NNR Revisited \\ Sh656
Saharon Shelah \thanks {\null\newline I would like to thank Alice Leonhardt for the beautiful typing. \null\newline First Typed - 97/June/17 \null\newline Written - \S1 in 97/April/21-22} \null\newline Latest Revision - 2000/Mar/14 \endthanks
656.tex
% processed by citealice (August 1999) on Tue Mar 14 11:57:42 CST 2000

% To: Saharon Shelah <shelah@math.huji.ac.il>,        Andrzej Roslanowski <roslanow@e-math.ams.org>
% Subject: Sh656
% Date: Tue, 14 Mar 2000 09:33:30 -0500 (EST)
% From: Alice Leonhardt <leonhard@math.rutgers.edu>
% Mime-Version: 1.0
% Content-Description: revisions
% X-sliced-and-diced-by: 'savemail' 0.3, Feb 1999

\ifx\shlhetal\undefinedcontrolsequence\let\shlhetal\relax\fi


\input amstex
% % \input mathdefs
\expandafter\ifx\csname mathdefs.tex\endcsname\relax
  \expandafter\gdef\csname mathdefs.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to
  \string\input{mathdefs.tex} twice.   This does not make sense.} 
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi


%mathdefs.tex v1.3.2

%%% Changes from v1.0: footnote macros, warning for duplicated tags,
%%%   control sequences \( and \verbatimtags.
%%% From v1.2: \pretags, redefinition of \( using \ifinner, multi-part
%%%   equation numbering, control sequences \[, \references, and
%%%   \resetbracket. 
%%% From v1.3: \rm in \lastpart; write root of multi-part tag to .tgs 

%See file texdefs.doc for documentation.

\catcode`\X=12\catcode`\@=11

%Minor control sequences:
\def\n@wcount{\alloc@0\count\countdef\insc@unt}
\def\n@wwrite{\alloc@7\write\chardef\sixt@@n}
\def\n@wread{\alloc@6\read\chardef\sixt@@n}
\def\r@s@t{\relax}\def\v@idline{\par}\def\@mputate#1/{#1}
\def\l@c@l#1X{\firstpart.#1}\def\gl@b@l#1X{#1}\def\t@d@l#1X{{}}

%Creation of tag families and output of assignments and citations:
\def\crossrefs#1{\ifx\all#1\let\tr@ce=\all\else\def\tr@ce{#1,}\fi
   \n@wwrite\cit@tionsout\openout\cit@tionsout=\jobname.cit 
   \write\cit@tionsout{\tr@ce}\expandafter\setfl@gs\tr@ce,}
\def\setfl@gs#1,{\def\@{#1}\ifx\@\empty\let\next=\relax
   \else\let\next=\setfl@gs\expandafter\xdef
   \csname#1tr@cetrue\endcsname{}\fi\next}
\def\m@ketag#1#2{\expandafter\n@wcount\csname#2tagno\endcsname
     \csname#2tagno\endcsname=0\let\tail=\all\xdef\all{\tail#2,}
   \ifx#1\l@c@l\let\tail=\r@s@t\xdef\r@s@t{\csname#2tagno\endcsname=0\tail}\fi
   \expandafter\gdef\csname#2cite\endcsname##1{\expandafter
     \ifx\csname#2tag##1\endcsname\relax?\else\csname#2tag##1\endcsname\fi
     \expandafter\ifx\csname#2tr@cetrue\endcsname\relax\else
     \write\cit@tionsout{#2tag ##1 cited on page \folio.}\fi}
   \expandafter\gdef\csname#2page\endcsname##1{\expandafter
     \ifx\csname#2page##1\endcsname\relax?\else\csname#2page##1\endcsname\fi
     \expandafter\ifx\csname#2tr@cetrue\endcsname\relax\else
     \write\cit@tionsout{#2tag ##1 cited on page \folio.}\fi}
   \expandafter\gdef\csname#2tag\endcsname##1{\expandafter
      \ifx\csname#2check##1\endcsname\relax
      \expandafter\xdef\csname#2check##1\endcsname{}%
      \else\immediate\write16{Warning: #2tag ##1 used more than once.}\fi
      \multit@g{#1}{#2}##1/X%
      \write\t@gsout{#2tag ##1 assigned number \csname#2tag##1\endcsname\space
      on page \number\count0.}%
   \csname#2tag##1\endcsname}}
\def\multit@g#1#2#3/#4X{\def\t@mp{#4}\ifx\t@mp\empty%
      \global\advance\csname#2tagno\endcsname by 1 
      \expandafter\xdef\csname#2tag#3\endcsname
      {#1\number\csname#2tagno\endcsnameX}%
   \else\expandafter\ifx\csname#2last#3\endcsname\relax
      \expandafter\n@wcount\csname#2last#3\endcsname
      \global\advance\csname#2tagno\endcsname by 1 
      \expandafter\xdef\csname#2tag#3\endcsname
      {#1\number\csname#2tagno\endcsnameX}
      \write\t@gsout{#2tag #3 assigned number \csname#2tag#3\endcsname\space
      on page \number\count0.}\fi
   \global\advance\csname#2last#3\endcsname by 1
   \def\t@mp{\expandafter\xdef\csname#2tag#3/}%
   \expandafter\t@mp\@mputate#4\endcsname
   {\csname#2tag#3\endcsname\lastpart{\csname#2last#3\endcsname}}\fi}
\def\t@gs#1{\def\all{}\m@ketag#1e\m@ketag#1s\m@ketag\t@d@l p
   \m@ketag\gl@b@l r \n@wread\t@gsin
   \openin\t@gsin=\jobname.tgs \re@der \closein\t@gsin
   \n@wwrite\t@gsout\openout\t@gsout=\jobname.tgs }
\outer\def\localtags{\t@gs\l@c@l}
\outer\def\globaltags{\t@gs\gl@b@l}
\outer\def\newlocaltag#1{\m@ketag\l@c@l{#1}}
\outer\def\newglobaltag#1{\m@ketag\gl@b@l{#1}}

%Reading in tag information:
\newif\ifpr@ 
\def\m@kecs #1tag #2 assigned number #3 on page #4.%
   {\expandafter\gdef\csname#1tag#2\endcsname{#3}
   \expandafter\gdef\csname#1page#2\endcsname{#4}
   \ifpr@\expandafter\xdef\csname#1check#2\endcsname{}\fi}
\def\re@der{\ifeof\t@gsin\let\next=\relax\else
   \read\t@gsin to\t@gline\ifx\t@gline\v@idline\else
   \expandafter\m@kecs \t@gline\fi\let \next=\re@der\fi\next}
\def\pretags#1{\pr@true\pret@gs#1,,}
\def\pret@gs#1,{\def\@{#1}\ifx\@\empty\let\n@xtfile=\relax
   \else\let\n@xtfile=\pret@gs \openin\t@gsin=#1.tgs \message{#1} \re@der 
   \closein\t@gsin\fi \n@xtfile}

%Sections and subsections; local numbering:
\newcount\sectno\sectno=0\newcount\subsectno\subsectno=0
\newif\ifultr@local \def\ultralocal{\ultr@localtrue}
\def\firstpart{\number\sectno}
\def\lastpart#1{\ifcase#1 \or a\or b\or c\or d\or e\or f\or g\or h\or 
   i\or k\or l\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or 
   x\or y\or z \fi}
\def\closeup{\vskip-\bigskipamount}
\def\resetall{\global\advance\sectno by 1\subsectno=0
   \gdef\firstpart{\number\sectno}\r@s@t}
\def\resetsub{\global\advance\subsectno by 1
   \gdef\firstpart{\number\sectno.\number\subsectno}\r@s@t}
\def\newsection#1\par{\resetall\vskip0pt plus.3\vsize\penalty-250
   \vskip0pt plus-.3\vsize\bigskip\bigskip
   \message{#1}\leftline{\bf#1}\nobreak\bigskip}
\def\subsection#1\par{\ifultr@local\resetsub\fi
   \vskip0pt plus.2\vsize\penalty-250\vskip0pt plus-.2\vsize
   \bigskip\smallskip\message{#1}\leftline{\bf#1}\nobreak\medskip}

%Verbatim tags:
\def\t@gsoff#1,{\def\@{#1}\ifx\@\empty\let\next=\relax\else\let\next=\t@gsoff
   \def\@@{p}\ifx\@\@@\else
   \expandafter\gdef\csname#1cite\endcsname##1{\zeigen{##1}}
   \expandafter\gdef\csname#1page\endcsname##1{?}
   \expandafter\gdef\csname#1tag\endcsname##1{\zeigen{##1}}\fi\fi\next}
\def\verbatimtags{\ifx\all\relax\else\expandafter\t@gsoff\all,\fi}
\def\zeigen#1{\hbox{$\langle$}#1\hbox{$\rangle$}}


%Equation numbering:
\def\(#1){\edef\dot@g{\ifmmode\ifinner(\hbox{\noexpand\etag{#1}})
   \else\noexpand\eqno(\hbox{\noexpand\etag{#1}})\fi
   \else(\noexpand\ecite{#1})\fi}\dot@g}

%Reference numbering:
\newif\ifbr@ck
\def\eat#1{}
\def\[#1]{\br@cktrue[\br@cket#1'X]}
\def\br@cket#1'#2X{\def\temp{#2}\ifx\temp\empty\let\next\eat
   \else\let\next\br@cket\fi
   \ifbr@ck\br@ckfalse\br@ck@t#1,X\else\br@cktrue#1\fi\next#2X}
\def\br@ck@t#1,#2X{\def\temp{#2}\ifx\temp\empty\let\neext\eat
   \else\let\neext\br@ck@t\def\temp{,}\fi
   \def\teemp{#1}\ifx\teemp\empty\else\rcite{#1}\fi\temp\neext#2X}
\def\resetbr@cket{\gdef\[##1]{[\rtag{##1}]}}
\def\references{\resetbr@cket\newsection References\par}

%Footnotes:
\newtoks\symb@ls\newtoks\s@mb@ls\newtoks\p@gelist\n@wcount\ftn@mber
    \ftn@mber=1\newif\ifftn@mbers\ftn@mbersfalse\newif\ifbyp@ge\byp@gefalse
\def\defm@rk{\ifftn@mbers\n@mberm@rk\else\symb@lm@rk\fi}
\def\n@mberm@rk{\xdef\m@rk{{\the\ftn@mber}}%
    \global\advance\ftn@mber by 1 }
\def\rot@te#1{\let\temp=#1\global#1=\expandafter\r@t@te\the\temp,X}
\def\r@t@te#1,#2X{{#2#1}\xdef\m@rk{{#1}}}
\def\b@@st#1{{$^{#1}$}}\def\str@p#1{#1}
\def\symb@lm@rk{\ifbyp@ge\rot@te\p@gelist\ifnum\expandafter\str@p\m@rk=1 
    \s@mb@ls=\symb@ls\fi\write\f@nsout{\number\count0}\fi \rot@te\s@mb@ls}
\def\byp@ge{\byp@getrue\n@wwrite\f@nsin\openin\f@nsin=\jobname.fns 
    \n@wcount\currentp@ge\currentp@ge=0\p@gelist={0}
    \re@dfns\closein\f@nsin\rot@te\p@gelist
    \n@wread\f@nsout\openout\f@nsout=\jobname.fns }
\def\m@kelist#1X#2{{#1,#2}}
\def\re@dfns{\ifeof\f@nsin\let\next=\relax\else\read\f@nsin to \f@nline
    \ifx\f@nline\v@idline\else\let\t@mplist=\p@gelist
    \ifnum\currentp@ge=\f@nline
    \global\p@gelist=\expandafter\m@kelist\the\t@mplistX0
    \else\currentp@ge=\f@nline
    \global\p@gelist=\expandafter\m@kelist\the\t@mplistX1\fi\fi
    \let\next=\re@dfns\fi\next}
\def\symbols#1{\symb@ls={#1}\s@mb@ls=\symb@ls} 
\def\bigsymbol{\textstyle}
\symbols{\bigsymbol\ast,\dagger,\ddagger,\sharp,\flat,\natural,\star}
\def\ftnumbers{\ftn@mberstrue} \def\ftsymbols{\ftn@mbersfalse}
\def\paginal{\byp@ge} \def\resetftnumbers{\ftn@mber=1}
\def\ftnote#1{\defm@rk\expandafter\expandafter\expandafter\footnote
    \expandafter\b@@st\m@rk{#1}}

%Miscellaneous macros:
\long\def\jump#1\endjump{}
\def\ssum{\mathop{\lower .1em\hbox{$\textstyle\Sigma$}}\nolimits}
\def\down#1{_{{}_{\scriptstyle #1}}}
\def\qed{\nobreak\kern 1em \vrule height .5em width .5em depth 0em}
\def\newneq{\hbox{\rlap{\hbox to 1\wd9{\hss$=$\hss}}\raise .1em 
   \hbox to 1\wd9{\hss$\scriptscriptstyle/$\hss}}}
\def\subsetne{\setbox9 = \hbox{$\subset$}\mathrel{\hbox{\rlap
   {\lower .4em \newneq}\raise .13em \hbox{$\subset$}}}}
\def\supsetne{\setbox9 = \hbox{$\subset$}\mathrel{\hbox{\rlap
   {\lower .4em \newneq}\raise .13em \hbox{$\supset$}}}}

%Blackboard bold:
\def\vbar{\mathchoice{\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height4.1ptdepth-.35ptwidth.6pt\kern-.6pt}
   {\vrule height3.1ptdepth-.25ptwidth.5pt\kern-.5pt}}
\def\f@dge{\mathchoice{}{}{\mkern.5mu}{\mkern.8mu}}
\def\b@c#1#2{{\rm \mkern#2mu\vbar\mkern-#2mu#1}}
\def\b@b#1{{\rm I\mkern-3.5mu #1}}
\def\b@a#1#2{{\rm #1\mkern-#2mu\f@dge #1}}
\def\bb#1{{\count4=`#1 \advance\count4by-64 \ifcase\count4\or\b@a A{11.5}\or
   \b@b B\or\b@c C{5}\or\b@b D\or\b@b E\or\b@b F \or\b@c G{5}\or\b@b H\or
   \b@b I\or\b@c J{3}\or\b@b K\or\b@b L \or\b@b M\or\b@b N\or\b@c O{5} \or
   \b@b P\or\b@c Q{5}\or\b@b R\or\b@a S{8}\or\b@a T{10.5}\or\b@c U{5}\or
   \b@a V{12}\or\b@a W{16.5}\or\b@a X{11}\or\b@a Y{11.7}\or\b@a Z{7.5}\fi}}

\catcode`\X=11 \catcode`\@=12

% % \input citeadd
%   citeadd -- a few additions for 
% files from alice that were procesed with "citealice"

\expandafter\ifx\csname citeadd.tex\endcsname\relax
\expandafter\gdef\csname citeadd.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to
\string\input{citeadd.tex} twice.   This does not make sense.} 
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi





\def\sciteu{\sciteerror{undefined}}
\def\sciteuphantom{\complainaboutcitation{undefined}}

\def\scitet{\sciteerror{ambiguous}}
\def\scitetphantom{\complainaboutcitation{ambiguous}}




\def\sciteerror#1#2{{\mathortextbf{\scite{#2}}}\complainaboutcitation{#1}{#2}}
\def\mathortextbf#1{\hbox{\bf #1}}
\def\complainaboutcitation#1#2{%
\vadjust{\line{\llap{---$\!\!>$ }\qquad scite$\{$#2$\}$ #1\hfil}}}





\sectno=-1   % start with sect 0
\localtags
%\verbatimtags
\NoBlackBoxes
\define\mr{\medskip\roster}
\define\sn{\smallskip\noindent}
\define\mn{\medskip\noindent}
\define\bn{\bigskip\noindent}
\define\ub{\underbar}
\define\wilog{\text{without loss of generality}}
\define\ermn{\endroster\medskip\noindent}
\define\dbca{\dsize\bigcap}
\define\dbcu{\dsize\bigcup}
\define \nl{\newline}
\magnification=\magstep 1
\documentstyle {amsppt}
\pageheight{8.5truein}
\topmatter
\title{NNR Revisited \\
 Sh656} \endtitle
\author {Saharon Shelah \thanks {\null\newline I would like to thank 
Alice Leonhardt for the beautiful typing. \null\newline
 First Typed - 97/June/17 \null\newline
 Written - \S1 in 97/April/21-22} \null\newline
 Latest Revision - 2000/Mar/14 \endthanks} \endauthor 
% Previous Version - 99/Nov/11  
\affil{Institute of Mathematics\\
 The Hebrew University\\
 Jerusalem, Israel
 \medskip
 Rutgers University\\
 Mathematics Department\\
 New Brunswick, NJ  USA} \endaffil
\bn
\abstract We are interested in proving that if we use CS iterations of
forcing notions not adding reals that satisfies 
additional conditions then the limit
forcing does not add reals.  As a result we prove that we can amalgamate two
earlier methods and prove the consistency with ZFC + G.C.H. of two
statements gotten separately earlier: SH and non-club guessing.  We also
prove the consistency of further cases of ``strong failure of club 
guessing". \endabstract
\endtopmatter
\document  
% % \input alice2jlem
%% # Keywords  Input file to be used for texing Alice's files

\expandafter\ifx\csname alice2jlem.tex\endcsname\relax
  \expandafter\gdef\csname alice2jlem.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to
\string\input{alice2jlem.tex}  twice.   This does not make sense.}
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi



% % \input bib4plain
\expandafter\ifx\csname bib4plain.tex\endcsname\relax
  \expandafter\gdef\csname bib4plain.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to \string\input
  bib4plain.tex twice.   This does not make sense.}
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi


%  This file should be inputted if you want to use 
%  bibtex fom within plain TeX. 
\def\makeatletter{\catcode`\@11 }      % Not really need for standard
\def\makeatother{\catcode`\@12 }       % bibtex files, but these commands
\def\renewcommand{\newcommand}	       % are used in our literal-unsrt.bst
				       % and other files. 
%  The above lines were written by Martin.Goldstern@tuwien.ac.at
%  The rest is just the file btxmac.tex. 
%  (except for one small change below, regarding undefined citations)

