:
-\begin{lstlisting}
-  pose t := x + y.
-\end{lstlisting}
-is a valid tactic expression.
-
-The \ssrC{pose} tactic is also improved for the
-local definition of higher order terms.
-Local definitions of functions can use the same syntax as
-global ones. The tactic:
-\begin{lstlisting}
-  pose f x y := x + y.
-\end{lstlisting}
-adds to the context the defined constant:
-\begin{lstlisting}
-  f := fun x y : nat => x + y : nat -> nat -> nat
-\end{lstlisting}
-
-The \ssr{} \ssrC{pose} tactic also supports (co)fixpoints,
-by providing the local counterpart of the
-\ssrC{Fixpoint f := $\dots$ } and \ssrC{CoFixpoint f := $\dots$ } constructs.
-For instance, the following tactic:
-\begin{lstlisting}
-  pose fix f (x y : nat) {struct x} : nat :=
-      if x is S p then S (f p y) else 0.
-\end{lstlisting}
-defines a local fixpoint \ssrC{f}, which mimics the standard \ssrC{plus}
-operation on natural numbers.
-
-Similarly, local cofixpoints can be defined by a tactic of the form:
-\begin{lstlisting}
-  pose cofix f (arg : T) ...
-\end{lstlisting}
-
-The possibility to include wildcards in the body of the definitions
- offers a smooth
-way of defining local abstractions. The type of ``holes'' is
-guessed by type inference, and the holes are abstracted.
-For instance the tactic:
-\begin{lstlisting}
-  pose f := _ + 1.
-\end{lstlisting}
-is shorthand for:
-\begin{lstlisting}
-  pose f n := n + 1.
-\end{lstlisting}
-
-When the local definition of a function involves both arguments and
-holes, hole abstractions appear first. For instance, the
-tactic:
-\begin{lstlisting}
-  pose f x := x + _.
-\end{lstlisting}
-is shorthand for:
-\begin{lstlisting}
-  pose f n x := x + n.
-\end{lstlisting}
-
-
-The interaction of the \ssrC{pose} tactic with the interpretation of
-implicit arguments results in a powerful and concise syntax for local
-definitions involving dependent types.
-For instance, the tactic:
-\begin{lstlisting}
-  pose f x y := (x, y).
-\end{lstlisting}
-adds to the context the local definition:
-\begin{lstlisting}
-  pose f (Tx Ty : Type) (x : Tx) (y : Ty) := (x, y).
-\end{lstlisting}
-The generalization of wildcards makes the use of the \ssrC{pose} tactic
-resemble ML-like definitions of polymorphic functions.
-
-% The use of \ssrC{Prenex Implicits} declarations (see section
-% \ref{ssec:parampoly}), makes this feature specially convenient.
-% Note that this combines with the interpretation of wildcards, and that
-% it is possible to define:
-% \begin{lstlisting}
-%   pose g x y : prod _ nat := (x, y).
-% \end{lstlisting}
-% which is equivalent to:
-% \begin{lstlisting}
-%   pose g x (y : nat) := (x, y).
-% \end{lstlisting}
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Abbreviations}\label{ssec:set}
-\idx{set \dots{} := \dots{}}
-
-
-The \ssr{} \ssrC{set} tactic performs abbreviations: it introduces a
-defined constant for a subterm appearing in the goal and/or in the
-context.
-
-\ssr{} extends the \ssrC{set} tactic by supplying:
-\begin{itemize}
-\item an open syntax, similarly to the \ssrC{pose} tactic;
-\item a more aggressive matching algorithm;
-\item an improved interpretation of wildcards, taking advantage of the
-  matching algorithm;
-\item an improved occurrence selection mechanism allowing to abstract only
-  selected occurrences of a term.
-\end{itemize}
-
-The general syntax of this tactic is
-\begin{center}
-\ssrC{set} \ssrN{ident} \optional{\ssrC{:} \ssrN[1]{term}} \ssrC{:=} \optional{\ssrN{occ-switch}} \ssrN[2]{term}
-\end{center}
-\begin{center}
-\ssrN{occ-switch} ::= \ssrC{\{}[\ssrC{+}|\ssrC{-}] {\naturalnumber}$^*$ \ssrC{\}}
-\end{center}
-
-
-where:
-
-\begin{itemize}
-\item \ssrN{ident} is a fresh identifier chosen by the user.
-\item \ssrN[1]{term} is
-an optional type annotation. The type annotation \ssrN[1]{term} can be
-given in open syntax (no surrounding parentheses). If no \ssrN{occ-switch}
-(described hereafter) is present, it is also
-the case for \ssrN[2]{term}.
-On the other hand, in  presence of \ssrN{occ-switch}, parentheses
-surrounding \ssrN[2]{term} are mandatory.
-\item In the occurrence switch \ssrN{occ-switch}, if the first element
-  of the list is a {\naturalnumber}, this element should be a number, and not
-  an Ltac variable. The empty list \ssrC{\{\}} is not interpreted as a
-  valid occurrence switch.
-\end{itemize}
-% For example, the script:
-% \begin{lstlisting}
-%   Goal forall (f : nat -> nat)(x y : nat), f x + f x = f x.
-%   move=> f x y.
-% \end{lstlisting}
-
-The tactic:
-\begin{lstlisting}
-  set t := f _.
-\end{lstlisting}
-transforms the goal  \ssrC{f x + f x = f x} into \ssrC{t + t = t}, adding
-\ssrC{t := f x} to the context, and the tactic:
-\begin{lstlisting}
-  set t := {2}(f _).
-\end{lstlisting}
-transforms it into \ssrC{f x + t = f x}, adding \ssrC{t := f x} to the context.
-
-The type annotation \ssrN[1]{term} may
-contain wildcards, which will be filled with the appropriate value by
-the matching process.
-
-The tactic first tries to find a subterm of the goal matching
-\ssrN[2]{term} (and its type \ssrN[1]{term}),
-and stops at the first subterm it finds. Then the occurrences
-of this subterm selected by the optional \ssrN{occ-switch} are replaced
-by \ssrN{ident} and a definition \ssrN{ident} \ssrC{:=} {\term} is added to
-the context. If no \ssrN{occ-switch} is present, then all the
-occurrences are abstracted.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Matching}
-
-The matching algorithm compares a pattern \textit{term}
- with a subterm of the goal by comparing their heads
-and then pairwise unifying their arguments (modulo conversion). Head
-symbols match under the following conditions:
-
-\begin{itemize}
-\item If the head of \textit{term} is a constant, then it
-  should be syntactically equal to the head symbol of the subterm.
-\item If this head is a projection of a canonical structure,
-  then canonical structure equations are used for the matching.
-\item If the head of \textit{term} is \emph{not} a constant, the
-  subterm should have the same structure ($\lambda$ abstraction,
-  \ssrC{let}$\dots$\ssrC{in} structure \dots).
-\item If the head of \textit{term} is a hole, the subterm should have
-  at least as many arguments as  \textit{term}. For instance the tactic:
-\begin{lstlisting}
-  set t := _ x.
-\end{lstlisting}
-transforms the goal \ssrL-x + y = z- into \ssrC{t y = z} and adds
-\ssrC{t := plus x : nat -> nat} to the context.
-
-\item In the special case where \textit{term} is of the form
-  \ssrC{(let f := }$t_0$ \ssrC{in f) }$t_1\dots t_n$,
- then the pattern \textit{term} is treated
-as \ssrC{(_ }$t_1\dots t_n$\ssrC{)}. For each subterm in
-the goal having the form $(A\  u_1\dots u_{n'})$ with $n' \geq n$, the
-matching algorithm successively tries to find the largest
-partial application $(A\ u_1\dots u_{i'})$ convertible to the head
-$t_0$ of \textit{term}. For instance the following tactic:
-\begin{lstlisting}
-  set t := (let g y z := y.+1 + z in g) 2.
-\end{lstlisting}
-transforms the goal
-\begin{lstlisting}
-  (let f x y z := x + y + z in f 1) 2 3 = 6.
-\end{lstlisting}
-into \ssrC{t 3 = 6} and adds the local definition of \ssrC{t} to the
-context.
-\end{itemize}
-
-Moreover:
-\begin{itemize}
-\item Multiple holes in \textit{term} are treated as independent
-  placeholders. For instance, the tactic:
-\begin{lstlisting}
-  set t := _ + _.
-\end{lstlisting}
-transforms the goal \ssrC{x + y = z} into \ssrC{t = z} and pushes
-\ssrC{t := x + y : nat} in the context.
-\item The type of the subterm matched should fit the type
-  (possibly casted by some type annotations) of the pattern
-  \textit{term}.
-\item The replacement of the subterm found by the instantiated pattern
-  should not capture variables, hence the following script:
-\begin{lstlisting}
-  Goal forall x : nat, x + 1 = 0.
-  set u := _ + 1.
-\end{lstlisting}
-raises an error message, since \ssrC{x} is bound in the goal.
-\item Typeclass inference should fill in any residual hole, but
-matching should never assign a value to a global existential variable.
-
-\end{itemize}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Occurrence selection}\label{sssec:occselect}
-
-\ssr{} provides a generic syntax for the selection of occurrences by
-their position indexes. These \emph{occurrence switches} are shared by
-all
-\ssr{} tactics which require control on subterm selection like rewriting,
-generalization, \dots
-
-An \emph{occurrence switch} can be:
-\begin{itemize}
-\item A list \ssrC{\{} {\naturalnumber}$^*$ \ssrC{\}} of occurrences affected by the
-  tactic.
-For instance, the tactic:
-\begin{lstlisting}
-  set x := {1 3}(f 2).
-\end{lstlisting}
-transforms the goal \ssrC{f 2 + f 8 = f 2 + f 2} into
-\ssrC{x + f 8 = f 2  + x}, and adds the abbreviation
-\ssrC{x := f 2} in the
-context. Notice that some occurrences of a
-given term may be hidden to the user, for example because of a
-notation. The vernacular \ssrC{$\texttt{\textcolor{dkviolet}{Set }}$
-  Printing All} command displays all
-these hidden occurrences and should be used to find the correct
-coding of the occurrences to be selected\footnote{Unfortunately,
-even after a call to the Set Printing All command, some occurrences are
-still not displayed to the user, essentially the ones possibly hidden
-in the predicate of a dependent match structure.}. For instance, the
-following script:
-\begin{lstlisting}
-  Notation "a < b":= (le (S a) b).
-  Goal forall x y, x < y -> S x < S y.
-  intros x y; set t := S x.
-\end{lstlisting}
-generates the goal
-\ssrC{t <= y -> t < S y} since \ssrC{x < y} is now a notation for
-\ssrC{S x <= y}.
-\item A list \ssrC{\{}{\naturalnumber}$^+$\ssrC{\}}. This is equivalent to
-  \ssrC{\{} {\naturalnumber}$^+$ \ssrC{\}} but the list should start with a number, and
-  not with an Ltac variable.
-\item A list \ssrC{\{}{\naturalnumber}$^*$\ssrC{\}} of occurrences \emph{not} to be
-  affected by the tactic. For instance, the tactic:
-\begin{lstlisting}
-  set x := {-2}(f 2).
-\end{lstlisting}
-behaves like
-\begin{lstlisting}
-  set x := {1 3}(f 2).
-\end{lstlisting}
-on the goal \ssrL-f 2 + f 8 = f 2 + f 2- which has three occurrences of
-the the term \ssrC{f 2}
-\item In particular, the switch \ssrC{\{+\}} selects \emph{all} the
-  occurrences. This switch is useful to turn
-  off the default behavior of a tactic which automatically clears
-  some assumptions (see section \ref{ssec:discharge} for instance).
-\item The switch \ssrC{\{-\}} imposes that \emph{no} occurrences of the
-  term should be affected by the tactic. The tactic:
-\begin{lstlisting}
-  set x := {-}(f 2).
-\end{lstlisting}
-leaves the goal unchanged and adds the definition \ssrC{x := f 2} to the
-context. This kind of tactic may be used to take advantage of the
-power of the matching algorithm in a local definition, instead of
-copying large terms by hand.
-\end{itemize}
-
-
-It is important to remember that matching \emph{precedes} occurrence
-selection, hence the tactic:
-\begin{lstlisting}
-  set a := {2}(_ + _).
-\end{lstlisting}
-transforms the goal \ssrC{x + y = x + y + z} into \ssrC{x + y = a + z}
-and fails on the goal \\
-\ssrC{(x + y) + (z + z) = z + z} with the error message:
-\begin{lstlisting}
-  User error: only 1 < 2 occurrence of (x + y + (z + z))
-\end{lstlisting}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Localization}\label{ssec:loc}
-
-
-It is possible to define an abbreviation for a term appearing in the
-context of a goal thanks to the \ssrC{in} tactical.
-
-A tactic of the form:
-
-\begin{center}
-  \ssrC{set x :=} {\term} \ssrC{in} \ssrN[1]{fact}\ssrC{...}\ssrN[n]{fact}.
-\end{center}
-
-introduces a defined constant called \ssrC{x} in the context, and folds
-it in the facts \textit{fact$_1 \dots$ fact$_n$}
-The body of \ssrC{x} is the first subterm matching \textit{term} in
-\textit{fact$_1 \dots$ fact$_n$}.
-
-A tactic of the form:
-
-\begin{center}
-  \ssrC{set x :=} {\term} \ssrC{in} \ssrN[1]{fact}\ssrC{...}\ssrN[n]{fact} \ssrC{*.}
-\end{center}
-
-matches {\term} and then folds \ssrC{x} similarly in
-\textit{fact$_1 \dots$ fact$_n$}, but also folds \ssrC{x} in the goal.
-
-A goal \ssrL-x + t = 4-, whose context contains \ssrC{Hx : x = 3}, is left
-unchanged by the tactic:
-\begin{lstlisting}
-  set z := 3 in Hx.
-\end{lstlisting}
-but the context is extended with the definition \ssrC{z := 3} and \ssrC{Hx} becomes
-\ssrC{Hx : x = z}.
-On the same goal and context, the tactic:
-\begin{lstlisting}
-  set z := 3 in Hx *.
-\end{lstlisting}
-will moreover change the goal into \ssrL-x + t = S z-. Indeed, remember
-that \ssrC{4} is just a notation for \ssrC{(S 3)}.
-
-The use of the \ssrC{in} tactical is not limited to the localization of
-abbreviations: for a complete description of the \ssrC{in} tactical, see
-section \ref{ssec:profstack}.
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Basic tactics}\label{sec:book}
-
-
-
-A sizable fraction of proof scripts consists of steps that do not
-"prove" anything new, but instead perform menial bookkeeping tasks
-such as selecting the names of constants and assumptions or splitting
-conjuncts. Although they are logically trivial, bookkeeping steps are
-extremely important because they define the structure of the data-flow
-of a proof script. This is especially true for reflection-based
-proofs, which often involve large numbers of constants and
-assumptions.  Good bookkeeping consists in always explicitly declaring
-(i.e., naming) all new constants and assumptions in the script, and
-systematically pruning irrelevant constants and assumptions in the
-context. This is essential in the context of an interactive
-development environment (IDE), because it facilitates navigating the
-proof, allowing to instantly "jump back" to the point at which a
-questionable assumption was added, and to find relevant assumptions by
-browsing the pruned context.  While novice or casual \Coq{} users may
-find the automatic name selection feature convenient, the usage of
-such a feature severely undermines the readability and maintainability
-of proof scripts, much like automatic variable declaration in programming
-languages. The \ssr{} tactics are therefore designed to support
-precise bookkeeping and to eliminate name generation heuristics.
-The bookkeeping features of \ssr{} are implemented as tacticals (or
-pseudo-tacticals), shared across most \ssr{} tactics, and thus form
-the foundation of the \ssr{} proof language.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Bookkeeping}\label{ssec:profstack}
-\idx{move: \dots{}}
-\idx{move=> \dots{}}
-\idx{move: \dots{} => \dots{}}
-\idx{\dots{} in \dots{}}
-
-During the course of a proof \Coq{} always present the user with
-a \emph{sequent} whose general form is
-\begin{displaymath}\begin{array}{l}
-%\arrayrulecolor{dkviolet}
-c_i\ \ssrC{:}\ T_i \\
-\dots\\
-d_j\ \ssrC{:=}\ e_j\ \ssrC{:}\ T_j \\
-\dots\\
-F_k\ \ssrC{:}\ P_k \\
-\dots \\[3pt]
-\hline\hline\\[-8pt]
-\ssrC{forall}\ \ssrC{(}x_\ell\ \ssrC{:}\ T_\ell\ssrC{)}\ \dots,\\
-\ssrC{let}\ y_m\ \ssrC{:=}\ b_m\ \ssrC{in}\ \dots\ \ssrC{in}\\
-P_n\ \ssrC{->}\ \dots\ \ssrC{->}\ C
-\end{array}\end{displaymath}
-The \emph{goal} to be proved appears below the double line; above the line is
-the \emph{context} of the sequent, a set of declarations of
-\emph{constants}~$c_i$, \emph{defined constants}~$d_i$, and
-\emph{facts}~$F_k$ that can be used to prove the goal (usually, $T_i,
-T_j\;:\;\ssrC{Type}$ and $P_k\;:\;\ssrC{Prop}$). The various kinds of
-declarations can come in any order. The top part of the context
-consists of declarations produced by the \ssrC{Section} commands
-\ssrC{Variable}, \ssrC{Let}, and \ssrC{Hypothesis}. This \emph{section context}
-is never affected by the \ssr{} tactics: they only operate on
-the lower part --- the \emph{proof context}.
-As in the figure above, the goal often decomposes into a series of
-(universally) quantified \emph{variables}
-$\ssrC{(}x_\ell\;\ssrC{:}\;T_\ell\ssrC{)}$, local \emph{definitions}
-$\ssrC{let}\;y_m\:\ssrC{:=}\;b_m\;\ssrC{in}$, and \emph{assumptions}
-$P_n\;\ssrC{->}$, and a \emph{conclusion}~$C$ (as in the context, variables,
-definitions, and assumptions can appear in any order). The conclusion
-is what actually needs to be proved --- the rest of the goal can be
-seen as a part of the proof context that happens to be ``below the line''.
-
-However, although they are logically equivalent, there are fundamental
-differences between constants and facts on the one hand, and variables
-and assumptions on the others. Constants and facts are
-\emph{unordered}, but \emph{named} explicitly in the proof text;
-variables and assumptions are \emph{ordered}, but \emph{unnamed}: the
-display names of variables may change at any time because of
-$\alpha$-conversion.
-
-Similarly, basic deductive steps such as \ssrC{apply} can only operate on
-the goal because the Gallina terms that control their action (e.g.,
-the type of the lemma used by \ssrC{apply}) only provide unnamed bound
-variables.\footnote{Thus scripts that depend on bound variable names, e.g.,
-via \ssrC{intros} or \ssrC{with}, are inherently fragile.}  Since the proof
-script can only refer directly to the context, it must constantly
-shift declarations from the goal to the context and conversely in
-between deductive steps.
-
-In \ssr{} these moves are performed by two \emph{tacticals} `\ssrC{=>}'
-and `\ssrC{:}', so that the bookkeeping required by a deductive step can
-be directly associated to that step, and that tactics in an \ssr{}
-script correspond to actual logical steps in the proof rather than
-merely shuffle facts. Still, some isolated bookkeeping is unavoidable,
-such as naming variables and assumptions at the beginning of a proof.
-\ssr{} provides a specific \ssrC{move} tactic for this purpose.
-
-Now \ssrC{move} does essentially nothing: it is mostly a placeholder for
-`\ssrC{=>}' and `\ssrC{:}'. The `\ssrC{=>}' tactical moves variables, local
-definitions, and assumptions to the context, while the `\ssrC{:}'
-tactical moves facts and constants to the goal. For example, the proof
-of\footnote{The name \ssrC{subnK} reads as
-``right cancellation rule for \ssrC{nat} subtraction''.}
-\begin{lstlisting}
-  Lemma |*subnK*| : forall m n, n <= m -> m - n + n = m.
-\end{lstlisting}\noindent
-might start with
-\begin{lstlisting}
-  move=> m n le_n_m.
-\end{lstlisting}
-where \ssrC{move} does nothing, but \ssrL|=> m n le_m_n| changes the
-variables and assumption of the goal in the constants \ssrC{m n : nat}
-and the fact \ssrL|le_n_m : n <= m|, thus exposing the conclusion\\
- \ssrC{m - n + n = m}. 
-
-The `\ssrC{:}' tactical is the converse of `\ssrC{=>}': it removes facts
-and constants from the context by turning them into variables and assumptions.
-Thus
-\begin{lstlisting}
-  move: m le_n_m.
-\end{lstlisting}
-turns back \ssrC{m} and \ssrL|le_m_n| into a variable and an assumption, removing
-them from the proof context, and changing the goal to
-\begin{lstlisting}
-  forall m, n <= m -> m - n + n = m.
-\end{lstlisting}
-which can be proved by induction on \ssrC{n} using \ssrC{elim: n}.
-
-\noindent
-Because they are tacticals, `\ssrC{:}' and `\ssrC{=>}' can be combined, as in
-\begin{lstlisting}
-  move: m le_n_m => p le_n_p.
-\end{lstlisting}
-simultaneously renames \ssrL|m| and \ssrL|le_m_n| into \ssrL|p| and \ssrL|le_n_p|,
-respectively, by first turning them into unnamed variables, then
-turning these variables back into constants and facts.
-
-Furthermore, \ssr{} redefines the basic \Coq{} tactics \ssrC{case},
-\ssrC{elim}, and \ssrC{apply} so that they can take better advantage of
-'\ssrC{:}' and `\ssrC{=>}'. In there \ssr{} variants, these tactic operate
-on the first  variable or constant of the goal and they do not use or
-change the proof context. The `\ssrC{:}' tactical is used to operate on
-an element in the context. For instance the proof of \ssrC{subnK} could
-continue with
-\begin{lstlisting}
-  elim: n.
-\end{lstlisting}
-instead of \ssrC{elim n}; this has the advantage of
-removing \ssrC{n} from the context. Better yet, this \ssrC{elim} can be combined
-with previous \ssrC{move} and with the branching version of the \ssrC{=>} tactical
-(described in~\ref{ssec:intro}),
-to encapsulate the inductive step in a single command:
-\begin{lstlisting}
-  elim: n m le_n_m => [|n IHn] m => [_ | lt_n_m].
-\end{lstlisting}
-which breaks down the proof into two subgoals,
-\begin{lstlisting}
-  m - 0 + 0 = m
-\end{lstlisting}
-given \ssrC{m : nat}, and
-\begin{lstlisting}
-  m - S n + S n = m
-\end{lstlisting}
-given \ssrC{m n : nat}, \ssrL|lt_n_m : S n <= m|, and
-\begin{lstlisting}
-  IHn : forall m, n <= m -> m - n + n = m.
-\end{lstlisting}
-The '\ssrC{:}' and `\ssrC{=>}' tacticals can be explained very simply
-if one views the goal as a stack of variables and assumptions piled
-on a conclusion:
-\begin{itemize}
-\item {\tac} \ssrC{:} $a$ $b$ $c$ pushes the context constants $a$, $b$, $c$
-as goal variables \emph{before} performing {\tac}.
-\item {\tac} \ssrC{=>} $a$ $b$ $c$ pops the top three goal variables as
-context constants $a$, $b$, $c$, \emph{after} {\tac}
-has been performed.
-\end{itemize}
-These pushes and pops do not need to balance out as in the examples above,
-so
-\begin{lstlisting}
-  move: m le_n_m => p.
-\end{lstlisting}
-would rename \ssrC{m} into \ssrC{p}, but leave an extra assumption \ssrC{n <= p}
-in the goal.
-
-Basic tactics like \ssrC{apply} and \ssrC{elim} can also be used without the
-'\ssrC{:}' tactical: for example we can directly start a proof of \ssrC{subnK}
-by induction on the top variable \ssrC{m} with
-\begin{lstlisting}
-  elim=> [|m IHm] n le_n.
-\end{lstlisting}
-
-\noindent
-The general form of the localization tactical \ssrC{in} is also best
-explained in terms of the goal stack:
-
-\begin{center}
-  {\tac} \ssrC{in a H1 H2 *.}
-\end{center}
-
-is basically equivalent to
-
-\begin{center}
-  \ssrC{move: a H1 H2;} {\tac} \ssrC{=> a H1 H2.}
-\end{center}
-
-with two differences: the \ssrC{in} tactical will preserve the body of \ssrC{a} if
-\ssrC{a} is a defined constant, and if the `\ssrC{*}' is omitted it
-will use a temporary abbreviation to hide the statement of the goal
-from \ssrC{/*tactic*/}.
-
-The general form of the \ssrC{in} tactical can be used directly with
-the \ssrC{move}, \ssrC{case} and \ssrC{elim} tactics, so that one can write
-\begin{lstlisting}
-  elim: n => [|n IHn] in m le_n_m *.
-\end{lstlisting}
-instead of
-\begin{lstlisting}
-  elim: n m le_n_m => [|n IHn] m le_n_m.
-\end{lstlisting}
-This is quite useful for inductive proofs that involve many facts.
-
-\noindent See section \ref{ssec:gloc} for the general syntax and presentation
-of the \ssrC{in} tactical.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{The defective tactics}\label{ssec:basictac}
-
-In this section we briefly present the three basic tactics performing
-context manipulations and the main backward chaining tool.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{The \ssrC{move} tactic.}\label{sssec:move}
-\idx{move}
-
-The \ssrC{move} tactic, in its
-defective form, behaves like the primitive \ssrC{hnf} \Coq{} tactic. For
-example, such a defective:
-\begin{lstlisting}
-  move.
-\end{lstlisting}
-exposes the first assumption in the goal, i.e. its changes the goal
-\ssrC{\~ False} into \ssrC{False -> False}.
-
-More precisely, the \ssrC{move} tactic inspects the goal and does nothing
-(\ssrC{idtac}) if an introduction step is possible, i.e. if the
-goal is a product or a \ssrC{let}$\dots$\ssrC{in}, and performs \ssrC{hnf}
-otherwise.
-
-Of course this tactic is most often used in combination with the
-bookkeeping tacticals (see section \ref{ssec:intro} and
-\ref{ssec:discharge}). These combinations mostly subsume the \ssrC{intros},
-\ssrC{generalize}, \ssrC{revert}, \ssrC{rename}, \ssrC{clear} and
-\textcolor{dkblue}{\texttt{pattern}} tactics.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{The \ssrC{case} tactic.}
-\idx{case: \dots{}}
-
-The \ssrC{case} tactic performs
-\emph{primitive case analysis} on (co)inductive types; specifically,
-it destructs the top variable or assumption of the goal,
-exposing its constructor(s) and its arguments, as well as setting the value
-of its type family indices if it belongs to a type family
-(see section \ref{ssec:typefam}).
-
-The \ssr{} \ssrC{case} tactic has a special behavior on
-equalities.
-If the top assumption of the goal is an equality, the \ssrC{case} tactic
-``destructs'' it as a set of equalities between the constructor
-arguments of its left and right hand sides, as per the
-tactic \ssrC{injection}.
-For example, \ssrC{case} changes the goal
-\begin{lstlisting}
-  (x, y) = (1, 2) -> G.
-\end{lstlisting}
-into
-\begin{lstlisting}
-  x = 1 -> y = 2 -> G.
-\end{lstlisting}
-
-Note also that the case of \ssr{} performs \ssrC{False}
-elimination, even if no branch is generated by this case operation.
-Hence the command:
-\begin{lstlisting}
-  case.
-\end{lstlisting}
-on a goal of the form \ssrC{False -> G} will succeed and prove the goal.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{The \ssrC{elim} tactic.} 
-\idx{elim: \dots{}}
-
-The \ssrC{elim} tactic performs
-inductive elimination on inductive types.
-The defective:
-\begin{lstlisting}
-  elim.
-\end{lstlisting}
-tactic performs inductive elimination on a goal whose top assumption
-has an inductive type. For example on goal of the form:
-\begin{lstlisting}
-  forall n : nat, m <= n
-\end{lstlisting}
- in a context containing \ssrC{m : nat}, the
-\begin{lstlisting}
-  elim.
-\end{lstlisting}
-tactic produces two goals,
-\begin{lstlisting}
-  m <=  0
-\end{lstlisting}
-on one hand and
-\begin{lstlisting}
-  forall n : nat, m <= n -> m <= S n
-\end{lstlisting}
-on the other hand.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{The \ssrC{apply} tactic.}\label{sssec:apply}
-\idx{apply: \dots{}}
-
-The \ssrC{apply} tactic is the main
-backward chaining tactic of the proof system. It takes as argument any
-\ssrC{/*term*/} and applies it to the goal.
-Assumptions in the type of \ssrC{/*term*/} that don't directly match the
-goal may generate one or more subgoals.
-
-In fact the \ssr{} tactic:
-\begin{lstlisting}
-  apply.
-\end{lstlisting}
-is a synonym for:
-\begin{lstlisting}
-  intro top; first [refine top | refine (top _) |  refine (top _ _) | ...]; clear top.
-\end{lstlisting}
-where \ssrC{top} is fresh name, and the sequence of \ssrC{refine} tactics
-tries to catch the appropriate number of wildcards to be inserted.
-Note that this use of the \ssrC{refine} tactic implies that the tactic
-tries to match the goal up to expansion of
-constants and evaluation of subterms.
-
-\ssr{}'s \ssrC{apply} has a special behaviour on goals containing
-existential metavariables of sort \ssrC{Prop}.  Consider the
-following example:
-\begin{lstlisting}
-Goal (forall y, 1 < y -> y < 2 -> exists x : { n | n < 3 }, proj1_sig x > 0).
-move=> y y_gt1 y_lt2; apply: (ex_intro _ (exist _ y _)).
-  by apply: gt_trans _ y_lt2.
-by move=> y_lt3; apply: lt_trans y_gt1.
-\end{lstlisting}
-Note that the last \ssrC{_} of the tactic \ssrC{apply: (ex_intro _ (exist _ y _))}
-represents a proof that \ssrC{y < 3}. Instead of generating the following
-goal
-\begin{lstlisting}
-   0 <  (n:=3) (m:=y) ?54
-\end{lstlisting}
-\noindent the system tries to prove \ssrC{y < 3} calling the \ssrC{trivial}
-tactic. If it succeeds, let's say because the context contains
-\ssrC{H : y < 3}, then the system generates the following goal:
-\begin{lstlisting}
-   0 < proj1_sig (exist (fun n => n < 3) y H
-\end{lstlisting}
-\noindent Otherwise the missing proof is considered to be irrelevant, and
-is thus discharged generating the following goals:
-\begin{lstlisting}
-   y < 3
-   forall H : y < 3, proj1_sig (exist (fun n => n < 3) y H)
-\end{lstlisting}
-Last, the user can replace the \ssrC{trivial} tactic by defining
-an Ltac expression named \ssrC{ssrautoprop}.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Discharge}\label{ssec:discharge}
-\idx{\dots{} : \dots{}}
-
-The general syntax of the discharging tactical `\ssrC{:}' is:
-\begin{center}
-	{\tac} \optional{\ssrN{ident}} \ssrC{:} \ssrN[1]{d-item} $\dots$ \ssrN[n]{d-item} \optional{\ssrN{clear-switch}}
-\end{center}
-where $n > 0$, and \ssrN{d-item} and \ssrN{clear-switch} are defined as
-\begin{longtable}{rcl}
-\ssrN{d-item} & ::= & \optional{\ssrN{occ-switch} {\optsep} \ssrN{clear-switch}} {\term} \\
-\ssrN{clear-switch}& ::=& \ssrC{\{} \ssrN[1]{ident}\, \ldots\,  \ssrN[m]{ident} \ssrC{\}}
-\end{longtable}
-with the following requirements:
-\begin{itemize}
-\item {\tac} must be one of the four basic tactics described
-      in~\ref{ssec:basictac}, i.e., \ssrC{move}, \ssrC{case}, \ssrC{elim} or \ssrC{apply},
-      the \ssrC{exact} tactic (section \ref{ssec:termin}),
-      the \ssrC{congr} tactic (section \ref{ssec:congr}), or the application
-      of the \emph{view} tactical `\ssrC{/}' (section \ref{ssec:assumpinterp})
-      to one of \ssrC{move}, \ssrC{case}, or \ssrC{elim}.
-\item The optional \ssrN{ident} specifies \emph{equation generation}
-      (section \ref{ssec:equations}), and is only allowed if {\tac}
-      is \ssrC{move}, \ssrC{case} or \ssrC{elim}, or the application of the
-      view tactical `\ssrC{/}' (section \ref{ssec:assumpinterp})
-      to \ssrC{case} or \ssrC{elim}.
-\item An \ssrN{occ-switch} selects occurrences of {\term},
-      as in \ref{sssec:occselect}; \ssrN{occ-switch} is not allowed if
-      {\tac} is \ssrC{apply} or \ssrC{exact}.
-\item A clear item \ssrN{clear-switch} specifies facts and constants to be
-  deleted from the proof context (as per the \ssrC{clear} tactic).
-\end{itemize}
-The `\ssrC{:}' tactical first \emph{discharges} all the \ssrN{d-item}s,
-right to left, and then performs {\tac}, i.e., for each \ssrN{d-item},
-starting with $\ssrN[n]{d-item}$:
-\begin{enumerate}
-\item The \ssr{} matching algorithm described in section~\ref{ssec:set}
-      is used to find occurrences of {\term} in the goal,
-      after filling any holes `\ssrC{_}' in {\term}; however if {\tac}
-      is \ssrC{apply} or \ssrC{exact} a different matching algorithm,
-      described below, is used
-      \footnote{Also, a slightly different variant may be used for the first
-      \ssrN{d-item} of \ssrC{case} and \ssrC{elim}; see section~\ref{ssec:typefam}.}.
-\item~\label{enum:gen} These occurrences are replaced by a new
-  variable; in particular,
-      if {\term} is a fact, this adds an assumption to the goal.
-\item~\label{enum:clr} If {\term} is \emph{exactly} the name of a constant
-      or fact in the proof context, it is deleted from the context,
-      unless there is an \ssrN{occ-switch}.
-\end{enumerate}
-Finally, {\tac} is performed just after $\ssrN[1]{d-item}$ has been
-generalized ---
-that is, between steps \ref{enum:gen} and \ref{enum:clr} for $\ssrN[1]{d-item}$.
-The names listed in the final \ssrN{clear-switch} (if it is present)
-are cleared first, before $\ssrN[n]{d-item}$ is discharged.
-
-\noindent
-Switches affect the discharging of a \ssrN{d-item} as follows:
-\begin{itemize}
-\item An \ssrN{occ-switch} restricts generalization (step~\ref{enum:gen})
-      to a specific subset of the occurrences of {\term}, as per
-      \ref{sssec:occselect}, and prevents clearing (step~\ref{enum:clr}).
-\item All the names specified by a \ssrN{clear-switch} are deleted from the
-      context in step~\ref{enum:clr}, possibly in addition to {\term}.
-\end{itemize}
-For example, the tactic:
-\begin{lstlisting}
-  move: n {2}n (refl_equal n).
-\end{lstlisting}
-\begin{itemize}
-\item first generalizes \ssrC{(refl_equal n : n = n)};
-\item then generalizes the second occurrence of \ssrC{n}.
-\item finally generalizes all the other occurrences of \ssrC{n},
-      and clears \ssrC{n} from the proof context
-      (assuming \ssrC{n} is a proof constant).
-\end{itemize}
-Therefore this tactic changes any goal \ssrC{G} into
-\begin{lstlisting}
- forall n n0 : nat, n = n0 -> G.
-\end{lstlisting}
-where the name \ssrC{n0} is picked by the \Coq{} display function,
-and assuming \ssrC{n} appeared only in~\ssrC{G}.
-
-Finally, note that a discharge operation generalizes defined constants
-as variables, and not as local definitions. To override this behavior,
-prefix the name of the local definition with a \ssrC{@},
-like in \ssrC{move: @n}.
-
-This is in contrast with the behavior of the \ssrC{in} tactical (see section
-\ref{ssec:gloc}), which preserves local definitions by default.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Clear rules}
-
-The clear step will fail if {\term} is a proof constant that
-appears in other facts; in that case either the facts should be
-cleared explicitly with a \ssrN{clear-switch}, or the clear step should be
-disabled. The latter can be done by adding an \ssrN{occ-switch} or simply by
-putting parentheses around {\term}: both
-\begin{lstlisting}
-  move: (n).
-\end{lstlisting}
-and
-\begin{lstlisting}
-  move: {+}n.
-\end{lstlisting}
-generalize \ssrC{n} without clearing \ssrC{n} from the proof context.
-
-The clear step will also fail if the \ssrN{clear-switch} contains a
-\ssrN{ident} that is not in the \emph{proof} context.
-Note that \ssr{} never clears a section constant.
-
-If {\tac} is \ssrC{move} or \ssrC{case} and an equation \ssrN{ident} is given,
-then clear (step~\ref{enum:clr}) for $\ssrN[1]{d-item}$ is suppressed
-(see section \ref{ssec:equations}).
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Matching for \ssrC{apply} and \ssrC{exact}}\label{sss:strongapply}
-
-The matching algorithm for \ssrN{d-item}s of the \ssr{} \ssrC{apply} and
-\ssrC{exact} tactics
-exploits the type of $\ssrN[1]{d-item}$ to interpret
-wildcards in the other \ssrN{d-item} and to determine which occurrences of
-these should be generalized.
-Therefore, \ssrN{occur switch}es are not needed for \ssrC{apply} and \ssrC{exact}.
-
-Indeed, the \ssr{} tactic \ssrC{apply: H x} is equivalent to
-\begin{lstlisting}
-  refine (@H _ ... _ x); clear H x
-\end{lstlisting}
-with an appropriate number of wildcards between \ssrC{H} and~\ssrC{x}.
-
-Note that this means that matching for \ssrC{apply} and \ssrC{exact} has
-much more context to interpret wildcards; in particular it can accommodate
-the `\ssrC{_}' \ssrN{d-item}, which would always be rejected after `\ssrC{move:}'.
-For example, the tactic
-\begin{lstlisting}
-  apply: trans_equal (Hfg _) _.
-\end{lstlisting}
-transforms the goal \ssrC{f a = g b}, whose context contains
-\ssrC{(Hfg : forall x, f x = g x)}, into \ssrC{g a = g b}.
-This tactic is equivalent (see section \ref{ssec:profstack}) to:
-\begin{lstlisting}
-  refine (trans_equal (Hfg _) _).
-\end{lstlisting}
-and this is a common idiom for applying transitivity on the left hand side
-of an equation.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{The \ssrC{abstract} tactic}\label{ssec:abstract}
-\idx{abstract: \dots{}}
-
-The \ssrC{abstract} tactic assigns an abstract constant previously introduced with
-the \ssrC{[: name ]} intro pattern (see section~\ref{ssec:intro},
-page~\pageref{ssec:introabstract}).
-In a goal like the following:
-\begin{lstlisting}
- m : nat
- abs : <hidden>
- n : nat
- =============
- m < 5 + n
-\end{lstlisting}
-The tactic \ssrC{abstract: abs n} first generalizes the goal with respect to
-\ssrC{n} (that is not visible to the abstract constant \ssrC{abs}) and then
-assigns \ssrC{abs}.  The resulting goal is:
-\begin{lstlisting}
- m : nat
- n : nat
- =============
- m < 5 + n
-\end{lstlisting}
-Once this subgoal is closed, all other goals having \ssrC{abs} in their context
-see the type assigned to \ssrC{abs}.  In this case:
-\begin{lstlisting}
- m : nat
- abs : forall n, m < 5 + n
-\end{lstlisting}
-
-For a more detailed example the user should refer to section~\ref{sssec:have},
-page~\pageref{sec:havetransparent}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Introduction}\label{ssec:intro}
-\idx{\dots{} => \dots{}}
-
-The application of a tactic to a given goal can generate
-(quantified) variables, assumptions, or definitions, which the user may want to
-\emph{introduce} as new facts, constants or defined constants, respectively.
-If the tactic splits the goal into several subgoals,
-each of them may require the introduction of different constants and facts.
-Furthermore it is very common to immediately decompose
-or rewrite with an assumption instead of adding it to the context,
-as the goal can often be simplified and even
-proved after this.
-
-All these operations are performed by the introduction tactical
-`\ssrC{=>}', whose general syntax is
-\begin{center}
-	{\tac} \ssrC{=>} \ssrN[1]{i-item} $\dots$ \ssrN[n]{i-item}
-\end{center}
-where {\tac} can be any tactic, $n > 0$ and
-\begin{longtable}{rcl}
-  \ssrN{i-item}& ::=& \ssrN{i-pattern} {\optsep} \ssrN{s-item} {\optsep} \ssrN{clear-switch} {\optsep} \ssrC{/}{\term} \\
-  \ssrN{s-item}& ::=& \ssrC{/=} {\optsep} \ssrC{//} {\optsep} \ssrC{//=} \\
-  \ssrN{i-pattern}& ::=& \ssrN{ident} {\optsep} \ssrC{_} {\optsep} \ssrC{?} {\optsep} \ssrC{*} {\optsep} \optional{\ssrN{occ-switch}}\ssrC{->} {\optsep} \optional{\ssrN{occ-switch}}\ssrC{<-} {\optsep} \\
-	   &&      \ssrC{[} \ssrN[1]{i-item}$^*$ \ssrC{|} $\dots$ \ssrC{|} \ssrN[m]{i-item}$^*$ \ssrC{]} {\optsep} \ssrC{-} {\optsep} \ssrC{[:} \ssrN{ident}$^+$ \ssrC{]}
-\end{longtable}
-
-The `\ssrC{=>}' tactical first executes {\tac}, then the
-\ssrN{i-item}s, left to right, i.e., starting from $\ssrN[1]{i-item}$.  An
-\ssrN{s-item} specifies a simplification operation; a $\ssrN{clear
-switch}$ specifies context pruning as in~\ref{ssec:discharge}. The
-\ssrN{i-pattern}s can be seen as a variant of \emph{intro patterns}~\ref{intros-pattern}:
-each performs an introduction operation, i.e., pops some variables or
-assumptions from the goal.
-
-An \ssrN{s-item} can simplify the set of subgoals or the subgoal themselves:
-\begin{itemize}
-\item \ssrC{//} removes all the ``trivial'' subgoals that can be resolved by
-      the \ssr{} tactic \ssrC{done} described in~\ref{ssec:termin}, i.e., it
-      executes \ssrC{try done}.
-\item \ssrC{/=} simplifies the goal by performing partial evaluation, as
-   per the tactic \ssrC{simpl}.\footnote{Except \ssrC{/=} does not
-   expand the local definitions created by the \ssr{} \ssrC{in} tactical.}
-\item \ssrC{//=} combines both kinds of simplification; it is equivalent
-    to \ssrC{/= //}, i.e., \ssrC{simpl; try done}.
-\end{itemize}
-When an \ssrN{s-item} bears a \ssrN{clear-switch}, then the \ssrN{clear-switch} is
-executed \emph{after} the \ssrN{s-item}, e.g., \ssrL|{IHn}//| will solve
-some subgoals, possibly using the fact \ssrL|IHn|, and will erase \ssrL|IHn|
-from the context of the remaining subgoals.
-
-The last entry in the \ssrN{i-item} grammar rule, \ssrC{/}{\term},
-represents a view (see section~\ref{sec:views}). If $\ssrN[k+1]{i-item}$
-is a view \ssrN{i-item}, the view is applied to the assumption in top
-position once $\ssrN[1]{i-item} \dots \ssrN[k]{i-item}$ have been performed.
-
-The view is applied to the top assumption.
-
-\ssr{} supports the following \ssrN{i-pattern}s:
-\begin{itemize}
-\item \ssrN{ident} pops the top variable, assumption, or local definition into
-      a new constant, fact, or defined constant \ssrN{ident}, respectively.
-      Note that defined constants cannot be introduced when
-      $\delta$-expansion is required to expose the top variable or assumption.
-\item \ssrC{?} pops the top variable into an anonymous constant or fact,
-      whose name is picked by the tactic interpreter.
-      \ssr{} only generates names that
-      cannot appear later in the user script.\footnote{\ssr{} reserves
-      all identifiers of the form ``\ssrC{_x_}'', which is used for such
-      generated names.}
-\item \ssrC{_} pops the top variable into an anonymous constant that will be
-     deleted from
-     the proof context of all the subgoals produced by the \ssrC{=>} tactical.
-     They should thus never be displayed, except in an error message
-     if the constant is still actually used in the goal or context after
-     the last \ssrN{i-item} has been executed (\ssrN{s-item}s can erase goals
-     or terms where the constant appears).
-\item \ssrC{*} pops all the remaining apparent variables/assumptions
-     as anonymous constants/facts. Unlike \ssrC{?} and \ssrC{move} the \ssrC{*}
-     \ssrN{i-item} does not expand definitions in the goal to expose
-     quantifiers, so it may be useful to repeat a \ssrC{move=> *} tactic,
-     e.g., on the goal
-\begin{lstlisting}
-  forall a b : bool, a <> b
-\end{lstlisting}
-a first \ssrC{move=> *} adds only \ssrC{_a_ : bool} and \ssrC{_b_ : bool} to
-the context; it takes a second \ssrC{move=> *} to add
-\ssrC{_Hyp_ : _a_ = _b_}.
-\item $[\ssrN{occ-switch}]$\ssrC{->} (resp. $[\ssrN{occ-switch}]$\ssrC{<-})
-  pops the top assumption
-  (which should be a rewritable proposition) into an anonymous fact,
-  rewrites (resp. rewrites right to left) the goal with this fact
-  (using the \ssr{} \ssrC{rewrite} tactic described in section~\ref{sec:rw},
-  and honoring the optional occurrence selector),
-  and finally deletes the anonymous fact from the context.
-\item\ssrC{[ $\ssrN[1]{i-item}^*$ | $\dots$ | $\ssrN[m]{i-item}^*$ ]},
-   when it is the very \emph{first} \ssrN{i-pattern} after ${\tac}\;\ssrC{=>}$
-   tactical \emph{and} {\tac} is not a \ssrC{move}, is a \emph{branching}
-   \ssrN{i-pattern}. It executes
-   the sequence $\ssrN[i]{i-item}^*$ on the $i^{\rm th}$
-   subgoal produced by {\tac}. The execution of {\tac}
-   should thus generate exactly $m$
-   subgoals, unless the \ssrC{[$\dots$]} \ssrN{i-pattern} comes after an initial
-   \ssrC{//} or \ssrC{//=} \ssrN{s-item} that closes some of the goals produced by
-   {\tac}, in which case exactly $m$ subgoals should remain after the
-   \ssrN{s-item}, or we have the trivial branching \ssrN{i-pattern} \ssrC{[]},
-   which always does nothing, regardless of the number of remaining subgoals.
-\item\ssrC{[ $\ssrN[1]{i-item}^*$ | $\dots$ | $\ssrN[m]{i-item}^*$ ]}, when it is
-   \emph{not} the first \ssrN{i-pattern} or when {\tac} is a
-   \ssrC{move}, is a \emph{destructing} \ssrN{i-pattern}.  It starts by
-   destructing the top variable, using the \ssr{} \ssrC{case} tactic
-   described in~\ref{ssec:basictac}.  It then behaves as the
-   corresponding branching \ssrN{i-pattern}, executing the sequence
-   $\ssrN[i]{i-item}^*$ in the $i^{\rm th}$ subgoal generated by the case
-   analysis; unless we have the trivial destructing \ssrN{i-pattern}
-   \ssrC{[]}, the latter should generate exactly $m$ subgoals, i.e., the
-   top variable should have an inductive type with exactly $m$
-   constructors.\footnote{More precisely, it should have a quantified
-   inductive type with $a$ assumptions and $m - a$ constructors.}
-   While it is good style to use the $\ssrN[i]{i-item}^*$
-   to pop the variables and assumptions corresponding to each constructor,
-   this is not enforced by \ssr{}.
-\item\ssrC{-} does nothing, but counts as an intro pattern. It can also
-  be used to force the interpretation of
-  \ssrC{[ $\ssrN[1]{i-item}^*$ | $\dots$ | $\ssrN[m]{i-item}^*$ ]}
-   as a case analysis like in \ssrC{move=> -[H1 H2]}. It can also be used
-   to indicate explicitly the link between a view and a name like in
-   \ssrC{move=> /eqP-H1}. Last, it can serve as a separator between
-   views. Section~\ref{ssec:multiview} explains in which respect
-   the tactic \ssrC{move=> /v1/v2} differs from the tactic
-   \ssrC{move=> /v1-/v2}.
-\item\ssrC{[: $\ssrN{ident}^+$ ]} introduces in the context an abstract constant
-   for each \ssrN{ident}.  Its type has to be fixed later on by using
-   the \ssrC{abstract} tactic (see page~\pageref{ssec:abstract}).  Before then
-   the type displayed is \ssrC{<hidden>}.\label{ssec:introabstract}
-\end{itemize}
-Note that \ssr{} does not support the syntax
-$\ssrC{(}\ssrN{ipat}\ssrC{,}\dots\ssrC{,}\ssrN{ipat}\ssrC{)}$ for destructing
-intro-patterns.
-
-Clears are deferred until the end of the intro pattern. For
-example, given the goal:
-\begin{lstlisting}
-x, y : nat
-==================
-0 < x = true -> (0 < x) && (y < 2) = true
-\end{lstlisting}
-the tactic \ssrC{move=> \{x\} ->} successfully rewrites the goal and
-deletes \ssrC{x} and the anonymous equation. The goal is thus turned into:
-\begin{lstlisting}
-y : nat
-==================
-true && (y < 2) = true
-\end{lstlisting}
-If the cleared names are reused in the same intro pattern, a renaming
-is performed behind the scenes.
-
-Facts mentioned in a clear switch must be valid
-names in the proof context (excluding the section context).
-
-The rules for interpreting branching and destructing \ssrN{i-pattern}
-are motivated by the fact that it would be pointless to have a branching
-pattern if {\tac} is a \ssrC{move}, and in most of the remaining cases
-{\tac} is \ssrC{case} or \ssrC{elim}, which implies destruction.
-The rules above imply that
-\begin{lstlisting}
-  move=> [a b].
-  case=> [a b].
-  case=> a b.
-\end{lstlisting}
-are all equivalent, so which one to use is a matter of style;
-\ssrC{move} should be used for casual decomposition,
-such as splitting a pair, and \ssrC{case} should be used for actual decompositions,
-in particular for type families (see~\ref{ssec:typefam})
-and proof by contradiction.
-
-The trivial branching \ssrN{i-pattern} can be used to force the branching
-interpretation, e.g.,
-\begin{lstlisting}
-  case=> [] [a b] c.
-  move=> [[a b] c].
-  case;  case=> a b c.
-\end{lstlisting}
-are all equivalent.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Generation of equations}\label{ssec:equations}
-\idx{move eq : \dots{}}
-
-The generation of named equations option stores the definition of a
-new constant as an equation. The tactic:
-\begin{lstlisting}
-  move En: (size l) => n.
-\end{lstlisting}
-where \ssrC{l} is a list, replaces \ssrC{size l} by \ssrC{n} in the goal and
-adds the fact \ssrC{En : size l = n} to the context.
- This is quite different from:
-\begin{lstlisting}
-  pose n := (size l).
-\end{lstlisting}
-which generates a definition \ssrC{n := (size l)}. It is not possible to
-generalize or
-rewrite such a definition; on the other hand, it is automatically
-expanded during
-computation, whereas expanding the equation \ssrC{En} requires explicit
-rewriting.
-
-The use of this equation name generation option with a \ssrC{case} or an
-\ssrC{elim} tactic changes the status of the first \iitem{}, in order to
-deal with the possible parameters of the constants introduced.
-
-On the
-goal \ssrC{a <> b} where \ssrC{a, b} are natural numbers, the tactic:
-\begin{lstlisting}
-  case E : a => [|n].
-\end{lstlisting}
-generates two subgoals. The equation \ssrC{E : a = 0}  (resp. \ssrC{E : a =
-  S n}, and the constant \ssrC{n : nat}) has been added to
-the context of the goal \ssrC{0 <> b} (resp. \ssrC{S n  <> b}).
-
-If the user does not provide a branching \iitem{} as first \iitem{},
-or if the \iitem{} does not provide enough names for the arguments of
-a constructor,
-then the constants generated are introduced under fresh \ssr{} names.
-For instance, on the goal \ssrC{a <> b}, the tactic:
-\begin{lstlisting}
-  case E : a => H.
-\end{lstlisting}
-also generates two subgoals, both requiring a proof of \ssrC{False}.
- The hypotheses \ssrC{E : a = 0} and
-\ssrC{H : 0 = b} (resp. \ssrC{E : a = S _n_} and
-\ssrC{H : S _n_ = b}) have been added to the context of the first
-subgoal (resp. the second subgoal).
-
-Combining the generation of named equations mechanism with the
-\ssrC{case} tactic strengthens the power of a case analysis. On the other
-hand, when combined with the \ssrC{elim} tactic, this feature is mostly
-useful for
-debug purposes, to trace the values of decomposed parameters and
-pinpoint failing branches.
-
-% This feature is also useful
-% to analyse and debug generate-and-test style scripts that prove program
-% properties by generating a large set of input patterns and uniformly
-% solving the corresponding subgoals by computation and rewriting, e.g,
-
-% \begin{lstlisting}
-%   case: et => [|e' et]; first by case: s.
-%   case: e => //; case: b; case: w.
-% \end{lstlisting}
-% If the above sequence fails, then it's easy enough to replace the line
-% above with
-% \begin{lstlisting}
-%   case: et => [|e' et].
-%   case Ds: s; case De: e => //; case Db: b; case Dw: w=> [|s' w'] //=.
-% \end{lstlisting}
-% Then the first subgoal that appears will be the failing one, and the
-% equations \ssrC{Ds}, \ssrC{De}, \ssrC{Db}
-% and \ssrC{Dw} will pinpoint its branch. When the constructors of
-% the decomposed type have arguments (like \ssrC{w : (seq nat)}
-% above) these need to be
-% introduced in order to generate the equation, so there should
-% always be an explicit \iitem{} (\ssrC{\[|s' w'\]} above) that
-% assigns names to these arguments. If this \iitem{}
-% is omitted the arguments are introduced with default
-% name; this
-% feature should be
-% avoided except for quick debugging runs (it has some uses in complex tactic
-% sequences, however).
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Type families}\label{ssec:typefam}
-\idx{case: \dots{} / \dots{}}
-
-When the top assumption of a goal has an inductive type, two
-specific operations are possible: the case analysis performed by the
-\ssrC{case} tactic, and the application of an induction principle,
-performed by the \ssrC{elim} tactic. When this top assumption has an
-inductive type, which is moreover an instance of a type family, \Coq{}
-may need help from the user to specify which occurrences of the parameters
-of the type should be substituted.
-
-A specific \ssrC{/} switch indicates the type family parameters of the
-type of a \ditem{} immediately following this \ssrC{/} switch, using the
-syntax:
-
-\begin{center}
- \ssrC{[} \ssrC{case} {\optsep} \ssrC{elim} \ssrC{]:} \ssrN{d-item}$^+$ \ssrC{/} \ssrN{d-item}$^*$
-\end{center}
-
-The \ssrN{d-item}s on the right side of the \ssrC{/} switch are discharged
-as described in section \ref{ssec:discharge}. The case analysis or
-elimination will be done on the type of the top assumption after these
-discharge operations.
-
-Every \ssrN{d-item} preceding the \ssrC{/} is interpreted as arguments of this
-type, which should be an instance of an inductive type family. These terms are
-not actually generalized, but rather selected for substitution. Occurrence
-switches can be used to restrict the substitution.  If a {\term} is left
-completely implicit (e.g. writing just $\ssrC{\_}$), then a pattern is inferred
-looking at the type of the top assumption.  This allows for the compact syntax
-\ssrC{case: \{2\}\_ / eqP}, were \ssrC{\_} is interpreted as \ssrC{(\_ == \_)}. Moreover
-if the \ssrN{d-item}s list is too short, it is padded with an initial
-sequence of $\ssrC{\_}$ of the right length.
-
-Here is a small example on lists. We define first a function which
-adds an element at the end of a given list.
-\begin{lstlisting}
-  Require Import List.
-
-  Section LastCases.
-  Variable A : Type.
-
-  Fixpoint |*add_last*|(a : A)(l : list A): list A :=
-  match l with
-    |nil => a :: nil
-    |hd :: tl => hd :: (add_last a tl)
-  end.
-\end{lstlisting}
-Then we define an inductive predicate for
-case analysis on lists according to their last element:
-\begin{lstlisting}
-  Inductive |*last_spec*| : list A -> Type :=
-    | LastSeq0 : last_spec nil
-    | LastAdd s x : last_spec (add_last x s).
-
-  Theorem |*lastP*| : forall l : list A, last_spec l.
-\end{lstlisting}
-Applied to the goal:
-\begin{lstlisting}
-  Goal forall l : list A, (length l) * 2 = length (app l l).
-\end{lstlisting}
-the command:
-\begin{lstlisting}
-  move=> l; case: (lastP l).
-\end{lstlisting}
-generates two subgoals:
-\begin{lstlisting}
-  length nil * 2 = length (nil ++ nil)
-\end{lstlisting}
-and
-\begin{lstlisting}
-  forall (s : list A) (x : A),
-    length (add_last x s) * 2 = length (add_last x s ++ add_last x s)
-\end{lstlisting}
-both having \ssrC{l : list A} in their context.
-
-Applied to the same goal, the command:
-\begin{lstlisting}
-  move=> l; case: l / (lastP l).
-\end{lstlisting}
-generates the same subgoals but \ssrC{l} has been cleared from both
-contexts.
-
-Again applied to the same goal, the command:
-\begin{lstlisting}
-  move=> l; case: {1 3}l / (lastP l).
-\end{lstlisting}
-generates the subgoals \ssrL-length l * 2 = length (nil ++ l)- and
-\ssrL-forall (s : list A) (x : A), length l * 2 = length (add_last x s++l)-
-where the selected occurrences on the left of the \ssrC{/} switch have
-been substituted with \ssrC{l} instead of being affected by the case
-analysis.
-
-The equation name generation feature combined with a type family \ssrC{/}
-  switch generates an equation for the \emph{first} dependent d-item
-specified by the user.
-Again starting with the above goal, the command:
-\begin{lstlisting}
-  move=> l; case E: {1 3}l / (lastP l)=>[|s x].
-\end{lstlisting}
-adds \ssrC{E : l = nil} and \ssrC{E : l = add_last x s},
-respectively, to the context of the two subgoals it generates.
-
-There must be at least one \emph{d-item} to the left of the \ssrC{/}
-switch; this prevents any
-confusion with the view feature. However, the \ditem{}s to the right of
-the \ssrC{/} are optional, and if they are omitted the first assumption
-provides the instance of the type family.
-
-The equation always refers to the first \emph{d-item} in the actual
-tactic call, before any padding with initial $\ssrC{\_}$s. Thus, if an
-inductive type has two family parameters, it is possible to have
-\ssr{} generate an equation for the second one by omitting the pattern
-for the first; note however that this will fail if the type of the
-second parameter depends on the value of the first parameter.
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Control flow}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Indentation and bullets}\label{ssec:indent}
-
-A linear development of \Coq{} scripts gives little information on
-the structure of the proof. In addition, replaying a proof after some
-changes in the statement to be proved will usually not display information to
-distinguish between the various branches of case analysis for instance.
-
-To help the user in this organization of the proof script at
-development time, \ssr{} provides some bullets to highlight the
-structure of branching proofs. The available bullets are \ssrC{-},
-\ssrC{+} and \ssrC{*}. Combined with tabulation, this lets us highlight four
-nested levels of branching; the most we have ever
-needed is three. Indeed, the use of ``simpl and closing'' switches, of
-terminators (see above section \ref{ssec:termin}) and selectors (see
- section \ref{ssec:select}) is powerful enough
-to avoid most of the time more than two levels of indentation.
-
-Here is a fragment of such a structured script:
-
-\begin{lstlisting}
-case E1: (abezoutn _ _) => [[| k1] [| k2]].
-- rewrite !muln0 !gexpn0 mulg1 => H1.
-  move/eqP: (sym_equal F0); rewrite -H1 orderg1 eqn_mul1.
-  by case/andP; move/eqP.
-- rewrite muln0 gexpn0 mulg1 => H1.
-  have F1: t %| t * S k2.+1 - 1.
-    apply: (@dvdn_trans (orderg x)); first by rewrite F0; exact: dvdn_mull.
-    rewrite orderg_dvd; apply/eqP; apply: (mulgI x).
-    rewrite -{1}(gexpn1 x) mulg1 gexpn_add leq_add_sub //.
-    by move: P1; case t.
-  rewrite dvdn_subr in F1; last by exact: dvdn_mulr.
-  + rewrite H1 F0 -{2}(muln1 (p ^ l)); congr (_ * _).
-    by apply/eqP; rewrite -dvdn1.
-  + by move: P1; case: (t) => [| [| s1]].
-- rewrite muln0  gexpn0 mul1g => H1.
-...
-\end{lstlisting}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Terminators}\label{ssec:termin}
-\idx{by \dots{}}
-
-To further structure scripts, \ssr{}
-supplies \emph{terminating} tacticals to explicitly close off tactics.
-When replaying scripts, we then have the nice property that
-an error immediately occurs when a closed tactic fails to prove its
-subgoal.
-
-It is hence recommended practice that the proof of any subgoal should
-end with a tactic which \emph{fails if it does not solve the current
-  goal}, like \ssrC{discriminate}, \ssrC{contradiction} or \ssrC{assumption}.
-
-In fact, \ssr{} provides a generic tactical which turns any tactic into
-a closing one (similar to \ssrC{now}). Its general syntax is:
-
-\begin{center}
-  \ssrC{by} {\tac}\ssrC{.}
-\end{center}
-
-The Ltac expression:
-
-\begin{center}
-  \ssrC{by [}\ssrN[1]{tactic} \ssrC{| [}\ssrN[2]{tactic} \ssrC{| ...].}
-\end{center}
-
-is equivalent to:
-
-\begin{center}
-  \ssrC{[by} \ssrN[1]{tactic} \ssrC{| by} \ssrN[2]{tactic} \ssrC{| ...].}
-\end{center}
-
-and this form should be preferred to the former.
-
-In the script provided as example in section \ref{ssec:indent}, the
-paragraph corresponding to each sub-case ends with a tactic line prefixed
-with a \ssrC{by}, like in:
-
-\begin{center}
-  \ssrC{by apply/eqP; rewrite -dvdn1.}
-\end{center}
-
-
-The \ssrC{by} tactical is implemented using the user-defined,
-and extensible \ssrC{done} tactic. This \ssrC{done} tactic tries to solve
-the current goal by some trivial means and fails if it doesn't succeed.
-Indeed, the tactic expression:
-
-\begin{center}
-   \ssrC{by} {\tac}\ssrC{.}
-\end{center}
-
-is equivalent to:
-
-\begin{center}
-  {\tac}\ssrC{; done.}
-\end{center}
-
-Conversely, the tactic
-
-\begin{center}
-  \ssrC{by [ ].}
-\end{center}
-
-is equivalent to:
-
-\begin{center}
-  \ssrC{done.}
-\end{center}
-
-The default implementation of the \ssrC{done} tactic, in the {\tt
-  ssreflect.v} file, is:
-
-\begin{lstlisting}
-Ltac done :=
-  trivial; hnf; intros; solve
-   [ do ![solve [trivial | apply: sym_equal; trivial]
-         | discriminate | contradiction | split]
-   | case not_locked_false_eq_true; assumption
-   | match goal with H : ~ _ |- _ => solve [case H; trivial] end ].
-\end{lstlisting}
-
-The lemma \ssrC{|*not_locked_false_eq_true*|} is needed to discriminate
-\emph{locked} boolean predicates (see section \ref{ssec:lock}).
-The iterator tactical \ssrC{do} is presented in section
-\ref{ssec:iter}.
-This tactic can be customized by the user, for instance to include an
-\ssrC{auto} tactic.
-
-A natural and common way of closing a goal is to apply a lemma which
-is the \ssrC{exact} one needed for the goal to be solved. The defective
-form of the tactic:
-\begin{lstlisting}
-  exact.
-\end{lstlisting}
-is equivalent to:
-\begin{lstlisting}
-  do [done | by move=> top; apply top].
-\end{lstlisting}
-where \ssrC{top} is a fresh name affected to the top assumption of the goal.
-This applied form is supported by the  \ssrC{:} discharge tactical, and
-the tactic:
-\begin{lstlisting}
-  exact: MyLemma.
-\end{lstlisting}
-is equivalent to:
-\begin{lstlisting}
-  by apply: MyLemma.
-\end{lstlisting}
-(see section \ref{sss:strongapply} for the documentation of the \ssrC{apply:}
-combination).
-
-\textit{Warning} The list of tactics, possibly chained by
-semi-columns, that follows a \ssrC{by} keyword is considered as a
-parenthesized block
-applied to the current goal. Hence for example if the tactic:
-\begin{lstlisting}
-   by rewrite my_lemma1.
-\end{lstlisting}
-succeeds, then the tactic:
-\begin{lstlisting}
-   by rewrite my_lemma1; apply my_lemma2.
-\end{lstlisting}
-usually fails since it is equivalent to:
-\begin{lstlisting}
-   by (rewrite my_lemma1; apply my_lemma2).
-\end{lstlisting}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Selectors}\label{ssec:select}
-\idx{last \dots{} first}
-\idx{first \dots{} last}
-
-When composing tactics, the two tacticals \ssrC{first} and
-\ssrC{last} let the user restrict the application of a tactic to only one
-of the subgoals generated by the previous tactic. This
-covers the frequent cases where a tactic generates two subgoals one of
-which can be easily disposed of.
-
-This is an other powerful way of linearization of scripts, since it
-happens very often that a trivial subgoal can be solved in a less than
-one line tactic. For instance, the tactic:
-
-\begin{center}
-  \ssrN[1]{tactic}\ssrC{; last by} \ssrN[2]{tactic}\ssrC{.}
-\end{center}
-
-tries to solve the last subgoal generated by \ssrN[1]{tactic} using the
-\ssrN[2]{tactic}, and fails if it does not succeeds. Its analogous
-
-\begin{center}
-  \ssrN[1]{tactic}\ssrC{; first by} \ssrN[2]{tactic}.
-\end{center}
-
-tries to solve the first subgoal generated by \ssrN[1]{tactic} using the
-tactic \ssrN[2]{tactic}, and fails if it does not succeeds.
-
-
-\ssr{} also offers an extension of this facility, by supplying
-tactics to \emph{permute}  the subgoals generated by a tactic.
-The tactic:
-
-\begin{center}
-  {\tac}\ssrC{; last first.}
-\end{center}
-
-inverts the order of the subgoals generated by {\tac}. It is
-equivalent to:
-
-\begin{center}
-  {\tac}\ssrC{; first last.}
-\end{center}
-
-
-More generally, the tactic:
-
-\begin{center}
-  {\tac}\ssrC{; last }${\naturalnumber}$ \ssrC{first.}
-\end{center}
-
-where ${\naturalnumber}$ is
-a \Coq{} numeral, or and Ltac variable denoting
-a \Coq{} numeral, having the value $k$. It
-rotates the $n$ subgoals $G_1,
-\dots, G_n$ generated by {\tac}. The first subgoal becomes
-$G_{n + 1 - k}$ and the circular order of subgoals remains unchanged.
-
-Conversely, the tactic:
-
-  {\tac}\ssrC{; first }${\naturalnumber}$ \ssrC{last.}
-
-rotates the $n$ subgoals $G_1,
-\dots, G_n$ generated by \ssrC{tactic} in order that the first subgoal
-becomes $G_{k}$.
-
-Finally, the tactics \ssrC{last} and \ssrC{first} combine with the
-branching syntax of Ltac:
-if the tactic $\ssrN[0]{tactic}$ generates $n$
-subgoals on a given goal, then the tactic
-
-  $tactic_0$\ssrC{; last }${\naturalnumber}$ \ssrC{[}$tactic_1$\ssrC{|}$\dots$\ssrC{|}$tactic_m$\ssrC{] || }$tactic_{m+1}$\ssrC{.}
-
-where ${\naturalnumber}$ denotes the integer $k$ as above, applies $tactic_1$ to the
-$n -k + 1$-th goal, $\dots tactic_m$ to the $n -k + 2 - m$-th
-goal and $tactic_{m+1}$ to the others.
-
-For instance, the script:
-\begin{lstlisting}
-  Inductive test : nat ->  Prop :=
-    C1 : forall n, test n | C2 : forall n, test n |
-    C3 : forall n, test n | C4 : forall n, test n.
-
-  Goal forall n, test n -> True.
-  move=> n t; case: t; last 2 [move=> k| move=> l]; idtac.
-\end{lstlisting}
-
-creates a goal with four subgoals, the first and the last being
-\ssrC{nat -> True}, the second and the third being \ssrC{True} with
-respectively \ssrC{k : nat} and \ssrC{l : nat} in their context.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Iteration}\label{ssec:iter}
-\idx{do \dots{} [ \dots{} ]}
-
-\ssr{} offers an accurate control on the repetition of
-tactics, thanks to the \ssrC{do} tactical, whose general syntax is:
-
-\begin{center}
-	\ssrC{do} \optional{\ssrN{mult}} \ssrC{[} \ssrN[1]{tactic} \ssrC{|} $\dots$ \ssrC{|} \ssrN[n]{tactic} \ssrC{]}
-\end{center}
-where \ssrN{mult} is a \emph{multiplier}.
-
-Brackets can only be omitted if a single tactic is given \emph{and} a
-multiplier is present.
-
-A tactic of the form:
-
-\begin{center}
-  \ssrC{do [} \tac$_1$ \ssrC{|} $\dots$ \ssrC{|} \tac$_n$\ssrC{].}
-\end{center}
-
-is equivalent to the standard Ltac expression:
-
-\begin{center}
-  \ssrC{first [} \tac$_1$ \ssrC{|} $\dots$ \ssrC{|} \tac$_n$\ssrC{].}
-\end{center}
-
-
-The optional multiplier \ssrN{mult} specifies how many times
-the action of {\tac} should be repeated on the current subgoal.
-
-There are four kinds of multipliers:
-  \begin{itemize}
-  \item \ssrC{n!}: the step {\tac} is repeated exactly $n$ times
-    (where $n$ is a positive integer argument).
-  \item \ssrC{!}: the step {\tac} is repeated as many times as possible,
-    and done at least once.
-  \item \ssrC{?}: the step {\tac} is repeated as many times as possible,
-    optionally.
-  \item \ssrC{n?}: the step {\tac} is repeated up to $n$ times,
-    optionally.
-  \end{itemize}
-
-For instance, the tactic:
-
-\begin{center}
- {\tac} \ssrL+; do 1?rewrite mult_comm.+
-\end{center}
-
-rewrites at most one time the lemma \ssrC{mult_com} in all the subgoals
-generated by {\tac} , whereas the tactic:
-
-\begin{center}
-  {\tac} \ssrL+; do 2!rewrite mult_comm.+
-\end{center}
-
-rewrites exactly two times the lemma \ssrC{mult_com} in all the subgoals
-generated by {\tac}, and fails if this rewrite is not possible
-in some subgoal.
-
-Note that the combination of multipliers and \ssrC{rewrite} is so often
-used that multipliers are in fact integrated to the syntax of the \ssr{}
-\ssrC{rewrite} tactic, see section \ref{sec:rw}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Localization}\label{ssec:gloc}
-\idx{\dots{} in \dots{}}
-
-In sections \ref{ssec:loc} and \ref{ssec:profstack}, we have already
-presented the \emph{localization} tactical \ssrC{in}, whose general
-syntax is:
-\begin{center}
-	{\tac} \ssrC{in} \ssrN{ident}$^+$ \optional{\ssrC{*}}
-\end{center}
-
-where \ssrN{ident}$^+$ is a non empty list of fact
-names in the context. On the left side of \ssrC{in}, {\tac} can be
-\ssrC{move}, \ssrC{case}, \ssrC{elim}, \ssrC{rewrite}, \ssrC{set},
- or any tactic formed with the general iteration tactical \ssrC{do} (see
- section \ref{ssec:iter}).
-
-The operation described by {\tac} is performed in the facts
-listed in \ssrN{ident}$^+$ and in the goal if a \ssrC{*} ends
-the list.
-
-The \ssrC{in} tactical successively:
-\begin{itemize}
-\item generalizes the selected hypotheses, possibly ``protecting'' the
- goal if \ssrC{*} is not present,
-\item performs {\tac}, on the obtained goal,
-\item reintroduces the generalized facts, under the same names.
-\end{itemize}
-
-This defective form of the \ssrC{do} tactical is useful to avoid clashes
-between standard Ltac \ssrC{in} and the \ssr{} tactical \ssrC{in}.
-For example, in the following script:
-\begin{lstlisting}
-  Ltac |*mytac*| H := rewrite H.
-
-  Goal forall x y, x = y -> y = 3 -> x + y  = 6.
-  move=> x y H1 H2.
-  do [mytac H2] in H1 *.
-\end{lstlisting}
-the last tactic rewrites the hypothesis \ssrC{H2 : y = 3} both in
-\ssrC{H1 : x = y} and in the goal \ssrC{x + y = 6}.
-
-By default \ssrC{in} keeps the body of local definitions. To erase
-the body of a local definition during the generalization phase,
-the name of the local definition must be written between parentheses,
-like in \ssrC{rewrite H in H1 (def_n) $\;\;$H2}.
-
-From \ssr{} 1.5 the grammar for the \ssrC{in} tactical has been extended
-to the following one:
-
-\begin{center}
-  {\tac} \ssrC{in} \optional{ 
-	  \ssrN{clear-switch} {\optsep}
-	  \optional{\ssrC{@}}\ssrN{ident} {\optsep}
-          \ssrC{(}\ssrN{ident}\ssrC{)} {\optsep}
-	  \ssrC{(}\optional{\ssrC{@}}\ssrN{ident} \ssrC{:=} \ssrN{c-pattern}\ssrC{)}
-		}$^+$ \optional{\ssrC{*}}
-\end{center}
-
-In its simplest form the last option lets one rename hypotheses that can't be
-cleared (like section variables).  For example \ssrC{(y := x)} generalizes
-over \ssrC{x} and reintroduces the generalized
-variable under the name \ssrC{y} (and does not clear \ssrC{x}).\\
-For a more precise description the $\ssrC{(}[\ssrC{@}]\ssrN{ident}\ \ssrC{:=}\ \ssrN{c-pattern}\ssrC{)}$
-item refer to the ``Advanced generalization'' paragraph at page~\pageref{par:advancedgen}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Structure}\label{ssec:struct}
-
-Forward reasoning structures the script by explicitly specifying some
-assumptions to be added to the proof context. It is closely associated
-with the declarative style of proof, since an extensive use of these
-highlighted statements
-make the script closer to a (very detailed) text book proof.
-
-Forward chaining tactics allow to state an intermediate lemma and start a
-piece of script dedicated to the proof of this statement. The use of
-closing tactics (see section \ref{ssec:termin}) and of
-indentation makes syntactically explicit the portion of the script
-building the proof of the intermediate statement.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{The \ssrC{have} tactic.}
-\label{sssec:have}
-\idx{have: \dots{}}
-\idx{have: \dots{} := \dots{}}
-
-The main \ssr{} forward reasoning tactic is the \ssrC{have} tactic. It
-can be use in two modes: one starts a new (sub)proof for an
-intermediate result in the main proof, and the other
-provides explicitly a proof term for this intermediate step.
-
-In the first mode, the syntax of \ssrC{have} in its defective form is:
-
-  \ssrC{have: }{\term}\ssrC{.}
-
-This tactic supports open syntax for {\term}.
-Applied to a goal \ssrC{G}, it generates a first subgoal requiring a
-proof of {\term} in the context of \ssrC{G}. The  second generated
-subgoal is of the form {\term} \ssrC{-> G}, where {\term} becomes
-the new top assumption, instead of being introduced with a fresh
-name. At the proof-term level, the \ssrC{have} tactic creates a $\beta$
-redex, and introduces the lemma under a fresh name, automatically
-chosen.
-
-
-Like in the case of the \ssrC{pose} tactic (see section \ref{ssec:pose}),
-the types of the holes are abstracted in {\term}.
-For instance, the tactic:
-\begin{lstlisting}
-  have: _ * 0 = 0.
-\end{lstlisting}
-is equivalent to:
-\begin{lstlisting}
-  have: forall n : nat, n * 0 = 0.
-\end{lstlisting}
-The \ssrC{have} tactic also enjoys the same abstraction mechanism as the
-\ssrC{pose} tactic for the non-inferred implicit arguments. For instance,
-the tactic:
-\begin{lstlisting}
-  have: forall x y, (x, y) = (x, y + 0).
-\end{lstlisting}
-opens a new subgoal to prove that:
-
-\noindent\ssrC{forall (T : Type) (x : T) (y : nat), (x, y) = (x, y + 0)}
-
-
-The behavior of the defective \ssrC{have} tactic makes it possible to
-generalize it in the
-following general construction:
-\begin{center}
-	\ssrC{have} \ssrN{i-item}$^*$ \optional{\ssrN{i-pattern}}
-	\optional{\ssrN{s-item} {\optsep} \ssrN{binder}$^+$}
-	\optional{\ssrC{:} \ssrN[1]{term}}
-	\optional{\ssrC{:=} \ssrN[2]{term} {\optsep} \ssrC{by} {\tac}}
-\end{center}
-
-Open syntax is supported for $\ssrN[1]{term}$ and $\ssrN[2]{term}$. For the
-description of
-\iitem{}s and clear switches see section \ref{ssec:intro}.
-The first mode of the \ssrC{have} tactic, which opens a sub-proof for an
-intermediate result, uses tactics of the form:
-
-\begin{center}
-  \ssrC{have} \ssrN{clear-switch} \ssrN{i-item} \ssrC{:} {\term} \ssrC{by} {\tac}.
-\end{center}
-
-which behave like:\\
-
-\begin{center}
-  \ssrC{have:} {\term} \ssrC{; first by } {\tac}.
-\end{center}
-\begin{center}
-  \ssrC{  move=>} \ssrN{clear-switch} \ssrN{i-item}.
-\end{center}
-
-
-Note that the \ssrN{clear-switch} \emph{precedes} the
-\ssrN{i-item}, which allows to reuse a name of the context, possibly used
-by the proof of the assumption, to introduce the new assumption
-itself.
-
-The \ssrC{by} feature is especially convenient when the proof script of the
-statement is very short, basically when it fits in one line like in:
-\begin{lstlisting}
-  have H : forall x y, x + y = y + x by move=> x y; rewrite addnC.
-\end{lstlisting}
-
-The possibility of using \iitem{}s supplies a very concise
-syntax for the further use of the intermediate step. For instance,
-\begin{lstlisting}
-  have -> : forall x, x * a = a.
-\end{lstlisting}
-on a goal \ssrC{G}, opens a new subgoal asking for a proof of
-\ssrC{forall x, x * a = a}, and a second subgoal in which the lemma
- \ssrC{forall x, x * a = a} has been rewritten in the goal \ssrC{G}. Note
- that in this last subgoal, the intermediate result does not appear in
- the context.
-Note that, thanks to the deferred execution of clears, the following
-idiom is supported (assuming \ssrC{x} occurs in the goal only):
-\begin{lstlisting}
-  have {x} -> : x = y
-\end{lstlisting}
-
-An other frequent use of the intro patterns combined with \ssrC{have} is the
-destruction of existential assumptions like in the tactic:
-\begin{lstlisting}
-  have [x Px]: exists x : nat, x > 0.
-\end{lstlisting}
-which opens a new subgoal asking for a proof of \ssrC{exists x : nat, x >
-  0} and  a second subgoal in which the witness is introduced under
-the name \ssrC{x : nat}, and its property under the name \ssrC{Px : x > 0}.
-
-An alternative use of the \ssrC{have} tactic is to provide the explicit proof
-term for the intermediate lemma, using tactics of the form:
-
-\begin{center}
-  \ssrC{have} \optional{\ssrN{ident}} \ssrC{:=} {\term}.
-\end{center}
-
-This tactic creates a new assumption of type the type of
-{\term}. If the
-optional \ssrN{ident} is present, this assumption is introduced under
-the name \ssrN{ident}. Note that the body of the constant is lost for
-the user.
-
-Again, non inferred implicit arguments and explicit holes are abstracted. For
-instance, the tactic:
-\begin{lstlisting}
-  have H := forall x, (x, x) = (x, x).
-\end{lstlisting}
-adds to the context \ssrC{H : Type -> Prop}. This is a schematic example but
-the feature is specially useful when the proof term to give involves
-for instance a lemma with some hidden implicit arguments.
-
-After the \ssrN{i-pattern}, a list of binders is allowed.
-For example, if \ssrC{Pos_to_P} is a lemma that proves that
-\ssrC{P} holds for any positive, the following command:
-\begin{lstlisting}
-  have H x (y : nat) : 2 * x + y = x + x + y by auto.
-\end{lstlisting}
-will put in the context \ssrC{H : forall x, 2 * x = x + x}. A proof term
-provided after \ssrC{:=} can mention these bound variables (that are
-automatically introduced with the given names).
-Since the \ssrN{i-pattern} can be omitted, to avoid ambiguity, bound variables
-can be surrounded with parentheses even if no type is specified:
-\begin{lstlisting}
-  have (x) : 2 * x = x + x by auto.
-\end{lstlisting}
-
-The \ssrN{i-item}s and \ssrN{s-item} can be used to interpret the
-asserted hypothesis with views (see section~\ref{sec:views}) or
-simplify the resulting goals.
-
-The \ssrC{have} tactic also supports a \ssrC{suff} modifier which allows for
-asserting that a given statement implies the current goal without
-copying the goal itself. For example, given a goal \ssrC{G} the tactic
-\ssrC{have suff H : P} results in the following two goals:
-\begin{lstlisting}
-   |- P -> G
-   H : P -> G |- G
-\end{lstlisting}
-Note that \ssrC{H} is introduced in the second goal. The \ssrC{suff}
-modifier is not compatible with the presence of a list of binders.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Generating \ssrC{let in} context entries with \ssrC{have}}
-\label{sec:havetransparent}
-
-Since \ssr{} 1.5 the \ssrC{have} tactic supports a ``transparent'' modifier to
-generate \ssrC{let in} context entries: the \ssrC{@} symbol in front of the context
-entry name.  For example:
-
-\begin{lstlisting}
-have @i : 'I_n by apply: (Sub m); auto.
-\end{lstlisting}
-generates the following two context entry:
-\begin{lstlisting}
-i := Sub m proof_produced_by_auto : 'I_n
-\end{lstlisting}
-
-Note that the sub-term produced by \ssrC{auto} is in general huge and
-uninteresting, and hence one may want to hide it.
-
-For this purpose the \ssrC{[: name ]} intro pattern and the tactic
-\ssrC{abstract} (see page~\pageref{ssec:abstract}) are provided.
-Example:
-\begin{lstlisting}
-have [:blurb] @i : 'I_n by apply: (Sub m); abstract: blurb; auto.
-\end{lstlisting}
-generates the following two context entries:
-\begin{lstlisting}
-blurb : (m < n) (*1*)
-i := Sub m blurb : 'I_n
-\end{lstlisting}
-The type of \ssrC{blurb} can be cleaned up by its annotations by just simplifying
-it.  The annotations are there for technical reasons only.
-
-When intro patterns for abstract constants are used in conjunction
-with \ssrC{have} and an explicit term, they must be used as follows:
-
-\begin{lstlisting}
-have [:blurb] @i : 'I_n := Sub m blurb.
-  by auto.
-\end{lstlisting}
-
-In this case the abstract constant \ssrC{blurb} is assigned by using it
-in the term that follows \ssrC{:=} and its corresponding goal is left to
-be solved.  Goals corresponding to intro patterns for abstract constants
-are opened in the order in which the abstract constants are declared (not
-in the ``order'' in which they are used in the term).
-
-Note that abstract constants do respect scopes.  Hence, if a variable
-is declared after their introduction, it has to be properly generalized (i.e.
-explicitly passed to the abstract constant when one makes use of it).
-For example any of the following two lines:
-\begin{lstlisting}
-have [:blurb] @i k : 'I_(n+k) by apply: (Sub m); abstract: blurb k; auto.
-have [:blurb] @i k : 'I_(n+k) := apply: Sub m (blurb k); first by auto.
-\end{lstlisting}
-generates the following context:
-\begin{lstlisting}
-blurb : (forall k, m < n+k) (*1*)
-i := fun k => Sub m (blurb k) : forall k, 'I_(n+k)
-\end{lstlisting}
-
-Last, notice that the use of intro patterns for abstract constants is
-orthogonal to the transparent flag \ssrC{@} for \ssrC{have}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{The \ssrC{have} tactic and type classes resolution}
-\label{ssec:havetcresolution}
-
-Since \ssr{} 1.5 the \ssrC{have} tactic behaves as follows with respect to type
-classes inference.
-
-\begin{itemize}
-\item \ssrC{have foo : ty.}
-	Full inference for \ssrC{ty}.
-	The first subgoal demands a proof of such instantiated statement.
-\item \ssrC{have foo : ty := .}
-	No inference for \ssrC{ty}. Unresolved instances are quantified in
-	\ssrC{ty}.  The first subgoal demands a proof of such quantified
-	statement.  Note that no proof term follows \ssrC{:=}, hence two
-	subgoals are generated.
-\item \ssrC{have foo : ty := t.}
-	No inference for \ssrC{ty} and \ssrC{t}.
-\item \ssrC{have foo := t.}
-	No inference for \ssrC{t}. Unresolved instances are quantified in the
-	(inferred) type of \ssrC{t} and abstracted in \ssrC{t}.
-\end{itemize}
-
-The behavior of \ssr{} 1.4 and below (never resolve type classes)
-can be restored with the option \ssrC{Set SsrHave NoTCResolution}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Variants: the \ssrC{suff} and  \ssrC{wlog} tactics.}
-\label{ssec:wlog}
-\idx{suff: \dots{}}
-\idx{suffices: \dots{}}
-\idx{wlog: \dots{} / \dots{}}
-\idx{without loss: \dots{} / \dots{}}
-
-As it is often the case  in mathematical textbooks, forward
-reasoning may be used in slightly different variants.
-One of these variants is to show that the intermediate step $L$
-easily implies the initial goal $G$. By easily we mean here that
-the proof of $L \Rightarrow G$ is shorter than the one of $L$
-itself. This kind of reasoning step usually starts with:
-``It suffices to show that \dots''.
-
-This is such a frequent way of reasoning that \ssr{} has a variant of the
-\ssrC{have} tactic called \ssrC{suffices} (whose abridged name is
-\ssrC{suff}). The \ssrC{have} and \ssrC{suff} tactics are equivalent and
-have the same syntax but:
-\begin{itemize}
-\item the order of the generated subgoals is inversed
-\item but the optional clear item is still performed in the
-  \emph{second} branch. This means that the tactic:
-\begin{lstlisting}
-  suff {H} H : forall x : nat, x >= 0.
-\end{lstlisting}
-fails if the context of the current goal indeed contains an
-assumption named \ssrC{H}.
-\end{itemize}
-The rationale of this clearing policy is to make possible ``trivial''
-refinements of an assumption, without changing its name in the main
-branch of the reasoning.
-
-The \ssrC{have} modifier can follow the \ssrC{suff} tactic.
-For example, given a goal \ssrC{G} the tactic
-\ssrC{suff have H : P} results in the following two goals:
-\begin{lstlisting}
-  H : P |- G
-  |- (P -> G) -> G
-\end{lstlisting}
-Note that, in contrast with \ssrC{have suff}, the name \ssrC{H} has been introduced
-in the first goal.
-
-Another useful construct is reduction,
-showing that a particular case is in fact general enough to prove
-a general property. This kind of reasoning step usually starts with:
-``Without loss of generality, we can suppose that \dots''.
-Formally, this corresponds to the proof of a goal \ssrC{G} by introducing
-a cut \ssrN{wlog\_statement} \ssrC{-> G}. Hence the user shall provide a
-proof for both \ssrC{(}\ssrN{wlog\_statement} \ssrC{-> G) -> G} and
-\ssrN{wlog\_statement} \ssrC{-> G}. However, such cuts are usually rather
-painful to perform by hand, because the statement
-\ssrN{wlog\_statement} is tedious to write by hand, and somtimes even
-to read.
-
-\ssr{} implements this kind of reasoning step through the \ssrC{without loss}
-tactic, whose short name is \ssrC{wlog}. It offers support to describe
-the shape of the cut statements, by providing the simplifying
-hypothesis and by pointing at the elements of the initial goals which
-should be generalized. The general syntax of \ssrC{without loss} is:
-
-\begin{center}
-  \ssrC{wlog} \optional{\ssrC{suff}} \optional{\ssrN{clear-switch}} \optional{\ssrN{i-item}} \ssrC{:} \optional{\ssrN[1]{ident} $\dots$ \ssrN[n]{ident}} \ssrC{/} {\term}
-\end{center}
-
-where \ssrN[1]{ident} $\dots$ \ssrN[n]{ident} are identifiers for constants
-in the context of the goal. Open syntax is supported for {\term}.
-
-In its defective form:
-
-\begin{center}
-  \ssrC{wlog: /} {\term}.
-\end{center}
-
-on a goal \ssrC{G}, it creates two subgoals: a first one to prove the formula
-\ssrC{(}{\term} \ssrC{-> G) -> G} and a second one to prove the formula
-{\term} \ssrC{-> G}.
-
-:browse confirm wa
-If the optional list \ssrN[1]{ident} $\dots$ \ssrN[n]{ident} is present on the left
-side of \ssrC{/}, these constants are generalized in the premise
-\ssrC{(}{\term} \ssrC{-> G)} of the first subgoal. By default the body of
-local definitions  is erased. This behavior can be inhibited
-prefixing the name of the local definition with the \ssrC{@} character.
-
-In the second subgoal, the tactic:
-
-\begin{center}
-  \ssrC{move=>} \ssrN{clear-switch}\ssrC{} \ssrN{i-item}\ssrC{.}
-\end{center}
-
-is performed if at least one of these optional switches is present in
-the \ssrC{wlog} tactic.
-
-The \ssrC{wlog} tactic is specially useful when a symmetry argument
-simplifies a proof. Here is an example showing the beginning of the
-proof that quotient and reminder of natural number euclidean division
-are unique.
-\begin{lstlisting}
-  Lemma quo_rem_unicity: forall d q1 q2 r1 r2,
-     q1*d + r1 = q2*d + r2 -> r1 < d -> r2 < d -> (q1, r1) =  (q2, r2).
-  move=> d q1 q2 r1 r2.
-  wlog: q1 q2 r1 r2 / q1 <= q2.
-    by case (le_gt_dec q1 q2)=> H; last symmetry; eauto with arith.
-\end{lstlisting}
-
-The \ssrC{wlog suff} variant is simpler, since it cuts
-\ssrN{wlog\_statement} instead of \ssrN{wlog\_statement} \ssrC{-> G}. It thus
-opens the goals \ssrN{wlog\_statement} \ssrC{-> G} and \ssrN{wlog\_statement}.
-
-In its simplest form
-the \ssrC{generally have :...} tactic
-is equivalent to \ssrC{wlog suff :...} followed by \ssrC{last first}.
-When the \ssrC{have} tactic
-is used with the \ssrC{generally} (or \ssrC{gen}) modifier it accepts an
-extra identifier followed by a comma before the usual intro pattern.
