Merge branch 'trunk' into located_switch

9 files changed, 109 insertions, 81 deletions

diff --git a/doc/RecTutorial/coqartmacros.tex b/doc/RecTutorial/coqartmacros.tex
index 2a2c211963..72d7492690 100644
--- a/doc/RecTutorial/coqartmacros.tex
+++ b/doc/RecTutorial/coqartmacros.tex
@@ -149,7 +149,7 @@
 \newcommand{\PicAbst}[3]{\begin{bundle}{\bf abst}\chunk{#1}\chunk{#2}\chunk{#3}%
                         \end{bundle}}
 
-% the same in DeBruijn form
+% the same in de Bruijn form
 \newcommand{\PicDbj}[2]{\begin{bundle}{\bf abst}\chunk{#1}\chunk{#2}
                        \end{bundle}}
 
diff --git a/doc/refman/RefMan-cic.tex b/doc/refman/RefMan-cic.tex
index b9c17d8148..fdd2725810 100644
--- a/doc/refman/RefMan-cic.tex
+++ b/doc/refman/RefMan-cic.tex
@@ -79,8 +79,8 @@ An algebraic universe $u$ is either a variable (a qualified
 identifier with a number) or a successor of an algebraic universe (an
 expression $u+1$), or an upper bound of algebraic universes (an
 expression $max(u_1,...,u_n)$), or the base universe (the expression
-$0$) which corresponds, in the arity of sort-polymorphic inductive
-types (see Section \ref{Sort-polymorphism-inductive}),
+$0$) which corresponds, in the arity of template polymorphic inductive
+types (see Section \ref{Template-polymorphism}),
 to the predicative sort {\Set}. A graph of constraints between
 the universe variables is maintained globally. To ensure the existence
 of a mapping of the universes to the positive integers, the graph of
@@ -977,8 +977,8 @@ Inductive exType (P:Type->Prop) : Type :=
 %is recursive or not.  We shall write the type $(x:_R T)C$ if it is 
 %a recursive argument and $(x:_P T)C$ if the argument is not recursive.
 
-\paragraph[Sort-polymorphism of inductive types.]{Sort-polymorphism of inductive types.\index{Sort-polymorphism of inductive types}}
-\label{Sort-polymorphism-inductive}
+\paragraph[Template polymorphism.]{Template polymorphism.\index{Template polymorphism}}
+\label{Template-polymorphism}
 
 Inductive types declared in {\Type} are
 polymorphic over their arguments in {\Type}.
@@ -1120,6 +1120,10 @@ Check (fun (A:Prop) (B:Set) => prod A B).
 Check (fun (A:Type) (B:Prop) => prod A B).
 \end{coq_example}
 
+\Rem Template polymorphism used to be called ``sort-polymorphism of
+inductive types'' before universe polymorphism (see
+Chapter~\ref{Universes-full}) was introduced.
+
 \subsection{Destructors}
 The specification of inductive definitions with arities and
 constructors is quite natural.  But we still have to say how to use an
diff --git a/doc/refman/RefMan-ltac.tex b/doc/refman/RefMan-ltac.tex
index 9378529cbe..0346c4a555 100644
--- a/doc/refman/RefMan-ltac.tex
+++ b/doc/refman/RefMan-ltac.tex
@@ -1087,8 +1087,8 @@ Fail all:let n:= numgoals in guard n=2.
 Reset Initial.
 \end{coq_eval}
 
-\subsubsection[Proving a subgoal as a separate lemma]{Proving a subgoal as a separate lemma\tacindex{abstract}\comindex{Qed exporting}
-\index{Tacticals!abstract@{\tt abstract}}}
+\subsubsection[Proving a subgoal as a separate lemma]{Proving a subgoal as a separate lemma\tacindex{abstract}\tacindex{transparent\_abstract}\comindex{Qed exporting}
+\index{Tacticals!abstract@{\tt abstract}}\index{Tacticals!transparent\_abstract@{\tt transparent\_abstract}}}
 
 From the outside ``\texttt{abstract \tacexpr}'' is the same as
 {\tt solve \tacexpr}. Internally it saves an auxiliary lemma called
@@ -1114,13 +1114,24 @@ on. This can be obtained thanks to the option below.
 {\tt Set Shrink Abstract}
 \end{quote}
 