%% @texfile{
%%   author = "Karl Berry and Oren Patashnik",
%%   version = "0.99j",
%%   date = "14 Mar 1992",
%%   filename = "btxmac.tex",
%%   address = "Please use electronic mail",
%%   checksum = "834    4503   33061",
%%   email = "opbibtex@cs.stanford.edu",
%%   codetable = "ISO/ASCII",
%%   supported = "yes",
%%   docstring = "Defines macros that make BibTeX work with plain TeX",
%% }
% BibTeX-for-TeX macros, version 0.99j, for BibTeX 0.99c, TeX 3.0 or later.
% Copyright (C) 1990--92 by Karl Berry and Oren Patashnik; all rights reserved.
% You may copy this file provided: that it's accompanied by the
% "BibTeXing" document, whose text is contained in the file `btxdoc.tex';
% that any documentation you write for these macros also gives a
% reference for "BibTeXing"; and that either you make absolutely no
% changes to your copy, or if you do make changes, (1) you name the file
% something other than `btxmac.tex' and you remove all occurrences of
% `btxmac.tex' from the file, (2) you put, somewhere in the first twenty
% lines of the file, your name, along with an electronic address at which
% others who might use the file may reach you, and (3) you remove each
% occurrence of Oren's name and electronic address from this file.  These
% restrictions help ensure that all standard versions of these macros are
% identical, and that Oren doesn't get deluged with inappropriate e-mail.
%
% This file, btxmac.tex, contains TeX macros that allow BibTeX, a
% bibliography program that was originally designed for use with LaTeX,
% to work with plain TeX.  Please report any bugs (outright goofs,
% improvable macros, misfeatures, or unclear documentation) to Oren
% Patashnik (opbibtex@cs.stanford.edu).  These macros will become frozen
% shortly after BibTeX version 1.00 is released.
%
% AMS-TEX WARNING: We tried very hard, for version .99i of these macros,
% to make them compatible with AmS-TeX.  We succeeded to the extent
% that, if you use one of the standard bibliography styles, you probably
% won't notice any problems with version 0.99i of btxmac.tex.  But
% ultimately we failed, in that the inherent incompatibilities between
% plain TeX and AmS-TeX kept making these macros break, for certain
% inputs or certain styles.  Examples:  (1) AmS-TeX treats at-signs as
% special, in ways that plain TeX and LaTeX don't, so that, for example,
% you can't have any `@' characters in an argument to the \cite command,
% the way you can in TeX or LaTeX; (2) AmS-TeX decided that plain TeX's
% and LaTeX's macron-accent control sequence `\=' should be undefined;
% so you'll need to define `\=' to be `\B' to get the xampl.bib example
% suggested below to work with AmS-TeX; (3) AmS-TeX redefines the tie
% character `~' of plain TeX, and AmS-TeX's `amsppt' style redefines
% plain TeX's `\nobreak' macro, so that if you use an author-date style
% like `apalike' and you have a multiple-author reference for which the
% author-date style automatically produces a citation in the text like
% `(Jones et~al., 1992)' you will throw AmS-TeX's `amsppt' style into
% an infinite loop, exceeding its input stack size.  In practice, such
% incompatibilities surface infrequently; but it is now clear to us that
% it's not worth the effort (perhaps it's not even possible) to make the
% btxmac.tex macros robust when used both with plain TeX and Ams-TeX.
% If the BibTeX/AmS-TeX results attainable with the current btxmac.tex
% macros are sufficient, fine.  But if there's a demand for more robust
% BibTeX/AmS-TeX behavior, then someone who's very familiar with the
% AmS-TeX package should probably make an amsbtxmc.tex version of the
% macros (remembering to follow the copyright restrictions above).
% Until then, if you're an AmS-TeX user, or a LaTeX or plain TeX user
% sharing files with an AmS-TeX user, beware.
% END OF AMS-TEX WARNING.
%
% To use these macros you should be familiar with how BibTeX interacts
% with LaTeX, since BibTeX's interaction with TeX is very similar; that
% interaction is explained in the LaTeX manual.  It also helps to
% have read "BibTeXing", the documentation that accompanies BibTeX.
% Then, if you want, you should redefine any of the macros that begin
% with `\bbl' or `\biblabel' or `\print' that you need to get formatting
% different from the default (the default settings are designed to
% accompany a bibliography style like BibTeX's standard style `plain').
% The macros you might want to change are described briefly a few
% paragraphs hence.  [To get started without reading any documentation,
% try running the nine-line .tex file below through TeX and BibTeX.
% Remember the general scheme: Running (La)TeX writes information on
% the .aux (auxiliary) file; then running BibTeX reads information from
% the .aux, .bst (style), and .bib (database) files, and writes
% information (the bibliography) on a .bbl file; then running (La)TeX
% incorporates the bibliography; then running (La)TeX once more fixes
% the remaining forward references into the bibliography.  Thus, to get
% everything incorporated into your output, you'll have to run (La)TeX,
% BibTeX, (La)TeX, (La)TeX.  (Standup, sitdown, fight, fight, fight.)]
%
% These macros can stand alone or they can be \input into a macro
% package, like Eplain, that is sufficiently compatible with plain TeX.
% To use these macros to format the 0.99 version of the xampl.bib file
% that's distributed with BibTeX (that version of the file has no
% self-identification), you'll need to define \mbox, which is a LaTeX
% command, to be \hbox, as in the example below.
%
% Here's a nine-line plain TeX file for trying out btxmac.tex; of course
% you'll have to remove the comment characters at the beginning of each
% line, and, depending on your system, you might have to take steps so
% that BibTeX can "see" the files xampl.bib and plain.bst (BibTeX will
% give you two empty-field warning messages that you should ignore).
%
%     \def\mbox#1{\leavevmode\hbox{#1}}
%     \input btxmac
%     \noindent This cites Aamport's gnominious article~\cite{article-full}.
%     \medskip
%     \leftline{\bf References}
%     \nocite{*}   % put all database entries into the reference list
%     \bibliography{xampl}   % specify the database files; here, just xampl.bib
%     \bibliographystyle{plain}   % specify plain.bst as the style file
%     \bye
%
%
%   HISTORY
%
% Karl Berry wrote the original version of these macros in 1989 and
% 1990, for use in his `Eplain' package.  Oren Patashnik modified them
% slightly in July 1990, as part of the official BibTeX distribution.
%
%    1-Aug-90  Version 0.99a, not released to the general public.
%   14-Aug-90  0.99b, first general release.
%   26-Aug-90  0.99c, made \@undefinedmessage work with other macro packages.
%    6-Sep-90  0.99d, allowed for general formatting of bibliography labels,
%                     for general formatting of (in-text) citations, and for
%                     changing certain catcodes while reading the .aux file.
%   14-Nov-90  0.99e, changed the way \@setletters works, made some \new...'s
%                     non-outer, and changed the way Eplain reads this file.
%   12-Dec-90  0.99f, made \@resetnumerals change the `,' and `.' catcodes; and
%                     added \biblabelextrahang, \@getoptionalarg, and \bblsc.
%   11-Mar-91  0.99g, made a few minor changes required by the way Eplain reads
%                     this file, but no functional changes.
%   24-Apr-91  0.99h, inhibited the reading and writing of the .aux file if it
%                     isn't used or if the \noauxfile macro is defined, and
%                     removed some .aux-file-opening detritus; printed the
%                     cite-key of undefined citations in \tt font; changed the
%                     catcode of `_' inside \cite; and called \@resetnumerals
%                     from inside a group.
%   29-Feb-92  0.99i, made these macros semi-compatible with AmS-TeX; removed
%                     \@resetnumerals, \@setletters, \@tokstostring, and
%                     friends; changed the way \cite handles catcodes; changed
%                     \@getoptionalarg, and had \bibitem and \newcommand use
%                     it; added \@futurenonspacelet and (to facilitate the use
%                     of multiple reference lists) \bblfilebasename; changed
%                     \biblabelprint to use the new macros \biblabelprecontents
%                     and \biblabelpostcontents, and to, by default, right-
%                     justify numeric labels; and renamed \biblabelextrahang to
%                     the more descriptive \biblabelextraspace.
%   14-Mar-92  0.99j, made 0.99i's use of `\\' local to btxmac.tex.
%
%
% The LaTeX-related commands defined in this file include (a) the four
% commands that a user types (\bibliography, \bibliographystyle, \cite,
% and \nocite); (b) the three commands that BibTeX looks for in the .aux
% file (\bibdata, \bibstyle, and \citation---there is a fourth command
% that BibTeX looks for, but that command is related to LaTeX's \include
% facility, so these macros ignore that command); and (c) a LaTeX
% command (\newcommand) that's written by one of the four standard
% bibliography styles (alpha).  The definitions here are not exactly the
% same as the corresponding LaTeX definitions (those eight LaTeX
% definitions depend on a significant fraction of LaTeX itself).  But
% the only substantial differences are with \newcommand, which here,
% without complaining, lets you redefine a preexisting control sequence
% (in LaTeX, \newcommand won't let you redefine a preexisting command),
% and which here doesn't make the control sequences it defines \long (in
% LaTeX, that happens automatically); there may also be other minor
% differences.  To summarize: Unless you know what you're doing, you
% shouldn't define any control sequences with these eight names:
%
% \bibdata
% \bibliography
% \bibliographystyle
% \bibstyle
% \citation
% \cite
% \newcommand
% \nocite
%
% There are three other commands written by one or more of the four
% standard (plain, abbrv, alpha, unsrt) or four semistandard (acm,
% apalike, ieeetr, siam) bibliography styles; they take effect only
% within the bibliography, and are redefinable, as explained later:
%
% \em
% \newblock
% \sc
%
% There's one control sequence you might want to use (but not redefine)
% in redefining \biblabelprint:
%
% \biblabelwidth
%
% There are fifteen other control sequences (explained later in more detail)
% that the macros of this file will use if you define them---you should
% define them after the \input btxmac command but before the \bibliography
% command.  The first six begin with `\bbl' and affect fonts, spacing,
% perhaps other characteristics of the bibliography, and which .bbl files
% get read; the next five begin with `\biblabel' and determine how labels
% are formatted in the bibliography; and the last four begin with `\print'
% and determine how the in-text citations are formatted:
%
% \bblem
% \bblfilebasename
% \bblhook
% \bblnewblock
% \bblrm
% \bblsc
% \biblabelcontents
% \biblabelprecontents
% \biblabelprint
% \biblabelpostcontents
% \biblabelextraspace
% \printbetweencitations
% \printcitefinish
% \printcitenote
% \printcitestart
%
% If it's defined before the \input btxmac command, the control sequence
% below inhibits the reading and writing of the .aux file(s), and the
% issuing of related warning messages.  Any definition will do.  This
% feature might help when you're working on draft stages of a document:
%
% \noauxfile
%
%
% Here's another control sequence (it's described later) that you
% probably won't want to redefine unless you are writing another macro
% package; if you do redefine it, however, do it before the \input btxmac
% command (and notice that it has an `@' in its name):
%
% \@undefinedmessage
%
% Any other control sequence in this file that might conflict with
% something you've defined will have an `@' in its name, so such conflicts
% are unlikely; but if you're worried about a specific control sequence
% name, do a text search of this file to look for it.
%
%
% So to start things off we turn `@' into a letter (category code 11),
% keeping track of the old category code for future restoration.
% (Simply resetting it to 12 when we leave these macros is
% insufficient.)  The use of `\cite' as a temporary control sequence is
% a kludge, but it's a reasonably simple way to accomplish what we need
% without possibly overwriting something (without an `@' in its name)
% that might already be defined.
%
\edef\cite{\the\catcode`@}%
\catcode`@ = 11
\let\@oldatcatcode = \cite
\chardef\@letter = 11
\chardef\@other = 12
%
%
% Next come some things that will be useful later.
%
% Make an outer definition into an inner one (due to Chris Thompson).
% The arguments should be the control sequence to be defined, and the
% new of the \outer control sequence, as characters; the control
% sequence #1 is defined to be just the same as \csname#2\endcsname, but
% not \outer.  For example, \@innerdef\innernewcount{newcount} would
% define \innernewcount to be a non-outer version of \newcount.
%
\def\@innerdef#1#2{\edef#1{\expandafter\noexpand\csname #2\endcsname}}%
%
% We use \@innerdef to make some of our allocations local, because
% Eplain includes our code inside a conditional.  We put @'s in the
% names to minimize the (already small) chance of conflicts.
%
\@innerdef\@innernewcount{newcount}%
\@innerdef\@innernewdimen{newdimen}%
\@innerdef\@innernewif{newif}%
\@innerdef\@innernewwrite{newwrite}%
%
%
% Swallow one parameter.
%
\def\@gobble#1{}%
%
%
% Use TeX 3.0's \inputlineno to get the line number, for better error
% messages, but if we're using an old version of TeX, don't do anything.
%
\ifx\inputlineno\@undefined
   \let\@linenumber = \empty % Pre-3.0.
\else
   \def\@linenumber{\the\inputlineno:\space}%
\fi
%
%
% The following macro \@futurenonspacelet (from the TeXbook) behaves
% essentially like \futurelet except that it discards any implicit or
% explicit space tokens that intervene before a nonspace is scanned:
%
\def\@futurenonspacelet#1{\def\cs{#1}%
   \afterassignment\@stepone\let\@nexttoken=
}%
\begingroup % The grouping here avoids stepping on an outside use of `\\'.
\def\\{\global\let\@stoken= }%
\\ % now \@stoken is a space token (\\ is a control symbol, so that
   % space after it is seen).
\endgroup
\def\@stepone{\expandafter\futurelet\cs\@steptwo}%
\def\@steptwo{\expandafter\ifx\cs\@stoken\let\@@next=\@stepthree
   \else\let\@@next=\@nexttoken\fi \@@next}%
\def\@stepthree{\afterassignment\@stepone\let\@@next= }%
%
%
% \@getoptionalarg\CS gets an optional argument from the input, enclosed
% in brackets, then expands \CS.  We set \@optionalarg to \empty if we
% don't find one, otherwise to the text of the argument.  This assumes
% the brackets don't have a funny category code.
%
\def\@getoptionalarg#1{%
   \let\@optionaltemp = #1%
   \let\@optionalnext = \relax
   \@futurenonspacelet\@optionalnext\@bracketcheck
}%
%
% The \expandafter's in this macro let us avoid the use of \aftergroup,
% which is somewhat more expensive.
%
\def\@bracketcheck{%
   \ifx [\@optionalnext
      \expandafter\@@getoptionalarg
   \else
      \let\@optionalarg = \empty
      % We can't do the \temp after the \fi, because then the \temp gets
      % in the way of reading the optional argument from the input, if
      % we do have one.
      \expandafter\@optionaltemp
   \fi
}%
%
\def\@@getoptionalarg[#1]{%
   \def\@optionalarg{#1}%
   \@optionaltemp
}%
%
%
% From LaTeX.
%
\def\@nnil{\@nil}%
\def\@fornoop#1\@@#2#3{}%
%
\def\@for#1:=#2\do#3{%
   \edef\@fortmp{#2}%
   \ifx\@fortmp\empty \else
      \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}%
   \fi
}%
%
\def\@forloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@iforloop #3\@@#4{#5}\fi\fi
}%
%
\def\@iforloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@fornoop \else
      #4\relax\let\@nextwhile=\@iforloop\fi\@nextwhile#2\@@#3{#4}%
}%
%
%
% This macro tests if a file \jobname.#1 exists, and sets \if@fileexists
% appropriately.  If an optional argument is given, it is used as the
% root part of the filename instead of \jobname.
%
\@innernewif\if@fileexists
%
\def\@testfileexistence{\@getoptionalarg\@finishtestfileexistence}%
\def\@finishtestfileexistence#1{%
   \begingroup
      \def\extension{#1}%
      \immediate\openin0 =
         \ifx\@optionalarg\empty\jobname\else\@optionalarg\fi
         \ifx\extension\empty \else .#1\fi
         \space
      \ifeof 0
         \global\@fileexistsfalse
      \else
         \global\@fileexiststrue
      \fi
      \immediate\closein0
   \endgroup
}%
%
%
%% [[[start of BibTeX-specific stuff]]]
%
% Now come the four main LaTeX commands and their associated .aux
% commands.  Just as in LaTeX, \bibliographystyle defines the BibTeX
% style name (.bst file, that is), and \bibliography defines the
% database (.bib) file(s).  The corresponding .aux-file commands are
% \bibstyle and \bibdata, which are there only for BibTeX's (but not
% LaTeX's) use.
%
\def\bibliographystyle#1{%
   \@readauxfile
   \@writeaux{\string\bibstyle{#1}}%
}%
\let\bibstyle = \@gobble
%
% As well as writing the \bibdata command to tell BibTeX which .bib
% files to read, we read the .bbl file that BibTeX (or a person,
% conceivably) has produced.  We use \bblfilebasename as the root of the
% filename to read; this defaults to \jobname.
%
\let\bblfilebasename = \jobname
\def\bibliography#1{%
   \@readauxfile
   \@writeaux{\string\bibdata{#1}}%
   \@testfileexistence[\bblfilebasename]{bbl}%
   \if@fileexists
      % We just output a non-discardable item (the `whatsit' with the
      % \bibdata command).  This means that the glue that will be
      % inserted next (\parskip or \baselineskip, most likely) will be a
      % legal breakpoint.  Most likely, this is after some kind of
      % heading, where we don't want to allow a page break.  So:
      \nobreak
      \@readbblfile
   \fi
}%
\let\bibdata = \@gobble
%
% The \nocite{label,label,...} command writes its argument to \@auxfile,
% unless instructed not to, but produces no text in the document.  Both
% the \nocite and \cite commands produce \citation commands in the .aux file.
%
\def\nocite#1{%
   \@readauxfile
   \@writeaux{\string\citation{#1}}%
}%
%
\@innernewif\if@notfirstcitation
%
% \cite[note]{label,label,...} produces the citations for the labels as
% well.  If the optional argument `note' is present, it's added after
% the labels.  Since \cite calls \nocite to do its .aux-file writing,
% \cite doesn't need to call \@readauxfile (\nocite does).
%
\def\cite{\@getoptionalarg\@cite}%
%
% Typeset the citations for the labels in #1, followed by the note, if
% it exists.  To change the citation's format in the text, redefine one
% or more `\print...' macros, whose defaults appear later in this file.
%
\def\@cite#1{%
   % Remember the optional argument, in case one of the macros we call
   % below ends up looking for an optional argument itself.  For
   % example, if a \cite[note] triggers reading the .aux file, then the
   % [note] would be clobbered, since \@testfileexistence looks for an
   % optional arg.
   \let\@citenotetext = \@optionalarg
   % Start printing the text, beginning with a left bracket by default.
   \printcitestart
   % It's complicated, but because \nocite puts a `whatsit' onto the list,
   % \nocite should follow \printcitestart.  It's conceivable, but very
   % unlikely, that this `whatsit' will cause a problem (glue that doesn't
   % disappear when you want it to is the most likely symptom), requiring
   % a change either to \printcitestart or to the label that the .bst file
   % produces.
   \nocite{#1}%
   \@notfirstcitationfalse
   \@for \@citation :=#1\do
   {%
      \expandafter\@onecitation\@citation\@@
   }%
   \ifx\empty\@citenotetext\else
      \printcitenote{\@citenotetext}%
   \fi
   \printcitefinish
}%
%
\def\@onecitation#1\@@{%
   \if@notfirstcitation
      \printbetweencitations
   \fi
   %
   \expandafter \ifx \csname\@citelabel{#1}\endcsname \relax
      \if@citewarning
         \message{\@linenumber Undefined citation `#1'.}%
      \fi
      % Give it a dummy definition:
      \expandafter\gdef\csname\@citelabel{#1}\endcsname{%
% Change: marginal remark added, goldstrn@math.huji.ac.il, 
% goldstern@tuwien.ac.at, May 1996 mg
%  !!! change !!!
\strut
\vadjust{\vskip-\dp\strutbox
\vbox to 0pt{\vss\parindent0cm \leftskip=\hsize 
\advance\leftskip3mm
\advance\hsize 4cm\strut\openup-4pt 
\rightskip 0cm plus 1cm minus 0.5cm ?  #1 ?\strut}}
         {\tt
            \escapechar = -1
            \nobreak\hskip0pt
            \expandafter\string\csname#1\endcsname
            \nobreak\hskip0pt
         }%
      }%
   \fi
   % Now produce the text, whether it was undefined or not.
   \csname\@citelabel{#1}\endcsname
   \@notfirstcitationtrue
}%
%
% Given a label `foo', the macro `\b@foo' is supposed to
% hold the text that should be produced.
%
\def\@citelabel#1{b@#1}%
%
% So, how does a citation label get defined?  When we read the .bbl file
% (below), a \bibitem writes out a \@citedef command.  And when we read
% the \@citedef, we define \@citelabel{#1}, where #1 is the user's
% label.
%
\def\@citedef#1#2{\expandafter\gdef\csname\@citelabel{#1}\endcsname{#2}}%
%
%
% Reading the .bbl file also produces the typeset bibliography.  Please
% notice, however, that we do not produce the title for the references
% (e.g., `References'), as LaTeX does.  The formatting and spacing of
% that title, whether it should go into the headline, and so on, are all
% things determined by your format.  We cannot know those things in
% advance.  If you wish, you can define \bblhook to produce the title.
% Or just do it before the \bibliography command.
%
\def\@readbblfile{%
   % Define a counter to tell us which item number we are on, unless
   % we've already defined it (because the document has more than one
   % bibliography).
   \ifx\@itemnum\@undefined
      \@innernewcount\@itemnum
   \fi
   %
   \begingroup
      \def\begin##1##2{%
         % ##1 is just `thebibliography'.
         % ##2 is the widest label.
         % We set (new dimen) \biblabelwidth based on the widest label
         \setbox0 = \hbox{\biblabelcontents{##2}}%
         \biblabelwidth = \wd0
      }%
      \def\end##1{}% ##1 is `thebibliography' again.
      %
      % Here we have two possibilities:
      % \bibitem[typesetlabel]{citationlabel}
      % \bibitem{citationlabel}
      % If we have the second of these, the citations are numbered, starting
      % from one; we use our own count register \@itemnum for this.
      %
      \@itemnum = 0
      \def\bibitem{\@getoptionalarg\@bibitem}%
      \def\@bibitem{%
         \ifx\@optionalarg\empty
            \expandafter\@numberedbibitem
         \else
            \expandafter\@alphabibitem
         \fi
      }%
      \def\@alphabibitem##1{%
         % Need \xdef here for various reasons.
         \expandafter \xdef\csname\@citelabel{##1}\endcsname {\@optionalarg}%
         % Left-justify alpha labels, unless \biblabel{pre,post}contents
         % are already defined.
         \ifx\biblabelprecontents\@undefined
            \let\biblabelprecontents = \relax
         \fi
         \ifx\biblabelpostcontents\@undefined
            \let\biblabelpostcontents = \hss
         \fi
         \@finishbibitem{##1}%
      }%
      %
      \def\@numberedbibitem##1{%
         \advance\@itemnum by 1
         \expandafter \xdef\csname\@citelabel{##1}\endcsname{\number\@itemnum}%
         % Right-justify numeric labels, unless \biblabel{pre,post}contents
         % are already defined.
         \ifx\biblabelprecontents\@undefined
            \let\biblabelprecontents = \hss
         \fi
         \ifx\biblabelpostcontents\@undefined
            \let\biblabelpostcontents = \relax
         \fi
         \@finishbibitem{##1}%
      }%
      %
      \def\@finishbibitem##1{%
         \biblabelprint{\csname\@citelabel{##1}\endcsname}%
         \@writeaux{\string\@citedef{##1}{\csname\@citelabel{##1}\endcsname}}%
         \ignorespaces
      }%
      %
      % Do the printing (we're producing the bibliography, remember).
      %
      \let\em = \bblem
      \let\newblock = \bblnewblock
      \let\sc = \bblsc
      % Punctuation won't affect spacing;
      \frenchspacing
      % the penalties below are from LaTeX's [article,book,report].sty;
      \clubpenalty = 4000 \widowpenalty = 4000
      % the next two values come from LaTeX's \sloppy command;
      \tolerance = 10000 \hfuzz = .5pt
      \everypar = {\hangindent = \biblabelwidth
                      \advance\hangindent by \biblabelextraspace}%
      \bblrm
      % the \parskip is a guess at what looks good;
      \parskip = 1.5ex plus .5ex minus .5ex
      % and the space between label and text comes from LaTeX's \labelsep.
      \biblabelextraspace = .5em
      \bblhook
      %
      \input \bblfilebasename.bbl
   \endgroup
}%
%
% The widest label's width is useful for redefining \biblabelprint;
% you redefine \biblabelwidth, in effect, by redefining the
% \biblabelcontents macro that appears below.  And \biblabelextraspace,
% which is redefinable inside \bblhook, is added to \biblabelwidth to
% determine the amount of hanging indentation.
%
\@innernewdimen\biblabelwidth
\@innernewdimen\biblabelextraspace
%
% Now come the main macros that are related to the printing of the
% bibliography.  Since you might want to redefine them, they are given
% default definitions outside of \@readbblfile.
%
% The first one controls the printing of a bibliography entry's label.
% If you change it, make sure that it starts with something like
% \noindent or \indent or \leavevmode that puts TeX into horizontal mode
% (even if the label itself is empty); otherwise, the hanging
% indentation will get messed up in certain circumstances.
%
\def\biblabelprint#1{%
   \noindent
   \hbox to \biblabelwidth{%
      \biblabelprecontents
      \biblabelcontents{#1}%
      \biblabelpostcontents
   }%
   \kern\biblabelextraspace
}%
%
% If you are using numeric labels, and you want them left-justified
% (numeric labels by default are right-justified), do something like:
%     \def\biblabelprecontents{\relax}
%     \def\biblabelpostcontents{\hss}
%
% By default the labels are typeset in \bblrm, and enclosed in brackets.
%
\def\biblabelcontents#1{{\bblrm [#1]}}%
%
% The main text, too, is typeset using \bblrm, which is \rm by default.
%
\def\bblrm{\rm}%
%
% Emphasis for producing, e.g., titles, is done with \it by default.
%
\def\bblem{\it}%
%
% Some styles use a caps-and-small-caps font for author names.  LaTeX
% defines an \sc command but plain TeX doesn't, so we need one here.
% The definition below doesn't load the font unless it's needed, but it
% tries to load only the 10pt version, because it might not exist at
% other point sizes.
%
\def\bblsc{\ifx\@scfont\@undefined
              \font\@scfont = cmcsc10
           \fi
           \@scfont
}%
%
% The major parts of an entry are separated with \bblnewblock.  The
% numbers below are taken from LaTeX's `article' style.
%
\def\bblnewblock{\hskip .11em plus .33em minus .07em }%
%
% Here's where you stick any other bibliography-formatting goodies, or
% redefine the values above.
%
\let\bblhook = \empty
%
%
% Here are the four default definitions for formatting the in-text
% citations.  These are what you redefine (after your \input btxmac but
% before your \bibliography) to get parens instead of brackets, or
% superscripts, or footnotes, or whatever.
%
\def\printcitestart{[}%         left bracket
\def\printcitefinish{]}%        right bracket
\def\printbetweencitations{, }% comma, space
\def\printcitenote#1{, #1}%     comma, space, note (if it exists)
%
% That scheme is pretty flexible.  For example you could use
%     \def\printcitestart{\unskip $^\bgroup}
%     \def\printcitefinish{\egroup$}
%     \def\printbetweencitations{,}
%     \def\printcitenote#1{\hbox{\sevenrm\space (#1)}}
%     \font\eighttt = cmtt8
%     \scriptfont\ttfam = \eighttt
% to get superscripted in-text citations.  (The scriptfont stuff
% exists only to print an undefined citation; it's in cmtt8 because
% there is no cmtt7.)  To get something radically different, however,
% you'll have to define your own \cite command.
%
% When we read `\citation' from the .aux file, it means nothing.
%
\let\citation = \@gobble
%
%
% Now comes the stuff for dealing with LaTeX's \newcommand.  As
% mentioned earlier, this \newcommand will redefine a preexisting
% command; that's different from how LaTeX's \newcommand behaves.
%
\@innernewcount\@numparams
%
% \newcommand{\foo}[n]{text} defines the control sequence \foo to have
% n parameters, and replacement text `text'.
%
\def\newcommand#1{%
   \def\@commandname{#1}%
   \@getoptionalarg\@continuenewcommand
}%
%
% Figure out if this definition has parameters.
%
\def\@continuenewcommand{%
   % If no optional argument, we have zero parameters.  Otherwise, we
   % have that many.
   \@numparams = \ifx\@optionalarg\empty 0\else\@optionalarg \fi \relax
   \@newcommand
}%
%
% \@numparams is how many arguments this command has.  The name of the
% command is \@commandname.  The replacement text for the new macro is #1.
%
\def\@newcommand#1{%
   \def\@startdef{\expandafter\edef\@commandname}%
   \ifnum\@numparams=0
      \let\@paramdef = \empty
   \else
      \ifnum\@numparams>9
         \errmessage{\the\@numparams\space is too many parameters}%
      \else
         \ifnum\@numparams<0
            \errmessage{\the\@numparams\space is too few parameters}%
         \else
            \edef\@paramdef{%
               % This is disgusting, but \loop doesn't work inside \edef,
               % because \body isn't defined.
               \ifcase\@numparams
                  \empty  No arguments.
               \or ####1%
               \or ####1####2%
               \or ####1####2####3%
               \or ####1####2####3####4%
               \or ####1####2####3####4####5%
               \or ####1####2####3####4####5####6%
               \or ####1####2####3####4####5####6####7%
               \or ####1####2####3####4####5####6####7####8%
               \or ####1####2####3####4####5####6####7####8####9%
               \fi
            }%
         \fi
      \fi
   \fi
   \expandafter\@startdef\@paramdef{#1}%
}%
%
%% [[[end of BibTeX-specific stuff]]]
%
%
% Names of references (arguments given in the \cite and \nocite
% commands) and file names (arguments given in the \bibliography and
% \bibliographystyle commands) are recorded in \jobname.aux, called the
% \@auxfile in these macros.  Here's how they get read in.
%
\def\@readauxfile{%
   \if@auxfiledone \else % remember: \@auxfiledonetrue if \noauxfile is defined
      \global\@auxfiledonetrue
      \@testfileexistence{aux}%
      \if@fileexists
         \begingroup
            % Because we might be in horizontal mode when \@readauxfile
            % is called (if it's in response to a \cite or \nocite), we
            % want to ignore all the would-be spaces at the ends of
            % lines in the aux file.  Fortunately, it's highly unlikely
            % an end-of-line might actually be desired.
            % And because we don't change the category code of anything
            % but @, primitives like \gdef can't be used to define labels
            % in the aux file.  The solution adopted by btxmac.tex is to
            % write `\@citedef{LABEL}{DEFINITION}' to the aux file, and
            % use \csname on LABEL.
            \endlinechar = -1
            \catcode`@ = 11
            \input \jobname.aux
         \endgroup
      \else
         \message{\@undefinedmessage}%
         \global\@citewarningfalse
      \fi
      \immediate\openout\@auxfile = \jobname.aux
   \fi
}%
%
% The \@readauxfile macro does all that work the first time it's called.
% Since it's called once for every \cite, \nocite, \bibliography, and
% \bibliographystyle command that the user issues, we need to remember
% whether the work's been done.  It's considered done if we're not to do
% it---that is, if \noauxfile is defined.
%
\newif\if@auxfiledone
\ifx\noauxfile\@undefined \else \@auxfiledonetrue\fi
%
% It's conceivable you'd want to change how other characters are read;
% to do that, change their category code before doing \input btxmac.
%
%
% After reading the .aux file, \@readauxfile opens it for writing.
% The \@writeaux macro does the actual writing (as long as
% \noauxfile is undefined).
%
\@innernewwrite\@auxfile
\def\@writeaux#1{\ifx\noauxfile\@undefined \write\@auxfile{#1}\fi}%
%
%
% A macro package that uses btxmac.tex might define
% \@undefinedmessage (before doing an \input btxmac).
%
\ifx\@undefinedmessage\@undefined
   \def\@undefinedmessage{No .aux file; I won't give you warnings about
                          undefined citations.}%
\fi
%
% Even if citations are undefined, we want to complain only if
% \@citewarningtrue.  The default is to set \@citewarningtrue unless
% \noauxfile is defined.  Again, a macro package that uses
% btxmac.tex might want to redefine this.
%
\@innernewif\if@citewarning
\ifx\noauxfile\@undefined \@citewarningtrue\fi
%
%
% Finally, before leaving we restore @'s old category code.
%
\catcode`@ = \@oldatcatcode

  % This will define \cite and make sure it works as in latex


\def\widestnumber#1#2{}
  % Our amstex-ppt style does not know about \widestnumber



\def\rm{\fam0 \tenrm}


\def\fakesubhead#1\endsubhead{\bigskip\noindent{\bf#1}\par}


% % \input rsfs

% # Keywords: Script or Calligraphic (Caligraphic) letters with the RSFS Font

% The story so far:    July 1998 -- Saharon would like to have a
% ``nicer'' calligraphic font. In particualr, the leters S and P in
% the usual calligraphic font do not look ``special'' enough. 
% 
% I found out that ``rsfs'' (``Ralph Smith Formal Script'') may be
% what he wants.   I installed the mf file, the .tfm file, as well as
% a few pk files in ~/TeX/rsfs.    Let's hope that this is enough.
% Using amstex, all you have to do is to \input rsfs.tex 
% Files prepared with citealice willdothis automatically. 
%
%  Note:  for some reason xdvi calls MakeTeXpk, then Maketexpk
%  complains about wrong resolution, but still writes commands to
%  missfont.log...  
%



% we redefine a macro inside amstex's \Cal command , so that it calls
% our nice font ``rsfs'' rather than the usual calligraphic font. 
% Note thisworks for amstex only.   
% In plain tex, would have to add definitions of \Cal
% in latex... we should insteaduse mathrsfs.sty
% 



\font\textrsfs=rsfs10
\font\scriptrsfs=rsfs7
\font\scriptscriptrsfs=rsfs5


\newfam\rsfsfam
\textfont\rsfsfam=\textrsfs
\scriptfont\rsfsfam=\scriptrsfs
\scriptscriptfont\rsfsfam=\scriptscriptrsfs


\edef\oldcatcodeofat{\the\catcode`\@}
\catcode`\@11

\def\Cal@@#1{\noaccents@ \fam \rsfsfam #1}

\catcode`\@\oldcatcodeofat



\newpage

\head {Anotated Content} \endhead  \resetall 
 % \resetall 
\bn
\S0  $\quad$ Introduction
\mr
\item "{{}}"  [We give lengthy explanations of the problems and proofs for
No-New-Reals iterations (or proper forcing, CS iterations).]
\ermn
\S1 $\quad$ Preservation of not adding reals
\mr
\item "{{}}"  [We present sufficient conditions for CS iteration of proper
forcing not to add reals.  For this we define ``reasonable parameters
${\frak p}$" and we have two main demands.  One (clause (c) of Definition
\scite{1.6}) is a weakening of ``$\alpha$-proper for every $\alpha <
\omega_1$".  This time it has the form (on $Q_i$), ${\frak p}$-proper which
informally says that: 
if ${\frak p} \in N,Y \subseteq \{M \in N:M \text{ appropriate}\}$
is $\alpha$-large then for some $(N,Q_i)$-generic condition $q \ge p,q$ 
forces that
$\{M \in Y:M[{\underset\tilde {}\to G_Q}] \cap V = M\}$ is $\alpha$-large
(the meaning of $\alpha -$ large depends on ${\frak p}$), hence \wilog, \, 
${\frak p}$ has
length $\omega_1$.  The other main demand (clause (d) of Definition
\scite{1.6}) is a ``weak diamond preventive". \nl
We then show that $\alpha$-properness for $\alpha < \omega_1$ is sufficient
for the first main demand (in \scite{1.12}(3)).  The demand on the games for
${\frak p}$ helps to prove the preservation of ${\frak p}$-properness.]
\ermn
\S2 $\quad$ Delayed properness
\mr
\item "{{}}"  [The preservation theorem in the first section does not, for
standard ${\frak p}$, cover shooting a club $C \subseteq \omega_1$ running
away for $C_\delta \subseteq \delta = \sup(C_\delta),C_\delta$ small (see
\S3).  For this we will use $({\frak p},\alpha,\beta)$-proper for enough
pairs $\alpha \le \beta < \ell g({\frak p})$ (so starting from $\beta$-large
we get $\alpha$-large; for many $\alpha$ we can choose $\beta = \alpha$ 
but during the inductive proof we pass through cases of $\alpha < \beta$). \nl
Here we introduce various definitions and basic facts needed.  We discuss
axioms, version of the properties preserved by CS iterations and 
strengthening of the iteration Lemmas of \S1.]
\ermn
\S3 $\quad$ Example: shooting a thin club
\mr
\item "{{}}"  [We present the natural forcing showing $\kappa = 2$ is
interesting (not only $\kappa = \aleph_0$) (from \cite[Ch.VIII,\S4]{Sh:b}).
\nl
We show that the natural forcing (see above) for running away from $C_\delta
\subseteq \delta$, of small order type (see \cite[Ch.XVIII,\S2]{Sh:f}) falls
under our framework for delayed properness.  
We give examples: running away from
$\langle C_{\delta,0},C_{\delta,1}:\delta < \omega_1 \text{ limit} \rangle,
C_{\delta,0},C_{\delta,1}$ are disjoint closed subsets of $\delta$ 
with no restrictions on their order type so we
ask for $C,C \cap C_{\delta,0}$ or $C \cap C_{\delta,1}$ is bounded in
$\delta$ and more.]
\ermn
\S4 $\quad$ Second preservation of not adding reals
\mr
\item "{{}}"  [We give a sufficient condition for the limit not to add
reals.  We here are weakening the demand ``${\frak p}$-proper", using 
$({\frak p},\alpha,f(\alpha))$-proper instead 
$({\frak p},\alpha,\alpha)$-proper, what we called
delayed properness.  The price is that here ${\frak p}$ has
length of large cofinality, so essentially we catch our tails on a club of 
it.  Also the Lemma here covers the examples.]
\ermn
\S5 $\quad$ Problematic Forcing
\mr
\item "{{}}"  [We discuss further generalizations.]
\endroster
\newpage

\head {\S0 Introduction} \endhead  \resetall \sectno=0
 % \resetall 
\bigskip

We try to explain our problems and results.  If the explanations look opaque,
try to return to them after reading at least part of the proof.  Sections
\S0, \S1 are based on lectures in the logic seminar in the Hebrew University,
Spring 1997, whose participants I thank.  On the history see in
\cite[V,\S7,VIII,\S4,XVIII,\S1,\S2]{Sh:f}, \cite[\S3]{Sh:666}.
\definition{\stag{0.1} Definition}  1)  Let $K_0$ be the family of CS 
iterations, $\bar Q = \langle P_i,{\underset\tilde {}\to Q_i}:i < \alpha 
\rangle$, we denote $P_\alpha = \text{ Lim}(\bar Q)$. \nl
2) We say $\bar Q$ is proper if each $Q_i$ is (which means: every
$f \in ({}^\omega \omega)^{V^{P_{i+1}}} = ({}^\omega \omega)
^{(V^{P_i})^{O_i}}$ is bounded by some $g \in ({}^\omega \omega)^{V^{P_i}}$, 
hence $P_j/P_i$ is proper for 
$i < j \le \alpha$ (see \cite{Sh:b} or \cite{Sh:f}). \nl
3) We say $\bar Q$ is ${}^\omega \omega$-bounding if each $Q_i$ is
\footnote {where $Q$ is ${}^\omega \omega$ bounded (in the universe $V$) if
every $f \in ({}^\omega)^{V^Q}$ is bounded by some $g \in ({}^\omega
\omega)^V$}, 
hence $P_j/P_i$ is ${}^\omega \omega$-bounding for $i < j \le \alpha$. \nl
4) We say $\bar Q$ is NNR if $i < \alpha \Rightarrow P_{i+1}$ 
adds no reals. \nl
[Equivalently: $i < \alpha \Rightarrow Q_i$ adds no reals and $\beta <
\alpha \Rightarrow P_\beta$ adds not reals.]
\enddefinition
\bn
\ub{Discussion}:  It would be nice if also this (NNR) would be preserved in
limit.  But this is wrong for two known reasons, obstacles, explained below:
\mr
\item "{$\bigotimes_1$}"  weak diamond
\sn
\item "{$\bigotimes_2$}"  existence of clubs.
\ermn
Our aim here is to weaken the medicine one uses against $\bigotimes_2$ for CS
iterations of proper forcing notions.  Let us explain the ``obstacles". 
\bn
\ub{Concerning the weak diamond}:

Let $\bar \eta = \langle \eta_\delta:\delta < \omega_1,\delta \text{ limit}
\rangle,\eta_\delta = \langle \eta_\delta(n):n < \omega \rangle$ where
$\eta_\delta$ is an $\omega$-sequence of ordinals (strictly) increasing
with limit $\delta$.  Let $D$ be a non-principal ultrafilter on $\omega$.
\nl
For $f \in {}^{\omega_1}2,\delta < \omega_1$ limit, let Av$_D(f,\eta_\delta)
= \ell$ iff $\{n:f(\eta_\delta(n)) = \ell\} \in D$. \nl
\bn
\ub{\stag{0.2} Question[CH]}:  Given $\bar e = \langle e_\delta:
\delta < \omega_1 \text{ limit} \rangle,e_\delta \in \{0,1\}$ is there 
$f \in {}^{\omega_1}2$ such that for a club
of $\delta < \omega_1$ we have $e_\delta = \text{ Av}_D(f,\eta_D)$?
\mn
Naturally, trying to prove consistency we should use a CS iteration $\bar Q$,
for simplicity we assume
\mr
\item "{{}}"  $V \models GCH,\bar Q = \langle P_i,{\underset\tilde {}\to Q_i}:
i < \omega_2 \rangle$
\sn
\item "{{}}" $\quad \qquad \qquad Q_i = Q_{\bar e}$ (i.e.
${\underset\tilde {}\to Q_i} = Q_{\underset\tilde {}\to{\bar e}})$
where
$$
\align
Q_{\bar e} = \biggl\{f:&\text{ for some } \zeta < \omega_1,f \in {}^\zeta 2
\text{ and for every limit} \\
  &\,\delta \le \zeta \text{ we have Av}_D(f,\eta_\delta) = e_\delta \biggr\}.
\endalign
$$
\ermn
This is a very nice forcing notion - it is proper (even $< \omega_1$-proper,
see below) and NNR and for every
$\alpha < \omega_1,{\Cal I}_\alpha = \{f \in Q_{\bar e}:\alpha \subseteq
\text{ Dom}(f)\}$ is a dense open subset of it. \nl
But the weak diamond (\cite{DvSh:65} or see \cite[Ch.XII,\S1]{Sh:b} =
\cite[AP,\S1]{Sh:f}) tells us for this case that the answer is \ub{no},
that is: \nl
$\exists \bar e \forall f \in {}^{\omega_1}2 \exists^{\text{stat}} \delta
(e_\delta \ne \text{ Av}_D(f,\eta_\delta))$.

