diff -r 0b302c0b449a -r 469c26d19f8e Attic/UTM.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Attic/UTM.thy Wed Feb 06 02:39:53 2013 +0000 @@ -0,0 +1,5165 @@ +theory UTM +imports Main uncomputable recursive abacus UF GCD +begin + +section {* Wang coding of input arguments *} + +text {* + The direct compilation of the universal function @{text "rec_F"} can not give us UTM, because @{text "rec_F"} is of arity 2, + where the first argument represents the Godel coding of the TM being simulated and the second argument represents the right number (in Wang's coding) of the TM tape. + (Notice, left number is always @{text "0"} at the very beginning). However, UTM needs to simulate the execution of any TM which may + very well take many input arguments. Therefore, a initialization TM needs to run before the TM compiled from @{text "rec_F"}, and the sequential + composition of these two TMs will give rise to the UTM we are seeking. The purpose of this initialization TM is to transform the multiple + input arguments of the TM being simulated into Wang's coding, so that it can be consumed by the TM compiled from @{text "rec_F"} as the second + argument. + + However, this initialization TM (named @{text "t_wcode"}) can not be constructed by compiling from any resurve function, because every recursive + function takes a fixed number of input arguments, while @{text "t_wcode"} needs to take varying number of arguments and tranform them into + Wang's coding. Therefore, this section give a direct construction of @{text "t_wcode"} with just some parts being obtained from recursive functions. + +\newlength{\basewidth} +\settowidth{\basewidth}{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx} +\newlength{\baseheight} +\settoheight{\baseheight}{$B:R$} +\newcommand{\vsep}{5\baseheight} + +The TM used to generate the Wang's code of input arguments is divided into three TMs + executed sequentially, namely $prepare$, $mainwork$ and $adjust$¡£According to the + convention, start state of ever TM is fixed to state $1$ while the final state is + fixed to $0$. + +The input and output of $prepare$ are illustrated respectively by Figure +\ref{prepare_input} and \ref{prepare_output}. + + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + [tbox/.style = {draw, thick, inner sep = 5pt}] + \node (0) {}; + \node (1) [tbox, text height = 3.5pt, right = -0.9pt of 0] {\wuhao $m$}; + \node (2) [tbox, right = -0.9pt of 1] {\wuhao $0$}; + \node (3) [tbox, right = -0.9pt of 2] {\wuhao $a_1$}; + \node (4) [tbox, right = -0.9pt of 3] {\wuhao $0$}; + \node (5) [tbox, right = -0.9pt of 4] {\wuhao $a_2$}; + \node (6) [right = -0.9pt of 5] {\ldots \ldots}; + \node (7) [tbox, right = -0.9pt of 6] {\wuhao $a_n$}; + \draw [->, >=latex, thick] (1)+(0, -4\baseheight) -- (1); +\end{tikzpicture}} +\caption{The input of TM $prepare$} \label{prepare_input} +\end{figure} + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + \node (0) {}; + \node (1) [draw, text height = 3.5pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; + \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; + \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; + \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; + \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; + \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; + \node (7) [right = -0.9pt of 6] {\ldots \ldots}; + \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_n$}; + \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $0$}; + \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$}; + \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $1$}; + \draw [->, >=latex, thick] (10)+(0, -4\baseheight) -- (10); +\end{tikzpicture}} +\caption{The output of TM $prepare$} \label{prepare_output} +\end{figure} + +As shown in Figure \ref{prepare_input}, the input of $prepare$ is the same as the the input +of UTM, where $m$ is the Godel coding of the TM being interpreted and $a_1$ through $a_n$ are the $n$ input arguments of the TM under interpretation. The purpose of $purpose$ is to transform this initial tape layout to the one shown in Figure \ref{prepare_output}, +which is convenient for the generation of Wang's codding of $a_1, \ldots, a_n$. The coding procedure starts from $a_n$ and ends after $a_1$ is encoded. The coding result is stored in an accumulator at the end of the tape (initially represented by the $1$ two blanks right to $a_n$ in Figure \ref{prepare_output}). In Figure \ref{prepare_output}, arguments $a_1, \ldots, a_n$ are separated by two blanks on both ends with the rest so that movement conditions can be implemented conveniently in subsequent TMs, because, by convention, +two consecutive blanks are usually used to signal the end or start of a large chunk of data. The diagram of $prepare$ is given in Figure \ref{prepare_diag}. + + +\begin{figure}[h!] +\centering +\scalebox{0.9}{ +\begin{tikzpicture} + \node[circle,draw] (1) {$1$}; + \node[circle,draw] (2) at ($(1)+(0.3\basewidth, 0)$) {$2$}; + \node[circle,draw] (3) at ($(2)+(0.3\basewidth, 0)$) {$3$}; + \node[circle,draw] (4) at ($(3)+(0.3\basewidth, 0)$) {$4$}; + \node[circle,draw] (5) at ($(4)+(0.3\basewidth, 0)$) {$5$}; + \node[circle,draw] (6) at ($(5)+(0.3\basewidth, 0)$) {$6$}; + \node[circle,draw] (7) at ($(6)+(0.3\basewidth, 0)$) {$7$}; + \node[circle,draw] (8) at ($(7)+(0.3\basewidth, 0)$) {$0$}; + + + \draw [->, >=latex] (1) edge [loop above] node[above] {$S_1:L$} (1) + ; + \draw [->, >=latex] (1) -- node[above] {$S_0:S_1$} (2) + ; + \draw [->, >=latex] (2) edge [loop above] node[above] {$S_1:R$} (2) + ; + \draw [->, >=latex] (2) -- node[above] {$S_0:L$} (3) + ; + \draw [->, >=latex] (3) edge[loop above] node[above] {$S_1:S_0$} (3) + ; + \draw [->, >=latex] (3) -- node[above] {$S_0:R$} (4) + ; + \draw [->, >=latex] (4) edge[loop above] node[above] {$S_0:R$} (4) + ; + \draw [->, >=latex] (4) -- node[above] {$S_0:R$} (5) + ; + \draw [->, >=latex] (5) edge[loop above] node[above] {$S_1:R$} (5) + ; + \draw [->, >=latex] (5) -- node[above] {$S_0:R$} (6) + ; + \draw [->, >=latex] (6) edge[bend left = 50] node[below] {$S_1:R$} (5) + ; + \draw [->, >=latex] (6) -- node[above] {$S_0:R$} (7) + ; + \draw [->, >=latex] (7) edge[loop above] node[above] {$S_0:S_1$} (7) + ; + \draw [->, >=latex] (7) -- node[above] {$S_1:L$} (8) + ; + \end{tikzpicture}} +\caption{The diagram of TM $prepare$} \label{prepare_diag} +\end{figure} + +The purpose of TM $mainwork$ is to compute the Wang's encoding of $a_1, \ldots, a_n$. Every bit of $a_1, \ldots, a_n$, including the separating bits, is processed from left to right. +In order to detect the termination condition when the left most bit of $a_1$ is reached, +TM $mainwork$ needs to look ahead and consider three different situations at the start of +every iteration: +\begin{enumerate} + \item The TM configuration for the first situation is shown in Figure \ref{mainwork_case_one_input}, + where the accumulator is stored in $r$, both of the next two bits + to be encoded are $1$. The configuration at the end of the iteration + is shown in Figure \ref{mainwork_case_one_output}, where the first 1-bit has been + encoded and cleared. Notice that the accumulator has been changed to + $(r+1) \times 2$ to reflect the encoded bit. + \item The TM configuration for the second situation is shown in Figure + \ref{mainwork_case_two_input}, + where the accumulator is stored in $r$, the next two bits + to be encoded are $1$ and $0$. After the first + $1$-bit was encoded and cleared, the second $0$-bit is difficult to detect + and process. To solve this problem, these two consecutive bits are + encoded in one iteration. In this situation, only the first $1$-bit needs + to be cleared since the second one is cleared by definition. + The