-When set, all lemmas generated through \texttt{abstract {\tacexpr}} are
-quantified only over the variables that appear in the term constructed by
-\texttt{\tacexpr}.
+When set, all lemmas generated through \texttt{abstract {\tacexpr}}
+and \texttt{transparent\_abstract {\tacexpr}} are quantified only over the
+variables that appear in the term constructed by \texttt{\tacexpr}.
 
 \begin{Variants}
 \item \texttt{abstract {\tacexpr} using {\ident}}.\\
   Give explicitly the name of the auxiliary lemma.
+  Use this feature at your own risk; explicitly named and reused subterms
+  don't play well with asynchronous proofs.
+\item \texttt{transparent\_abstract {\tacexpr}}.\\
+  Save the subproof in a transparent lemma rather than an opaque one.
+  Use this feature at your own risk; building computationally relevant terms
+  with tactics is fragile.
+\item \texttt{transparent\_abstract {\tacexpr} using {\ident}}.\\
+  Give explicitly the name of the auxiliary transparent lemma.
+  Use this feature at your own risk; building computationally relevant terms
+  with tactics is fragile, and explicitly named and reused subterms
+  don't play well with asynchronous proofs.
 \end{Variants}
 
 \ErrMsg \errindex{Proof is not complete}
@@ -1231,7 +1242,7 @@ This will automatically print the same trace as {\tt Info \num} at each tactic c
 
 The current value for the {\tt Info Level} option can be checked using the {\tt Test Info Level} command.
 
-\subsection[Interactive debugger]{Interactive debugger\optindex{Ltac Debug}}
+\subsection[Interactive debugger]{Interactive debugger\optindex{Ltac Debug}\optindex{Ltac Batch Debug}}
 
 The {\ltac} interpreter comes with a step-by-step debugger. The
 debugger can be activated using the command
@@ -1262,6 +1273,17 @@ r $n$: & advance $n$ steps further\\
 r {\qstring}: & advance up to the next call to ``{\tt idtac} {\qstring}''\\
 \end{tabular}
 
+A non-interactive mode for the debugger is available via the command
+
+\begin{quote}
+{\tt Set Ltac Batch Debug.}
+\end{quote}
+
+This option has the effect of presenting a newline at every prompt,
+when the debugger is on.  The debug log thus created, which does not
+require user input to generate when this option is set, can then be
+run through external tools such as \texttt{diff}.
+
 \subsection[Profiling {\ltac} tactics]{Profiling {\ltac} tactics\optindex{Ltac Profiling}\comindex{Show Ltac Profile}\comindex{Reset Ltac Profile}}
 
 It is possible to measure the time spent in invocations of primitive tactics as well as tactics defined in {\ltac} and their inner invocations. The primary use is the development of complex tactics, which can sometimes be so slow as to impede interactive usage. The reasons for the performence degradation can be intricate, like a slowly performing {\ltac} match or a sub-tactic whose performance only degrades in certain situations. The profiler generates a call tree and indicates the time spent in a tactic depending its calling context. Thus it allows to locate the part of a tactic definition that contains the performance bug.
diff --git a/doc/refman/RefMan-oth.tex b/doc/refman/RefMan-oth.tex
index 56ce753cd6..3daaac88b1 100644
--- a/doc/refman/RefMan-oth.tex
+++ b/doc/refman/RefMan-oth.tex
@@ -697,8 +697,8 @@ which can be any valid path.
 
 \subsection[\tt Add LoadPath {\str} as {\dirpath}.]{\tt Add LoadPath {\str} as {\dirpath}.\comindex{Add LoadPath}\label{AddLoadPath}}
 
-This command is equivalent to the command line option {\tt -Q {\dirpath}
-  {\str}}. It adds the physical directory {\str} to the current {\Coq}
+This command is equivalent to the command line option {\tt -Q {\str}
+  {\dirpath}}. It adds the physical directory {\str} to the current {\Coq}
 loadpath and maps it to the logical directory {\dirpath}.
 
 \begin{Variants}
@@ -707,8 +707,8 @@ Performs as {\tt Add LoadPath {\str} as {\dirpath}} but for the empty directory
 \end{Variants}
 
 \subsection[\tt Add Rec LoadPath {\str} as {\dirpath}.]{\tt Add Rec LoadPath {\str} as {\dirpath}.\comindex{Add Rec LoadPath}\label{AddRecLoadPath}}
-This command is equivalent to the command line option {\tt -R {\dirpath}
-  {\str}}. It adds the physical directory {\str} and all its
+This command is equivalent to the command line option {\tt -R {\str}
+  {\dirpath}}. It adds the physical directory {\str} and all its
 subdirectories to the current {\Coq} loadpath.
 
 \begin{Variants}
@@ -960,6 +960,22 @@ time of writing this documentation, the default value is 50).
 \subsection[\tt Test Printing Depth.]{\tt Test Printing Depth.\optindex{Printing Depth}}
 This command displays the current nesting depth used for display.
 