In fact this holds for any function Av$':\dbcu_{\delta < \omega_1}
{}^\delta 2 \rightarrow \{0,1\}$.
\bn
\ub{Why is $Q_{\bar e}$ NNR}?

Let $\langle N_i:i \le \omega^2 \rangle$ be increasing continuously with 
$i,Q_{\bar e} \in N_0$ where $N_i \prec ({\Cal H}(\chi),\in)$ is countable,
$\bar N \restriction (i+1) \in N_{i+1}$; 
let $\delta_i = N_i \cap \omega_1$.  So
$\langle \delta_i:i \le \omega^2 \rangle$ is strictly increasing continuously.
But $\{\eta_{\delta_{\omega^2}}(n):n < \omega\} \subset \delta_{\omega^2}$ has
order type $\omega$, so $W = \{i < \omega^2:\exists n(\delta_i \le
\eta_{\delta_{\omega^2}}(n) < \delta_{i+1})\}$ has order type $\omega$.
\sn
So we can find $\ell_n < \omega$ such that $\dsize \bigwedge_{n < \omega} \,
\dsize \bigwedge_{m < \omega} \omega \times n + \ell_n +m \notin W$.  
Let $p \in Q_{\bar e} \cap N_0$, let 
$\underset\tilde {}\to \tau:\omega \rightarrow \text{ Ord},
\underset\tilde {}\to \tau \in N_0$.  We choose by induction on $n,p_n \in
Q_{\bar e}$ such that $p \le p_n,
p_{n-1} \le p_n,p_n \in N_{\omega n + \ell_n +1},p_n$ force some value to
$\underset\tilde {}\to \tau(n)$ and on
$[\delta_{\omega m + \ell_m},\delta_{\omega(m+1)+ \ell_{m+1}}) \
\cap \text{ Rang}(\eta_\delta) \backslash
\text{ Dom}(p)$ which agrees with $e_{\delta_{\omega^2}}$. \nl
So
\mr
\item "{$(*)$}"  the desired demand on $\bar Q_i$ (guaranteeing $P_\alpha$
in NNR) should exclude the $Q_{\bar e}$'s. 
\endroster
\bigskip

\definition{\stag{0.3} Definition}  Let $K_1$ be the class of proper 
${}^\omega \omega$-bounding iteration $\bar Q \in K_0$, let $K_2$ be 
the class of NNR iterations $\bar Q \in K_1$, and let $K_3$ be the 
class of $\bar Q \in K_2$ such that
\mr
\item "{$(*)$}"  if $\chi$ is large enough (so $\bar a \in {\Cal H}(\chi)$),
\nl
$N \prec ({\Cal H}(\chi),\in)$ countable, \nl
$\bar Q \in N$, \nl
$i \in \ell g(\bar Q) \cap N$, \nl
$p \in P_{i+1} \cap N$, \nl
$q_0,q_1 \in P_i$ are $(N,P_i)$-generic \nl
(i.e. $q_\ell \Vdash N[{\underset\tilde {}\to G_{P_i}}] \cap V = N$) and 
moreover \nl
$q_\ell \Vdash ``{\underset\tilde {}\to G_{P_i}} \cap N = G^*"$ and \nl
$p \restriction i \le q_\ell$ \nl
\ub{then} we can find $q'_0,q'_1,G^{**}$ such that $p \restriction (i+1),
q_\ell \le q'_\ell \in P_{i+1}$, \nl
$q'_\ell \Vdash ``{\underset\tilde {}\to G_P} \cap N = G^{**}"$, \nl
$q'_\ell$ is $(N,P_{i+1})$-generic, so $G^{**} \subseteq P_{i+1} \cap N$
is generic over $N$.
\endroster
\enddefinition
\bn
\ub{This tries to say}:

We know ${\underset\tilde {}\to G_{P_i}} \cap N$ (as being $G^*$) and we
are looking at $N[G^*]$ (formally, only its isomorphism type).  So we know 
${\underset\tilde {}\to Q^N_i}[G^*_i]$.

We would like to find $G' \subseteq {\underset\tilde {}\to Q^N_i}[G^*]$ 
generic over $N[G^*]$, so that $G^*,G'$ will determine $G^{**}$.  But we 
need a guarantee that $G'$ will have an upper bound in
${\underset\tilde {}\to Q_i}[G_{P_i}]$.  If we know $G_{P_i}$, fine;
but in a sense, we are given 2 candidates (by $q_0,q_1$ and can increase
then to $q'_0 \restriction i,q'_1 \restriction i$) and have to find $G'$ 
``accepted" by both.

This (weak diamond) obstacle was overcome, with a price, i.e. more than
being in $K_3$, \cite[V,\S7]{Sh:f} using $\aleph_1$-completeness systems
and \cite[XVIII,\S4]{Sh:f} for 2-completeness systems (phrased in
\cite[XVIII,\S2]{Sh:f} without them).

Unfortunately, this does not suffice: there is an extreme example where e.g. 
if for $q_0,q_1$ incompatible in $Q_i,{\underset\tilde {}\to E_i}$ a 
$Q_i$-name of a club, for some $\alpha(q_0,q_1)$ we have:

$$
q_\ell \le q'_\ell
$$

$$
q_\ell \Vdash ``{\underset\tilde {}\to E_i} \cap \delta
= E^\delta_i \Rightarrow E^{\delta_0}_i \cap E^{\delta_1}_1 \backslash
\alpha(q_0,q_1) \text{ is finite}".
$$
\mn
This represents the reason, the obstacle which we shall call $\bigotimes_2$,
it is 
overcomed (with a price) either with $(< \omega_1)$-properness or by a kind
of ``finite powers are proper" (see below).
\definition{\stag{0.4} Definition}  1) $Q$ is $\alpha$-proper when:
\mr
\item "{{}}"  if $\bar N = \langle N_i:i \le \alpha \rangle$ is $\prec$-
increasing continuously, $\alpha \in N_0$ \nl
$N_i \prec ({\Cal H}(\chi),\in)$ countable, 
$\bar N \restriction (i+1) \in N_{i+1}$, \nl
$Q \in N_0$ and $p \in Q \cap N_0$ \nl
\ub{then} there is $q,p \le q \in Q_i$ such that $q$ is $(N_i,q)$-generic 
for $i \le \alpha$.
\ermn
2) A forcing notion $Q$ is $(<^+ \omega_1)$-proper if the above holds for any
$\alpha < \omega_1$ even omitting ``$\alpha \in N_0$".  We say $Q$ is
$(< \omega_1)$-proper if $Q$ is $\alpha$-proper for any $\alpha <
\omega_1$.  \nl
So $(< \omega_1)$-proper is an antidote to such problems, i.e. against
``reason $\otimes_2$".  
Okay for specializing a Aronszajn tree and many others, \ub{but} it seems to 
me since \cite{Sh:177} too strong: it kills the following:
\bn
\ub{\stag{0.5} Question}:  Let $\bar C = \langle C_\delta:\delta < \omega_1,
\delta \text{ limit} \rangle,C_\delta \subseteq \delta = 
\sup(C_\delta)$, otp$(C_\delta) = \omega$ or at least $< \delta$, is there a
club $E$ of $\omega_1$ such that $\delta < \omega_1 \Rightarrow \delta >
\sup(C_\delta \cap E)$? (i.e. is this consistent with CH).
\mn
We consider

$$
\align
Q^1_{\bar C} = \biggl\{ \bar f:&\text{ for some non-limit } \alpha < \omega_1
\text{ we have } f \in {}^\alpha 2,f^{-1}(\{1\}) \text{ closed} \\
  &\text{and } \delta < \alpha \text{ limit } \Rightarrow \sup(f^{-1}(\{1\}) 
\cap C_\delta) < \delta \biggr\}.
\endalign
$$
\mn
This is the natural forcing for adding a club such that $\dsize \bigwedge
_\delta[C_\delta \cap E$ bounded in $\delta$].  So $E$ ``runs away" from each
$C_\delta$.  This forcing notion is NOT $\omega$-proper: if 
$\langle N_i:i \le \omega \rangle$ satisfies 
$C_{N_\omega \cap \omega_1} = \{N_i \cap \omega_1:i < \omega\}$, 
\ub{then} no $f \in Q$ is $(N_i,G)$-generic, for infinitely many
$i$'s.

A solution (\cite[XVIII,\S2]{Sh:f}) was to 
demand ``essentially" e.g. $P_i \times P_i$ is proper for
$i < \ell g(\bar Q)$.  While this is fine for $Q^1_i$, this seems to exclude
specializing an Aronszajn tree by not adding reals.  We will deal with a
condition implied by both $(< \omega_1)$-proper and (essentially) ``the 
square of the forcing notion is proper".
\enddefinition
\bn
\ub{Continuing explanation}:

So for CS iteration $\bar Q$ of proper forcing the ``reasons", ``dangers" 
for adding reals may come from:
\mr
\item "{$\bigotimes_0$}"  (0-reason, danger) some $Q_i$ adds reals and 
\sn
\item "{$\bigotimes_1$}"  weak diamond.
\ermn
Against this, we will assume something like (Definition \scite{0.3}): 
many times in some sense
$q_0,q_1 \in P_i$ are $(N,P_i)$-generic, $p \in {\underset\tilde {}\to Q_i} 
\cap N,q_\ell \Vdash_{P_i} ``{\underset\tilde {}\to G_{P_i}} \cap N = G^*"$ 
and for some $G',q'_0 \ge q_0,q'_1 \ge q_1$ in $P_{i+1}$ we have 
$G'\subseteq ({\underset\tilde {}\to Q_i} 
\cap N)[G^*]$ and $q'_\ell \Vdash_{P_i} 
``{\underset\tilde {}\to G_{Q_i}} \cap N[G^*] = G'"$ and $p \in G'$. \nl
It is simpler in the proof to allow $q_\ell(\ell < n)$ for some $n < \omega$;
but in addition we have the obstacle:
\mr
\item "{$\bigotimes_2$}"   adding almost disjoint clubs 
(\cite[Ch.XVIII,\S1]{Sh:f}).
\ermn
There were two medicines:
\mr
\item "{$(\alpha)$}"  $\alpha$-proper for every $\alpha < \omega_1$
\sn
\item "{$(\beta)$}"  something like $P_i \times P_i$ is proper.
\ermn
In the proofs we have a situation:
\mr
\item "{$(*)$}"  $\bar Q \in N_0 \in N$ \nl
$N_0 \prec ({\Cal H}(\chi),\in)$ and $N \prec ({\Cal H}(\chi),\in)$ are
countable \nl
$q_\ell$ is $(N,P_i)$-generic and $(N_0,P_i)$-generic \nl
$q_\ell$ forces that ${\underset\tilde {}\to G_{P_i}} \cap
N = G_\ell$, (for $\ell < 2$) \nl
$G^* = G_1 \cap N_0 = G_2 \cap N_0$ \nl
$i,j,p \in N_0[G^*],i \le j \le \ell g(\bar Q)$ \nl
$p \in P_j,p \restriction i \in G^*$ (possibly more).
\ermn
We would like to find $G' \subseteq P^N_j/G^*$ generic over $N_0$ such that
$q_0$ and $q_1$ both forces that it has an upper bound in $P_j/
{\underset\tilde {}\to G_{P_i}}$.  If $j=i+1$ this means
$G' \subseteq {\underset\tilde {}\to Q_i}[G^*]$ is generic over 
$N_0$ such that $q_0,q_1$ both force that $G'$ has an upper bound in 
${\underset\tilde {}\to Q_i}[G_{P_i}]$. \nl
It is natural to demand $G' \in N$, otherwise the two possible generic
extensions (for $q_0$ and $q_1$) become not related.  For the case $j=i+1$,
a ``weak diamond medicine" should help us.  But we need it for every $j$,
naturally we prove it by induction on $j$, and the successor case can be
reduced to the case $j=i+1$. 
\mn
But to continue in a limit we need $G' \in N$ and more: for some intermediate
$N_1,N_0 \in N_1 \in N$ also $\dsize \bigwedge_\ell [q_\ell \Vdash
N_1[{\underset\tilde {}\to G_{P_i}}] \cap V = N_1]$.  
So the clubs of elementary submodels which $q_0,q_1$ induce on
$\{M \prec N:M \in N\}$ should have non-trivial intersection.  This is a
major point and it has always appeared in some form.  Here the medicine
against $\bigotimes_2$ should help, in some way there will be many possible
$N_1$'s; but its help has a price: we have to carry it during the induction.
On the other hand the models playing the role of $N_1$ may change, we may
``consume it and discard it".

Note that the discussion is on two levels. Necessary limitations of universes
with CH on the one hand, \ub{and} how we try to carry the inductive proof
on appropriate iterations on the other hand; the connection though is
quite tight.
\mn
So we shall try for $j \in \ell g(\bar Q)
\cap N_0$ to extend the situation with $i$ being replaced by $j$ while
$G^*$ is being increased to $G^{**}$.  We shall prove by induction 
suitable facts, with $G^{**}$ the
object we are really interested in. \nl
We are given $q_1,q_2 \in P_i$ and would like to find suitable
$q'_1,q'_2 \in P_j$ such that $q'_\ell \restriction i = q_\ell$ (otherwise
in limit why is there an upper bound?) 
\sn
So the real action occur for $j$ limit, so we choose $\zeta_n \in N \cap
[i,j)$ such that $\zeta_0=i,\zeta_n < \zeta_{n+1}$ (sometimes better to have
$i$ and each $\zeta_n$ non-limit) and $\dbcu_{n < \omega} \zeta_n = \sup
(j \cap N)$.
\mn
You can think of:
\mr
\item "{{}}"    in each case of limit $j$, proving the inductive statement,
we choose a ``surrogate" for $N$ called $N_1$, during the induction it
serves like $N$ in the limit dealing with $\zeta_0,\zeta_1,\ldots$ using
the induction hypothesis on $N_1$ we get $G^{**}$ which may not be in $N_1$
but is in $N$.
\endroster
\mn
So we try to choose by induction on $n,q_{0,n},q_{1,n},G^*_n$ such that:
$q_{\ell,n} \in P_{\zeta_n}$ is $(N,P_{\zeta_n})$-generic, $q_{\ell,0} =
q_\ell,q_{\ell,n+1} \restriction \zeta_n = q_{\ell,n},G^*_n \in N_1,
G^*_n \subseteq P_{\zeta_n} \cap N$ is generic over $N$ and
$q_{\ell,n} \Vdash ``{\underset\tilde {}\to G_{P_{\zeta_n}}} \cap N = G^*_n$.
The construction of the $G^*_n$ should use little information on the actual
$q_{\ell,n}$ so that the choices of the $G^*_n$ can be carried say inside
$N_1$ so that $\langle G^*_n:n < \omega \rangle \in N$.  In fact several
models will play a role like $N_1$. \nl
By the proof of the preservation of ${}^\omega \omega$-bounding we can choose
some $N_1$ and demand ``$q_{\ell,n}$ gives to each $P_{\zeta_n}$-name of an
ordinal ${\underset\tilde {}\to \tau_n} \in N_1$, only finitely many
possibilities".

Now how does $(< \omega_1)$-proper help?  \nl
We can assume in the beginning that
$\langle N_{1,\gamma}:\gamma \in A \rangle \in N$ is $\prec$-increasing
continuously, $N_0 \prec N_{1,\gamma} \prec N,\langle N_{1,\gamma}:\gamma \le
\beta \rangle \in N_{\beta +1}$ with $A =: (j+1) \cap N \backslash i$ 
and assume $q_\ell$
is $(N_{1,\gamma},P_i)$-generic for $\gamma \in A$ (similarly for $q'_0,
q'_1,j$ in the conclusion) and demand $q_{\ell,n}$ is $(N_{1,\gamma},
P_{\zeta_n})$-generic for $n < \omega$ and $\gamma \in A \backslash \zeta_n$.
We are ignoring several points
including how the induction change and having $\ell < 2$ rather than
$\ell < n (< \omega)$ which complicates life.
\sn
How does ``$Q \times Q$ is proper" help? \nl
We demand things like ``$(q_0,q_1)$ is $(N_1,P_i \times P_i)$-generic" 
so this gives
many common $N_1$'s, but to preserve this we need more complicated situations.
Instead of a ``tower" of models of countable length, we have a finite tower
of models (say of length 5) where on the bottom we are computing $G^{**} \cap
P_{\zeta_n}$ and as we go up less and less is demanded. \nl
The medicine in the present work is ${\frak p}$-properness
where ``$Q$ is ${\frak p}$-proper"
say that if $Y$ is a large family of $M \prec N$ and $p \in Q \cap N,Q \in
N$ then for some $q$ we have $p \le q$ and $q$ is $(N,Q)$-generic and
$q \Vdash ``\{M \in Y:M[{\underset\tilde {}\to G_Q}] \cap V=M\}$ is large".
(The idea of the finite tower is retained in the proof).  This is quite 
obvious in hindsight.
\mn
Why is it important to be inside $N$?  Otherwise, we could
forget about $N$ and we have $q_0,q_1$.  We know they have a common candidate
but we need to increase them to know if and in limit by the regular having
bounded in $P_i$. \nl
``We need a real not a name of a real." \nl
\sn
Note that a sufficient condition for ${\frak p}$-properness for $Q$, if ${\frak p}$ is
standard, is homogeneity.
\bn
\ub{Notation}:  ${\frak p}$ denote a reasonable parameter.
\sn
We thank Todd Eisworth for many corrections; he has continued this work.
\newpage

\head {\S1 Preservation of not adding reals} \endhead  \resetall \sectno=1
 % \resetall 
\bigskip

\definition{\stag{1.1} Definition}  We say ${\frak p} = (\bar \chi,\bar R,
\bar{\Cal E},\bar D) = (\bar \chi^{\frak p},\bar R^{\frak p},
\bar{\Cal E}^{\frak p},\bar D^{\frak p})$ is a reasonable parameter if 
for some ordinal $\alpha^*$ called $\ell g({\frak p})$ we have:
\mr
\item "{$(a)$}"  $\bar \chi = \langle \chi_\alpha:\alpha < \alpha^* \rangle,
\chi_\alpha$ a regular cardinal, ${\Cal H}((\dbcu_{\beta < \alpha} 
\chi_\beta)^+) \in {\Cal H}(\chi_\alpha)$
\sn
\item "{$(b)$}"  $\bar R = \langle R_\alpha:\alpha < \alpha^* \rangle,
R_\alpha \in {\Cal H}(\chi_\alpha)$; \nl
we could have asked ``$R_\alpha$ a relation on ${\Cal H}(\chi_\alpha)$", no
real difference for our purpose; in a sense it codes a club of
$[{\Cal H}(\chi_\alpha)]^{\le \aleph_0}$.
\sn
\item "{$(c)$}"  $\bar{\Cal E} = \langle 
{\Cal E}_\alpha:\alpha < \alpha^* \rangle$ where
${\Cal E}_\alpha \subseteq [{\Cal H}(\chi_\alpha)]^{\le \aleph_0}$ is
stationary
\sn
\item "{$(d)$}"  $\bar D = \langle D_\alpha:\alpha < \alpha^* \rangle,
D_\alpha$ is a function with domain ${\Cal E}_\alpha,a \in {\Cal E}_\alpha
\Rightarrow D_\alpha(a)$ is a pseudo-filter on $a$, i.e.
$D_\alpha(a)$ is a family of subsets of $a$ closed under supersets, non-empty
if $\alpha > 0$ \nl
(and 
let $D^-_\alpha(a) = (D_\alpha(a))^- = {\Cal P}(a) \backslash D_\alpha(a)$) 
\sn
\item "{$(e)$}"  for $\alpha < \alpha^*$ we let 
${\frak p}^{[\alpha]} =: \langle \bar \chi \restriction
\alpha,\bar R \restriction (\alpha + 1),\bar{\Cal E} \restriction \alpha,
\bar D \restriction \alpha \rangle$, so it belongs 
to ${\Cal H}(\chi_\alpha)$. \nl
[Why $\bar R \restriction (\alpha +1)$?  This makes it an easy demand on
${\Cal E}_\alpha:N \in {\Cal E}_\alpha \Rightarrow R_\alpha \in N$].
\sn
\item "{$(f)$}"  if $a \in {\Cal E}_\alpha$, \ub{then} for some countable
$N \prec ({\Cal H}(\chi_\alpha),\in)$ we have: \nl
$a$ is the universe of $N$, so we may write
$D_\alpha(N)$ instead of $D_\alpha(a)$ and $N \in {\Cal E}_\alpha$ instead of
$|N| \in {\Cal E}_\alpha$
\sn
\item "{$(g)$}"  if $\alpha < \alpha^*$ and $N \in {\Cal E}_\alpha$, then
${\frak p}^{[\alpha]} \in N$
\sn
\item "{$(h)$}"  for $N \in {\Cal E}_\alpha$ and $X \subseteq N$ we have: \nl
$X \in D_\alpha(N)$ iff $(\dbcu_{\beta < \alpha} {\Cal E}_\beta) \cap X \cap
N \in D_\alpha(a)$
\sn
\item "{$(i)$}"  if $N \in {\Cal E}_\alpha,X \in D_\alpha(N),\beta \in 
\alpha \cap N$ and $y \in N \cap {\Cal H}(\chi^{\frak p}_\beta)$,
\ub{then} for some $M \in {\Cal E}_\beta \cap X$ we have $X \cap M \in 
D_\beta(M)$ and $y \in M$.
\endroster
\enddefinition
\bigskip

\remark{\stag{1.1A} Remark}  1) Note that 
$\langle {\Cal E}_\alpha:\alpha < \ell g({\frak p}) \rangle$ are 
pairwise disjoint by clause (g) (and clause (e)) so $D(N)$ can be well defined
as $D_\alpha(N)$ for the unique $\alpha$ such that $N \in {\Cal E}_\alpha$). 
\nl
2) Clearly only $D_\alpha(N) \cap {\Cal P} (\dbcu_{\beta < \alpha}
{\Cal E}_\beta)$ matters. \nl
3) Note that the most natural case is ``$D(N)$ is $\{X \subseteq N:X \ne
\emptyset \text{ mod } D^{\text{fil}}\}$" for some filter $D^{\text{fil}}$ on
$N$. \nl
4) Natural cases are:  $D_\alpha(a)$ is a filter on $a$, and
$D_\alpha(a) = \{b \subseteq a:a \backslash b \notin D'_\alpha(a)\}$ for a
filter $D'_\alpha(a)$ on $a$; we say $D_\alpha$ is dual to $D'_\alpha$ or
$D_\alpha = (D'_\alpha)^+$. \nl
5) We may add in clause (i) of \scite{1.1} that some $X' \in X \cap M$ belongs
to $N \cap D_\beta(M)$.  No beginning harm but not necessary at present.
\endremark
\bigskip

\definition{\stag{1.2} Definition}  1) In \scite{1.1} we say $\bar D$ 
is standard if for every $\alpha < \alpha^*$ 
and $N \in {\Cal E}_\alpha$ we have

$$
\align
D_\alpha(N) = \biggl\{ X \subseteq N:&\text{ for every } \gamma \in N \cap
\alpha \text{ and} \\
  &y \in N \cap \cup\{{\Cal H}(\chi_\beta):\beta \in \alpha \cap N\} 
   \text{ for some} \\
  &\beta \in N \cap \alpha \backslash \gamma \text{ and} \\
  &M \in X \cap {\Cal E}_\beta \text{ we have } y \in M, \\
  &X \cap M \in D_\beta(M) \biggr\}.
\endalign
$$
\mn
2) We say ${\frak p}$ is simple if $\alpha \le \beta < \alpha^* \Rightarrow
\alpha \le_{\frak p} \beta$ where we let $\alpha \le_{\frak p} \beta$ be
the following partial order on $\alpha^*:\alpha \le_{\frak p} \beta$ iff
$\alpha \le \beta < \alpha^* = \ell g({\frak p})$ and we have 
$N \in {\Cal E}^\beta \and \alpha \in N \Rightarrow M =: N \cap 
{\Cal H}(\chi^{\frak p}_\alpha) \in {\Cal E}^{\frak p}_\alpha$ and
$N \in {\Cal E}^\beta \and \alpha \in N \and Y \in D_\beta(N) \Rightarrow
Y \cap \dbcu_{\gamma < \alpha} {\Cal E}^{\frak p}_\gamma \in 
D_\alpha(M)$.
\nl
3) We say ${\frak p}$ is standard if (${\frak p}$ is a reasonable parameter
such that) $\bar D^{\frak p}$ is standard. \nl
4) If $N \prec ({\Cal H}(\chi),\in)$ and $N \cap {\Cal H}(\chi^{\frak p}
_\alpha) \in {\Cal E}_\alpha$, (hence $\alpha \in N,{\frak p} \restriction
\alpha \in N,R^{\frak p}_\alpha \in N$), \ub{then} we let 
$D_\alpha(N) = D^{\frak p}_\alpha (N)$ be 
$D^{\frak p}_\alpha(N \cap {\Cal H}(\chi^{\frak p}_\alpha))$. 
\enddefinition
\bn
\ub{\stag{1.3} Convention}:  If $\bar D = \bar D^{\frak p}$ is standard, 
we may omit it.  If ${\frak p}$ clear from the content, we may write
${\Cal E}_\alpha$ instead ${\Cal E}^{\frak p}_\alpha$.
\bigskip

\definition{\stag{1.4} Definition}  1) We say that ${\frak p}$ is a winner or 
a $\Game$-winner if:
\mr
\item "{{}}"  for every $\alpha < \ell g({\frak p}),\alpha > 0$ and $N \in 
{\Cal E}^{\frak p}_\alpha$, in the game \nl
$\Game_\alpha(N) = \Game_\alpha(N,{\frak p})$ (defined below)
\sn
\item "{{}}"  the chooser player has a winning strategy where:
\ermn
2) $\Game_\alpha(N,{\frak p})$ is the following game:
\mr
\item "{{}}"  a play lasts $\omega$ moves, in the $n$-th move
\sn
\item "{{}}"  \ub{the challenger} $\quad$ choose $X_n \in D_\alpha(N)$
such that $m < n \Rightarrow X_n \subseteq X_m$
\sn
\item "{{}}" \ub{the chooser} chooses 
$\quad M_n \in X_n$ and $Y_n \subseteq M_n
\cap X_n,Y_n \in D(M_n) \cap N$
\sn
\item "{{}}"  \ub{the challenger} $\quad$ chooses $Z_n \subseteq Y_n$
such that $Z_n \in D(M_n)$.
\ermn
In the end the chooser wins if $\cup\{\{M_n\} \cup Z_n:n < \omega\} \in 
D_\alpha(N)$. \nl
3) Assume $N \in N' \prec ({\Cal H}(\chi),\in)$ and ${\frak p} \restriction
\alpha \in N'$, and, of course, $N \prec N'$ are countable.  The game
$\Game'_\alpha(N,N',{\frak p})$ is defined similarly but during the $n$-th
move, we demand that all the chosen objects belong to $N'$ and in 
the end the chooser also
chooses $X'_n \subseteq X_n,X'_n \in D(N) \cap N'$ and the challenger in the
next move has to satisfy $X_{n+1} \subseteq X'_n$.  Omitting $N'$ we mean:
for any such $N'$ the demand holds. \nl
4) ${\frak p}$ is a non-$\Game$-loser if for $\alpha < \ell g({\frak p}),
\alpha > 0,N \in {\Cal E}_\alpha$ the challenger has no winning strategy
in $\Game_\alpha(N,{\frak p})$. \nl
5) ``$\Game'_\alpha$-winner" or ``non-$\Game'_\alpha$-loser" means we 
(in part (1) or part (4)) use $\Game'_\alpha(N,{\frak p})$. 
We say that ``the chooser/challenger wins the game $\Game_\alpha(N)$" if he
has a winning strategy and so ``the chooser/challenger does not win the
game $\Game_\alpha(N)$" says the negation.  (Similarly for the other games in
this paper). \nl
6) Omitting $\alpha$ means for every $\alpha < \ell g({\frak p})$.
\enddefinition
\bigskip

\demo{\stag{1.5} Observation}  1) If ${\frak p}$ is a reasonable parameter with
the standard $\bar D$, \ub{then} ${\frak p}$ is a winner. \nl
2) If ${\frak p}$ is a $\Game_\alpha$-winner \ub{then} ${\frak p}$ is a
$\Game'_\alpha$-winner; if ${\frak p}$ is a $\Game$-winner, \ub{then}
${\frak p}$ is a $\Game'$-winner; similarly for a non-loser.
\enddemo
\bigskip

\demo{Proof}  Straight.
\enddemo
\bigskip

\definition{\stag{1.5A} Definition}  Assume ${\frak p}$ is a reasonable
parameter, $N \in {\Cal E}^{\frak p}_\alpha,y \in N$ and 
$P \in N$ is a forcing notion.  We let 
${\Cal M}_P[{\underset\tilde {}\to G_P},N,y]
=: \{M \in N:P,y \in M \text{ and } {\underset\tilde {}\to G_P} 
\cap M \text{ is a subset of } P \cap M \text{ generic over } M\}$.
If $P$ is clear from the context, we may omit it; note that ${\Cal M}_P
[G,N,y] = {\Cal M}_P[G \cap N,N,y]$ so we may write $G \cap N$ instead of $G$.
If $y = \emptyset$ we may omit it.
\enddefinition
\bigskip

\definition{\stag{1.6} Definition}  We say $\bar Q$ is 
${\frak p}-NNR^0_{\aleph_0}$ iteration if:
\mr
\item "{$(a)$}"  $\bar Q = \langle P_i,{\underset\tilde {}\to Q_i}:i < j(*)
\rangle$ is a CS iteration of proper forcing notions which belongs to 
${\Cal H}(\chi^{\frak p}_0)$, even ${\Cal P}(P_{j(*)}) \in {\Cal H}
(\chi^{\frak p}_0)$
\sn
\item "{$(b)$}"  forcing with $P_{j(*)} = \text{ Lim}(\bar Q)$ does not add
reals
\sn
\item "{$(c)$}"  [long properness]
if $i \le j \le j(*),\alpha < \ell g({\frak p}),
N \in {\Cal E}_\alpha,\{i,j,\bar Q\} \in N$, the condition $q \in P_i$ 
is $(N,P_i)$-generic forcing ${\underset\tilde {}\to G_{P_i}} \cap N = G$ 
and $p \in P_j \cap N,p \restriction i \in G$ and $Y \subseteq 
{\Cal M}_{P_i}[G,N,y]$ where $y = \langle \bar Q,i,j \rangle$ and $Y \in 
D_\alpha(N)$ \nl
(note that from $\bar Q,G$ the ordinal $i$ is reconstructible); 

\ub{then} there are $G'',q'$ such that:
{\roster
\itemitem{ $(\alpha)$ }  $q' \in P_j,p \le q'$ and $q \le q' \restriction i$
\sn
\itemitem{ $(\beta)$ }  $q'$ is $(N,P_j)$-generic
\sn
\itemitem{ $(\gamma)$ }  $q'$ forces ${\underset\tilde {}\to G_{P_j}} \cap
N = G''$
\sn
\itemitem{ $(\delta)$ }  $Y \cap {\Cal M}_{P_j}[G'',N,y] \in D_\alpha(N)$
\endroster}
(in \S2 this is called ${\frak p}$-proper) 
\sn
\item "{$(d)$}"  [anti-w.d.]  assume
$i \le j \le j(*),\alpha < \ell g({\frak p}),N_0 \in
N_1 \in {\Cal E}_\alpha,N_0 \in \dbcu_{\beta < \alpha} {\Cal E}_\beta$,
otp$(N_0 \cap [i,j)) < \alpha$ and \footnote{so naturally 
$\ell g({\frak p}) = \omega_1$} $n < \omega$, for $\ell < n$ we have
$q_\ell \in P_i$ is $(N_1,P_i)$-generic and $q_\ell$ forces 
${\underset\tilde {}\to G_{P_i}} \cap N_1 = G^\ell,\dsize \bigwedge_{\ell <n}
[G^\ell \cap N_0 = G^*]$ where 
$G^* \subseteq P_i \cap N_0$ is generic over $N_0$ and
$Y =: \dbca_{\ell <n} {\Cal M}[G^\ell,N_1] \in D_\alpha(N)$ and $p \in P_j
\cap N_0,p \restriction i \in G^*$. \ub{Then} for some $G^{**} \subseteq
P_j \cap N_0$ generic over $N_0$ we have $p \in G^{**} \in N_1$ and
$\dsize \bigwedge_{\ell < n} \,\,\dsize \bigvee_{q \in G^\ell}[q \Vdash
``G^{**}$ has an upper bound in $P_j/{\underset\tilde {}\to G_{P_i}}"]$.
\endroster
\enddefinition
\bigskip

\remark{Remark}  We may like to phrase clause (c) as a condition on each
${\underset\tilde {}\to Q_i}$, for this see Definition \scite{2.6},
\scite{2.8}, \scite{2.14}; this is a slight loss if we deal with the case
$i < j,i$ non limit $\Rightarrow P_j/P_i$ is proper.  As no need arise here
we ignore this.
\endremark
\bigskip

\proclaim{\stag{1.7} Main Claim}  Assume $\bar Q$ is a CS iteration,
$\bar Q \in {\Cal H}(\chi^{\frak p}_0)$ and ${\Cal P}(\text{Lim }
\bar Q) \in {\Cal H}(\chi^{\frak p}_0)),{\frak p}$ a reasonable parameter
of length $\omega_1,\delta = \ell g(\bar Q)$ is a limit ordinal and 
for every $\alpha < \delta,\bar Q \restriction \alpha$ is 
a ${\frak p}-NNR^0_{\aleph_0}$ iteration and ${\frak p}$ is a $\Game$-winner.

\ub{Then} $\bar Q$ is a ${\frak p}-NNR^0_{\aleph_0}$ iteration.
\endproclaim
\bigskip

\demo{Proof}
\sn
\ub{Proof of clause (a) of Definition \scite{1.6}}:

Trivial.
\mn
\ub{Proof of clause (b) of Definition \scite{1.6}}:

Follows from clause (d) of Definition \scite{1.6} proved below.
\mn
\ub{Proof of clause (d)}:

Let $i,j,\alpha,N_0,N_1,n,q_0,\dotsc,q_{n-1},G^\ell,G^*,p$ be as in the
assumptions of clause (d).  Let $\alpha' = \text{ otp}(N_0 \cap [i,j))$, and
$\alpha' < \omega_1$ so $\alpha' \in N_1$.
If $j < j(*)$ use ``$\bar Q \restriction j$ is a ${\frak p}-NNR^0_{\aleph_0}$ 
iteration", so assume $j = j(*)$; if $i=j$ the conclusion is trivial so 
assume $i<j$.  Let $i_m \in N_0 \cap j$ be such that $i_0 = i,i_m < i_{m+1}$ 
and $\langle i_m:m < \omega \rangle \in N_1$ and $\dbcu_{m < \omega} i_m 
= \sup(N_0 \cap j)$.
Choose $M_\ell$ for $\ell < 5$ such that $y^* =: \{i,j,\alpha,\alpha',
\bar Q,N_0,\langle i_m:m < \omega \rangle\} \in M_\ell \in 
{\Cal E}_{\alpha'} \cap N_1 \cap \dbca_{\ell < n} {\Cal M}[G^\ell,N_1],
M_0 \in M_1 \in M_2 \in M_3 \in M_4$ and $\dbca_{\ell < n} {\Cal M}
[G^\ell,M_0,y^*] \in D_\alpha(M_0)$.

Now for $\ell < n$ we can choose $q'_\ell \in G^\ell \cap M_4$ which forces
(for $P_{i_0} = P_i$) a value to ${\underset\tilde {}\to G_{P_{i_0}}} \cap
M_3$ which necessarily is $G^\ell \cap M_3$ and necessarily $q'_\ell \le 
q_\ell$ and, of course, $q'_\ell$ is $(M_k,P_{i_0})$-generic forcing
${\underset\tilde {}\to G_{P_{i_0}}} \cap M_k = G^\ell \cap M_k$ for $k =
0,1,2,3$ and ${\underset\tilde {}\to G_{P_{i_0}}} \cap N_0 = G^*$.

Let $\langle {\Cal I}^*_m:m < \omega \rangle \in M_0$ list 
the maximal antichains of $P_j$ that belongs to $N_0$.  Now we choose 
by induction on $m < \omega$, the objects $r_m,G^*_m,p_m,n_m,G^\ell_m$
(for $\ell < n_m$) and $y_m$ such that:
\mr
\item "{$(a)$}"  $r_m \in P_{i_m} \cap M_4$
\sn
\item "{$(b)$}"  Dom$(r_m) \subseteq [i,i_m)$
\sn
\item "{$(c)$}"  $r_{m+1} \restriction i_m = r_m$
\sn
\item "{$(d)$}"  $q'_\ell \cup r_m (\in P_{i_m})$ is $(M_k,P_{i_m})$-generic
for $k=0,1,2,3$ and is $(N_0,P_{i_m})$-generic \nl
(note that $q'_\ell,r_m$ have disjoint domains)
\sn
\item "{$(e)$}"    for every predense subset ${\Cal I}$ of $P_{i_m}$ which
belongs to $M_2$, for some finite ${\Cal J} \subseteq {\Cal I} \cap M_2$
the set ${\Cal J}$ is predense above $q'_\ell \cup r_m$ for each $\ell < n$
\sn
\item "{$(f)$}"  $n_m < \omega$ and for $\ell < n_m$ we have: \nl
$G^\ell_m$ is a subset of $P_{i_m} \cap M_0$ generic over $M_0,G^\ell_m \in 
M_1$
\sn
\item "{$(g)$}"  if $\ell < n_{m+1}$ then 
$G^\ell_{m+1} \cap P_{i_m} \in \{G^k_m:k < n_m\}$
\sn
\item "{$(h)$}"  $n_0 = n,G^\ell_0 = G^\ell \cap M_0$
\sn
\item "{$(i)$}"  $q'_\ell \cup r_m \Vdash_{P_{i_m}} 
``{\underset\tilde {}\to G_{P_{i_m}}} \cap M_0 \in \{G^\ell_m:\ell < n_m\}"$
\sn
\item "{$(j)$}"  $G^*_m$ is a subset of $P_{i_m} \cap N_0$ generic over
$N_0$
\sn
\item "{$(k)$}"  $G^*_m \subseteq G^\ell_m$ for $\ell < n_m$
\sn
\item "{$(l)$}"  $p_m \in P_j \cap N_0,p_m \restriction i_m \in G^*_m,
p_{m+1} \in {\Cal I}^*_m,p_0 = p,p_m \le p_{m +1}$
\sn
\item "{$(m)$}"  $Y_m =: \dbca_{\ell < n_m} {\Cal M}[G^\ell_m,M_0,y^*] \in
D_{\alpha'}(M_0)$ where 
$y^* =: \{N_0,\langle i_k:k < \omega \rangle,\bar Q,i,j\}$.
\ermn
\ub{Why is this sufficient?}

During the construction above we choose inductively members of $M_4$ and
all the parameters used are from $M_4$, so if we choose
a well ordering $<^*$ of $M_4$ and always choose the $<^*$-first object the
construction is determined.  Clearly there is such $<^* \in N_1$.  Now
\mr
\item "{$(\alpha)$}"  $r = \dbcu_m r_m$ (i.e. the unique 
$r \in P_j$ satisfying $m < \omega \Rightarrow r \restriction i_m
= r_m$) belongs to $P_j$ and to $N_1$,
\sn
\item "{$(\beta)$}"  $G^{**} = \{p' \in P_j \cap N_0:
\dsize \bigvee_{m < \omega}[p' \le p_m]\}$ belongs to $M_4$ and is a subset of
$P_j \cap N_0$ generic over $N_0$, \nl
(by the choice of $\langle {\Cal I}^*_m:m < \omega \rangle$ and clause (k))
and $q'_\ell \cup r$ is above $G^{**}$ \nl
(in $P_j$).
\ermn
So we are done.
\enddemo
\bn
\ub{Why can we carry out the construction?}

For $m=0$ there is no problem.  So assume we have it for $m$ and we shall
choose for $m+1$.
\bn
\ub{Stage A}:  Choose $p_{m+1} \in N_0 \cap {\Cal I}^*_m$ such that
$p_m \le p_{m+1}$ and $p_{m+1} \restriction i_m \in G^*_m$. \nl
No problem.
\bn
\ub{Stage B}:  Choose $G^*_{m+1} \subseteq P_{i_{m+1}} \cap N_0$ generic over
$N_0,G^*_m \subseteq G^*_{m+1} \in M_0,p_{m+1} \restriction i_{m+1} \in
G^*_{m+1}$ and $\dsize \bigwedge_{\ell < n_m} \dsize \bigvee_{r \in G^\ell_m}
[r \Vdash_{P_{i_m}} ``G^*_{m+1}$ has an upper bound in $P_{i_{m+1}}/
{\underset\tilde {}\to G_{P_{i_m}}}]$.
\mn
This is easy by ``$\bar Q \restriction i_{m+1}$ is a
${\frak p}-NNR^0_{\aleph_0}$ iteration" applied with 
$i_m,i_{m+1},\alpha',p_{m+1} \restriction i_m,G^*_m,\langle
G^\ell_m:\ell < n_m \rangle,N_0,M_0$ here standing for $i,j,\alpha,p,G^*,
\langle G^\ell:\ell < n \rangle,N_0,N_1$ there (i.e. we use clause (d) of the 
Definition \scite{1.6}); we are using otp$(N_0 \cap [i_m,i_{m+1})) < 
\text{otp}(N_0 \cap [i_m,j)) = \alpha'$.
\bn
\ub{Stage C}:  

Now ${\underset\tilde {}\to G_{P_{i_m}}}
\cap M_1$ is a $P_{i_m}$-name of an object from $V$ (as $P_{i_m}$ is proper
not adding reals), so ${\Cal I} =: \{p \in P_{i_m}:p 
\text{ forces a value to } {\underset\tilde {}\to G_{P_{i_m}}} \cap M_1
(\in V)\}$ is a dense open subset of $P_{i_m}$ and ${\Cal I} \in M_2$.  By
clause (e) in the induction hypothesis there is a finite 
${\Cal J} \subseteq {\Cal I}
\cap M_2$ such that: $\ell < n \Rightarrow {\Cal J}$ is predense above
$q'_\ell \cup r_m$.  Without loss of generality ${\Cal J}$ is minimal.  Let
$n_{m+1} = |{\Cal J}|$.
\sn
Let ${\Cal J} = \{p^\ell_m:\ell < n_{m+1}\}$ let $p^\ell_m \Vdash
``{\underset\tilde {}\to G_{P_{i_m}}} \cap M_1 = H^\ell_m"$.  So $H^\ell_m \in
M_2$, and as ${\Cal J}$ is minimal, $H^\ell_m \cap M_0 \in \{G^\ell_m:\ell <
n_m\}$ so let $H^\ell_m \cap M_0 = G^{h(\ell)}_m$ where 
$h:n_{m+1} \rightarrow n_m$.

Let $Y =: \dbca_{\ell < m_n} {\Cal M}[G^\ell_m,M_0,y^*] \in 
D_{\alpha'}(M_0)$. \nl
Now we choose by induction on $\ell \le n_{m+1}$ a condition 
$r^\ell_m \in M_1$ such that:
\mr
\item "{$(\alpha)$}"  $r^\ell_m \in P_{i_{m+1}} \cap M_1,r^\ell_m \restriction
i_m \in H^\ell_m$
\sn
\item "{$(\beta)$}"  $r^\ell_m$ is $(M_0,P_{i_m})$-generic and force a value
to ${\underset\tilde {}\to G_{P_{i_m}}} \cap M_0$ called $G^\ell_{m+1}$
\sn
\item "{$(\gamma)$}"  $r^\ell_m$ is above $G^*_{m+1}$ (chosen in the previous
stage), moreover above $p^{h(\ell)}_m$
\sn
\item "{$(\delta)$}"  $Y \cap \dbca_{k < \ell} {\Cal M}[G^k_{m+1},M_0,y^*] \in
D_{\alpha'}(M_0)$.
\ermn
For the induction step, apply clause (c) in Definition \scite{1.6} (as
$\bar Q \restriction i_{m+1}$ is a \nl
${\frak p}-NNR^0_{\aleph_0}$-iteration) with $i_m,i_{m+1},\alpha',M_0$, large 
enough member of $H^\ell_m,p^{h(\ell)}_m$, \nl
$Y^\ell_m = Y \cap \dbca_{k < \ell}{\Cal M}[G^k_{m+1},M_0,y^*]$ 
here standing for $i,j,\alpha,N,q,p,Y$ there, noting $Y^\ell_m \in M_1$.
\bn
\ub{Stage D}:  

We can choose $r_{m+1}$ as required such that $\{r^\ell_m:\ell < n_{m+1}\}$
is predense over it by \cite[Ch.XVIII,2.6]{Sh:f} (can first do it for each 
$r^\ell_m$ separately and then put them together). \nl
So we have finished proving clause (d).
\bigskip

\demo{\ub{Proof of clause (c)}}  

We prove this by induction on $\alpha$. \nl
Let $i,j,\alpha,N,p,q,Y$ be as there.  If $j < j(*)$ we can apply ``$Q
\restriction j$ is a ${\frak p}-NNR^0_{\aleph_0}$ iteration", so \wilog \,
$j = j(*)$.  If $i=j$ the statement is trivial so assume $i<j$.  Choose
$i_n$ for $n < \omega$ such that $i_0 = i,i_n \in N \cap j,i_n < i_{n+1}$ and
$\dbcu_{n < \omega} i_n = \sup(j \cap N)$.
Let $\langle (y_n,\beta_n):n < \omega \rangle$ list the pairs $(y,\beta) \in
N \times (\alpha \cap N)$ such that 
$y \in {\Cal H}(\chi^{\frak p}_\beta)$.
Let {\bf St} be a winning strategy for the chooser in the game
$\Game_\alpha(N)$. \nl
Let $\langle {\Cal I}_n:n < \omega \rangle$ list the dense open subsets of
$P_j$ which belongs to $N$.

Now we choose by induction on $n < \omega$, the objects $q_n,
{\underset\tilde {}\to p_m},{\underset\tilde {}\to M_n},
{\underset\tilde {}\to Y_m}$ such that:
\mr
\item "{$(a)$}"  $q_n \in P_{i_n},q_0 = q$
\sn
\item "{$(b)$}"  $q_n$ is $(N,P_{i_n})$-generic
\sn
\item "{$(c)$}"  $q_{n+1} \restriction i_n = q_n$
\sn
\item "{$(d)$}"  ${\underset\tilde {}\to p_n}$ is a $P_{i_n}$-name of a member
of $(P_j/{\underset\tilde {}\to G_{P_{i_n}}}) \cap N$
\sn
\item "{$(e)$}"  ${\underset\tilde {}\to p_n}$ is forced to belong to
${\Cal I}_n$
\sn
\item "{$(f)$}"  ${\underset\tilde {}\to M_n}$ is a $P_{i_n}$-name of a
member of ${\Cal E}_{\beta_n} \cap N$
\sn
\item "{$(g)$}"  if $G_j \subseteq P_j$ is generic over $V,q_n \in
G_j,{\underset\tilde {}\to p_n}[G_j \cap P_{i_n}] \in G_j,M =
{\underset\tilde {}\to M_n}[G_j]$, \ub{then}
{\roster
\itemitem{ $(\alpha)$ }  $G_j \cap M$ is a subset of $Q \cap M$ generic
over $M$
\sn
\itemitem{ $(\beta)$ }  ${\Cal M}[G_j \cap M,M,y^*] \cap Y \in D_{\beta_n}
[M]$
\sn
\itemitem{ $(\gamma)$ }  ${\underset\tilde {}\to p_n}[G_j]$ belongs to $M$
\endroster}
\item "{$(h)$}"  $\langle Y_m \cap {\Cal M}[{\underset\tilde {}\to G_{i_m}},
{\underset\tilde {}\to M_m},y^*],{\underset\tilde {}\to Y_m},
P_{i_m},{\underset\tilde {}\to M_m}:m \le n \rangle$ is forced by $q_n$ to
be an initial segment of a play of the game $\Game_\alpha(N)$ in which 
the chooser uses the fixed winning strategy {\bf St}.
\ermn
The proof is straight by the induction hypothesis on $\beta$ and
``$\bar Q \restriction i_n$ is a ${\frak p}-NNR^0_{\aleph_0}$-iteration"
remembering that ${\underset\tilde {}\to M_n},{\underset\tilde {}\to Y_n}$
are $P_{i_n}$-names but of objects in $V$. \nl
Alternatively, see more in the proof in \S4 or see below.
\hfill$\square_{\scite{1.7}}$
\enddemo
\bigskip

\remark{\stag{1.8} Remark}  1) We could have used the ``adding no reals" and
clause (d) in the proof of clause (c) in order to weaken ``winner" to ``not
loser"; also we can use $\Game'_\alpha(N,N',P)$, see \S4.  Also
the ``$N_0 \in \dbcu_{\beta < \alpha} {\Cal E}_\beta"$ can be replaced by
$``N_0 \in {\Cal E}'_0"$ with ${\Cal E}'_0 \subseteq [{\Cal H}(\chi^{\frak p}
_0)]^{\aleph_0}$ stationary; also we can put extra restrictions on 
$G^*$ (and $G^{**}$), e.g. ${\Cal M}[G^*,N_0,y^*]$ large. \nl
2) Of course, the use of $\langle \chi_\alpha:\alpha < \ell g({\frak p})
\rangle$ is not really necessary, we could have used subsets of ${\Cal P}
(P_{\ell g(\bar Q)})$, (so ${\Cal E}_\alpha$ is changed accordingly) as 
all the properties depend just on $N_\ell \cap {\Cal P}(P_{\ell g(\bar Q)})$.
We feel the present way is more transparent. \nl
3) The proof of clause (c) being preserved can be applied to $\bar Q$ 
satisfying (a) + (c) of Definition \scite{1.6} (so possibly adding reals), 
but then we have to replace $D_\alpha(N)$ by a definition of such pseudo 
filters with the winning strategy being absolute enough, e.g. for standard 
$\bar D$. \nl
4) We can also replace $D_\alpha(a)$ by a partial ordered set $L_\alpha(a)$ 
and a function $v_a:L_\alpha(a) \rightarrow {\Cal P}(a)$ (i.e. by 
$(L_a,v_a)$). \nl
5) In the case we adopt remark (3), then remark (1) (non-losing) becomes less
clear, as the universe changes during the proof.  We may consider to
weaken ``the chooser has a winning strategy" in $\Game'_\alpha(N)$
(game depends on $\bar D$), e.g. to not losing in a game with finitely many
boards, possibly splitting in each move (no real novelty in the proof), but
it is not clear how interesting this is.  But if $D_\alpha(N)$ is inductively
defined as sums over an ultrafilter (which is preserved), it just seems 
that not losing is enough.
\endremark
\bigskip

\definition{\stag{1.8A} Definition}  $\bar Q$ a CS iteration will be called
${\frak p}$-proper if clauses (a) + (c) of Definition \scite{1.6} holds.
\enddefinition
\bn
\ub{\stag{1.8B} Question}:  Is this notion of interest in iterations adding
reals?
\bigskip

\definition{\stag{1.9} Definition}  Let $\kappa \in [2,\omega]$.  We say 
that $\bar Q$ is a ${\frak p}-NNR^0_\kappa$-iteration if: from Definition
\scite{1.6} we have:
\mr
\item "{$(a)$}"  as there
\sn
\item "{$(b)$}"  as there
\sn
\item "{$(c)$}"  [long properness] as there
\sn
\item "{$(d)$}"  [$\kappa$-anti w.d.]  as in (d) there but 
$1+n < \kappa$ and $N_0 \in {\Cal E}_0$.
\endroster
\enddefinition
\bigskip

\proclaim{\stag{1.10} Main Claim}  Assume $2 \le \kappa < \omega,
\bar Q$ is a CS iteration, ${\Cal P}(\text{Lim}(\bar Q)) \subseteq 
{\Cal H}(\chi^{\frak p}_0),{\frak p}$ a reasonable parameter of length
$\omega_1$ which is a non-$\Game'$-loser, $\delta = \ell g(\bar Q)$ 
a limit ordinal and $i < \delta \Rightarrow \bar Q \restriction i$ is a
${\frak p}-NNR_\kappa$-iteration.  \ub{Then} $\bar Q$ is a
${\frak p}-NNR^0_\kappa$-iteration.
\endproclaim
\bigskip

\remark{Remark}  1) As it is just combining the proof of \scite{1.7} and
\cite[Ch.XVIII,\S2]{Sh:f} (or \nl
\cite[Ch.VIII,4.10]{Sh:b}-\cite[Ch.VIII,4.10]{Sh:f}) we elaborate less. \nl
2) See more in \S4.
\endremark
\bigskip

\demo{Proof}  Similar to the proof of \scite{1.7}, with some changes (as
in \cite[Ch.VIII,\S4]{Sh:b}, \cite[Ch.VIII,\S4]{Sh:f}, 
\cite[Ch.XVIII,2.10C]{Sh:f}).  During the proof of clause (d) we add to
clauses (a)-(m): 
\mr
\item "{$(n)$}"  $n_m$ is a power of 2, say $2^{n^*_m}$ and so we can rename
$\{G^\ell_m:\ell < n_m\}$ as $\{G^\eta_m:\eta \in {}^{n^*_m}2\}$
\sn
\item "{$(o)$}"  
{\roster
\itemitem{ $(\alpha)$ }  $M_\eta \in M_1 \cap {\Cal E}_{\beta_\eta}$ for 
$\eta \in {}^{(n^*_m \ge)}2$ where $j_0 = \text{ otp}([i,j) \cap N_0)$ and if
$\eta = \nu \char 94 <i>$ then $j_\eta = \text{ otp}([i,j) \cap M_\nu)$
\sn
\itemitem{ $(\beta)$ }  $M_{<>} = N_0$
\sn
\itemitem{ $(\gamma)$ }  $M_\eta \in M_{\eta \char 94 \langle 0 \rangle} \cap
M_{\eta \char 94 \langle 1 \rangle}$
\sn
\itemitem{ $(\delta)$ }  $\eta \triangleleft \nu_1 \in {}^{n^*_m}2 \and
\eta \triangleleft \nu_2 \in {}^{n^*_m}2 \Rightarrow G^{\nu_1}_m \cap M_\eta
= G^{\nu_2}_m \cap M_\eta$ so we call it $K^\eta_m$
\sn
\itemitem{ $(\varepsilon)$ }  $M_{\eta \char 94 \langle 0 \rangle} =
M_{\eta \char 94 \langle 1 \rangle}$ when $\eta \in {}^{n^*_m}2$ call it
$N_\eta$
\sn
\itemitem{ $(\zeta)$ }  $N_\eta \in {\Cal E}_{\ell g(\eta)}$ for
$\eta \in {}^{(m^*_m >)}2$
\sn
\itemitem{ $(\eta)$ }  $Y^\eta_m = {\Cal M}[K^{\eta \char 94 \langle 0 
\rangle}_m,N_\eta] \cap {\Cal M}[K^{\eta \char 94 \langle 1 \rangle}_m,
N_\eta] \in D_{j_{\eta \char 94 <0>}}(N_\eta)$.
\endroster}
\endroster
\hfill$\square_{\scite{1.10}}$
\enddemo
\bigskip

\demo{\stag{1.11} Conclusion}  Let $\bar Q$ be a CS iteration and ${\frak p}$
a non-$\Game'$-loser reasonable parameter, 
$2 \le n(*) \le \aleph_0,\ell g({\frak p}) = \omega_1$. \nl
\ub{Then} $\bar Q$ is a ${\frak p}-NNR^0_{n(*)}$-iteration iff for each $i <
\ell g(\bar Q)$
\mr
\item "{$(*)_i$}"  ${\underset\tilde {}\to Q_i}$ is a proper forcing and
$P_i,{\underset\tilde {}\to Q_i}$ satisfies clauses (d) + (c) of the
Definition ``${\frak p}-NNR^0_{n(*)}$-iteration" with $i,i+1$ here standing
for $i,j$ there.
\endroster
\enddemo
\bigskip

\demo{Proof}  By induction on $j = \ell g(\bar Q)$.  
For $j=0$ there is nothing, for $j$ limit use \scite{1.7} (or \scite{1.10})
and if $j$ successor just read the definitions.
\hfill$\square_{\scite{1.11}}$
\enddemo
\bigskip

We point out here that clause (c) of Definitions \scite{1.6} and \scite{1.9}
really follows from earlier properties which play parallel roles.
\proclaim{\stag{1.12} Claim}  1) Assume that ${\frak p}$ is a standard
reasonable parameter, and is standard and $\alpha < \ell g({\frak p}),N \in
{\Cal E}^{\frak p}_\alpha,Y \in D_\alpha(N)$ and $\delta \le \omega_1
\cap N$ a limit ordinal. \nl
\ub{Then} we can find sequences $\bar N = \langle N_i:i < \delta \rangle,
\langle \gamma_i:i < \delta \rangle$ such that:
\mr
\item "{$(a)$}"  $N_i \in N$ (for $i < \delta$) is countable,
$N \cap \alpha \subseteq N_i$ and $N_i \in Y$
\sn
\item "{$(b)$}"  $N_i \subseteq \dbcu_{\beta \in \alpha \cap N}
({\Cal H}(\chi^{\frak p}_\beta),\in)$, and $\beta \in \alpha \cap N_i
\Rightarrow N_i \restriction {\Cal H}(\chi^{\frak p}_\beta) \prec
({\Cal H}(\chi^{\frak p}_\beta),\in)$
\sn
\item "{$(c)$}"  $i < j \Rightarrow N_i \subseteq N_j$
\sn
\item "{$(d)$}"  for $i$ limit $N_i = \dbcu_{j<i} N_j$ and 
$N \cap \dbcu \{{\Cal H}(\chi^{\frak p}_\beta):\beta \in \alpha \cap
N\} = \dbcu_{j < \delta} N_j$ so we can stipulate $N_\delta = N$
\sn
\item "{$(e)$}"  $\beta \in \alpha \cap N \Rightarrow \langle {\Cal H}
(\chi^{\frak p}_\beta) \cap N_j:j \le i \rangle \in N_{i+1}$
\sn
\item "{$(f)$}"  $\gamma_i \in N_i \cap \alpha,N_i \cap {\Cal H}
(\chi^{\frak p}_{\gamma_i}) \in {\Cal E}_{\gamma_i} \cap Y,
\bar \gamma \restriction (i+1) \in N_{i+1}$
\sn
\item "{$(g)$}"  if $i \le \delta$ is a limit ordinal, $(i = \delta \and
\beta \in \alpha \cap N_i) \vee (i < \delta \and \beta \in \gamma_i \cap N_i)$
(so $\beta \in {\Cal H}(\chi^{\frak p}_\beta))$
and $y \in {\Cal H}(\chi^{\frak p}_\beta)$, \ub{then} for some 
$j < i,\gamma_j = \beta,y \in N_j$.  Moreover
\sn
\item "{$(g)^+$}"  if $i \le \delta$ is a limit ordinal, then
$\{N_j \cap {\Cal H}(\chi^{\frak p}_{\gamma_j}):j < i \text{ and }
\gamma_j < \gamma_i\} \in D_{\gamma_i}(N) = D_{\gamma_i}(N \cap {\Cal H}
(\chi^{\frak p}_{\gamma_i}))$
\sn
\item "{$(h)$}"  if $\delta < N \cap \omega_1$ then $\delta \in N_0$, if
$\delta = N \cap \omega_1$ then $i < \delta \Rightarrow i \in N_i$.
\ermn
2) If ${\frak p}$ is a standard, reasonable parameter, $\bar Q$ is a 
CS iteration, $\ell g(\bar Q) = \alpha +1,\bar Q
\restriction \alpha$ is ${\frak p}-NNR^0_{k(*)}$-iteration and \nl
$\Vdash_{P_\alpha} ``{\underset\tilde {}\to Q_\alpha}$ is proper and
$(<^+ \omega_1)$-proper", \ub{then} trying to apply \scite{1.10} for
$\bar Q$,
clauses (a),(c) (called ${\frak p}$-proper in Definition \scite{2.3} below) 
holds. \nl
3) If $\ell g({\frak p}) = \omega_1$, \ub{then} in part (2) it suffices to ask
$\Vdash_{P_i} ``\alpha < \omega_1 \Rightarrow {\underset\tilde {}\to Q_i}$ is
$\alpha$-proper", that is $\Vdash_{P_i} ``{\underset\tilde {}\to Q_i}$ is
$(< \omega_1)$-proper".
\endproclaim
\bigskip

\demo{Proof}  1) By induction on $\beta \in (\alpha \cap N) \cup
\{\alpha\}$ we prove that there is \nl
$\langle N_j:j < \beta \cap N \rangle
\in N$ satisfying the relevant requirements. \nl
2) By Definition of $(< \omega_1)$-proper, if $G_\alpha \subseteq P_\alpha$
is generic over $\alpha,Y =: {\Cal M}[G_\alpha,N,y^*] \in 
D_\alpha(N)$, let $\langle N_i:i < \delta \rangle$ be as in $(1)$ for
$\delta = N \cap \omega_1$, without loss of generality 
$p \in N_0 \cap {\underset\tilde {}\to Q_\alpha}[G_\alpha]$, 
let $q \ge p$ be $(N_i,{\underset\tilde {}\to Q_\alpha}[G_\alpha])$-generic 
for every $i < \delta$ (formally look at $\bar N' = \langle N'_i:i < \delta 
\rangle,N'_i = N_i \restriction {\Cal H}(\chi^{\frak p}_0)$ and apply to it 
the $(< \omega_1)$-proper). \nl
3) Follows as $\alpha \in N \Rightarrow \delta = \omega \alpha \in N \cap
\omega_1$. \hfill$\square_{\scite{1.12}}$
\enddemo
\bn
\ub{\stag{1.13} Discussion}:  1) This includes as special cases
\cite[Ch.V,\S5,\S7]{Sh:b}.  There is no direct comparison with 
\cite[Ch.VIII,\S4]{Sh:b}, \cite[Ch.VIII,\S4]{Sh:f}, but we can make the notion
somewhat more complicated, to include the theorem there in our context,
i.e., what is not included is a generalization there not really needed
for the examples discussed there (see here in \S3 below).
The condition there involves having many sequences $\langle N_\alpha:
\alpha \le \delta \rangle$ such that if $p_0,p_1 \in P,p_\ell$ is
$(N_i,P_\alpha)$-generic for $i,p_\ell \Vdash ``
{\underset\tilde {}\to G_P} \cap N_0 = G^*"$, then there is $G' \subseteq
\text{ Gen}({\underset\tilde {}\to Q_i}[G^*],N_0[G^*]),G' \in N_0,
P_i \nVdash_{P_i} ``{\underset\tilde {}\to G_0}$ has no bound in
${\underset\tilde {}\to Q_\alpha}"$.  This speaks on a family of sequences
from $[{\Cal H}(\chi)]^{\aleph_0}$ rather than members. \nl
2) For \cite[Ch.XVIII,\S2]{Sh:f}, the comparison is not so easy.  
Our problem is to
``carry" good $(N,P_i,\langle G_\ell:\ell < n \rangle),G_\ell \in
\text{ Gen}(N,P_i)$ with a bound, such that we can ``increase $i$" and we
can find $N',y \in N' \in N,N' \prec N$ such that $(N',P_i,\langle G_i \cap
N:\ell < n \rangle)$ is good enough.  In \cite[Ch.XVIII]{Sh:f} we are carrying
genericity in some $P_{\bar \alpha},\bar \alpha \in \text{ trind}(i)$, here
much less.  But what we need is the implication ``if $(N,P_i,\bar G)$ is good
we can extend it to good $(N,P_{i+1},\bar G')$", so making good weaker 
generates an incomparable notion and clearly there are other variants. 
We can consider other such notions (see \S5).  \nl
2A) We can give alternative proofs of consistency of the questions in
\cite[Ch.XVIII,\S1]{Sh:f} by the present iteration theorem (see \S3 below). \nl
3)  We can in \scite{1.12}(2) weaken the assumption ``$(< \omega_1)$-proper"
to things of the form: if $\bar N = \langle N_i \le \delta \rangle,p \in Q
\cap N_0,Q \in N_0$ then there is $q \ge p$ such that $q \Vdash ``
\text{for many } i < \delta,{\underset\tilde {}\to G_Q} \cap N_i \in
\text{ Gen}(N_i,Q)"$.  In particular the condition applies to the forcing
notions considered in \cite[Ch.XVIII,\S1]{Sh:f}. 
\newpage

\head {\S2 Delayed properness} \endhead  \resetall \sectno=2
 % \resetall 
\bn
In this section we prove little, but the notions introduced are used in
\S3, \S4.   We concentrate here on simple parameters so the reader may assume
it all the time.  We give two versions, the simpler one is version 2 for
which simplicity is a very natural demand.
\demo{\stag{2.1} Observation}  1) Assume
\mr
\item "{$(a)$}"  $\bar \chi = \langle \chi_\alpha:\alpha < \alpha^* \rangle$
increases fast enough, so $\dbcu_{\beta < \alpha} {\Cal H}(\chi_\beta) \in
{\Cal H}(\chi_\alpha)$
\sn
\item "{$(b)$}"  ${\Cal E}_\alpha \subseteq \{N:N$ a countable elementary
submodel of $({\Cal H}(\chi_\alpha),\in)\}$ \nl
is stationary
\sn
\item "{$(c)$}"  $R_\alpha \in {\Cal H}(\chi_\alpha)$ and $N \in
{\Cal E}_\alpha$ implies \nl
$\langle \chi_\beta:\beta < \alpha \rangle \in N, \langle R_\beta:\beta \le
\alpha \rangle \in N$ and $\langle {\Cal E}_\beta:\beta < \alpha \rangle
\in N$.
\ermn
\ub{Then} there is a standard reasonable parameter ${\frak p}$ with
$\ell g({\frak p}) = \alpha^*,{\Cal E}^{\frak p}_\alpha = {\Cal E}_\alpha$,\nl
$R^{\frak p}_\alpha = R_\alpha$ (and $\bar D^{\frak p}$ standard). \nl
2) If in addition clause (d) below hold, \ub{then} ${\frak p}$ is a simple
standard reasonable parameter where
\mr
\item "{$(d)$}"  $\beta \in N \in {\Cal E}_\alpha,\beta < \alpha
\Rightarrow N \cap {\Cal H}(\chi_\beta) \in {\Cal E}_\beta$.
\ermn
3) If $\chi_\alpha = (\beth_{2 \alpha +1})^+$ for $\alpha < \alpha^*,
R_\alpha \in {\Cal H}(\chi_\alpha)$, then $\chi_\alpha$ increases fast
enough.  If $\langle \chi_\alpha:\alpha < \alpha^* \rangle,\langle R_\alpha:
\alpha < \alpha^* \rangle$ are as in part (1), $\chi \le \chi_0,{\Cal E}
\subseteq [{\Cal H}(\chi)]^{\le \aleph_0}$ stationary and we let ${\Cal E}
_\alpha = \{N:N$ a countable elementary submodel of $({\Cal H}(\chi_\alpha),
\in)$ and $\langle \chi_\beta:\beta < \alpha \rangle,\langle R_\beta:\beta
\le \alpha \rangle,{\Cal E}$ belong to $N$ and $N \cap {\Cal H}(\chi) \in
{\Cal E}\}$, \ub{then} the assumptions of parts (1)
and (2) above holds (hence their conclusions).
\enddemo
\bigskip

\demo{Proof}  Straight.
\enddemo
\bigskip

\definition{\stag{2.2} Definition}  Let ${\frak p}$ be a reasonable
parameter and $\alpha \le \beta < \ell g({\frak p})$. \nl
1) For $N \in {\Cal E}_\beta$ such that $\alpha \in N$ we define a game
$\Game_{\alpha,\beta}(N) = \Game_{\alpha,\beta}(N,{\frak p})$ as follows.
\nl
A play lasts $\omega$-moves.  In the $n$-th move:
\mr
\item "{$(a)$}"  the \ub{challenger chooses} $X_n \in D_\beta(N)$ such that
\nl
$m < n \Rightarrow X_n \subseteq X_m$
\sn
\item "{$(b)$}"  the \ub{chooser} chooses $\alpha_n \in \alpha \cap N$
\sn
\item "{$(c)$}"  the \ub{challenger} chooses $\beta'_n \in \beta \cap
N$ and $y'_n \in N \cap {\Cal H}(\chi^{\frak p}_{\alpha_n})$
\sn
\item "{$(d)$}"  the \ub{chooser} chooses $\beta_n \in \beta \cap N
\backslash \beta'_n$ and $M_n \in X_n \cap {\Cal E}_{\beta_n}$ 
and $y_n \in M_n \cap {\Cal H}(\chi^{\frak p}_{\alpha_n})$ satisfying 
$\alpha_n \le \beta_n,y'_n \in M_n,\alpha_n \in M_n$ and $Y_n \in D_{\beta_n}
(M_n)$ such that $Y_n \subseteq X_n$ and $Y_n \in N$
\sn
\item "{$(e)$}"  the \ub{challenger} chooses $M'_n \in Y_n \cap
{\Cal E}_{\alpha_n} \cap (M_n \cup \{M_n \cap {\Cal H}
(\chi^{\frak p}_{\alpha_n})\})$ satisfying $y_n,y'_n \in M'_n$ and
chooses
$Z_n \in D_{\alpha_n}(M'_n) = D_{\alpha_n}(M'_n \cap {\Cal H}(\chi
^{\frak p}_{\alpha_n}))$ such that $Z_n \subseteq Y_n$.
\ermn
In the end the chooser wins the play if

$$
\bigcup \{Z_n \cup \{M'_n\}:n < \omega\} \in D^+_\alpha(N) \,
(= D^+_\alpha(N \cap {\Cal H}(\chi^{\frak p}_\alpha)))
$$
\mn
1A) We call $\Game_{\alpha,\beta}(N) = \Game_{\alpha,\beta}(N,{\frak p})$
version 1; version 2 means that in clause (e) we add the requirement
$M'_n = M_n \cap {\Cal H}(\chi^{\frak p}_{\alpha_n})$ and in clause (d) we
require $\alpha_n \le_{\frak p} \beta_n$.  If we do not mention the version
it means that it holds for both.
\nl
2) Assume $N \in N' \prec ({\Cal H}(\chi),\in)$.
We define a game $\Game'_{\alpha,\beta}(N,N',{\frak p})$ similarly but
replace a) - e) by
\mr
\item "{$(a)'$}"  the \ub{challenger} chooses $X_n \in D_\beta(N)$ such
that $m < n \Rightarrow X_n \subseteq X'_m$
\sn
\item "{$(b)'$}"  the \ub{chooser} chooses $\alpha_n \in \alpha \cap
N$ and $X'_n \subseteq X_n$ such that $X'_n \in D_\beta(N)$
\sn
\item "{$(c)'$}"  like (c)
\sn
\item "{$(d)'$}"  like (d) but we replace ``$Y_n \in N$" by ``$Y_n \in N'$"
\sn
\item "{$(e)'$}"  like (e) but add $Z_n \in N'$.
\ermn
Note: so every proper initial segment of a play belongs to $N'$.
\enddefinition
\bigskip

\definition{\stag{2.3} Definition}  Let ${\frak p}$ be a reasonable
parameter. \nl
1) For $\alpha \le \beta < \ell g({\frak p})$ we say ${\frak p}$ is an
$\Game_{\alpha,\beta}$-winner [non-$\Game_{\alpha,\beta}$-loser] when for
some $x \in {\Cal H}(\chi^{\frak p}_\beta)$ we have: 
\mr
\item "{{}}"  if $\{x,\alpha\} \in N \in {\Cal E}_\beta$, \ub{then} the 
chooser wins the game $\Game_{\alpha,\beta}(N,{\frak p})$ 
[the challenger does not win in the game $\Game_{\alpha,\beta}
(N,{\frak p})$]. 
\ermn
2)  We can replace $\Game_{\alpha,\beta}$ by $\Game'_{\alpha,\beta}$. \nl
3)  For any function $f:\ell g({\frak p}) \rightarrow {\Cal P}(\ell g
({\frak p}))$ we can replace $\alpha,\beta$ by $f$ meaning: for every 
$\alpha < \ell g({\frak p})$ and $\beta \in f(\alpha)$, we have 
${\frak p}$ is a $\Game_{\alpha,\beta}$-winner. \nl
4) Let ${\Cal F}^{\frak p}$ be the family of functions from
$\ell g({\frak p})$ to ${\Cal P}(\ell g({\frak p}))$ such that for each
$\alpha < \ell g({\frak p}),f(\alpha)$ is a nonempty subset of
$[\alpha,\ell g({\frak p}))$.  Let ${\Cal F}^{\frak p}_{\text{club}}$ be
the set of $f \in {\Cal F}^{\frak p}$ such that for each $\alpha < \ell g
({\frak p}),f(\alpha)$ is a club of $\ell g({\frak p})$.  Let
${\Cal F}^{\frak p}_{\text{end}}$ be the set of $f \in {\Cal F}^{\frak p}$
such that for each $\alpha < \ell g({\frak p}),f(\alpha)$ is an end segment
of $\ell g({\frak p})$, we then may identify $f(\alpha)$ with Min$(f
(\alpha))$. \nl
5) We call $f$ decreasing continuous if $\alpha_1 < \alpha_2 < \ell g
({\frak p}) \Rightarrow f(\alpha_2) \subseteq f(\alpha_1)$ and for limit
$\alpha < \ell g({\frak p})$ we have $f(\alpha) = \dbca_{\gamma < \alpha}
f(\gamma)$.  Let $f \le g$ mean $(\forall \alpha < \ell g({\frak p}))
(g(\alpha) \subseteq f(\alpha))$.

There are obvious monotonicity properties.
\enddefinition
\bigskip

\proclaim{\stag{2.4} Claim}  Assume ${\frak p}$ is a reasonable
parameter. \nl
1) If $\alpha \le_{\frak p} \alpha',\alpha' \le \alpha \le \beta = \beta' 
< \ell g({\frak p})$, and ${\frak p}$ is a $\Game_{\alpha,\beta}$-winner, 
\ub{then} it is
$\Game_{\alpha',\beta'}$-winner.  Similarly for $\Game'$-winner, 
non-$\Game$-loser, non-$\Game'$-loser. \nl
2)  If ${\frak p}$ is a $\Game_{\alpha,\beta}$-winner, \ub{then} 
${\frak p}$ is a $\Game'_{\alpha,\beta}$-winner and a 
non-$\Game_{\alpha,\beta}$-loser.
If ${\frak p}$ is a $\Game'_{\alpha,\beta}$-winner or 
non-$\Game_{\alpha,\beta}$-loser, \ub{then} ${\frak p}$ is
non-$\Game'_{\alpha,\beta}$-loser. \nl
3) Assume $f,g \in {\Cal F}^{\frak p}$ and $f \le g$.  
If ${\frak p}$ is a $\Game_f$-winner [or $\Game'_f$-winner] 
[ or non-$\Game_f$-loser] [or non-$\Game_f$-loser], \ub{then} 
${\frak p}$ is a $\Game_g$-winner [or
$\Game'_g$-winner] [or non-$\Game_g$-loser] [or non-$\Game'_g$-loser].
\endproclaim
\bigskip

\demo{Proof} Straight (we are using the simplicity of ${\frak p}$).
\enddemo
\bigskip

\proclaim{\stag{2.5} Claim}  1) Assume ${\frak p}$ is a standard reasonable
parameter.  \ub{Then} ${\frak p}$ is a winner (see Definition \scite{1.4}). 
\nl
2)  If ${\frak p}$ is a reasonable parameter and it is a winner, \ub{then}
${\frak p}$ is a $\Game_{\alpha,\alpha}$-winner. \nl
3)  If ${\frak p}$ is a reasonable parameter and it is a winner and
$\alpha \le \beta < \ell g({\frak p})$ \ub{then} ${\frak p}$
is a $\Game_{\alpha,\beta}$-winner (hence $\Game_f$-winner for $f$ as in
\scite{2.3}(3)). \nl
4) Similarly with $\Game'$-winner, $\Game'_{\alpha,\beta}$ winner and/or
with the ``non loser" cases.
\endproclaim
\bigskip

\demo{Proof}  Straight.
\enddemo
\bigskip

\definition{\stag{2.6} Definition}  1) Let ${\frak p}$ be a reasonable
parameter and $P$ be a proper forcing notion not adding reals, ${\Cal P}(P)
\in {\Cal H}(\chi^{\frak p}_0)$ and $G_P \subseteq P$ is generic over
$V$.

\ub{Then} we interpret ${\frak p}$ in $V^P$ as ${\frak p}' = 
{\frak p}^{V[G]}$, or we may write ${\frak p}^P$ defined as follows:
\mr
\item "{$(a)$}"  $\chi^{{\frak p}'}_\alpha = \chi^{\frak p}_\alpha$
\sn
\item "{$(b)$}"  $R^{{\frak p}'}_\alpha = \langle R_\alpha,P,G_P \rangle$
\sn
\item "{$(c)$}"  ${\Cal E}^{{\frak p}'}_\alpha = \{N[G_P]:N \in {\Cal E}
_\alpha$ and $P \in N$ and $N[G_P] \cap V = N\}$
\sn
\item "{$(d)$}"  $D_\alpha(N[G_P]) = \bigl\{ \{M[G_P] \in {\Cal E}_\alpha:
M \in Y \cap \dbcu_{\beta < \alpha} {\Cal E}^{\frak p}_\beta\}:Y \in
D_\alpha(N) \bigr\}$.
\endroster
\enddefinition
\bigskip

\proclaim{\stag{2.7} Claim}  Let ${\frak p},P$ be as in Definition
\scite{1.6}. \nl
1) In Definition \scite{2.6}, ${\frak p}^{V[G_P]}$ is
a reasonable parameter in $V[G_P]$.  \nl
2) 
If ${\frak p}$ is, in $V$, a $\Game$-winner (or
non-$\Game$-loser or $\Game'$-winner or non-$\Game'$-loser), \ub{then}
${\frak p}^{V[G_P]}$ is so (in $V[G_P]$). \nl
3) If ${\frak p}$ is, in $V$, a $\Game_{\alpha,\beta}$-winner (or a non
$\Game_{\alpha,\beta}$-lower or $\Game'_{\alpha,\beta}$-winner or
$\Game'_{\alpha,\beta}$-non loser), \ub{then} ${\frak p}^{V[G_P]}$ is so
in $V[G_p]$.
\endproclaim
\bigskip

\demo{Proof}  Straight (we use $P$ is a proper forcing notion not adding
reals).
\enddemo
\bigskip

\centerline {$* \qquad * \qquad *$}
\bigskip

\definition{\stag{2.8} Definition}  Let ${\frak p}$ be a reasonable
parameter. \nl
1) For $\alpha \le \beta < \ell g({\frak p})$, we say a forcing notion $Q$
is $({\frak p},\alpha,\beta)$-proper if: \nl
$({\Cal P}(Q) \in {\Cal H}(\chi^{\frak p}_0)$ and):
\mr
\item "{$(*)$}"  for some $x \in {\Cal H}(\chi_\beta)$ \ub{if} $N \in {\Cal E}
_\beta,\{x,Q,\alpha\} \in N,p \in N \cap Q$ and \nl
$Y \in D_\alpha(N)$, \ub{then} for some $q$ we have:
{\roster
\itemitem{ (a) }  $p \le q \in Q$
\sn
\itemitem{ (b) }  $q$ is $(N,Q)$-generic
\sn
\itemitem{ (c) }  version (1): for some $N' \in ({\Cal E}_\alpha \cap N \cap
Y) \cup \{N \cap {\Cal H}(\chi^{\frak p}_\alpha)\}$ satisfying $\alpha =
\beta \Rightarrow N' = N$ we have 
$q \Vdash_Q ``{\Cal M}[{\underset\tilde {}\to G_Q},N',y^*] \cap Y \in
D_\alpha(N')"$ where $y^* = \langle x,p,Q \rangle$ hence necessarily $q$
is $(N',Q)$-generic.
\endroster}
(Version 2: similarly but $N' = N \restriction {\Cal H}(\chi^{\frak p}
_\alpha)$ and naturally \nl
\,\,\,we demand $\alpha \le_{\frak p} \beta$).
\ermn
2) We say $Q$ is $({\frak p},f)$-proper if (${\frak p},f$ are as above and) 
for every $\alpha < \ell g({\frak p})$ and $\beta \in f(\alpha)$ we have 
$Q$ is $({\frak p},\alpha,\beta)$-proper. \nl
3) We say $Q$ is ${\frak p}$-proper if $Q$ is 
$({\frak p},\alpha,\alpha)$-proper for 
$\alpha < \ell g({\frak p})$.  We say $Q$ is almost ${\frak p}$-proper 
if $Q$ is $({\frak p},f)$-proper for some $f \in {\Cal F}^{\frak p}$ 
(as above).
\enddefinition
\bigskip

\proclaim{\stag{2.9} Claim}  Assume ${\frak p}$ is a simple reasonable
parameter. \nl
1) If $\alpha' \le \alpha \le \beta \le \beta' < \ell g({\frak p})$ 
(for version 2 we demand $\alpha' \le_{\frak p} \beta'$ and $\alpha 
\le_{\frak p} \beta$) and $Q$ is as $({\frak p},\alpha,\beta)$-proper 
forcing notion, \ub{then} $Q$ is a
$({\frak p},\alpha',\beta')$-proper forcing notion. \nl
2) Assume $f,f'$ are in ${\Cal F}^{\frak p}$ and $f \le f'$.
If $Q$ is a $({\frak p},f)$-proper forcing notion, \ub{then} $Q$ is a 
$({\frak p},f')$-proper forcing notion.
\endproclaim
\bigskip

\demo{Proof}  Straight.
\enddemo
\bn
\ub{\stag{2.10} Discussion}:  We may like to consider $({\frak p},f)$-proper
for iterations which may add reals.  Then we have to replace $D_\alpha(N)$
by a definition which is absolute enough, (and the non-loser versions have
to be absolute enough). \nl
It is natural to restrict ourselves to ${\frak p}$-closed $Y$, see below.
\bigskip

\definition{\stag{2.11} Definition}  Let ${\frak p}$ be a simple reasonable
parameter and $N \in {\Cal E}_\alpha$.  Now $Y \subseteq N$ is called
${\frak p}$-closed if (see \scite{3.9}):
\mr
\item "{$(a)$}"  $Y \subseteq N \cap \dbcu_{\beta < \alpha} {\Cal E}_\beta$
\sn
\item "{$(b)$}"  if $M \in N \cap {\Cal E}_\beta$ and $\beta < \alpha$
(hence $\beta \in \alpha \cap M \subseteq \alpha \cap N$) and
$\gamma \in M \cap \beta$ and $M \cap {\Cal H}(\chi^{\frak p}_\gamma) \in
{\Cal E}_\gamma$ (this requirement is redundant if ${\frak p}$ is simple or
just $\gamma \le_{\frak p} \beta$), \nl
\ub{then} 
$M \cap {\Cal H}(\chi^{\frak p}_\gamma) \in Y \Leftrightarrow M \in Y$
\sn
\item "{$(c)$}"  if $\beta < \alpha,M_\ell \in N \cap {\Cal E}_\beta$
(hence $\beta \in \alpha \cap N$) and $M_\ell \subseteq M_{\ell +1}$ for
$\ell < \omega$ and 
$M = \dbcu_{\ell < \omega} M_\ell \in N \cap {\Cal E}_\beta$, 
and even $\langle M_\ell:\ell < \omega \rangle \in N$, \ub{then}
$\dsize \bigwedge_\ell M_\ell \in Y \Rightarrow M \in Y$.
\endroster 
\enddefinition
\bn
We may consider
\definition{\stag{2.12} Definition}  Let ${\frak p}$ be a reasonable parameter
and $\Xi \subseteq \{(\alpha,\beta):\alpha \le \beta < \ell g({\frak p})\}$ 
and we understand $N \in {\Cal E}_\alpha \Rightarrow \Xi \cap ((\alpha + 1)
\times (\alpha +1) \in N)$ and let $\alpha_{\frak p}(N) =$ the $\alpha$ such
that $N \in {\Cal E}^{\frak p}_\alpha$ if it exists.

