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-\achapter{\texttt{Natural} : proofs in natural language}
-\aauthor{Yann Coscoy}
-
-\asection{Introduction}
-
-\Natural~ is a package allowing the writing of proofs in natural
-language. For instance, the proof in \Coq~of the induction principle on pairs
-of natural numbers looks like this:
-
-\begin{coq_example*}
-Require Natural.
-\end{coq_example*}
-\begin{coq_example}
-Print nat_double_ind.
-\end{coq_example}
-
-Piping it through the \Natural~pretty-printer gives:
-
-\comindex{Print Natural}
-\begin{coq_example}
-Print Natural nat_double_ind.
-\end{coq_example}
-
-\asection{Activating \Natural}
-
-To enable the printing of proofs in natural language, you should
-type under \texttt{coqtop} or \texttt{coqtop -full} the command
-
-\begin{coq_example*}
-Require Natural.
-\end{coq_example*}
-
-By default, proofs are transcripted in english. If you wish to print them 
-in French, set the French option by
-
-\comindex{Set Natural}
-\begin{coq_example*}
-Set Natural French.
-\end{coq_example*}
-
-If you want to go back to English, type in
-
-\begin{coq_example*}
-Set Natural English.
-\end{coq_example*}
-
-Currently, only \verb=French= and \verb=English= are available.
-
-You may see for example the natural transcription of the proof of
-the induction principle on pairs of natural numbers:
-
-\begin{coq_example*}
-Print Natural nat_double_ind.
-\end{coq_example*}
-
-You may  also show in natural language the current proof in progress:
-
-\comindex{Show Natural}
-\begin{coq_example}
-Goal (n:nat)(le O n).
-Induction n.
-Show Natural Proof.
-\end{coq_example}
-
-\subsection*{Restrictions}
-
-For \Natural, a proof is an object of type a proposition (i.e. an
-object of type something of type {\tt Prop}). Only proofs are written
-in natural language when typing {\tt Print Natural \ident}.  All other
-objects (the objects of type something which is of type {\tt Set} or
-{\tt Type}) are written as usual $\lambda$-terms.
-
-\asection{Customizing \Natural}
-
-The transcription of proofs in natural language is mainly a paraphrase of
-the formal proofs, but some specific hints in the transcription
-can be given.
-Three kinds of customization are available.
-
-\asubsection{Implicit proof steps}
-
-\subsubsection*{Implicit lemmas}
-
-Applying a given lemma or theorem \verb=lem1= of statement, say $A
-\Rightarrow B$, to an hypothesis, say $H$ (assuming $A$) produces the
-following kind of output translation:
-
-\begin{verbatim}
-...
-Using lem1 with H we get B.
-...
-\end{verbatim}
-
-But sometimes, you may prefer not to see the explicit invocation to
-the lemma. You may prefer to see:
-
-\begin{verbatim}
-...
-With H we have A.
-...
-\end{verbatim}
-
-This is possible by declaring the lemma as implicit. You should type:
-
-\comindex{Add Natural}
-\begin{coq_example*}
-Add Natural Implicit lem1.
-\end{coq_example*}
-
-By default, the lemmas \verb=proj1=, \verb=proj2=, \verb=sym_equal=
-and \verb=sym_eqT= are declared implicit. To remove a lemma or a theorem
-previously declared as implicit, say \verb=lem1=, use the command
-
-\comindex{Remove Natural}
-\begin{coq_example*}
-Remove Natural Implicit lem1.
-\end{coq_example*}
-
-To test if the lemma or theorem \verb=lem1= is, or is not,
-declared as implicit, type
-
-\comindex{Test Natural}
-\begin{coq_example*}
-Test Natural Implicit for lem1.
-\end{coq_example*}
-
-\subsubsection*{Implicit proof constructors}
-
-Let \verb=constr1= be a proof constructor of a given inductive
-proposition (or predicate)
-\verb=Q= (of type \verb=Prop=). Assume \verb=constr1= proves 
-\verb=(x:A)(P x)->(Q x)=. Then, applying \verb=constr1= to an hypothesis,
-say \verb=H= (assuming \verb=(P a)=) produces the following kind of output:
-
-\begin{verbatim}
-...
-By the definition of Q, with H we have (Q a).
-...
-\end{verbatim}
-
-But sometimes, you may prefer not to see the explicit invocation to
-this constructor. You may prefer to see:
-
-\begin{verbatim}
-...
-With H we have (Q a).
-...
