(*<*)

theory Slides2

imports "../thys/UTM" "../thys/Abacus_Defs"

begin

declare [[show_question_marks = false]]

hide_const (open) s 

abbreviation

  "update2 p a \<equiv> update a p"

notation (latex output)

  Cons ("_::_" [48,47] 68) and

  append ("_@_" [65, 64] 65) and

  Oc ("1") and

  Bk ("0") and

  exponent ("_\<^bsup>_\<^esup>" [107] 109) and

  inv_begin0 ("I\<^isub>0") and

  inv_begin1 ("I\<^isub>1") and

  inv_begin2 ("I\<^isub>2") and

  inv_begin3 ("I\<^isub>3") and

  inv_begin4 ("I\<^isub>4") and 

  inv_begin ("I\<^bsub>cbegin\<^esub>") and

  inv_loop1 ("J\<^isub>1") and

  inv_loop0 ("J\<^isub>0") and

  inv_end1 ("K\<^isub>1") and

lemma inv_begin_print:

  shows "s = 0 \<Longrightarrow> inv_begin n (s, tp) = inv_begin0 n tp" and

        "s = 1 \<Longrightarrow> inv_begin n (s, tp) = inv_begin1 n tp" and 

        "s = 2 \<Longrightarrow> inv_begin n (s, tp) = inv_begin2 n tp" and 

        "s = 3 \<Longrightarrow> inv_begin n (s, tp) = inv_begin3 n tp" and 

        "s = 4 \<Longrightarrow> inv_begin n (s, tp) = inv_begin4 n tp" and

        "s \<notin> {0,1,2,3,4} \<Longrightarrow> inv_begin n (s, l, r) = False"

apply(case_tac [!] tp)

by (auto)

lemma inv1: 

  shows "0 < (n::nat) \<Longrightarrow> Turing_Hoare.assert_imp (inv_begin0 n) (inv_loop1 n)"

unfolding Turing_Hoare.assert_imp_def

unfolding inv_loop1.simps inv_begin0.simps

apply(auto)

apply(rule_tac x="1" in exI)

apply(auto simp add: replicate.simps)

done

lemma inv2: 

  shows "0 < n \<Longrightarrow> inv_loop0 n = inv_end1 n"

apply(rule ext)

apply(case_tac x)

apply(simp add: inv_end1.simps)

done

lemma measure_begin_print:

  shows "s = 2 \<Longrightarrow> measure_begin_step (s, l, r) = length r" and

        "s = 3 \<Longrightarrow> measure_begin_step (s, l, r) = (if r = [] \<or> r = [Bk] then 1 else 0)" and

        "s = 4 \<Longrightarrow> measure_begin_step (s, l, r) = length l" and 

        "s \<notin> {2,3,4} \<Longrightarrow> measure_begin_step (s, l, r) = 0"

by (simp_all)

lemma inv_begin01:

  assumes "n > 1"

  shows "inv_begin0 n (l, r) = (n > 1 \<and> (l, r) = (Oc \<up> (n - 2), [Oc, Oc, Bk, Oc]))"

using assms by auto                          

lemma inv_begin02:

  assumes "n = 1"

  shows "inv_begin0 n (l, r) = (n = 1 \<and> (l, r) = ([], [Bk, Oc, Bk, Oc]))"

using assms by auto

(*>*)

text_raw {*

  \renewcommand{\slidecaption}{ITP, 24 July 2013}

  \newcommand{\bl}[1]{#1}                        

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}

  \frametitle{%

  \begin{tabular}{@ {}c@ {}}

  \\[-3mm]

  {\fontsize{20}{20}\selectfont{}Mechanising Turing Machines and}\\[-2mm] 

  {\fontsize{20}{20}\selectfont{}Computability Theory in Isabelle}\\[-3mm] 

  \end{tabular}}

  \bigskip\medskip\medskip

  \begin{center}

  \begin{tabular}{c@ {\hspace{1cm}}c}

  \includegraphics[scale=0.29]{jian.jpg}  &

  \includegraphics[scale=0.034]{xingyuan.jpg} \\

  Jian Xu & Xingyuan Zhang\\[-3mm]

  \end{tabular}\\[3mm]

  \small PLA University of Science and Technology

  \end{center}

  \begin{center}

  Christian Urban\\[0mm]

  \small King's College London

  \end{center}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}[c]

  \frametitle{Why Turing Machines?}%

  \begin{itemize}

  \item At the beginning, it was just a student project

  about computability.\smallskip

  \begin{center}

  \begin{tabular}{c}

  \includegraphics[scale=0.3]{boolos.jpg}\\[-2mm]

  \footnotesize Computability and Logic (5th.~ed)\\[-2mm]

  \footnotesize Boolos, Burgess and Jeffrey\\[2mm]

  \end{tabular}

  \end{center}

  \item found an inconsistency in the definition of halting computations

  (Chap.~3 vs Chap.~8)

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}[c]

  \begin{itemize}

  \item Norrish formalised computability theory in HOL starting

  \begin{itemize}

  \item \textcolor{gray}{for technical reasons we could not follow him}

  \item \textcolor{gray}{some proofs use TMs (Wang tilings)}

  \end{itemize}

  \bigskip

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}[c]

  \frametitle{Turing Machines}%

  \begin{itemize}

  \item tapes are lists and contain @{text 0}s or @{text 1}s only

  \begin{center}

\begin{tikzpicture}[scale=1]

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  \draw[fill]     (-1.65,0.1) rectangle (-1.35,0.4);

  \draw (-0.25,0.8) -- (-0.25,-0.8);

  \draw[<->] (-1.25,-0.7) -- (0.75,-0.7);

  \node [anchor=base] at (-0.85,-0.5) {\small left list\;\;\;\;\mbox{}};

  \node [anchor=base] at (0.40,-0.5) {\small \mbox{}\;\;\;\;right list};

  \node [anchor=base] at (0.1,0.7) {\small \;\;head};

  \node [anchor=base] at (-2.2,0.2) {\ldots};

  \node [anchor=base] at ( 2.3,0.2) {\ldots};

  \end{tikzpicture}

  \item @{text steps} function:

  \begin{quote}\rm

  What does the TM claclulate after it has executed @{text n} steps?

  \end{quote}\pause

  forever, i.e.~have a @{text Nop}-action

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}[c]

  \begin{itemize}

  \begin{center}

  \end{tabular}

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}[c]

  \begin{itemize}

  \begin{center}

  \node [anchor=base] at (2.2,0.1) {\small$\Rightarrow$};

  \node [anchor=base] at (5.6,0.1) {\small$\Rightarrow$};

  \node [anchor=base] at (10.5,0.1) {\small$\Rightarrow$};

  \node [anchor=base] at (2.2,-0.8) {\small$\overbrace{@{text cbegin}}^{}$};

  \node [anchor=base] at (5.6,-0.8) {\small$\overbrace{@{text cloop}}^{}$};

  \node [anchor=base] at (10.5,-0.8) {\small$\overbrace{@{text cend}}^{}$};

  \begin{scope}[shift={(0.5,0)}]

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  \draw[fill]     ( 0.85,0.1) rectangle (1.15,0.4);

  \end{scope}

  \end{tikzpicture}

  \end{center}\bigskip

  \footnotesize

  \begin{center}

  \begin{tabular}{@ {\hspace{-8mm}}c@ {\hspace{2mm}}c@ {\hspace{2mm}}c}

  \begin{tabular}[t]{@ {}l@ {}}

  @{text cbegin} @{text "\<equiv>"}\\

  \hspace{2mm}@{text "["}@{text "(W\<^bsub>0\<^esub>, 0), (R, 2), (R, 3),"}\\

  \hspace{2mm}\phantom{@{text "["}}@{text "(R, 2), (W\<^bsub>1\<^esub>, 3), (L, 4),"}\\

  \hspace{2mm}\phantom{@{text "["}}@{text "(L, 4), (L, 0)"}@{text "]"} 

  \end{tabular}

  &

  \begin{tabular}[t]{@ {}l@ {}}

  @{text cloop} @{text "\<equiv>"}\\

  \hspace{2mm}@{text "["}@{text "(R, 0), (R, 2), (R, 3),"}\\

  \hspace{2mm}\phantom{@{text "["}}@{text "(W\<^bsub>0\<^esub>, 2), (R, 3), (R, 4),"}\\

  \hspace{2mm}\phantom{@{text "["}}@{text "(W\<^bsub>1\<^esub>, 5), (R, 4), (L, 6),"}\\

  \hspace{2mm}\phantom{@{text "["}}@{text "(L, 5), (L, 6), (L, 1)"}@{text "]"} 

  \end{tabular}

  &

  \begin{tabular}[t]{@ {}l@ {}}

  @{text cend} @{text "\<equiv>"}\\

  \hspace{2mm}@{text "["}@{text "(L, 0), (R, 2), (W\<^bsub>1\<^esub>, 3),"}\\

  \hspace{2mm}\phantom{@{text "["}}@{text "(L, 4), (R, 2), (R, 2),"}\\

  \hspace{2mm}\phantom{@{text "["}}@{text "(L, 5), (W\<^bsub>0\<^esub>, 4), (R, 0),"}\\

  \hspace{2mm}\phantom{@{text "["}}@{text "(L, 5)"}@{text "]"} 

  \end{tabular}

  \end{tabular}

  \end{center}