configuration at the end of the iteration + is shown in Figure \ref{mainwork_case_two_output}. + Notice that the accumulator has been changed to + $(r+1) \times 4$ to reflect the two encoded bits. + \item The third situation corresponds to the case when the last bit of $a_1$ is reached. + The TM configurations at the start and end of the iteration are shown in + Figure \ref{mainwork_case_three_input} and \ref{mainwork_case_three_output} + respectively. For this situation, only the read write head needs to be moved to + the left to prepare a initial configuration for TM $adjust$ to start with. +\end{enumerate} +The diagram of $mainwork$ is given in Figure \ref{mainwork_diag}. The two rectangular nodes +labeled with $2 \times x$ and $4 \times x$ are two TMs compiling from recursive functions +so that we do not have to design and verify two quite complicated TMs. + + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + \node (0) {}; + \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; + \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; + \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; + \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; + \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; + \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; + \node (7) [right = -0.9pt of 6] {\ldots \ldots}; + \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$}; + \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$}; + \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $1$}; + \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $0$}; + \node (12) [right = -0.9pt of 11] {\ldots \ldots}; + \node (13) [draw, right = -0.9pt of 12, thick, inner sep = 5pt] {\wuhao $0$}; + \node (14) [draw, text height = 3.9pt, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $r$}; + \draw [->, >=latex, thick] (13)+(0, -4\baseheight) -- (13); +\end{tikzpicture}} +\caption{The first situation for TM $mainwork$ to consider} \label{mainwork_case_one_input} +\end{figure} + + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + \node (0) {}; + \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; + \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; + \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; + \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; + \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; + \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; + \node (7) [right = -0.9pt of 6] {\ldots \ldots}; + \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$}; + \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$}; + \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$}; + \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $0$}; + \node (12) [right = -0.9pt of 11] {\ldots \ldots}; + \node (13) [draw, right = -0.9pt of 12, thick, inner sep = 5pt] {\wuhao $0$}; + \node (14) [draw, text height = 2.7pt, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $(r+1) \times 2$}; + \draw [->, >=latex, thick] (13)+(0, -4\baseheight) -- (13); +\end{tikzpicture}} +\caption{The output for the first case of TM $mainwork$'s processing} +\label{mainwork_case_one_output} +\end{figure} + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + \node (0) {}; + \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; + \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; + \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; + \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; + \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; + \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; + \node (7) [right = -0.9pt of 6] {\ldots \ldots}; + \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$}; + \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$}; + \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$}; + \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $1$}; + \node (12) [draw, right = -0.9pt of 11, thick, inner sep = 5pt] {\wuhao $0$}; + \node (13) [right = -0.9pt of 12] {\ldots \ldots}; + \node (14) [draw, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $0$}; + \node (15) [draw, text height = 3.9pt, right = -0.9pt of 14, thick, inner sep = 5pt] {\wuhao $r$}; + \draw [->, >=latex, thick] (14)+(0, -4\baseheight) -- (14); +\end{tikzpicture}} +\caption{The second situation for TM $mainwork$ to consider} \label{mainwork_case_two_input} +\end{figure} + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + \node (0) {}; + \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; + \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; + \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; + \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; + \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; + \node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; + \node (7) [right = -0.9pt of 6] {\ldots \ldots}; + \node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$}; + \node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$}; + \node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$}; + \node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $0$}; + \node (12) [draw, right = -0.9pt of 11, thick, inner sep = 5pt] {\wuhao $0$}; + \node (13) [right = -0.9pt of 12] {\ldots \ldots}; + \node (14) [draw, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $0$}; + \node (15) [draw, text height = 2.7pt, right = -0.9pt of 14, thick, inner sep = 5pt] {\wuhao $(r+1) \times 4$}; + \draw [->, >=latex, thick] (14)+(0, -4\baseheight) -- (14); +\end{tikzpicture}} +\caption{The output for the second case of TM $mainwork$'s processing} +\label{mainwork_case_two_output} +\end{figure} + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + \node (0) {}; + \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; + \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; + \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; + \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $1$}; + \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; + \node (6) [right = -0.9pt of 5] {\ldots \ldots}; + \node (7) [draw, right = -0.9pt of 6, thick, inner sep = 5pt] {\wuhao $0$}; + \node (8) [draw, text height = 3.9pt, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $r$}; + \draw [->, >=latex, thick] (7)+(0, -4\baseheight) -- (7); +\end{tikzpicture}} +\caption{The third situation for TM $mainwork$ to consider} \label{mainwork_case_three_input} +\end{figure} + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + \node (0) {}; + \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; + \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; + \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; + \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $1$}; + \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; + \node (6) [right = -0.9pt of 5] {\ldots \ldots}; + \node (7) [draw, right = -0.9pt of 6, thick, inner sep = 5pt] {\wuhao $0$}; + \node (8) [draw, text height = 3.9pt, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $r$}; + \draw [->, >=latex, thick] (3)+(0, -4\baseheight) -- (3); +\end{tikzpicture}} +\caption{The output for the third case of TM $mainwork$'s processing} +\label{mainwork_case_three_output} +\end{figure} + +\begin{figure}[h!] +\centering +\scalebox{0.9}{ +\begin{tikzpicture} + \node[circle,draw] (1) {$1$}; + \node[circle,draw] (2) at ($(1)+(0.3\basewidth, 0)$) {$2$}; + \node[circle,draw] (3) at ($(2)+(0.3\basewidth, 0)$) {$3$}; + \node[circle,draw] (4) at ($(3)+(0.3\basewidth, 0)$) {$4$}; + \node[circle,draw] (5) at ($(4)+(0.3\basewidth, 0)$) {$5$}; + \node[circle,draw] (6) at ($(5)+(0.3\basewidth, 0)$) {$6$}; + \node[circle,draw] (7) at ($(2)+(0, -7\baseheight)$) {$7$}; + \node[circle,draw] (8) at ($(7)+(0, -7\baseheight)$) {$8$}; + \node[circle,draw] (9) at ($(8)+(0.3\basewidth, 0)$) {$9$}; + \node[circle,draw] (10) at ($(9)+(0.3\basewidth, 0)$) {$10$}; + \node[circle,draw] (11) at ($(10)+(0.3\basewidth, 0)$) {$11$}; + \node[circle,draw] (12) at ($(11)+(0.3\basewidth, 0)$) {$12$}; + \node[draw] (13) at ($(6)+(0.3\basewidth, 0)$) {$2 \times x$}; + \node[circle,draw] (14) at ($(13)+(0.3\basewidth, 0)$) {$j_1$}; + \node[draw] (15) at ($(12)+(0.3\basewidth, 0)$) {$4 \times x$}; + \node[draw] (16) at ($(15)+(0.3\basewidth, 0)$) {$j_2$}; + \node[draw] (17) at ($(7)+(0.3\basewidth, 0)$) {$0$}; + + \draw [->, >=latex] (1) edge[loop left] node[above] {$S_0:L$} (1) + ; + \draw [->, >=latex] (1) -- node[above] {$S_1:L$} (2) + ; + \draw [->, >=latex] (2) -- node[above] {$S_1:R$} (3) + ; + \draw [->, >=latex] (2) -- node[left] {$S_1:L$} (7) + ; + \draw [->, >=latex] (3) edge[loop above] node[above] {$S_1:S_0$} (3) + ; + \draw [->, >=latex] (3) -- node[above] {$S_0:R$} (4) + ; + \draw [->, >=latex] (4) edge[loop above] node[above] {$S_0:R$} (4) + ; + \draw [->, >=latex] (4) -- node[above] {$S_1:R$} (5) + ; + \draw [->, >=latex] (5) edge[loop above] node[above] {$S_1:R$} (5) + ; + \draw [->, >=latex] (5) -- node[above] {$S_0:S_1$} (6) + ; + \draw [->, >=latex] (6) edge[loop above] node[above] {$S_1:L$} (6) + ; + \draw [->, >=latex] (6) -- node[above] {$S_0:R$} (13) + ; + \draw [->, >=latex] (13) -- (14) + ; + \draw (14) -- ($(14)+(0, 6\baseheight)$) -- ($(1) + (0, 6\baseheight)$) node [above,midway] {$S_1:L$} + ; + \draw [->, >=latex] ($(1) + (0, 6\baseheight)$) -- (1) + ; + \draw [->, >=latex] (7) -- node[above] {$S_0:R$} (17) + ; + \draw [->, >=latex] (7) -- node[left] {$S_1:R$} (8) + ; + \draw [->, >=latex] (8) -- node[above] {$S_0:R$} (9) + ; + \draw [->, >=latex] (9) -- node[above] {$S_0:R$} (10) + ; + \draw [->, >=latex] (10) -- node[above] {$S_1:R$} (11) + ; + \draw [->, >=latex] (10) edge[loop above] node[above] {$S_0:R$} (10) + ; + \draw [->, >=latex] (11) edge[loop above] node[above] {$S_1:R$} (11) + ; + \draw [->, >=latex] (11) -- node[above] {$S_0:S_1$} (12) + ; + \draw [->, >=latex] (12) -- node[above] {$S_0:R$} (15) + ; + \draw [->, >=latex] (12) edge[loop above] node[above] {$S_1:L$} (12) + ; + \draw [->, >=latex] (15) -- (16) + ; + \draw (16) -- ($(16)+(0, -4\baseheight)$) -- ($(1) + (0, -18\baseheight)$) node [below,midway] {$S_1:L$} + ; + \draw [->, >=latex] ($(1) + (0, -18\baseheight)$) -- (1) + ; + \end{tikzpicture}} +\caption{The diagram of TM $mainwork$} \label{mainwork_diag} +\end{figure} + +The purpose of TM $adjust$ is to encode the last bit of $a_1$. The initial and final configuration +of this TM are shown in Figure \ref{adjust_initial} and \ref{adjust_final} respectively. +The diagram of TM $adjust$ is shown in Figure \ref{adjust_diag}. + + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + \node (0) {}; + \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; + \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; + \node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; + \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $1$}; + \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; + \node (6) [right = -0.9pt of 5] {\ldots \ldots}; + \node (7) [draw, right = -0.9pt of 6, thick, inner sep = 5pt] {\wuhao $0$}; + \node (8) [draw, text height = 3.9pt, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $r$}; + \draw [->, >=latex, thick] (3)+(0, -4\baseheight) -- (3); +\end{tikzpicture}} +\caption{Initial configuration of TM $adjust$} \label{adjust_initial} +\end{figure} + +\begin{figure}[h!] +\centering +\scalebox{1.2}{ +\begin{tikzpicture} + \node (0) {}; + \node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; + \node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; + \node (3) [draw, text height = 2.9pt, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $r+1$}; + \node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $0$}; + \node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; + \node (6) [right = -0.9pt of 5] {\ldots \ldots}; + \draw [->, >=latex, thick] (1)+(0, -4\baseheight) -- (1); +\end{tikzpicture}} +\caption{Final configuration of TM $adjust$} \label{adjust_final} +\end{figure} + + +\begin{figure}[h!] +\centering +\scalebox{0.9}{ +\begin{tikzpicture} + \node[circle,draw] (1) {$1$}; + \node[circle,draw] (2) at ($(1)+(0.3\basewidth, 0)$) {$2$}; + \node[circle,draw] (3) at ($(2)+(0.3\basewidth, 0)$) {$3$}; + \node[circle,draw] (4) at ($(3)+(0.3\basewidth, 0)$) {$4$}; + \node[circle,draw] (5) at ($(4)+(0.3\basewidth, 0)$) {$5$}; + \node[circle,draw] (6) at ($(5)+(0.3\basewidth, 0)$) {$6$}; + \node[circle,draw] (7) at ($(6)+(0.3\basewidth, 0)$) {$7$}; + \node[circle,draw] (8) at ($(4)+(0, -7\baseheight)$) {$8$}; + \node[circle,draw] (9) at ($(8)+(0.3\basewidth, 0)$) {$9$}; + \node[circle,draw] (10) at ($(9)+(0.3\basewidth, 0)$) {$10$}; + \node[circle,draw] (11) at ($(10)+(0.3\basewidth, 0)$) {$11$}; + \node[circle,draw] (12) at ($(11)+(0.3\basewidth, 0)$) {$0$}; + + + \draw [->, >=latex] (1) -- node[above] {$S_1:R$} (2) + ; + \draw [->, >=latex] (1) edge[loop above] node[above] {$S_0:S_1$} (1) + ; + \draw [->, >=latex] (2) -- node[above] {$S_1:R$} (3) + ; + \draw [->, >=latex] (3) edge[loop above] node[above] {$S_0:R$} (3) + ; + \draw [->, >=latex] (3) -- node[above] {$S_1:R$} (4) + ; + \draw [->, >=latex] (4) -- node[above] {$S_1:L$} (5) + ; + \draw [->, >=latex] (4) -- node[right] {$S_0:L$} (8) + ; + \draw [->, >=latex] (5) -- node[above] {$S_0:L$} (6) + ; + \draw [->, >=latex] (5) edge[loop above] node[above] {$S_1:S_0$} (5) + ; + \draw [->, >=latex] (6) -- node[above] {$S_1:R$} (7) + ; + \draw [->, >=latex] (6) edge[loop above] node[above] {$S_0:L$} (6) + ; + \draw (7) -- ($(7)+(0, 6\baseheight)$) -- ($(2) + (0, 6\baseheight)$) node [above,midway] {$S_0:S_1$} + ; + \draw [->, >=latex] ($(2) + (0, 6\baseheight)$) -- (2) + ; + \draw [->, >=latex] (8) edge[loop left] node[left] {$S_1:S_0$} (8) + ; + \draw [->, >=latex] (8) -- node[above] {$S_0:L$} (9) + ; + \draw [->, >=latex] (9) edge[loop above] node[above] {$S_0:L$} (9) + ; + \draw [->, >=latex] (9) -- node[above] {$S_1:L$} (10) + ; + \draw [->, >=latex] (10) edge[loop above] node[above] {$S_0:L$} (10) + ; + \draw [->, >=latex] (10) -- node[above] {$S_0:L$} (11) + ; + \draw [->, >=latex] (11) edge[loop above] node[above] {$S_1:L$} (11) + ; + \draw [->, >=latex] (11) -- node[above] {$S_0:R$} (12) + ; + \end{tikzpicture}} +\caption{Diagram of TM $adjust$} \label{adjust_diag} +\end{figure} +*} + + +definition rec_twice :: "recf" + where + "rec_twice = Cn 1 rec_mult [id 1 0, constn 2]" + +definition rec_fourtimes :: "recf" + where + "rec_fourtimes = Cn 1 rec_mult [id 1 0, constn 4]" + +definition abc_twice :: "abc_prog" + where + "abc_twice = (let (aprog, ary, fp) = rec_ci rec_twice in + aprog [+] dummy_abc ((Suc 0)))" + +definition abc_fourtimes :: "abc_prog" + where + "abc_fourtimes = (let (aprog, ary, fp) = rec_ci rec_fourtimes in + aprog [+] dummy_abc ((Suc 0)))" + +definition twice_ly :: "nat list" + where + "twice_ly = layout_of abc_twice" + +definition fourtimes_ly :: "nat list" + where + "fourtimes_ly = layout_of abc_fourtimes" + +definition t_twice :: "tprog" + where + "t_twice = change_termi_state (tm_of (abc_twice) @ (tMp 1 (start_of twice_ly (length abc_twice) - Suc 0)))" + +definition t_fourtimes :: "tprog" + where + "t_fourtimes = change_termi_state (tm_of (abc_fourtimes) @ + (tMp 1 (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)))" + + +definition t_twice_len :: "nat" + where + "t_twice_len = length t_twice div 2" + +definition t_wcode_main_first_part:: "tprog" + where + "t_wcode_main_first_part \ + [(L, 1), (L, 2), (L, 7), (R, 3), + (R, 4), (W0, 3), (R, 4), (R, 5), + (W1, 6), (R, 5), (R, 13), (L, 6), + (R, 0), (R, 8), (R, 