diff options
Diffstat (limited to 'doc/refman')
33 files changed, 110 insertions, 26 deletions
diff --git a/doc/refman/AddRefMan-pre.tex b/doc/refman/AddRefMan-pre.tex index eee41a6798..856a823de0 100644 --- a/doc/refman/AddRefMan-pre.tex +++ b/doc/refman/AddRefMan-pre.tex @@ -4,6 +4,7 @@ \setheaders{Presentation of the Addendum} %END LATEX \chapter*{Presentation of the Addendum} +%HEVEA\cutname{addendum.html} Here you will find several pieces of additional documentation for the \Coq\ Reference Manual. Each of this chapters is concentrated on a diff --git a/doc/refman/AsyncProofs.tex b/doc/refman/AsyncProofs.tex index 1609e4a041..30039d4898 100644 --- a/doc/refman/AsyncProofs.tex +++ b/doc/refman/AsyncProofs.tex @@ -1,4 +1,5 @@ \achapter{Asynchronous and Parallel Proof Processing} +%HEVEA\cutname{async-proofs.html} \aauthor{Enrico Tassi} \label{pralitp} diff --git a/doc/refman/CanonicalStructures.tex b/doc/refman/CanonicalStructures.tex index 275e1c2d55..8961b00964 100644 --- a/doc/refman/CanonicalStructures.tex +++ b/doc/refman/CanonicalStructures.tex @@ -1,4 +1,5 @@ \achapter{Canonical Structures} +%HEVEA\cutname{canonical-structures.html} \aauthor{Assia Mahboubi and Enrico Tassi} \label{CS-full} diff --git a/doc/refman/Cases.tex b/doc/refman/Cases.tex index a95d8114ff..7ad895f9d8 100644 --- a/doc/refman/Cases.tex +++ b/doc/refman/Cases.tex @@ -1,4 +1,5 @@ \achapter{Extended pattern-matching} +%HEVEA\cutname{cases.html} %BEGIN LATEX \defaultheaders %END LATEX diff --git a/doc/refman/Classes.tex b/doc/refman/Classes.tex index 7e07868a38..22c75b4fc8 100644 --- a/doc/refman/Classes.tex +++ b/doc/refman/Classes.tex @@ -6,6 +6,7 @@ \newcommand\tele[1]{\overrightarrow{#1}} \achapter{\protect{Type Classes}} +%HEVEA\cutname{type-classes.html} \aauthor{Matthieu Sozeau} \label{typeclasses} diff --git a/doc/refman/Coercion.tex b/doc/refman/Coercion.tex index 16006a6adf..ec46e1eb58 100644 --- a/doc/refman/Coercion.tex +++ b/doc/refman/Coercion.tex @@ -1,4 +1,5 @@ \achapter{Implicit Coercions} +%HEVEA\cutname{coercions.html} \aauthor{Amokrane Saïbi} \label{Coercions-full} diff --git a/doc/refman/Extraction.tex b/doc/refman/Extraction.tex index 499239b6f3..83e866e9f3 100644 --- a/doc/refman/Extraction.tex +++ b/doc/refman/Extraction.tex @@ -1,4 +1,5 @@ \achapter{Extraction of programs in Objective Caml and Haskell} +%HEVEA\cutname{extraction.html} \label{Extraction} \aauthor{Jean-Christophe Filliâtre and Pierre Letouzey} \index{Extraction} diff --git a/doc/refman/Micromega.tex b/doc/refman/Micromega.tex index 4daf98f87a..2617142f5a 100644 --- a/doc/refman/Micromega.tex +++ b/doc/refman/Micromega.tex @@ -1,4 +1,5 @@ \achapter{Micromega: tactics for solving arithmetic goals over ordered rings} +%HEVEA\cutname{micromega.html} \aauthor{Frédéric Besson and Evgeny Makarov} \newtheorem{theorem}{Theorem} diff --git a/doc/refman/Misc.tex b/doc/refman/Misc.tex index e953d2f709..ab00fbfe37 100644 --- a/doc/refman/Misc.tex +++ b/doc/refman/Misc.tex @@ -1,4 +1,5 @@ \achapter{\protect{Miscellaneous extensions}} +%HEVEA\cutname{miscellaneous.html} \asection{Program derivation} diff --git a/doc/refman/Nsatz.tex b/doc/refman/Nsatz.tex index 70e36a5ee9..1401af10f6 100644 --- a/doc/refman/Nsatz.tex +++ b/doc/refman/Nsatz.tex @@ -1,4 +1,5 @@ \achapter{Nsatz: tactics for proving equalities in integral domains} +%HEVEA\cutname{nsatz.html} \aauthor{Loïc Pottier} The tactic \texttt{nsatz} proves goals of the