We say $\bar Q$ is an $(\kappa_1,\Xi)$-anti w.d.-iteration for ${\frak p}$
as in clauses (a), (b), (d) of Definitions \scite{1.6}, \scite{1.9} only we 
replace clause (d) by
\mr
\item "{$(d)_\Xi$}"  [$(\kappa,\Xi)$-anti w.d.] like old clause (d) but 
for some $(\alpha,\beta) \in \Xi,
N_0 \in N_1$ are countable elementary submodels of $({\Cal H}(\chi),\in),
{\frak p} \in N_0,N_0 \cap {\Cal H}(\chi^{\frak p}_\alpha) \in 
{\Cal E}_\alpha,N_1 \cap {\Cal H}(\chi^{\frak p}_\beta) \in {\Cal E}_\beta,
\alpha \in N_0,\beta \in N_1$ and the rest as before.
\endroster
\enddefinition
\bn
We omit ``for ${\frak p}$" if ${\frak p}$ is clear from the context.
\proclaim{\stag{2.13} Lemma}  Let ${\frak p}$ be a reasonable parameter. \nl
1) If $\Xi \ne \emptyset$ and $\bar Q$ is an
$NNR^0_{\aleph_0,\Xi}$-iteration, \ub{then} $P_{\ell g(\bar Q)} =
\text{ Lim}(\bar Q)$ does not add reals and is 
$({\frak p},\Xi)$-proper. 
\endproclaim
\bigskip

\demo{Proof}  Easy.
\enddemo
\bigskip

As the sets ${\Cal H}(\chi_\alpha)$ may change with forcing, we may prefer to
use ${\Cal E}_\alpha \subseteq [\chi_\alpha]^{\le \aleph_0}$, for this we
define:
\definition{\stag{2.14} Definition}  1) We call ${\frak p}$ an o.b. (ordinal 
based) parameter if \nl
${\frak p} = (\bar \chi^{\frak p},\bar R^{\frak p},
\bar{\Cal E}^{\frak p},\bar D^{\frak p})$ and for some ordinal $\alpha^*$
called $\ell g({\frak p})$ we have:
\mr
\item "{$(a)$}"  $\bar \chi = \langle \chi_\alpha:\alpha < \alpha^* \rangle,
\chi_\alpha$ is a regular cardinal and ${\Cal H}\bigl( (\dbcu_{\beta <
\alpha} \chi_\beta)^+ \bigr) \in {\Cal H}(\chi_\alpha)$
\sn
\item "{$(b)$}"  $\bar R = \langle R_\alpha:\alpha < \alpha^* \rangle,
R_\alpha$ an $n(R_\alpha)$-place relation on some bounded subset of
$\chi_\alpha$ (we could have asked ``on $\chi_\alpha$", no real difference)
\sn
\item "{$(c)$}"  $\bar{\Cal E} = \langle {\Cal E}_\alpha:\alpha < \alpha^*
\rangle,{\Cal E}_\alpha \subseteq [\chi_\alpha]^{\le \aleph_0}$ is
stationary
\sn
\item "{$(d)$}"  $\bar D = \langle D_\alpha:\alpha < \alpha^* \rangle$ and
$D_\alpha$ is a function with domain ${\Cal E}_\alpha$ and for each $a \in
{\Cal E}_\alpha,D_\alpha(a)$ is a family of subsets of $a$, closed
under supersets, non-empty if $\alpha > 0$ 
(let $D^-_\alpha(a) = (D_\alpha(a))^- = {\Cal P}(a) \backslash D_\alpha(a))$
\sn
\item "{$(e)$}"  let ${\frak p}^{[\alpha]} = \langle \bar \chi \restriction
\alpha,\bar R \restriction (\alpha +1),\bar{\Cal E} \restriction \alpha,
\bar D \restriction \alpha \rangle$
\sn
\item "{$(f)$}"  if $a \in {\Cal E}_\alpha$ and $X \subseteq {\Cal P}(a)$,
\ub{then}: \nl
$X \in D_\alpha(a) 
\Leftrightarrow X \cap \dbcu_{\beta < \alpha} {\Cal E}_\beta \in D_\alpha(a)$.
\ermn
2) We say that an o.b. parameter ${\frak p}$ is simple if
\mr
\item "{$(g)$}"  if $a \in {\Cal E}_\alpha$ and $X \in D_\alpha(a)$ and
$\beta \in \alpha \cap a$, \ub{then} $a \cap \chi_\beta \in {\Cal E}_\beta$.
\ermn
3) For ${\frak p}$ as above let ${\frak q} =: {\frak p}^V$ be defined by

$$
\alpha^{*,{\frak q}} = \alpha^{*,{\frak p}}
$$
\mn
and we define by induction on $\alpha < \alpha^{*,{\frak p}}$:

$$
\chi^{\frak q}_\alpha = \chi^{\frak p}_\alpha
$$

$$
R^{\frak q}_\alpha = R^{\frak p}_\alpha
$$

$$
{\Cal E}^{\frak q}_\alpha = \{N \prec ({\Cal H}(\chi^{\frak q}_\alpha),\in):
N \text{ is countable}, N \cap \chi^{\frak p}_\alpha \in 
{\Cal E}^{\frak p}_\alpha,{\frak q}^{[\alpha]} \in N\}
$$
\mn
(note that ${\frak q}^{[\alpha]}$ is well defined by the induction 
hypothesis)

$$
\align
D^{\frak q}_\alpha(N) = \biggl\{ Y':&Y' \subseteq \dbcu_{\beta < \alpha}
{\Cal E}^{\frak q}_\beta \text{ and for some } y \in N \cap \dbcu_{\beta <
\alpha} {\Cal H}(\chi^{\frak p}_\beta) \\
  &\text{and } Y \in D^{\frak p}_\alpha(N \cap \chi^{\frak p}_\alpha) 
\text{ we have} \\
  &Y' \supseteq \{M:M \in N \cap \dbcu_{\beta < \alpha} {\Cal E}^{\frak q}
_\beta \text{ and } y \in M \text{ and } M \cap \chi^{\frak p}_\alpha \in Y\} 
\biggr\}.
\endalign
$$
\mn
4) For such ${\frak p}$ we say $\bar Q$ is a $NNR^0_\kappa$-iteration for
${\frak p}$ if it is an $NNR^0_\kappa$-iteration for ${\frak p}^V$. 
We say ${\frak p}$ is simple if ${\frak p}^V$ is. \nl
Similarly for $\Game$-winner, non-$\Game$-loser, etc.
\enddefinition
\bigskip

\proclaim{\stag{2.14A} Claim}  Assume ${\frak p}$ is an o.b. [simple]
parameter in the universe $V$. \nl
1) If $P \in {\Cal H}(\chi^{\frak p}_0)$ is a proper forcing notion (or at
least preserve the stationarity of ${\Cal E}^{\frak p}_\alpha$ for each
$\alpha < \ell g({\frak p})$), then $\Vdash_P ``{\frak p}$ is an o.b. [simple]
parameter". \nl
2) If forcing with $P$ adds no reals, then also $\Game$-winner,
non-$\Game$-loser, etc., are preserved. \nl
3) ${\frak p}^V$ is a reasonable parameter.
\endproclaim
\bigskip

\definition{\stag{2.15} Definition}  Let ${\frak p}$ be an o.b. parameter. \nl
1) We say $Q$ is an $NNR^0_\kappa$-forcing for ${\frak p}$ or a
${\frak p}-NNR^0_\kappa$-forcing notion when: ($Q$ is
a forcing notion in a universe $V$ and) if for some transitive class
$V_0,{\frak p} \in V_0$ and $NNR^0_\kappa$-iteration $\bar Q = \langle P_i,
{\underset\tilde {}\to Q_j}:i \le \alpha,j < \alpha \rangle$, we have
$V^{P_\alpha}_0 = V$, \ub{then} we can let ${\underset\tilde {}\to Q_\alpha} 
= Q$ and get $\bar Q'$, an $NNR^0_\kappa$-iteration (i.e. $V = V_0[G_\alpha],
G_\alpha \subseteq P_\alpha$ is generic over $V_0$ and there is a
$P_\alpha$-name ${\underset\tilde {}\to Q_\alpha}$ such that $\langle P_i,
{\underset\tilde {}\to Q_i}:i \le \alpha \rangle$ is an $NNR^0_\kappa$-
iteration and $Q = {\underset\tilde {}\to Q_\alpha}[G_\alpha]$). 
In particular $Q$ is proper and does not add reals. \nl
2)  If we omit ``for ${\frak p}$" we mean for any ${\frak p}$ which makes
sense.  Alternatively, we can put a family of ${\frak p}$'s. \nl
3)  We add ``over $x$" if this holds whenever $x \in V_0$.  We can use the
same definition for other versions of $NNR$. \nl
4)  $Ax^\alpha_\lambda({\frak p},\kappa,0)$ means: if $Q$ is an
$\aleph_2$-e.c.c. (see \cite[Ch.VII,\S1]{Sh:f}), ($\aleph_2$-pic if $\lambda
= \aleph_2$; see \cite[Ch.VIII,\S2]{Sh:f}), an $NNR^0_\kappa$-forcing notion 
for ${\frak p},{\Cal I}_\beta$
a dense open subset of $Q$ for $\beta < \beta^* < \lambda$ and
${\underset\tilde {}\to S_i}$ is a $Q_i$-name of a stationary subset of
$\omega_1$ for $i < i^* < \alpha$, \ub{then} for some directed $G \subseteq Q$
we have: \nl
$\beta < \beta^* \Rightarrow G \cap {\Cal I}_\beta \ne \emptyset,
{\underset\tilde {}\to S_i}[G] = \{\gamma < \omega_1:(\exists r \in G)
(r \Vdash_Q ``\gamma \in {\underset\tilde {}\to S_i"})\}$ is a stationary
subset of $\omega_1$. \nl

Now we can deduce conclusions on preservation of being 
an $NNR^0_\kappa$-forcing notion for ${\frak p}$ and consistency of axioms.
\enddefinition
\bigskip

\demo{\stag{2.16} Conclusion}  1) if ${\frak p}$ is o.b. parameter,
non-$\Game'_{\text{id}}$-loser, $\ell g({\frak p}) = \omega_1$ and 
$\bar Q$ is an CS iteration such that $\alpha < \ell g(\bar Q) \Rightarrow
\Vdash_{P_\alpha} ``{\underset\tilde {}\to Q_\alpha}$ is an
$NNR^0_\kappa$-forcing notion for \nl
${\frak p}"$, \ub{then} $\bar Q$ is an
$NNR^0_\kappa$-iteration for ${\frak p}$ (note \scite{1.7} applies). \nl
2) Assume $CH + \mu = \mu^{< \mu} \ge \lambda$, if ${\frak p}$ is a
non$-\Game'_{\text{id}}$-loser o.b. parameter, $\chi^{\frak p}_0 > 2^\lambda$,
\ub{then} for some $\aleph_2$-e.c.c., (if $\lambda = \aleph_2,\aleph_2$-pic)
$NNR^0_\kappa$-forcing notion $P,|P| = \mu$ we have $\Vdash_P ``Ax_\lambda
({\frak p},\kappa,0)"$. \nl
3) The parallel of \scite{1.11} holds.
\enddemo
\bigskip

\demo{Proof}  Straight.
\enddemo
\newpage

\head {\S3 Examples: shooting a thin club} \endhead  \resetall \sectno=3
 % \resetall 
\bigskip

We would like here to see how some examples fit our framework.  We already
know that $(< \omega_1)$-proper forcings are ${\frak p}$-proper (by
\scite{1.12}).  We first deal with a forcing notion for which the
$\kappa$ (in $NNR^x_\kappa$) is $< \aleph_0$ (in Definition \scite{3.1}).
Second, we deal with shooting clubs of $\omega_1$ running away
from some $C_\delta \subseteq \delta = \sup(C_\delta)$ which are small (see
\cite[Ch.XVIII,\S1]{Sh:f}).  Those are the most natural non-$\omega$-proper
forcing notions not adding reals, (though this may depend on the set theory).
\bigskip

The prototype for \cite[Ch.VIII,\S4]{Sh:b}, \cite[Ch.VIII,\S4]{Sh:f} is
\scite{3.1}(1) below.
\definition{\stag{3.1} Definition}  1) Let $\bar C = \langle (C_\delta,
n_\delta):\delta < \omega_1 \text{ limit} \rangle$ where $C_\delta$ is an
unbounded subset of $\delta$ of order type $\omega$ and $1 \le n_\delta <
\omega$ (e.g. $n_\delta = 1$).  Let $\bar u = \langle u_\delta:\delta <
\omega_1 \text{ limit} \rangle$, where $u_\delta \in [2n_\delta +1]
^{n_\delta}$, (if $n_\delta =1$, then $u_\delta = \{k_\delta\},k_\delta \in
\{0,1,2\}$).  Then we define

$$
\align
Q = Q_{\bar C,\bar u} = \biggl\{ f:&\text{ for some } \alpha < \omega_1,f
\text{ is a function from } \alpha \\
  &\text{ to } \omega \text{ such that for every limit ordinal } \delta \le
\alpha, \\
  &\text{ for some }k < 2n_\delta +1,k \notin u_\delta \text{ and for every}\\
  &\, i \in C_\delta \text{ large enough we have } f(i) = k \biggr\}
\endalign
$$
\mn
ordered by inclusion. \nl
2) Assume $\bar C = \langle C_\delta:\delta < \omega_1 \text{ limit} \rangle,
C_\delta$ a closed subset of $\delta$ of order type $< \omega \times \delta$
and for $\delta_1 < \delta_2$ limit, sup$(C_{\delta_1} \cap C_{\delta_2}) <
\delta_1$, and for limit $\delta^*$ we have $\{C_\delta \cap \delta^*:\delta
< \omega_1\}$ is countable.  Assume further $\bar \kappa = \langle
\kappa_\delta:\delta < \omega_1 \text{ limit} \rangle,\kappa_\delta \in
\{2,3,\dotsc,\aleph_0\},\bar D = \langle D_\delta:\delta < \omega_1 \rangle,
D_\delta$ is a family of subsets of Dom$(D_\delta)$, such that the
intersection of any $< \kappa_\delta$ of them is non-empty.

Let $\bar f = \langle f_{\delta,x}:\delta < \omega_1,x \in
\text{ Dom}(D_\delta) \rangle$ satisfy $f_{\delta,x}:C_\delta
\rightarrow \omega$ and \nl
$\bar A = \langle A_\delta:\delta < \omega_1 \rangle$ satisfy
$A_\delta \in D_\delta$.  \ub{Then} we define 
$Q = Q_{\bar C,\bar D,\bar \kappa,\bar f,\bar A}$ as

$$
\align
\biggl\{f:&\text{ for some } \alpha < \omega_1,f \text{ is a function from }
\alpha \text{ to } \omega \\
  &\text{ such that for every limit } \delta \le \omega \text{ for some} \\
  &\, x \in A_\delta \text{ we have } f_{\delta,x} \subseteq^* f
\text{ i.e. for every large enough} \\
  &\, i \in C_\delta \text{ we have } f_{\delta,x}(i) = f(i) \biggr\}
\endalign
$$
\mn
ordered by inclusion.
\enddefinition
\bigskip

\proclaim{\stag{3.2} Claim}  1) The forcing notion $Q$ from \scite{3.1}(1)
is proper, does not add reals, and ia $\alpha$-proper for $\alpha <
\omega_1$ and even is $(< \omega_1)$-proper, also $\Bbb D$-complete for
some simple 2-completeness system, hence causes no problem to the demand (d) 
in Definition \scite{1.7} which means:
\mr
\item "{$(*)$}"   if $\bar Q$ is a CS iteration, $\ell g(\bar Q) = \alpha +1,
\bar Q \restriction \alpha$ is NNR$^0_2$-iteration and
$\Vdash_{P_\alpha} ``{\underset\tilde {}\to Q_\alpha} = Q_{\bar c,u}", \,\,
(\bar c,\bar u)$ as above $\in V$ \ub{then} $\bar Q$ satisfies demand ((a),
(b) and) (d)) of Definition \scite{1.7}.  
\ermn
2) Similarly for the forcing notion $Q$ from \scite{3.1}(2).
\endproclaim
\bigskip

\demo{Proof}  1) See \cite[Ch.VIII,\S4]{Sh:b} or \cite[Ch.VIII,\S4]{Sh:f}. \nl
2) Similarly.
\enddemo
\bigskip

\definition{\stag{3.3} Definition}  Assume $\bar C = \langle C_\delta:C_\delta
< \omega_1 \rangle,C_\delta$ an unbounded subset of the limit ordinal
$\delta$ (think of the case $C_\delta$ of order type $< \delta$ but not
necessarily).  Let

$$
\align
Q_{\bar C} = \biggl\{ c:&\text{ for some } \alpha < \omega_1,c
\text{ is a closed subset of } \alpha \\
  &\text{ and for every limit ordinal } \delta \le \alpha \text{ we have}: \\
  &\, \delta = \sup(c \cap \delta) \Rightarrow c \cap C_\delta \text{ bounded 
in } \delta \biggr\}
\endalign
$$

$$
c_1 < c_2 \text{ iff } c_1 \text{ is an initial segment of } c_2.
$$
\enddefinition
\bigskip

\remark{Remark}  1) See \cite[Ch.XVIII,\S1]{Sh:f}, note $Q_{\bar C}$ may be not
$\omega$-proper.  We first deal with the simple case: otp$(C_\delta) 
= \omega$. \nl
2) Note that $c \in Q_{\bar c}$ is a closed subset of $\alpha$, but not
necessarily a closed subset of $\omega_1$.
\endremark
\bigskip

\proclaim{\stag{3.4} Claim}  Assume ${\frak p}$ is a simple reasonable
parameter, $\bar C$ as in \scite{3.3}, $\dsize \bigwedge_\delta 
{\text {\rm otp\/}}(C_\delta) = \omega$, let $f \in {\Cal F}^{\frak p}$ 
be $f(0) = 0,f(\beta) = 1+ \beta$, \ub{then} $Q_{\bar C}$ is $({\frak p},
f)$-proper.
\endproclaim
\bn
We shall prove more in \scite{3.6} below.
\remark{\stag{3.5} Remark}  1) Note that if $\dsize \bigwedge_\delta
\text{ otp}(C_\delta) < \delta$, we can split the analysis by, for each
ordinal $\gamma < \omega_1$ such that $S_\gamma = 
\{\delta:\text{otp}(C_\delta) = \gamma\}$ is stationary, that is restricting 
ourselves to $\{N:N \cap \omega_1 \in S_\gamma\}$.
\endremark
\bigskip

\proclaim{\stag{3.6} Claim}  1) Assume
\mr
\item "{$(a)$}"  ${\frak p}$ is a simple reasonable parameter
\sn
\item "{$(b)$}"  ${\frak p}$ is non-$\Game_f$-loser, $f \in 
{\Cal F}^{\frak p}$
\sn
\item "{$(c)$}"  $\gamma(*) < \omega_1$
\sn
\item "{$(d)$}"  $\bar C = \langle C_\delta:\delta < \omega_1 \rangle,
C_\delta \subseteq \delta = \sup(C_\delta)$
\sn
\item "{$(e)$}"  {\rm otp}$(C_\delta) \le \omega^{\gamma(*)}$ for every
$\delta$
\ermn
Define $g \in {\Cal F}^{\frak p}$ by $g(0) = 0,g(1) = f(1) + \gamma(*),
g(\alpha +1) = f(g(\alpha)) + \gamma(*) +1$ (and necessarily, for limit
$\alpha,g(\alpha) = \underset{\beta < \alpha} {}\to \sup g(\beta)$). \nl
\ub{Then} for every $\alpha < \ell g({\frak p})$, the forcing notion
$Q = Q_{\bar C}$ is $({\frak p},\alpha,g(\alpha))$-proper (version 1). \nl
2) We can in part (1) get ``version 2" when: if $g(\delta) = \delta,N \in
{\Cal E}_\alpha,C \subseteq \omega_1 \cap N = \sup(C)$, {\rm otp}$(C) \le 
\omega^{\gamma(*)}$ (for example $C = C_\alpha \cap N$ for some $\alpha$)
and $Y \in D_\alpha(N)$ \ub{then} 
$Y' = \{M \in Y:M \cap \omega_1 \notin C\} \in D_\alpha(N)$, which is 
a mild condition. \nl
3) If we weaken clause $(b)$ to $(b)_{f,g}$ below, \ub{then} for 
$\alpha < \ell g({\frak p})$, the forcing notion $Q_{\bar C}$ is 
$({\frak p},\alpha,f(\gamma + \alpha))$-proper, where \nl
$(b)_{f,g} \, f \in {\Cal F}^{\frak p}$ and $f(f(\alpha)) =
f(\alpha)$ for every $\alpha < \ell g({\frak p})$ and 
${\frak p}$ is a non-$\Game'_f$-loser.
\endproclaim
\bigskip

\demo{Proof}  First observe
\mr
\item "{$(*)$}"  for each $\alpha < \omega_1$ the set ${\Cal I}^*_\alpha =
\{p \in Q_{\bar C}:\text{ there is } \beta \in p 
\text{ which is } \ge \alpha\}$ is a dense subset of $Q$.
\ermn
1) Similar to the proof of 2) noting version (2) is harder on the chooser,
and using the extra freedom we can avoid the need for the extra 
assumption from (2). \nl
2) We prove this by induction on $\alpha$, let $\beta = g(\alpha)$.

Let $N \in {\Cal E}_\beta$ be countable, 
$p \in N \cap Q_{\bar C}$ (and $Q_{\bar C},\alpha,
\beta \in N$) and $Y \in D_\beta(N)$ be given.  Let $\delta = \delta_N =: 
N \cap \omega_1$.
\enddemo
\bn
\ub{Case 1}:  $\alpha = 0$.

Comprehending the demand in Definition \scite{2.8}(1) it just means 
``$Q_{\bar C}$ is proper".  Let
$\langle {\Cal I}_n:n < \omega \rangle$ list the dense open subsets of
$Q_{\bar C}$ that belongs to $N$ and we shall define by induction on 
$n$, a condition $p_n$ such that
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  $p_0 = p,p_n \in N$
\sn
\item "{$(ii)$}"  $p_n \le p_{n+1} \in {\Cal I}_n$
\sn
\item "{$(iii)$}"  the set $p_{n+1} \cup \{\sup p_{n+1}\} \backslash
(p_n \cup \{\sup p_n\})$ is disjoint to $C_\delta$.
\ermn
If we succeed to carry out the induction clearly we are done as 
$\dbcu_{n < \omega} p_n$ is as required, noting for each $\alpha < N \cap
\omega_1,{\Cal I}^*_\alpha \in \{{\Cal I}_n:n < \omega\}$, hence 
$(\exists \beta \in (\dbcu_{n < \omega} p_n))[\alpha \le \beta < \omega_1]$.  
Also there is no problem to choose
$p_0$.  So assume $p_n$ has been chosen and we shall choose $p_{n+1}$ as
required.  There is a function $F_n \in N$ with domain $Q_{\bar C}$ such that:
$q \in Q_{\bar C} \Rightarrow q \le F_n(q) \in {\Cal I}_n$.

For $\alpha < \omega_1$ let $q^{[\alpha]} = q \cup \{\sup(q)\} \cup 
\{\sup(q) + 1 + \alpha\}$, so clearly $q \le q^{[\alpha]} \in Q_{\bar C}$ 
and the function $(q,\alpha) \mapsto q^{[\alpha]}$ belongs to $N$.  Define 
a function $H:\omega_1 \rightarrow \omega_1$ by $H(\alpha) = 
\sup(F_n(p^{[\alpha]}_n))$,
clearly it is well defined, belongs to $N$ and let $C =: \{\beta < \omega_1:
\beta \text{ a limit ordinal, moreover } \omega \beta = \beta \text{ and }
(\forall \alpha < \beta)(H(\alpha) < \beta) \text{ and sup}(p_n) < \beta\}$,
so $C$ is a club of $\omega_1$ which belongs to $N$ and $\gamma(*) \in N$ 
hence we can find $\beta^* \in C$ such that otp$(\beta^* \cap C)$ is 
divisible by $\omega^{\gamma(*)}$, but
otp$(C_\delta \cap \beta^*) < \omega^{\gamma(*)}$ hence for some $\beta \in C$
we have sup$(C_\delta \cap \beta) < \beta$.  Let $p_{n+1} = F_n
(p^{[\sup(C_\delta \cap \beta)+1]})$; it is as required.
\bn
\ub{Case 2}:  $\alpha =1$.

Easily $Y' =: \{M \in Y:M \cap \omega_1 \notin C_\delta\} \in 
D_{g(1)}(N)$ as $\beta \ge g(1) = f(1) + \gamma(*)$, just prove this by
induction on $\gamma(*)$.  Let
$\langle {\Cal I}_n:n < \omega \rangle$ list the dense open subsets of
$Q_{\bar C}$ which belong to $N$ and let $\delta = \dbcu_{n < \omega} 
\alpha_n$ and $\alpha_n < \alpha_{n+1} < \delta$.  We now simulate a strategy 
for the challenger in the game $\Game_{\alpha,\beta}(N)$, together with 
choosing in the $n$-th move, (in the end of it) also the challenger chooses
$p_{n+1} \in Q_{\bar C} \cap N$ such that $p_0 = p,p_n \le p_{n+1} \in
{\Cal I}_n$ and during the play letting $Z_n = \emptyset$ 
(in fact also the chooser has to use $Y_n = \emptyset$) and $p_{n+1}$ is
$(M_n,Q_{\bar C})$-generic and sup$(p_{n+1}) > \alpha_n$ and the set
$(p_{n+1} \cup \{\sup p_{n+1}\}) \backslash (p_n \cup \{\sup p_n\})$ 
is disjoint to
$C_\delta$.  This is possible by Case 1 and its proof because $M_n \cap
\omega_1 \notin C_\delta$ which holds as $M_n \in Y'$.  As this is a legal
strategy for the challenger, so it cannot be a winning strategy hence for
some such play the chooser wins hence $\{M_n:n < \omega\} \in D_1(N)$, 
remember $\alpha=1$.  Now
$q = \dbcu_{n < \omega} p_n$ is well defined, and sup$(q) = \delta$ and
$q \cap C_\delta \subseteq p \cup \{\sup(p)\}$ and $q \Vdash_{Q_{\bar C}} 
``\{M_n:n < \omega\} \subseteq {\Cal M}
[{\underset\tilde {}\to G_{Q_{\bar C}}},N]"$, so $q$ is as required as the
chooser has won the play.
\bn
\ub{Case 3}:  $\alpha > 1,\alpha$ successor.

Similar to Case 2 only we use the induction hypothesis instead of using
Case 1.
\bn
\ub{Case 4}:  $\alpha$ a limit ordinal. \nl
Similar to Case 2. \nl
Easy.  \hfill$\square_{\scite{3.6}}$
\bigskip

\definition{\stag{3.7} Definition}  1) Assume
\mr
\item "{$(a)$}"  $S \subseteq \omega_1$ is stationary
\sn
\item "{$(b)$}"  $f \in {}^{\omega_1}(\omega_1)$
\sn
\item "{$(c)$}"  $\bar C = \langle C_\delta:\delta < \omega_1 \rangle$
\sn
\item "{$(d)$}"  $C_\delta$ is an unbounded subset of $\delta$.
\ermn
We define when ``$\bar C$ obeys $f$ on $S$" for $f \in {}^{\omega_1}\omega_1$
a $({\Cal D}_{\omega_1} + S,\gamma)$-th function (see part (2)) by induction 
on $\gamma$.  For $f$ being $({\Cal D}_{\omega_1} + S,\gamma)$-function, 
$\gamma < \omega_1$ this 
means $\{\delta \in S:\text{ otp}(C_\delta) \le \omega^{1+f(\delta)}\}
= S$ mod ${\Cal D}_{\omega_1}$.  In general it means that for some $g:
\omega_1 \rightarrow \omega_1$ and pressing down function on $h$ on $S$, for
every $\zeta < \omega_1$ for which $h^{-1}\{\zeta\}$ is stationary,
for some $\beta < \gamma$ and $f_\beta$, a
$({\Cal D} + h^{-1}\{\zeta\},\beta)$-th function, we have 
$\langle C_{g(\delta)} \cap
\delta:\delta \in h^{-1}\{\zeta\} \rangle$ obeys $f_\beta$. \nl
2) We say $f \in {}^{\omega_1}(\omega_1)$ is a $({\Cal D}_{\omega_1} + S,
\gamma)$-th function when: 
$S \Vdash_{({\Cal D}^+_{\omega_1},\supseteq)}$ ``in 
$V[\underset\tilde {}\to G]$, \nl
$\{x \in V^{\omega_1}/\underset\tilde {}\to G:
V^{\omega_1}/G \models x$ an ordinal $< f/ \underset\tilde {}\to G\}$ 
has order type $\gamma$".
\enddefinition
\bigskip

\proclaim{\stag{3.8} Claim}  Assume
\mr
\item "{$(a)$}"  ${\frak p}$ is a simple reasonable parameter,
$\ell g({\frak p})$ of uncountable cofinality
\sn
\item "{$(b)$}"  $S \in {\Cal D}^+_{\omega_1}$ and $N \in \dbcu_\alpha 
{\Cal E}^{\frak p}_\alpha \Rightarrow N \cap \omega_1 \in S$
\sn
\item "{$(c)$}"  ${\frak p}$ is a non-$\Game_{\alpha,\alpha}$-loser 
(or just non $\Game'_{\alpha,\alpha}$-loser) for all
$\alpha \in C^*,C^*$ a club of $\ell g({\frak p})$,
\sn
\item "{$(d)$}"  $\bar C$ obeys $f$ on $S$ which is a
$({\Cal D}_{\omega_1} +S,\gamma)$-function
\sn
\item "{$(e)$}"  $0 < \text{ Min}(C),g(\alpha) = \text{ Min}
(C^* \backslash \alpha)$.
\ermn
\ub{Then} $Q_{\bar C}$ is $({\frak p},g)$-proper.
\endproclaim
\bigskip

\demo{Proof}  Similar.  
\enddemo
\bigskip

\centerline {$* \qquad * \qquad *$}
\bigskip

Another example (which could have been done in \cite[Ch.XVIII]{Sh:f}) is:
\definition{\stag{3.9} Definition}  1) We say $\bar{\Cal D} = 
\langle {\Cal D}_\delta:\delta < \omega_1 \text{ limit} \rangle$ is an 
$\omega_1$-filter-sequence if:
\mr
\item "{$(a)$}"  ${\Cal D}_\delta$ is a filter on $\delta$, containing the
co-bounded subsets
\sn
\item "{$(b)$}"  ${\Cal D}_\delta$ is a $P$-filter and 
some $C_\delta \in D_\delta$ has order type $\omega$ ($P$-filter means that
${\Cal D}_\delta$ contains all co-finite sets and if $A_n \in D_\delta$ 
for $n < \omega$
then for some $A \in D_\delta$ we have 
$n < \omega \Rightarrow |A \backslash A_n| < \aleph_0$) 
(can generalize as in \scite{3.1}(2) but did not)
\sn
\item "{$(c)$}"  for every club $C \subseteq \omega_1$ and $\alpha < 
\omega_1$, the set $A^\alpha_C[\bar{\Cal D}]$ is stationary, where, 
by induction on $\alpha$ we define: \nl
$A^\alpha_C[\bar{\Cal D}] = \{\delta < \omega_1:\delta 
\text{ is a limit ordinal, } \delta \in C 
\text{ and for every } \beta < \alpha \text{ we have}$ \nl
$\delta = \sup(\delta \cap A^\beta_C[\bar{\Cal D}])$, moreover 
$\delta \cap A^\beta_C[\bar{\Cal D}] \in {\Cal D}_\delta\}$.
\ermn
2) We say a reasonable parameter ${\frak p}$ obeys $\bar{\Cal D}$ \ub{if} 
for each $\alpha < \ell g({\frak p})$ and $N \in {\Cal E}_\alpha$ we have:
$\bar{\Cal D} \in N$ and we have

$$
\align
{\Cal D}^{\frak p}_\alpha(N) = \biggl\{Y:&Y \subseteq N \cap
\dbcu_{\beta < \alpha} {\Cal E}_\beta \text{ is closed (see \scite{2.11})
and if} \\
  &\alpha > 0 \text{ then there are} \\
  &\bar \beta = \langle \beta_n:n < \omega \rangle,\bar M =
\langle M_n:n < \omega \rangle \text{ satisfying} \\
  &(a) \quad \beta_n \in N \cap \alpha \text{ and} \\
  &(b) \quad \text{ either } \dsize \bigwedge_n [\alpha = \beta_n +1] \\
  &\qquad \text{ \ub{or} } \beta_n < \beta_{n+1},
          \sup_{n < \omega} \beta_n = \sup(\alpha \cap N) \\
  &(c) \quad M_n \in Y \cap {\Cal E}_{\beta_n},M_n \in M_{n+1}, \\
  &\qquad \dbcu_{n < \omega} M_n = N \cap \dbcu_{\beta \in \alpha \cap N} 
          {\Cal H}(\chi_\beta) \\
  &(d) \quad \{M_n \cap \omega_1:n < \omega\} \in {\Cal D}_{N \cap \omega_1}
 \biggr\}.
\endalign
$$
\mn
3) We say a forcing notion $Q$ is a $\bar{\Cal D}-NNR^0_\kappa$-forcing 
if for 
every reasonable parameter ${\frak p}$ which obeys $\bar{\Cal D}$ 
we have: $Q$ is an $NNR^0_\kappa$-forcing over ${\frak p}$
(see \scite{2.15}). \nl
4) For a $P$-filter ${\Cal D}$, we say a reasonable parameter 
${\frak p}$ obeys ${\Cal D}$ if: \nl
for every $N \in{\Cal E}_\alpha$

$$
\align
D^{\frak p}_\alpha(N) = \biggl\{ Y:&Y \subseteq N \cap \dbcu_{\beta < \alpha}
{\Cal E}_\beta \text{ is closed (see \scite{2.11}) and if } \alpha > 0
\text{ then} \\
  &\text{for some } \bar \beta,\bar M \text{ satisfying clauses } 
(a), (b), (c) \\
  &\text{ of part } (2) \text{ and} \\
  &(d)' \qquad \{n:M_n \in Y\} \in {\Cal D} \\
  &\qquad \qquad (*) {\Cal D} \text{ is a } P \text{ -filter on } 
\omega \biggr\}.
\endalign
$$
\mn
5) In the parts 1) - 4) above we may replace the word filter by ultrafilter
if the ${\Cal D}$'s are ultrafilter (so we have ``$\omega_1$-ultrafilter
sequence").
\enddefinition
\bigskip