+\subsection[\tt Unset Printing Compact Contexts.]{\tt Unset Printing Compact Contexts.\optindex{Printing Compact Contexts}}
+This command resets the displaying of goals contexts to non compact
+mode (default at the time of writing this documentation). Non compact
+means that consecutive variables of different types are printed on
+different lines.
+
+\subsection[\tt Set Printing Compact Contexts.]{\tt Set Printing Compact Contexts.\optindex{Printing Compact Contexts}}
+This command sets the displaying of goals contexts to compact mode.
+The printer tries to reduce the vertical size of goals contexts by
+putting several variables (even if of different types) on the same
+line provided it does not exceed the printing width (See {\tt Set
+  Printing Width} above).
+
+\subsection[\tt Test Printing Compact Contexts.]{\tt Test Printing Compact Contexts.\optindex{Printing Compact Contexts}}
+This command displays the current state of compaction of goal d'isolat.
+
 \subsection[\tt Set Printing Dependent Evars Line.]{\tt Set Printing Dependent Evars Line.\optindex{Printing Dependent Evars Line}}
 This command enables the printing of the ``{\tt (dependent evars: \ldots)}''
 line when {\tt -emacs} is passed.
diff --git a/doc/refman/RefMan-pre.tex b/doc/refman/RefMan-pre.tex
index f36969e821..0441f952df 100644
--- a/doc/refman/RefMan-pre.tex
+++ b/doc/refman/RefMan-pre.tex
@@ -529,7 +529,7 @@ intensive computations.
 
 Christine Paulin implemented an extension of inductive types allowing
 recursively non uniform parameters. Hugo Herbelin implemented
-sort-polymorphism for inductive types.
+sort-polymorphism for inductive types (now called template polymorphism).
 
 Claudio Sacerdoti Coen improved the tactics for rewriting on arbitrary
 compatible equivalence relations. He also generalized rewriting to
diff --git a/doc/refman/RefMan-pro.tex b/doc/refman/RefMan-pro.tex
index 6d54b4de15..0760d716e3 100644
--- a/doc/refman/RefMan-pro.tex
+++ b/doc/refman/RefMan-pro.tex
@@ -90,25 +90,12 @@ memory overflow.
 
   Defines the proved term as a transparent constant.
 
-\item {\tt Save.}
-\comindex{Save}
-
-  This is a deprecated equivalent to {\tt Qed}.
-
 \item {\tt Save {\ident}.}
   
   Forces the name of the original goal to be {\ident}.  This command
   (and the following ones) can only be used if the original goal has
   been opened using the {\tt Goal} command.
 
-\item {\tt Save Theorem {\ident}.} \\
- {\tt Save Lemma {\ident}.} \\
- {\tt Save Remark {\ident}.}\\
- {\tt Save Fact {\ident}.}
- {\tt Save Corollary {\ident}.}
- {\tt Save Proposition {\ident}.}
-
-  Are equivalent to {\tt Save {\ident}.} 
 \end{Variants}
 
 \subsection[\tt Admitted.]{\tt Admitted.\comindex{Admitted}\label{Admitted}}
@@ -118,7 +105,7 @@ the current proof and declare the initial goal as an axiom.
 \subsection[\tt Proof {\term}.]{\tt Proof {\term}.\comindex{Proof}
 \label{BeginProof}}
 This command applies in proof editing mode. It is equivalent to {\tt
-  exact {\term}; Save.} That is, you have to give the full proof in
+  exact {\term}. Qed.} That is, you have to give the full proof in
 one gulp, as a proof term (see Section~\ref{exact}).
 
 \variant {\tt Proof.}
diff --git a/doc/refman/RefMan-tac.tex b/doc/refman/RefMan-tac.tex
index 3f12411863..fc3fdd0025 100644
--- a/doc/refman/RefMan-tac.tex
+++ b/doc/refman/RefMan-tac.tex
@@ -1275,15 +1275,18 @@ in the list of subgoals remaining to prove.
 