-\end{verbatim}
-
-This is possible by declaring the constructor as implicit. You should
-type, as before:
-
-\comindex{Add Natural Implicit}
-\begin{coq_example*}
-Add Natural Implicit constr1.
-\end{coq_example*}
-
-By default, the proposition (or predicate) constructors
-
-\verb=conj=, \verb=or_introl=, \verb=or_intror=, \verb=ex_intro=,
-\verb=exT_intro=, \verb=refl_equal=, \verb=refl_eqT= and \verb=exist=
-
-\noindent are declared implicit. Note that declaring implicit the
-constructor of a datatype (i.e. an inductive type of type \verb=Set=)
-has no effect.
-
-As above, you can remove or test a constant declared implicit.
-
-\subsubsection*{Implicit inductive constants}
-
-Let \verb=Ind= be an inductive type (either a proposition (or a
-predicate) -- on \verb=Prop= --, or a datatype -- on \verb=Set=).
-Suppose the proof proceeds by induction on an hypothesis \verb=h=
-proving \verb=Ind= (or more generally \verb=(Ind A1 ... An)=). The
-following kind of output is produced:
-
-\begin{verbatim}
-...
-With H, we will prove A by induction on the definition of Ind.
-Case 1. ...
-Case 2. ...
-...
-\end{verbatim}
-
-But sometimes, you may prefer not to see the explicit invocation to
-\verb=Ind=. You may prefer to see:
-
-\begin{verbatim}
-...
-We will prove A by induction on H.
-Case 1. ...
-Case 2. ...
-...
-\end{verbatim}
-
-This is possible by declaring the inductive type as implicit. You should
-type, as before:
-
-\comindex{Add Natural Implicit}
-\begin{coq_example*}
-Add Natural Implicit Ind.
-\end{coq_example*}
-
-This kind of parameterization works for any inductively defined
-proposition (or predicate) or datatype. Especially, it works whatever
-the definition is recursive or purely by cases.
-
-By default, the data type \verb=nat= and the inductive connectives
-\verb=and=, \verb=or=, \verb=sig=, \verb=False=, \verb=eq=,
-\verb=eqT=, \verb=ex= and \verb=exT= are declared implicit.
-
-As above, you can remove or test a constant declared implicit.  Use
-{\tt Remove Natural Contractible $id$} or {\tt Test Natural
-Contractible for $id$}.
-
-\asubsection{Contractible proof steps}
-
-\subsubsection*{Contractible lemmas or constructors}
-
-Some lemmas, theorems or proof constructors of inductive predicates are
-often applied in a row and you obtain an output of this kind:
-
-\begin{verbatim}
-...
-Using T with H1 and H2 we get P.
-      * By H3 we have Q.
-      Using T with theses results we get R.
-...
-\end{verbatim}
-
-where \verb=T=, \verb=H1=, \verb=H2= and \verb=H3= prove statements
-of the form \verb=(X,Y:Prop)X->Y->(L X Y)=, \verb=A=, \verb=B= and \verb=C=
-respectively (and thus \verb=R= is \verb=(L (L A B) C)=).
-
-You may obtain a condensed output of the form
-
-\begin{verbatim}
-...
-Using T with H1, H2, and H3 we get R.
-...
-\end{verbatim}
-
-by declaring \verb=T= as contractible:
-
-\comindex{Add Natural Contractible}
-\begin{coq_example*}
-Add Natural Contractible T.
-\end{coq_example*}
-
-By default, the lemmas \verb=proj1=, \verb=proj2= and the proof
-constructors \verb=conj=, \verb=or_introl=, \verb=or_intror= are
-declared contractible. As for implicit notions, you can remove or
-test a lemma or constructor declared contractible.
-
-\subsubsection*{Contractible induction steps}
-
-Let \verb=Ind= be an inductive type. When the proof proceeds by
-induction in a row, you may obtain an output of this kind:
-
-\begin{verbatim}
-...
-We have (Ind A (Ind B C)).
-We use definition of Ind in a study in two cases.
-Case 1: We have A.
-Case 2: We have (Ind B C).
-  We use definition of Ind in a study of two cases.
-  Case 2.1: We have B.
-  Case 2.2: We have C.
-...
-\end{verbatim}
-
-You may prefer to see
-
-\begin{verbatim}
-...
-We have (Ind A (Ind B C)).
-We use definition of Ind in a study in three cases.
-Case 1: We have A.
-Case 2: We have B.
-Case 3: We have C.
-...