\proclaim{\stag{3.10} Claim}  1) If $\diamondsuit_{\aleph_1}$ (or much less),
\ub{then} there is an $\omega_1$-ultrafilter sequence. \nl
2) If $\bar{\Cal D}$ is an $\omega_1$-filter sequence and
$\langle (\chi_\alpha,{\Cal E}_\alpha):\alpha < \alpha^* \rangle$ is $a$ in
Definition \scite{1.1}, \ub{then} there is a
reasonable parameter ${\frak p}$ of length $\omega_1$ obeying $\bar{\Cal D}$ 
which is a non-$\Game$-loser $\chi^{\frak p}_\alpha = \chi_\alpha,
{\Cal E}^{\frak p}_\alpha = {\Cal E}_\alpha$. \nl
3) If $\diamondsuit_{\aleph_1}$ and $\langle (\chi_\alpha,{\Cal E}_\alpha):
\alpha < \alpha^* \rangle$ is as in Definition \scite{1.1} and ${\Cal D}$ is
a $P$-filter on $\omega$, \ub{then} some
reasonable parameter ${\frak p}$ is $P$-filter like, 
non-$\Game_{\text{id}}$-loser with $\chi^{\frak p}_\alpha = \chi_\alpha,
{\Cal E}^{\frak p}_\alpha = {\Cal E}_\alpha,\alpha^{*,{\frak p}} = \alpha^*$.
Similarly for ultrafilters.
\nl
4) Instead of $\diamondsuit_{\aleph_1}$ it is enough to assume CH and
that for some $\langle C_\delta:\delta < \omega_1 \text{ limit} \rangle$ 
and normal filter $D$ on $\omega_1$ and for every club $C$ of $\omega_1,
\{\delta:\delta > \sup (C_\delta \backslash C)\} \in D$.
\endproclaim
\bigskip

\demo{Proof}  Straight (in part (4), we define the $C_\delta$'s
simultaneously).
\enddemo
\bigskip

\proclaim{\stag{3.11} Claim}  1) If ${\frak p}$ is a reasonable parameter
obeying ${\Cal D}$, a $P$-filter on $\omega$ [or $P$-ultrafilter on
$\omega$], \ub{then} for some $\bar{\Cal D},
\bar{\Cal D}$ is an $\omega_1$-filter-sequence [or $\omega_1$-ultrafilter-
sequence] and ${\frak p}$ obeys $\bar{\Cal D}$. \nl
2) If ${\frak p}$ is a $P$-point filter (or ultrafilter), 
then ${\frak p}$ is a non-$\Game$-loser.
\endproclaim
\bigskip

\demo{Proof}  Should be clear.
\enddemo
\bigskip

\remark{\stag{3.12} Remark}  We may consider higher order types than
$\omega$.
\endremark
\bigskip

\proclaim{\stag{3.13} Claim}  1) Assume
\mr
\item "{$(a)$}"  $\bar C = \langle C_{\delta,\ell}:\ell < k_\delta,\delta <
\omega_1 \text{ limit}\rangle,1 + \kappa \le k_\delta \le \omega,
C_{\delta,\ell}$ is a closed unbounded subset of $\delta$ and 
$\ell < m < k_\delta \Rightarrow C_{\delta,\ell} \cap C_{\delta,m} 
= \emptyset$ 
\sn
\item "{$(b)$}"  $Q = Q_{\bar C} = \{C:C$ is a closed bounded subset of
$\omega_1$ such that for every limit $\delta < \sup(C)$ and for every $\ell
< k_\delta$ except $< 1 + \kappa$ many, $\delta > \sup
(C \cap C_{\delta,\ell})\}$
\sn
\item "{$(c)$}"  ${\frak p}$ obeys the $P$-ultrafilter ${\Cal D}$ and is a
reasonable parameter.
\ermn
\ub{Then} $Q$ is a ${\frak p}-NNR^0_\kappa$ forcing notion. \nl
2) In part (1) if we add: $N \in {\Cal E}_\alpha \and D^{\frak p}_\alpha(N) =
\{Y:\{n:M_n \in Y\} \in {\Cal D}_N\},{\Cal D}_N$ a $P$-point ultrafilter 
(see above) and $\ell < k_\delta \Rightarrow \{n < \omega:M_n \cap \omega_1
\in C_{\delta,\ell}\} = \emptyset \text{ mod } {\Cal D}_N$, 
\ub{then} we can allow $C_{\delta,0} = C_{\delta,1}$. \nl
3) Assume
\mr
\item "{$(a)$}"  $D_\delta$ is a family of subsets of Dom$(D_\delta)$, the
intersection $Y$ of any $<1 + \kappa$ of them satisfies
{\roster
\itemitem{ $(*)$ }  $\exists n(\exists y_1,\dotsc,y_n \in Y)[\delta > \sup
(\dbca^n_{\ell =1} C_{\delta,y_\ell})]$
\endroster}
\sn
\item "{$(b)$}"  $\bar C = \langle C_{\delta,x}:x \in \text{ Dom}(D_\delta)$
and $\delta$ is a limit ordinal $< \omega_1 \rangle$
\sn
\item "{$(c)$}"  $\langle C_{\delta,x}:x \in \text{ Dom}(D_\delta) \rangle$
is a sequence of pairwise disjoint subsets of $\delta$
\sn
\item "{$(d)$}"  $\bar X \in \dsize \prod_{\delta < \omega_1} \text{Dom}
(D_\delta)$
\sn
\item "{$(e)$}"  $Q_{\bar C,\bar X,\bar D} = \{C:C$ is a closed bounded
subset of $\omega_1$ such that for every limit $\delta \le \sup(C)$ we have
$(\exists x \in X_\delta)(\delta > \sup(C \cap C_{\delta,x}))\}$ \nl
ordered by:  end extension
\sn
\item "{$(f)$}"  $\bar{\Cal D}$ is a $P$-ultrafilter sequence.
\ermn
\ub{Then} $Q$ is a ${\frak p}-NNR^0_\kappa$ forcing notion ${\frak p}$
which obeys $\bar{\Cal D}$. 
\endproclaim
\bigskip

\demo{Proof}  Straight. \nl
1) So let $V = V^{P_\alpha}_0,\Vdash_{P_\alpha} ``Q_\alpha =
Q_{\underset\tilde {}\to {\bar C}}$ is as above" and in $V_0,N_0 \in
{\Cal E}_0,N_0 \in N_1 \in {\Cal E}_\alpha = \{M_n:n < \omega\},D_{N_0}$ as
in Definition \scite{3.13}(2), $p \in P_{\alpha +1} \cap N,G_m \subseteq
N_1 \cap P_\alpha$ is generic over $N_1$ for 
$m < k < 1 + \kappa$ (so $k$ is now fixed).  
So for each $\ell$ we have ${\underset\tilde {}\to C_{N_0 \cap \omega_1,\ell}}
[G_m]$ is a closed subset of $\delta$ and for $\ell_1 < \ell_2 <
k_{\delta_{N_0}}$ we have 
${\underset\tilde {}\to C_{N_0 \cap \omega_1,\ell_1}}[G_m]
\cap {\underset\tilde {}\to C_{N_0 \cap \omega_1,\ell_2}}[G_m] = \emptyset$
so for some $\ell(m) \in \{0,1,\dotsc,k_{\delta_{N_0}}-1\}$ we have 
$\ell \ne \ell(m) \Rightarrow
{\underset\tilde {}\to C_{N_0 \cap \omega_1,\ell}}[G_m] = \emptyset$ mod
$D_{N_0}$.  Now let $B = \{n:$if $\ell < k_\delta,\ell \notin \{\ell(m):
m < k\}$ and $\ell < n$ then $M_n \cap \omega_1 \notin
{\underset\tilde {}\to C_{N_0 \cap \omega_1,\ell}}[G_m]$ for $m < k$ and
$p \in M_n\}$ belongs to $D_{N_0}$.  Clearly, \wilog \, $\langle M_n:n <
\omega \rangle \in N_1,D_{N_0} \in N_1,B \in N_1$.  Let $B = \{n_i:i <
\omega\},n_i$ increasing with $i$ and $\langle {\Cal I}_n:n < \omega 
\rangle$ list the dense open subsets of $P_{\alpha +1}$ which belongs 
to $N_0$.  We can choose $p_i$ by induction on $i < \omega$ such that:

$$
p_i \in N_{n_i} \cap P_{\alpha +1}
$$

$$
p_i \restriction \alpha \in \dbca_{m < k} G_m
$$

$$
p_i \in \bigcap \{{\Cal I}_n:
n < n_i,{\Cal I}_n \in N_{n_i} \text{ and } i > 0\}
$$

$$
p \le p_i
$$

$$
p_i \le p_{i+1}
$$

$$
p_{i+1} \backslash p_i \text{ is disjoint to } \cup
\{{\underset\tilde {}\to C_{N_0 \cap \omega_1,\ell}}[G_m]:\ell < k_0,\ell <
n_i \text{ and } m < k \Rightarrow \ell \ne \ell(m)\}.
$$
\mn
This is possible as
\mr
\item "{$\bigotimes$}"  $\biggl( \dsize \bigcup \Sb m < k \\
\ell < k \and \ell \ne \ell(m) \endSb
{\underset\tilde {}\to C_{N_0 \cap \omega_1,\ell}}[G_m] \biggr) \cap
\omega_1 \cap M_n \text{ is a bounded subset of } M_n \cap \omega_1$.
\ermn
2), 3), 4)  Similarly. \hfill$\square_{\scite{3.13}}$
\enddemo
\bigskip

\remark{\stag{3.14} Remark}  1) The forcing notion from \scite{3.13}(1) is
$NNR^0_{\aleph_0}$-forcing notion for every ${\frak p}$,
non-$\Game_{\text{id}}$-loser. \nl
2) If ${\frak p}$ is a reasonable parameter obeying $\bar{\Cal D}$, an
$\omega_1$-filter sequence, then any $(< \omega_1)$-proper forcing is
${\frak p}$-proper.
\endremark
\bigskip

\demo{Proof}   Similar to earlier proofs.
\enddemo
\newpage

\head {\S4 Second preservation of not adding reals} \endhead  \resetall \sectno=4
 % \resetall 
\bigskip

We shall concentrate on the simple case.
\definition{\stag{4.1} Definition}  Let ${\frak p}$ be a reasonable
parameter.  We say that $\bar Q$ is a ${\frak p}-NNR^1_\kappa$ 
iteration (where $2 \le \kappa \le \aleph_0$ as we omit
$\kappa = \aleph_1$ for convenience) if:
\mr
\item "{$(a)$}"  cf$(\ell g({\frak p})) > \ell g(\bar Q)$ (for simplicity)
\sn
\item "{$(b)$}"  $\bar Q$ is a countable support iteration of proper forcing
notions \nl
[in \cite{Sh:311} we will 
relax the requirement that each iterand is proper] \nl
such that $i < \ell g(\bar Q) \Rightarrow P_i$ adds no reals \nl
(follows by other parts (close (d)) even for $P_{i+1},i < \ell g(\bar Q)$
\sn
\item "{$(c)$}"  [long properness]
for each $i$, for some $f_i \in {\Cal F}^{\frak p}_{\text{club}}$ 
we have $\Vdash_{P_i} ``{\underset\tilde {}\to Q_i}$ is
$({\frak p},f_i)$-proper, $f_i$ increasing continuous, see Definition
\scite{2.8}(2) + \scite{2.8}(4)
\sn
\item "{$(d)$}"  [$\kappa$-anti w.d.]  if 
$f_i(\alpha) \le \beta < \ell g({\frak p}),\{\bar Q,i,
\alpha,\beta\} \in N_0 \in {\Cal E}_\alpha,N_0 \in N_1 \in {\Cal E}_\beta,
k < 1 + \kappa,j = i + 1$ and for $\ell =0,1,\dotsc,k-1$ we have $q_\ell 
\in P_i$ is $(N_1,P_i)$-generic and is $(N_0,P_i)$-generic, $q_\ell 
\Vdash ``{\underset\tilde {}\to G_{P_i}} \cap N_1 = G^\ell",G^\ell \cap
N_0 = G^*,p^* \in P_j \cap N_0$ satisfies $p^* \restriction i \in
G^{**}$ and lastly $Y =: \dbca_{\ell < k} {\Cal M}[G^\ell,N_1,y^*] \in 
D_\beta(N_1)$, \ub{then} for some $G^{**} \in N_1,G^{**} \subseteq N_0 
\cap P_j$ is generic over $N_0,G^* \subseteq G^{**}$ and $p^* \in G^{**}$ and
$\dsize \bigwedge_{\ell <k} \, \dsize \bigvee_{r \in G^\ell} r 
\Vdash_{P_i} ``G^{**} \restriction
{\underset\tilde {}\to Q_i} (= \{p(i):p \in G^{**}\})$ has an upper bound
in ${\underset\tilde {}\to Q_i}"$.
\endroster
\enddefinition
\bigskip

\remark{\stag{4.2} Remark}  1) $\kappa$ is the amount of 
``$\Bbb D$-completeness", in other words what versions of weak diamond 
we kill by our iteration.
\mn
2) \ub{Question}:  Why do we ask for $f_i \in V$ and not a $P_i$-name
${\underset\tilde {}\to f_i}$ of such a function?
\sn
\ub{Answer}:  If $i < \ell g(\bar Q) \Rightarrow P_i \models 
\text{ cf}(\ell g({\frak p}))$-c.c., then we can find $f'_i \in 
{\Cal F}^{\frak p},f_i \ge {\underset\tilde {}\to f_i}$; so as we are
assuming, for conveniency cf$(\ell g({\frak p})) > |P_\alpha|$ there is
no point at present for $f_i$ to be a $P_i$-name. \nl
2A) In clause (d) we have implicitly used:
\mr
\item "{$(*)$}"  if $\alpha \le \beta' < \beta$ then clause (d) for
$(\alpha,\beta)$ and $k$ implies clause (d) for $(\alpha,\beta')$ and $k$.
\ermn
This holds by clause (i) of Definition \scite{1.1}. \nl
We could replace $f_i$ by a club $E_i$ of $\ell g({\frak p})$, letting
$f_i(\alpha) = E_i \backslash \alpha$. \nl
3) No real reason to ``use some $\kappa$", but also no real 
reason to use 2.  For each $\ell$ to find $G^{**}$ means essentially 
``forcing with $Q_i$ adds no reals".  The point is the common solution. \nl
4) In clause (c) for a club $C$ of $\ell g({\frak p})$ we
catch our tail, that is $f_i(\alpha) \cap C = C \backslash \alpha$ for a club of
$\alpha < \ell g({\frak p})$.  We could use less, no real point
now. \nl
5) In clause (d) much of the freedom/variation will be due to the 
decision how ``similar" are $\langle G^\ell:\ell < k \rangle$ such that 
$G^{**}$ exists.  Here we demand
\mr
\item "{$(\alpha)$}"  $Y \in D_\beta(N_1)$.
\ermn 
In \cite[Ch.VIII,\S2]{Sh:f} it is essentially required that
\mr
\item "{$(\beta)$}"  $G^0 \times G^1 \times \cdots \times G^{k-1} \subseteq
(P_i \times \cdots \times P_i) \cap N_1$ ($k$ times) is \nl
generic over $N_1$.
\ermn
In \cite[Ch.V]{Sh:f}
\mr
\item "{$(\gamma)$}"  the common $Y$ is a predetermined increasing 
sequence of models.
\ermn
We have a trade-off; clause $(\beta)$ makes demand (d) in
\scite{4.1} easier, but the parallel of (c) harder compared to clause 
$(\alpha)$. \nl
This explains why the present work doesn't supercede Ch.XVIII,\S2. \nl
6) It is harder to win with a ``slower" function, so the
assumption above is the strongest possible - although practically makes no
difference, probably.
\endremark
\bigskip

\proclaim{\stag{4.3} Claim}  Let $\bar Q$ be a ${\frak p}-NNR^1_\kappa$
iteration (see Definition \scite{4.1}), $\kappa \in \{2,\aleph_0\}$ and
${\frak p}$ is a reasonable parameter and ${\frak p}$ is a
$\Game_f$-winner for some $f \in {\Cal F}^{\frak p}_{\text{club}}$ or at least
$\Game'_f$-non loser. \nl
1) Forcing with $P_{\ell g(\bar Q)} = \text{ Lim}(\bar Q)$ 
does not add reals (so
consequently no $\omega$-sequences, as we are assuming properness). \nl
2) If $i \le j \le \ell g(\bar Q)$, \ub{then}
\mr
\item "{$(b)'$}"  $P_j/P_i$ is proper
\sn
\item "{$(c)'$}"  $P_j/P_i$ is $({\frak p},f_{i,j})$-proper, where \nl
$f_{i,j} \in {\Cal F}^{\frak p}_{\text{club}}$ is increasing continuous and
is computable from the
$f_\varepsilon \in {\Cal F}^{\frak p}$ for $\varepsilon \in [i,j)$, 
\sn
\item "{$(d)'$}"  we have the parallel of clause (d), this time 
without assuming that $j = i+1$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  By induction on $\ell g(\bar Q)$.  For notational simplicity
we assume
\mr
\item "{$\boxtimes$}"  all $f_i$ are even in ${\Cal F}^{\frak p}_{\text{end}}$
so consider them as functions from $\ell g({\frak p})$ to $\ell g({\frak p})$
(increasing continuous), and we demand also the $f_{i,j}$ are like that,
increasing continuous and moreover $f_{i,j}(f_{i,j}(\alpha)) = f_{i,j}
(\alpha)$ and they are $\ge f^*,f^* \in {\Cal F}^{\frak p}_{\text{end}}$
increasing continuous, ${\frak p}$ is $\Game_{f^*}$-winner or at least
$\Game'_{f^*}$-non-loser.
\endroster
\enddemo
\bn
\ub{Case 1}:  \ub{$\ell g(\bar Q) = 0$}

Trivial.
\bn
\ub{Case 2}:  \ub{$\ell g(\bar Q) = i(*) + 1$}
\mn
\ub{Part (1)}:  $P_{i(*)}$ adds no reals by the induction hypothesis,
${\underset\tilde {}\to Q_{i(*)}}$ adds no reals by clause (d) in
Definition \scite{4.1}, hence $P_{i(*)+1} = P_{i(*)} * 
{\underset\tilde {}\to Q_{i(*)}}$ adds no reals.
\mn
\ub{Part (2)}:  Clause $(b)'$ is known, (namely $P_j/P_i$ is proper).
\mn
\ub{Clause $(c)'$}:

Given $i \le j \le \ell g(\bar Q)$ if $j < i(*)+1$ the conclusion follows
by the induction hypothesis.  So assume
$j =i(*) +1$.  If $i=j$, the required demand is trivial, so assume $i < j$.  
If $i=i(*)$, use
clause (c) of Definition \scite{4.1} for $i$.  So assume $i < i(*)$.  Let
$f_{i,j,0} = f_{i(*)},f_{i,j,m+1} = f_{i(*)} \circ f_{i,j(*)} \circ
f_{i,j,m}$ and $f_{i,j}(\alpha) = \underset{m < \omega} {}\to \sup \,
f_{i,j,m}(\alpha)$.  Check that they are as required in $\boxtimes$. For
proving ``$P_j/P_i$ is $({\frak p},f_{i,j})$-proper" assume
\mr
\item "{$(a)$}"   $N \prec ({\Cal H}(\chi),\in)$ is countable
\sn
\item "{$(b)$}"  $\{\bar Q,i,j,\alpha,\beta,f_{i,i(*)},f_{i(*)},
f_{i,j}\} \in N$
\sn
\item "{$(c)$}"  $\alpha \le f_{i,j}(\alpha) \le \beta < \ell g({\frak p})$
\sn
\item "{$(d)$}"  $q \in P_i \text{ is } (N,P_i) \text{-generic}$
\sn
\item "{$(e)$}"  $p \in N \cap P_j,p \restriction i \le q$
\sn
\item "{$(f)$}"  $Y \in D_\beta(N)$
\sn
\item "{$(g)$}"  $q \Vdash 
``Y \subseteq {\Cal M}[{\underset\tilde {}\to G_{P_i}},N]"$.
\ermn
First we deal with version 2 and assume that ${\frak p}$ is simple.
Choose $y^* \in N$ which codes enough; clearly 
$\beta' =: f_{i(*)}(\alpha)$ belongs to $N$.  So 
$f_{i,i(*)}(\beta') \le \beta$, hence
by the induction hypothesis there are $q',Y'$ such that:
$p \restriction i(*) \le q,q \le q' \in P_{i(*)},q'$ is 
$(N,P_{i(*)})$-generic, $Y' \subseteq Y,Y'\in D_{\beta'}(N)$ and 
$q' \Vdash ``Y' \subseteq {\Cal M}
[{\underset\tilde {}\to G_{P_i(*)}},N,y^*]"$. 
\mn
Next, we apply clause (c) in the Definition \scite{4.1} 
for $i(*)$, so there are $q'',Y''$ such that $p \le q'',
q' \le q'' \in P_{i(*)+1} = P_j,q''$ is $(N,P_j)$-generic,
$Y'' \subseteq Y',Y'' \in D_\alpha(N)$ and $q'' \Vdash ``Y \subseteq {\Cal M}
[{\underset\tilde {}\to G_{P_j}},N,y^*]"$.

The proof for version 1 is similar.
\mn
\ub{Clause $(d)'$}: \nl

Recall that we have demanded 
$f_{i,j}(f_{i,j}(\alpha)) = f_{i,j}(\alpha)$ (see $\boxtimes$ at the 
beginning of the proof).

Let $N_0,N_1,G^*,\alpha,\beta,i,j,p,k,q_\ell$ (for $\ell < k$), $G^\ell$
(for $\ell < k$) and $G^*$ be as in the assumption of clause (d)
(from Definition \scite {4.1}) be given. \nl
Without loss of generality $i < j,i < i(*),j = i(*) +1$ (as in the proof of
clause $(c)$ the other cases are trivial). \nl
First, choose $G^{**}$ for $G^*,\alpha,\beta,i,i(*),G^{**} \in N_1$.
For each $\ell < k$, if for some $s_\ell \in G^\ell$ we have

$$
s_\ell \Vdash_{P_i} ``\text{in } P_{i(*)}/{\underset\tilde {}\to G_{P_i}}
\text{ there is an upper bound to } G^{**}"
$$
\mn
then (possibly increasing $s_\ell$, recalling $G_\ell$ is generic over
$N^*$) there are $s_\ell \in G^\ell,r_\ell \in P_{i(*)} \cap N_1$ such that
$s_\ell$ forces that $r_\ell$ is an upper bound to $G^{**}$ and without
loss of generality
$r_\ell \restriction i \le s_\ell$.  Now \wilog \, $[G_{\ell_1} =
G_{\ell_2} \Rightarrow s_{\ell_1} = s_{\ell_2}]$ and $[G_{\ell_1} \ne
G_{\ell_2} \Rightarrow s_{\ell_1},s_{\ell_2}$ incompatible].  Now choose
$r \in P_{i(*)} \cap N_1$ with domain $\subseteq i(*) \backslash i$ as 
follows:
Dom$(r) = \dbcu_{\ell < k} \text{ Dom}(r_\ell) \backslash i$ and $r(\alpha)$
is $r_\ell(\alpha)$ if $s_\ell \in {\underset\tilde {}\to G_{P_i}},\ell < k$
and is $\emptyset_{P_\alpha}$ if this occurs for no $\ell$.  Renaming
$r \in N_i \cap P_{i(*)}$, Dom$(r) \subseteq i(*) \backslash i$ and
$s_\ell \in G^\ell,r_\ell = s_\ell \cup r$ is above $G^{**}$ in
$P_{i(*)}$.  Let $\beta_\ell = f_{i,j,1 +\ell}(\alpha)$ for 
$\ell \le k$.
We choose by induction on $\ell \le k$ the objects 
$Y_\ell,q'_\ell,M_\ell$ such that

$$
Y_0 = Y
$$

$$
M_0 = N_1
$$

$$
N_0 \in M_{\ell +1}
$$

$$
M_{\ell +1} \in M_\ell \cap {\Cal E}_{\beta_{k-\ell}}
$$

$$
Y_{\ell +1} \subseteq Y_\ell
$$

$$
Y_\ell \in D_{\beta_\ell}(M_\ell)
$$

$$
M_{\ell +1} \in Y_\ell
$$

$$
q_\ell \le q'_\ell \in P_{i(*)}
$$

$$
q'_\ell \text{ is } (M_{\ell +1},P_{i(*)}) \text{-generic}
$$

$$
q'_\ell \text{ forces a value to } {\underset\tilde {}\to G_{P_{i(*)}}}
\cap M_{\ell +1}
$$

$$
q'_\ell \text{ is } (N_0,P_{i(*)}) \text{-generic}
$$

$$
q'_\ell \Vdash ``Y_{\ell +1} \subseteq {\Cal M}
[{\underset\tilde {}\to G_{P_{i(*)}}},M_\ell]"
$$

$$
q'_\ell \restriction i \in G^\ell.
$$
\mn
(In the older version we have to increase $\beta \, \omega$ times if $k$ not
given, arriving to a fixed point.)  
\sn
No problem, now apply clause (d) of the definition for $i(*),N_0,M_k$, \nl
$\langle q'_\ell:\ell < k \rangle,Y_k,G^{**}$ and get $G^{***}$ as required.
\mn
\ub{Comment}:  The $Y$ transfers information between the various generics.
In \nl
\cite[Ch.V]{Sh:f}, in the first proof after $\omega$ steps we are lost, 
\ub{but} having the common tower of models help us.
\bn
\ub{Case 3}:  $\delta = \ell g(\bar Q)$ is limit. 
\sn
\ub{Part (1)}:  Follows from clause (d) in part (2) and part (2) is proved
below.
\bn
\ub{Part (2)}:  Let $f_{i,j}$ be fast enough (you can collect the demands
used below). \nl
Clause $(b)'$ is obvious.  We first prove clause $(d)'$ and later $(c)'$.
\mn
\ub{Clause $(d)'$}:  So again \wilog \, $i < j = \delta$.  \nl
Let $N_0,N_1,p,G^*,\alpha,\beta,k < 1 + \kappa$ and $G_\ell,q_\ell$ for
$\ell < k$ be as in the assumption of clause (d) in Definition 
\scite{4.1} but for $i < j$.\nl
We for simplicity use $\kappa = \aleph_0$.  We choose $\gamma \in N_0,
\alpha < \gamma < \beta$ such that $\gamma$ is larger enough, 
in particular: $i \le i' < j' < j \Rightarrow f_{i',j'}(\gamma) = \gamma$. \nl
Choose $\langle i_m:m < \omega \rangle \in N_1$ such that:

$$
i_0 = i
$$

$$
i_m < i_{m+1}
$$

$$
\sup_{m < \omega} i_m = \sup(j \cap N_0).
$$
\mn
Choose $y^* \in N \cap {\Cal H}(\chi_\gamma)$ coding enough.  We choose
$M_0,M_1,M_2,M_3,M_4 \in N_1 \cap {\Cal E}_\gamma \cap Y$ such that

$$
N_0 \in M_0 \in M_1 \in M_2 \in M_3 \in M_4
$$

$$
Y \cap M_m \in D_\gamma(M_m) \text{ for } m < 5.
$$
\mn
Choose $q'_\ell \in G^\ell \cap M_4$ above $G^\ell \cap M_3$ so \nl
$q'_\ell$ is $(M_0,P_i)$-generic,$(M_1,P_i)$-generic,$(M_2,P_i)$-generic,
$(M_3,P_i)$-generic. \nl
Let $\langle {\Cal I}_m:m < \omega\rangle \in M_0$ list the dense open 
subsets of $P_j$ from $N_0$.
\bigskip

\definition{Explanation}  Now we shall use the diagonal argument, choose
${\underset\tilde {}\to G_P} \cap N_0,p_m \in P_{i_m} \cap N_0,r_m \in
P_{i_n}$ as usual, using in the $m$-th step $f_{i_m,i_{m+1}} +$ relevant 
$(d)'$ things.  We fulfill the above in $M_4$, so in the end can find a
solution in $N_1$, by using a canonical construction, e.g. each time the 
$<^*_\chi$-choice.

But to carry this we need to have finitely many candidates for
${\underset\tilde {}\to G_{P_{i_m}}} \cap M_0$ with a common $Y_m$. \nl
(Note: if $Q \times \ldots \times Q$ is proper, like in Ch.XVIII,\S2 we 
can get such a common $Y_m$, i.e. the present result is stronger for this
stage.)  \nl
To get this in the inductive step, we need in step $m-1$ that for $M_1$ we 
just have finitely many candidates for
${\underset\tilde {}\to G_{P_{i_m}}} \cap M_1$, and in turn to get this in
the step $m-1$ we use that in step $m-2$ for $M_2$ we have:
from every maximal antichain we choose a finite subset.  To get this we use
that for $M_3$ we just ask $M_3 [{\underset\tilde {}\to G_{P_{i_{m-3}}}}] 
\cap V = M_3$.  So along the
way $N_0,M_0,M_1,M_2,M_3$ our induction demands go down, but slowly, so that
in each step $m$, advancing for say $M_0$, we have to preserve less than
really knowing ${\underset\tilde {}\to G_{P_{i_m}}} \cap M_0$, and are 
helped by our demand on $M_1$, just like in \cite[Ch.XVIII]{Sh:f}.  
So compared to \cite[Ch.V]{Sh:f}, we have a finite tower.