 \item{\tt assert ( {\ident} := {\term} )}
 
-  This behaves as {\tt assert ({\ident} :\ {\type});[exact
-      {\term}|idtac]} where {\type} is the type of {\term}. This is
-  deprecated in favor of {\tt pose proof}.
+  This behaves as {\tt assert ({\ident} :\ {\type}) by exact {\term}}
+  where {\type} is the type of {\term}. This is deprecated in favor of
+  {\tt pose proof}.
+
+  If the head of {\term} is {\ident}, the tactic behaves as
+  {\tt specialize \term}.
 
   \ErrMsg \errindex{Variable {\ident} is already declared}
 
-\item \texttt{pose proof {\term} as {\intropattern}\tacindex{pose proof}}
+\item \texttt{pose proof {\term} \zeroone{as {\intropattern}}\tacindex{pose proof}}
 
-  This tactic behaves like \texttt{assert T as {\intropattern} by
+  This tactic behaves like \texttt{assert T \zeroone{as {\intropattern}} by
   exact {\term}} where \texttt{T} is the type of {\term}.
 
   In particular, \texttt{pose proof {\term} as {\ident}} behaves as
@@ -1326,8 +1329,8 @@ in the list of subgoals remaining to prove.
   following subgoals: {\tt U -> T} and \texttt{U}.  The subgoal {\tt U
     -> T} comes first in the list of remaining subgoal to prove.
 
-\item {\tt specialize ({\ident} \term$_1$ \dots\ \term$_n$)\tacindex{specialize}} \\
-      {\tt specialize {\ident} with \bindinglist}
+\item {\tt specialize ({\ident} \term$_1$ \dots\ \term$_n$)\tacindex{specialize} \zeroone{as \intropattern}}\\
+      {\tt specialize {\ident} with {\bindinglist} \zeroone{as \intropattern}}
 
       The tactic {\tt specialize} works on local hypothesis \ident.
       The premises of this hypothesis (either universal
@@ -1338,14 +1341,19 @@ in the list of subgoals remaining to prove.
       second form, all instantiation elements must be given, whereas
       in the first form the application to \term$_1$ {\ldots}
       \term$_n$ can be partial. The first form is equivalent to
-      {\tt assert (\ident' := {\ident} {\term$_1$} \dots\ \term$_n$);
-           clear \ident; rename \ident' into \ident}.
+      {\tt assert ({\ident} := {\ident} {\term$_1$} \dots\ \term$_n$)}.
+
+      With the {\tt as} clause, the local hypothesis {\ident} is left
+      unchanged and instead, the modified hypothesis is introduced as
+      specified by the {\intropattern}.
 
       The name {\ident} can also refer to a global lemma or
       hypothesis. In this case, for compatibility reasons, the
       behavior of {\tt specialize} is close to that of {\tt
         generalize}: the instantiated statement becomes an additional
-      premise of the goal.
+      premise of the goal. The {\tt as} clause is especially useful
+      in this case to immediately introduce the instantiated statement
+      as a local hypothesis.
 
   \begin{ErrMsgs}
   \item \errindexbis{{\ident} is used in hypothesis \ident'}{is used in hypothesis}
@@ -2179,32 +2187,24 @@ the given bindings to instantiate parameters or hypotheses of {\term}.
 \label{injection}
 \tacindex{injection}
 
-The {\tt injection} tactic is based on the fact that constructors of
-inductive sets are injections. That means that if $c$ is a constructor
-of an inductive set, and if $(c~\vec{t_1})$ and $(c~\vec{t_2})$ are two
-terms that are equal then $~\vec{t_1}$ and $~\vec{t_2}$ are equal
-too.
+The {\tt injection} tactic exploits the property that constructors of
+inductive types are injective, i.e. that if $c$ is a constructor
+of an inductive type and $c~\vec{t_1}$ and $c~\vec{t_2}$ are equal
+then $\vec{t_1}$ and $\vec{t_2}$ are equal too.
 