-The identifier will name the new hypothesis in its more general form,
-while the intro pattern will be used to process its instance.  For example:
-\begin{lstlisting}
-  Lemma simple n (ngt0 : 0 < n ) : P n.
-  gen have ltnV, /andP[nge0 neq0] : n ngt0 / (0 <= n) && (n != 0).
-\end{lstlisting}
-The first subgoal will be
-\begin{lstlisting}
-  n : nat
-  ngt0 : 0 < n
-  ====================
-  (0 <= n) && (n != 0)
-\end{lstlisting}
-while the second one will be
-\begin{lstlisting}
-  n : nat
-  ltnV : forall n, 0 < n -> (0 <= n) && (n != 0)
-  nge0 : 0 <= n
-  neqn0 : n != 0
-  ====================
-  P n
-\end{lstlisting}
-
-\paragraph{Advanced generalization}\label{par:advancedgen}
-The complete syntax for the items on the left hand side of the \ssrC{/}
-separator is the following one:
-\begin{center}
-\ssrN{clear-switch} {\optsep} \optional{\ssrC{@}} \ssrN{ident} {\optsep} \ssrC{(}\optional{\ssrC{@}}\ssrN{ident} \ssrC{:=} \ssrN{c-pattern}\ssrC{)}
-\end{center}
-Clear operations are intertwined with generalization operations. This
-helps in particular avoiding dependency issues while generalizing some facts.
-
-\noindent
-If an \ssrN{ident} is prefixed with the \ssrC{@} prefix mark, then a
-let-in redex is created, which keeps track if its body (if any). The
-syntax \ssrC{(}\ssrN{ident}\ssrC{:=}\ssrN{c-pattern}\ssrC{)} allows to
-generalize an arbitrary term using a given name.  Note that its simplest
-form \ssrC{(x := y)} is just a renaming of \ssrC{y} into \ssrC{x}. In
-particular, this can be useful in order to simulate the generalization
-of a section variable, otherwise not allowed. Indeed renaming does not
-require the original variable to be cleared.
-
-
-\noindent
-The syntax \ssrC{(@x := y)} generates a let-in abstraction but with the following
-caveat: \ssrC{x} will not bind \ssrC{y}, but its body, whenever \ssrC{y} can be
-unfolded. This cover the case of both local and global definitions, as
-illustrated in the following example:
-
-\begin{lstlisting}
-Section Test.
-Variable x : nat.
-Definition addx z := z + x.
-Lemma test : x <= addx x.
-wlog H : (y := x) (@twoy := addx x) / twoy = 2 * y.
-\end{lstlisting}
-\noindent
-The first subgoal is:
-\begin{lstlisting}
-   (forall y : nat, let twoy := y + y in twoy = 2 * y -> y <= twoy) ->
-   x <= addx x
-\end{lstlisting}
-\noindent
-To avoid unfolding the term captured by the pattern \ssrC{add x} one
-can use the pattern \ssrC{id (addx x)}, that would produce the following first
-subgoal:
-\begin{lstlisting}
-   (forall y : nat, let twoy := addx y in twoy = 2 * y -> y <= twoy) ->
-   x <= addx x
-\end{lstlisting}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Rewriting}\label{sec:rw}
-\idx{rewrite \dots{}}
-
-The generalized use of reflection implies that most of the
-intermediate results handled are properties of effectively computable
-functions. The most efficient mean of establishing such results are
-computation and simplification of expressions involving such
-functions, i.e., rewriting. \ssr{} therefore includes an extended
-\ssrC{rewrite} tactic, that unifies and combines most of the rewriting
-functionalities.
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{An extended \ssrC{rewrite} tactic}\label{ssec:extrw}
-The main features of the \ssrC{rewrite} tactic are:
-\begin{itemize}
-\item  It can perform an entire series of such operations in any
-  subset of the goal and/or context;
-\item It allows to perform rewriting,
-  simplifications, folding/unfolding of definitions, closing of goals;
-\item Several rewriting operations can be chained in a single tactic;
-\item Control over the occurrence at which rewriting is to be performed is
-  significantly enhanced.
-\end{itemize}
-
-
-The general form of an \ssr{} rewrite tactic is:
-
-\begin{center}
-   \ssrC{rewrite} \ssrN{rstep}$^+$\ssrC{.}
-\end{center}
-
-The combination of a rewrite tactic with the \ssrC{in} tactical (see
-section \ref{ssec:loc}) performs rewriting in both the context and the
-goal.
-
-A rewrite step \ssrN{rstep} has the general form:
-
-\begin{center}
-  \optional{\ssrN{r-prefix}}\ssrN{r-item}
-\end{center}
-
-where:
-
-\begin{longtable}{rcl}
-\ssrN{r-prefix} & ::= &
-	\optional{\ssrC{-}} \optional{\ssrN{mult}} \optional{\ssrN{occ-switch} {\optsep} \ssrN{clear-switch}} \optional{\ssrC{[}\ssrN{r-pattern}\ssrC{]}}\\
-\ssrN{r-pattern} & ::= &
-{\term} {\optsep} \ssrC{in} \optional{\ssrN{ident} \ssrC{in}} {\term} {\optsep} \optional{{\term} \ssrC{in} {\optsep} {\term} \ssrC{as} } \ssrN{ident} \ssrC{in} {\term}\\
-\ssrN{r-item} & ::= &
-\optional{\ssrC{/}}{\term} {\optsep} \ssrN{s-item} \\
-\end{longtable}
-
-
-% \begin{eqnarray*}
-% \ssrN{r-prefix} & ::= &
-% [\ssrC{-}]\ [\ssrN{mult}][\ssrN{occ-switch} | \ssrN{cl-item}][{\term}]\\
-% \ssrN{r-item} & ::= &
-% [\ssrC{-}]{\term}\ |\ [\ssrC{-}]\ssrC{[}\ssrN[1]{term}\ssrC{]}\ssrC{/(}\ssrN[2]{term}\ssrC{)} \ |\
-% \ssrN{simpl switch} \ |\  \\
-% && \ssrN{eq-term} \ |\  \ssrC{(} \ssrN[1]{eq-term}\ssrC{,}\dots
-% \ssrC{,}\ssrN[n]{eq-term} \ssrC{)} \ |\  \ssrC{(_ :}\ssrN{eq-term} \ssrC{)}
-% \end{eqnarray*}
-
-
-An \ssrN{r-prefix} contains annotations to qualify where and how   the
-rewrite operation should be performed:
-\begin{itemize}
-\item The optional initial \ssrC{-} indicates the direction of the rewriting
-  of \ssrN{r-item}: if present the direction is right-to-left and it is
-  left-to-right otherwise.
-\item The multiplier \ssrN{mult} (see section \ref{ssec:iter})
-  specifies if and how the rewrite operation should be repeated.
-\item A rewrite operation matches the occurrences of a \emph{rewrite
-  pattern}, and replaces these occurrences by an other term, according
-  to the given \ssrN{r-item}.
-  The optional \emph{redex switch} $\ssrC{[}\ssrN{r-pattern}\ssrC{]}$, which
-  should always be surrounded by brackets, gives explicitly this
-  rewrite pattern. In its simplest form, it is a regular term.
-  If no explicit redex switch
-  is present the rewrite pattern to be matched is inferred from the
-  \ssrN{r-item}.
-\item This optional {\term}, or
-  the \ssrN{r-item}, may be preceded by an
-  occurrence switch (see section \ref{ssec:select}) or a clear item
-  (see section \ref{ssec:discharge}), these two possibilities being
-  exclusive. An occurrence switch selects the occurrences of the
-  rewrite pattern which should be affected by the rewrite operation.
-\end{itemize}
-
-
-An \ssrN{r-item} can be:
-
-
-\begin{itemize}
-\item A \emph{simplification r-item}, represented by a
-  \ssrN{s-item} (see section \ref{ssec:intro}).
-% In some cases, \ssrN{r-prefix}es are not supported.
-  Simplification operations are
-  intertwined with the possible other rewrite operations specified by
-  the list of r-items.
-\item A \emph{folding/unfolding r-item}. The tactic:
-
-  \ssrC{rewrite /}{\term}
-
-unfolds the head constant of \textit{term} in every occurrence of the
-first matching of \textit{term} in the goal. In particular, if
-\ssrC{my_def} is a (local or global) defined constant, the tactic:
-\begin{lstlisting}
-  rewrite /my_def.
-\end{lstlisting}
-is analogous to:
-\begin{lstlisting}
-  unfold my_def.
-\end{lstlisting}
-Conversely:
-\begin{lstlisting}
-  rewrite -/my_def.
-\end{lstlisting}
-is equivalent to:
-\begin{lstlisting}
-  fold my_def.
-\end{lstlisting}
-%\emph{Warning} The combination of redex switch with unfold
-%\ssrN{r-item} is not yet implemented.
-
-When an unfold r-item is combined with a redex pattern, a conversion
-operation is performed. A tactic of the form:
-
-\begin{center}
-  \ssrC{rewrite -[}\ssrN[1]{term}\ssrC{]/}\ssrN[2]{term}\ssrC{.}
-\end{center}
-
-is equivalent to:
-
-
-\begin{center}
-  \ssrC{change} \ssrN[1]{term} \ssrC{with} \ssrN[2]{term}\ssrC{.}
-\end{center}
-
-
-If \ssrN[2]{term} is a single constant and \ssrN[1]{term} head symbol
-is not \ssrN[2]{term}, then the head symbol of \ssrN[1]{term} is
-repeatedly unfolded until \ssrN[2]{term} appears.
-
-\begin{lstlisting}
-  Definition double x := x + x.
-  Definition ddouble x := double (double x).
-  Lemma ex1 x : ddouble x = 4 * x.
-  rewrite [ddouble _]/double.
-\end{lstlisting}
-
-The resulting goal is:
-
-\begin{lstlisting}
-  double x + double x = 4 * x
-\end{lstlisting}
-
-\emph{Warning} The \ssr{} terms containing holes are \emph{not}
-typed as abstractions in this context. Hence the following script:
-\begin{lstlisting}
-  Definition f := fun x y => x + y.
-  Goal forall x y, x +  y = f y x.
-  move=> x y.
-  rewrite -[f y]/(y + _).
-\end{lstlisting}
-raises the error message
-\begin{verbatim}
-   User error: fold pattern (y + _) does not match redex (f y)
-\end{verbatim}
-but the script obtained by replacing the last line with:
-\begin{lstlisting}
-  rewrite -[f y x]/(y + _).
-\end{lstlisting}
-is valid.
-
-
-\item A term, which can be:
-  \begin{itemize}
-    \item A term whose type has the form:
-      $$\ssrC{forall}\ (x_1\ :\ A_1)\dots(x_n\ :\ A_n),\ eq\ term_1\ term_2$$
-      where $eq$ is the Leibniz equality or a registered setoid
-      equality. %In the case of setoid relations, the only supported
-      %r-prefix is the directional \ssrC{-}.
-    \item A list of terms $(t_1,\dots,t_n)$, each $t_i$ having a type of the
-      form: $$\ssrC{forall}\ (x_1\ :\ A_1)\dots(x_n\ :\ A_n),\ eq\ term_1\ term_2$$ where
-      $eq$ is the Leibniz equality or a registered setoid
-      equality. The tactic:
-
-        \centerline{\ssrC{rewrite} \ssrN{r-prefix}\ssrC{(}$t_1$\ssrC{,}$\dots$\ssrC{,}$t_n$\ssrC{).}}
-
-      is equivalent to:
-
-  \centerline{\ssrC{do [rewrite} \ssrN{r-prefix} $t_1$ \ssrC{|} $\dots$ \ssrC| rewrite} \ssrN{r-prefix} $t_n$\ssrC{].}
-
-    \item An anonymous rewrite lemma
-      \ssrC{(_ :} {\term}), where \textit{term} has again the form:
-      $$\ssrC{forall}\ (x_1\ :\ A_1)\dots(x_n\ :\ A_n),\ eq\ term_1\ term_2$$
-      The tactic:
-
-        \centerline{\ssrC{rewrite (_ :} {\term}\ssrC{)}}
-
-      is in fact synonym of:
-
-        \centerline{\ssrC{cutrewrite (}{\term}\ssrC{).}}
-
-
-  \end{itemize}
-
-\end{itemize}
-
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Remarks and examples}\label{ssec:rwex}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Rewrite redex selection}
-The general strategy of \ssr{}
-is to grasp as many redexes as possible and to let the user select the
-ones to be rewritten thanks to the improved syntax for the control of
-rewriting.
-
-This may be a source of incompatibilities between the two \ssrC{rewrite}
-tactics.
-
-In a rewrite tactic of the form:
-
-  \ssrC{rewrite} \ssrN{occ-switch}\ssrC{[}\ssrN[1]{term}\ssrC{]}\ssrN[2]{term}.
-
-\ssrN[1]{term} is the explicit rewrite redex and
-\ssrN[2]{term} is the
-rewrite rule. This execution of this tactic unfolds as follows:
-
-\begin{itemize}
-\item First \ssrN[1]{term} and \ssrN[2]{term} are $\beta\iota$ normalized. Then
-  \ssrN[2]{term} is put in head normal form if the Leibniz equality
-  constructor \ssrC{eq} is not the head symbol. This may involve $\zeta$
-  reductions.
-\item Then, the matching algorithm (see section \ref{ssec:set})
-  determines the first subterm of the goal matching the rewrite pattern.
-  The rewrite pattern is
-  given by \ssrN[1]{term}, if an explicit redex pattern switch is provided, or by
-  the type of \ssrN[2]{term} otherwise. However, matching skips over
-  matches that would lead to trivial rewrites. All the
-  occurrences of this subterm in the goal are candidates for rewriting.
-\item Then only the occurrences coded by \ssrN{occ-switch} (see again
-  section \ref{ssec:set}) are finally selected for rewriting.
-\item The left hand side of $\ssrN[2]{term}$ is unified with the subterm found
-  by the matching algorithm, and if this succeeds, all the selected
-  occurrences in the goal are replaced by the right hand side of
-  $\ssrN[2]{term}$.
-\item Finally the goal is $\beta\iota$ normalized.
-\end{itemize}
-
-In the case $\ssrN[2]{term}$ is a list of terms, the first top-down (in
-the goal) left-to-right (in the list) matching rule gets selected.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Chained rewrite steps}
-
-
-The possibility to chain rewrite operations in a single tactic makes
-scripts more compact and gathers in a single command line a bunch
-of surgical
-operations  which would be described by a one sentence in a pen and
-paper proof.
-
-Performing rewrite and simplification operations in a single tactic
-enhances significantly the concision of scripts. For instance the
-tactic:
-\begin{lstlisting}
-  rewrite /my_def {2}[f _]/= my_eq //=.
-\end{lstlisting}
-unfolds \ssrC{my_def} in the goal, simplifies the second occurrence of the
-first subterm matching pattern \ssrC{[f _]}, rewrites \ssrC{my_eq},
-simplifies the whole goal and closes trivial goals.
-
-Here are some concrete examples of chained rewrite operations, in the
-proof of basic results on natural numbers arithmetic:
-
-\begin{lstlisting}
-  Lemma |*addnS*| : forall m n, m + n.+1 = (m + n).+1.
-  Proof. by move=> m n; elim: m. Qed.
-
-  Lemma |*addSnnS*| : forall m n, m.+1 + n = m + n.+1.
-  Proof. move=> *; rewrite addnS; apply addSn. Qed.
-
-  Lemma |*addnCA*| : forall m n p, m + (n + p) = n + (m + p).
-  Proof. by move=> m n; elim: m => [|m Hrec] p; rewrite ?addSnnS -?addnS. Qed.
-
-  Lemma |*addnC*| : forall m n, m + n = n + m.
-  Proof. by move=> m n; rewrite -{1}[n]addn0 addnCA addn0. Qed.
-\end{lstlisting}
-
-Note the use of the \ssrC{?} switch for parallel rewrite operations in
-the proof of \ssrC{|*addnCA*|}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Explicit redex switches are matched first}
-If an \ssrN{r-prefix} involves a \emph{redex switch}, the first step is to
-find a subterm matching this redex pattern, independently from the left hand
-side \ssrC{t1} of the equality the user wants to rewrite.
-
-For instance, if \ssrL-H : forall t u, t + u = u + t- is in the context of a
-goal \ssrL-x + y = y + x-, the tactic:
-\begin{lstlisting}
-  rewrite [y + _]H.
-\end{lstlisting}
-transforms the goal into \ssrL-x + y = x + y-.
-
-Note that if this first pattern matching is not compatible with the
-\emph{r-item}, the rewrite fails, even if the goal contains a correct
-redex matching both the redex switch and the left hand side of the
-equality. For instance, if \ssrL-H : forall t u, t + u * 0 = t- is
-in the context of a goal \ssrL-x + y * 4 + 2 * 0 = x + 2 * 0-, then tactic:
-\begin{lstlisting}
-  rewrite [x + _]H.
-\end{lstlisting}
-raises the error message:
-\begin{verbatim}
-  User error: rewrite rule H doesn't match redex (x + y * 4)
-\end{verbatim}
-while the tactic:
-\begin{lstlisting}
-  rewrite (H _ 2).
-\end{lstlisting}
-transforms the goal into \ssrL-x + y * 4 = x + 2 * 0-.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Occurrence switches and redex switches}
-The tactic:
-\begin{lstlisting}
-  rewrite {2}[_ + y + 0](_: forall z, z + 0 = z).
-\end{lstlisting}
-transforms the goal:
-\begin{lstlisting}
-  x + y + 0 = x + y + y + 0 + 0 + (x + y + 0)
-\end{lstlisting}
-into:
-\begin{lstlisting}
-  x + y + 0 = x + y + y + 0 + 0 + (x + y)
-\end{lstlisting}
-and generates a second subgoal:
-\begin{lstlisting}
-  forall z : nat, z + 0 = z
-\end{lstlisting}
-The second subgoal is generated by the use of an anonymous lemma in
-the rewrite tactic. The effect of the tactic on the initial goal is to
-rewrite this lemma at the second occurrence of the first matching
-\ssrL-x + y + 0- of the explicit rewrite redex \ssrL-_ + y + 0-.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Occurrence selection and repetition}
-Occurrence selection has priority over repetition switches. This means
-the repetition of a rewrite tactic specified by a multiplier
-will perform matching each time an elementary rewrite operation is
-performed. Repeated rewrite tactics apply to every subgoal generated
-by the previous tactic, including the previous instances of the
-repetition. For example:
-\begin{lstlisting}
-  Goal forall x y z : nat,  x + 1 = x + y + 1.
-  move=> x y z.
-\end{lstlisting}
-creates a goal \ssrC{ x + 1 = x + y + 1}, which is turned into \ssrC{z = z}
-by the additional tactic:
-\begin{lstlisting}
-  rewrite 2!(_ : _ + 1 = z).
-\end{lstlisting}
-In fact, this last tactic generates \emph{three} subgoals,
-respectively
-\ssrC{ x + y + 1 = z}, \ssrC{ z = z} and \ssrC{x + 1 = z}. Indeed, the second
-rewrite operation specified with the \ssrC{2!} multiplier applies to
-the two subgoals generated by the first rewrite.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Multi-rule rewriting}
-The \ssrC{rewrite} tactic can be provided a \emph{tuple} of rewrite rules,
-or more generally a tree of such rules, since this tuple can feature
-arbitrary inner parentheses. We call \emph{multirule} such a
-generalized rewrite rule. This feature is of special interest when it
-is combined with  multiplier switches, which makes the \ssrC{rewrite}
-tactic iterates the rewrite operations prescribed by the rules on the
-current goal. For instance, let us define two triples \ssrC{multi1} and
-\ssrC{multi2} as:
-\begin{lstlisting}
-  Variables (a b c : nat).
-
-  Hypothesis eqab : a = b.
-
-  Hypothesis eqac : a = c.
-\end{lstlisting}
-
-Executing the tactic:
-\begin{lstlisting}
-  rewrite (eqab, eqac)
-\end{lstlisting}
-on the goal:
-\begin{lstlisting}
-  =========
-  a = a
-\end{lstlisting}
-turns it into \ssrC{b = b}, as rule \ssrC{eqab} is the first to apply among
-the ones gathered in the tuple passed to the \ssrC{rewrite}
-tactic. This multirule \ssrC{(eqab, eqac)} is actually a \Coq{} term and we
-can name it with a definition:
-\begin{lstlisting}
-Definition |*multi1*| := (eqab, eqac).
-\end{lstlisting}
-In this case, the tactic \ssrC{rewrite multi1} is a synonym for
-\ssrC{(eqab, eqac)}. More precisely, a multirule rewrites
-the first subterm to which one of the rules applies in a left-to-right
-traversal of the goal, with the first rule from the multirule tree in
-left-to-right order. Matching is performed according to the algorithm
-described in Section~\ref{ssec:set}, but literal matches have
-priority. For instance if we add a definition and a new multirule to
-our context:
-
-\begin{lstlisting}
-  Definition |*d*| := a.
-
-  Hypotheses eqd0 : d = 0.
-
-  Definition |*multi2*| := (eqab, eqd0).
-\end{lstlisting}
-then executing the tactic:
-\begin{lstlisting}
-   rewrite multi2.
-\end{lstlisting}
-on the goal:
-\begin{lstlisting}
-  =========
-  d = b
-\end{lstlisting}
-turns it into \ssrC{0 = b}, as rule \ssrC{eqd0} applies without unfolding
-the definition of \ssrC{d}.  For repeated rewrites the selection process
-is repeated anew. For instance, if we define:
-
-\begin{lstlisting}
-   Hypothesis eq_adda_b : forall x, x + a = b.
-
-   Hypothesis eq_adda_c : forall x, x + a = c.
-
-   Hypothesis eqb0 : b = 0.
-
-   Definition |*multi3*| := (eq_adda_b, eq_adda_c, eqb0).
-\end{lstlisting}
-then executing the tactic:
-\begin{lstlisting}
-  rewrite 2!multi3.
-\end{lstlisting}
-on the goal:
-\begin{lstlisting}
-  =========
-  1 + a = 12 + a
-\end{lstlisting}
-turns it into \ssrC{0 = 12 + a}: it uses \ssrC{eq_adda_b} then \ssrC{eqb0} on the
-left-hand side only. Now executing the tactic \ssrC{rewrite !multi3}
-turns the same goal into \ssrC{0 = 0}.
-
-The grouping of rules inside a multirule does not affect the selection
-strategy but can make it easier to include one rule set in another or
-to (universally) quantify over the parameters of a subset of rules (as
-there is special code that will omit unnecessary quantifiers for rules that
-can be syntactically extracted). It is also possible to
-reverse the direction of a rule subset, using a special dedicated syntax:
-the tactic \ssrC{rewrite (=~ multi1)} is equivalent to
-\ssrC{rewrite multi1_rev} with:
-\begin{lstlisting}
-  Hypothesis eqba : b = a.
-
-  Hypothesis eqca : c = a.
-
-  Definition |*multi1_rev*| := (eqba, eqca).
-\end{lstlisting}
-except that the constants \ssrC{eqba, eqab, mult1_rev} have not been created.
-
-Rewriting with multirules is useful to implement simplification or
-transformation procedures, to be applied on terms of small to medium
-size. For instance, the library \ssrL{ssrnat} --- available in the
-external math-comp library --- provides two implementations for
-arithmetic operations on natural numbers: an elementary one and a tail
-recursive version, less inefficient but also less convenient for
-reasoning purposes. The library also provides one lemma per such
-operation, stating that both versions return the same values when
-applied to the same arguments:
-
-\begin{lstlisting}
-  Lemma |*addE*| : add =2 addn.
-  Lemma |*doubleE*| : double =1 doublen.
-  Lemma |*add_mulE*| n m s : add_mul n m s = addn (muln n m) s.
-  Lemma |*mulE*| : mul =2 muln.
-  Lemma |*mul_expE*| m n p : mul_exp m n p = muln (expn m n) p.
-  Lemma |*expE*| : exp =2 expn.
-  Lemma |*oddE*| : odd =1 oddn.
-\end{lstlisting}
-
-The operation on the left hand side of each lemma is the efficient
-version, and the corresponding naive implementation is on the right
-hand side. In order to reason conveniently on expressions involving
-the efficient operations, we gather all these rules in the
-definition \ssrC{|*trecE*|}:
-\begin{lstlisting}
- Definition |*trecE*| := (addE, (doubleE, oddE), (mulE, add_mulE, (expE, mul_expE))).
-\end{lstlisting}
-The tactic:
-\begin{lstlisting}
-  rewrite !trecE.
-\end{lstlisting}
-restores the naive versions of each operation in a goal involving the
-efficient ones, e.g. for the purpose of a correctness proof.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Wildcards vs abstractions}
-     The \ssrC{rewrite} tactic supports r-items containing holes. For example
-      in the tactic $(1)$:
-\begin{lstlisting}
-  rewrite (_ : _ * 0 = 0).
-\end{lstlisting}
-      the term \ssrC{_ * 0 = 0} is interpreted as \ssrC{forall n : nat, n * 0 = 0}.
-      Anyway this tactic is \emph{not} equivalent to the tactic $(2)$:
-\begin{lstlisting}
-  rewrite (_ : forall x, x * 0 = 0).
-\end{lstlisting}
-      The tactic $(1)$ transforms the goal
-      \ssrL-(y * 0) + y * (z * 0) = 0- into \ssrC{y * (z * 0) = 0}
-      and generates a new subgoal to prove the statement \ssrC{y * 0 = 0},
-      which is the \emph{instance} of the\\ \ssrC{forall x, x * 0 = 0}
-      rewrite rule that
-      has been used to perform the rewriting. On the other hand, tactic
-      $(2)$ performs the same rewriting on the current goal but generates a
-      subgoal to prove \ssrC{forall x, x * 0 = 0}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{When \ssr{} \ssrC{rewrite} fails on standard \Coq{} licit rewrite}
-In a few cases, the \ssr{} \ssrC{rewrite} tactic fails
-rewriting some redexes which standard \Coq{} successfully rewrites.
-There are two main cases:
-
-\begin{itemize}
-\item \ssr{} never accepts to rewrite indeterminate patterns like:
-\begin{lstlisting}
-  Lemma |*foo*| : forall x : unit, x = tt.
-\end{lstlisting}
-\ssr{} will however accept the $\eta\zeta$ expansion of this rule:
-\begin{lstlisting}
-  Lemma |*fubar*| : forall x : unit, (let u := x in u) = tt.
-\end{lstlisting}
-\item In standard \Coq{}, suppose that we work in the following context:
-\begin{lstlisting}
-  Variable g : nat -> nat.
-  Definition |*f*| := g.
-\end{lstlisting}
-then rewriting \ssrC{H : forall x, f x = 0} in the goal
-\ssrC{g 3 + g 3 = g 6} succeeds
-and transforms the goal into \ssrC{0 + 0 = g 6}.
-
-This rewriting is not possible in \ssr{} because there is no
-occurrence of the head symbol \ssrC{f} of the rewrite rule in the
-goal. Rewriting with \ssrC{H} first requires unfolding the occurrences of
-\ssrC{f} where the substitution is to be performed (here there is a single
-such occurrence), using tactic \ssrC{rewrite /f} (for a global
-replacement of \ssrC{f} by \ssrC{g}) or \ssrC{rewrite $\ \ssrN{pattern}$/f}, for a
-finer selection.
-\end{itemize}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Existential metavariables and rewriting}
-\label{ssec:rewcaveats}
-The \ssrC{rewrite} tactic will not instantiate existing existential
-metavariables when matching a redex pattern.
-
-If a rewrite rule generates a goal
-with new existential metavariables, these will be generalized as for \ssrC{apply}
-(see page~\pageref{sssec:apply}) and corresponding new goals will be generated.
-For example, consider the following script:
-
-\begin{lstlisting}
-  Lemma |*ex3*| (x : 'I_2) y (le_1 : y < 1) (E : val x = y) : Some x = insub y.
-  rewrite insubT ?(leq_trans le_1)// => le_2.
-\end{lstlisting}
-
-Since \ssrC{insubT} has the following type:
-
-\begin{lstlisting}
-  forall T P (sT : subType P) (x : T) (Px : P x), insub x = Some (Sub x Px)
-\end{lstlisting}
-
-and since the implicit argument corresponding to the \ssrC{Px} abstraction is not
-supplied by the user, the resulting goal should be \ssrC{Some x = Some (Sub y
-$\;\;?_{Px}$)}. Instead, \ssr{} \ssrC{rewrite} tactic generates the two following
-goals:
-\begin{lstlisting}
-  y < 2
-  forall Hyp0 : y < 2, Some x = Some (Sub y Hyp0)
-\end{lstlisting}
-The script closes the former with \ssrC{?(leq_trans le_1)//}, then it introduces
-the new generalization naming it \ssrC{le_2}.
-
-\begin{lstlisting}
-  x : 'I_2
-  y : nat
-  le_1 : y < 1
-  E : val x = y
-  le_2 : y < 2
-  ============================
-   Some x = Some (Sub y le_2)
-\end{lstlisting}
-
-As a temporary limitation, this behavior is available only if the rewriting
-rule is stated using Leibniz equality (as opposed to setoid relations).
-It will be extended to other rewriting relations in the future.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Locking, unlocking} \label{ssec:lock}
-
-As program proofs tend to generate large goals, it is important to be
-able to control the partial evaluation performed by the simplification
-operations that are performed by the tactics. These evaluations can
-for example come from a \ssrC{/=} simplification switch, or from rewrite steps
-which may expand large terms while performing conversion. We definitely
-want to avoid repeating large subterms of the goal in
-the proof script. We do this by
-``clamping down'' selected function symbols in the goal, which
-prevents them from
-being considered in simplification or rewriting steps. This clamping
-is accomplished by using the occurrence switches (see section
-\ref{sssec:occselect}) together with ``term tagging'' operations.
-
-\ssr{} provides two levels of tagging.
-
-The first one uses auxiliary definitions to introduce a provably equal
-copy of any term \ssrC{t}. However this copy is (on purpose)
-\emph{not convertible} to \ssrC{t} in the \Coq{} system\footnote{This is
-  an implementation feature: there is not such obstruction in the
-  metatheory}. The job is done by the following construction:
-
-\begin{lstlisting}
-  Lemma |*master_key*| : unit. Proof. exact tt. Qed.
-  Definition |*locked*| A := let: tt := master_key in fun x : A => x.
-  Lemma |*lock*| : forall A x, x = locked x :> A.
-\end{lstlisting}
-Note that the definition of \ssrC{|*master_key*|} is explicitly opaque.
-The equation \ssrC{t = locked t} given by the \ssrC{lock} lemma can be used
-for selective rewriting, blocking on the fly the reduction in the
-term \ssrC{t}.
-For example the script:
-\begin{lstlisting}
-  Require Import List.
-  Variable A : Type.
-
-  Fixpoint |*my_has*| (p : A -> bool)(l : list A){struct l} : bool:=
-    match l with
-      |nil => false
-      |cons x l => p x || (my_has p l)
-    end.
-
-  Goal forall a x y l, a x = true -> my_has a ( x :: y :: l) = true.
-  move=> a x y l Hax.
-\end{lstlisting}
-where \ssrL{||} denotes the boolean disjunction, results in a goal
-\ssrC{my_has a ( x :: y :: l) = true}. The tactic:
-\begin{lstlisting}
-  rewrite {2}[cons]lock /= -lock.
-\end{lstlisting}
-turns it into \ssrC{a x || my_has a (y :: l) = true}.
-Let us now start by reducing the initial goal without blocking reduction.
-The script:
-\begin{lstlisting}
-  Goal forall a x y l, a x = true -> my_has a ( x :: y :: l) = true.
-  move=> a x y l Hax /=.
-\end{lstlisting}
-creates a goal \ssrC{(a x) || (a y) || (my_has a l) = true}. Now the
-tactic:
-\begin{lstlisting}
-  rewrite {1}[orb]lock orbC -lock.
-\end{lstlisting}
-where \ssrC{orbC} states the commutativity of \ssrC{orb}, changes the
-goal into\\ \ssrC{(a x) || (my_has a l) || (a y) = true}: only the
-arguments of the second disjunction where permuted.
-
-
-It is sometimes desirable to globally prevent a definition from being
-expanded by simplification; this is done by adding \ssrC{locked} in the
-definition.
-
-For instance, the function \ssrC{|*fgraph_of_fun*|} maps a function whose
-domain and codomain are finite types to a concrete representation of
-its (finite) graph. Whatever implementation of this transformation we
-may use, we want it to be hidden to simplifications and tactics, to
-avoid the collapse of the graph object:
-\begin{lstlisting}
-  Definition |*fgraph_of_fun*| :=
-    locked
-    (fun (d1 :finType) (d2 :eqType) (f : d1 -> d2) => Fgraph (size_maps f _)).
-\end{lstlisting}
-
-We provide a special tactic \ssrC{unlock} for unfolding such definitions
-while removing ``locks'', e.g., the tactic:
-
-  \ssrC{unlock} \ssrN{occ-switch}\ssrC{fgraph_of_fun}.
-
-replaces the occurrence(s) of \ssrC{fgraph_of_fun} coded by the \ssrN{occ-switch}
-with \ssrC{(Fgraph (size_maps _ _))} in the goal.
-
-We found that it was usually preferable to prevent the expansion of
-some functions by the partial evaluation switch ``/='', unless
-this allowed the evaluation of a condition. This is possible thanks to
-an other mechanism of term tagging, resting on the following
-\emph{Notation}:
-\begin{lstlisting}
-  Notation "'nosimpl' t" := (let: tt := tt in t).
-\end{lstlisting}
-
-The term \ssrC{(nosimpl t)} simplifies to t \emph{except} in a
-definition. More precisely,
-given:
-\begin{lstlisting}
-  Definition |*foo*| := (nosimpl bar).
-\end{lstlisting}
-the term \ssrC{foo (or (foo t'))} will \emph{not} be expanded by the
-\emph{simpl} tactic unless it is in a forcing context (e.g., in
-\ssrC{match foo t' with $\dots$ end}, \ssrC{foo t'} will be reduced if this allows
-\ssrC{match} to be reduced). Note that \ssrC{nosimpl bar} is simply notation
-for a term that reduces to \ssrC{bar}; hence \ssrC{unfold foo} will replace
-    \ssrC{foo} by \ssrC{bar}, and \ssrC{fold foo} will replace \ssrC{bar} by
-    \ssrC{foo}.
-
-\emph{Warning} The \ssrC{nosimpl} trick only works if no reduction is
-apparent in \ssrC{t}; in particular, the declaration:
-\begin{lstlisting}
-  Definition |*foo*| x := nosimpl (bar x).
-\end{lstlisting}
-will usually not work. Anyway, the common practice is to tag only the
-function, and to use the following definition, which blocks the
-reduction as expected:
-\begin{lstlisting}
-  Definition |*foo*| x := nosimpl bar x.
-\end{lstlisting}
-
-
-A standard example making this technique shine is the case of
-arithmetic operations. We define for instance:
-\begin{lstlisting}
-  Definition |*addn*| := nosimpl plus.
-\end{lstlisting}
-The operation \ssrC{addn} behaves exactly like plus, except that
-\ssrC{(addn (S n) m)} will not
-simplify spontaneously to \ssrC{(S (addn n m))} (the two terms, however, are
-inter-convertible). In addition, the unfolding step:
-\begin{lstlisting}
-rewrite /addn
-\end{lstlisting}
-will replace \ssrC{addn} directly with \ssrC{plus}, so the \ssrC{nosimpl} form
-is essentially invisible.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Congruence}\label{ssec:congr}
-
-Because of the way matching interferes with type families parameters,
-the tactic:
-\begin{lstlisting}
-  apply: my_congr_property.
-\end{lstlisting}
-will generally fail to perform congruence simplification, even on
-rather simple cases. We therefore provide a
-more robust alternative in which the function is supplied:
-$$\ssrC{congr}\ [\ssrN{int}]\ {\term}$$
-
-This tactic:
-\begin{itemize}
-\item checks that the goal is a Leibniz equality
-\item matches both sides of this equality with ``{\term} applied to
-  some arguments'', inferring the right number of arguments from the goal
-  and the type of {\term}. This may
-  expand some definitions or fixpoints.
-\item generates the subgoals corresponding to pairwise equalities of
-  the arguments present in the goal.
-\end{itemize}
-
-The goal can be a non dependent product \ssrC{P -> Q}.
-In that case, the system asserts the equation \ssrC{P = Q}, uses it to solve
-the goal, and calls the \ssrC{congr} tactic on the remaining goal
-\ssrC{P = Q}. This can be useful for instance to perform a transitivity
-step, like in the following situation:
-\begin{lstlisting}
-  x, y, z : nat
-  ===============
-  x = y -> x = z
-\end{lstlisting}
-the tactic \ssrC{congr (_ = _)} turns this goal into:
-
-\begin{lstlisting}
-  x, y, z : nat
-  ===============
-  y = z
-\end{lstlisting}
-which can also be obtained starting from:
-\begin{lstlisting}
-  x, y, z : nat
-  h : x = y
-  ===============
-  x = z
-\end{lstlisting}
-and using the tactic \ssrC{congr (_ = _): h}.
-
-The optional \ssrN{int} forces the number of arguments for which the
-tactic should generate equality proof obligations.
-
-This tactic supports equalities between applications with dependent
-arguments. Yet dependent arguments should have exactly the same
-parameters on both sides, and these parameters should appear as first
-arguments.
-
-The following script:
-\begin{lstlisting}
-  Definition f n := match n with 0 => plus | S _ => mult end.
-  Definition g (n m : nat) := plus.
-
-  Goal forall x y, f 0 x y = g 1 1 x y.
-  by move=> x y; congr plus.
-  Qed.
-\end{lstlisting}
-shows that the \ssrC{congr} tactic matches \ssrC{plus} with \ssrC{f 0} on the
-left hand side and \ssrC{g 1 1} on the right hand side, and solves the goal.
-
-The script:
-\begin{lstlisting}
-  Goal forall n m, m <= n -> S m + (S n - S m) = S n.
-  move=> n m Hnm; congr S; rewrite -/plus.
-\end{lstlisting}
-generates the subgoal \ssrC{m + (S n - S m) = n}. The tactic
-\ssrC{rewrite -/plus} folds back the expansion of \ssrC{plus} which was
-necessary for matching both sides of the equality with an application
-of \ssrC{S}.
-
-Like most \ssr{} arguments, {\term} can contain wildcards.
-The script:
-\begin{lstlisting}
-  Goal forall x y, x + (y * (y + x - x)) = x * 1 + (y + 0) * y.
-  move=> x y; congr ( _ + (_ * _)).
-\end{lstlisting}
-generates three subgoals, respectively \ssrC{x = x * 1}, \ssrC{y = y + 0}
-and \ssrC{ y + x - x = y}.
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Contextual patterns}
-\label{ssec:rewp}
-
-The simple form of patterns used so far, ${\term}s$ possibly containing
-wild cards, often require an additional \ssrN{occ-switch} to be specified.
-While this may work pretty fine for small goals, the use of polymorphic
-functions and dependent types may lead to an invisible duplication of functions
-arguments. These copies usually end up in types hidden by the implicit
-arguments machinery or by user defined notations.  In these situations
-computing the right occurrence numbers is very tedious because they must be
-counted on the goal as printed after setting the \ssrC{Printing All} flag.
-Moreover the resulting script is not really informative for the reader, since
-it refers to occurrence numbers he cannot easily see.
-
-Contextual patterns mitigate these issues allowing to specify occurrences
-according to the context they occur in.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Syntax}
-
-The following table summarizes the full syntax of
-\ssrN{c-pattern} and the corresponding subterm(s) identified
-by the pattern.
-In the third column we use s.m.r. for
-``the subterms matching the redex'' specified in the second column.
-
-\begin{center}
-%\begin{tabularx}{\textwidth}{>{\arraybackslash}m{0.30\textwidth}|>{\arraybackslash}m{0.21\textwidth}|>{\arraybackslash}m{0.39\textwidth}}
-\begin{tabular}{llp{10em}}
-\ssrN{c-pattern} & redex & subterms affected \\
-\hline
-{\term} & {\term} & all occurrences of {\term}\\
-\hline
-$\ssrN{ident}\ \ssrC{in}\ {\term}$ &
-  subterm of {\term} selected by \ssrN{ident} &
-  all the subterms identified by \ssrN{ident} in all
-  the occurrences of {\term} \\
-\hline
-$\ssrN[1]{term}\ \ssrC{in}\ \ssrN{ident}\ \ssrC{in}\ \ssrN[2]{term}$ & $\ssrN[1]{term}$ &
-  in all  s.m.r. in all the subterms identified by \ssrN{ident} in all
-  the occurrences of $\ssrN[2]{term}$ \\
-\hline
-$\ssrN[1]{term}\ \ssrC{as}\ \ssrN{ident}\ \ssrC{in}\ \ssrN[2]{term}$ & $\ssrN[1]{term}$ &
-  in all the subterms identified by \ssrN{ident} in all
-  the occurrences of $\ssrN[2]{term}[\ssrN[1]{term}/\ssrN{ident}]$\\
-\hline
-%\end{tabularx}
-\end{tabular}
-\end{center}
-
-The \ssrC{rewrite} tactic supports two more patterns obtained
-prefixing the first two with \ssrC{in}. The intended meaning is that the
-pattern identifies all subterms of the specified context. The
-\ssrC{rewrite} tactic will infer a pattern for the redex looking at the
-rule used for rewriting.
-
-\begin{center}
-\begin{tabularx}{\textwidth}{>{\arraybackslash}m{0.30\textwidth}|>{\arraybackslash}m{0.21\textwidth}|>{\arraybackslash}m{0.39\textwidth}}
-\ssrN{r-pattern} & redex & subterms affected \\
-\hline
-$\ssrC{in}\ {\term}$ & inferred from rule &
-  in all  s.m.r. in all occurrences of {\term}\\
-\hline
-$\ssrC{in}\ \ssrN{ident}\ \ssrC{in}\ {\term}$ & inferred from rule &
-  in all  s.m.r. in all the subterms identified by \ssrN{ident} in all
-  the occurrences of {\term} \\
-\hline
-\end{tabularx}
-\end{center}
-
-The first \ssrN{c-pattern} is the simplest form matching any
-context but selecting a specific redex and has been described in the
-previous sections. We have seen so far that the possibility of
-selecting a redex using a term with holes is already a powerful mean of redex
-selection. Similarly, any {\term}s provided by the
-user in the more complex forms of \ssrN{c-pattern}s presented in the
-tables above can contain holes.
-
-For a quick glance at what can be expressed with the last
-\ssrN{r-pattern} consider the goal \ssrC{a = b} and the tactic
-\begin{lstlisting}
-  rewrite [in X in _ = X]rule.
-\end{lstlisting}
-It rewrites all occurrences of the left hand side of \ssrC{rule} inside
-\ssrC{b} only (\ssrC{a}, and the hidden type of the equality, are ignored).
-Note that the variant \ssrC{rewrite [X in _ = X]rule} would have
-rewritten \ssrC{b} exactly (i.e., it would only work if \ssrC{b} and the
-left hand side of \ssrC{rule} can be unified).
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Matching contextual patterns}
-
-The \ssrN{c-pattern}s and \ssrN{r-pattern}s involving
-{\term}s with holes are matched
-against the goal in order to find a closed instantiation. This
-matching proceeds as follows:
-
-\begin{center}
-\begin{tabularx}{\textwidth}{>{\arraybackslash}m{0.30\textwidth}|>{\arraybackslash}m{0.65\textwidth}}
-\ssrN{c-pattern} & instantiation order and place for $\ssrN[i]{term}$ and redex\\
-\hline
-{\term} & {\term} is matched against the goal, redex is unified with
-    the instantiation of {\term}\\
-\hline
-$\ssrN{ident}\ \ssrC{in}\ {\term}$ &
-    {\term} is matched against the goal, redex is
-    unified with the subterm of the
-    instantiation of {\term} identified by \ssrN{ident}\\
-\hline
-$\ssrN[1]{term}\ \ssrC{in}\ \ssrN{ident}\ \ssrC{in}\ \ssrN[2]{term}$ &
-    $\ssrN[2]{term}$ is matched against the goal, $\ssrN[1]{term}$ is
-    matched against the subterm of the
-    instantiation of $\ssrN[1]{term}$ identified by \ssrN{ident},
-    redex is unified with the instantiation of $\ssrN[1]{term}$\\
-\hline
-$\ssrN[1]{term}\ \ssrC{as}\ \ssrN{ident}\ \ssrC{in}\ \ssrN[2]{term}$ &
-    $\ssrN[2]{term}[\ssrN[1]{term}/\ssrN{ident}]$
-    is matched against the goal,
-    redex is unified with the instantiation of $\ssrN[1]{term}$\\
-\hline
-\end{tabularx}
-\end{center}
-
-In the following patterns, the redex is intended to be inferred from the
-rewrite rule.
-
-\begin{center}
-\begin{tabularx}{\textwidth}{>{\arraybackslash}m{0.30\textwidth}|>{\arraybackslash}m{0.65\textwidth}}
-\ssrN{r-pattern} & instantiation order and place for $\ssrN[i]{term}$ and redex\\
-\hline
-$\ssrC{in}\ \ssrN{ident}\ \ssrC{in}\ {\term}$ &
-    {\term} is matched against the goal, the redex is
-    matched against the subterm of the
-    instantiation of {\term} identified by \ssrN{ident}\\
-\hline
-$\ssrC{in}\ {\term}$ & {\term} is matched against the goal, redex is
-   matched against the instantiation of {\term}\\
-\hline
-\end{tabularx}
-\end{center}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Examples}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection{Contextual pattern in \ssrC{set} and the \ssrC{:} tactical}
-
-As already mentioned in section~\ref{ssec:set} the \ssrC{set} tactic
-takes as an argument a term in open syntax. This term is interpreted
-as the simplest for of \ssrN{c-pattern}. To void confusion in the grammar,
-open syntax is supported only for the simplest form of patterns, while
- parentheses are required around more complex patterns.
-
-\begin{lstlisting}
-set t := (X in _ = X).
-set t := (a + _ in X in _ = X).
-\end{lstlisting}
-
-Given the goal \ssrC{a + b + 1 = b + (a + 1)} the first tactic
-captures \ssrC{b + (a + 1)}, while the latter \ssrC{a + 1}.
-
-Since the user may define an infix notation for \ssrC{in} the former
-tactic may result ambiguous. The disambiguation rule implemented is
-to prefer patterns over simple terms, but to interpret a pattern with
-double parentheses as a simple term. For example
-the following tactic would capture any occurrence of the term `\ssrC{a in A}'.
-
-\begin{lstlisting}
-set t := ((a in A)).
-\end{lstlisting}
-
-Contextual pattern can also be used as arguments of the \ssrC{:} tactical.
-For example:
-\begin{lstlisting}
-elim: n (n in _ = n) (refl_equal n).
-\end{lstlisting}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection{Contextual patterns in \ssrC{rewrite}}
-As a more comprehensive example consider the following goal:
-\begin{lstlisting}
-    (x.+1 + y) + f (x.+1 + y) (z + (x + y).+1) = 0
-\end{lstlisting}
-The tactic \ssrC{rewrite [in f _ _]addSn} turns it into:
-\begin{lstlisting}
-    (x.+1 + y) + f (x + y).+1 (z + (x + y).+1) = 0
-\end{lstlisting}
-since the simplification rule \ssrC{addSn} is applied only under the \ssrC{f} symbol.
-Then we simplify also the first addition and expand \ssrC{0} into \ssrC{0+0}.
-\begin{lstlisting}
-    rewrite addSn -[X in _ = X]addn0.
-\end{lstlisting}
-obtaining:
-\begin{lstlisting}
-    (x + y).+1 + f (x + y).+1 (z + (x + y).+1) = 0 + 0
-\end{lstlisting}
-Note that the right hand side of \ssrC{addn0} is undetermined, but the
-rewrite pattern specifies the redex explicitly. The right hand side of
-\ssrC{addn0} is unified with the term identified by \ssrC{X}, \ssrC{0} here.
-
-The following pattern does not specify a redex, since it
-identifies an entire region, hence the rewrite rule has to be instantiated
-explicitly. Thus the tactic:
-\begin{lstlisting}
-    rewrite -{2}[in X in _ = X](addn0 0).
-\end{lstlisting}
-changes the goal as follows:
-\begin{lstlisting}
-    (x + y).+1 + f (x + y).+1 (z + (x + y).+1) = 0 + (0 + 0)
-\end{lstlisting}
-The following tactic is quite tricky:
-\begin{lstlisting}
-    rewrite [_.+1 in X in f _ X](addnC x.+1).
-\end{lstlisting}
-and the resulting goals is:
-\begin{lstlisting}
-    (x + y).+1 + f (x + y).+1 (z + (y + x.+1)) = 0 + (0 + 0)
-\end{lstlisting}
-The explicit redex \ssrC{_.+1} is important since its head
-constant \ssrC{S} differs from the head constant inferred from
-\ssrC{(addnC x.+1)} (that is \ssrC{addn}, denoted \ssrC{+} here).
-Moreover, the pattern \ssrC{f _ X} is important to rule out the first occurrence
-of \ssrC{(x + y).+1}. Last, only the subterms of \ssrC{f _ X} identified by \ssrC{X} are
-rewritten, thus the first argument of \ssrC{f} is skipped too.
-Also note the pattern \ssrC{_.+1} is interpreted in the context
-identified by \ssrC{X}, thus it gets instantiated to \ssrC{(y + x).+1} and
-not \ssrC{(x + y).+1}.
-
-The last rewrite pattern allows to specify exactly the shape of the term
-identified by \ssrC{X}, that is thus unified with the left hand side of the
-rewrite rule.
-\begin{lstlisting}
-    rewrite [x.+1 + y as X in f X _]addnC.
-\end{lstlisting}
-The resulting goal is:
-\begin{lstlisting}
-    (x + y).+1 + f (y + x.+1) (z + (y + x.+1)) = 0 + (0 + 0)
-\end{lstlisting}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Patterns for recurrent contexts}
-
-The user can define shortcuts for recurrent contexts corresponding to the
-\ssrN{ident} \ssrC{in} {\term} part. The notation scope identified
-with \ssrC{\%pattern} provides a special notation `\ssrC{(X in t)}' the user
-must adopt to define context shortcuts.
-
-The following example is taken from \ssrC{ssreflect.v} where the
-\ssrC{LHS} and \ssrC{RHS} shortcuts are defined.
-
-\begin{lstlisting}
-Notation RHS := (X in _ = X)%pattern.
-Notation LHS := (X in X = _)%pattern.
-\end{lstlisting}
-
-Shortcuts defined this way can be freely used in place of the
-trailing \ssrN{ident} \ssrC{in} {\term} part of any contextual
-pattern.
-Some examples follow:
-
-\begin{lstlisting}
-set rhs := RHS.
-rewrite [in RHS]rule.
-case: (a + _ in RHS).
-\end{lstlisting}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Views and reflection}\label{sec:views}
-
-The bookkeeping facilities presented in section \ref{sec:book} are
-crafted to ease simultaneous introductions and generalizations of facts and
-casing,
-naming $\dots$ operations. It also a common practice to make a stack
-operation immediately followed by an \emph{interpretation} of the fact
-being pushed,
-that is, to apply a lemma to this fact before passing it
-to a tactic for decomposition, application and so on.
-
-
-% possibly
-
-% Small scale reflection consists in using a two levels
-% approach locally when developing formal proofs. This means that a
-% fact, which may be an assumption, or the goal itself, will often be
-% \emph{interpreted} before being passed to a tactic
-% for decomposition, application and so on.
-
-\ssr{} provides a convenient, unified syntax to combine these
-interpretation operations with the proof stack operations. This
-\emph{view mechanism} relies on the combination of the \ssrC{/} view
-switch with bookkeeping tactics and tacticals.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Interpreting eliminations}
-\idx{elim/\dots{}}
-
-The view syntax combined with the \ssrC{elim} tactic specifies an
-elimination scheme to
-be used instead of the default, generated, one. Hence the \ssr{} tactic:
-\begin{lstlisting}
-  elim/V.
-\end{lstlisting}
-is a synonym for:
-\begin{lstlisting}
-  intro top; elim top using V; clear top.
-\end{lstlisting}
-where \ssrC{top} is a fresh name and \ssrC{V} any second-order lemma.
-
-Since an elimination view supports the two bookkeeping tacticals of
-discharge and introduction (see section \ref{sec:book}), the \ssr{} tactic:
-\begin{lstlisting}
-  elim/V: x => y.
-\end{lstlisting}
-is a synonym for:
-\begin{lstlisting}
-  elim x using V; clear x; intro y.
-\end{lstlisting}
-where \ssrC{x} is a variable in the context, \ssrC{y} a fresh name and \ssrC{V}
-any second order lemma; \ssr{} relaxes the syntactic restrictions of
-the \Coq{} \ssrC{elim}. The first pattern following \ssrC{:} can be a \ssrC{_}
-wildcard if the conclusion of the view \ssrC{V} specifies a pattern for
-its last argument (e.g., if \ssrC{V} is a functional induction lemma
-generated by the \ssrC{Function} command).
-
-The elimination view mechanism is compatible with the equation name
-generation (see section \ref{ssec:equations}).
-
-The following script illustrate a toy example of this feature. Let us
-define a function adding an element at the end of a list:
-\begin{lstlisting}
-  Require Import List.
-
-  Variable d : Type.
-
-  Fixpoint |*add_last*|(s : list d) (z : d) {struct s} : list d :=
-    match s with
-    | nil =>  z :: nil
-    | cons x s' => cons x (add_last s' z)
-    end.
-\end{lstlisting}
-
-One can define an alternative, reversed, induction principle on inductively
-defined \ssrC{list}s, by proving the following lemma:
-
-\begin{lstlisting}
-  Lemma |*last_ind_list*| : forall (P : list d -> Type),
-  P nil ->
-  (forall (s : list d) (x : d), P s -> P (add_last s x)) -> forall s : list d, P s.
-\end{lstlisting}
-
-Then the combination of elimination views with equation names result
-in a concise syntax for reasoning inductively using the user
-defined elimination scheme. The script:
-\begin{lstlisting}
-  Goal forall (x : d)(l : list d), l = l.
-  move=> x l.
-  elim/last_ind_list E : l=> [| u v]; last first.
-\end{lstlisting}
-generates two subgoals: the first one to prove \ssrC{nil = nil} in a
-context featuring \ssrC{E : l = nil} and the second to prove
-\ssrC{add_last u v = add_last u v}, in a context containing
-\ssrC{E : l = add_last u v}.
-
-User provided eliminators (potentially generated with the
-\ssrC{Function} \Coq{}'s command) can be combined with the type family switches
-described in section~\ref{ssec:typefam}. Consider an eliminator
-\ssrC{foo_ind} of type:
-
-  \ssrC{foo_ind : forall $\dots$, forall x : T, P p$_1$ $\dots$ p$_m$}
-
-and consider the tactic
-
-  \ssrC{elim/foo_ind: e$_1$ $\dots$ / e$_n$}
-
-The \ssrC{elim/} tactic distinguishes two cases:
-\begin{description}
-\item[truncated eliminator] when \ssrC{x} does not occur in \ssrC{P p$_1 \dots$ p$_m$}
-  and the type of \ssrC{e$_n$} unifies with \ssrC{T} and \ssrC{e$_n$} is not \ssrC{_}.
-  In that case, \ssrC{e$_n$} is passed to the eliminator as the last argument
-  (\ssrC{x} in \ssrC{foo_ind}) and \ssrC{e$_{n-1} \dots$ e$_1$} are used as patterns
-  to select in the goal the occurrences that will be bound by the
-  predicate \ssrC{P}, thus it must be possible to unify the sub-term of
-  the goal matched by \ssrC{e$_{n-1}$} with \ssrC{p$_m$}, the one matched by
-  \ssrC{e$_{n-2}$} with \ssrC{p$_{m-1}$} and so on.
-\item[regular eliminator] in all the other cases. Here it must be
-  possible to unify the term matched by
-  \ssrC{e$_n$} with \ssrC{p$_m$}, the one matched by
-  \ssrC{e$_{n-1}$} with \ssrC{p$_{m-1}$} and so on. Note that
-  standard eliminators have the shape \ssrC{$\dots$forall x, P $\dots$ x}, thus
-  \ssrC{e$_n$} is the pattern identifying the eliminated term, as expected.
-\end{description}
-As explained in  section~\ref{ssec:typefam}, the initial prefix of
-\ssrC{e$_i$} can be omitted.
-
-Here an example of a regular, but non trivial, eliminator:
-\begin{lstlisting}
-  Function |*plus*| (m n : nat) {struct n} : nat :=
-     match n with 0 => m | S p => S (plus m p) end.
-\end{lstlisting}
-The type of \ssrC{plus_ind} is
-\begin{lstlisting}
-plus_ind : forall (m : nat) (P : nat -> nat -> Prop),
-  (forall n : nat, n = 0 -> P 0 m) ->
-  (forall n p : nat, n = p.+1 -> P p (plus m p) -> P p.+1 (plus m p).+1) ->
-    forall n : nat, P n (plus m n)
-\end{lstlisting}
-Consider the following goal
-\begin{lstlisting}
-  Lemma |*exF*| x y z: plus (plus x y) z = plus x (plus y z).
-\end{lstlisting}
-The following tactics are all valid and perform the same elimination
-on that goal.
-\begin{lstlisting}
-  elim/plus_ind: z / (plus _ z).
-  elim/plus_ind: {z}(plus _ z).
-  elim/plus_ind: {z}_.
-  elim/plus_ind: z / _.
-\end{lstlisting}
-In the two latter examples, being the user provided pattern a wildcard, the
-pattern inferred from the type of the eliminator is used instead.  For both
-cases it is \ssrC{(plus _ _)} and matches the subterm \ssrC{plus (plus x y)$\;$z} thus
-instantiating the latter \ssrC{_} with \ssrC{z}.  Note that the tactic
-\ssrC{elim/plus_ind: y / _} would have resulted in an error, since \ssrC{y} and \ssrC{z}
-do no unify but the type of the eliminator requires the second argument of
-\ssrC{P} to be the same as the second argument of \ssrC{plus} in the second
-argument of \ssrC{P}.
-
-Here an example of a truncated eliminator. Consider the goal
-\begin{lstlisting}
-  p : nat_eqType
-  n : nat
-  n_gt0 : 0 < n
-  pr_p : prime p
- =================
-  p %| \prod_(i <- prime_decomp n | i \in prime_decomp n) i.1 ^ i.2 ->
-    exists2 x : nat * nat, x \in prime_decomp n & p = x.1
-\end{lstlisting}
-and the tactic
-\begin{lstlisting}
-elim/big_prop: _ => [| u v IHu IHv | [q e] /=].
-\end{lstlisting}
-where the type of the eliminator is
-\begin{lstlisting}
-big_prop: forall (R : Type) (Pb : R -> Type) (idx : R) (op1 : R -> R -> R),
-  Pb idx ->
-  (forall x y : R, Pb x -> Pb y -> Pb (op1 x y)) ->
-  forall (I : Type) (r : seq I) (P : pred I) (F : I -> R),
-  (forall i : I, P i -> Pb (F i)) ->
-    Pb (\big[op1/idx]_(i <- r | P i) F i)
-\end{lstlisting}
-Since the pattern for the argument of \ssrC{Pb} is not specified, the inferred one
-is used instead: \ssrC{(\\big[_/_]_(i <- _ | _ i) _ i)}, and after the
-introductions, the following goals are generated.
-\begin{lstlisting}
-subgoal 1 is:
- p %| 1 -> exists2 x : nat * nat, x \in prime_decomp n & p = x.1
-subgoal 2 is:
- p %| u * v -> exists2 x : nat * nat, x \in prime_decomp n & p = x.1
-subgoal 3 is:
- (q, e) \in prime_decomp n -> p %| q ^ e ->
-   exists2 x : nat * nat, x \in prime_decomp n & p = x.1
-\end{lstlisting}
-Note that the pattern matching algorithm instantiated all the variables
-occurring in the pattern.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Interpreting assumptions}\label{ssec:assumpinterp}
-\idx{move/\dots{}}
-
-Interpreting an assumption in the context of a proof is applying it a
-correspondence lemma before generalizing, and/or decomposing it.
-For instance, with the extensive use of boolean reflection (see
-section \ref{ssec:boolrefl}), it is
-quite frequent to need to decompose the logical interpretation of (the
-boolean expression of) a
-fact, rather than the fact itself.
-This can be achieved by a combination of \ssrC{move : _ => _}
-switches, like in the following script, where \ssrC{||} is a notation for
-the boolean disjunction:
-\begin{lstlisting}
-  Variables P Q : bool -> Prop.
-  Hypothesis |*P2Q*| : forall a b, P (a || b) -> Q a.
-
-  Goal forall a, P (a || a) -> True.
-  move=> a HPa; move: {HPa}(P2Q _ _ HPa) => HQa.
-\end{lstlisting}
-which transforms the hypothesis \ssrC{HPn : P n} which has been
-introduced from the initial statement into \ssrC{HQn : Q n}.
-This operation is so common that the tactic shell has
-specific syntax for it.
-The following scripts:
-\begin{lstlisting}
-  Goal forall a, P (a || a) -> True.
-  move=> a HPa; move/P2Q: HPa => HQa.
-\end{lstlisting}
-or more directly:
-\begin{lstlisting}
-  Goal forall a, P (a || a) -> True.
-  move=> a; move/P2Q=> HQa.
-\end{lstlisting}
-are equivalent to the former one. The former script shows how to
-interpret a fact (already in the context), thanks to the discharge
-tactical (see section \ref{ssec:discharge}) and the latter, how to
-interpret the top assumption of a goal. Note
-that the number of wildcards to be inserted to find the correct
-application of the view lemma to the hypothesis has been automatically
-inferred.
-
-The view mechanism is compatible with the \ssrC{case} tactic and with the
-equation name generation mechanism (see section \ref{ssec:equations}):
-\begin{lstlisting}
-  Variables P Q: bool -> Prop.
-  Hypothesis |*Q2P*| : forall a b, Q (a || b) -> P a \/ P b.
-
-  Goal forall a b, Q (a || b) -> True.
-  move=> a b; case/Q2P=> [HPa | HPb].
-\end{lstlisting}
-creates two new subgoals whose contexts no more contain
-\ssrC{HQ : Q (a || b)} but respectively \ssrC{HPa : P a} and
-\ssrC{HPb : P b}. This view tactic
-performs:
-\begin{lstlisting}
-  move=> a b HQ; case: {HQ}(Q2P _ _ HQ) => [HPa | HPb].
-\end{lstlisting}
-
-The term on the right of the \ssrC{/} view switch is called a \emph{view
-  lemma}. Any \ssr{} term coercing to a product type can be used as a
-view lemma.
-
-
-The examples we have given so far explicitly provide the direction of the
-translation to be performed. In fact, view lemmas need not to be
-oriented. The view mechanism is able to detect which
-application is relevant for the current goal. For instance, the
-script:
-\begin{lstlisting}
-  Variables P Q: bool -> Prop.
-  Hypothesis |*PQequiv*| : forall a b, P (a || b) <-> Q a.
-
-  Goal forall a b, P (a || b) -> True.
-  move=> a b; move/PQequiv=> HQab.
-\end{lstlisting}
-has the same behavior as the first example above.
-
-The view mechanism can insert automatically a \emph{view hint} to
-transform the double implication into the expected simple implication.
-The last script is in fact equivalent to:
-\begin{lstlisting}
-  Goal forall a b, P (a || b) -> True.
-  move=> a b; move/(iffLR (PQequiv _ _)).
-\end{lstlisting}
-where:
-\begin{lstlisting}
-  Lemma |*iffLR*| : forall P Q, (P <-> Q) -> P -> Q.
-\end{lstlisting}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Specializing assumptions}
-\idx{move/\dots{}}
-
-The special case when the \emph{head symbol} of the view lemma is a
-wildcard is used to interpret an assumption by \emph{specializing}
-it. The view mechanism hence offers the possibility to
-apply a higher-order assumption to some given arguments.
-
-For example, the script:
-\begin{lstlisting}
-  Goal forall z, (forall x y, x + y = z -> z = x) -> z = 0.
-  move=> z; move/(_ 0 z).
-\end{lstlisting}
-changes the goal into:
-\begin{lstlisting}
-  (0 + z = z -> z = 0) -> z = 0
-\end{lstlisting}
-
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Interpreting goals}\label{ssec:goalinterp}
-
-In a similar way, it is also often convenient to interpret a goal by changing
-it into an equivalent proposition. The view mechanism of \ssr{} has a
-special syntax \ssrC{apply/} for combining simultaneous goal
-interpretation operations and
-bookkeeping steps in a single tactic.
-
-With the hypotheses of section \ref{ssec:assumpinterp}, the following
-script, where \ssrL+~~+ denotes the boolean negation:
-\begin{lstlisting}
-  Goal forall a, P ((~~ a) || a).
-  move=> a; apply/PQequiv.
-\end{lstlisting}
-transforms the goal into \ssrC{Q (~~ a)}, and is equivalent to:
-\begin{lstlisting}
-  Goal forall a, P ((~~ a) || a).
-  move=> a; apply: (iffRL (PQequiv _ _)).
-\end{lstlisting}
-where \ssrC{iffLR} is the analogous of \ssrC{iffRL} for the converse
-implication.
-
-Any \ssr{} term whose type coerces to a double implication can be used
-as a view for goal interpretation.
-
-Note that the goal interpretation view mechanism supports both
-\ssrC{apply} and \ssrC{exact} tactics. As expected, a goal interpretation
-view command \ssrC{exact/$term$} should solve the current goal or it will
-fail.
-
-
-\emph{Warning} Goal interpretation view tactics are \emph{not} compatible
-with the bookkeeping tactical \ssrC{=>} since this would be redundant with
-the \ssrC{apply:} {\term} \ssrC{=> _} construction.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Boolean reflection}\label{ssec:boolrefl}
-In the Calculus of Inductive Construction, there is
-an obvious distinction between logical propositions and boolean values.
-On the one hand, logical propositions are objects
-of \emph{sort} \ssrC{Prop} which is the carrier of intuitionistic
-reasoning. Logical connectives in \ssrC{Prop} are \emph{types}, which give precise
-information on the structure of their proofs; this information is
-automatically exploited by \Coq{} tactics. For example, \Coq{} knows that a
-proof of \ssrL+A \/ B+ is either a proof of \ssrC{A} or a proof of \ssrC{B}.
-The tactics \ssrC{left} and \ssrC{right} change the goal \ssrL+A \/ B+
-to \ssrC{A} and \ssrC{B}, respectively; dualy, the tactic \ssrC{case} reduces the goal
-\ssrL+A \/ B => G+ to two subgoals \ssrC{A => G} and \ssrC{B => G}.
-
-On the other hand, \ssrC{bool} is an inductive \emph{datatype}
-with two constructors \ssrC{true} and \ssrC{false}.
-Logical connectives on \ssrC{bool} are \emph{computable functions}, defined by
-their truth tables, using case analysis:
-\begin{lstlisting}
-  Definition (b1 || b2) := if b1 then true else b2.
-\end{lstlisting}
-Properties of such connectives are also established using case
-analysis: the tactic \ssrC{by case: b} solves the goal
-\begin{lstlisting}
-  b || ~~ b = true
-\end{lstlisting}
-by replacing \ssrC{b} first by \ssrC{true} and then by \ssrC{false}; in either case,
-the resulting subgoal reduces by computation to the trivial
-\ssrC{true = true}.
-
-Thus, \ssrC{Prop} and \ssrC{bool} are truly complementary: the former
-supports robust natural deduction, the latter allows brute-force
-evaluation.
-\ssr{} supplies
-a generic mechanism to have the best of the two worlds and move freely
-from a propositional version of a
-decidable predicate to its boolean version.
-
-First, booleans are injected into propositions
-using the coercion mechanism:
-\begin{lstlisting}
-  Coercion |*is_true*| (b : bool) := b = true.
-\end{lstlisting}
-This allows any boolean formula~\ssrC{b} to be used in a context
-where \Coq{} would expect a proposition, e.g., after \ssrC{Lemma $\dots$ : }.
-It is then interpreted as \ssrC{(is_true b)}, i.e.,
-the  proposition \ssrC{b = true}. Coercions are elided by the pretty-printer,
-so they are essentially transparent to the user.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{The \ssrC{reflect} predicate}\label{ssec:reflpred}
-
-To get all the benefits of the boolean reflection, it is in fact
-convenient to introduce the following inductive predicate
-\ssrC{reflect} to relate propositions and booleans:
-
-\begin{lstlisting}
-  Inductive |*reflect*| (P: Prop): bool -> Type :=
-    | Reflect_true: P => reflect P true
-    | Reflect_false: ~P => reflect P false.
-\end{lstlisting}
-
-The statement \ssrC{(reflect P b)} asserts that \ssrC{(is_true b)}
-and \ssrC{P} are logically equivalent propositions.
-
-For instance, the following lemma:
-\begin{lstlisting}
-  Lemma |*andP*|: forall b1 b2, reflect (b1 /\ b2) (b1 && b2).
-\end{lstlisting}
-relates the boolean conjunction \ssrC{&&} to
-the logical one \ssrL+/\+.
-Note that in \ssrC{andP}, \ssrC{b1} and \ssrC{b2} are two boolean variables and
-the proposition \ssrL+b1 /\ b2+ hides two coercions.
-The conjunction of \ssrC{b1} and \ssrC{b2} can then be viewed
-as \ssrL+b1 /\ b2+  or as \ssrC{b1 && b2}.
-
-
-Expressing logical equivalences through this family of inductive types
-makes possible to take benefit from \emph{rewritable equations}
-associated to the case analysis of \Coq{}'s inductive types.
-
-Since the equivalence predicate is defined in \Coq{} as:
-\begin{lstlisting}
-  Definition |*iff*| (A B:Prop) := (A -> B) /\ (B -> A).
-\end{lstlisting}
-where \ssrC{/\\} is a notation for \ssrC{and}:
-\begin{lstlisting}
-  Inductive |*and*| (A B:Prop) : Prop :=
-    conj : A -> B -> and A B
-\end{lstlisting}
-
-This make case analysis very different according to the way an
-equivalence property has been defined.
-
-
-For instance, if we have proved the lemma:
-\begin{lstlisting}
-  Lemma |*andE*|: forall b1 b2, (b1 /\ b2) <-> (b1 && b2).
-\end{lstlisting}
-let us compare the respective behaviours of \ssrC{andE} and \ssrC{andP} on a
-goal:
-\begin{lstlisting}
-  Goal forall b1 b2, if (b1 && b2) then b1 else ~~(b1||b2).
-\end{lstlisting}
-
-The command:
-\begin{lstlisting}
-  move=> b1 b2; case (@andE b1 b2).
-\end{lstlisting}
-generates a single subgoal:
-\begin{lstlisting}
-  (b1 && b2 -> b1 /\ b2) -> (b1 /\ b2 -> b1 && b2) ->
-                   if b1 && b2 then b1 else ~~ (b1 || b2)
-\end{lstlisting}
-
-while the command:
-\begin{lstlisting}
-  move=> b1 b2; case (@andP b1 b2).
-\end{lstlisting}
-generates two subgoals, respectively \ssrL+b1 /\ b2 -> b1+ and
-\ssrL+~ (b1 /\ b2) -> ~~ (b1 || b2)+.
-
-
-
-Expressing reflection relation through the \ssrC{reflect} predicate
-is hence a very convenient way to deal with classical reasoning, by
-case analysis. Using the \ssrC{reflect} predicate allows moreover to
-program rich specifications inside
-its two constructors, which will be automatically taken into account
-during destruction. This formalisation style gives far more
-efficient specifications than quantified (double) implications.
-
-
-A naming convention in \ssr{} is to postfix the name of view lemmas with \ssrC{P}.
-For example, \ssrC{orP} relates  \ssrC{||} and \ssrL+\/+, \ssrC{negP} relates
-\ssrL+~~+ and \ssrL+~+.
-
-The view mechanism is compatible with \ssrC{reflect} predicates.
-
-For example, the script
-\begin{lstlisting}
-  Goal forall a b : bool, a -> b -> a /\\ b.
-  move=> a b Ha Hb; apply/andP.
-\end{lstlisting}
-changes the goal \ssrL+a /\ b+ to \ssrC{a && b} (see section \ref{ssec:goalinterp}).
-
-Conversely, the script
-\begin{lstlisting}
-  Goal forall a b : bool, a /\ b -> a.
-  move=> a b; move/andP.
-\end{lstlisting}
-changes the goal \ssrL+a /\ b -> a+ into \ssrC{a && b -> a} (see section
-\ref{ssec:assumpinterp}).
-
-
-The same tactics can also be used to perform the converse
-operation, changing a boolean conjunction into a logical one. The view
-mechanism guesses the direction of the
-transformation to be used i.e., the constructor of the \ssrC{reflect}
-predicate which should be chosen.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{General mechanism for interpreting goals and assumptions}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Specializing assumptions}
-\idx{move/\dots{}}
-
-The \ssr{}
-tactic:
-
-  \ssrC{move/(_} \ssrN[1]{term} $\dots$ \ssrN[n]{term}\ssrC{)}
-
-\noindent
-is equivalent to the tactic:
-
-  \ssrC{intro top; generalize (top} \ssrN[1]{term} $\dots$ \ssrN[n]{term}\ssrC{); clear top.}
-
-\noindent
-where \ssrC{top} is a fresh name for introducing the top assumption of
-the current goal.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Interpreting assumptions}
-\label{sssec:hypview}
-The general form of an assumption view tactic is:
-
-\begin{center}
- \optional{\ssrC{move} {\optsep} \ssrC{case}} \ssrC{/} \ssrN[0]{term}
-\end{center}
-
-The term \ssrN[0]{term}, called the \emph{view lemma} can be:
-\begin{itemize}
-\item a (term coercible to a) function;
-\item a (possibly quantified) implication;
-\item a (possibly quantified) double implication;
-\item a (possibly quantified) instance of the \ssrC{reflect} predicate
-  (see section \ref{ssec:reflpred}).
-\end{itemize}
-
-Let \ssrC{top} be the top assumption in the goal.
-
-There are three steps in the behaviour of an assumption view tactic:
-\begin{itemize}
-\item It first introduces \ssrL+top+.
-\item If the type of \ssrN[0]{term} is neither a double implication nor
-  an instance of the \ssrC{reflect} predicate, then the tactic
-  automatically generalises a term of the form:
-  
-\begin{center}
-  \ssrC{(}\ssrN[0]{term} \ssrN[1]{term} $\dots$ \ssrN[n]{term}\ssrC{)}
-\end{center}
-  
-  where the terms \ssrN[1]{term} $\dots$ \ssrN[n]{term} instantiate the
-  possible quantified variables of \ssrN[0]{term}, in order for
-  \ssrC{(}\ssrN[0]{term} \ssrN[1]{term} $\dots$ \ssrN[n]{term} \ssrC{top)} to be well typed.
-\item If the type of $\ssrN[0]{term}$ is an equivalence, or
-  an instance of the \ssrC{reflect} predicate, it generalises a term of
-  the form:
-  \begin{center}
-  (\ssrN[vh]{term} (\ssrN[0]{term} \ssrN[1]{term} $\dots$ \ssrN[n]{term}))
-  \end{center}
-  where the term \ssrN[vh]{term} inserted is called an
-  \emph{assumption interpretation view hint}.
-\item It finally clears \ssrC{top}.
-\end{itemize}
-For a \ssrC{case/}\ssrN[0]{term} tactic, the generalisation step is
-replaced by a case analysis step.
-
-\emph{View hints} are declared by the user (see section
-\ref{ssec:vhints}) and are stored in the \ssrC{Hint View} database.
-The proof engine automatically
-detects from the shape of the top assumption \ssrC{top} and of the view
-lemma $\ssrN[0]{term}$ provided to the tactic the appropriate view hint in
-the database to be inserted.
-
-If $\ssrN[0]{term}$ is a double implication, then the view hint \ssrC{A} will
-be one of the defined view hints for implication. These hints are by
-default the ones present in the file {\tt ssreflect.v}:
-\begin{lstlisting}
-  Lemma |*iffLR*| : forall P Q, (P <-> Q) -> P -> Q.
-\end{lstlisting}
-which transforms a double  implication into the left-to-right one, or:
-\begin{lstlisting}
-  Lemma |*iffRL*| : forall P Q, (P <-> Q) -> Q -> P.
-\end{lstlisting}
-which produces the converse implication. In both cases, the two first
-\ssrC{Prop} arguments are implicit.
-
-If $\ssrN[0]{term}$ is an instance of the \ssrC{reflect} predicate, then \ssrC{A}
-will be one of the defined view hints  for the \ssrC{reflect}
-predicate,  which are by
-default the ones present in the file {\tt ssrbool.v}.
-These hints are not only used for choosing the appropriate direction of
-the translation, but they also allow complex transformation, involving
-negations.
- For instance the hint:
-\begin{lstlisting}
-  Lemma |*introN*| : forall (P : Prop) (b : bool), reflect P b -> ~ P -> ~~ b.
-\end{lstlisting}
-makes the following script:
-\begin{lstlisting}
-  Goal forall a b : bool, a -> b -> ~~ (a && b).
-  move=> a b Ha Hb. apply/andP.
-\end{lstlisting}
-transforms the goal into \ssrC{ \~ (a /\ b)}.
-In fact\footnote{The current state of the proof shall be displayed by
-  the \ssrC{Show Proof} command of \Coq{} proof mode.}
-this last script does not exactly use the hint \ssrC{introN}, but the
-more general hint:
-\begin{lstlisting}
-  Lemma |*introNTF*| : forall (P : Prop) (b c : bool),
-      reflect P b -> (if c then ~ P else P) -> ~~ b = c
-\end{lstlisting}
-The lemma \ssrL+|*introN*|+ is an instantiation of \ssrC{introNF} using
- \ssrC{c := true}.
-
-Note that views, being part of \ssrN{i-pattern}, can be used to interpret
-assertions too. For example the following script asserts \ssrC{a \&\& b}
-but actually used its propositional interpretation.
-\begin{lstlisting}
-  Lemma |*test*| (a b : bool) (pab : b && a) : b.
-  have /andP [pa ->] : (a && b) by rewrite andbC.
-\end{lstlisting}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection*{Interpreting goals}
-\idx{apply/\dots{}}
-
-A goal interpretation view tactic of the form:
-
-\begin{center}
-	\ssrC{apply/} \ssrN[0]{term}
-\end{center}
-applied to a goal \ssrC{top} is interpreted in the following way:
-\begin{itemize}
-\item If the type of $\ssrN[0]{term}$ is not an instance of the
-  \ssrC{reflect} predicate, nor an equivalence,
-  then the term $\ssrN[0]{term}$ is applied to the current goal \ssrC{top},
-  possibly inserting implicit arguments.
-\item If the type of $\ssrN[0]{term}$ is an instance of the \ssrC{reflect}
-  predicate or an equivalence, then
-a \emph{goal interpretation view hint} can possibly be inserted, which
-corresponds to the application of a term
-\ssrC{($\ssrN[vh]{term}$ ($\ssrN[0]{term}$ _ $\dots$ _))} to the current
-goal, possibly inserting implicit arguments.
-\end{itemize}
-
-Like assumption interpretation view hints, goal interpretation ones
-are user defined lemmas stored (see section \ref{ssec:vhints}) in the
-\ssrC{Hint View} database bridging
-the possible gap between the type of $\ssrN[0]{term}$ and the type of the
-goal.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Interpreting equivalences}
-\idx{apply/\dots{}/\dots{}}
-
-Equivalent boolean propositions are simply \emph{equal} boolean terms.
-A special construction helps the user to prove boolean equalities by
-considering them as logical double implications (between their coerced
-versions), while
-performing at the same time logical operations on both sides.
-
-The syntax of double views is:
-\begin{center}
- \ssrC{apply/} \ssrN[l]{term} \ssrC{/} \ssrN[r]{term}
-\end{center}
-
-The term \ssrN[l]{term} is the view lemma applied to the left hand side of the
-equality, \ssrN[r]{term} is the one applied to the right hand side.
-
-In this context, the identity view:
-\begin{lstlisting}
-Lemma |*idP*| : reflect b1 b1.
-\end{lstlisting}
-is useful, for example the tactic:
-\begin{lstlisting}
-  apply/idP/idP.
-\end{lstlisting}
-transforms the goal
-\ssrL+~~ (b1 || b2)= b3+
- into two subgoals, respectively
- \ssrL+~~  (b1 || b2) -> b3+ and \\
-\ssrL+b3 -> ~~  (b1 || b2).+
-
-The same goal can be decomposed in several ways, and the user may
-choose the most convenient interpretation. For instance, the tactic:
-\begin{lstlisting}
-  apply/norP/idP.
-\end{lstlisting}
-applied on the same goal \ssrL+~~ (b1 || b2) = b3+ generates the subgoals
-\ssrL+~~  b1 /\ ~~  b2 -> b3+ and\\
-\ssrL+b3 -> ~~  b1 /\ ~~  b2+.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Declaring new \ssrC{Hint View}s}\label{ssec:vhints}
-\idxC{Hint View}
-
-The database of hints for the view mechanism is extensible via a
-dedicated vernacular command. As library {\tt ssrbool.v} already
-declares a corpus of hints, this feature is probably useful only for
-users who define their own logical connectives. Users can declare
-their own hints following the syntax used in {\tt ssrbool.v}:
-
-\begin{center}
-  \ssrC{Hint View for} {\tac} \ssrC{/} \ssrN{ident} \optional{\ssrC{|}{\naturalnumber}}
-\end{center}
-
-  where {\tac}$\in \{$\ssrC{move, apply}$\}$, \ssrN{ident} is the
-name of the lemma to be declared as a hint, and ${\naturalnumber}$ a natural
-number. If \ssrL+move+ is used as {\tac}, the hint is declared for
-assumption interpretation tactics, \ssrL+apply+ declares hints for goal
-interpretations.
-Goal interpretation view hints are declared for both simple views and
-left hand side views. The optional natural number ${\naturalnumber}$ is the
-number of implicit arguments to be considered for the declared hint
-view lemma \ssrC{name_of_the_lemma}.
-
-The command:
-
-\begin{center}
-  \ssrC{Hint View for apply//} \ssrN{ident}\optional{\ssrC{|}{\naturalnumber}}.
-\end{center}
-
-with a double slash \ssrL+//+, declares hint views for right hand sides of
-double views.
-
-
-\noindent See the files {\tt ssreflect.v} and {\tt ssrbool.v} for examples.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Multiple views}\label{ssec:multiview}
-
-The hypotheses and the goal can be interpreted applying multiple views in
-sequence. Both \ssrC{move} and \ssrC{apply} can be followed by an arbitrary number
-of \ssrC{/}$\ssrN[i]{term}$. The main difference between the following two tactics
-\begin{lstlisting}
-  apply/v1/v2/v3.
-  apply/v1; apply/v2; apply/v3.
-\end{lstlisting}
-is that the former applies all the views to the principal goal.
-Applying a view with hypotheses generates new goals, and the second line
-would apply the view \ssrC{v2} to all the goals generated by \ssrC{apply/v1}.
-Note that the NO-OP intro pattern \ssrC{-} can be used to separate two
-views, making the two following examples equivalent:
-\begin{lstlisting}
-  move=> /v1; move=> /v2.
-  move=> /v1-/v2.
-\end{lstlisting}
-
-The tactic \ssrC{move} can be used together with the \ssrC{in}
-tactical to pass a given hypothesis to a lemma. For example, if
-\ssrC{P2Q : P -> Q } and \ssrC{Q2R : Q -> R}, the following
-tactic turns the hypothesis \ssrC{p : P} into \ssrC{P : R}.
-\begin{lstlisting}
-  move/P2Q/Q2R in p.
-\end{lstlisting}
-
-If the list of views is of length two, \ssrC{Hint View}s for interpreting
-equivalences are indeed taken into account, otherwise only single
-\ssrC{Hint View}s are used.
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{\ssr{} searching tool}
-\idxC{Search \dots{}}
-
-\ssr{} proposes an extension of the \ssrC{Search} command. Its syntax is:
-
-\begin{center}
-	\ssrC{Search} \optional{\ssrN{pattern}} \optional{\optional{\ssrC{\-}} \optional{\ssrN{string}\optional{\ssrC{\%}\ssrN{key}} {\optsep} \ssrN{pattern}}}$^*$ \optional{\ssrC{in} \optional{\optional{\ssrC{\-}} \ssrN{name} }$^+$}
-\end{center}
-
-% \begin{lstlisting}
-%   Search [[\~]\ssrN{string}]$^*$ [\ssrN{pattern}] [[$\ssrN[1]{pattern} \dots $ $\ssrN[n]{pattern}$]] $[[$inside$|$outside$]$ $M_1 \dots M_n$].
-% \end{lstlisting}
-
-% This tactic returns the list of defined constants matching the
-%  given criteria:
-% \begin{itemize}
-% \item \ssrL+[[-]\ssrN{string}]$^*$+ is an open sequence of strings, which sould
-%   all appear in the name of the returned constants. The optional \ssrL+-+
-%   prefixes strings that are required  \emph{not} to appear.
-% % \item \ssrN{pattern} should be a subterm of the
-% %   \emph{conclusion} of the lemmas found by the command. If a lemma features
-% %   an occurrence
-% %   of this pattern only in one or several of its assumptions, it will not be
-% %   selected by the searching tool.
-% \item
-% \ssrL=[$\ssrN{pattern}^+$]=
-% is a list of \ssr{} terms, which may
-% include types, that are required to appear in the returned constants.
-% Terms with holes should be surrounded by parentheses.
-% \item $\ssrC{in}\ [[\ssrC{\-}]M]^+$ limits the search to the signature
-%   of open modules given in the list, but the ones preceeded by the
-%   $\ssrC{\-}$ flag. The
-%   command:
-% \begin{lstlisting}
-%   Search in M.
-% \end{lstlisting}
-% is hence a way of obtaining the complete signature of the module \ssrL{M}.
-% \end{itemize}
-where \ssrN{name} is the name of an open module.
-This command search returns the list of lemmas:
-\begin{itemize}
-\item whose \emph{conclusion} contains a subterm matching the optional
-  first \ssrN{pattern}. A $\ssrC{-}$ reverses the test, producing the list
-  of lemmas whose conclusion does not contain any subterm matching
-  the pattern;
-\item whose name contains the given string. A $\ssrC{-}$ prefix reverses
-  the test, producing the list of lemmas whose name does not contain the
-  string. A string that contains symbols or
-is followed by a scope \ssrN{key}, is interpreted as the constant whose
-notation involves that string (e.g., \ssrL=+= for \ssrL+addn+), if this is
-unambiguous; otherwise the diagnostic includes the output of the
-\ssrC{Locate} vernacular command.
-
-\item whose statement, including assumptions and types, contains a
-  subterm matching the next patterns. If a pattern is prefixed by
-  $\ssrC{-}$, the test is reversed;
-\item contained in the given list of modules, except the ones in the
-  modules prefixed by a $\ssrC{-}$.
-\end{itemize}
-
-Note that:
-\begin{itemize}
-\item As for regular terms, patterns can feature scope
-  indications. For instance, the command:
-\begin{lstlisting}
-  Search _ (_ + _)%N.
-\end{lstlisting}
-lists all the lemmas whose statement (conclusion or hypotheses)
-involve an application of the binary operation denoted by the infix
-\ssrC{+} symbol in the \ssrC{N} scope (which is \ssr{} scope for natural numbers).
-\item Patterns with holes should be surrounded by parentheses.
-\item Search always volunteers the expansion of the notation, avoiding the
-  need to execute Locate independently. Moreover, a string fragment
-  looks for any notation that contains fragment as
-  a substring. If the \ssrL+ssrbool+ library is imported, the command:
-\begin{lstlisting}
-  Search "~~".
-\end{lstlisting}
-answers :
-\begin{lstlisting}
-"~~" is part of notation (~~ _)
-In bool_scope, (~~ b) denotes negb b
-negbT  forall b : bool, b = false -> ~~ b
-contra  forall c b : bool, (c -> b) -> ~~ b -> ~~ c
-introN  forall (P : Prop) (b : bool), reflect P b -> ~ P -> ~~ b
-\end{lstlisting}
- \item A diagnostic is issued if there are different matching notations;
-  it is an error if all matches are partial.
-\item Similarly, a diagnostic warns about multiple interpretations, and
-  signals an error if there is no default one.
-\item The command  \ssrC{Search in M.}
-is a way of obtaining the complete signature of the module \ssrL{M}.
-\item Strings and pattern indications can be interleaved, but the
-  first indication has a special status if it is a pattern, and only
-  filters the conclusion of lemmas:
-\begin{itemize}
-  \item The command :
-    \begin{lstlisting}
-      Search (_ =1 _) "bij".
-    \end{lstlisting}
-lists all the lemmas whose conclusion features a '$\ssrC{=1}$' and whose
-name contains the string \verb+bij+.
-\item The command :
-    \begin{lstlisting}
-      Search  "bij" (_ =1 _).
-    \end{lstlisting}
-lists all the lemmas whose statement, including hypotheses, features a
-'$\ssrC{=1}$' and whose name contains the string \verb+bij+.