form diff --git a/doc/refman/Omega.tex b/doc/refman/Omega.tex index 1610305e75..8025fbe29f 100644 --- a/doc/refman/Omega.tex +++ b/doc/refman/Omega.tex @@ -1,5 +1,6 @@ \achapter{Omega: a solver of quantifier-free problems in Presburger Arithmetic} +%HEVEA\cutname{omega.html} \aauthor{Pierre Crégut} \label{OmegaChapter} diff --git a/doc/refman/Polynom.tex b/doc/refman/Polynom.tex index 77d5928345..d9b8b8c522 100644 --- a/doc/refman/Polynom.tex +++ b/doc/refman/Polynom.tex @@ -1,4 +1,5 @@ \achapter{The \texttt{ring} and \texttt{field} tactic families} +%HEVEA\cutname{ring.html} \aauthor{Bruno Barras, Benjamin Gr\'egoire, Assia Mahboubi, Laurent Th\'ery\footnote{based on previous work from Patrick Loiseleur and Samuel Boutin}} diff --git a/doc/refman/Program.tex b/doc/refman/Program.tex index f60908da6c..1e204dc83d 100644 --- a/doc/refman/Program.tex +++ b/doc/refman/Program.tex @@ -1,4 +1,5 @@ \achapter{\Program{}} +%HEVEA\cutname{program.html} \label{Program} \aauthor{Matthieu Sozeau} \index{Program} diff --git a/doc/refman/RefMan-cic.tex b/doc/refman/RefMan-cic.tex index ad795d4064..0dbfe05d48 100644 --- a/doc/refman/RefMan-cic.tex +++ b/doc/refman/RefMan-cic.tex @@ -2,6 +2,7 @@ \label{Cic} \index{Cic@\textsc{CIC}} \index{Calculus of Inductive Constructions}} +%HEVEA\cutname{cic.html} The underlying formal language of {\Coq} is a {\em Calculus of Inductive Constructions} (\CIC) whose inference rules are presented in diff --git a/doc/refman/RefMan-com.tex b/doc/refman/RefMan-com.tex index 45230fb6e5..9790111f14 100644 --- a/doc/refman/RefMan-com.tex +++ b/doc/refman/RefMan-com.tex @@ -2,6 +2,7 @@ \ttindex{coqtop} \ttindex{coqc} \ttindex{coqchk}} +%HEVEA\cutname{commands.html} There are three \Coq~commands: \begin{itemize} diff --git a/doc/refman/RefMan-ext.tex b/doc/refman/RefMan-ext.tex index 713f344cbe..b27a4dc943 100644 --- a/doc/refman/RefMan-ext.tex +++ b/doc/refman/RefMan-ext.tex @@ -1,4 +1,5 @@ \chapter[Extensions of \Gallina{}]{Extensions of \Gallina{}\label{Gallina-extension}\index{Gallina}} +%HEVEA\cutname{gallina-ext.html} {\gallina} is the kernel language of {\Coq}. We describe here extensions of the Gallina's syntax. @@ -279,15 +280,78 @@ of the chapter devoted to coercions. \label{prim-proj} The option {\tt Set Primitive Projections} turns on the use of primitive -projections when defining subsequent records. Primitive projections +projections when defining subsequent records (even through the {\tt + Inductive} and {\tt CoInductive} commands). Primitive projections extended the Calculus of Inductive Constructions with a new binary term constructor {\tt r.(p)} representing a primitive projection p applied to a record object {\tt r} (i.e., primitive projections are always applied). Even if the record type has parameters, these do not appear at applications of the projection, considerably reducing the sizes of terms when manipulating parameterized records and typechecking time. On the -user level, primitive projections are a transparent replacement -for the usual defined ones. +user level, primitive projections can be used as a replacement for the +usual defined ones, although there are a few notable differences. + +The internally omitted parameters can be reconstructed at printing time +even though they are absent in the actual AST manipulated by the kernel. This +can be obtained by setting the {\tt Printing Primitive Projection Parameters} +flag. Another compatibility printing can be activated thanks to the +{\tt Printing Primitive Projection Compatibility} option which governs the +printing of pattern-matching over primitive records. + +\subsubsection{Primitive Record Types} +When the {\tt Set Primitive Projections} option is on, definitions of +record types change meaning. When a type is declared with primitive +projections, its {\tt match} construct is disabled (see +\ref{primproj:compat} though). To eliminate the (co-)inductive type, one +must use its defined primitive projections. + +There are currently two ways to introduce primitive records types: +\begin{itemize} +\item Through the {\tt Record} command, in which case the type has to be + non-recursive. The defined type enjoys eta-conversion definitionally, + that is the generalized form of surjective pairing for records: + {\tt $r$ = Build\_R ($r$.($p_1$) .. $r$.($p_n$))}. Eta-conversion allows to define + dependent elimination for these types as well. +\item Through the {\tt Inductive} and {\tt CoInductive} commands, when + the body of the definition is a record declaration of the form {\tt + Build\_R \{ $p_1$ : $t_1$; .. ; $p_n$ : $t_n$ \}}. In this case the types can be + recursive and eta-conversion is disallowed. These kind of record types + differ from their traditional versions in the sense that dependent + elimination is not available for them and only non-dependent case analysis + can be defined. +\end{itemize} + +\subsubsection{Reduction} + +The basic reduction rule of a primitive projection is {\tt $p_i$ + (Build\_R $t_1$ .. $t_n$) $\rightarrow_{\iota}$ $t_i$}. However, to take the $\delta$ flag into +account, projections can be in two states: folded or unfolded. An +unfolded primitive projection application obeys the rule above, while +the folded version delta-reduces to the unfolded version. This allows to +precisely mimic the usual unfolding rules of constants. Projections +obey the usual {\tt simpl} flags of the {\tt Arguments} command in particular. + +There is currently no way to input unfolded primitive projections at the +user-level, and one must use the {\tt Printing Primitive Projection + Compatibility} to display unfolded primitive projections as matches +and distinguish them from folded ones. + +\subsubsection{Compatibility Projections and {\tt match}} +\label{primproj:compat} +To ease compatibility with ordinary record types, each primitive +projection is also defined as a ordinary constant taking parameters and +an object of the record type as arguments, and whose body is an +application of the unfolded primitive projection of the same name. These +constants are used when elaborating partial applications of the +projection. One can distinguish them from applications of the primitive +projection if the {\tt Printing Primitive Projection Parameters} option +is off: for a primitive projection application, parameters are printed +as underscores while for the compatibility projections they are printed +as usual. + +Additionally, user-written {\tt match} constructs on primitive records +are desugared into substitution of the projections, they cannot be +printed back as {\tt match} constructs. % - r.