 If {\term} is a proof of a statement of conclusion
  {\tt {\term$_1$} = {\term$_2$}},
-then {\tt injection} applies injectivity as deep as possible to
-derive the equality of all the subterms of {\term$_1$} and {\term$_2$}
-placed in the same positions. For example, from {\tt (S
-  (S n))=(S (S (S m)))} we may derive {\tt n=(S m)}.  To use this
-tactic {\term$_1$} and {\term$_2$} should be elements of an inductive
-set and they should be neither explicitly equal, nor structurally
-different. We mean by this that, if {\tt n$_1$} and {\tt n$_2$} are
-their respective normal forms, then:
-\begin{itemize}
-\item {\tt n$_1$} and {\tt n$_2$} should not be syntactically equal,
-\item there must not exist any pair of subterms {\tt u} and {\tt w},
-  {\tt u} subterm of {\tt n$_1$} and {\tt w} subterm of {\tt n$_2$} ,
-  placed in the same positions and having different constructors as
-  head symbols.
-\end{itemize}
-If these conditions are satisfied, then, the tactic derives the
-equality of all the subterms of {\term$_1$} and {\term$_2$} placed in
-the same positions and puts them as antecedents of the current goal.
+then {\tt injection} applies the injectivity of constructors as deep as possible to
+derive the equality of all the subterms of {\term$_1$} and {\term$_2$} at positions
+where {\term$_1$} and {\term$_2$} start to differ.
+For example, from {\tt (S p, S n) = (q, S (S m)} we may derive {\tt S
+  p = q} and {\tt n = S m}. For this tactic to work, {\term$_1$} and
+{\term$_2$} should be typed with an inductive
+type and they should be neither convertible, nor having a different
+head constructor. If these conditions are satisfied, the tactic
+derives the equality of all the subterms of {\term$_1$} and
+{\term$_2$} at positions where they differ and adds them as
+antecedents to the conclusion of the current goal.
 
 \Example Consider the following goal:
 
@@ -2243,6 +2243,7 @@ context using \texttt{intros until \ident}.
 \item \errindex{Not a projectable equality but a discriminable one}
 \item \errindex{Nothing to do, it is an equality between convertible terms}
 \item \errindex{Not a primitive equality}
+\item \errindex{Nothing to inject}
 \end{ErrMsgs}
 
 \begin{Variants}
@@ -2618,9 +2619,9 @@ as the ones described in Section~\ref{Tac-induction}.
 In the syntax of the tactic, the identifier {\ident} is the name given
 to the induction hypothesis. The natural number {\num} tells on which
 premise of the current goal the induction acts, starting
-from 1 and counting both dependent and non dependent
-products. Especially, the current lemma must be composed of at least
-{\num} products.
+from 1, counting both dependent and non dependent
+products, but skipping local definitions. Especially, the current
+lemma must be composed of at least {\num} products.
 
 Like in a {\tt fix} expression, the induction
 hypotheses have to be used on structurally smaller arguments.
diff --git a/doc/refman/RefMan-tus.tex b/doc/refman/RefMan-tus.tex
index 797b0adedd..7e5bb81a90 100644
--- a/doc/refman/RefMan-tus.tex
+++ b/doc/refman/RefMan-tus.tex
@@ -288,8 +288,8 @@ constructors:
 \item $(\texttt{VAR}\;id)$, a reference to a global identifier called  $id$;
 \item $(\texttt{Rel}\;n)$, a bound variable, whose binder is the $nth$
       binder up in the term;
-\item $\texttt{DLAM}\;(x,t)$, a deBruijn's binder on the term $t$;
-\item $\texttt{DLAMV}\;(x,vt)$, a deBruijn's binder on all the terms of 
+\item $\texttt{DLAM}\;(x,t)$, a de Bruijn's binder on the term $t$;
+\item $\texttt{DLAMV}\;(x,vt)$, a de Bruijn's binder on all the terms of
       the vector $vt$;
 \item $(\texttt{DOP0}\;op)$, a unary operator $op$;
 \item $\texttt{DOP2}\;(op,t_1,t_2)$, the application of a binary
@@ -299,7 +299,7 @@ vector of terms $vt$.
 \end{itemize}
 