-
-\end{itemize}
-
-\end{itemize}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Synopsis and Index}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection*{Parameters}
-
-\begin{minipage}[c]{\textwidth}\renewcommand{\footnoterule}{}
-\begin{longtable}{lcl}
-\ssrN{d-tactic} && one of the
-  \ssrC{elim}, \ssrC{case}, \ssrC{congr}, \ssrC{apply}, \ssrC{exact}
-  and \ssrC{move} \ssr{} tactics \\
-\ssrN{fix-body} && standard \Coq{} \textit{fix\_body}\\
-\ssrN{ident} && standard \Coq{} identifier\\
-\ssrN{int} && integer literal \\
-\ssrN{key} && notation scope\\
-\ssrN{name} && module name\\
-${\naturalnumber}$ && \ssrN{int} or Ltac variable denoting a standard \Coq{} numeral\footnote{The name of this Ltac variable should not be the name of a tactic which can be followed by a bracket
-  \ssrL+[+, like  \ssrL+do+, \ssrL+ have+,\dots}\\
-\ssrN{pattern} && synonym for {\term}\\
-\ssrN{string} && standard \Coq{} string\\
-{\tac} && standard \Coq{} tactic or \ssr{} tactic\\
-{\term} & \hspace{1cm} & Gallina term, possibly containing wildcards\\
-%\ssrN{view} && global constant\\
-\end{longtable}
-\end{minipage}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection*{Items and switches}
-
-\begin{longtable}{lclr}
-\ssrN{binder}  & {\ident} {\optsep} \ssrC{(} {\ident} \optional{\ssrC{:} {\term} } \ssrC{)} & binder& p. \pageref{ssec:pose}\\
-\\
-\ssrN{clear-switch} & \ssrC{\{} {\ident}$^+$ \ssrC{\}} & clear switch & p. \pageref{ssec:discharge}\\
-\\
-\ssrN{c-pattern} & \optional{{\term} \ssrC{in} {\optsep} {\term} \ssrC{as}} {\ident} \ssrC{in} {\term} & context pattern & p. \pageref{ssec:rewp} \\
-\\
-\ssrN{d-item} & \optional{\ssrN{occ-switch} {\optsep} \ssrN{clear-switch}} \optional{{\term} {\optsep} \ssrC{(}\ssrN{c-pattern}\ssrC{)}} & discharge item & p. \pageref{ssec:discharge}\\
-\\
-\ssrN{gen-item}  & \optional{\ssrC{@}}{\ident} {\optsep} \ssrC{(}{\ident}\ssrC{)} {\optsep} \ssrC{(}\optional{\ssrC{@}}{\ident} \ssrC{:=} \ssrN{c-pattern}\ssrC{)} & generalization item & p. \pageref{ssec:struct}\\
-\\
-\ssrN{i-pattern} & {\ident} {\optsep} \ssrC{_} {\optsep} \ssrC{?} {\optsep} \ssrC{*} {\optsep} \optional{\ssrN{occ-switch}}\ssrC{->} {\optsep} \optional{\ssrN{occ-switch}}\ssrC{<-} {\optsep} &  intro pattern & p. \pageref{ssec:intro}\\
-& \ssrC{[} \ssrN{i-item}$^*$ \ssrC{|} $\dots$ \ssrC{|} \ssrN{i-item}$^*$ \ssrC{]} {\optsep} \ssrC{-} {\optsep} \ssrC{[:} {\ident}$^+$\ssrC{]} &\\
-\\
-\ssrN{i-item} & \ssrN{clear-switch} {\optsep} \ssrN{s-item} {\optsep} \ssrN{i-pattern} {\optsep} \ssrC{/}{\term} & intro item & p. \pageref{ssec:intro}\\
-\\
-\ssrN{int-mult} & \optional{{\naturalnumber}} \ssrN{mult-mark} & multiplier & p. \pageref{ssec:iter}\\
-\\
-\ssrN{occ-switch} & \ssrC{\{} \optional{\ssrC{+} {\optsep} \ssrC{-}} {\naturalnumber}$^*$\ssrC{\}} & occur. switch &  p. \pageref{sssec:occselect}\\
-\\
-\ssrN{mult} & \optional{{\naturalnumber}} \ssrN{mult-mark} & multiplier & p. \pageref{ssec:iter}\\
-\\
-\ssrN{mult-mark} &  \ssrC{?} {\optsep} \ssrC{!} & multiplier mark &  p. \pageref{ssec:iter}\\
-\\
-\ssrN{r-item} &  \optional{\ssrC{/}} {\term} {\optsep} \ssrN{s-item} & rewrite item & p. \pageref{ssec:extrw}\\
-\\
-\ssrN{r-prefix} & \optional{\ssrC{-}} \optional{\ssrN{int-mult}} \optional{\ssrN{occ-switch} {\optsep} \ssrN{clear-switch}} \optional{\ssrC{[}\ssrN{r-pattern}\ssrC{]}} & rewrite prefix & p. \pageref{ssec:extrw}\\
-\\
-\ssrN{r-pattern} & {\term} {\optsep} \ssrN{c-pattern} {\optsep} \ssrC{in} \optional{{\ident} \ssrC{in}} {\term} & rewrite pattern & p. \pageref{ssec:extrw}\\
-\\
-\ssrN{r-step} & \optional{\ssrN{r-prefix}}\ssrN{r-item} & rewrite step & p. \pageref{ssec:extrw}\\
-\\
-\ssrN{s-item} & \ssrC{/=} {\optsep} \ssrC{//} {\optsep} \ssrC{//=} & simplify switch & p. \pageref{ssec:intro}\\
-\\
-\end{longtable}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection*{Tactics}
-\emph{Note}: \ssrC{without loss} and \ssrC{suffices} are synonyms for \ssrC{wlog} and
-\ssrC{suff} respectively.
-
-\begin{longtable}{llr}
-\ssrC{move} & \textcolor{dkblue}{\texttt{idtac}} or \ssrC{hnf}& p. \pageref{ssec:profstack} \\
-\ssrC{apply} & application & p. \pageref{ssec:basictac}\\
-\ssrC{exact} &&\\
-\ssrC{abstract} && p. \pageref{ssec:abstract}, \pageref{sec:havetransparent}\\
-\\
-\ssrC{elim} & induction & p. \pageref{ssec:basictac}\\
-\ssrC{case} & case analysis  & p. \pageref{ssec:basictac}\\
-\\
-\ssrC{rewrite} \ssrN{rstep}$^+$ & rewrite& p. \pageref{ssec:extrw}\\
-\\
-\ssrC{have} \ssrN{i-item}$^*$ \optional{\ssrN{i-pattern}} \optional{\ssrN{s-item} {\optsep} \ssrN{binder}$^+$} \optional{\ssrC{:} {\term}} \ssrC{:=} {\term} & forward & p. \pageref{ssec:struct}\\
-\ssrC{have} \ssrN{i-item}$^*$ \optional{\ssrN{i-pattern}} \optional{\ssrN{s-item}{\optsep} \ssrN{binder}$^+$} \ssrC{:} {\term} \optional{\ssrC{by} {\tac}} & chaining & \\
-\ssrC{have suff} \optional{\ssrN{clear-switch}} \optional{\ssrN{i-pattern}} \optional{\ssrC{:} {\term}} \ssrC{:=} {\term} & & \\
-\ssrC{have suff} \optional{\ssrN{clear-switch}} \optional{\ssrN{i-pattern}} \ssrC{:} {\term} \optional{\ssrC{by} {\tac}} & & \\
-\ssrC{gen have} \optional{{\ident}\ssrC{,}} \optional{\ssrN{i-pattern}} \ssrC{:} \ssrN{gen-item}$^+$ \ssrC{/} {\term} \optional{\ssrC{by} {\tac}} & & \\
-\\
-\ssrC{wlog} \optional{\ssrC{suff}} \optional{\ssrN{i-item}} \ssrC{:} \optional{\ssrN{gen-item}{\optsep} \ssrN{clear-switch}}$^*$ \ssrC{/} {\term} & specializing &  p. \pageref{ssec:struct} \\
-\\
-\ssrC{suff} \ssrN{i-item}$^*$ \optional{\ssrN{i-pattern}} \optional{\ssrN{binder}$^+$} \ssrC{:} {\term} \optional{\ssrC{by} {\tac}} & backchaining & p. \pageref{ssec:struct}\\
-\ssrC{suff} \optional{\ssrC{have}} \optional{\ssrN{clear-switch}} \optional{\ssrN{i-pattern}} \ssrC{:} {\term} \optional{\ssrC{by} {\tac}} & & \\
-\\
-\ssrC{pose} {\ident} \ssrC{:=} {\term} & local definition& p. \pageref{ssec:pose}\\
-\ssrC{pose} {\ident} \ssrN{binder}$^+$ \ssrC{:=} {\term} & \rlap{local function definition}& \\
-\ssrC{pose fix} \ssrN{fix-body} & \rlap{local fix definition} & \\
-\ssrC{pose cofix} \ssrN{fix-body} & \rlap{local cofix definition} &  \\
-\\
-\ssrC{set} {\ident} \optional{\ssrC{:} {\term}} \ssrC{:=} \optional{\ssrN{occ-switch}} \optional{{\term}{\optsep} \ssrC{(}\ssrN{c-pattern}\ssrC{)}} & abbreviation&p. \pageref{ssec:set}\\
-\\
-\ssrC{unlock} \optional{\ssrN{r-prefix}]{\ident}}$^*$ & unlock & p. \pageref{ssec:lock}\\
-\\
-\ssrC{congr} \optional{\naturalnumber} {\term} & congruence& p. \pageref{ssec:congr}\\
-\end{longtable}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection*{Tacticals}
-
-\begin{longtable}{lclr}
-\ssrN{d-tactic} \optional{\ident} \ssrC{:} \ssrN{d-item}$^{+}$ \optional{\ssrN{clear-switch}} & & discharge &  p. \pageref{ssec:discharge}\\
-\\
-{\tac} \ssrC{=>} \ssrN{i-item}$^+$ && introduction & p. \pageref{ssec:intro}\\
-\\
-{\tac} \ssrC{in} \optional{\ssrN{gen-item} {\optsep} \ssrN{clear-switch}}$^+$  \optional{\ssrC{*}} && localization & p. \pageref{ssec:gloc}\\
-\\
-\ssrC{do} \optional{\ssrN{mult}} \ssrC{[} \nelist{\tac}{|} \ssrC{]}&& iteration & p. \pageref{ssec:iter}\\
-\ssrC{do} \ssrN{mult} {\tac} &&& \\
-\\
-{\tac} \ssrC{ ; first} \optional{\naturalnumber} \ssrC{[}\nelist{\tac}{|}\ssrC{]} && selector & p. \pageref{ssec:select}\\
-{\tac} \ssrC{ ; last}  \optional{\naturalnumber} \ssrC{[}\nelist{\tac}{|}\ssrC{]} \\
-{\tac} \ssrC{ ; first} \optional{\naturalnumber} \ssrC{last} && subgoals & p. \pageref{ssec:select}\\
-{\tac} \ssrC{; last} \optional{\naturalnumber} \ssrC{first} && rotation & \\
-\\
-\ssrC{by [} \nelist{\tac}{|} \ssrC{]} && closing & p. \pageref{ssec:termin}\\
-\ssrC{by []} \\
-\ssrC{by} {\tac} \\
-\end{longtable}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection*{Commands}
-\begin{longtable}{lclr}
-\ssrL+Hint View for+ \optional{\ssrL+move+ {\it |} \ssrL+apply+} {\tt /} {\ident} \optional{{\tt|} {\naturalnumber}} && view hint
-declaration & p. \pageref{ssec:vhints}\\
-\\
-\ssrL+Hint View for apply//+ {\ident} \optional{{\tt|}{\naturalnumber}} && right hand side double
- & p. \pageref{ssec:vhints}\\
-&&  view hint declaration &\\
-\\
-%\ssrL+Import Prenex Implicits+ && enable prenex implicits &
-%p. \pageref{ssec:parampoly}\\
-%\\
-\ssrL+Prenex Implicits+ {\ident}$^+$ & \hspace{.6cm} & prenex implicits decl.
- & p. \pageref{ssec:parampoly}\\
-
-\end{longtable}
-
-\iffalse
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Changes}
-
-\subsection{\ssr{} version 1.3}
-All changes are retrocompatible extensions but for:
-\begin{itemize}
-\item Occurrences in the type family switch now refer only to the goal, while
-      before they used to refer also to the types in the abstractions of the
-      predicate used by the eliminator. This bug used to affect lemmas like
-      \ssrC{boolP}. See the relative comments in \ssrC{ssrbool.v}.
-\item Clear switches can only mention existing hypothesis and
-      otherwise fail. This can in particular affect intro patterns
-      simultaneously applied to several goals.
-      % commit: 2686
-\item A bug in the \ssrC{rewrite} tactic allowed to
-      instantiate existential metavariables occurring in the goal.
-      This is not the case any longer (see section~\ref{ssec:rewcaveats}).
-\item The \ssrC{fold} and \ssrC{unfold} \ssrN{r-items} for \ssrC{rewrite} used to
-      fail silently when used in combination with a \ssrN{r-pattern} matching no
-      goal subterm. They now fail. The old behavior can be obtained using
-      the \ssrC{?} multiplier (see section~\ref{ssec:extrw}).
-\item \Coq{} 8.2 users with a statically linked toplevel must comment out the\\
-      \ssrC{Declare ML Module "ssreflect".}\\
-      line at the beginning of \ssrC{ssreflect.v} to compile the 1.3 library.
-\end{itemize}
-New features:
-\begin{itemize}
-\item Contextual \ssrC{rewrite} patterns.
-      The context surrounding the redex can now be used to specify which
-      redex occurrences should be rewritten (see section~\ref{ssec:rewp}).\\
-      \ssrC{rewrite [in X in _ = X]addnC.}
-      % commit: 2690, 2689, 2718, 2733
-\item Proof irrelevant interpretation of goals with existential metavariables.
-      Goals containing an existential metavariable of sort \ssrC{Prop} are
-      generalized over it, and a new goal for the missing subproof is
-      generated (see page~\pageref{sssec:apply} and
-      section~\ref{ssec:rewcaveats}).\\
-      \ssrC{apply: (ex_intro _ (@Ordinal _ y _)).}\\
-      \ssrC{rewrite insubT.}
-      % commit: 2553, 2544, 2543, 2733
-\item Views are now part of \ssrN{i-pattern} and can thus be used
-      inside intro patterns (see section~\ref{ssec:intro}).\\
-      \ssrC{move=> a b /andP [Ha Hb].}
-      % commit: 2720
-\item Multiple views for \ssrC{move}, \ssrC{move $\dots$ in} and \ssrC{apply}
-      (see section~\ref{ssec:multiview}).\\
-      \ssrC{move/v1/v2/v3.}\\
-      \ssrC{move/v1/v2/v3 in H.}\\
-      \ssrC{apply/v1/v2/v3.}
-      % commit: 2720
-\item \ssrC{have} and \ssrC{suff} idiom with view (see section~\ref{sssec:hypview}).
-\begin{lstlisting}
-  Lemma |*test*| (a b : bool) (pab : a && b) : b.
-  have {pab} /= /andP [pa ->] // : true && (a && b) := pab.
-\end{lstlisting}
-      % commit: 2726
-\item \ssrC{have suff}, \ssrC{suff have} and \ssrC{wlog suff} forward reasoning
-      tactics (see section~\ref{ssec:struct}).\\
-      \ssrC{have suff H : P.}
-      % commit: 2633
-\item Binders support in \ssrC{have} (see section~\ref{sssec:have}).\\
-      \ssrC{have H x y (r : R x y) : P x -> Q y.}
-      % commit: 2633
-\item Deferred clear switches. Clears are deferred to the end of the
-      intro pattern. In the meanwhile, cleared variables are still
-      part of the context, thus the goal can mention them, but are
-      renamed to non accessible dummy names (see section~\ref{ssec:intro}).\\
-      \ssrC{suff: G \\x H = K; first case/dprodP=> \{G H\} [[G H -> -> defK]].}
-      % commit: 2660
-\item Relaxed alternation condition in intro patterns. The
-      \ssrN{i-item} grammar rule is simplified (see section~\ref{ssec:intro}).\\
-       \ssrC{move=> a \{H\} /= \{H1\} // b c /= \{H2\}.}
-      % commit: 2713
-\item Occurrence selection for \ssrC{->} and \ssrC{<-} intro pattern
-       (see section~\ref{ssec:intro}).\\
-      \ssrC{move=> a b H \{2\}->.}
-      % commit: 2714
-\item Modifiers for the discharging '\ssrC{:}' and \ssrC{in} tactical to override
-      the default behavior when dealing with local definitions (let-in):
-      \ssrC{@f} forces the body of \ssrC{f} to be kept, \ssrC{(f)} forces the body of
-      \ssrC{f} to be dropped  (see sections~\ref{ssec:discharge}
-      and~\ref{ssec:gloc}).\\
-      \ssrC{move: x y @f z.}\\
-      \ssrC{rewrite rule in (f) $\;\;$H.}
-      %commit: 2659, 2710
-\item Type family switch in \ssrC{elim} and \ssrC{case}
-      can contain patterns with occurrence switch
-      (see section~\ref{ssec:typefam}).\\
-      \ssrC{case: \{2\}(_ == x) / eqP.}
-      % commit: 2593, 2598, 2539, 2538, 2527, 2529
-\item Generic second order predicate support for \ssrC{elim}
-      (see section~\ref{sec:views}).\\
-      \ssrC{elim/big\_prop: _}
-      % commit: 2767
-\item The \ssrC{congr} tactic now also works on products (see
-      section~\ref{ssec:congr}).
-\begin{lstlisting}
-  Lemma |*test*| x (H : P x) : P y.
-  congr (P _): H.
-\end{lstlisting}
-      % commit: 2608
-\item Selectors now support Ltac variables
-      (see section~\ref{ssec:select}).\\
-      \ssrC{let n := 3 in tac; first n last.}
-      % commit: 2725
-\item Deprecated use of \ssrC{Import Prenex Implicits} directive.
-      It must be replaced with the \Coq{} \ssrC{Unset Printing
-        Implicit Defensive} vernacular command.
-\item New synonym \ssrC{Canonical} for \ssrC{Canonical Structure}.
-\end{itemize}
-\subsection{\ssr{} version 1.4}
-New features:
-\begin{itemize}
-\item User definable recurrent contexts (see section~\ref{ssec:rewp}).\\
-      \ssrC{Notation RHS := (X in _ = X)\%pattern}
-\item Contextual patterns in
-      \ssrC{set} and `\ssrC{:}' (see section~\ref{ssec:rewp}).\\
-      \ssrC{set t := (a + _ in RHS)}
-\item NO-OP intro pattern (see section~\ref{ssec:intro}).\\
-      \ssrC{move=> /eqP-H /fooP-/barP}
-\item \ssrC{if $\ {\term}\ $ isn't $\ \ssrN{pattern}\ $ then $\ {\term}\ $
-  else $\ {\term}\ $} notation (see section~\ref{ssec:patcond}).\\
-      \ssrC{if x isn't Some y then simple else complex y}
-\end{itemize}
-\subsection{\ssr{} version 1.5}
-Incompatibilities:
-\begin{itemize}
-\item The \ssrC{have} tactic now performs type classes resolution.  The old
-	behavior can be restored with \ssrC{Set SsrHave NoTCResolution}
-\end{itemize}
-Fixes:
-\begin{itemize}
-\item The \ssrC{let foo := type of t in} syntax of standard \ssrC{Ltac} has
-	been made compatible with \ssr{} and can be freely used even if
-	the \ssr{} plugin is loaded
-\end{itemize}
-New features:
-\begin{itemize}
-\item Generalizations supported in have (see section~\ref{ssec:struct}).\\
-      \ssrC{generally have hx2px, pa : a ha / P a.}
-\item Renaming and patterns in wlog (see section~\ref{ssec:struct} and
-  page \pageref{par:advancedgen}).\\
-      \ssrC{wlog H : (n := m)$\;$ (x := m + _)$\;$ / T x}.\\
-      \ssrC{wlog H : (n := m)$\;$ (@ldef := secdef m)$\;$ / T x}.
-\item Renaming, patterns and clear switches in \ssrC{in}
-      tactical (see section~\ref{ssec:gloc}).\\
-      \ssrC{$\dots$ in H1 \{H2\} (n := m).}
-\item Handling of type classes in \ssrC{have}
-	(see page~\pageref{ssec:havetcresolution}).\\
-	\ssrC{have foo : ty. (* TC inference for ty *)}\\
-	\ssrC{have foo : ty := . (* no TC inference for ty *)}\\
-	\ssrC{have foo : ty := t. (* no TC inference for ty and t *)}\\
-	\ssrC{have foo := t. (* no TC inference for t *)}
-\item Transparent flag for \ssrC{have} to generate a \ssrC{let in} context entry
-	(see page~\pageref{sec:havetransparent}).\\
-        \ssrC{have @i : 'I\_n by apply: (Sub m); auto.}
-\item Intro pattern \ssrC{[: foo bar ]} to create abstract variables
-	(see page~\pageref{ssec:introabstract}).
-\item Tactic \ssrC{abstract:} to assign an abstract variable
-	(see page~\pageref{ssec:abstract}).\\
-  \ssrC{have [: blurb ] @i : 'I\_n by apply: (Sub m); abstract: blurb; auto.}\\
-  \ssrC{have [: blurb ] i : 'I\_n := Sub m  blurb; first by auto.}
-
-\end{itemize}
-
-\fi
diff --git a/doc/refman/Reference-Manual.tex b/doc/refman/Reference-Manual.tex
index 5122b2a36..231742e56 100644
--- a/doc/refman/Reference-Manual.tex
+++ b/doc/refman/Reference-Manual.tex
@@ -105,7 +105,6 @@ Options A and B of the licence are {\em not} elected.}
 \lstset{language=SSR}
 \lstset{moredelim=[is][]{|*}{*|}}
 \lstset{moredelim=*[is][\itshape\rmfamily]{/*}{*/}}
-\include{RefMan-ssr}
 
 \part{User extensions}
 %%SUPPRIME \include{RefMan-tus.v}% Writing tactics
diff --git a/doc/sphinx/index.rst b/doc/sphinx/index.rst
index b3f160df1..4566db494 100644
--- a/doc/sphinx/index.rst
+++ b/doc/sphinx/index.rst
@@ -26,6 +26,7 @@ Table of contents
 
    proof-engine/tactics
    proof-engine/detailed-tactic-examples
+   proof-engine/ssreflect-proof-language
 
 .. toctree::
    :caption: User extensions
diff --git a/doc/sphinx/proof-engine/ssreflect-proof-language.rst b/doc/sphinx/proof-engine/ssreflect-proof-language.rst
new file mode 100644
index 000000000..61dffa024
--- /dev/null
+++ b/doc/sphinx/proof-engine/ssreflect-proof-language.rst
@@ -0,0 +1,5497 @@
+.. include:: ../replaces.rst
+
+.. _thessreflectprooflanguage:
+
+------------------------------
+ The |SSR| proof language
+------------------------------
+
+:Source: https://coq.inria.fr/distrib/current/refman/ssreflect.html
+:Converted by: Enrico Tassi
+
+Author: Georges Gonthier, Assia Mahboubi, Enrico Tassi
+
+
+Introduction
+------------
+
+This chapter describes a set of tactics known as |SSR| originally
+designed to provide support for the so-called *small scale reflection*
+proof methodology. Despite the original purpose this set of tactic is
+of general interest and is available in |Coq| starting from version 8.7.
+
+|SSR| was developed independently of the tactics described in
+Chapter :ref:`tactics`. Indeed the scope of the tactics part of |SSR| largely
+overlaps with the standard set of tactics. Eventually the overlap will
+be reduced in future releases of |Coq|.
+
+Proofs written in |SSR| typically look quite different from the
+ones written using only tactics as per Chapter :ref:`tactics`. We try to
+summarise here the most “visible” ones in order to help the reader
+already accustomed to the tactics described in Chapter :ref:`tactics` to read
+this chapter.
+
+The first difference between the tactics described in this chapter and the
+tactics described in Chapter :ref:`tactics` is the way hypotheses are managed
+(we call this *bookkeeping*). In Chapter :ref:`tactics` the most common
+approach is to avoid moving explicitly hypotheses back and forth between the
+context and the conclusion of the goal. On the contrary in |SSR| all
+bookkeeping is performed on the conclusion of the goal, using for that
+purpose a couple of syntactic constructions behaving similar to tacticals
+(and often named as such in this chapter). The ``:`` tactical moves hypotheses
+from the context to the conclusion, while ``=>`` moves hypotheses from the
+conclusion to the context, and in moves back and forth an hypothesis from the
+context to the conclusion for the time of applying an action to it.
+
+While naming hypotheses is commonly done by means of an ``as`` clause in the
+basic model of Chapter :ref:`tactics`, it is here to ``=>`` that this task is
+devoted. Tactics frequently leave new assumptions in the conclusion, and are
+often followed by ``=>`` to explicitly name them. While generalizing the
+goal is normally not explicitly needed in Chapter :ref:`tactics`, it is an
+explicit operation performed by ``:``.
+
+Beside the difference of bookkeeping model, this chapter includes
+specific tactics which have no explicit counterpart in Chapter :ref:`tactics`
+such as tactics to mix forward steps and generalizations as generally
+have or without loss.
+
+|SSR| adopts the point of view that rewriting, definition
+expansion and partial evaluation participate all to a same concept of
+rewriting a goal in a larger sense. As such, all these functionalities
+are provided by the rewrite tactic.
+
+|SSR| includes a little language of patterns to select subterms in
+tactics or tacticals where it matters. Its most notable application is
+in the rewrite tactic, where patterns are used to specify where the
+rewriting step has to take place.
+
+Finally, |SSR| supports so-called reflection steps, typically
+allowing to switch back and forth between the computational view and
+logical view of a concept.
+
+To conclude it is worth mentioning that |SSR| tactics can be mixed
+with non |SSR| tactics in the same proof, or in the same Ltac
+expression. The few exceptions to this statement are described in
+section :ref:`compatibility_issues_ssr`.
+
+
+Acknowledgments
+~~~~~~~~~~~~~~~
+
+The authors would like to thank Frédéric Blanqui, François Pottier and
+Laurence Rideau for their comments and suggestions.
+
+
+Usage
+-----
+
+
+Getting started
+~~~~~~~~~~~~~~~
+
+To be available, the tactics presented in this manual need the
+following minimal set of libraries to loaded: ``ssreflect.v``,
+``ssrfun.v`` and ``ssrbool.v``.
+Moreover, these tactics come with a methodology
+specific to the authors of |SSR| and which requires a few options
+to be set in a different way than in their default way. All in all,
+this corresponds to working in the following context:
+
+.. coqtop:: all
+
+   From Coq Require Import ssreflect ssrfun ssrbool.
+   Set Implicit Arguments.
+   Unset Strict Implicit.
+   Unset Printing Implicit Defensive.
+
+.. _compatibility_issues_ssr:
+
+
+Compatibility issues
+~~~~~~~~~~~~~~~~~~~~
+
+Requiring the above modules creates an environment which is mostly
+compatible with the rest of |Coq|, up to a few discrepancies:
+
+
++ New keywords (``is``) might clash with variable, constant, tactic or
+  tactical names, or with quasi-keywords in tactic or vernacular
+  notations.
++ New tactic(al)s names (``last``, ``done``, ``have``, ``suffices``,
+  ``suff``, ``without loss``, ``wlog``, ``congr``, ``unlock``)
+  might clash with user tactic names.
++ Identifiers with both leading and trailing ``_``, such as ``_x_``, are
+  reserved by |SSR| and cannot appear in scripts.
++ The extensions to the rewrite tactic are partly incompatible with those
+  available in current versions of |Coq|; in particular: ``rewrite .. in
+  (type of k)`` or ``rewrite .. in *`` or any other variant of ``rewrite``
+  will not work, and the |SSR| syntax and semantics for occurrence selection
+  and rule chaining is different. Use an explicit rewrite direction
+  (``rewrite <- …`` or ``rewrite -> …``) to access the |Coq| rewrite tactic.
++ New symbols (``//``, ``/=``, ``//=``) might clash with adjacent
+  existing symbols.
+  This can be avoided by inserting white spaces.
++ New constant and theorem names might clash with the user theory.
+  This can be avoided by not importing all of |SSR|:
+
+  .. coqtop:: in
+
+     From Coq Require ssreflect.
+     Import ssreflect.SsrSyntax.
+
+  Note that the full
+  syntax of |SSR|’s rewrite and reserved identifiers are enabled
+  only if the ssreflect module has been required and if ``SsrSyntax`` has
+  been imported. Thus a file that requires (without importing) ssreflect
+  and imports ``SsrSyntax``, can be required and imported without
+  automatically enabling |SSR|’s extended rewrite syntax and
+  reserved identifiers.
++ Some user notations (in particular, defining an infix ``;``) might
+  interfere with the "open term", parenthesis free, syntax of tactics
+  such as have, set and pose.
++ The generalization of if statements to non-Boolean conditions is turned off
+  by |SSR|, because it is mostly subsumed by Coercion to ``bool`` of the
+  ``sumXXX`` types (declared in ``ssrfun.v``) and the ``if`` *term* ``is`` *pattern* ``then``
+  *term* ``else`` *term* construct (see :ref:`pattern_conditional_ssr`).  To use the
+  generalized form, turn off the |SSR| Boolean if notation using the command:
+  ``Close Scope boolean_if_scope``.
++ The following two options can be unset to disable the incompatible
+  rewrite syntax and allow reserved identifiers to appear in scripts.
+
+  .. coqtop:: in
+
+     Unset SsrRewrite.
+     Unset SsrIdents.
+
+
+|Gallina| extensions
+--------------------
+
+Small-scale reflection makes an extensive use of the programming
+subset of |Gallina|, |Coq|’s logical specification language. This subset
+is quite suited to the description of functions on representations,
+because it closely follows the well-established design of the ML
+programming language. The |SSR| extension provides three additions
+to |Gallina|, for pattern assignment, pattern testing, and polymorphism;
+these mitigate minor but annoying discrepancies between |Gallina| and
+ML.
+
+
+Pattern assignment
+~~~~~~~~~~~~~~~~~~
+
+The |SSR| extension provides the following construct for
+irrefutable pattern matching, that is, destructuring assignment:
+
+.. prodn::
+   term += let: @pattern := @term in @term
+
+Note the colon ``:`` after the ``let`` keyword, which avoids any ambiguity
+with a function definition or |Coq|’s basic destructuring let. The let:
+construct differs from the latter in that
+
+
++ The pattern can be nested (deep pattern matching), in particular,
+  this allows expression of the form:
+
+.. coqtop:: in
+
+  let: exist (x, y) p_xy := Hp in … .
+
++ The destructured constructor is explicitly given in the pattern, and
+  is used for type inference.
+
+  .. example::
+
+    .. coqtop:: reset
+
+       From Coq Require Import ssreflect.
+       Set Implicit Arguments.
+       Unset Strict Implicit.
+       Unset Printing Implicit Defensive.
+
+    .. coqtop:: all
+
+       Definition f u := let: (m, n) := u in m + n.
+       Check f.
+
+    Using :g:`let:` Coq infers a type for :g:`f`,
+    whereas with a usual ``let`` the same term requires an extra type
+    annotation in order to type check.
+
+    .. coqtop:: reset all
+
+       Definition f u := let (m, n) := u in m + n.
+
+
+The ``let:`` construct is just (more legible) notation for the primitive
+|Gallina| expression
+
+.. coqtop:: in
+
+   match term with pattern => term end.
+
+The |SSR| destructuring assignment supports all the dependent
+match annotations; the full syntax is
+
+.. prodn::
+   term += let: @pattern {? as @ident} {? in @pattern} := @term {? return @term} in @term
+
+where the second :token:`pattern` and the second :token:`term` are *types*.
+
+When the ``as`` and ``return`` keywords are both present, then :token:`ident` is bound
+in both the second :token:`pattern` and the second :token:`term`; variables
+in the optional type :token:`pattern` are bound only in the second term, and
+other variables in the first  :token:`pattern` are bound only in the third
+:token:`term`, however.
+
+
+.. _pattern_conditional_ssr:
+
+Pattern conditional
+~~~~~~~~~~~~~~~~~~~
+
+The following construct can be used for a refutable pattern matching,
+that is, pattern testing:
+
+.. prodn::
+   term += if @term is @pattern then @term else @term
+
+Although this construct is not strictly ML (it does exist in variants
+such as the pattern calculus or the ρ-calculus), it turns out to be
+very convenient for writing functions on representations, because most
+such functions manipulate simple data types such as Peano integers,
+options, lists, or binary trees, and the pattern conditional above is
+almost always the right construct for analyzing such simple types. For
+example, the null and all list function(al)s can be defined as follows:
+
+.. example::
+
+    .. coqtop:: reset
+
+       From Coq Require Import ssreflect.
+       Set Implicit Arguments.
+       Unset Strict Implicit.
+       Unset Printing Implicit Defensive.
+       Section Test.
+
+   .. coqtop:: all
+
+      Variable d: Set.
+      Fixpoint null (s : list d) :=
+        if s is nil then true else false.
+      Variable a : d -> bool.
+      Fixpoint all (s : list d) : bool :=
+        if s is cons x s' then a x && all s' else true.
+
+The pattern conditional also provides a notation for destructuring
+assignment with a refutable pattern, adapted to the pure functional
+setting of |Gallina|, which lacks a ``Match_Failure`` exception.
+
+Like ``let:`` above, the ``if…is`` construct is just (more legible) notation
+for the primitive |Gallina| expression:
+
+.. coqtop:: in
+
+   match term with pattern => term | _ => term end.
+
+Similarly, it will always be displayed as the expansion of this form
+in terms of primitive match expressions (where the default expression
+may be replicated).
+
+Explicit pattern testing also largely subsumes the generalization of
+the if construct to all binary data types; compare:
+
+.. coqtop:: in
+
+   if term is inl _ then term else term.
+
+and:
+
+.. coqtop:: in
+
+   if term then term else term.
+
+The latter appears to be marginally shorter, but it is quite
+ambiguous, and indeed often requires an explicit annotation
+``(term : {_} + {_})`` to type-check, which evens the character count.
+
+Therefore, |SSR| restricts by default the condition of a plain if
+construct to the standard ``bool`` type; this avoids spurious type
+annotations.
+
+.. example::
+
+   .. coqtop:: all
+
+      Definition orb b1 b2 := if b1 then true else b2.
+
+As pointed out in section :ref:`compatibility_issues_ssr`,
+this restriction can be removed with
+the command:
+
+``Close Scope boolean_if_scope.``
+
+Like ``let:`` above, the ``if-is-then-else``
+construct supports
+the dependent match annotations:
+
+.. prodn::
+   term += if @term is @pattern as @ident in @pattern return @term then @term else @term
+
+As in ``let:`` the variable :token:`ident` (and those in the type pattern)
+are bound in the second :token:`term`; :token:`ident` is also bound in the
+third :token:`term` (but not in the fourth :token:`term`), while the
+variables in the first :token:`pattern` are bound only in the third
+:token:`term`.
+
+Another variant allows to treat the ``else`` case first:
+
+.. prodn::
+   term += if @term isn't @pattern then @term else @term
+
+Note that :token:`pattern` eventually binds variables in the third
+:token:`term` and not in the second :token:`term`.
+
+.. _parametric_polymorphism_ssr:
+
+Parametric polymorphism
+~~~~~~~~~~~~~~~~~~~~~~~
+
+Unlike ML, polymorphism in core |Gallina| is explicit: the type
+parameters of polymorphic functions must be declared explicitly, and
+supplied at each point of use. However, |Coq| provides two features to
+suppress redundant parameters:
+
+
++ Sections are used to provide (possibly implicit) parameters for a
+  set of definitions.
++ Implicit arguments declarations are used to tell |Coq| to use type
+  inference to deduce some parameters from the context at each point of
+  call.
+
+
+The combination of these features provides a fairly good emulation of
+ML-style polymorphism, but unfortunately this emulation breaks down
+for higher-order programming. Implicit arguments are indeed not
+inferred at all points of use, but only at points of call, leading to
+expressions such as
+
+.. example::
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+      Section Test.
+      Variable T : Type.
+      Variable null : forall T : Type, T -> bool.
+      Variable all : (T -> bool) -> list T -> bool.
+
+   .. coqtop:: all
+
+      Definition all_null (s : list T) := all (@null T) s.
+
+Unfortunately, such higher-order expressions are quite frequent in
+representation functions, especially those which use |Coq|'s
+``Structures`` to emulate Haskell type classes.
+
+Therefore, |SSR| provides a variant of |Coq|’s implicit argument
+declaration, which causes |Coq| to fill in some implicit parameters at
+each point of use, e.g., the above definition can be written:
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+     Variable T : Type.
+     Variable null : forall T : Type, T -> bool.
+     Variable all : (T -> bool) -> list T -> bool.
+
+
+  .. coqtop:: all undo
+
+     Prenex Implicits null.
+     Definition all_null (s : list T) := all null s.
+
+Better yet, it can be omitted entirely, since ``all_null s`` isn’t much of
+an improvement over ``all null s``.
+
+The syntax of the new declaration is
+
+.. cmd:: Prenex Implicits {+ @ident}.
+
+Let us denote :math:`c_1` … :math:`c_n` the list of identifiers given to a
+``Prenex Implicits`` command.  The command checks that each ci is the name of
+a functional constant, whose implicit arguments are prenex, i.e., the first
+:math:`n_i > 0` arguments of :math:`c_i` are implicit; then it assigns
+``Maximal Implicit`` status to these arguments.
+
+As these prenex implicit arguments are ubiquitous and have often large
+display strings, it is strongly recommended to change the default
+display settings of |Coq| so that they are not printed (except after
+a ``Set Printing All command``). All |SSR| library files thus start
+with the incantation
+
+.. coqtop:: all undo
+
+   Set Implicit Arguments.
+   Unset Strict Implicit.
+   Unset Printing Implicit Defensive.
+
+
+Anonymous arguments
+~~~~~~~~~~~~~~~~~~~
+
+When in a definition, the type of a certain argument is mandatory, but
+not its name, one usually use “arrow” abstractions for prenex
+arguments, or the ``(_ : term)`` syntax for inner arguments. In |SSR|,
+the latter can be replaced by the open syntax ``of term`` or
+(equivalently) ``& term``, which are both syntactically equivalent to a
+``(_ : term)`` expression.
+
+For instance, the usual two-constructor polymorphic type list, i.e.
+the one of the standard List library, can be defined by the following
+declaration:
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Inductive list (A : Type) : Type := nil | cons of A & list A.
+
+
+Wildcards
+~~~~~~~~~
+
+The terms passed as arguments to |SSR| tactics can contain
+*holes*, materialized by wildcards ``_``. Since |SSR| allows a more
+powerful form of type inference for these arguments, it enhances the
+possibilities of using such wildcards. These holes are in particular
+used as a convenient shorthand for abstractions, especially in local
+definitions or type expressions.
+
+Wildcards may be interpreted as abstractions (see for example sections
+:ref:`definitions_ssr` and ref:`structure_ssr`), or their content can be
+inferred from the whole context of the goal (see for example section
+:ref:`abbreviations_ssr`).
+
+
+.. _definitions_ssr:
+
+Definitions
+~~~~~~~~~~~
+
+The pose tactic allows to add a defined constant to a proof context.
+|SSR| generalizes this tactic in several ways. In particular, the
+|SSR| pose tactic supports *open syntax*: the body of the
+definition does not need surrounding parentheses. For instance:
+
+.. coqtop:: reset
+
+   From Coq Require Import ssreflect.
+   Set Implicit Arguments.
+   Unset Strict Implicit.
+   Unset Printing Implicit Defensive.
+   Lemma test : True.
+   Proof.
+
+.. coqtop:: in
+
+   pose t := x + y.
+
+is a valid tactic expression.
+
+The pose tactic is also improved for the local definition of higher
+order terms. Local definitions of functions can use the same syntax as
+global ones. For example the tactic ``pose`` supoprts parameters:
+
+.. example::
+
+   .. coqtop:: all
+
+      Lemma test : True.
+      pose f x y := x + y.
+
+The |SSR| pose tactic also supports (co)fixpoints, by providing
+the local counterpart of the ``Fixpoint f := …`` and ``CoFixpoint f := …``
+constructs. For instance, the following tactic:
+
+.. coqtop:: in
+
+   pose fix f (x y : nat) {struct x} : nat :=
+     if x is S p then S (f p y) else 0.
+
+defines a local fixpoint ``f``, which mimics the standard plus operation
+on natural numbers.
+
+Similarly, local cofixpoints can be defined by a tactic of the form:
+
+.. coqtop:: in
+
+   pose cofix f (arg : T) := … .
+
+The possibility to include wildcards in the body of the definitions
+offers a smooth way of defining local abstractions. The type of
+“holes” is guessed by type inference, and the holes are abstracted.
+For instance the tactic:
+
+.. coqtop:: in
+
+   pose f := _ + 1.
+
+is shorthand for:
+
+.. coqtop:: in
+
+   pose f n := n + 1.
+
+When the local definition of a function involves both arguments and
+holes, hole abstractions appear first. For instance, the tactic:
+
+.. coqtop:: in
+
+   pose f x := x + _.
+
+is shorthand for:
+
+.. coqtop:: in
+
+   pose f n x := x + n.
+
+The interaction of the pose tactic with the interpretation of implicit
+arguments results in a powerful and concise syntax for local
+definitions involving dependent types. For instance, the tactic:
+
+.. coqtop:: in
+
+   pose f x y := (x, y).
+
+adds to the context the local definition:
+
+.. coqtop:: in
+
+   pose f (Tx Ty : Type) (x : Tx) (y : Ty) := (x, y).
+
+The generalization of wildcards makes the use of the pose tactic
+resemble ML-like definitions of polymorphic functions.
+
+
+.. _abbreviations_ssr:
+
+
+Abbreviations
+~~~~~~~~~~~~~
+
+The |SSR| set tactic performs abbreviations: it introduces a
+defined constant for a subterm appearing in the goal and/or in the
+context.
+
+|SSR| extends the set tactic by supplying:
+
+
++ an open syntax, similarly to the pose tactic;
++ a more aggressive matching algorithm;
++ an improved interpretation of wildcards, taking advantage of the
+  matching algorithm;
++ an improved occurrence selection mechanism allowing to abstract only
+  selected occurrences of a term.
+
+
+The general syntax of this tactic is
+
+.. tacn:: set @ident {? : @term } := {? @occ_switch } @term
+   :name: set (ssreflect)
+
+.. prodn::
+   occ_switch ::= { {? + %| - } {* @num } }
+
+where:
+
++ :token:`ident` is a fresh identifier chosen by the user.
++ term 1 is an optional type annotation. The type annotation term 1
+  can be given in open syntax (no surrounding parentheses). If no
+  :token:`occ_switch` (described hereafter) is present,
+  it is also the case for the second :token:`term`.
+  On the other hand, in presence of :token:`occ_switch`, parentheses
+  surrounding the second :token:`term` are mandatory.
++ In the occurrence switch :token:`occ_switch`, if the first element of the
+  list is a natural, this element should be a number, and not an Ltac
+  variable. The empty list {} is not interpreted as a valid occurrence
+  switch.
+
+The tactic:
+
+.. example::
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+      Axiom f : nat -> nat.
+
+   .. coqtop:: all
+
+      Lemma test x :  f x + f x = f x.
+      set t := f _.
+
+   .. coqtop:: none
+
+      Undo.
+
+   .. coqtop:: all
+
+      set t := {2}(f _).
+
+
+The type annotation may contain wildcards, which will be filled
+with the appropriate value by the matching process.
+
+The tactic first tries to find a subterm of the goal matching
+the second :token:`term`
+(and its type), and stops at the first subterm it finds. Then
+the occurrences of this subterm selected by the optional :token:`occ_switch`
+are replaced by :token:`ident` and a definition ``ident := term``
+is added to the
+context. If no :token:`occ_switch` is present, then all the occurrences are
+abstracted.
+
+
+Matching
+````````
+
+The matching algorithm compares a pattern ``term`` with a subterm of the
+goal by comparing their heads and then pairwise unifying their
+arguments (modulo conversion). Head symbols match under the following
+conditions:
+
+
++ If the head of ``term`` is a constant, then it should be syntactically
+  equal to the head symbol of the subterm.
++ If this head is a projection of a canonical structure, then
+  canonical structure equations are used for the matching.
++ If the head of term is *not* a constant, the subterm should have the
+  same structure (λ abstraction,let…in structure …).
++ If the head of ``term`` is a hole, the subterm should have at least as
+  many arguments as ``term``.
+
+.. example::
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+
+   .. coqtop:: all
+
+      Lemma test (x y z : nat) :  x + y = z.
+      set t := _ x.
+
++ In the special case where ``term`` is of the form
+  ``(let f := t0 in f) t1 … tn`` , then the pattern ``term`` is treated
+  as ``(_ t1 … tn)``. For each
+  subterm in the goal having the form ``(A u1 … um)`` with m ≥ n, the
+  matching algorithm successively tries to find the largest partial
+  application ``(A u1 … uj)`` convertible to the head ``t0`` of ``term``.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+
+     .. coqtop:: all
+
+        Lemma test : (let f x y z := x + y + z in f 1) 2 3 = 6.
+        set t := (let g y z := S y + z in g) 2.
+
+  The notation ``unkeyed`` defined in ``ssreflect.v`` is a shorthand for
+  the degenerate term ``let x := … in x``.
+
+Moreover:
+
++ Multiple holes in ``term`` are treated as independent placeholders.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+
+     .. coqtop:: all
+
+        Lemma test x y z : x + y = z.
+        set t := _ + _.
+
++ The type of the subterm matched should fit the type (possibly casted
+  by some type annotations) of the pattern ``term``.
++ The replacement of the subterm found by the instantiated pattern
+  should not capture variables. In the example above ``x`` is bound
+  and should not be captured.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+
+     .. coqtop:: all
+
+        Lemma test : forall x : nat, x + 1 = 0.
+        set t := _ + 1.
+
++ Typeclass inference should fill in any residual hole, but matching
+  should never assign a value to a global existential variable.
+
+
+.. _occurrence_selection_ssr:
+
+Occurrence selection
+````````````````````
+
+|SSR| provides a generic syntax for the selection of occurrences
+by their position indexes. These *occurrence switches* are shared by
+all |SSR| tactics which require control on subterm selection like
+rewriting, generalization, …
+
+An *occurrence switch* can be:
+
++ A list natural numbers ``{+ n1 … nm}``
+  of occurrences affected by the tactic.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+        Axiom f : nat -> nat.
+
+     .. coqtop:: all
+
+        Lemma test : f 2 + f 8 = f 2 + f 2.
+        set x := {+1 3}(f 2).
+
+  Notice that some occurrences of a given term may be
+  hidden to the user, for example because of a notation. The vernacular
+  ``Set Printing All`` command displays all these hidden occurrences and
+  should be used to find the correct coding of the occurrences to be
+  selected [#1]_.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+
+     .. coqtop:: all
+
+        Notation "a < b":= (le (S a) b).
+        Lemma test x y : x < y -> S x < S y.
+        set t := S x.
+
++ A list of natural numbers between ``{n1 … nm}``.
+  This is equivalent to the previous ``{+ n1 … nm}`` but the list
+  should start with a number, and not with an Ltac variable.
++ A list ``{- n1 … nm}`` of occurrences *not* to be affected by the
+  tactic.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+        Axiom f : nat -> nat.
+
+     .. coqtop:: all
+
+        Lemma test : f 2 + f 8 = f 2 + f 2.
+        set x := {-2}(f 2).
+
+
+  Note that, in this goal, it behaves like ``set x := {1 3}(f 2).``
++ In particular, the switch ``{+}`` selects *all* the occurrences. This
+  switch is useful to turn off the default behavior of a tactic which
+  automatically clears some assumptions (see section :ref:`discharge_ssr` for
+  instance).
++ The switch ``{-}`` imposes that *no* occurrences of the term should be
+  affected by the tactic. The tactic: ``set x := {-}(f 2).`` leaves the goal
+  unchanged and adds the definition ``x := f 2`` to the context. This kind
+  of tactic may be used to take advantage of the power of the matching
+  algorithm in a local definition, instead of copying large terms by
+  hand.
+
+It is important to remember that matching *preceeds* occurrence
+selection.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+
+     .. coqtop:: all
+
+        Lemma test x y z : x + y = x + y + z.
+        set a := {2}(_ + _).
+
+Hence, in the following goal, the same tactic fails since there is
+only one occurrence of the selected term.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+
+     .. coqtop:: all
+
+        Lemma test x y z : (x + y) + (z + z) = z + z.
+        set a := {2}(_ + _).
+
+
+.. _basic_localization_ssr:
+
+Basic localization
+~~~~~~~~~~~~~~~~~~
+
+It is possible to define an abbreviation for a term appearing in the
+context of a goal thanks to the in tactical.
+
+A tactic of the form:
+
+.. tacv:: set @ident := @term in {+ @ident}
+
+introduces a defined constant called ``x`` in the context, and folds it in
+the context entries mentioned on the right hand side of ``in``.
+The body of ``x`` is the first subterm matching these context entries
+(taken in the given order).
+
+A tactic of the form:
+
+.. tacv:: set @ident := @term in {+ @ident} *
+
+matches term and then folds ``x`` similarly in all the given context entries
+but also folds ``x`` in the goal.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+
+     .. coqtop:: all undo
+
+        Lemma test x t (Hx : x = 3) : x + t = 4.
+        set z := 3 in Hx.
+
+If the localization also mentions the goal, then the result is the following one:
+
+  .. example::
+
+     .. coqtop:: all
+
+        Lemma test x t (Hx : x = 3) : x + t = 4.
+        set z := 3 in Hx * .
+
+Indeed, remember that 4 is just a notation for (S 3).
+
+The use of the ``in`` tactical is not limited to the localization of
+abbreviations: for a complete description of the in tactical, see
+section :ref:`bookkeeping_ssr` and :ref:`localization_ssr`.
+
+
+.. _basic_tactics_ssr:
+
+Basic tactics
+-------------
+
+A sizable fraction of proof scripts consists of steps that do not
+"prove" anything new, but instead perform menial bookkeeping tasks
+such as selecting the names of constants and assumptions or splitting
+conjuncts. Although they are logically trivial, bookkeeping steps are
+extremely important because they define the structure of the data-flow
+of a proof script. This is especially true for reflection-based
+proofs, which often involve large numbers of constants and
+assumptions. Good bookkeeping consists in always explicitly declaring
+(i.e., naming) all new constants and assumptions in the script, and
+systematically pruning irrelevant constants and assumptions in the
+context. This is essential in the context of an interactive
+development environment (IDE), because it facilitates navigating the
+proof, allowing to instantly "jump back" to the point at which a
+questionable assumption was added, and to find relevant assumptions by
+browsing the pruned context. While novice or casual |Coq| users may find
+the automatic name selection feature convenient, the usage of such a
+feature severely undermines the readability and maintainability of
+proof scripts, much like automatic variable declaration in programming
+languages. The |SSR| tactics are therefore designed to support
+precise bookkeeping and to eliminate name generation heuristics. The
+bookkeeping features of |SSR| are implemented as tacticals (or
+pseudo-tacticals), shared across most |SSR| tactics, and thus form
+the foundation of the |SSR| proof language.
+
+
+.. _bookkeeping_ssr:
+
+Bookkeeping
+~~~~~~~~~~~
+
+During the course of a proof |Coq| always present the user with a
+*sequent* whose general form is::
+
+  ci : Ti
+  …
+  dj := ej : Tj
+  …
+  Fk : Pk
+  …
+  =================
+  forall (xl : Tl ) …,
+  let ym := bm in … in
+  Pn -> … -> C
+
+The *goal* to be proved appears below the double line; above the line
+is the *context* of the sequent, a set of declarations of *constants*
+``ci`` , *defined constants* d i , and *facts* ``Fk`` that can be used to
+prove the goal (usually, ``Ti`` , ``Tj : Type`` and ``Pk : Prop``).
+The various
+kinds of declarations can come in any order. The top part of the
+context consists of declarations produced by the Section
+commands ``Variable``, ``Let``, and ``Hypothesis``.
+This *section context* is never
+affected by the |SSR| tactics: they only operate on the lower part
+— the *proof context*. As in the figure above, the goal often
+decomposes into a series of (universally) quantified *variables*
+``(xl : Tl)``, local *definitions*
+``let ym := bm in``, and *assumptions*
+``P n ->``,
+and a *conclusion* ``C`` (as in the context, variables, definitions, and
+assumptions can appear in any order). The conclusion is what actually
+needs to be proved — the rest of the goal can be seen as a part of the
+proof context that happens to be “below the line”.
+
+However, although they are logically equivalent, there are fundamental
+differences between constants and facts on the one hand, and variables
+and assumptions on the others. Constants and facts are *unordered*,
+but *named* explicitly in the proof text; variables and assumptions
+are *ordered*, but *unnamed*: the display names of variables may
+change at any time because of α-conversion.
+
+Similarly, basic deductive steps such as apply can only operate on the
+goal because the |Gallina| terms that control their action (e.g., the
+type of the lemma used by ``apply``) only provide unnamed bound variables.
+[#2]_ Since the proof script can only refer directly to the context, it
+must constantly shift declarations from the goal to the context and
+conversely in between deductive steps.
+
+In |SSR| these moves are performed by two *tacticals* ``=>`` and
+``:``, so that the bookkeeping required by a deductive step can be
+directly associated to that step, and that tactics in an |SSR|
+script correspond to actual logical steps in the proof rather than
+merely shuffle facts. Still, some isolated bookkeeping is unavoidable,
+such as naming variables and assumptions at the beginning of a
+proof. |SSR| provides a specific ``move`` tactic for this purpose.
+
+Now ``move`` does essentially nothing: it is mostly a placeholder for
+``=>`` and ``:``. The ``=>`` tactical moves variables, local definitions,
+and assumptions to the context, while the ``:`` tactical moves facts and
+constants to the goal.
+
+.. example::
+
+   For example, the proof of [#3]_
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+
+   .. coqtop:: all
+
+      Lemma subnK : forall m n, n <= m -> m - n + n = m.
+
+   might start with
+
+   .. coqtop:: all
+
+      move=> m n le_n_m.
+
+   where move does nothing, but ``=> m n le_m_n`` changes
+   the variables and assumption of the goal in the constants
+   ``m n : nat`` and the fact ``le_n_m : n <=  m``, thus exposing the
+   conclusion ``m - n + n = m``.
+
+   The ``:`` tactical is the converse of ``=>``, indeed it removes facts and
+   constants from the context by turning them into variables and
+   assumptions.
+
+   .. coqtop:: all
+
+      move: m le_n_m.
+
+   turns back ``m`` and ``le_m_n`` into a variable and an assumption,
+   removing them from the proof context, and changing the goal to
+   ``forall m, n <= m -> m - n + n = m``
+   which can be proved by induction on ``n`` using ``elim: n``.
+
+Because they are tacticals, ``:`` and ``=>`` can be combined, as in
+
+.. coqtop: in
+
+   move: m le_n_m => p le_n_p.
+
+simultaneously renames ``m`` and ``le_m_n`` into ``p`` and ``le_n_p``,
+respectively, by first turning them into unnamed variables, then
+turning these variables back into constants and facts.
+
+Furthermore, |SSR| redefines the basic |Coq| tactics ``case``, ``elim``,
+and ``apply`` so that they can take better advantage of
+``:`` and ``=>``. In there
+|SSR| variants, these tactic operate on the first variable or
+constant of the goal and they do not use or change the proof context.
+The ``:`` tactical is used to operate on an element in the context.
+
+.. example::
+
+   For instance the proof of ``subnK`` could continue with ``elim: n``.
+   Instead of ``elim n`` (note, no colon), this has the advantage of
+   removing n from the context. Better yet, this ``elim`` can be combined
+   with previous move and with the branching version of the ``=>`` tactical
+   (described in :ref:`introduction_ssr`),
+   to encapsulate the inductive step in a single
+   command:
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma subnK : forall m n, n <= m -> m - n + n = m.
+     move=> m n le_n_m.
+     elim: n m le_n_m => [|n IHn] m => [_ | lt_n_m].
+
+  which breaks down the proof into two subgoals, the second one
+  having in its context
+  ``lt_n_m : S n <= m`` and
+  ``IHn : forall m, n <= m -> m - n + n = m``.
+
+The ``:`` and ``=>`` tacticals can be explained very simply if one views
+the goal as a stack of variables and assumptions piled on a conclusion:
+
++ ``tactic : a b c`` pushes the context constants ``a``, ``b``, ``c`` as goal
+  variables *before* performing tactic.
++ ``tactic => a b c`` pops the top three goal variables as context
+  constants ``a``, ``b``, ``c``, *after* tactic has been performed.
+
+These pushes and pops do not need to balance out as in the examples
+above, so ``move: m le_n_m => p``
+would rename ``m`` into ``p``, but leave an extra assumption ``n <= p``
+in the goal.
+
+Basic tactics like apply and elim can also be used without the ’:’
+tactical: for example we can directly start a proof of ``subnK`` by
+induction on the top variable ``m`` with
+
+.. coqtop:: in
+
+   elim=> [|m IHm] n le_n.
+
+The general form of the localization tactical in is also best
+explained in terms of the goal stack:
+
+.. coqtop:: in
+
+   tactic in a H1 H2 *.
+
+is basically equivalent to
+
+.. coqtop:: in
+
+   move: a H1 H2; tactic => a H1 H2.
+
+
+with two differences: the in tactical will preserve the body of a ifa
+is a defined constant, and if the ``*`` is omitted it will use a
+temporary abbreviation to hide the statement of the goal from
+``tactic``.
+
+The general form of the in tactical can be used directly with the
+``move``, ``case`` and ``elim`` tactics, so that one can write
+
+.. coqtop:: in
+
+   elim: n => [|n IHn] in m le_n_m *.
+
+instead of
+
+.. coqtop:: in
+
+   elim: n m le_n_m => [|n IHn] m le_n_m.
+
+This is quite useful for inductive proofs that involve many facts.
+
+See section :ref:`localization_ssr` for
+the general syntax and presentation of the in
+tactical.
+
+
+.. _the_defective_tactics_ssr:
+
+The defective tactics
+~~~~~~~~~~~~~~~~~~~~~
+
+In this section we briefly present the three basic tactics performing
+context manipulations and the main backward chaining tool.
+
+
+The move tactic.
+````````````````
+
+The move tactic, in its defective form, behaves like the primitive ``hnf``
+|Coq| tactic. For example, such a defective:
+
+.. tacn:: move
+   :name: move
+
+exposes the first assumption in the goal, i.e. its changes the
+goal ``not False`` into ``False -> False``.
+
+More precisely, the ``move`` tactic inspects the goal and does nothing
+(``idtac``) if an introduction step is possible, i.e. if the goal is a
+product or a ``let…in``, and performs ``hnf`` otherwise.
+
+Of course this tactic is most often used in combination with the
+bookkeeping tacticals (see section :ref:`introduction_ssr` and :ref:`discharge_ssr`). These
+combinations mostly subsume the ``intros``, ``generalize``, ``revert``, ``rename``,
+``clear`` and ``pattern`` tactics.
+
+
+The case tactic
+```````````````
+
+The ``case`` tactic performs *primitive case analysis* on (co)inductive
+types; specifically, it destructs the top variable or assumption of
+the goal, exposing its constructor(s) and its arguments, as well as
+setting the value of its type family indices if it belongs to a type
+family (see section :ref:`type_families_ssr`).
+
+The |SSR| case tactic has a special behavior on equalities. If the
+top assumption of the goal is an equality, the case tactic “destructs”
+it as a set of equalities between the constructor arguments of its
+left and right hand sides, as per the tactic injection. For example,
+``case`` changes the goal
+
+.. coqtop:: in
+
+   (x, y) = (1, 2) -> G.
+
+into
+
+.. coqtop:: in
+
+   x = 1 -> y = 2 -> G.
+
+Note also that the case of |SSR| performs ``False`` elimination, even
+if no branch is generated by this case operation. Hence the command:
+``case.`` on a goal of the form ``False -> G`` will succeed and
+prove the goal.
+
+
+The elim tactic
+```````````````
+
+The ``elim`` tactic performs inductive elimination on inductive types. The
+defective:
+
+.. tacn:: elim
+   :name: elim (ssreflect)
+
+tactic performs inductive elimination on a goal whose top assumption
+has an inductive type.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test m : forall n : nat, m <= n.
+     elim.
+
+
+.. _apply_ssr:
+
+The apply tactic
+````````````````
+
+The ``apply`` tactic is the main backward chaining tactic of the proof
+system. It takes as argument any :token:`term` and applies it to the goal.
+Assumptions in the type of :token:`term` that don’t directly match the goal
+may generate one or more subgoals.
+
+In fact the |SSR| tactic:
+
+.. tacn:: apply
+   :name: apply (ssreflect)
+
+is a synonym for:
+
+.. coqtop:: in
+
+   intro top; first [refine top | refine (top _) | refine (top _ _) | …]; clear top.
+
+where ``top`` is fresh name, and the sequence of refine tactics tries to
+catch the appropriate number of wildcards to be inserted. Note that
+this use of the refine tactic implies that the tactic tries to match
+the goal up to expansion of constants and evaluation of subterms.
+
+|SSR|’s apply has a special behaviour on goals containing
+existential metavariables of sort Prop.
+
+.. example::
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+      Axiom lt_trans : forall a b c, a < b -> b < c -> a < c.
+
+   .. coqtop:: all
+
+      Lemma test : forall y, 1 < y -> y < 2 -> exists x : { n | n < 3 }, 0 < proj1_sig x.
+      move=> y y_gt1 y_lt2; apply: (ex_intro _ (exist _ y _)).
+        by apply: lt_trans y_lt2 _.
+      by move=> y_lt3; apply: lt_trans y_gt1.
+
+   Note that the last ``_`` of the tactic
+   ``apply: (ex_intro _ (exist _ y _))``
+   represents a proof that ``y < 3``. Instead of generating the goal
+
+   .. coqtop:: in
+
+      0 < proj1_sig (exist (fun n : nat => n < 3) y ?Goal).
+
+   the system tries to prove ``y < 3`` calling the trivial tactic.
+   If it succeeds, let’s say because the context contains
+   ``H : y < 3``, then the
+   system generates the following goal:
+
+   .. coqtop:: in
+
+      0 < proj1_sig (exist (fun n => n < 3) y H).
+
+   Otherwise the missing proof is considered to be irrelevant, and is
+   thus discharged generating the two goals shown above.
+
+   Last, the user can replace the trivial tactic by defining an Ltac
+   expression named ``ssrautoprop``.
+
+
+.. _discharge_ssr:
+
+Discharge
+~~~~~~~~~
+
+The general syntax of the discharging tactical ``:`` is:
+
+.. tacn:: @tactic {? @ident } : {+ @d_item } {? @clear_switch }
+
+.. prodn::
+   d_item ::= {? @occ_switch %| @clear_switch } @term
+.. prodn::
+   clear_switch ::= { {+ @ident } }
+
+with the following requirements:
+
++ :token:`tactic` must be one of the four basic tactics described in :ref:`the_defective_tactics_ssr`,
+  i.e., ``move``, ``case``, ``elim`` or ``apply``, the ``exact``
+  tactic (section :ref:`terminators_ssr`),
+  the ``congr`` tactic (section :ref:`congruence_ssr`),
+  or the application of the *view*
+  tactical ‘/’ (section :ref:`interpreting_assumptions_ssr`) to one of move, case, or elim.
++ The optional :token:`ident` specifies *equation generation* (section :ref:`generation_of_equations_ssr`),
+  and is only allowed if tactic is ``move``, ``case`` or ``elim``, or the
+  application of the view tactical ‘/’ (section :ref:`interpreting_assumptions_ssr`) to ``case`` or ``elim``.
++ An :token:`occ_switch` selects occurrences of :token:`term`, as in :ref:`abbreviations_ssr`; :token:`occ_switch`
+  is not allowed if :token:`tactic` is ``apply`` or ``exact``.
++ A clear item :token:`clear_switch` specifies facts and constants to be
+  deleted from the proof context (as per the clear tactic).
+
+
+The ``:`` tactical first *discharges* all the :token:`d_item`, right to left,
+and then performs tactic, i.e., for each :token:`d_item`, starting with the last one :
+
+
+#. The |SSR| matching algorithm described in section :ref:`abbreviations_ssr` is
+   used to find occurrences of term in the goal, after filling any holes
+   ‘_’ in term; however if tactic is apply or exact a different matching
+   algorithm, described below, is used [#4]_.
+#. These occurrences are replaced by a new variable; in particular, if
+   term is a fact, this adds an assumption to the goal.
+#. If term is *exactly* the name of a constant or fact in the proof
+   context, it is deleted from the context, unless there is an
+   :token:`occ_switch`.
+
+
+Finally, tactic is performed just after the first :token:`d_item`
+has been generalized
+— that is, between steps 2 and 3. The names listed in
+the final :token:`clear_switch` (if it is present) are cleared first, before
+:token:`d_item` n is discharged.
+
+Switches affect the discharging of a :token:`d_item` as follows:
+
+
++ An :token:`occ_switch` restricts generalization (step 2) to a specific subset
+  of the occurrences of term, as per section :ref:`abbreviations_ssr`, and prevents clearing (step
+  3).
++ All the names specified by a :token:`clear_switch` are deleted from the
+  context in step 3, possibly in addition to term.
+
+
+For example, the tactic:
+
+.. coqtop:: in
+
+   move: n {2}n (refl_equal n).
+
++ first generalizes ``(refl_equal n : n = n)``;
++ then generalizes the second occurrence of ``n``.
++ finally generalizes all the other occurrences of ``n``, and clears ``n``
+  from the proof context (assuming n is a proof constant).
+
+Therefore this tactic changes any goal ``G`` into
+
+.. coqtop::
+
+   forall n n0 : nat, n = n0 -> G.
+
+where the name ``n0`` is picked by the |Coq| display function, and assuming
+``n`` appeared only in ``G``.
+
+Finally, note that a discharge operation generalizes defined constants
+as variables, and not as local definitions. To override this behavior,
+prefix the name of the local definition with a ``@``, like in ``move: @n``.
+
+This is in contrast with the behavior of the in tactical (see
+section :ref:`localization_ssr`), which preserves local
+definitions by default.
+
+
+Clear rules
+```````````
+
+The clear step will fail if term is a proof constant that appears in
+other facts; in that case either the facts should be cleared
+explicitly with a :token:`clear_switch`, or the clear step should be disabled.
+The latter can be done by adding an :token:`occ_switch` or simply by putting
+parentheses around term: both
+``move: (n).``
+and
+``move: {+}n.``
+generalize ``n`` without clearing ``n`` from the proof context.
+
+The clear step will also fail if the :token:`clear_switch` contains a :token:`ident` that
+is not in the *proof* context. Note that |SSR| never clears a
+section constant.
+
+If tactic is ``move`` or ``case`` and an equation :token:`ident` is given, then clear
+(step 3) for :token:`d_item` is suppressed (see section :ref:`generation_of_equations_ssr`).
+
+
+Matching for apply and exact
+````````````````````````````
+
+The matching algorithm for :token:`d_item` of the |SSR|
+``apply`` and ``exact``
+tactics exploits the type of the first :token:`d_item` to interpret
+wildcards in the
+other :token:`d_item` and to determine which occurrences of these should be
+generalized. Therefore, occur switches are not needed for apply and
+exact.
+
+Indeed, the |SSR| tactic ``apply: H x`` is equivalent to
+``refine (@H _ … _ x); clear H x``
+with an appropriate number of wildcards between ``H`` and ``x``.
+
+Note that this means that matching for ``apply`` and ``exact`` has much more
+context to interpret wildcards; in particular it can accommodate the
+``_`` :token:`d_item`, which would always be rejected after ``move:``.
+
+.. example::
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+      Axiom f : nat -> nat.
+      Axiom g : nat -> nat.
+
+   .. coqtop:: all
+
+      Lemma test (Hfg : forall x, f x = g x) a b : f a = g b.
+      apply: trans_equal (Hfg _) _.
+
+This tactic is equivalent (see section
+:ref:`bookkeeping_ssr`) to:
+``refine (trans_equal (Hfg _) _).``
+and this is a common idiom for applying transitivity on the left hand
+side of an equation.
+
+
+.. _abstract_ssr:
+
+The abstract tactic
+```````````````````
+
+The ``abstract`` tactic assigns an ``abstract`` constant previously
+introduced with the ``[: name ]`` intro pattern
+(see section :ref:`introduction_ssr`).
+
+In a goal like the following::
+
+  m : nat
+  abs : <hidden>
+  n : nat
+  =============
+  m < 5 + n
+
+The tactic ``abstract: abs n`` first generalizes the goal with respect ton
+(that is not visible to the abstract constant abs) and then assigns
+abs. The resulting goal is::
+
+  m : nat
+  n : nat
+  =============
+  m < 5 + n
+
+Once this subgoal is closed, all other goals having abs in their
+context see the type assigned to ``abs``. In this case::
+
+  m : nat
+  abs : forall n, m < 5 + n
+  =============
+  …
+
+For a more detailed example the reader should refer to
+section :ref:`structure_ssr`.
+
+
+.. _introduction_ssr:
+
+Introduction in the context
+~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The application of a tactic to a given goal can generate (quantified)
+variables, assumptions, or definitions, which the user may want to
+*introduce* as new facts, constants or defined constants,
+respectively. If the tactic splits the goal into several subgoals,
+each of them may require the introduction of different constants and
+facts. Furthermore it is very common to immediately decompose or
+rewrite with an assumption instead of adding it to the context, as the
+goal can often be simplified and even proved after this.
+
+All these operations are performed by the introduction tactical ``=>``,
+whose general syntax is
+
+.. tacn:: @tactic => {+ @i_item }
+   :name: =>
+
+.. prodn::
+   i_item ::= @i_pattern %| @s_item %| @clear_switch %| /@term
+
+.. prodn::
+   s_item ::= /= %| // %| //=
+
+.. prodn::
+   i_pattern ::= @ident %| _ %| ? %| * %| {? @occ_switch } -> %| {? @occ_switch }<- %| [ {?| @i_item } ] %| - %| [: {+ @ident } ]
+
+The ``=>`` tactical first executes tactic, then the :token:`i_item` s,
+left to right. An :token:`s_item` specifies a
+simplification operation; a :token:`clear_switch`
+h specifies context pruning as in :ref:`discharge_ssr`.
+The :token:`i_pattern` s can be seen as a variant of *intro patterns*
+:ref:`tactics`: each performs an introduction operation, i.e., pops some
+variables or assumptions from the goal.
+
+An :token:`s_item` can simplify the set of subgoals or the subgoal themselves:
+
++ ``//`` removes all the “trivial” subgoals that can be resolved by the
+  |SSR| tactic ``done`` described in :ref:`terminators_ssr`, i.e.,
+  it executes ``try done``.
++ ``/=`` simplifies the goal by performing partial evaluation, as per the
+  tactic ``simpl`` [#5]_.
++ ``//=`` combines both kinds of simplification; it is equivalent to
+  ``/= //``, i.e., ``simpl; try done``.
+
+
+When an :token:`s_item` bears a :token:`clear_switch`, then the
+:token:`clear_switch` is executed
+*after* the :token:`s_item`, e.g., ``{IHn}//`` will solve some subgoals,
+possibly using the fact ``IHn``, and will erase ``IHn`` from the context
+of the remaining subgoals.
+
+The last entry in the :token:`i_item` grammar rule, ``/``:token:`term`,
+represents a view (see section :ref:`views_and_reflection_ssr`).
+If the next :token:`i_item` is a view, then the view is
+applied to the assumption in top position once all the
+previous :token:`i_item` have been performed.
+
+The view is applied to the top assumption.
+
+|SSR| supports the following :token:`i_pattern` s:
+
+:token:`ident`
+  pops the top variable, assumption, or local definition into
+  a new constant, fact, or defined constant :token:`ident`, respectively.
+  Note that defined constants cannot be introduced when δ-expansion is
+  required to expose the top variable or assumption.
+``?``
+  pops the top variable into an anonymous constant or fact, whose name
+  is picked by the tactic interpreter. |SSR| only generates names that cannot
+  appear later in the user script [#6]_.
+``_``
+  pops the top variable into an anonymous constant that will be deleted
+  from the proof context of all the subgoals produced by the ``=>`` tactical.
+  They should thus never be displayed, except in an error message if the
+  constant is still actually used in the goal or context after the last
+  :token:`i_item` has been executed (:token:`s_item` can erase goals or
+  terms where the constant appears).
+``*``
+  pops all the remaining apparent variables/assumptions as anonymous
+  constants/facts. Unlike ``?`` and ``move`` the ``*``
+  :token:`i_item` does not
+  expand definitions in the goal to expose quantifiers, so it may be useful
+  to repeat a ``move=> *`` tactic, e.g., on the goal::
+
+    forall a b : bool, a <> b
+
+  a first ``move=> *`` adds only ``_a_ : bool`` and ``_b_ : bool``
+  to the context; it takes a second ``move=> *`` to add ``_Hyp_ : _a_ = _b_``.
+:token:`occ_switch` ``->``
+  (resp. :token:`occ_switch` ``<-``)
+  pops the top assumption (which should be a rewritable proposition) into an
+  anonymous fact, rewrites (resp. rewrites right to left) the goal with this
+  fact (using the |SSR| ``rewrite`` tactic described in section
+  :ref:`rewriting_ssr`, and honoring the optional occurrence selector), and
+  finally deletes the anonymous fact from the context.
+``[`` :token:`i_item` * ``| … |`` :token:`i_item` * ``]``
+  when it is the
+  very *first* :token:`i_pattern` after tactic ``=>`` tactical *and* tactic
+  is not a move, is a *branching*:token:`i_pattern`. It executes the sequence
+  :token:`i_item`:math:`_i` on the i-th   subgoal produced by tactic. The
+  execution of tactic should thus generate exactly m subgoals, unless the
+  ``[…]`` :token:`i_pattern` comes after an initial ``//`` or ``//=``
+  :token:`s_item` that closes some of the goals produced by ``tactic``, in
+  which case exactly m subgoals should remain after thes- item, or we have
+  the trivial branching :token:`i_pattern` [], which always does nothing,
+  regardless of the number of remaining subgoals.
+``[`` :token:`i_item` * ``| … |`` :token:`i_item` * ``]``
+  when it is *not*
+  the first :token:`i_pattern` or when tactic is a ``move``, is a
+  *destructing* :token:`i_pattern`. It starts by destructing the top
+  variable, using the |SSR| ``case`` tactic described in
+  :ref:`the_defective_tactics_ssr`. It then behaves as the corresponding
+  branching :token:`i_pattern`, executing the
+  sequence:token:`i_item`:math:`_i`  in the i-th subgoal generated by the
+  case analysis; unless we have the trivial destructing :token:`i_pattern`
+  ``[]``, the latter should generate exactly m subgoals, i.e., the top
+  variable should have an inductive type with exactly m constructors [#7]_.
+  While it is good style to use the :token:`i_item` i * to pop the variables
+  and assumptions corresponding to each constructor, this is not enforced by
+  |SSR|.
+``-``
+  does nothing, but counts as an intro pattern. It can also be used to
+  force the interpretation of ``[`` :token:`i_item` * ``| … |``
+  :token:`i_item` * ``]`` as a case analysis like in ``move=> -[H1 H2]``. It
+  can also be used to indicate explicitly the link between a view and a name
+  like in ``move=> /eqP-H1``.  Last, it can serve as a separator between
+  views.  Section :ref:`views_and_reflection_ssr` [#9]_ explains in which
+  respect the tactic ``move=> /v1/v2`` differs from the tactic ``move=>
+  /v1-/v2``.
+``[:`` :token:`ident` ``…]``
+  introduces in the context an abstract constant
+  for each :token:`ident`.  Its type has to be fixed later on by using the
+  ``abstract`` tactic.  Before then the type displayed is ``<hidden>``.
+
+
+Note that |SSR| does not support the syntax ``(ipat, …, ipat)`` for
+destructing intro-patterns.
+
+Clears are deferred until the end of the intro pattern.
+
+.. example::
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect ssrbool.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+
+   .. coqtop:: all
+
+      Lemma test x y : Nat.leb 0 x = true -> (Nat.leb 0 x) && (Nat.leb y 2) = true.
+      move=> {x} ->.
+
+If the cleared names are reused in the same intro pattern, a renaming
+is performed behind the scenes.
+
+Facts mentioned in a clear switch must be valid names in the proof
+context (excluding the section context).
+
+The rules for interpreting branching and destructing :token:`i_pattern` are
+motivated by the fact that it would be pointless to have a branching
+pattern if tactic is a ``move``, and in most of the remaining cases
+tactic is ``case`` or ``elim``, which implies destruction.
+The rules above imply that:
+
++ ``move=> [a b].``
++ ``case=> [a b].``
++ ``case=> a b.``
+
+are all equivalent, so which one to use is a matter of style; ``move`` should
+be used for casual decomposition, such as splitting a pair, and ``case``
+should be used for actual decompositions, in particular for type families
+(see :ref:`type_families_ssr`) and proof by contradiction.
+
+The trivial branching :token:`i_pattern` can be used to force the branching
+interpretation, e.g.:
+
++ ``case=> [] [a b] c.``
++ ``move=> [[a b] c].``
++ ``case; case=> a b c.``
+
+are all equivalent.
+
+
+.. _generation_of_equations_ssr:
+
+Generation of equations
+~~~~~~~~~~~~~~~~~~~~~~~
+
+The generation of named equations option stores the definition of a
+new constant as an equation. The tactic:
+
+.. coqtop:: in
+
+  move En: (size l) => n.
+
+where ``l`` is a list, replaces ``size l`` by ``n`` in the goal and
+adds the fact ``En : size l = n`` to the context.
+This is quite different from:
+
+.. coqtop:: in
+
+  pose n := (size l).
+
+which generates a definition ``n := (size l)``. It is not possible to
+generalize or rewrite such a definition; on the other hand, it is
+automatically expanded during computation, whereas expanding the
+equation ``En`` requires explicit rewriting.
+
+The use of this equation name generation option with a ``case`` or an
+``elim`` tactic changes the status of the first :token:`i_item`, in order to
+deal with the possible parameters of the constants introduced.
+
+.. example::
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+
+   .. coqtop:: all
+
+      Lemma test (a b :nat) : a <> b.
+      case E : a => [|n].
+
+If the user does not provide a branching :token:`i_item` as first
+:token:`i_item`, or if the :token:`i_item` does not provide enough names for
+the arguments of a constructor, then the constants generated are introduced
+under fresh |SSR| names.
+
+.. example::
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+
+   .. coqtop:: all
+
+      Lemma test (a b :nat) : a <> b.
+      case E : a => H.
+      Show 2.
+
+Combining the generation of named equations mechanism with thecase
+tactic strengthens the power of a case analysis. On the other hand,
+when combined with the elim tactic, this feature is mostly useful for
+debug purposes, to trace the values of decomposed parameters and
+pinpoint failing branches.
+
+
+.. _type_families_ssr:
+
+Type families
+~~~~~~~~~~~~~
+
+When the top assumption of a goal has an inductive type, two specific
+operations are possible: the case analysis performed by the ``case``
+tactic, and the application of an induction principle, performed by
+the ``elim`` tactic. When this top assumption has an inductive type, which
+is moreover an instance of a type family, |Coq| may need help from the
+user to specify which occurrences of the parameters of the type should
+be substituted.
+
+A specific ``/`` switch indicates the type family parameters of the type
+of a :token:`d_item` immediately following this ``/`` switch,
+using the syntax:
+
+.. tacv:: case: {+ @d_item } / {+ @d_item }
+.. tacv:: elim: {+ @d_item } / {+ @d_item }
+
+The :token:`d_item` on the right side of the ``/`` switch are discharged as
+described in section :ref:`discharge_ssr`. The case analysis or elimination
+will be done on the type of the top assumption after these discharge
+operations.
+
+Every :token:`d_item` preceding the ``/`` is interpreted as arguments of this
+type, which should be an instance of an inductive type family. These terms
+are not actually generalized, but rather selected for substitution.
+Occurrence switches can be used to restrict the substitution. If a term is
+left completely implicit (e.g. writing just ``_``), then a pattern is
+inferred looking at the type of the top assumption. This allows for the
+compact syntax:
+
+.. coqtop:: in
+
+  case: {2}_ / eqP.
+
+were ``_`` is interpreted as ``(_ == _)`` since
+``eqP T a b : reflect (a = b) (a == b)`` and reflect is a type family with
+one index.
+
+Moreover if the :token:`d_item` list is too short, it is padded with an
+initial sequence of ``_`` of the right length.
+
+.. example::
+
+   Here is a small example on lists. We define first a function which
+   adds an element at the end of a given list.
+
+   .. coqtop:: reset
+
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+
+   .. coqtop:: all
+
+      Require Import List.
+      Section LastCases.
+      Variable A : Type.
+      Implicit Type l : list A.
+      Fixpoint add_last a l : list A :=
+        match l with
+        | nil => a :: nil
+        | hd :: tl => hd :: (add_last a tl) end.
+
+   Then we define an inductive predicate for case analysis on lists
+   according to their last element:
+
+   .. coqtop:: all
+
+      Inductive last_spec : list A -> Type :=
+      | LastSeq0 : last_spec nil
+      | LastAdd s x : last_spec (add_last x s).
+
+      Theorem lastP : forall l : list A, last_spec l.
+      Admitted.
+
+   We are now ready to use ``lastP`` in conjunction with ``case``.
+
+   .. coqtop:: all
+
+      Lemma test l : (length l) * 2 = length (l ++ l).
+      case: (lastP l).
+
+   Applied to the same goal, the command:
+   ``case: l / (lastP l).``
+   generates the same subgoals but ``l`` has been cleared from both contexts.
+
+   Again applied to the same goal, the command.
+
+   .. coqtop:: none
+
+      Abort.
+
+   .. coqtop:: all
+
+      Lemma test l : (length l) * 2 = length (l ++ l).
+      case: {1 3}l / (lastP l).
+
+   Note that selected occurrences on the left of the ``/``
+   switch have been substituted with l instead of being affected by
+   the case analysis.
+
+The equation name generation feature combined with a type family /
+switch generates an equation for the *first* dependent :token:`d_item`
+specified by the user. Again starting with the above goal, the
+command:
+
+.. example::
+
+   .. coqtop:: none
+
+      Abort.
+
+   .. coqtop:: all
+
+      Lemma test l : (length l) * 2 = length (l ++ l).
+      case E: {1 3}l / (lastP l) => [|s x].
+      Show 2.
+
+
+There must be at least one :token:`d_item` to the left of the / switch; this
+prevents any confusion with the view feature. However, the :token:`d_item`
+to the right of the ``/`` are optional, and if they are omitted the first
+assumption provides the instance of the type family.
+
+The equation always refers to the first :token:`d_item` in the actual tactic
+call, before any padding with initial ``_``. Thus, if an inductive type
+has two family parameters, it is possible to have|SSR| generate an
+equation for the second one by omitting the pattern for the first;
+note however that this will fail if the type of the second parameter
+depends on the value of the first parameter.
+
+
+Control flow
+------------
+
+
+.. _indentation_ssr:
+
+Indentation and bullets
+~~~~~~~~~~~~~~~~~~~~~~~
+
+A linear development of |Coq| scripts gives little information on the
+structure of the proof. In addition, replaying a proof after some
+changes in the statement to be proved will usually not display
+information to distinguish between the various branches of case
+analysis for instance.
+
+To help the user in this organization of the proof script at development
+time, |SSR| provides some bullets to highlight the structure of branching
+proofs. The available bullets are ``-``, ``+`` and ``*``.  Combined with
+tabulation, this lets us highlight four nested levels of branching; the most
+we have ever needed is three. Indeed, the use of “simpl and closing”
+switches, of terminators (see above section :ref:`terminators_ssr`) and
+selectors (see section :ref:`selectors_ssr`) is powerful enough to avoid most
+of the time more than two levels of indentation.
+
+Here is a fragment of such a structured script::
+
+    case E1: (abezoutn _ _) => [[| k1] [| k2]].
+    - rewrite !muln0 !gexpn0 mulg1 => H1.
+      move/eqP: (sym_equal F0); rewrite -H1 orderg1 eqn_mul1.
+      by case/andP; move/eqP.
+    - rewrite muln0 gexpn0 mulg1 => H1.
+      have F1: t %| t * S k2.+1 - 1.
+        apply: (@dvdn_trans (orderg x)); first by rewrite F0; exact: dvdn_mull.
+        rewrite orderg_dvd; apply/eqP; apply: (mulgI x).
+        rewrite -{1}(gexpn1 x) mulg1 gexpn_add leq_add_sub //.
+        by move: P1; case t.
+      rewrite dvdn_subr in F1; last by exact: dvdn_mulr.
+      + rewrite H1 F0 -{2}(muln1 (p ^ l)); congr (_ * _).
+        by apply/eqP; rewrite -dvdn1.
+      + by move: P1; case: (t) => [| [| s1]].
+    - rewrite muln0 gexpn0 mul1g => H1.
+    ...
+
+
+.. _terminators_ssr:
+
+Terminators
+~~~~~~~~~~~
+
+To further structure scripts, |SSR| supplies *terminating*
+tacticals to explicitly close off tactics. When replaying scripts, we
+then have the nice property that an error immediately occurs when a
+closed tactic fails to prove its subgoal.
+
+It is hence recommended practice that the proof of any subgoal should
+end with a tactic which *fails if it does not solve the current goal*,
+like discriminate, contradiction or assumption.
+
+In fact, |SSR| provides a generic tactical which turns any tactic
+into a closing one (similar to now). Its general syntax is:
+
+.. tacn:: by @tactic
+   :name: by
+
+The Ltac expression:
+
+.. coqtop:: in
+
+   by [@tactic | [@tactic | …].
+
+is equivalent to:
+
+.. coqtop:: in
+
+   [by @tactic | by @tactic | ...].
+
+and this form should be preferred to the former.
+
+In the script provided as example in section :ref:`indentation_ssr`, the
+paragraph corresponding to each sub-case ends with a tactic line prefixed
+with a ``by``, like in:
+
+.. coqtop:: in
+
+   by apply/eqP; rewrite -dvdn1.
+
+The by tactical is implemented using the user-defined, and extensible
+done tactic. This done tactic tries to solve the current goal by some
+trivial means and fails if it doesn’t succeed. Indeed, the tactic
+expression:
+
+.. coqtop:: in
+
+   by tactic.
+
+is equivalent to:
+
+.. coqtop:: in
+
+   tactic; done.
+
+Conversely, the tactic
+
+.. coqtop::
+
+   by [ ].
+
+is equivalent to:
+
+.. coqtop::
+
+   done.
+
+The default implementation of the done tactic, in the ``ssreflect.v``
+file, is:
+
+.. coqtop:: in
+
+   Ltac done :=
+     trivial; hnf; intros; solve
+      [ do ![solve [trivial | apply: sym_equal; trivial]
+            | discriminate | contradiction | split]
+      | case not_locked_false_eq_true; assumption
+      | match goal with H : ~ _ |- _ => solve [case H; trivial] end ].
+
+The lemma ``not_locked_false_eq_true`` is needed to discriminate
+*locked* boolean predicates (see section :ref:`locking_ssr`). The iterator
+tactical do is presented in section :ref:`iteration_ssr`. This tactic can be
+customized by the user, for instance to include an ``auto`` tactic.
+
+A natural and common way of closing a goal is to apply a lemma which
+is the exact one needed for the goal to be solved. The defective form
+of the tactic:
+
+.. coqtop:: in
+
+   exact.
+
+is equivalent to:
+
+.. coqtop:: in
+
+   do [done | by move=> top; apply top].
+
+where top is a fresh name affected to the top assumption of the goal.
+This applied form is supported by the : discharge tactical, and the
+tactic:
+
+.. coqtop:: in
+
+   exact: MyLemma.
+
+is equivalent to:
+
+.. coqtop:: in
+
+   by apply: MyLemma.
+
+(see section :ref:`discharge_ssr` for the documentation of the apply: combination).
+
+Warning The list of tactics, possibly chained by semi-columns, that
+follows a by keyword is considered as a parenthesized block applied to
+the current goal. Hence for example if the tactic:
+
+.. coqtop:: in
+
+   by rewrite my_lemma1.
+
+succeeds, then the tactic:
+
+.. coqtop:: in
+
+   by rewrite my_lemma1; apply my_lemma2.
+
+usually fails since it is equivalent to:
+
+.. coqtop:: in
+
+   by (rewrite my_lemma1; apply my_lemma2).
+
+
+
+.. _selectors_ssr:
+
+Selectors
+~~~~~~~~~
+
+When composing tactics, the two tacticals ``first`` and ``last`` let the user
+restrict the application of a tactic to only one of the subgoals
+generated by the previous tactic. This covers the frequent cases where
+a tactic generates two subgoals one of which can be easily disposed
+of.
+
+This is an other powerful way of linearization of scripts, since it
+happens very often that a trivial subgoal can be solved in a less than
+one line tactic. For instance, the tactic:
+
+.. tacn:: @tactic ; last by @tactic
+   :name: last
+
+tries to solve the last subgoal generated by the first
+tactic using the given second tactic , and fails if it does not succeeds.
+Its analogous
+
+.. tacn:: @tactic ; first by @tactic
+   :name: first
+
+tries to solve the first subgoal generated by the first tactic using the
+second given tactic, and fails if it does not succeeds.
+
+|SSR| also offers an extension of this facility, by supplying
+tactics to *permute* the subgoals generated by a tactic. The tactic:
+
+.. tacv:: @tactic; last first
+
+inverts the order of the subgoals generated by tactic. It is
+equivalent to:
+
+.. tacv:: @tactic; first last
+
+More generally, the tactic:
+
+.. tacn:: @tactic; last @natural first
+   :name: last first
+
+where :token:`natural` is a |Coq| numeral, or and Ltac variable
+denoting a |Coq|
+numeral, having the value k. It rotates the n subgoals G1 , …, Gn
+generated by tactic. The first subgoal becomes Gn + 1 − k and the
+circular order of subgoals remains unchanged.
+
+Conversely, the tactic:
+
+.. tacn:: @tactic; first @natural last
+   :name: first last
+
+rotates the n subgoals G1 , …, Gn generated by tactic in order that
+the first subgoal becomes Gk .
+
+Finally, the tactics ``last`` and ``first`` combine with the branching syntax
+of Ltac: if the tactic generates n subgoals on a given goal,
+then the tactic
+
+.. coqtop:: in
+
+   tactic ; last k [ tactic1 |…| tacticm ] || tacticn.
+
+where natural denotes the integer k as above, applies tactic1 to the n
+−k + 1-th goal, … tacticm to the n −k + 2 − m-th goal and tactic n
+to the others.
+
+.. example::
+
+   Here is a small example on lists. We define first a function which
+   adds an element at the end of a given list.
+
+   .. coqtop:: reset
+
+      Abort.
+      From Coq Require Import ssreflect.
+      Set Implicit Arguments.
+      Unset Strict Implicit.
+      Unset Printing Implicit Defensive.
+
+   .. coqtop:: all
+
+      Inductive test : nat -> Prop :=
+      | C1 n of n = 1 : test n
+      | C2 n of n = 2 : test n
+      | C3 n of n = 3 : test n
+      | C4 n of n = 4 : test n.
+
+      Lemma example n (t : test n) : True.
+      case: t; last 2 [move=> k| move=> l]; idtac.
+
+
+.. _iteration_ssr:
+
+Iteration
+~~~~~~~~~
+
+|SSR| offers an accurate control on the repetition of tactics,
+thanks to the do tactical, whose general syntax is:
+
+.. tacn:: do {? @mult } ( @tactic | [ {+| @tactic } ] )
+   :name: do
+
+where :token:`mult` is a *multiplier*.
+
+Brackets can only be omitted if a single tactic is given *and* a
+multiplier is present.
+
+A tactic of the form:
+
+.. coqtop:: in
+
+   do [ tactic 1 | … | tactic n ].
+
+is equivalent to the standard Ltac expression:
+
+.. coqtop:: in
+
+   first [ tactic 1 | … | tactic n ].
+
+The optional multiplier :token:`mult` specifies how many times the action of
+tactic should be repeated on the current subgoal.
+
+There are four kinds of multipliers:
+
+.. prodn::
+   mult ::= @num ! %| ! %| @num ? %| ?
+
+Their meaning is:
+
++ ``n!`` the step tactic is repeated exactly n times (where n is a
+  positive integer argument).
++ ``!`` the step tactic is repeated as many times as possible, and done
+  at least once.
++ ``?`` the step tactic is repeated as many times as possible,
+  optionally.
++ ``n?`` the step tactic is repeated up to n times, optionally.
+
+
+For instance, the tactic:
+
+.. coqtop:: in
+
+   tactic; do 1? rewrite mult_comm.
+
+rewrites at most one time the lemma ``mult_com`` in all the subgoals
+generated by tactic , whereas the tactic:
+
+.. coqtop:: in
+
+   tactic; do 2! rewrite mult_comm.
+
+rewrites exactly two times the lemma ``mult_com`` in all the subgoals
+generated by tactic, and fails if this rewrite is not possible in some
+subgoal.
+
+Note that the combination of multipliers and rewrite is so often used
+that multipliers are in fact integrated to the syntax of the
+|SSR| rewrite tactic, see section :ref:`rewriting_ssr`.
+
+
+.. _localization_ssr:
+
+Localization
+~~~~~~~~~~~~
+
+In sections :ref:`basic_localization_ssr` and :ref:`bookkeeping_ssr`, we have
+already presented the *localization* tactical in, whose general syntax is:
+
+.. tacn:: @tactic in {+ @ident} {? * }
+   :name: in
+
+where :token:`ident` is a name in the
+context. On the left side of ``in``,
+:token:`tactic` can be ``move``, ``case``, ``elim``, ``rewrite``, ``set``,
+or any tactic formed with the general iteration tactical ``do`` (see section
+:ref:`iteration_ssr`).
+
+The operation described by tactic is performed in the facts listed after
+``in`` and in the goal if a ``*`` ends the list of names.
+
+The ``in`` tactical successively:
+
++ generalizes the selected hypotheses, possibly “protecting” the goal
+  if ``*`` is not present,
++ performs :token:`tactic`, on the obtained goal,
++ reintroduces the generalized facts, under the same names.
+
+This defective form of the ``do`` tactical is useful to avoid clashes
+between standard Ltac in and the |SSR| tactical in.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Ltac mytac H := rewrite H.
+
+     Lemma test x y (H1 : x = y) (H2 : y = 3) : x + y = 6.
+     do [mytac H2] in H1 *.
+
+  the last tactic rewrites the hypothesis ``H2 : y = 3`` both in
+  ``H1 : x = y`` and in the goal ``x + y = 6``.
+
+By default ``in`` keeps the body of local definitions. To erase the body
+of a local definition during the generalization phase, the name of the
+local definition must be written between parentheses, like in
+``rewrite H in H1 (def_n) H2.``
+
+From |SSR| 1.5 the grammar for the in tactical has been extended
+to the following one:
+
+.. tacv:: @tactic in {+ @clear_switch | {? @  } @ident | ( @ident ) | ( {? @  } @ident := @c_pattern ) } {? * }
+
+In its simplest form the last option lets one rename hypotheses that
+can’t be cleared (like section variables). For example ``(y := x)``
+generalizes over ``x`` and reintroduces the generalized variable under the
+name ``y`` (and does not clear ``x``).
+For a more precise description this form of localization refer
+to :ref:`advanced_generalization_ssr`.
+
+
+.. _structure_ssr:
+
+Structure
+~~~~~~~~~
+
+Forward reasoning structures the script by explicitly specifying some
+assumptions to be added to the proof context. It is closely associated
+with the declarative style of proof, since an extensive use of these
+highlighted statements make the script closer to a (very detailed)
+text book proof.
+
+Forward chaining tactics allow to state an intermediate lemma and start a
+piece of script dedicated to the proof of this statement. The use of closing
+tactics (see section :ref:`terminators_ssr`) and of indentation makes
+syntactically explicit the portion of the script building the proof of the
+intermediate statement.
+
+
+The have tactic.
+````````````````
+
+The main |SSR| forward reasoning tactic is the ``have`` tactic. It can
+be use in two modes: one starts a new (sub)proof for an intermediate
+result in the main proof, and the other provides explicitly a proof
+term for this intermediate step.
+
+In the first mode, the syntax of have in its defective form is:
+
+.. tacn:: have : @term
+   :name: have
+
+This tactic supports open syntax for :token:`term`. Applied to a goal ``G``, it
+generates a first subgoal requiring a proof of ``term`` in the context of
+``G``. The second generated subgoal is of the form ``term -> G``, where term
+becomes the new top assumption, instead of being introduced with a
+fresh name. At the proof-term level, the have tactic creates a β
+redex, and introduces the lemma under a fresh name, automatically
+chosen.
+
+Like in the case of the ``pose`` tactic (see section :ref:`definitions_ssr`), the types of
+the holes are abstracted in term.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test : True.
+     have: _ * 0 = 0.
+
+  The invokation of ``have`` is equivalent to:
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Lemma test : True.
+
+  .. coqtop:: all
+
+     have: forall n : nat, n * 0 = 0.
+
+The have tactic also enjoys the same abstraction mechanism as the ``pose``
+tactic for the non-inferred implicit arguments. For instance, the
+tactic:
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Lemma test : True.
+
+  .. coqtop:: all
+
+     have: forall x y, (x, y) = (x, y + 0).
+
+  opens a new subgoal where the type of ``x`` is quantified.
+
+The behavior of the defective have tactic makes it possible to
+generalize it in the following general construction:
+
+.. tacn:: have {* @i_item } {? @i_pattern } {? @s_item | {+ @binder } } {? : @term } {? := @term | by @tactic }
+
+Open syntax is supported for both :token:`term`. For the description
+of :token:`i_item` and :token:`s_item` see section
+:ref:`introduction_ssr`. The first mode of the
+have tactic, which opens a sub-proof for an intermediate result, uses
+tactics of the form:
+
+.. tacv:: have @clear_switch @i_item : @term by @tactic
+
+which behave like:
+
+.. coqtop:: in
+
+   have: term ; first by tactic.
+   move=> clear_switch i_item.
+
+Note that the :token:`clear_switch` *precedes* the:token:`i_item`, which
+allows to reuse
+a name of the context, possibly used by the proof of the assumption,
+to introduce the new assumption itself.
+
+The``by`` feature is especially convenient when the proof script of the
+statement is very short, basically when it fits in one line like in:
+
+.. coqtop:: in
+
+   have H23 : 3 + 2 = 2 + 3 by rewrite addnC.
+
+The possibility of using :token:`i_item` supplies a very concise syntax for
+the further use of the intermediate step. For instance,
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test a : 3 * a - 1 = a.
+     have -> : forall x, x * a = a.
+
+  Note how the second goal was rewritten using the stated equality.
+  Also note that in this last subgoal, the intermediate result does not
+  appear in the context.
+
+Thanks to the deferred execution of clears, the following idiom is
+also supported (assuming x occurs in the goal only):
+
+.. coqtop:: in
+
+   have {x} -> : x = y.
+
+An other frequent use of the intro patterns combined with ``have`` is the
+destruction of existential assumptions like in the tactic:
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test : True.
+     have [x Px]: exists x : nat, x > 0.
+     Focus 2.
+
+An alternative use of the ``have`` tactic is to provide the explicit proof
+term for the intermediate lemma, using tactics of the form:
+
+.. tacv:: have {? @ident } := term
+
+This tactic creates a new assumption of type the type of :token:`term`.
+If the
+optional :token:`ident` is present, this assumption is introduced under the
+name :token:`ident`. Note that the body of the constant is lost for the user.
+
+Again, non inferred implicit arguments and explicit holes are
+abstracted.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test : True.
+     have H := forall x, (x, x) = (x, x).
+
+  adds to the context ``H : Type -> Prop.`` This is a schematic example but
+  the feature is specially useful when the proof term to give involves
+  for instance a lemma with some hidden implicit arguments.
+
+After the :token:`i_pattern`, a list of binders is allowed.
+
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect Omega.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test : True.
+     have H x (y : nat) : 2 * x + y = x + x + y by omega.
+
+A proof term provided after ``:=`` can mention these bound variables
+(that are automatically introduced with the given names).
+Since the :token:`i_pattern` can be omitted, to avoid ambiguity,
+bound variables can be surrounded
+with parentheses even if no type is specified:
+
+.. coqtop:: in
+
+   have (x) : 2 * x = x + x by omega.
+
+The :token:`i_item` and :token:`s_item` can be used to interpret the asserted
+hypothesis with views (see section :ref:`views_and_reflection_ssr`) or simplify the resulting
+goals.
+
+The ``have`` tactic also supports a ``suff`` modifier which allows for
+asserting that a given statement implies the current goal without
+copying the goal itself.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test : True.
+     have suff H : 2 + 2 = 3.
+     Focus 2.
+
+  Note that H is introduced in the second goal.
+
+The ``suff`` modifier is not
+compatible with the presence of a list of binders.
+
+.. _generating_let_ssr:
+
+Generating let in context entries with have
+```````````````````````````````````````````
+
+Since |SSR| 1.5 the ``have`` tactic supports a “transparent” modifier
+to generate let in context entries: the ``@`` symbol in front of the
+context entry name.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect Omega.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Set Printing Depth 15.
+
+  .. coqtop:: all
+
+     Inductive Ord n := Sub x of x < n.
+     Notation "'I_ n" := (Ord n) (at level 8, n at level 2, format "''I_' n").
+     Arguments Sub {_} _ _.
+
+     Lemma test n m (H : m + 1 < n) : True.
+     have @i : 'I_n by apply: (Sub m); omega.
+
+Note that the sub-term produced by ``omega`` is in general huge and
+uninteresting, and hence one may want to hide it.
+For this purpose the ``[: name ]`` intro pattern and the tactic
+``abstract`` (see page :ref:`abstract_ssr`) are provided.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect Omega.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+     Inductive Ord n := Sub x of x < n.
+     Notation "'I_ n" := (Ord n) (at level 8, n at level 2, format "''I_' n").
+     Arguments Sub {_} _ _.
+
+  .. coqtop:: all
+
+     Lemma test n m (H : m + 1 < n) : True.
+     have [:pm] @i : 'I_n by apply: (Sub m); abstract: pm; omega.
+
+  The type of ``pm`` can be cleaned up by its annotation ``(*1*)`` by just
+  simplifying it. The annotations are there for technical reasons only.
+
+When intro patterns for abstract constants are used in conjunction
+with have and an explicit term, they must be used as follows:
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect Omega.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+     Inductive Ord n := Sub x of x < n.
+     Notation "'I_ n" := (Ord n) (at level 8, n at level 2, format "''I_' n").
+     Arguments Sub {_} _ _.
+
+  .. coqtop:: all
+
+     Lemma test n m (H : m + 1 < n) : True.
+     have [:pm] @i : 'I_n := Sub m pm.
+       by omega.
+
+In this case the abstract constant ``pm`` is assigned by using it in
+the term that follows ``:=`` and its corresponding goal is left to be
+solved. Goals corresponding to intro patterns for abstract constants
+are opened in the order in which the abstract constants are declared
+(not in the “order” in which they are used in the term).
+
+Note that abstract constants do respect scopes. Hence, if a variable
+is declared after their introduction, it has to be properly
+generalized (i.e. explicitly passed to the abstract constant when one
+makes use of it).
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect Omega.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+     Inductive Ord n := Sub x of x < n.
+     Notation "'I_ n" := (Ord n) (at level 8, n at level 2, format "''I_' n").
+     Arguments Sub {_} _ _.
+
+  .. coqtop:: all
+
+     Lemma test n m (H : m + 1 < n) : True.
+     have [:pm] @i k : 'I_(n+k) by apply: (Sub m); abstract: pm k; omega.
+
+Last, notice that the use of intro patterns for abstract constants is
+orthogonal to the transparent flag ``@`` for have.
+
+
+The have tactic and type classes resolution
+```````````````````````````````````````````
+
+Since |SSR| 1.5 the have tactic behaves as follows with respect to
+type classes inference.
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect Omega.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+     Axiom ty : Type.
+     Axiom t : ty.
+
+     Goal True.
+
++ .. coqtop:: in undo
+
+     have foo : ty.
+
+  Full inference for ``ty``. The first subgoal demands a
+  proof of such instantiated statement.
+
++ coqtop::
+
+     have foo : ty := .
+
+  No inference for ``ty``. Unresolved instances are
+  quantified in ``ty``. The first subgoal demands a proof of such quantified
+  statement. Note that no proof term follows ``:=``, hence two subgoals are
+  generated.
+
++ .. coqtop:: in undo
+
+     have foo : ty := t.
+
+  No inference for ``ty`` and ``t``.
+
++ .. coqtop:: in undo
+
+     have foo := t.
+
+  No inference for ``t``. Unresolved instances are
+  quantified in the (inferred) type of ``t`` and abstracted in ``t``.
+
+
+The behavior of |SSR| 1.4 and below (never resolve type classes)
+can be restored with the option
+
+.. cmd:: Set SsrHave NoTCResolution.
+
+
+Variants: the suff and wlog tactics
+```````````````````````````````````
+
+As it is often the case in mathematical textbooks, forward reasoning
+may be used in slightly different variants. One of these variants is
+to show that the intermediate step L easily implies the initial goal
+G. By easily we mean here that the proof of L ⇒ G is shorter than the
+one of L itself. This kind of reasoning step usually starts with: “It
+suffices to show that …”.
+
+This is such a frequent way of reasoning that |SSR| has a variant
+of the ``have`` tactic called ``suffices`` (whose abridged name is ``suff``).
+The
+``have`` and ``suff`` tactics are equivalent and have the same syntax but:
+
+
++ the order of the generated subgoals is inversed
++ but the optional clear item is still performed in the *second*
+  branch. This means that the tactic:
+
+  .. coqtop:: in
+
+     suff {H} H : forall x : nat, x >= 0.
+
+  fails if the context of the current goal indeed contains an
+  assumption named ``H``.
+
+
+The rationale of this clearing policy is to make possible “trivial”
+refinements of an assumption, without changing its name in the main
+branch of the reasoning.
+
+The ``have`` modifier can follow the ``suff`` tactic.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect Omega.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Axioms G P : Prop.
+
+  .. coqtop:: all
+
+     Lemma test : G.
+     suff have H : P.
+
+  Note that, in contrast with ``have suff``, the name H has been introduced
+  in the first goal.
+
+Another useful construct is reduction, showing that a particular case
+is in fact general enough to prove a general property. This kind of
+reasoning step usually starts with: “Without loss of generality, we
+can suppose that …”. Formally, this corresponds to the proof of a goal
+G by introducing a cut wlog_statement -> G. Hence the user shall
+provide a proof for both (wlog_statement -> G) -> G and
+wlog_statement -> G. However, such cuts are usually rather
+painful to perform by
+hand, because the statement wlog_statement is tedious to write by hand,
+and sometimes even to read.
+
+|SSR| implements this kind of reasoning step through the without
+loss tactic, whose short name is ``wlog``. It offers support to describe
+the shape of the cut statements, by providing the simplifying
+hypothesis and by pointing at the elements of the initial goals which
+should be generalized. The general syntax of without loss is:
+
+.. tacn:: wlog {? suff } {? @clear_switch } {? @i_item } : {* @ident } / @term
+   :name: wlog
+
+where each :token:`ident` is a constant in the context
+of the goal. Open syntax is supported for :token:`term`.
+
+In its defective form:
+
+.. tacv:: wlog: / @term
+
+
+on a goal G, it creates two subgoals: a first one to prove the
+formula (term -> G) -> G and a second one to prove the formula
+term -> G.
+
+If the optional list of :token:`itent` is present
+on the left side of ``/``, these constants are generalized in the
+premise (term -> G) of the first subgoal. By default the body of local
+definitions is erased. This behavior can be inhibited prefixing the
+name of the local definition with the ``@`` character.
+
+In the second subgoal, the tactic:
+
+.. coqtop:: in
+
+   move=> clear_switch i_item.
+
+is performed if at least one of these optional switches is present in
+the ``wlog`` tactic.
+
+The ``wlog`` tactic is specially useful when a symmetry argument
+simplifies a proof. Here is an example showing the beginning of the
+proof that quotient and reminder of natural number euclidean division
+are unique.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect Omega.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma quo_rem_unicity d q1 q2 r1 r2 :
+       q1*d + r1 = q2*d + r2 -> r1 < d -> r2 < d -> (q1, r1) = (q2, r2).
+     wlog: q1 q2 r1 r2 / q1 <= q2.
+       by case (le_gt_dec q1 q2)=> H; last symmetry; eauto with arith.
+
+The ``wlog suff`` variant is simpler, since it cuts wlog_statement instead
+of wlog_statement -> G. It thus opens the goals wlog_statement -> G
+and wlog_statement.
+
+In its simplest form the ``generally have : …`` tactic is equivalent to
+``wlog suff : …`` followed by last first. When the ``have`` tactic is used
+with the ``generally`` (or ``gen``) modifier it accepts an extra identifier
+followed by a comma before the usual intro pattern. The identifier
+will name the new hypothesis in its more general form, while the intro
+pattern will be used to process its instance.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrfun ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+     Axiom P : nat -> Prop.
+     Axioms eqn leqn : nat -> nat -> bool.
+     Notation "a != b" := (eqn a b) (at level 70) : this_scope.
+     Notation "a <= b" := (leqn a b) (at level 70) : this_scope.
+     Open Scope this_scope.
+
+  .. coqtop:: all
+
+     Lemma simple n (ngt0 : 0 < n ) : P n.
+     gen have ltnV, /andP[nge0 neq0] : n ngt0 / (0 <= n) && (n != 0).
+     Focus 2.
+
+
+.. _advanced_generalization_ssr:
+
+Advanced generalization
++++++++++++++++++++++++
+
+The complete syntax for the items on the left hand side of the ``/``
+separator is the following one:
+
+.. tacv wlog … : {? @clear_switch | {? @  } @ident | ( {? @  } @ident := @c_pattern) } / @term
+
+Clear operations are intertwined with generalization operations. This
+helps in particular avoiding dependency issues while generalizing some
+facts.
+
+If an :token:`ident` is prefixed with the ``@`` mark, then a let-in redex is
+created, which keeps track if its body (if any). The syntax
+``( ident := c_pattern)`` allows to generalize an arbitrary term using a
+given name. Note that its simplest form ``(x := y)`` is just a renaming of
+``y`` into ``x``. In particular, this can be useful in order to simulate the
+generalization of a section variable, otherwise not allowed. Indeed
+renaming does not require the original variable to be cleared.
+
+The syntax ``(@x := y)`` generates a let-in abstraction but with the
+following caveat: ``x`` will not bind ``y``, but its body, whenever ``y`` can be
+unfolded. This cover the case of both local and global definitions, as
+illustrated in the following example.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Section Test.
+     Variable x : nat.
+     Definition addx z := z + x.
+     Lemma test : x <= addx x.
+     wlog H : (y := x) (@twoy := addx x) / twoy = 2 * y.
+
+  To avoid unfolding the term captured by the pattern add x one can use
+  the pattern ``id (addx x)``, that would produce the following first
+  subgoal
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect Omega.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+     Variable x : nat.
+     Definition addx z := z + x.
+     Lemma test : x <= addx x.
+
+  .. coqtop:: all
+
+     wlog H : (y := x) (@twoy := id (addx x)) / twoy = 2 * y.
+
+
+.. _rewriting_ssr:
+
+Rewriting
+---------
+
+The generalized use of reflection implies that most of the
+intermediate results handled are properties of effectively computable
+functions. The most efficient mean of establishing such results are
+computation and simplification of expressions involving such
+functions, i.e., rewriting. |SSR| therefore includes an
+extended ``rewrite`` tactic, that unifies and combines most of the
+rewriting functionalities.
+
+
+An extended rewrite tactic
+~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The main features of the rewrite tactic are:
+
++ It can perform an entire series of such operations in any subset of
+  the goal and/or context;
++ It allows to perform rewriting, simplifications, folding/unfolding
+  of definitions, closing of goals;
++ Several rewriting operations can be chained in a single tactic;
++ Control over the occurrence at which rewriting is to be performed is
+  significantly enhanced.
+
+The general form of an |SSR| rewrite tactic is:
+
+.. tacn:: rewrite {+ @rstep }
+   :name: rewrite (ssreflect)
+
+The combination of a rewrite tactic with the in tactical (see section
+:ref:`localization_ssr`) performs rewriting in both the context and the goal.
+
+A rewrite step :token:`rstep` has the general form:
+
+.. prodn::
+   rstep ::= {? @r_prefix } @r_item
+
+.. prodn::
+   r_prefix ::= {? - } {? @mult } {? @occ_switch %| @clear_switch } {? [ @r_pattern ] }
+
+.. prodn::
+   r_pattern ::= @term %| in {? @ident in } @term %| %( @term in %| @term as %) @ident in @term
+
+.. prodn::
+   r_item ::= {? / } @term %| @s_item
+
+An :token:`r_prefix` contains annotations to qualify where and how the rewrite
+operation should be performed:
+
++ The optional initial ``-`` indicates the direction of the rewriting of
+  :token:`r_item`:
+  if present the direction is right-to-left and it is left-to-right otherwise.
++ The multiplier :token:`mult` (see section :ref:`iteration_ssr`)
+  specifies if and how the
+  rewrite operation should be repeated.
++ A rewrite operation matches the occurrences of a *rewrite pattern*,
+  and replaces these occurrences by an other term, according to the
+  given :token:`r_item`. The optional *redex switch* ``[r_pattern]``,
+  which should
+  always be surrounded by brackets, gives explicitly this rewrite
+  pattern. In its simplest form, it is a regular term. If no explicit
+  redex switch is present the rewrite pattern to be matched is inferred
+  from the :token:`r_item`.
++ This optional term, or the :token:`r_item`, may be preceded by an occurrence
+  switch (see section :ref:`selectors_ssr`) or a clear item
+  (see section :ref:`discharge_ssr`),
+  these two possibilities being exclusive. An occurrence switch selects
+  the occurrences of the rewrite pattern which should be affected by the
+  rewrite operation.
+
+
+An :token:`r_item` can be:
+
+
++ A *simplification* :token:`r_item`,
+  represented by a :token:`s_item` (see section
+  :ref:`introduction_ssr`). Simplification operations are intertwined with the possible
+  other rewrite operations specified by the list of :token:`r_item`.
++ A *folding/unfolding* :token:`r_item`. The tactic:
+  ``rewrite /term`` unfolds the
+  head constant of term in every occurrence of the first matching of
+  term in the goal. In particular, if ``my_def`` is a (local or global)
+  defined constant, the tactic: ``rewrite /my_def.`` is analogous to:
+  ``unfold my_def``.
+  Conversely: ``rewrite -/my_def.`` is equivalent to: ``fold my_def``.
+  When an unfold :token:`r_item` is combined with a
+  redex pattern, a conversion
+  operation is performed. A tactic of the form:
+  ``rewrite -[term1]/term2.``
+  is equivalent to: ``change term1 with term2.`` If ``term2`` is a
+  single constant and ``term1`` head symbol is not ``term2``, then the head
+  symbol of ``term1`` is repeatedly unfolded until ``term2`` appears.
+
+  .. example::
+
+     .. coqtop:: reset
+
+        From Coq Require Import ssreflect.
+        Set Implicit Arguments.
+        Unset Strict Implicit.
+        Unset Printing Implicit Defensive.
+
+     .. coqtop:: all
+
+        Definition double x := x + x.
+        Definition ddouble x := double (double x).
+        Lemma test x : ddouble x = 4 * x.
+        rewrite [ddouble _]/double.
+
+     *Warning* The |SSR|
+     terms containing holes are *not* typed as abstractions in this
+     context. Hence the following script fails.
+
+     .. coqtop:: none
+
+        Abort.
+
+     .. coqtop:: all
+
+        Definition f := fun x y => x + y.
+        Lemma test x y : x + y = f y x.
+        rewrite -[f y]/(y + _).
+
+     but the following script succeeds
+
+     .. coqtop:: none
+
+        Restart.
+
+     .. coqtop:: all
+
+        rewrite -[f y x]/(y + _).
+
++ A :token:`term`, which can be:
+
+    + A term whose type has the form:
+      ``forall (x1 : A1 )…(xn : An ), eq term1 term2`` where
+      ``eq`` is the Leibniz equality or a registered setoid
+      equality.
+    + A list of terms ``(t1 ,…,tn)``, each ``ti`` having a type above.
+      The tactic: ``rewrite r_prefix (t1 ,…,tn ).``
+      is equivalent to: ``do [rewrite r_prefix t1 | … | rewrite r_prefix tn ].``
+    + An anonymous rewrite lemma ``(_ : term)``, where term has a type as above.  tactic: ``rewrite (_ : term)`` is in fact synonym of: ``cutrewrite (term).``.
+
+
+Remarks and examples
+~~~~~~~~~~~~~~~~~~~~
+
+Rewrite redex selection
+```````````````````````
+
+The general strategy of |SSR| is to grasp as many redexes as
+possible and to let the user select the ones to be rewritten thanks to
+the improved syntax for the control of rewriting.
+
+This may be a source of incompatibilities between the two rewrite
+tactics.
+
+In a rewrite tactic of the form:
+
+.. coqtop:: in
+
+   rewrite occ_switch [term1]term2.
+
+``term1`` is the explicit rewrite redex and ``term2`` is the rewrite rule.
+This execution of this tactic unfolds as follows:
+
+
++ First ``term1`` and ``term2`` are βι normalized. Then ``term2``
+  is put in head
+  normal form if the Leibniz equality constructor ``eq`` is not the head
+  symbol. This may involve ζ reductions.
++ Then, the matching algorithm (see section :ref:`abbreviations_ssr`)
+  determines the
+  first subterm of the goal matching the rewrite pattern. The rewrite
+  pattern is given by ``term1``, if an explicit redex pattern switch is
+  provided, or by the type of ``term2`` otherwise. However, matching skips
+  over matches that would lead to trivial rewrites. All the occurrences
+  of this subterm in the goal are candidates for rewriting.
++ Then only the occurrences coded by :token:`occ_switch` (see again section
+  :ref:`abbreviations_ssr`) are finally selected for rewriting.
++ The left hand side of ``term2`` is unified with the subterm found by
+  the matching algorithm, and if this succeeds, all the selected
+  occurrences in the goal are replaced by the right hand side of ``term2``.
++ Finally the goal is βι normalized.
+
+
+In the case ``term2`` is a list of terms, the first top-down (in the
+goal) left-to-right (in the list) matching rule gets selected.
+
+
+Chained rewrite steps
+`````````````````````
+
+The possibility to chain rewrite operations in a single tactic makes
+scripts more compact and gathers in a single command line a bunch of
+surgical operations which would be described by a one sentence in a
+pen and paper proof.
+
+Performing rewrite and simplification operations in a single tactic
+enhances significantly the concision of scripts. For instance the
+tactic:
+
+.. coqtop:: in
+
+   rewrite /my_def {2}[f _]/= my_eq //=.
+
+
+unfolds ``my_def`` in the goal, simplifies the second occurrence of the
+first subterm matching pattern ``[f _]``, rewrites ``my_eq``, simplifies the
+goals and closes trivial goals.
+
+Here are some concrete examples of chained rewrite operations, in the
+proof of basic results on natural numbers arithmetic.
+
+.. example::
+
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Axiom addn0 : forall m, m + 0 = m.
+     Axiom addnS : forall m n, m + S n = S (m + n).
+     Axiom addSnnS : forall m n, S m + n = m + S n.
+
+     Lemma addnCA m n p : m + (n + p) = n + (m + p).
+     by elim: m p => [ | m Hrec] p; rewrite ?addSnnS -?addnS.
+     Qed.
+
+     Lemma addnC n m : m + n = n + m.
+     by rewrite -{1}[n]addn0 addnCA addn0.
+     Qed.
+
+Note the use of the ``?`` switch for parallel rewrite operations in the
+proof of ``addnCA``.
+
+
+Explicit redex switches are matched first
+`````````````````````````````````````````
+
+If an :token:`r_prefix` involves a *redex switch*, the first step is to find a
+subterm matching this redex pattern, independently from the left hand
+side of the equality the user wants to rewrite.
+
+
+.. example::
+
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test (H : forall t u, t + u = u + t) x y : x + y = y + x.
+     rewrite [y + _]H.
+
+Note that if this first pattern matching is not compatible with the
+:token:`r_item`, the rewrite fails, even if the goal contains a
+correct redex matching both the redex switch and the left hand side of
+the equality.
+
+.. example::
+
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test (H : forall t u, t + u * 0 = t) x y : x + y * 4 + 2 * 0 = x + 2 * 0.
+     rewrite [x + _]H.
+
+  Indeed the left hand side of ``H`` does not match
+  the redex identified by the pattern ``x + y * 4``.
+
+
+Occurrence switches and redex switches
+``````````````````````````````````````
+
+.. example::
+
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test x y : x + y + 0 = x + y + y + 0 + 0 + (x + y + 0).
+     rewrite {2}[_ + y + 0](_: forall z, z + 0 = z).
+
+The second subgoal is generated by the use of an anonymous lemma in
+the rewrite tactic. The effect of the tactic on the initial goal is to
+rewrite this lemma at the second occurrence of the first matching
+``x + y + 0`` of the explicit rewrite redex ``_ + y + 0``.
+
+
+Occurrence selection and repetition
+```````````````````````````````````
+
+Occurrence selection has priority over repetition switches. This means
+the repetition of a rewrite tactic specified by a multiplier will
+perform matching each time an elementary rewrite operation is
+performed. Repeated rewrite tactics apply to every subgoal generated
+by the previous tactic, including the previous instances of the
+repetition.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Lemma test x y (z : nat) : x + 1 = x + y + 1.
+     rewrite 2!(_ : _ + 1 = z).
+
+This last tactic generates *three* subgoals because
+the second rewrite operation specified with the ``2!`` multiplier
+applies to the two subgoals generated by the first rewrite.
+
+
+Multi-rule rewriting
+````````````````````
+
+The rewrite tactic can be provided a *tuple* of rewrite rules, or more
+generally a tree of such rules, since this tuple can feature arbitrary
+inner parentheses. We call *multirule* such a generalized rewrite
+rule. This feature is of special interest when it is combined with
+multiplier switches, which makes the rewrite tactic iterates the
+rewrite operations prescribed by the rules on the current goal.
+
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+     Section Test.
+
+  .. coqtop:: all
+
+     Variables (a b c : nat).
+     Hypothesis eqab : a = b.
+     Hypothesis eqac : a = c.
+
+     Lemma test : a = a.
+     rewrite (eqab, eqac).
+
+  Indeed rule ``eqab`` is the first to apply among the ones
+  gathered in the tuple passed to the rewrite tactic. This multirule
+  ``(eqab, eqac)`` is actually a |Coq| term and we can name it with a
+  definition:
+
+  .. coqtop:: none
+
+     Abort.
+
+  .. coqtop:: all
+
+     Definition multi1 := (eqab, eqac).
+
+  In this case, the tactic ``rewrite multi1`` is a synonym for
+  ``rewrite (eqab, eqac)``.
+
+More precisely, a multirule rewrites the first subterm to which one of
+the rules applies in a left-to-right traversal of the goal, with the
+first rule from the multirule tree in left-to-right order. Matching is
+performed according to the algorithm described in
+Section :ref:`abbreviations_ssr`, but
+literal matches have priority.
+
+.. example::
+
+   .. coqtop:: all
+
+      Definition d := a.
+      Hypotheses eqd0 : d = 0.
+      Definition multi2 := (eqab, eqd0).
+
+      Lemma test : d = b.
+      rewrite multi2.
+
+   Indeed rule ``eqd0`` applies without unfolding the
+   definition of ``d``.
+
+For repeated rewrites the selection process is
+repeated anew.
+
+.. example::
+
+  .. coqtop:: none
+
+     Abort.
+
+  .. coqtop:: all
+
+     Hypothesis eq_adda_b : forall x, x + a = b.
+     Hypothesis eq_adda_c : forall x, x + a = c.
+     Hypothesis eqb0 : b = 0.
+     Definition multi3 := (eq_adda_b, eq_adda_c, eqb0).
+
+     Lemma test : 1 + a = 12 + a.
+     rewrite 2!multi3.
+
+  It uses ``eq_adda_b`` then ``eqb0`` on the left-hand
+  side only. Without the bound ``2`` one would obtain ``0 = 0``.
+
+The grouping of rules inside a multirule does not affect the selection
+strategy but can make it easier to include one rule set in another or
+to (universally) quantify over the parameters of a subset of rules (as
+there is special code that will omit unnecessary quantifiers for rules
+that can be syntactically extracted). It is also possible to reverse
+the direction of a rule subset, using a special dedicated syntax: the
+tactic rewrite ``(=~ multi1)`` is equivalent to ``rewrite multi1_rev``.
+
+.. example::
+
+  .. coqtop:: none
+
+     Abort.
+
+  .. coqtop:: all
+
+     Hypothesis eqba : b = a.
+     Hypothesis eqca : c = a.
+     Definition multi1_rev := (eqba, eqca).
+
+except that the constants ``eqba``, ``eqab``, ``mult1_rev``
+have not been created.
+
+Rewriting with multirules is useful to implement simplification or
+transformation procedures, to be applied on terms of small to medium
+size. For instance the library `ssrnat` (Mathematical Components library)
+provides two implementations
+for arithmetic operations on natural numbers: an elementary one and a
+tail recursive version, less inefficient but also less convenient for
+reasoning purposes. The library also provides one lemma per such
+operation, stating that both versions return the same values when
+applied to the same arguments:
+
+.. coqtop:: in
+
+     Lemma addE : add =2 addn.
+     Lemma doubleE : double =1 doublen.
+     Lemma add_mulE n m s : add_mul n m s = addn (muln n m) s.
+     Lemma mulE : mul =2 muln.
+     Lemma mul_expE m n p : mul_exp m n p = muln (expn m n) p.
+     Lemma expE : exp =2 expn.
+     Lemma oddE : odd =1 oddn.
+
+The operation on the left hand side of each lemma is the efficient
+version, and the corresponding naive implementation is on the right
+hand side. In order to reason conveniently on expressions involving
+the efficient operations, we gather all these rules in the definition
+``trecE``:
+
+.. coqtop:: in
+
+   Definition trecE := (addE, (doubleE, oddE), (mulE, add_mulE, (expE, mul_expE))).
+
+The tactic: ``rewrite !trecE.``
+restores the naive versions of each operation in a goal involving the
+efficient ones, e.g. for the purpose of a correctness proof.
+
+
+Wildcards vs abstractions
+`````````````````````````
+
+The rewrite tactic supports :token:`r_items` containing holes. For example in
+the tactic ``rewrite (_ : _ * 0 = 0).``
+the term ``_ * 0 = 0`` is interpreted as ``forall n : nat, n * 0 = 0.``
+Anyway this tactic is *not* equivalent to
+``rewrite (_ : forall x, x * 0 = 0).``.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test y z : y * 0 + y * (z * 0) = 0.
+     rewrite (_ : _ * 0 = 0).
+
+  while the other tactic results in
+
+  .. coqtop:: none
+
+     Undo.
+
+  .. coqtop:: all
+
+     rewrite (_ : forall x, x * 0 = 0).
+
+  The first tactic requires you to prove the instance of the (missing)
+  lemma that was used, while the latter requires you prove the quantified
+  form.
+
+When |SSR| rewrite fails on standard |Coq| licit rewrite
+````````````````````````````````````````````````````````
+
+In a few cases, the |SSR| rewrite tactic fails rewriting some
+redexes which standard |Coq| successfully rewrites. There are two main
+cases:
+
+
++ |SSR| never accepts to rewrite indeterminate patterns like:
+
+  .. coqtop:: in
+
+     Lemma foo (x : unit) : x = tt.
+
+  |SSR| will however accept the
+  ηζ expansion of this rule:
+
+  .. coqtop:: in
+
+     Lemma fubar (x : unit) : (let u := x in u) = tt.
+
++ The standard rewrite tactic provided by |Coq| uses a different algorithm
+  to find instances of the rewrite rule.
+
+  .. example::
+
+    .. coqtop:: reset
+
+       From Coq Require Import ssreflect.
+       Set Implicit Arguments.
+       Unset Strict Implicit.
+       Unset Printing Implicit Defensive.
+
+       Section Test.
+
+    .. coqtop:: all
+
+       Variable g : nat -> nat.
+       Definition f := g.
+       Axiom H : forall x, g x = 0.
+
+       Lemma test : f 3 + f 3 = f 6.
+       (* we call the standard rewrite tactic here *)
+       rewrite -> H.
+
+    This rewriting is not possible in |SSR| because
+    there is no occurrence of the head symbol ``f`` of the rewrite rule in the
+    goal.
+
+    .. coqtop:: none
+
+       Undo.
+
+    .. coqtop:: all
+
+       rewrite H.
+
+    Rewriting with ``H`` first requires unfolding the occurrences of
+    ``f``
+    where the substitution is to be performed (here there is a single such
+    occurrence), using tactic ``rewrite /f`` (for a global replacement of
+    f by g) or ``rewrite pattern/f``, for a finer selection.
+
+    .. coqtop:: none
+
+       Undo.
+
+    .. coqtop:: all
+
+       rewrite /f H.
+
+    alternatively one can override the pattern inferred from ``H``
+
+    .. coqtop:: none
+
+       Undo.
+
+    .. coqtop:: all
+
+       rewrite [f _]H.
+
+
+Existential metavariables and rewriting
+```````````````````````````````````````
+
+The rewrite tactic will not instantiate existing existential
+metavariables when matching a redex pattern.
+
+If a rewrite rule generates a goal with new existential metavariables
+in the ``Prop`` sort, these will be generalized as for ``apply``
+(see :ref:`apply_ssr`) and
+corresponding new goals will be generated.
+
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrfun ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+
+  .. coqtop:: all
+
+     Axiom leq : nat -> nat -> bool.
+     Notation "m <= n" := (leq m n) : nat_scope.
+     Notation "m < n"  := (S m <= n) : nat_scope.
+     Inductive Ord n := Sub x of x < n.
+     Notation "'I_ n" := (Ord n) (at level 8, n at level 2, format "''I_' n").
+     Arguments Sub {_} _ _.
+     Definition val n (i : 'I_n) := let: Sub a _ := i in a.
+     Definition insub n x :=
+       if @idP (x < n) is ReflectT _ Px then Some (Sub x Px) else None.
+     Axiom insubT : forall n x Px, insub n x = Some (Sub x Px).
+
+     Lemma test (x : 'I_2) y : Some x = insub 2 y.
+     rewrite insubT.
+
+  Since the argument corresponding to Px is not supplied by the user, the
+  resulting goal should be ``Some x = Some (Sub y ?Goal).``
+  Instead, |SSR| ``rewrite`` tactic hides the existential variable.
+
+  As in :ref:`apply_ssr`, the ``ssrautoprop`` tactic is used to try to
+  solve the existential variable.
+
+  .. coqtop:: none
+
+     Abort.
+
+  .. coqtop:: all
+
+     Lemma test (x : 'I_2) y (H : y < 2) : Some x = insub 2 y.
+     rewrite insubT.
+
+
+As a temporary limitation, this behavior is available only if the
+rewriting rule is stated using Leibniz equality (as opposed to setoid
+relations). It will be extended to other rewriting relations in the
+future.
+
+
+.. _locking_ssr:
+
+Locking, unlocking
+~~~~~~~~~~~~~~~~~~
+
+As program proofs tend to generate large goals, it is important to be
+able to control the partial evaluation performed by the simplification
+operations that are performed by the tactics. These evaluations can
+for example come from a ``/=`` simplification switch, or from rewrite
+steps which may expand large terms while performing conversion. We
+definitely want to avoid repeating large subterms of the goal in the
+proof script. We do this by “clamping down” selected function symbols
+in the goal, which prevents them from being considered in
+simplification or rewriting steps. This clamping is accomplished by
+using the occurrence switches (see section:ref:`abbreviations_ssr`)
+together with “term tagging” operations.
+
+|SSR| provides two levels of tagging.
+
+The first one uses auxiliary definitions to introduce a provably equal
+copy of any term t. However this copy is (on purpose) *not
+convertible* to t in the |Coq| system [#8]_. The job is done by the
+following construction:
+
+.. coqtop:: in
+
+   Lemma master_key : unit. Proof. exact tt. Qed.
+   Definition locked A := let: tt := master_key in fun x : A => x.
+   Lemma lock : forall A x, x = locked x :> A.
+
+Note that the definition of *master_key* is explicitly opaque. The
+equation ``t = locked t`` given by the ``lock`` lemma can be used for
+selective rewriting, blocking on the fly the reduction in the term ``t``.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrfun ssrbool List.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Variable A : Type.
+     Fixpoint has (p : A -> bool) (l : list A) : bool :=
+       if l is cons x l then p x || (has p l) else false.
+
+     Lemma test p x y l (H : p x = true) : has p ( x :: y :: l) = true.
+     rewrite {2}[cons]lock /= -lock.
+
+It is sometimes desirable to globally prevent a definition from being
+expanded by simplification; this is done by adding locked in the
+definition.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+      Definition lid := locked (fun x : nat => x).
+
+      Lemma test : lid 3 = 3.
+      rewrite /=.
+      unlock lid.
+
+We provide a special tactic unlock for unfolding such definitions
+while removing “locks”, e.g., the tactic:
+
+.. tacn:: unlock {? @occ_switch } @ident
+
+replaces the occurrence(s) of :token:`ident` coded by the
+:token:`occ_switch` with the corresponding body.
+
+We found that it was usually preferable to prevent the expansion of
+some functions by the partial evaluation switch ``/=``, unless this
+allowed the evaluation of a condition. This is possible thanks to an
+other mechanism of term tagging, resting on the following *Notation*:
+
+.. coqtop:: in
+
+   Notation "'nosimpl' t" := (let: tt := tt in t).
+
+The term ``(nosimpl t)`` simplifies to ``t`` *except* in a definition.
+More precisely, given:
+
+.. coqtop:: in
+
+   Definition foo := (nosimpl bar).
+
+the term ``foo`` (or ``(foo t’)``) will *not* be expanded by the *simpl*
+tactic unless it is in a forcing context (e.g., in ``match foo t’ with …
+end``, ``foo t’`` will be reduced if this allows ``match`` to be reduced).
+Note that ``nosimpl bar`` is simply notation for a term that reduces to
+``bar``; hence ``unfold foo`` will replace ``foo`` by ``bar``, and
+``fold foo`` will replace ``bar`` by ``foo``.
+
+*Warning* The ``nosimpl`` trick only works if no reduction is apparent in
+``t``; in particular, the declaration:
+
+.. coqtop:: in
+
+   Definition foo x := nosimpl (bar x).
+
+will usually not work. Anyway, the common practice is to tag only the
+function, and to use the following definition, which blocks the
+reduction as expected:
+
+.. coqtop:: in
+
+   Definition foo x := nosimpl bar x.
+
+A standard example making this technique shine is the case of
+arithmetic operations. We define for instance:
+
+.. coqtop:: in
+
+   Definition addn := nosimpl plus.
+
+The operation ``addn`` behaves exactly like ``plus``, except that
+``(addn (S n) m)`` will not simplify spontaneously to
+``(S (addn n m))`` (the two terms, however, are inter-convertible).
+In addition, the unfolding step: ``rewrite /addn``
+will replace ``addn`` directly with ``plus``, so the ``nosimpl`` form is
+essentially invisible.
+
+
+.. _congruence_ssr:
+
+Congruence
+~~~~~~~~~~
+
+Because of the way matching interferes with type families parameters,
+the tactic:
+
+.. coqtop:: in
+
+   apply: my_congr_property.
+
+will generally fail to perform congruence simplification, even on
+rather simple cases. We therefore provide a more robust alternative in
+which the function is supplied:
+
+.. tacn:: congr {? @num } @term
+   :name: congr
+
+This tactic:
++ checks that the goal is a Leibniz equality
++ matches both sides of this equality with “term applied to some arguments”, inferring the right number of arguments from the goal and the type of term. This may expand some definitions or fixpoints.
++ generates the subgoals corresponding to pairwise equalities of the arguments present in the goal.
+
+The goal can be a non dependent product ``P -> Q``. In that case, the
+system asserts the equation ``P = Q``, uses it to solve the goal, and
+calls the ``congr`` tactic on the remaining goal ``P = Q``. This can be useful
+for instance to perform a transitivity step, like in the following
+situation.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test (x y z : nat) (H : x = y) : x = z.
+     congr (_ = _) : H.
+     Abort.
+
+     Lemma test (x y z : nat) : x = y -> x = z.
+     congr (_ = _).
+
+The optional :token:`num` forces the number of arguments for which the
+tactic should generate equality proof obligations.
+
+This tactic supports equalities between applications with dependent
+arguments. Yet dependent arguments should have exactly the same
+parameters on both sides, and these parameters should appear as first
+arguments.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Definition f n :=
+       if n is 0 then plus else mult.
+     Definition g (n m : nat) := plus.
+
+     Lemma test x y : f 0 x y = g 1 1 x y.
+     congr plus.
+
+  This script shows that the ``congr`` tactic matches ``plus``
+  with ``f 0`` on the left hand side and ``g 1 1`` on the right hand
+  side, and solves the goal.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test n m (Hnm : m <= n) : S m + (S n - S m) = S n.
+     congr S; rewrite -/plus.
+
+  The tactic ``rewrite -/plus`` folds back the expansion of plus
+  which was necessary for matching both sides of the equality with
+  an application of ``S``.
+
+Like most |SSR| arguments, term can contain wildcards.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test x y : x + (y * (y + x - x)) = x * 1 + (y + 0) * y.
+     congr ( _ + (_ * _)).
+
+.. _contextual_patterns_ssr:
+
+Contextual patterns
+-------------------
+
+The simple form of patterns used so far, terms possibly containing
+wild cards, often require an additional :token:`occ_switch` to be specified.
+While this may work pretty fine for small goals, the use of
+polymorphic functions and dependent types may lead to an invisible
+duplication of functions arguments. These copies usually end up in
+types hidden by the implicit arguments machinery or by user defined
+notations. In these situations computing the right occurrence numbers
+is very tedious because they must be counted on the goal as printed
+after setting the Printing All flag. Moreover the resulting script is
+not really informative for the reader, since it refers to occurrence
+numbers he cannot easily see.
+
+Contextual patterns mitigate these issues allowing to specify
+occurrences according to the context they occur in.
+
+
+Syntax
+~~~~~~
+
+The following table summarizes the full syntax of :token:`c_pattern` and the
+corresponding subterm(s) identified by the pattern. In the third
+column we use s.m.r. for “the subterms matching the redex” specified
+in the second column.
+
+.. list-table::
+   :header-rows: 1
+
+   * - :token:`c_pattern`
+     - redex
+     - subterms affected
+
+   * - ``term``
+     - ``term``
+     - all occurrences of ``term``
+
+   * - ``ident in term``
+     - subterm of ``term`` selected by ``ident``
+     - all the subterms identified by ``ident`` in all the
+       occurrences of ``term``
+
+   * - ``term1 in ident in term2``
+     - ``term1`` in all s.m.r.
+     - in all the subterms identified by
+       ``ident`` in all the occurrences of ``term2``
+   * - ``term1 as ident in term2``
+     - ``term 1``
+     - in all the subterms identified by ``ident`
+       in all the occurrences of ``term2[term 1 /ident]``
+
+The rewrite tactic supports two more patterns obtained prefixing the
+first two with in. The intended meaning is that the pattern identifies
+all subterms of the specified context. The ``rewrite`` tactic will infer a
+pattern for the redex looking at the rule used for rewriting.
+
+.. list-table::
+   :header-rows: 1
+
+   * - :token:`r_pattern`
+     - redex
+     - subterms affected
+
+   * - ``in term``
+     - inferred from rule
+     - in all s.m.r. in all occurrences of ``term``
+
+   * - ``in ident in term``
+     - inferred from rule
+     - in all s.m.r. in all the subterms identified by ``ident``
+       in all the occurrences of ``term``
+
+The first :token:`c_pattern` is the simplest form matching any context but
+selecting a specific redex and has been described in the previous
+sections. We have seen so far that the possibility of selecting a
+redex using a term with holes is already a powerful mean of redex
+selection. Similarly, any terms provided by the user in the more
+complex forms of :token:`c_patterns`
+presented in the tables above can contain
+holes.
+
+For a quick glance at what can be expressed with the last
+:token:`r_pattern`
+consider the goal ``a = b`` and the tactic
+
+.. coqtop:: in
+
+   rewrite [in X in _ = X]rule.
+
+It rewrites all occurrences of the left hand side of ``rule``
+inside ``b``  only (``a``, and the hidden type of the equality, are ignored). Note that the
+variant ``rewrite [X in _ = X]rule`` would have rewritten ``b``
+exactly (i.e., it would only work if ``b`` and the left hand side
+of rule can be unified).
+
+
+Matching contextual patterns
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The :token:`c_pattern` and :token:`r_pattern` involving terms
+with holes are matched
+against the goal in order to find a closed instantiation. This
+matching proceeds as follows:
+
+.. list-table::
+   :header-rows: 1
+
+   * - :token:`c_pattern`
+     - instantiation order and place for ``term_i`` and redex
+
+   * - ``term``
+     - ``term`` is matched against the goal, redex is unified with
+       the instantiation of ``term``
+
+   * - ``ident in term``
+     - ``term`` is matched against the goal, redex is unified with the
+       subterm of the instantiation of ``term`` identified by
+       ``ident``
+
+   * - ``term1 in ident in term2``
+     - ``term2`` is matched against the goal, ``term1``
+       is matched against the subterm of the instantiation of
+       ``term1`` identified by ``ident``, redex is unified with
+       the instantiation of ``term1``
+
+   * - ``term1 as ident in term2``
+     - ``term2[term1/ident]`` is matched against
+       the goal, redex is unified with the instantiation of ``term1``
+
+In the following patterns, the redex is intended to be inferred from
+the rewrite rule.
+
+.. list-table::
+   :header-rows: 1
+
+   * - :token:`r_pattern`
+     - instantiation order and place for ``term_i`` and redex
+
+   * - ``in ident in term``
+     - ``term`` is matched against the goal, the redex is matched against
+       the subterm of the instantiation of ``term`` identified by
+       ``ident``
+
+   * - ``in term``
+     - ``term`` is matched against the goal, redex is matched against the
+       instantiation of ``term``
+
+
+Examples
+~~~~~~~~
+
+
+Contextual pattern in set and the : tactical
+````````````````````````````````````````````
+
+As already mentioned in section :ref:`abbreviations_ssr` the ``set``
+tactic takes as an
+argument a term in open syntax. This term is interpreted as the
+simplest for of :token:`c_pattern`. To void confusion in the grammar, open
+syntax is supported only for the simplest form of patterns, while
+parentheses are required around more complex patterns.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test a b : a + b + 1 = b + (a + 1).
+     set t := (X in _ = X).
+     rewrite {}/t.
+     set t := (a + _ in X in _ = X).
+
+
+Since the user may define an infix notation for ``in`` the former tactic
+may result ambiguous. The disambiguation rule implemented is to prefer
+patterns over simple terms, but to interpret a pattern with double
+parentheses as a simple term. For example the following tactic would
+capture any occurrence of the term ``a in A``.
+
+.. coqtop:: in
+
+   set t := ((a in A)).
+
+Contextual pattern can also be used as arguments of the ``:`` tactical.
+For example:
+
+.. coqtop:: in
+
+   elim: n (n in _ = n) (refl_equal n).
+
+
+Contextual patterns in rewrite
+``````````````````````````````
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Notation "n .+1" := (Datatypes.S n) (at level 2, left associativity,
+                          format "n .+1") : nat_scope.
+
+     Axiom addSn : forall m n, m.+1 + n = (m + n).+1.
+     Axiom addn0 : forall m, m + 0 = m.
+     Axiom addnC : forall m n, m + n = n + m.
+
+     Lemma test x y z f : (x.+1 + y) + f (x.+1 + y) (z + (x + y).+1) = 0.
+     rewrite [in f _ _]addSn.
+
+  Note: the simplification rule ``addSn`` is applied only under the ``f``
+  symbol.
+  Then we simplify also the first addition and expand 0 into 0+0.
+
+  .. coqtop:: all
+
+     rewrite addSn -[X in _ = X]addn0.
+
+  Note that the right hand side of ``addn0`` is undetermined, but the
+  rewrite pattern specifies the redex explicitly. The right hand side
+  of ``addn0`` is unified with the term identified by ``X``, ``0`` here.
+
+
+  The following pattern does not specify a redex, since it identifies an
+  entire region, hence the rewrite rule has to be instantiated
+  explicitly. Thus the tactic:
+
+  .. coqtop:: all
+
+     rewrite -{2}[in X in _ = X](addn0 0).
+
+  The following tactic is quite tricky:
+
+  .. coqtop:: all
+
+     rewrite [_.+1 in X in f _ X](addnC x.+1).
+
+  The explicit redex ``_.+1`` is important since its head constant ``S``
+  differs from the head constant inferred from
+  ``(addnC x.+1)`` (that is ``+``).
+  Moreover, the pattern ``f _ X`` is important to rule out
+  the first occurrence of ``(x + y).+1``.
+  Last, only the subterms of ``f _ X``
+  identified by ``X`` are rewritten, thus the first argument of
+  ``f`` is skipped too.
+  Also note the pattern ``_.+1`` is interpreted in the context
+  identified by ``X``, thus it gets instantiated to
+  ``(y + x).+1`` and not ``(x + y).+1``.
+
+  The last rewrite pattern allows to specify exactly the shape of the
+  term identified by X, that is thus unified with the left hand side of
+  the rewrite rule.
+
+  .. coqtop:: all
+
+     rewrite [x.+1 + y as X in f X _]addnC.
+
+
+Patterns for recurrent contexts
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The user can define shortcuts for recurrent contexts corresponding to
+the ``ident in term`` part. The notation scope identified with
+``%pattern``
+provides a special notation ``(X in t)`` the user must adopt
+in order to define
+context shortcuts.
+
+The following example is taken from ``ssreflect.v`` where the
+``LHS`` and ``RHS`` shortcuts are defined.
+
+.. coqtop:: in
+
+   Notation RHS := (X in _ = X)%pattern.
+   Notation LHS := (X in X = _)%pattern.
+
+Shortcuts defined this way can be freely used in place of the trailing
+``ident in term`` part of any contextual pattern. Some examples follow:
+
+.. coqtop:: in
+
+   set rhs := RHS.
+   rewrite [in RHS]rule.
+   case: (a + _ in RHS).
+
+
+.. _views_and_reflection_ssr:
+
+Views and reflection
+--------------------
+
+The bookkeeping facilities presented in section :ref:`basic_tactics_ssr` are
+crafted to ease simultaneous introductions and generalizations of facts and
+operations of casing, naming etc. It also a common practice to make a stack
+operation immediately followed by an *interpretation* of the fact
+being pushed, that is, to apply a lemma to this fact before passing it
+to a tactic for decomposition, application and so on.
+
+|SSR| provides a convenient, unified syntax to combine these
+interpretation operations with the proof stack operations. This *view
+mechanism* relies on the combination of the ``/`` view switch with
+bookkeeping tactics and tacticals.
+
+
+Interpreting eliminations
+~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The view syntax combined with the ``elim`` tactic specifies an elimination
+scheme to be used instead of the default, generated, one. Hence the
+|SSR| tactic:
+
+.. coqtop:: in
+
+   elim/V.
+
+is a synonym for:
+
+.. coqtop:: in
+
+   intro top; elim top using V; clear top.
+
+where top is a fresh name and V any second-order lemma.
+
+Since an elimination view supports the two bookkeeping tacticals of
+discharge and introduction (see section :ref:`basic_tactics_ssr`),
+the |SSR| tactic:
+
+.. coqtop:: in
+
+   elim/V: x => y.
+
+is a synonym for:
+
+.. coqtop:: in
+
+   elim x using V; clear x; intro y.
+
+where ``x`` is a variable in the context, ``y`` a fresh name and ``V``
+any second order lemma; |SSR| relaxes the syntactic restrictions of the |Coq|
+``elim``. The first pattern following ``:`` can be a ``_`` wildcard if the
+conclusion of the view ``V`` specifies a pattern for its last argument
+(e.g., if ``V`` is a functional induction lemma generated by the
+``Function`` command).
+
+The elimination view mechanism is compatible with the equation name
+generation (see section :ref:`generation_of_equations_ssr`).
+
+
+.. example::
+
+   The following script illustrate a toy example of this feature. Let us
+   define a function adding an element at the end of a list:
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect List.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Variable d : Type.
+     Fixpoint add_last(s : list d) (z : d) {struct s} : list d :=
+       if s is cons x s' then cons x (add_last s' z) else z :: nil.
+
+  One can define an alternative, reversed, induction principle on
+  inductively defined lists, by proving the following lemma:
+
+  .. coqtop:: all
+
+     Axiom last_ind_list : forall P : list d -> Prop,
+       P nil -> (forall s (x : d), P s -> P (add_last s x)) ->
+         forall s : list d, P s.
+
+  Then the combination of elimination views with equation names result
+  in a concise syntax for reasoning inductively using the user defined
+  elimination scheme.
+
+  .. coqtop:: all
+
+     Lemma test (x : d) (l : list d): l = l.
+     elim/last_ind_list E : l=> [| u v]; last first.
+
+
+User provided eliminators (potentially generated with the ``Function``
+|Coq|’s command) can be combined with the type family switches described
+in section :ref:`type_families_ssr`.
+Consider an eliminator ``foo_ind`` of type:
+
+.. coqtop:: in
+
+   foo_ind : forall …, forall x : T, P p1 … pm.
+
+and consider the tactic:
+
+.. coqtop:: in
+
+   elim/foo_ind: e1 … / en.
+
+The ``elim/`` tactic distinguishes two cases:
+
+:truncated eliminator:  when ``x`` does not occur in ``P p1 … pm`` and the
+  type of ``en`` unifies with ``T`` and ``en`` is not ``_``.
+  In that case, ``en`` is
+  passed to the eliminator as the last argument (``x`` in ``foo_ind``) and
+  ``en−1 … e1`` are used as patterns to select in the goal the occurrences that
+  will be bound by the predicate ``P``, thus it must be possible to unify
+  the sub-term of the goal matched by ``en−1`` with ``pm`` , the one matched
+  by ``en−2`` with ``pm−1`` and so on.
+:regular eliminator: in all the other cases. Here it must be possible
+  to unify the term matched by ``en`` with ``pm`` , the one matched by
+  ``en−1``
+  with ``pm−1`` and so on. Note that standard eliminators have the shape
+  ``…forall x, P … x``, thus ``en`` is the pattern identifying the
+  eliminated term, as expected.
+
+
+As explained in section :ref:`type_families_ssr`, the initial prefix of
+``ei`` can be omitted.
+
+Here an example of a regular, but non trivial, eliminator.
+
+.. example::
+
+  Here is a toy example illustrating this feature.
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect FunInd.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Function plus (m n : nat) {struct n} : nat :=
+       if n is S p then S (plus m p) else m.
+
+     About plus_ind.
+
+     Lemma test x y z : plus (plus x y) z = plus x (plus y z).
+
+  The following tactics are all valid and perform the same elimination
+  on this goal.
+
+  .. coqtop:: in
+
+     elim/plus_ind: z / (plus _ z).
+     elim/plus_ind: {z}(plus _ z).
+     elim/plus_ind: {z}_.
+     elim/plus_ind: z / _.
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect FunInd.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+     Function plus (m n : nat) {struct n} : nat :=
+       if n is S p then S (plus m p) else m.
+
+     About plus_ind.
+
+     Lemma test x y z : plus (plus x y) z = plus x (plus y z).
+
+  .. coqtop:: all
+
+     elim/plus_ind: z / _.
+
+  The two latter examples feature a wildcard pattern: in this case,
+  the resulting pattern is inferred from the type of the eliminator.
+  In both these examples, it is ``(plus _ _)``, which matches the subterm
+  ``plus (plus x y) z`` thus instantiating the last ``_`` with ``z``.
+  Note that the tactic:
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect FunInd.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+     Function plus (m n : nat) {struct n} : nat :=
+       if n is S p then S (plus m p) else m.
+
+     About plus_ind.
+
+     Lemma test x y z : plus (plus x y) z = plus x (plus y z).
+
+  .. coqtop:: all
+
+     elim/plus_ind: y / _.
+
+  triggers an error: in the conclusion
+  of the ``plus_ind`` eliminator, the first argument of the predicate
+  ``P`` should be the same as the second argument of ``plus``, in the
+  second argument of ``P``, but ``y`` and ``z`` do no unify.
+
+Here an example of a truncated eliminator:
+
+.. example::
+
+  Consider the goal:
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect FunInd.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: in
+
+     Lemma test p n (n_gt0 : 0 < n) (pr_p : prime p) :
+       p %| \prod_(i <- prime_decomp n | i \in prime_decomp n) i.1 ^ i.2 ->
+         exists2 x : nat * nat, x \in prime_decomp n & p = x.1.
+     Proof.
+     elim/big_prop: _ => [| u v IHu IHv | [q e] /=].
+
+
+  where the type of the ``big_prop`` eliminator is
+
+  .. coqtop:: in
+
+     big_prop: forall (R : Type) (Pb : R -> Type)
+       (idx : R) (op1 : R -> R -> R), Pb idx ->
+       (forall x y : R, Pb x -> Pb y -> Pb (op1 x y)) ->
+       forall (I : Type) (r : seq I) (P : pred I) (F : I -> R),
+       (forall i : I, P i -> Pb (F i)) ->
+         Pb (\big[op1/idx]_(i <- r | P i) F i).
+
+  Since the pattern for the argument of Pb is not specified, the
+  inferred one is used instead: ``big[_/_]_(i <- _ | _ i) _ i``,
+  and after the introductions, the following goals are generated:
+
+  .. coqtop:: in
+
+     subgoal 1 is:
+       p %| 1 -> exists2 x : nat * nat, x \in prime_decomp n & p = x.1
+     subgoal 2 is:
+       p %| u * v -> exists2 x : nat * nat, x \in prime_decomp n & p = x.1
+     subgoal 3 is:
+       (q, e) \in prime_decomp n -> p %| q ^ e ->
+         exists2 x : nat * nat, x \in prime_decomp n & p = x.1.
+
+  Note that the pattern matching algorithm instantiated all the
+  variables occurring in the pattern.
+
+
+.. _interpreting_assumptions_ssr:
+
+Interpreting assumptions
+~~~~~~~~~~~~~~~~~~~~~~~~
+
+Interpreting an assumption in the context of a proof consists in
+applying it a lemma before generalizing, and/or decomposing this
+assumption. For instance, with the extensive use of boolean reflection
+(see section :ref:`views_and_reflection_ssr`.4), it is quite frequent
+to need to decompose the logical interpretation of (the boolean
+expression of) a fact, rather than the fact itself. This can be
+achieved by a combination of ``move : _ => _`` switches, like in the
+following example, where ``||`` is a notation for the boolean
+disjunction.
+
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Variables P Q : bool -> Prop.
+     Hypothesis P2Q : forall a b, P (a || b) -> Q a.
+
+     Lemma test a : P (a || a) -> True.
+     move=> HPa; move: {HPa}(P2Q HPa) => HQa.
+
+  which transforms the hypothesis ``HPa : P a`` which has been introduced
+  from the initial statement into ``HQa : Q a``.
+  This operation is so common that the tactic shell has specific
+  syntax for it. The following scripts:
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+     Variables P Q : bool -> Prop.
+     Hypothesis P2Q : forall a b, P (a || b) -> Q a.
+
+     Lemma test a : P (a || a) -> True.
+
+  .. coqtop:: all undo
+
+     move=> HPa; move/P2Q: HPa => HQa.
+
+  or more directly:
+
+  .. coqtop:: all undo
+
+     move/P2Q=> HQa.
+
+  are equivalent to the former one. The former script shows how to
+  interpret a fact (already in the context), thanks to the discharge
+  tactical (see section :ref:`discharge_ssr`) and the latter, how to interpret the top
+  assumption of a goal. Note that the number of wildcards to be inserted
+  to find the correct application of the view lemma to the hypothesis
+  has been automatically inferred.
+
+The view mechanism is compatible with the ``case`` tactic and with the
+equation name generation mechanism (see section :ref:`generation_of_equations_ssr`):
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Variables P Q: bool -> Prop.
+     Hypothesis Q2P : forall a b, Q (a || b) -> P a \/ P b.
+
+     Lemma test a b : Q (a || b) -> True.
+     case/Q2P=> [HPa | HPb].
+
+  This view tactic performs:
+
+  .. coqtop:: in
+
+     move=> HQ; case: {HQ}(Q2P HQ) => [HPa | HPb].
+
+The term on the right of the ``/`` view switch is called a *view lemma*.
+Any |SSR| term coercing to a product type can be used as a view
+lemma.
+
+The examples we have given so far explicitly provide the direction of
+the translation to be performed. In fact, view lemmas need not to be
+oriented. The view mechanism is able to detect which application is
+relevant for the current goal.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Variables P Q: bool -> Prop.
+     Hypothesis PQequiv : forall a b, P (a || b) <-> Q a.
+
+     Lemma test a b : P (a || b) -> True.
+     move/PQequiv=> HQab.
+
+  has the same behavior as the first example above.
+
+  The view mechanism can insert automatically a *view hint* to transform
+  the double implication into the expected simple implication. The last
+  script is in fact equivalent to:
+
+  .. coqtop:: in
+
+     Lemma test a b : P (a || b) -> True.
+     move/(iffLR (PQequiv _ _)).
+
+  where:
+
+  .. coqtop:: in
+
+     Lemma iffLR P Q : (P <-> Q) -> P -> Q.
+
+
+Specializing assumptions
+````````````````````````
+
+The special case when the *head symbol* of the view lemma is a
+wildcard is used to interpret an assumption by *specializing* it. The
+view mechanism hence offers the possibility to apply a higher-order
+assumption to some given arguments.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test z : (forall x y, x + y = z -> z = x) -> z = 0.
+     move/(_ 0 z).
+
+
+Interpreting goals
+~~~~~~~~~~~~~~~~~~
+
+In a similar way, it is also often convenient to
+changing a goal by turning it into an equivalent proposition. The view
+mechanism of |SSR| has a special syntax ``apply/`` for combining  in a
+single tactic simultaneous goal interpretation operations and
+bookkeeping steps.
+
+
+.. example::
+   The following example use the ``~~`` prenex notation for boolean negation:
+
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Variables P Q: bool -> Prop.
+     Hypothesis PQequiv : forall a b, P (a || b) <-> Q a.
+
+     Lemma test a : P ((~~ a) || a).
+     apply/PQequiv.
+
+  thus in this case, the tactic ``apply/PQequiv`` is equivalent to
+  ``apply: (iffRL (PQequiv _ _))``, where ``iffRL`` is tha analogue of
+  ``iffRL`` for the converse implication.
+
+Any |SSR| term whose type coerces to a double implication can be
+used as a view for goal interpretation.
+
+Note that the goal interpretation view mechanism supports both ``apply``
+and ``exact`` tactics. As expected, a goal interpretation view command
+exact/term should solve the current goal or it will fail.
+
+*Warning* Goal interpretation view tactics are *not* compatible with
+the bookkeeping tactical ``=>`` since this would be redundant with the
+``apply: term => _`` construction.
+
+
+Boolean reflection
+~~~~~~~~~~~~~~~~~~
+
+In the Calculus of Inductive Construction, there is an obvious
+distinction between logical propositions and boolean values. On the
+one hand, logical propositions are objects of *sort* ``Prop`` which is
+the carrier of intuitionistic reasoning. Logical connectives in
+``Prop`` are *types*, which give precise information on the structure
+of their proofs; this information is automatically exploited by |Coq|
+tactics.  For example, |Coq| knows that a proof of ``A \/ B`` is
+either a proof of ``A`` or a proof of ``B``.  The tactics ``left`` and
+``right`` change the goal ``A \/ B`` to ``A`` and ``B``, respectively;
+dually, the tactic ``case`` reduces the goal ``A \/ B => G`` to two
+subgoals ``A => G`` and ``B => G``.
+
+On the other hand, bool is an inductive *datatype* with two
+constructors true and false. Logical connectives on bool are
+*computable functions*, defined by their truth tables, using case
+analysis:
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Definition orb (b1 b2 : bool) := if b1 then true else b2.
+
+Properties of such connectives are also established using case
+analysis
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test b : b || ~~ b = true.
+     by case: b.
+
+  Once ``b`` is replaced by ``true`` in the first goal and by ``false`` in the
+  second one, the goals reduce by computations to the trivial ``true = true``.
+
+Thus, ``Prop`` and ``bool`` are truly complementary: the former supports
+robust natural deduction, the latter allows brute-force
+evaluation. |SSR| supplies a generic mechanism to have the best of
+the two worlds and move freely from a propositional version of a
+decidable predicate to its boolean version.
+
+First, booleans are injected into propositions using the coercion
+mechanism:
+
+.. coqtop:: in
+
+   Coercion is_true (b : bool) := b = true.
+
+This allows any boolean formula ``b`` to be used in a context where |Coq|
+would expect a proposition, e.g., after ``Lemma … : ``. It is then
+interpreted as ``(is_true b)``, i.e., the proposition ``b = true``. Coercions
+are elided by the pretty-printer, so they are essentially transparent
+to the user.
+
+The reflect predicate
+~~~~~~~~~~~~~~~~~~~~~
+
+To get all the benefits of the boolean reflection, it is in fact
+convenient to introduce the following inductive predicate ``reflect`` to
+relate propositions and booleans:
+
+.. coqtop:: in
+
+   Inductive reflect (P: Prop): bool -> Type :=
+   | Reflect_true : P -> reflect P true
+   | Reflect_false : ~P -> reflect P false.
+
+The statement ``(reflect P b)`` asserts that ``(is_true b)`` and ``P`` are
+logically equivalent propositions.
+
+For instance, the following lemma:
+
+.. coqtop:: in
+
+     Lemma andP: forall b1 b2, reflect (b1 /\ b2) (b1 && b2).
+
+relates the boolean conjunction to the logical one ``/\``. Note that in
+``andP``, ``b1`` and ``b2`` are two boolean variables and the
+proposition ``b1 /\ b2`` hides two coercions. The conjunction of
+``b1`` and ``b2`` can then be viewed as ``b1 /\ b2`` or as ``b1 && b2``.
+
+Expressing logical equivalences through this family of inductive types
+makes possible to take benefit from *rewritable equations* associated
+to the case analysis of |Coq|’s inductive types.
+
+Since the equivalence predicate is defined in |Coq| as:
+
+.. coqtop:: in
+
+   Definition iff (A B:Prop) := (A -> B) /\ (B -> A).
+
+where ``/\`` is a notation for ``and``:
+
+.. coqtop:: in
+
+   Inductive and (A B:Prop) : Prop := conj : A -> B -> and A B.
+
+This make case analysis very different according to the way an
+equivalence property has been defined.
+
+.. coqtop:: in
+
+   Lemma andE (b1 b2 : bool) : (b1 /\ b2) <-> (b1 && b2).
+
+Let us compare the respective behaviours of ``andE`` and ``andP``.
+
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+     Axiom andE : forall (b1 b2 : bool), (b1 /\ b2) <-> (b1 && b2).
+
+  .. coqtop:: all
+
+     Lemma test (b1 b2 : bool) : if (b1 && b2) then b1 else ~~(b1||b2).
+
+  .. coqtop:: all undo
+
+     case: (@andE b1 b2).
+
+  .. coqtop:: all undo
+
+     case: (@andP b1 b2).
+
+Expressing reflection relation through the ``reflect`` predicate is hence
+a very convenient way to deal with classical reasoning, by case
+analysis. Using the ``reflect`` predicate allows moreover to program rich
+specifications inside its two constructors, which will be
+automatically taken into account during destruction. This
+formalisation style gives far more efficient specifications than
+quantified (double) implications.
+
+A naming convention in |SSR| is to postfix the name of view lemmas
+with ``P``. For example, ``orP`` relates ``||`` and ``\/``,
+``negP`` relates ``~~`` and ``~``.
+
+The view mechanism is compatible with reflect predicates.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test (a b : bool) (Ha : a) (Hb : b) : a /\ b.
+     apply/andP.
+
+  Conversely
+
+  .. coqtop:: none
+
+     Abort.
+
+  .. coqtop:: all
+
+     Lemma test (a b : bool) : a /\ b -> a.
+     move/andP.
+
+The same tactics can also be used to perform the converse operation,
+changing a boolean conjunction into a logical one. The view mechanism
+guesses the direction of the transformation to be used i.e., the
+constructor of the reflect predicate which should be chosen.
+
+
+General mechanism for interpreting goals and assumptions
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Specializing assumptions
+````````````````````````
+
+The |SSR| tactic:
+
+.. coqtop:: in
+
+   move/(_ term1 … termn).
+
+is equivalent to the tactic:
+
+.. coqtop:: in
+
+   intro top; generalize (top term1 … termn); clear top.
+
+where ``top`` is a fresh name for introducing the top assumption of the
+current goal.
+
+
+Interpreting assumptions
+````````````````````````
+
+The general form of an assumption view tactic is:
+
+.. tacv:: [move | case] / @term
+
+The term , called the *view lemma* can be:
+
+
++ a (term coercible to a) function;
++ a (possibly quantified) implication;
++ a (possibly quantified) double implication;
++ a (possibly quantified) instance of the reflect predicate (see
+  section :ref:`views_and_reflection_ssr`).
+
+
+Let ``top`` be the top assumption in the goal.
+
+There are three steps in the behaviour of an assumption view tactic:
+
++ It first introduces ``top``.
++ If the type of :token:`term` is neither a double implication nor an
+  instance of the reflect predicate, then the tactic automatically
+  generalises a term of the form: ``term term1 … termn`` where the
+  terms ``term1 … termn`` instantiate the possible quantified variables of
+  ``term`` , in order for ``(term term1 … termn top)`` to be well typed.
++ If the type of ``term`` is an equivalence, or an instance of the
+  reflect predicate, it generalises a term of the form:
+  ``(termvh (term term1 … termn ))`` where the term ``termvh``
+  inserted is called an
+  *assumption interpretation view hint*.
++ It finally clears top.
+
+
+For a ``case/term`` tactic, the generalisation step is replaced by a
+case analysis step.
+
+*View hints* are declared by the user (see section:ref:`views_and_reflection_ssr`.8) and are
+stored in the Hint View database. The proof engine automatically
+detects from the shape of the top assumption ``top`` and of the view lemma
+``term`` provided to the tactic the appropriate view hint in the
+database to be inserted.
+
+If ``term`` is a double implication, then the view hint will be one of
+the defined view hints for implication. These hints are by default the
+ones present in the file ``ssreflect.v``:
+
+.. coqtop:: in
+
+   Lemma iffLR : forall P Q, (P <-> Q) -> P -> Q.
+
+which transforms a double implication into the left-to-right one, or:
+
+.. coqtop:: in
+
+   Lemma iffRL : forall P Q, (P <-> Q) -> Q -> P.
+
+which produces the converse implication. In both cases, the two
+first Prop arguments are implicit.
+
+If ``term`` is an instance of the ``reflect`` predicate, then ``A`` will be one
+of the defined view hints for the ``reflec``t predicate, which are by
+default the ones present in the file ``ssrbool.v``. These hints are not
+only used for choosing the appropriate direction of the translation,
+but they also allow complex transformation, involving negations.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: in
+
+     Lemma introN : forall (P : Prop) (b : bool), reflect P b -> ~ P -> ~~b.
+
+  .. coqtop:: all
+
+     Lemma test (a b : bool) (Ha : a) (Hb : b) : ~~ (a && b).
+     apply/andP.
+
+  In fact this last script does not
+  exactly use the hint ``introN``, but the more general hint:
+
+  .. coqtop:: in
+
+     Lemma introNTF : forall (P : Prop) (b c : bool),
+       reflect P b -> (if c then ~ P else P) -> ~~ b = c.
+
+  The lemma ` `introN`` is an instantiation of introNF using c := true.
+
+Note that views, being part of :token:`i_pattern`, can be used to interpret
+assertions too. For example the following script asserts ``a && b`` but
+actually used its propositional interpretation.
+
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test (a b : bool) (pab : b && a) : b.
+     have /andP [pa ->] : (a && b) by rewrite andbC.
+
+Interpreting goals
+
+A goal interpretation view tactic of the form:
+
+.. tacv:: apply/@term
+
+applied to a goal ``top`` is interpreted in the following way:
+
++ If the type of ``term`` is not an instance of the ``reflect`` predicate,
+  nor an equivalence, then the term ``term`` is applied to the current
+  goal ``top``, possibly inserting implicit arguments.
++ If the type of ``term`` is an instance of the reflect predicate or an
+  equivalence, then a *goal interpretation view hint* can possibly be
+  inserted, which corresponds to the application of a term
+  ``(termvh (term _ … _))`` to the current goal, possibly inserting implicit arguments.
+
+
+Like assumption interpretation view hints, goal interpretation ones
+are user defined lemmas stored (see section :ref:`views_and_reflection_ssr`) in the ``Hint View``
+database bridging the possible gap between the type of ``term`` and the
+type of the goal.
+
+
+Interpreting equivalences
+~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Equivalent boolean propositions are simply *equal* boolean terms. A
+special construction helps the user to prove boolean equalities by
+considering them as logical double implications (between their coerced
+versions), while performing at the same time logical operations on
+both sides.
+
+The syntax of double views is:
+
+.. tacv:: apply/@term/@term
+
+The first term is the view lemma applied to the left hand side of the
+equality, while the second term is the one applied to the right hand side.
+
+In this context, the identity view can be used when no view has to be applied:
+
+.. coqtop:: in
+
+   Lemma idP : reflect b1 b1.
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test (b1 b2 b3 : bool) : ~~ (b1 || b2) = b3.
+     apply/idP/idP.
+
+  The same goal can be decomposed in several ways, and the user may
+  choose the most convenient interpretation.
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+
+  .. coqtop:: all
+
+     Lemma test (b1 b2 b3 : bool) : ~~ (b1 || b2) = b3.
+     apply/norP/idP.
+
+
+.. _declaring_new_hints_ssr:
+
+Declaring new Hint Views
+~~~~~~~~~~~~~~~~~~~~~~~~
+
+The database of hints for the view mechanism is extensible via a
+dedicated vernacular command. As library ``ssrbool.v`` already declares a
+corpus of hints, this feature is probably useful only for users who
+define their own logical connectives. Users can declare their own
+hints following the syntax used in ``ssrbool.v``:
+
+.. cmd:: Hint View for move / @ident {? | @num }
+.. cmd:: Hint View for apply / @ident {? | @num }
+
+The :token:`ident` is the name of the lemma to be
+declared as a hint. If `move` is used as
+tactic, the hint is declared for assumption interpretation tactics,
+`apply` declares hints for goal interpretations. Goal interpretation
+view hints are declared for both simple views and left hand side
+views. The optional natural number is the number of implicit
+arguments to be considered for the declared hint view lemma.
+
+The command:
+
+.. cmd:: Hint View for apply//@ident {? | @num }
+
+with a double slash ``//``, declares hint views for right hand sides of
+double views.
+
+See the files ``ssreflect.v`` and ``ssrbool.v`` for examples.
+
+
+Multiple views
+~~~~~~~~~~~~~~
+
+The hypotheses and the goal can be interpreted applying multiple views
+in sequence. Both move and apply can be followed by an arbitrary
+number of ``/term``. The main difference between the following two
+tactics
+
+.. coqtop:: in
+
+   apply/v1/v2/v3.
+   apply/v1; apply/v2; apply/v3.
+
+is that the former applies all the views to the principal goal.
+Applying a view with hypotheses generates new goals, and the second
+line would apply the view ``v2`` to all the goals generated by ``apply/v1``.
+
+Note that the NO-OP intro pattern ``-`` can be used to separate two views,
+making the two following examples equivalent:
+
+.. coqtop:: in
+
+   move=> /v1; move=> /v2.
+   move=> /v1 - /v2.
+
+The tactic ``move`` can be used together with the ``in`` tactical to
+pass a given hypothesis to a lemma.
+
+
+.. example::
+
+  .. coqtop:: reset
+
+     From Coq Require Import ssreflect ssrbool.
+     Set Implicit Arguments.
+     Unset Strict Implicit.
+     Unset Printing Implicit Defensive.
+     Section Test.
+     Variables P Q R : Prop.
+
+  .. coqtop:: all
+
+     Variable P2Q : P -> Q.
+     Variable Q2R : Q -> R.
+
+     Lemma test (p : P) : True.
+     move/P2Q/Q2R in p.
+
+If the list of views is of length two, ``Hint Views`` for interpreting
+equivalences are indeed taken into account, otherwise only single
+``Hint Views`` are used.
+
+
+|SSR| searching tool
+--------------------
+
+|SSR| proposes an extension of the Search command. Its syntax is:
+
+.. cmd:: Search {? @pattern }  {* {? - } %( @string %| @pattern %) {? % @ident} } {? in {+ {? - } @qualid } }
+
+where :token:`qualid` is the name of an open module. This command search returns
+the list of lemmas:
+
+
++ whose *conclusion* contains a subterm matching the optional first
+  pattern. A - reverses the test, producing the list of lemmas whose
+  conclusion does not contain any subterm matching the pattern;
++ whose name contains the given string. A ``-`` prefix reverses the test,
+  producing the list of lemmas whose name does not contain the string. A
+  string that contains symbols or is followed by a scope key, is
+  interpreted as the constant whose notation involves that string (e.g.,
+  `+` for `addn`), if this is unambiguous; otherwise the diagnostic
+  includes the output of the ``Locate`` vernacular command.
++ whose statement, including assumptions and types, contains a subterm
+  matching the next patterns. If a pattern is prefixed by ``-``, the test is
+  reversed;
++ contained in the given list of modules, except the ones in the
+  modules prefixed by a ``-``.
+
+
+Note that:
+
+
++ As for regular terms, patterns can feature scope indications. For
+  instance, the command: ``Search _ (_ + _)%N.`` lists all the lemmas whose
+  statement (conclusion or hypotheses) involve an application of the
+  binary operation denoted by the infix ``+`` symbol in the ``N`` scope (which is
+  |SSR| scope for natural numbers).
++ Patterns with holes should be surrounded by parentheses.
++ Search always volunteers the expansion of the notation, avoiding the
+  need to execute Locate independently. Moreover, a string fragment
+  looks for any notation that contains fragment as a substring. If the
+  ``ssrbool.v`` library is imported, the command: ``Search "~~".`` answers :
+
+  .. example::
+
+    .. coqtop:: reset
+
+       From Coq Require Import ssreflect ssrbool.
+       Set Implicit Arguments.
+       Unset Strict Implicit.
+       Unset Printing Implicit Defensive.
+
+    .. coqtop:: all
+
+       Search "~~".
+
++ A diagnostic is issued if there are different matching notations; it
+  is an error if all matches are partial.
++ Similarly, a diagnostic warns about multiple interpretations, and
+  signals an error if there is no default one.
++ The command ``Search in M.`` is a way of obtaining the complete
+  signature of the module ``M``.
++ Strings and pattern indications can be interleaved, but the first
+  indication has a special status if it is a pattern, and only filters
+  the conclusion of lemmas:
+
+    + The command : ``Search (_ =1 _) "bij".`` lists all the lemmas whose
+      conclusion features a ``=1`` and whose name contains the string ``bij``.
+    + The command : ``Search "bij" (_ =1 _).`` lists all the lemmas whose
+      statement, including hypotheses, features a ``=1`` and whose name
+      contains the string ``bij``.
+
+Synopsis and Index
+------------------
+
+Parameters
+~~~~~~~~~~
+
+|SSR| tactics
+
+.. prodn::
+   d_tactic ::= elim %| case %| congr %| apply %| exact %| move
+
+Notation scope
+
+.. prodn:: key ::= @ident
+
+Module name
+
+.. prodn:: name ::= @qualid
+
+Natural number
+
+.. prodn:: natural ::= @num %| @ident
+
+where :token:`ident` is an Ltac variable denoting a standard |Coq| numeral
+(should not be the name of a tactic which can be followed by a
+bracket ``[``, like ``do``, ``have``,…)
+
+Pattern
+
+.. prodn:: pattern ::= @term
+
+Items and switches
+~~~~~~~~~~~~~~~~~~
+
+.. prodn:: binder ::= @ident %| ( @ident {? : @term } )
+
+binder see :ref:`abbreviations_ssr`.
+
+.. prodn:: clear_switch ::= { {+ @ident } }
+
+clear switch see :ref:`discharge_ssr`
+
+.. prodn:: c_pattern ::= {? @term in %| @term as } @ident in @term
+
+context pattern see :ref:`contextual_patterns_ssr`
+
+.. prodn:: d_item ::= {? @occ_switch %| @clear_switch } {? @term %| ( @c_pattern ) }
+
+discharge item see :ref:`discharge_ssr`
+
+.. prodn:: gen_item ::= {? @  } @ident %| ( @ident ) %| ( {? @  } @ident := @c_pattern )
+
+generalization item see :ref:`structure_ssr`
+
+.. prodn:: i_pattern ::= @ident %| _ %| ? %| * %| {? @occ_switch } -> %| {? @occ_switch } <- %| [ {*| {* @i_item } } %| - %| [: {+ @ident } ]
+
+intro pattern :ref:`introduction_ssr`
+
+.. prodn:: i_item ::= @clear_switch %| @s_item %| @i_pattern %| / @term
+
+intro item  see :ref:`introduction_ssr`
+
+.. prodn:: int_mult ::= {? @num } @mult_mark
+
+multiplier  see :ref:`iteration_ssr`
+
+.. prodn:: occ_switch ::= { {? + %| - } {* @natural } }
+
+occur. switch see :ref:`occurrence_selection_ssr`
+
+.. prodn:: mult ::= {? @natural } @mult_mark
+
+multiplier see :ref:`iteration_ssr`
+
+.. prodn:: mult_mark ::= ? %| !
+
+multiplier mark see :ref:`iteration_ssr`
+
+.. prodn:: r_item ::= {? / } @term %| @s_item
+
+rewrite item see :ref:`rewriting_ssr`
+
+.. prodn:: r_prefix ::= {? - } {? @int_mult } {? @occ_switch %| @clear_switch } {? [ @r_pattern ] }
+
+rewrite prefix see :ref:`rewriting_ssr`
+
+.. prodn:: r_pattern ::= @term %| @c_pattern %| in {? @ident in } @term
+
+rewrite pattern see :ref:`rewriting_ssr`
+
+.. prodn:: r_step ::= {? @r_prefix } @r_item
+
+rewrite step see :ref:`rewriting_ssr`
+
+.. prodn:: s_item ::= /= %| // %| //=
+
+simplify switch see :ref:`introduction_ssr`
+
+Tactics
+~~~~~~~
+
+*Note*: ``without loss`` and ``suffices`` are synonyms for ``wlog`` and ``suff``
+respectively.
+
+.. tacn:: move
+
+idtac or hnf see  :ref:`bookkeeping_ssr`
+
+.. tacn:: apply
+.. tacn:: exact
+
+application see :ref:`the_defective_tactics_ssr`
+
+.. tacn:: abstract
+
+ see :ref:`abstract_ssr` and :ref:`generating_let_ssr`
+
+.. tacn:: elim
+
+induction see :ref:`the_defective_tactics_ssr`
+
+.. tacn:: case
+
+case analysis see :ref:`the_defective_tactics_ssr`
+
+.. tacn:: rewrite {+ @r_step }
+
+rewrite  see :ref:`rewriting_ssr`
+
+.. tacn:: have {* @i_item } {? @i_pattern } {? @s_item %| {+ @binder } } {? : @term } := @term
+.. tacv:: have {* @i_item } {? @i_pattern } {? @s_item %| {+ @binder } } : @term {? by @tactic }
+.. tacn:: have suff {? @clear_switch } {? @i_pattern } {? : @term } := @term
+.. tacv:: have suff {? @clear_switch } {? @i_pattern } : @term {? by @tactic }
+.. tacv:: gen have {? @ident , } {? @i_pattern } : {+ @gen_item } / @term {? by @tactic }
+
+forward  chaining see :ref:`structure_ssr`
+
+
+.. tacn:: wlog {? suff } {? @i_item } : {* @gen_item %| @clear_switch } / @term
+
+specializing see :ref:`structure_ssr`
+
+.. tacn:: suff {* @i_item } {? @i_pattern } {+ @binder } : @term {? by @tactic }
+.. tacv:: suff {? have } {? @clear_switch } {? @i_pattern } : @term {? by @tactic }
+
+backchaining see :ref:`structure_ssr`
+
+.. tacn:: pose @ident := @term
+
+local definition :ref:`definitions_ssr`
+
+.. tacv:: pose @ident {+ @binder } := @term
+
+local function definition
+
+.. tacv:: pose fix @fix_body
+
+local fix definition
+
+.. tacv:: pose cofix @fix_body
+
+local cofix definition
+
+.. tacn:: set @ident {? : @term } := {? @occ_switch } %( @term %| ( @c_pattern) %)
+
+abbreviation see :ref:`abbreviations_ssr`
+
+.. tacn:: unlock {* {? @r_prefix } @ident }
+
+unlock see :ref:`locking_ssr`
+
+.. tacn:: congr {? @num } @term
+
+congruence :ref:`congruence_ssr`
+
+
+Tacticals
+~~~~~~~~~
+
+.. prodn:: tactic += @d_tactic {? @ident } : {+ @d_item } {? @clear_switch }
+
+discharge :ref:`discharge_ssr`
+
+.. prodn:: tactic += @tacitc => {+ @i_item }
+
+introduction see :ref:`introduction_ssr`
+
+.. prodn:: tactic += @tactic in {+ @gen_item %| @clear_switch } {? * }
+
+localization see :ref:`localization_ssr`
+
+.. prodn:: tactic += do {? @mult } %( @tactic %| [ {+| @tactic } ] %)
+
+iteration  see :ref:`iteration_ssr`
+
+.. prodn:: tactic += @tactic ; %( first %| last %) {? @num } %( @tactic %| [ {+| @tactic } ] %)
+
+selector  see :ref:`selectors_ssr`
+
+.. prodn:: tactic += @tactic ; %( first %| last %) {? @num }
+
+rotation see :ref:`selectors_ssr`
+
+.. prodn:: tactic += by %( @tactic %| [ {*| @tactic } ] %)
+
+closing see :ref:`terminators_ssr`
+
+Commands
+~~~~~~~~
+
+.. cmd:: Hint View for %( move %| apply %) / @ident {? | @num }
+
+view hint declaration see :ref:`declaring_new_hints_ssr`
+
+.. cmd:: Hint View for apply // @ident {? @num }
+
+right hand side double , view hint declaration see :ref:`declaring_new_hints_ssr`
+
+.. cmd:: Prenex Implicits {+ @ident }
+
+prenex implicits declaration see :ref:`parametric_polymorphism_ssr`
+
+.. rubric:: Footnotes
+
+.. [#1] Unfortunately, even after a call to the Set Printing All command,
+  some occurrences are still not displayed to the user, essentially the
+  ones possibly hidden in the predicate of a dependent match structure.
+.. [#2] Thus scripts that depend on bound variable names, e.g., via intros
+  or with, are inherently fragile.
+.. [#3] The name ``subnK`` reads as “right cancellation rule for nat
+  subtraction”.
+.. [#4] Also, a slightly different variant may be used for the first :token:`d_item`
+  of case and elim; see section :ref:`type_families_ssr`.
+.. [#5] Except /= does not expand the local definitions created by the
+  |SSR| in tactical.
+.. [#6] |SSR| reserves all identifiers of the form “_x_”, which is
+  used for such generated names.
+.. [#7] More precisely, it should have a quantified inductive type with a
+  assumptions and m − a constructors.
+.. [#8] This is an implementation feature: there is not such obstruction
+  in the metatheory
+.. [#9] The current state of the proof shall be displayed by the Show
+  Proof command of |Coq| proof mode.
diff --git a/doc/sphinx/replaces.rst b/doc/sphinx/replaces.rst
index 2c38219b9..d4f6835ef 100644
--- a/doc/sphinx/replaces.rst
+++ b/doc/sphinx/replaces.rst
@@ -59,6 +59,7 @@
 .. |p_i| replace:: `p`\ :math:`_{i}`
 .. |p_n| replace:: `p`\ :math:`_{n}`
 .. |Program| replace:: :strong:`Program`
+.. |SSR| replace:: :smallcaps:`SSReflect`
 .. |t_1| replace:: `t`\ :math:`_{1}`
 .. |t_i| replace:: `t`\ :math:`_{i}`
 .. |t_m| replace:: `t`\ :math:`_{m}`