Returning to the proof we choose by induction on 
$m < \omega$ the objects $r_m,G^*_m,p_m,n_m$, \nl
$\langle G^\ell_m:\ell < n_m \rangle,Y_m$ such that
\mr
\item "{$(a)$}"  $r_m \in P_{i_m} \cap M_4$
\sn
\item "{$(b)$}"  Dom$(r_m) \subseteq [i,i_m)$
\sn
\item "{$(c)$}"  $r_{m+1} \restriction i_m = r_m$ \nl
(comment: $r_m$'s act as a generic for $N_0$)
\sn
\item "{$(d)$}"  $q'_\ell \cup r_m \in P_{i_m}$ is $(M_0,P_{i_m})$-generic,
$(M_1,P_{i_m})$-generic, $(M_2,P_{i_m})$-generic, $(M_3,P_{i_m})$-generic
\sn
\item "{$(e)$}"  if $\ell < k,{\Cal J} \subseteq P_{i_m}$ is dense open and
${\Cal J} \in M_2$, \ub{then} for some finite \nl
${\Cal J} \subseteq {\Cal I}
\cap M_2,{\Cal J}$ is predense above $q'_\ell \cup r_m$ \nl
(like in the proof of ${}^\omega \omega$-bounding)
\sn
\item "{$(f)$}"  $n_m < \omega$, and \nl
$\ell < n_m \Rightarrow G^\ell_m$ is a subset of $P_{i_m} \cap M_0$ generic
over $M_0$, \nl
and $G^\ell_m \in M_1$
\sn
\item "{$(g)$}"  $G^\ell_{m+1} \cap P_{i_m} \in \{G^\ell_m:\ell < n_m\}$
\sn
\item "{$(h)$}"  $n_0 =k,G^\ell_0 = G^\ell \cap M_1$
\sn
\item "{$(i)$}"  $q_\ell \cup r_m \Vdash ``{\underset\tilde {}\to G_{P_{i,m}}}
\cap M_1 \in \{G^\ell_m:\ell < n_m\}"$
\sn
\item "{$(j)$}"  $G^*_m$ is a subset of $P_{i_m} \cap N_0$ generic over
$N_0$
\sn
\item "{$(k)$}"  $G^*_m \subseteq G^\ell_m$, so $G^*_m \subseteq G^*_{m+1},
G^*_0 = G^*$
\sn
\item "{$(l)$}"  $p_m \in P_j \cap N$
\sn
\item "{$(m)$}"  $p_0 = p$
\sn
\item "{$(n)$}"   $p_m \restriction i_m \in G^*_m$
\sn
\item "{$(o)$}"  $p_m \le p_{m+1} \in {\Cal I}_m$
\sn
\item "{$(p)$}"  $Y_m \subseteq {\Cal M}[G^\ell_m,M_0,y^*]$
\sn
\item "{$(q)$}"  $Y_m \in D_\gamma(M_0)$.
\ermn
Why is this sufficient?  If we can carry out the induction, then without
loss of generality the construction belongs to $N_1$. \nl
So $G^{**} = \{s \in P_j \cap N_0:\dsize \bigvee_{n < \omega} s \le p_m\}$ is
as required, as $q'_\ell =: q_\ell \cup \dbcu_m r_m \in P_j \cap N_1$, and
is above $G^{**}$ and $p \le q'_\ell$.
\enddefinition
\bn
\ub{Induction}:
\sn
\ub{$m=0$}:  Trivial.
\bn
\ub{$m+1$}: \ub{Stage A}:  Choosing $p_{m+1}$ is trivial, the demands are:
$p_{m+1} \ge p_m,p_{m+1} \restriction i_m \in G^*_m$ and $p_{m + 1} \in
{\Cal I}_m$.
\bn
\ub{Stage B}:  Choosing $G^*_{m+1}$: apply the induction hypothesis using
clause $(d)'$ of what we are proving with $i_m,i_{m+1},\gamma,f_{i_m,i_{m+1}}
(\gamma),N_0,M_1$ here standing for $i,j,\alpha,\beta,N_0,N_1$ there.
\bn
\ub{Stage C}:  Let $\{H^\ell_m:\ell < n_{m+1}\}$ list the possibilities of
${\underset\tilde {}\to G_{P_{i_m}}} \cap M_1$ 
(by clause (e) this exists). \nl
Without loss of generality $H^\ell_m \cap M_0 = G^{h(\ell)}_m$, for some
function $h = h_m:n_{m+1} \rightarrow n_m$. \nl
We choose $s^\ell_m \in P_{i_{m+1}} \cap M_1$, above $G^*_{m+1}$, such that
$s^\ell_m \restriction i_m \in G^{h(\ell)}_m$.  Now repeat the argument of
the successor stage of shrinking $Y$ (but now we have a fixed $\gamma$!).

So we can find $t^\ell_m \in P_{i_{m+1}} \cap M_1$, above $s^\ell_m,
t^\ell_m \restriction i_m \in H^\ell_m,t^\ell_n \Vdash
``{\underset\tilde {}\to G_{P_{i_{m+1}}}} \cap M_0 =: G^\ell_{m+1}"$ such that

$$
Y_{m+1} =: \dbca_{i < n_{m+1}} {\Cal M}[G^\ell_{m+1},M_0,y^*] 
\in D_\gamma(M_0).
$$
\mn
The rest is as in \S1 so we have finished 
proving clause $(d)'$ in the case $\ell g(\bar Q)$ is
a limit ordinal (which is last).
\bn
\ub{Clause $(c)'$}:  Again, \wilog \, $i < j = \delta$.  So assume
$f_{i,j}(\alpha) \le \beta,\{i,j,\alpha,\beta\} \in 
N^* \in {\Cal E}_\beta,q$ is 
$(N^*,P_i)$-generic, $Y^* \in D_\beta (N^*),q \in P_\alpha$ and
$q \Vdash ``Y^* \subseteq {\Cal M}
[{\underset\tilde {}\to G_{P_i}},N^*,y^*]"$ are given.  We prove the desired
conclusion by induction on $\alpha$.  For each $\alpha$, we would like to 
simulate a play of $\Game_{\alpha,\beta}(N^*)$, supplying the challenger 
with a strategy.  For this we apply the proof of clause $(d)'$. \nl
Choose $N_0,N_1,M_0,\dotsc,M_4,q_0,G_0,G^*$ 
(and $k=1$) as there for some $\alpha'
< \gamma' < \beta'$ as there such that $\beta < \alpha'$ 
such that $N^*,q,Y^* \in N_0$ (easy to find). 
\bn
\ub{Explanation}:  Note: as ${\frak p}$ is simple, we can use 
$N \prec ({\Cal H}(\chi),\in),N \cap {\Cal H}
(\chi_\beta) \in {\Cal E}_\beta$ with no mention - there were other places we
could have said so.  During the construction this time we demand $p_m \in N^*
\cap P_j$, so a generic for $N_0$ is not necessarily created.  But still
$p_m \le p_{m+1},p_m \restriction i_m \in G^*_m$.  Now $p_m$ will be played
by the chooser.  Now
$g(1 + \alpha)$ will be a fixed point of $f_{i_m,i_m + 1}$.  So we can add the
demand $N^*[G^*_m \cap N^*] \cap V = N^*$, i.e. $G^*_m$ is generic over $N^*$
and

$$
{\Cal M}[G^*_m \cap N^*,N^*,y^*] \in D_{\delta(1 + \alpha)}(N^*).
$$
\mn
The challenger chooses

$$
X_{m+1} = {\Cal M}[G^*_m \cap N^*,N^*,y^*] \cap \dbcu_{\xi < \gamma
(1+ \alpha)} {\Cal E}_\xi \cap \{M:p_n \in M\} \in N_0.
$$
\mn
Now the chooser chooses $\alpha_m,\beta'_m$ and the challenger chooses
$\beta_n \ge \beta'_m,f_{i_m,i_{m+1}}(\alpha)$ in $N^*_0 \cap j$ and the
chooser chooses $M^*_m$ such that $p_m \in M_{m+1}$.
\bn
\ub{Comment}:  The game was defined with this point in mind. \nl
Now the chooser chooses $Y_n \in D_{\beta_n}(M_0),Y_n \subseteq X_n \cap M_0,
Y_n \in N^*_0$ (check definition of game!) \nl
Now we play $Z_n$ for the challenger as follows:

there is $p_{m+1} \ge p_m,(M^*_{m+1},P_{i_{m+1}})$-generic

$p_{m+1} \restriction i_m \in G^*_m$

and $p'_{m+1}$ forces ${\underset\tilde {}\to G_{P_{i_{m+1}}}} \cap N^*$ and

forces $Z_n = Y_n \cap {\Cal M}[{\underset\tilde {}\to G_{P_{i_{m+1}}}},
M^*_m,y^*] \in D_{\alpha_n}(M^*_m)$. \nl
As the challenger does not have a winning strategy, there is a play he wins,
giving $\dbcu_{n < \omega} G^*_n \cap N^*_0$ with a bound.
\bn
\ub{Continuation of the Proof}:  Choose also $\langle i'_m:m < \omega 
\rangle \in N_0$ such that $i_m \in N^*,i_0 = i,i_m < i_{m+1}$,
sup$\{i_m:m < \omega\} = \text{ sup}(N^* \cap j)$ and this time we let
$\langle {\Cal I}'_m:m < \omega \rangle$ list the dense open subsets of
$P_j$ from $N^*$.  For $\bold m < \omega$ let ${\Cal T}_{\bold m}$ be the
set of the finite sequences ${\frak x}$ from $M_4$ coding $\langle
r_{{\frak x},m}:m \le \bold m \rangle,\langle G_{{\frak x},m}:m \le
\bold m \rangle,\langle p_{{\frak x},m}:m \le \bold m \rangle,\langle
n_{{\frak x},m}:m \le \bold m \rangle,\langle G^\ell_{{\frak x},m}:\ell \le
n_{{\frak x},m},m \le \bold m \rangle,\langle Y_{{\frak x},m}:m \le \bold m
\rangle$ and also $(X_{{\frak x},m},\alpha_{{\frak x},m},\beta'_{{\frak x},m},
\beta_{{\frak x},m},M_{{\frak x},n},y'_{{\frak x},m},
M'_{{\frak x},m},y_{{\frak x},m})$ for $m \le \bold m$ and 
$Z_{{\frak x},m}$ for $m < \bold m$ satisfying:
\sn
clauses (a)-(k),(m),(n),(p),(q) from the proof of (d)$'$ above and
\mr
\item "{$(l)'$}"  $p_m \in _j \cap N^*$
\sn
\item "{$(o)'$}"  $p_m \le p_{m+1} \in {\Cal I}'_n$
\sn
\item "{$(r)'$}"  $r_{{\frak x},m}$ is $(N^*,G_{i_m})$-generic for 
$m \le \bold m$ 
\sn
\item "{$(s)'$}"  $\langle (X_{{\frak x},m},\alpha_{{\frak x},m},
\beta'_{{\frak x},m},\beta_{{\frak x},m},M_{{\frak x},m},Y_{{\frak x},m},
M'_{{\frak x},m},Z_{{\frak x},m'}):m \le \bold m \rangle$ 
belongs to $N$ and is an initial segment of a play of the game
$\Game'_{\alpha,\beta}(N^*,{\frak p})$ or just $\Game'_{\alpha,\beta}(N^*,N,
{\frak p})$, note that in the $\bold m$-th move the challenger has not yet
chose $Z_{{\frak x},\bold m}$, (see clause (e) of Definition \scite{2.2}(1)
\sn
\item "{$(t)'$}"  $Z_{{\frak x},m} \subseteq Y_{m+1}$
\sn
\item "{$(u)'$}"  $y_{{\frak x},m}$ code $p_m,\langle i_m:m < \omega
\rangle$
\sn
\item "{$(v)'$}"  $f_{i_m,i_{m+1}}(\alpha_m) \le \beta'_m$ for $m \le
\bold m$.
\ermn
Let ${\frak x} \triangleleft {\frak y}$ has the natural meaning for ${\frak x}
\in {\Cal T}_{{\bold m}_1},{\frak y} \in {\Cal T}_{{\bold m}_2},
\bold m_1 < \bold m_2$. \nl
Note
\mr
\item "{$\boxtimes_1$}"  ${\Cal T}_{\bold m} \subseteq N$ for $\bold m <
\omega$
\sn
\item "{$\boxtimes_2$}" ${\Cal T_0} \ne \emptyset$
\sn
\item "{$\boxtimes_3$}"  if $\bold x \in {\Cal T}_{\bold m}$, \ub{then}
considering the game $\Game_{\alpha,\beta}(N^*,{\frak p}),{\frak x}$ is an initial
segment of a play of it (see clause $(s)'$ above).
\ermn
Assume
\mr
\item "{$(*)$}"   $M'_{\bold m} \in Y_{{\frak x},\bold m} \cap 
{\Cal E}_{\alpha_{\bold m}} \cap (M_{{\frak x},\bold m} \cup 
\{M_{{\frak x},\bold m} \cap {\Cal H}(\chi^{\frak p}_{\bold m})\}$
satisfying $y_{\bold m},y'_{\bold m} \in M'_{\bold m}$ and $Z_{\bold m}
\subseteq {\Cal D}_{\alpha_{{\frak x},m}}(M'_m),Z_{\bold m} \subseteq 
Y_{\bold m}$ (hence $Z_{\bold m} \subseteq X_{{\frak x},\bold m})$ and
$X_{\bold m +1} \in D_\beta(N^*) \cap X_{{\frak x},\bold m}$ such that
$Z_{\bold m} \subseteq X_{\bold m +1}$ and $\alpha_{\bold m +1} \in 
\alpha \cap N^*$ and any $\beta'_{\bold m +1} \in \beta \cap N^* \backslash
p_{i_{\bold m +1},i_{\bold m_r}}(\alpha_{\bold m +1}),y'_{\bold m +1} \in
N \cap {\Cal H}(\chi^{\frak p}_{\alpha_{\bold m +1}})$ 
and any $\beta_{\bold m}
\in \beta \cap N \backslash \beta'_n \backslash \alpha_n$ and
$M_{\bold m +1} \in X_{\bold m + 1} \cap {\Cal E}_{\beta_{\bold m + 1}},
y_{\bold m + 1} \in M_{\bold m + 1} \cap {\Cal H}(\chi^{\frak p}
_{\alpha_{\bold m}})$ satisfying $y'_{\bold m + 1} \in M_{\bold m + 1}$ and
$Y_{\bold m + 1} \in N \cap D_{X_{\bold m + 1}}(M_{\bold m + 1})$ and any
$M'_{\bold m + 1} \in Y_{\bold m + 1} \cap {\Cal E}_{\alpha_n} \cap
(M_{\bold m + 1} \cup \{M_{\bold m + 1} \cap {\Cal H}(\chi^{\frak p}
_{\alpha_m})\}$
satisfying $y_{\bold m + 1},y'_{\bold m +1} \in M'_{\bold m + 1}$.
\ermn
\ub{Then}
\mr
\item "{$(**)$}"  \ub{there} is ${\frak y} \in {\Cal T}_{\bold m +1}$ such that ${\frak x}
\triangleleft {\frak y}$ and $(Z_{{\frak y},\bold m},
X_{{\frak y},\bold m + 1}$, $\alpha_{{\frak y},\bold m +1},
\beta'_{{\frak y},m + 1},y'_{{\frak y},\bold m + 1}$, \nl
$\beta_{{\frak y},\bold m + 1},y_{{\frak y},\bold m +1},
M_{{\frak y},\bold m + 1},Y_{{\frak y},\bold m + 1},M'_{{\frak y},\bold m+1})$
is equal to \nl
$(Z_{\bold m},X_{\bold m + 1},\alpha_{\bold m + 1},
\beta'_{\bold m +1}$, 
$y'_{\bold m + 1},\beta_{\bold m + 1},y_{\bold m + 1}$, 
$M_{\bold m + 1},Y_{\bold m + 1},M'_{\bold m + 1})$. \nl
Why?  Because $f_{i_m,i_{m+1}}(\alpha_{\bold m}) \le \beta_{\bold m}$, hence
we know that $P_{i_{\bold m + 1}}/P_{i_{\bold m}}$ is $({\frak p},\alpha_n,
\beta_n)$-proper and let $G_{i_{\bold m}} \subseteq P_{i_m}$ be generic
over $V,r_{\bold m} \in G_{i_{\bold m}}$ to the model
$M'_{{\frak x},\bold m}$ and the set $Y_{{\frak x},m}$.
So now we can describe a strategy for the challenger in the game
$\Game_{\alpha,\beta}(N^*,{\frak p})$ (or $\Game'_{\alpha,\beta}(N^*,N,
{\frak p})$) delaying his choice of $M'_m,Z_m$ to the $(m+1)$-th move, he
just chose on the side ${\frak x}_m \in {\Cal T}_m$ which ``code" they play
so far, and preserve ${\frak x}_m \triangleleft {\frak x}_{m+1}$.
\ermn
By $\boxtimes_3$ this is O.K. - all possible choices of the chooser are
allowed, that is this gives a well defined strategy for the challenger (will
he have some free choice, this does not hurt).  But the chosen does not lose
the game, so there is such a choice $\langle {\frak x}_{\bold m}:\bold
m < \omega \rangle$ with $\cup\{(M'_{{\frak x}_{m+1},\bold m}\} \cup
Y_{{\frak x}_{m+1},\bold m}:\bold m < \omega\} \in D_\alpha(N)$ and
$\dbcu_{m < \omega} r_m$ is as required.
\bn
\centerline {$* \qquad * \qquad *$}
\bn
The adaptation for the proof when $\kappa=2$ should be clear.
\bigskip

\demo{Proof}  Similar to the proof of \scite{1.7}, with some changes.  We
choose $\langle j_n:n < \omega \rangle \in M_0$ such that $j < j_n < j_{n+1}
< h^*(j,i),h^*(j_n,i_m) < j_{n+1}$
\mr
\item "{$(l)$}"  $n_m$ is a power of 2, say $2^{n^*_m}$ and so we can rename
$\{G^\ell_m:\ell < n_m\}$ as $\{G^\eta_m:\eta \in 2^{n^*_m}\}$
\sn
\item "{$(m)$}"  
{\roster
\itemitem{ $(\alpha)$ }  $M_\eta \in M_1$ for $\eta \in {}^{(n^*_m \ge)}2$
\sn
\itemitem{ $(\beta)$ }  $M_{<>} = N_0$
\sn
\itemitem{ $(\gamma)$ }  $M_\eta \in M_{\eta \char 94 \langle 0 \rangle} \cap
M_{\eta \char 94 \langle 1 \rangle}$
\sn
\itemitem{ $(\delta)$ }  $\eta \triangleleft \nu_1 \in {}^{n^*_m}2,
\eta \triangleleft \nu_2 \in {}^{n^*_m}2 \Rightarrow G^{\nu_1}_m \cap M_\eta
= G^{\nu_2}_m \cap M_\eta$ so we call it $K^\eta_m$
\sn
\itemitem{ $(\varepsilon)$ }  $M_{\eta \char 94 \langle 0 \rangle} =
M_{\eta \char 94 \langle 1 \rangle}$ when $\eta \in {}^{n^*_m}2$ call it
$N_\eta$
\sn
\itemitem{ $(\zeta)$ }  $N_\eta \in {\Cal E}_{\ell g(\eta)}$ for
$\eta \in {}^{(m^*_m >)}2$
\sn
\itemitem{ $(\eta)$ }  $Y_\eta = {\Cal M}[K_{\eta \char 94 \langle 0 \rangle},
N_\eta] \cap {\Cal M}[K_{\eta \char 94 \langle 1 \rangle},N_\eta] \in
D_{j_{\ell g(\eta)}}(N_\eta)$.
\endroster}
\endroster
\enddemo
\bn
\ub{Discussion}:  We may be interested in non-proper forcing, say 
semi-proper and UP ones (see \cite[Ch.X,XI,XV]{Sh:f} and \cite{Sh:311}).  
Here the change from (reasonable) parameter ${\frak p} = 
{\frak p}^V$ to ${\frak p}^{V[G]}$ is more serious as $\{N \cap \chi_\alpha:
N \in {\Cal E}^{{\frak p}^{V[G]}}\}$ is in general not equal to
$\{N \cap \chi_\alpha:N \in {\Cal E}^{\frak p}_\alpha\}$.  We intend to spell
it out in \cite{Sh:311}.

We may be interested in combining this work with \cite{Sh:587}, 
\cite{Sh:F259}.  Intend to
deal with it later (see also \cite{Sh:655}).
\newpage

\head {\S5 Problematic forcing} \endhead  \resetall \sectno=5
 % \resetall 
\bn
\ub{\stag{5.1} Discussion}:  1) In the examples 
the ``barely adding reals but not so"
may occur on some stationary ${\Cal E} \subseteq [\chi^*]^{\aleph_0}$ and
otherwise just properness is asked.  We do not bother to do this in the
examples. \nl
2) We may like to put the present lemmas and \cite[Ch.XVIII,\S2]{Sh:f} and
more together.  The way is clear, we concentrate on $\kappa \in \aleph_0$.

So instead of a ``tower" with six levels we have one with $n(*)+1$ levels.
\bigskip

\definition{\stag{5.2} Definition}  1) We say $\bar Q$ is an
${\frak p}-NNR^0_{\aleph_0,{\bar Pr},\bar \Xi}$-iteration for ${\frak p}$
is:
\mr
\item "{$(a)$}"  $\bar Q$ is a CS iteration of proper notions forcing, 
${\frak p}$-proper
\sn
\item "{$(b)$}"  forcing with Lim$(\bar Q) = P_{\ell g(\bar Q)}$ does not
add reals
\sn
\item "{$(c)$}"  $\bar Pr = \langle Pr_\ell:\ell < n(*) \rangle,\bar \Xi =
\langle \Xi_\ell:\ell \le n(*) \rangle,\Xi_\ell \subseteq \ell g({\frak p})
\times \ell g({\frak p})$
\sn
\item "{$(d)$}"  $Pr_\ell(N,\bar G,P)$ implies: $P$ a forcing notion,
$N$ is a countable elementary submodel of $({\Cal H}(\chi),\in),
\bar G = \langle G_m:m < k \rangle,k < \omega,G_m \subseteq N \cap P$ is
generic over $N$
\sn
\item "{$(e)$}"  if $\ell = 0,Pr_\ell(N,\bar G,P)$, then
$\dsize \bigwedge_\ell G_\ell = G_0$
\sn
\item "{$(f)$}"  if $\ell = n(*)-1$, \ub{then} $Pr_\ell(N,\bar G,P)$, iff
the demand in clause (d) holds
\sn
\item "{$(g)$}"  if Pr$_\ell(N,\bar G,P)$ and 
$\bar G \triangleleft \bar G',\ell g(\bar G')$
finite and Rang$(\bar G) = \text{ Rang}(\bar G')$ \ub{then} $Pr_\ell(N,
\bar G',P)$
\sn
\item "{$(h)$}"  $Pr_0(N,\bar G,P)$ holds iff (the condition in clause (c)
and) $\dsize \bigwedge_\ell G_\ell = G_0$
\sn
\item "{$(i)$}"  if $\ell < n(*),i < j \le \ell g(\bar Q),N_0 \prec N_1
\prec ({\Cal H}(\chi),\in),{\frak p} \in N_0,N_0,N_1$ are countable and for
some $(\alpha,\beta) \in \Xi_\ell \cap N_0$ we have $N_0 \cap {\Cal H}
(\chi^{\frak p}_\alpha) \in {\Cal E}_\alpha,N_1 \cap {\Cal H}(\chi^{\frak p}
_\beta) \in {\Cal E}_\beta,Pr_\ell(N_0,\bar G,P_i),Pr_{\ell +1}(N_1,\bar H,
P_i),\ell g(\bar G) = \ell g(\bar H),G_k \subseteq H_k,\bar p = \langle p_k:
k < \ell g(\bar G) \rangle,p_\ell \in N_0 \cap P_j,p_k \restriction i \in
G_k,\ell = 0 \Rightarrow p_k = p_0$, \ub{then} we can find $\bar G^+ =
\langle G^+_k:k < \ell g(\bar G) \rangle \in N_1$ such that $Pr(N_0,\bar G^+,
P_j)$ and $p_k \in G^+_k$
\sn
\item "{$(j)$}"  assume $Pr_\ell(N,\bar G,P_i),i_n < i_{n+1},i_n \in N,i_0=i,
\sup_{n < \omega} i_n = \sup(N \cap j),(\alpha,\beta) \in N \cap \Xi_\ell,N'=
N \cap {\Cal H}(\chi^{\frak p}_\alpha) \in {\Cal E}_\alpha$, \ub{then} in
the following game the $Pr^+_\ell$-player has a winning strategy (or just
does not lose).
\ermn
Before the $n$-th move $\bar G^n$ such that $Pr_\ell(N,\bar G^n,P_{i_n})$ is
chosen $\ell g(\bar G^n) = \ell g(\bar Q)$ with $\bar G^0 = \bar G$.  In the
$n$-th move the challenger chooses $\bar p^n = \langle p^n_k:k < \ell g
(\bar G) \rangle,p^n_k \in N_0 \cap P_{in+1},P^n_k \restriction i_n \in
G^n_k,\ell = 0 \Rightarrow p^n_k = p^n_0$ and then the chooser chooses
$\bar G^{n+1}$ is above such that $p^n_k \in G^{n+1}_k$.  In the end of the
play the chooser wins if $Pr_\ell(N,\bar G^*,P_j)$ where $G^*_k = \{p \in
N \cap P_j:\text{for every } n,p \restriction i_n \in G^n_k\}$.
\enddefinition
\bn
\ub{\stag{5.3} Comment}:  0) Of course, 
we can also use a reasonable parameter for induction. \nl
1) In \cite[Ch.XVIII,\S2]{Sh:f} for $NNR^2_\kappa$ 
we can use weaken clause (5):
$t$ a tree with 2 branches if $\bar Q \in N \prec ({\Cal H}(\chi),\in)$, if
$i^* < \alpha,P_{i^*,i} =: \{(q,p_0,p_1) \in P_i:p_0 \restriction i^* =
p_1 \restriction i^*\}$, then for $i < j \le \delta,i$ non-limit we have:
$P_{i^*,i}/P_{i^*,j}$ is ${\Cal E}_2$-proper.

In light of the previous proofs this is straight. \nl
2) We can replace $D_\alpha(N)$ by $D_{\alpha,i,j}(N)$ for $i < j \le
\delta^*,\delta^*$ the supremum of the length of the iterations we consider.

%% you may want to move the following lines up a bit
\newpage

\shlhetal
    
REFERENCES.  
\bibliographystyle{lit-plain}
\bibliography{lista,listb,listx,listf,liste}


\enddocument%%

   

\bye


656.bbl
\def\germ{\frak} \def\scr{\cal}
  \ifx\documentclass\undefinedcs\def\rm{\fam0\tenrm}\fi%f**k-amstex!
  \def\defaultdefine#1#2{\expandafter\ifx\csname#1\endcsname\relax
  \expandafter\def\csname#1\endcsname{#2}\fi} \defaultdefine{Bbb}{\bf}
  \defaultdefine{frak}{\bf} \defaultdefine{mathbb}{\bf}
  \defaultdefine{mathcal}{\cal} \defaultdefine{beth}{BETH} \def\bbfI{{\Bbb I}}
  \def\mbox{\hbox} \def\text{\hbox} \def\om{\omega} \def\Cal#1{{\bf #1}}
  \def\pcf{pcf} \defaultdefine{cf}{cf} \defaultdefine{reals}{{\Bbb R}}
  \defaultdefine{real}{{\Bbb R}} \def\restriction{{|}} \def\club{CLUB}
  \def\w{\omega} \def\exist{\exists} \def\se{{\germ se}} \def\bb{{\bf b}}
  \def\equivalence{\equiv} \let\lt< \let\gt> \def\cite#1{[#1]}
  \def\implies{\Rightarrow}
\begin{thebibliography}{DvSh 65}
\makeatletter \renewcommand{\@biblabel}[1]{[#1]} \makeatother

\bibitem[DvSh 65]{DvSh:65}Keith~J. Devlin and Saharon Shelah.
\newblock {A weak version of $\diamondsuit $ which follows from $2^{\aleph
  _{0}}<2^{\aleph _{1}}$}.
\newblock {\em {Israel Journal of Mathematics}}, {\bf 29}:239--247, 1978.

\bibitem[Sh 311]{Sh:311}Saharon Shelah.
\newblock {A more general iterable condition ensuring $\aleph_1$ is not
  collapsed, II}.
\newblock {\em {preprint}}.

\bibitem[Sh 655]{Sh:655}Saharon Shelah.
\newblock {Iteration of $\lambda$-complete forcing notions not collapsing
  $\lambda^+$.}
\newblock {\em {in preparation}}.
 {\tt math.LO/9906024}

\bibitem[Sh 587]{Sh:587}Saharon Shelah.
\newblock {Not collapsing cardinals $\leq\kappa$ in $(<\kappa)$--support
  iterations}.
\newblock {\em {Israel Journal of Mathematics}}, {\bf accepted}.
 {\tt math.LO/9707225}

\bibitem[Sh 666]{Sh:666}Saharon Shelah.
\newblock {On what I do not understand (and have something to say)}.
\newblock {\em {Fundamenta Mathematicae}}, {\bf to appear}.
 {\tt math.LO/9906113}

\bibitem[Sh:b]{Sh:b}Saharon Shelah.
\newblock {\em {Proper forcing}}, volume 940 of {\em {Lecture Notes in
  Mathematics}}.
\newblock {Springer-Verlag, Berlin-New York, xxix+496 pp}, 1982.

\bibitem[Sh 177]{Sh:177}Saharon Shelah.
\newblock {More on proper forcing}.
\newblock {\em {The Journal of Symbolic Logic}}, {\bf 49}:1034--1038, 1984.

\bibitem[Sh:f]{Sh:f}Saharon Shelah.
\newblock {\em {Proper and improper forcing}}.
\newblock {Perspectives in Mathematical Logic}. {Springer}, 1998.

\bibitem[Sh:F259]{Sh:F259}{Shelah, Saharon}.
\newblock {Not collapsing cardinals $\leq\kappa$ in $(<\kappa)$--support
  iterations II}.

\end{thebibliography}