 In this meta-language, bound variables are represented using the
-so-called deBrujin's indexes. In this representation, an occurrence of
+so-called de Bruijn's indexes. In this representation, an occurrence of
 a bound variable is denoted by an integer, meaning the number of
 binders that must be traversed to reach its own
 binder\footnote{Actually, $(\texttt{Rel}\;n)$ means that $(n-1)$ binders
@@ -339,7 +339,7 @@ on the terms of the meta-language:
 \fun{val Generic.dependent : 'op term -> 'op term -> bool}
     {Returns true if the first term is a sub-term of the second.}
 %\fun{val Generic.subst\_var : identifier -> 'op term -> 'op term}
-%    { $(\texttt{subst\_var}\;id\;t)$ substitutes the deBruijn's index
+%    { $(\texttt{subst\_var}\;id\;t)$ substitutes the de Bruijn's index
 %     associated to $id$ to every occurrence of the term
 %    $(\texttt{VAR}\;id)$ in $t$.}
 \end{description}
@@ -482,7 +482,7 @@ following constructor functions:
 
 \begin{description}
 \fun{val Term.mkRel          : int -> constr}
-    {$(\texttt{mkRel}\;n)$ represents deBrujin's index $n$.}
+    {$(\texttt{mkRel}\;n)$ represents de Bruijn's index $n$.}
 
 \fun{val Term.mkVar : identifier -> constr} 
     {$(\texttt{mkVar}\;id)$
@@ -545,7 +545,7 @@ following constructor functions:
 
 \fun{val Term.mkProd : name ->constr ->constr -> constr}
     {$(\texttt{mkProd}\;x\;A\;B)$ represents the product $(x:A)B$.
-     The free ocurrences of $x$ in $B$ are represented by deBrujin's
+     The free ocurrences of $x$ in $B$ are represented by de Bruijn's
      indexes.}
 
 \fun{val Term.mkNamedProd : identifier -> constr -> constr -> constr}
@@ -553,14 +553,14 @@ following constructor functions:
      but the bound occurrences of $x$ in $B$ are denoted by 
      the identifier $(\texttt{mkVar}\;x)$. The function automatically 
      changes each occurrences of this identifier into the corresponding 
-     deBrujin's index.}
+     de Bruijn's index.}
 
 \fun{val Term.mkArrow : constr -> constr -> constr}
     {$(\texttt{arrow}\;A\;B)$ represents the type $(A\rightarrow B)$.}
 
 \fun{val Term.mkLambda       : name -> constr -> constr -> constr}
     {$(\texttt{mkLambda}\;x\;A\;b)$ represents the lambda abstraction 
-     $[x:A]b$. The free ocurrences of $x$ in $B$ are represented by deBrujin's
+     $[x:A]b$. The free ocurrences of $x$ in $B$ are represented by de Bruijn's
      indexes.}
 
 \fun{val Term.mkNamedLambda : identifier -> constr -> constr -> constr}
@@ -666,7 +666,7 @@ use the primitive \textsl{Case} described in Chapter \ref{Cic}
 \item Restoring type coercions and synthesizing the implicit arguments
 (the one denoted by question marks in
 {\Coq} syntax: see Section~\ref{Coercions}).
-\item Transforming the named bound variables into deBrujin's indexes.
+\item Transforming the named bound variables into de Bruijn's indexes.
 \item Classifying the global names into the different classes of
 constants (defined constants, constructors, inductive types, etc).
 \end{enumerate}
@@ -707,7 +707,7 @@ Once all the existential variables have been defined the derivation is
 completed, and a construction can be generated from the proof tree,
 replacing each of the existential variables by its definition.  This
 is exactly what happens when one of the commands
-\texttt{Qed}, \texttt{Save} or \texttt{Defined} is invoked
+\texttt{Qed} or \texttt{Defined} is invoked
 (see Section~\ref{Qed}). The saved theorem becomes a defined constant,
 whose body is the proof object generated.
 
diff --git a/doc/tutorial/Tutorial.tex b/doc/tutorial/Tutorial.tex
index 0d537256bb..30b6304c16 100644
--- a/doc/tutorial/Tutorial.tex
+++ b/doc/tutorial/Tutorial.tex
@@ -385,13 +385,11 @@ apply H; [ assumption | apply H0; assumption ].
 
 Let us now save lemma \verb:distr_impl::
 \begin{coq_example}
-Save.
+Qed.
 \end{coq_example}
 
-Here \verb:Save: needs no argument, since we gave the name \verb:distr_impl: 
-in advance;
-it is however possible to override the given name by giving a different 
-argument to command \verb:Save:.
+Here \verb:Qed: needs no argument, since we gave the name \verb:distr_impl: 
+in advance.
 
 Actually, such an easy combination of tactics \verb:intro:, \verb:apply:
 and \verb:assumption: may be found completely automatically by an automatic