-\end{verbatim}
-
-This is possible by declaring \verb=Ind= as contractible:
-
-\begin{coq_example*}
-Add Natural Contractible T.
-\end{coq_example*}
-
-By default, only \verb=or= is declared as a contractible inductive
-constant.
-As for implicit notions, you can remove or test an inductive notion declared
-contractible.
-
-\asubsection{Transparent definitions}
-
-``Normal'' definitions are all constructions except proofs and proof constructors.
-
-\subsubsection*{Transparent non inductive normal definitions}
-
-When using the definition of a non inductive constant, say \verb=D=, the
-following kind of output is produced:
-
-\begin{verbatim}
-...
-We have proved C which is equivalent to D.
-...
-\end{verbatim}
-
-But you may prefer to hide that D comes from the definition of C as
-follows:
-
-\begin{verbatim}
-...
-We have prove D.
-...
-\end{verbatim}
-
-This is possible by declaring \verb=C= as transparent:
-
-\comindex{Add Natural Transparent}
-\begin{coq_example*}
-Add Natural Transparent D.
-\end{coq_example*}
-
-By default, only \verb=not= (normally written \verb=~=) is declared as
-a non inductive transparent definition.
-As for implicit and contractible definitions, you can remove or test a
-non inductive definition declared transparent.
-Use \texttt{Remove Natural Transparent} \ident or 
-\texttt{Test Natural Transparent for} \ident.
-
-\subsubsection*{Transparent inductive definitions}
-
-Let \verb=Ind= be an inductive proposition (more generally: a
-predicate \verb=(Ind x1 ... xn)=). Suppose the definition of
-\verb=Ind= is non recursive and built with just
-one constructor proving something like \verb=A -> B -> Ind=.
-When coming back to the definition of \verb=Ind= the
-following kind of output is produced:
-
-\begin{verbatim}
-...
-Assume Ind (H).
-      We use H with definition of Ind.
-      We have A and B.
-      ...
-\end{verbatim}
-
-When \verb=H= is not used a second time in the proof, you may prefer
-to hide that \verb=A= and \verb=B= comes from the definition of
-\verb=Ind=. You may prefer to get directly:
-
-\begin{verbatim}
-...
-Assume A and B.
-...
-\end{verbatim}
-
-This is possible by declaring \verb=Ind= as transparent:
-
-\begin{coq_example*}
-Add Natural Transparent Ind.
-\end{coq_example*}
-
-By default, \verb=and=, \verb=or=, \verb=ex=, \verb=exT=, \verb=sig=
-are declared as inductive transparent constants.  As for implicit and
-contractible constants, you can remove or test an inductive
-constant declared transparent.
-
-As for implicit and contractible constants, you can remove or test an
-inductive constant declared transparent.
-
-\asubsection{Extending the maximal depth of nested text}
-
-The depth of nested text is limited. To know the current depth, do:
-
-\comindex{Set Natural Depth}
-\begin{coq_example}
-Set Natural Depth.
-\end{coq_example}
-
-To change the maximal depth of nested text (for instance to 125) do:
-
-\begin{coq_example}
-Set Natural Depth 125.
-\end{coq_example}
-
-\asubsection{Restoring the default parameterization}
-
-The command \verb=Set Natural Default= sets back the parameterization tables of
-\Natural~ to their default values, as listed in the above sections.
-Moreover, the language is set back to English and the max depth of
-nested text is set back to its initial value.
-
-\asubsection{Printing the current parameterization}
-
-The commands {\tt Print Natural Implicit}, {\tt Print Natural
-Contractible} and {\tt Print \\ Natural Transparent} print the list of
-constructions declared {\tt Implicit}, {\tt Contractible},
-{\tt Transparent} respectively.
-
-\asubsection{Interferences with \texttt{Reset}}
-
-The customization of \texttt{Natural} is dependent of the \texttt{Reset}
-command. If you reset the environment back to a point preceding an
-\verb=Add Natural ...= command, the effect of the command will be
-erased. Similarly, a reset back to a point before a 
-\verb=Remove Natural ... = command invalidates the removal.
-
-\asection{Error messages}
-
-An error occurs when trying to \verb=Print=, to \verb=Add=, to
-\verb=Test=, or to \verb=remove= an undefined ident. Similarly, an
-error occurs when trying to set a language unknown from \Natural.
-Errors may also occur when trying to parameterize the printing of
-proofs: some parameterization are effectively forbidden.
-Note that to \verb=Remove= an ident absent from a table or to
-\verb=Add= to a table an already present ident does not lead to an
-error.
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End: