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\title{Modal operators for a constructive account of well quasi-orders}

\author{Ingo Blechschmidt}
\date{November 30th, 2024}

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      \hil{Modal operators for \\ a constructive account of
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  \bf
  \colorbox{black}{November 30th, 2024}%
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    Ingo Blechschmidt \\
    University of Antwerp
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{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{2.1em}\includegraphics[height=\paperheight]{sea-of-clouds-1}\end{minipage}}
\begin{frame}{Well quasi-orders}
  \textbf{Def.} Let~$(X,{\leq})$ be a quasi-order.
  \vspace*{-0.3em}
  \begin{itemize}
    \item A sequence~$\alpha : \NN \to X$ is \hil{good} iff
    there exist~$i < j$ with~$\alpha\,i \leq \alpha\,j$.
    \item The quasi-order~$X$ is \hil{well\only<4->{\sinf}} iff every sequence~$\NN \to X$ is good.
  \end{itemize}
  \pause
  \vspace*{-0.9em}

  \begin{columns}[t]
    \begin{column}{0.46\textwidth}
      \begin{block}{Natural numbers\phantom{y}}
        \textbf{Prop.} $(\NN, {\leq})$ is well\only<4->{\sinf}.
        \smallskip

        \emph{Proof.} Let~$\alpha : \NN \to \NN$. By~\badbox{\textsc{lem}}, there is a
        \bad{minimum}~$\alpha\,i$.
        Set~$j \defeq i + 1$. \qed

        \hfill\color{gray}offensive?
      \end{block}
      \only<2-4>{
        \vspace*{-2em}
        \[ \astikznodetransparentlycircled{xm}{7}\!,
        \quad \astikznodetransparentlycircled{x0}{4}\!,
        \quad \astikznodecircled{t1}{mypurple}{3}\!,
        \quad \astikznodetransparentlycircled{x1}{1}\!,
        \quad \astikznodecircled{t2}{mypurple}{8}\!,
        \quad \astikznodetransparentlycircled{x2}{2}\!,
        \quad \ldots \]
      {\centering\begin{tikzpicture}[remember picture,overlay]
        \node[draw=mypurple, circle, thick, inner sep=0.1em] (t3) {\scriptsize$\leq$};
        \path[draw=mypurple,thick]
          (t1)
          to [out=-90, in=180] (t3)
          to [out=0, in=-90] (t2);
      \end{tikzpicture}\par}}
    \end{column}

    \pause

    \begin{column}{0.50\textwidth}
      \begin{block}{Key stability results}
        \justifying
        \only<1-5>{Assuming~\badbox{\textsc{lem}}
        and~\badbox{\textsc{dc}}}\only<6->{Constructively}, \ldots

        \hil{Dickson:} \tabto{1.8cm} If~$X$ and~$Y$ are well\only<4-5>{\sinf}\only<6->{\ind}, so is~$X \times Y$. \\
        \hil{Higman:}  \tabto{1.8cm} If~$X$ is well\only<4-5>{\sinf}\only<6->{\ind}, so is~$X^\star$. \\
        \hil{Kruskal:} \tabto{1.8cm} If~$X$ is well\only<4-5>{\sinf}\only<6->{\ind}, so is~$\mathrm{Tree}(X)$.
      \end{block}
    \end{column}
  \end{columns}
  \pause
  \pause

  \textbf{Def.} A quasi-order~$X$ is \hil{well\ind} iff~$\mathsf{Good} \mid
  [\,]$, where~$\mathsf{Good}\,x_0 \ldots x_{n-1}$ iff~$\exists i < j\_ x_i \leq x_j$.
  \pause
  \pause

  \small
  With \bad{bar induction},\tabto{3.10cm} $\text{well}\ind \Leftarrow \text{well}\sinf$. \\[-0.2em]
  Constructively,\tabto{3.10cm} $\text{well}\ind \Rightarrow \text{well}\sinf$.
  \only<8>{
    \bigskip
    \begin{columns}[c]
      \begin{column}{0.01\textwidth}
	\includegraphics[height=2.4em]{question-mark}
      \end{column}
      \begin{column}{0.9\textwidth}
      Is there a procedure for reinterpreting \hil{classical proofs}
      regarding well\sinf{} as \newline \hil{blueprints for constructive proofs} regarding well\ind?
      \end{column}
    \end{columns}
  }
  \pause
  \pause
  Moreover, if~$X$ is well\ind, then \ldots
  \begin{itemize}
  \vspace*{-0.7em}
    \small
    \item for every \emph{partial} function~$\alpha$,\tabto{5.25cm}
    if~$\forall n\_ \neg\neg(\alpha\,n\,{\downarrow})$,
    then~$\neg\neg \exists i < j\_ \alpha\,i\,{\downarrow} \wedge
    \alpha\,j\,{\downarrow} \wedge \alpha\,i \leq \alpha\,j$. \\[-2.0em]\
    \item for every \emph{multivalued} function~$\alpha$,\tabto{5.25cm}
    $\exists i < j\_ \exists x \in \alpha\,i\_
    \exists y \in \alpha\,j\_ x \leq y$.
  \end{itemize}
  \pause

  \vspace*{-0.7em}
  \hil{Central insight:} A quasi-order $X$ is well\ind iff $\hil{$\necessary$}\,\forall\alpha : \NN \to
  X\_ \exists i < j\_ \alpha\,i \leq \alpha\,j$.
\end{frame}}

{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\includegraphics[width=\paperwidth]{staircase-more-faded}\end{minipage}}
\begin{frame}{The modal multiverse of constructive forcing}
  \textbf{Def.} A statement~$\varphi$ holds \ldots
  \begin{itemize}
    \small
    \item \hil{everywhere} \tabto{2.15cm}($\necessary\varphi$)\tabto{3.08cm} iff it holds in every topos
    (over the current base).
    \item \hil{somewhere} \tabto{2.15cm}($\possible\varphi$)\tabto{3.08cm} iff it holds in some positive topos.
    \item \hil{proximally} \tabto{2.15cm}($\xpossible\varphi$)\tabto{3.08cm} iff it holds in some positive overt topos.
  \end{itemize}
  \bigskip

  \begin{tikzpicture}[overlay]
    \node[anchor=south east,inner sep=0] (image) at (14.8,0.6) {
      \includegraphics[width=0.2\textwidth]{branching}
    };
  \end{tikzpicture}\vspace*{-2.3em}
  \pause

  \colorbox{white}{\bignumber{1} A quasiorder is well\ind iff \emph{everywhere}, every sequence is good.}

  \mbox{\qquad
  \colorbox{white}{\bignumber{2} A relation is well-founded iff \emph{everywhere}, there is
  no infinite descending chain.}}

  \qquad
  \qquad
  \colorbox{white}{\bignumber{3} A ring element is nilpotent iff all prime ideals \emph{everywhere} contain it.}

  \mbox{\qquad
  \qquad
  \qquad
  \colorbox{white}{\bignumber{4} For every inhabited set~$X$, \emph{proximally} there is an
  enumeration~$\NN \twoheadrightarrow X$.}}

  \qquad
  \qquad
  \qquad
  \qquad
  \colorbox{white}{\bignumber{5} For every ring, \emph{proximally} there is a maximal ideal.}

  \qquad
  \qquad
  \qquad
  \qquad
  \qquad
  \colorbox{white}{\bignumber{6} \emph{Somewhere}, the law of excluded middle holds.}
\end{frame}}

{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{-4.68em}\includegraphics[width=1.2\paperwidth]{wqo-faded}\end{minipage}}
\begin{frame}{Answering a question by Berardi--Buriola--Schuster}
  %Stefano Berardi, Gabriele Buriola and Peter Schuster established that the
  %following notion is useful for structuring constructive arguments:

  \textbf{Def.} A quasi-order~$X$ is \hil{well\impl} iff (approximately) for
  every monotone predicate~$B$,
  \[ \text{if $B \mid_{\text{incr}} [\,]$, then $B \mid [\,]$.} \]
  \justifying
  {\small``Assume that no matter how the empty list evolves to an \emph{increasing}
  list~$\sigma$, eventually~$B\,\sigma$. Then no matter how the empty
  list evolves to an \emph{arbitrary} list~$\tau$, eventually~$B\,\tau$.''\par}
  \pause

  \hil{Equivalently:} If everywhere every \emph{increasing} infinite sequence~$\alpha : \NN
  \to X$ has a finite prefix validating~$B$, then so does every
  \emph{arbitrary} infinite sequence everywhere.
  \pause

  \textbf{Prop.} $\text{well\impl} \Rightarrow \text{well\ind}$.

  \emph{Proof.} Trivially~$\mathsf{Good} \mid_{\text{incr}} [\,]$,
  hence~$\mathsf{Good} \mid [\,]$ by assumption. \qed
  \pause

  \textbf{Thm.} $\text{well\impl} \Leftarrow \text{well\ind}$.

  \small
  \emph{Proof.} Let~$x_0,x_1,\ldots$ be an infinite sequence (in an arbitrary topos).
  \emph{Somewhere,} \textsc{lem} holds and implies that there is an increasing infinite subsequence~$x_{i_0},
  x_{i_1}, \ldots$. \emph{There} we have, by assumption, a finite prefix
  with~$B\,x_{i_0}\ldots x_{i_n}$. As~$B$ is monotone, \emph{there}
  we also have~$B\,x_0x_1\ldots x_{i_n}$. So \emph{somewhere} there is a finite
  prefix validating~$B$. Hence there actually is a finite prefix
  validating~$B$. \qed
\end{frame}}

\backupstart
{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{-2.2em}\includegraphics[width=\paperwidth]{agda-wqo}\end{minipage}}
\begin{frame}
  \vspace*{\paperheight}\vspace*{-2.5em}\centering
  \emph{Agda formalization in progress.}
\end{frame}}
\backupend

\end{document}