(p) and (p r) elaborate to native projection application, and % the parameters cannot be mentioned. The following arguments are @@ -305,13 +369,6 @@ for the usual defined ones. % - [pattern x at n], [rewrite x at n] and in general abstraction and selection % of occurrences may fail due to the disappearance of parameters. -The internally omitted parameters can be reconstructed at printing time -even though they are absent in the actual AST manipulated by the kernel. This -can be obtained by setting the {\tt Printing Primitive Projection Parameters} -flag. Another compatibility printing can be activated thanks to the -{\tt Printing Primitive Projection Compatibility} option which governs the -printing of pattern-matching over primitive records. - \section{Variants and extensions of {\mbox{\tt match}} \label{Extensions-of-match} \index{match@{\tt match\ldots with\ldots end}}} @@ -1664,7 +1721,7 @@ to be given as if none arguments were implicit. By symmetry, this also affects printing. To restore parsing and normal printing of implicit arguments, use: \begin{quote} -{\tt Set Parsing Explicit.} +{\tt Unset Parsing Explicit.} \end{quote} \subsection{Canonical structures diff --git a/doc/refman/RefMan-gal.tex b/doc/refman/RefMan-gal.tex index ef12fe416a..df0cd2b825 100644 --- a/doc/refman/RefMan-gal.tex +++ b/doc/refman/RefMan-gal.tex @@ -1,5 +1,6 @@ \chapter{The \gallina{} specification language \label{Gallina}\index{Gallina}} +%HEVEA\cutname{gallina.html} \label{BNF-syntax} % Used referred to as a chapter label This chapter describes \gallina, the specification language of {\Coq}. diff --git a/doc/refman/RefMan-ide.tex b/doc/refman/RefMan-ide.tex index c6fbd1c538..75f3d18ded 100644 --- a/doc/refman/RefMan-ide.tex +++ b/doc/refman/RefMan-ide.tex @@ -1,5 +1,6 @@ \chapter[\Coq{} Integrated Development Environment]{\Coq{} Integrated Development Environment\label{Addoc-coqide} \ttindex{coqide}} +%HEVEA\cutname{coqide.html} The \Coq{} Integrated Development Environment is a graphical tool, to be used as a user-friendly replacement to \texttt{coqtop}. Its main diff --git a/doc/refman/RefMan-int.tex b/doc/refman/RefMan-int.tex index 2b9e4e6051..f802a35950 100644 --- a/doc/refman/RefMan-int.tex +++ b/doc/refman/RefMan-int.tex @@ -2,6 +2,7 @@ \setheaders{Introduction} %END LATEX \chapter*{Introduction} +%HEVEA\cutname{introduction.html} This document is the Reference Manual of version \coqversion{} of the \Coq\ proof assistant. A companion volume, the \Coq\ Tutorial, is provided diff --git a/doc/refman/RefMan-lib.tex b/doc/refman/RefMan-lib.tex index 4ebb484e7c..c8e8443026 100644 --- a/doc/refman/RefMan-lib.tex +++ b/doc/refman/RefMan-lib.tex @@ -1,4 +1,5 @@ \chapter[The {\Coq} library]{The {\Coq} library\index{Theories}\label{Theories}} +%HEVEA\cutname{stdlib.html} The \Coq\ library is structured into two parts: diff --git a/doc/refman/RefMan-ltac.tex b/doc/refman/RefMan-ltac.tex index 3ce1d4ecd8..574591185c 100644 --- a/doc/refman/RefMan-ltac.tex +++ b/doc/refman/RefMan-ltac.tex @@ -1,4 +1,5 @@ \chapter[The tactic language]{The tactic language\label{TacticLanguage}} +%HEVEA\cutname{ltac.html} %\geometry{a4paper,body={5in,8in}} @@ -1105,19 +1106,14 @@ 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}\tacindex{transparent\_abstract}\comindex{Qed exporting} +\subsubsection[Proving a subgoal as a separate lemma]{Proving a subgoal as a separate lemma\tacindex{abstract}\tacindex{transparent\_abstract} \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 {\ident}\texttt{\_subproof}\textit{n} where {\ident} is the name of the current goal and \textit{n} is chosen so that this is a fresh name. -Such auxiliary lemma is inlined in the final proof term -unless the proof is ended with ``\texttt{Qed exporting}''. In such -case the lemma is preserved. The syntax -``\texttt{Qed exporting }\ident$_1$\texttt{, ..., }\ident$_n$'' -is also supported. In such case the system checks that the names given by the -user actually exist when the proof is ended. +Such an auxiliary lemma is inlined in the final proof term. This tactical is useful with tactics such as \texttt{omega} or \texttt{discriminate} that generate huge proof terms. With that tool diff --git a/doc/refman/RefMan-modr.tex b/doc/refman/RefMan-modr.tex index 2019a529fe..7c672cf422 100644 --- a/doc/refman/RefMan-modr.tex +++ b/doc/refman/RefMan-modr.tex @@ -1,4 +1,5 @@ \chapter[The Module System]{The Module System\label{chapter:Modules}} +%HEVEA\cutname{modules.html} The module system extends the Calculus of Inductive Constructions providing a convenient way to structure large developments as well as diff --git a/doc/refman/RefMan-oth.tex b/doc/refman/RefMan-oth.tex index 8f43ebcfbc..60cd8b73a4 100644 --- a/doc/refman/RefMan-oth.tex +++ b/doc/refman/RefMan-oth.tex @@ -1,5 +1,6 @@ \chapter[Vernacular commands]{Vernacular commands\label{Vernacular-commands} \label{Other-commands}} +%HEVEA\cutname{vernacular.html} \section{Displaying} diff --git a/doc/refman/RefMan-pre.tex b/doc/refman/RefMan-pre.tex index 0c2a18eb2e..991c9745e9 100644 --- a/doc/refman/RefMan-pre.tex +++ b/doc/refman/RefMan-pre.tex @@ -2,6 +2,7 @@ \setheaders{Credits} %END LATEX \chapter*{Credits} +%HEVEA\cutname{credits.html} %\addcontentsline{toc}{section}{Credits} \Coq{}~ is a proof assistant for higher-order logic, allowing the diff --git a/doc/refman/RefMan-pro.tex b/doc/refman/RefMan-pro.tex index eb59ca584e..8f659ded35 100644 --- a/doc/refman/RefMan-pro.tex +++ b/doc/refman/RefMan-pro.tex @@ -1,5 +1,6 @@ \chapter[Proof handling]{Proof handling\index{Proof editing} \label{Proof-handling}} +%HEVEA\cutname{proof-handling.html} In \Coq's proof editing mode all top-level commands documented in Chapter~\ref{Vernacular-commands} remain available diff --git a/doc/refman/RefMan-sch.tex b/doc/refman/RefMan-sch.tex index 23a1c9b029..956f308512 100644 --- a/doc/refman/RefMan-sch.tex +++ b/doc/refman/RefMan-sch.tex @@ -1,4 +1,5 @@ \chapter{Proof schemes} +%HEVEA\cutname{schemes.html} \section{Generation of induction principles with {\tt Scheme}} \label{Scheme} diff --git a/doc/refman/RefMan-ssr.tex b/doc/refman/RefMan-ssr.tex index 61f7421c44..be199e0b24 100644 --- a/doc/refman/RefMan-ssr.tex +++ b/doc/refman/RefMan-ssr.tex @@ -1,4 +1,5 @@ \achapter{The SSReflect proof language} +%HEVEA\cutname{ssreflect.html} \aauthor{Georges Gonthier, Assia Mahboubi, Enrico Tassi} \newcommand{\ssr}{{\sc SSReflect}} @@ -42,7 +43,7 @@ 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. +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 @@ -79,19 +80,19 @@ 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. -\ssrC{} includes a little language of patterns to select subterms in tactics +\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 the so-called reflection steps, typically +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~\label{sec:compat}. +or in the same Ltac expression. The few exceptions +to this statement are described in section~\ref{sec:compat}. \iffalse %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -130,7 +131,7 @@ ProofGeneral provided in the distribution: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection*{Acknowledgments} -The authors would like to thank Fr\'ed\'eric Blanqui, Fran\,cois Pottier +The authors would like to thank Frédéric Blanqui, François Pottier and Laurence Rideau for their comments and suggestions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/doc/refman/RefMan-syn.tex b/doc/refman/RefMan-syn.tex index d8a353300f..eecb5ac7c0 100644 --- a/doc/refman/RefMan-syn.tex +++ b/doc/refman/RefMan-syn.tex @@ -1,4 +1,5 @@ \chapter[Syntax extensions and interpretation scopes]{Syntax extensions and interpretation scopes\label{Addoc-syntax}} +%HEVEA\cutname{syntax-extensions.html} In this chapter, we introduce advanced commands to modify the way {\Coq} parses and prints objects, i.e. the translations between the diff --git a/doc/refman/RefMan-tac.tex b/doc/refman/RefMan-tac.tex index 6e27357008..a2d45046b0 100644 --- a/doc/refman/RefMan-tac.tex +++ b/doc/refman/RefMan-tac.tex @@ -3,6 +3,7 @@ \chapter{Tactics \index{Tactics} \label{Tactics}} +%HEVEA\cutname{tactics.html} A deduction rule is a link between some (unique) formula, that we call the {\em conclusion} and (several) formulas that we call the {\em @@ -3309,7 +3310,7 @@ evaluating purely computational expressions (i.e. with little dead code). fine-tuned. It is specially interesting for full evaluation of algebraic objects. This includes the case of reflection-based tactics. -\item {\tt native\_compute} \tacindex{native\_compute} +\item {\tt native\_compute} \tacindex{native\_compute} \optindex{NativeCompute Profiling} This tactic evaluates the goal by compilation to \ocaml{} as described in \cite{FullReduction}. If \Coq{} is running in native code, it can be typically diff --git a/doc/refman/RefMan-tacex.tex b/doc/refman/RefMan-tacex.tex index cb8f916f13..7cdb1a5274 100644 --- a/doc/refman/RefMan-tacex.tex +++ b/doc/refman/RefMan-tacex.tex @@ -1,4 +1,5 @@ \chapter[Detailed examples of tactics]{Detailed examples of tactics\label{Tactics-examples}} +%HEVEA\cutname{tactic-examples.html} This chapter presents detailed examples of certain tactics, to illustrate their behavior. diff --git a/doc/refman/RefMan-uti.tex b/doc/refman/RefMan-uti.tex index f6371f8e5c..ed41e32161 100644 --- a/doc/refman/RefMan-uti.tex +++ b/doc/refman/RefMan-uti.tex @@ -1,4 +1,5 @@ \chapter[Utilities]{Utilities\label{Utilities}} +%HEVEA\cutname{tools.html} The distribution provides utilities to simplify some tedious works beside proof development, tactics writing or documentation. diff --git a/doc/refman/Setoid.tex b/doc/refman/Setoid.tex index 6c79284389..b7b343112f 100644 --- a/doc/refman/Setoid.tex +++ b/doc/refman/Setoid.tex @@ -1,6 +1,7 @@ \newtheorem{cscexample}{Example} \achapter{\protect{Generalized rewriting}} +%HEVEA\cutname{setoid.html} \aauthor{Matthieu Sozeau} \label{setoids} @@ -223,7 +224,7 @@ the following command. \comindex{Add Parametric Morphism} \begin{quote} - \texttt{Add Parametric Morphism} ($x_1 : \T_!$) \ldots ($x_k : \T_k$)\\ + \texttt{Add Parametric Morphism} ($x_1 : \T_1$) \ldots ($x_k : \T_k$) : (\textit{f $t_1$ \ldots $t_n$})\\ \texttt{~with signature} \textit{sig}\\ \texttt{~as id}.\\ diff --git a/doc/refman/Universes.tex b/doc/refman/Universes.tex index 6ea2537399..75fac9454a 100644 --- a/doc/refman/Universes.tex +++ b/doc/refman/Universes.tex @@ -1,4 +1,5 @@ \achapter{Polymorphic Universes} +%HEVEA\cutname{universes.html} \aauthor{Matthieu Sozeau} \label{Universes-full} |