9), (Nop, 8), + (R, 10), (W0, 9), (R, 10), (R, 11), + (W1, 12), (R, 11), (R, t_twice_len + 14), (L, 12)]" + +definition t_wcode_main :: "tprog" + where + "t_wcode_main = (t_wcode_main_first_part @ tshift t_twice 12 @ [(L, 1), (L, 1)] + @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])" + +fun bl_bin :: "block list \ nat" + where + "bl_bin [] = 0" +| "bl_bin (Bk # xs) = 2 * bl_bin xs" +| "bl_bin (Oc # xs) = Suc (2 * bl_bin xs)" + +declare bl_bin.simps[simp del] + +type_synonym bin_inv_t = "block list \ nat \ tape \ bool" + +fun wcode_before_double :: "bin_inv_t" + where + "wcode_before_double ires rs (l, r) = + (\ ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)" + +declare wcode_before_double.simps[simp del] + +fun wcode_after_double :: "bin_inv_t" + where + "wcode_after_double ires rs (l, r) = + (\ ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>Suc (Suc (Suc 2*rs))\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +declare wcode_after_double.simps[simp del] + +fun wcode_on_left_moving_1_B :: "bin_inv_t" + where + "wcode_on_left_moving_1_B ires rs (l, r) = + (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Oc # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr > Suc 0 \ mr > 0)" + +declare wcode_on_left_moving_1_B.simps[simp del] + +fun wcode_on_left_moving_1_O :: "bin_inv_t" + where + "wcode_on_left_moving_1_O ires rs (l, r) = + (\ ln rn. + l = Oc # ires \ + r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +declare wcode_on_left_moving_1_O.simps[simp del] + +fun wcode_on_left_moving_1 :: "bin_inv_t" + where + "wcode_on_left_moving_1 ires rs (l, r) = + (wcode_on_left_moving_1_B ires rs (l, r) \ wcode_on_left_moving_1_O ires rs (l, r))" + +declare wcode_on_left_moving_1.simps[simp del] + +fun wcode_on_checking_1 :: "bin_inv_t" + where + "wcode_on_checking_1 ires rs (l, r) = + (\ ln rn. l = ires \ + r = Oc # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_erase1 :: "bin_inv_t" + where +"wcode_erase1 ires rs (l, r) = + (\ ln rn. l = Oc # ires \ + tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +declare wcode_erase1.simps [simp del] + +fun wcode_on_right_moving_1 :: "bin_inv_t" + where + "wcode_on_right_moving_1 ires rs (l, r) = + (\ ml mr rn. + l = Bk\<^bsup>ml\<^esup> @ Oc # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr > Suc 0)" + +declare wcode_on_right_moving_1.simps [simp del] + +declare wcode_on_right_moving_1.simps[simp del] + +fun wcode_goon_right_moving_1 :: "bin_inv_t" + where + "wcode_goon_right_moving_1 ires rs (l, r) = + (\ ml mr ln rn. + l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc rs)" + +declare wcode_goon_right_moving_1.simps[simp del] + +fun wcode_backto_standard_pos_B :: "bin_inv_t" + where + "wcode_backto_standard_pos_B ires rs (l, r) = + (\ ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Bk # Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)" + +declare wcode_backto_standard_pos_B.simps[simp del] + +fun wcode_backto_standard_pos_O :: "bin_inv_t" + where + "wcode_backto_standard_pos_O ires rs (l, r) = + (\ ml mr ln rn. + l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc rs) \ mr > 0)" + +declare wcode_backto_standard_pos_O.simps[simp del] + +fun wcode_backto_standard_pos :: "bin_inv_t" + where + "wcode_backto_standard_pos ires rs (l, r) = (wcode_backto_standard_pos_B ires rs (l, r) \ + wcode_backto_standard_pos_O ires rs (l, r))" + +declare wcode_backto_standard_pos.simps[simp del] + +lemma [simp]: "<0::nat> = [Oc]" +apply(simp add: tape_of_nat_abv exponent_def tape_of_nat_list.simps) +done + +lemma tape_of_Suc_nat: " = replicate a Oc @ [Oc, Oc]" +apply(simp add: tape_of_nat_abv exp_ind tape_of_nat_list.simps) +apply(simp only: exp_ind_def[THEN sym]) +apply(simp only: exp_ind, simp, simp add: exponent_def) +done + +lemma [simp]: "length () = Suc a" +apply(simp add: tape_of_nat_abv tape_of_nat_list.simps) +done + +lemma [simp]: "<[a::nat]> = " +apply(simp add: tape_of_nat_abv tape_of_nl_abv exponent_def + tape_of_nat_list.simps) +done + +lemma bin_wc_eq: "bl_bin xs = bl2wc xs" +proof(induct xs) + show " bl_bin [] = bl2wc []" + apply(simp add: bl_bin.simps) + done +next + fix a xs + assume "bl_bin xs = bl2wc xs" + thus " bl_bin (a # xs) = bl2wc (a # xs)" + apply(case_tac a, simp_all add: bl_bin.simps bl2wc.simps) + apply(simp_all add: bl2nat.simps bl2nat_double) + done +qed + +declare exp_def[simp del] + +lemma bl_bin_nat_Suc: + "bl_bin () = bl_bin () + 2^(Suc a)" +apply(simp add: tape_of_nat_abv bin_wc_eq) +apply(simp add: bl2wc.simps) +done +lemma [simp]: " rev (a\<^bsup>aa\<^esup>) = a\<^bsup>aa\<^esup>" +apply(simp add: exponent_def) +done + +declare tape_of_nl_abv_cons[simp del] + +lemma tape_of_nl_rev: "rev () = ()" +apply(induct lm rule: list_tl_induct, simp) +apply(case_tac "list = []", simp add: tape_of_nl_abv tape_of_nat_list.simps) +apply(simp add: tape_of_nat_list_butlast_last tape_of_nl_abv_cons) +done +lemma [simp]: "a\<^bsup>Suc 0\<^esup> = [a]" +by(simp add: exp_def) +lemma tape_of_nl_cons_app1: "() = (Oc\<^bsup>Suc a\<^esup> @ Bk # ())" +apply(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps) +apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) +done + +lemma bl_bin_bk_oc[simp]: + "bl_bin (xs @ [Bk, Oc]) = + bl_bin xs + 2*2^(length xs)" +apply(simp add: bin_wc_eq) +using bl2nat_cons_oc[of "xs @ [Bk]"] +apply(simp add: bl2nat_cons_bk bl2wc.simps) +done + +lemma tape_of_nat[simp]: "() = Oc\<^bsup>Suc a\<^esup>" +apply(simp add: tape_of_nat_abv) +done +lemma tape_of_nl_cons_app2: "() = ( @ Bk # Oc\<^bsup>Suc b\<^esup>)" +proof(induct "length xs" arbitrary: xs c, + simp add: tape_of_nl_abv tape_of_nat_list.simps) + fix x xs c + assume ind: "\xs c. x = length xs \ = + @ Bk # Oc\<^bsup>Suc b\<^esup>" + and h: "Suc x = length (xs::nat list)" + show " = @ Bk # Oc\<^bsup>Suc b\<^esup>" + proof(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps) + fix a list + assume g: "xs = a # list" + hence k: " = @ Bk # Oc\<^bsup>Suc b\<^esup>" + apply(rule_tac ind) + using h + apply(simp) + done + from g and k show " = @ Bk # Oc\<^bsup>Suc b\<^esup>" + apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) + done + qed +qed + +lemma [simp]: "length () = Suc (Suc aa) + length ()" +apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) +done + +lemma [simp]: "bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista) @ [Bk, Oc]) = + bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)) + + 2* 2^(length (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)))" +using bl_bin_bk_oc[of "Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)"] +apply(simp) +done + +lemma [simp]: + "bl_bin () + (4 * rs + 4) * 2 ^ (length () - Suc 0) + = bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # ) + rs * (2 * 2 ^ (aa + length ()))" +apply(case_tac "list", simp add: add_mult_distrib, simp) +apply(simp add: tape_of_nl_cons_app2 add_mult_distrib) +apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) +done + +lemma tape_of_nl_app_Suc: "(()) = () @ [Oc]" +apply(induct list) +apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind) +apply(case_tac list) +apply(simp_all add:tape_of_nl_abv tape_of_nat_list.simps exp_ind) +done + +lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # @ [Oc]) + = bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ) + + 2^(length (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ))" +apply(simp add: bin_wc_eq) +apply(simp add: bl2nat_cons_oc bl2wc.simps) +using bl2nat_cons_oc[of "Oc # Oc\<^bsup>aa\<^esup> @ Bk # "] +apply(simp) +done +lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ) + (4 * 2 ^ (aa + length ()) + + 4 * (rs * 2 ^ (aa + length ()))) = + bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # ) + + rs * (2 * 2 ^ (aa + length ()))" +apply(simp add: tape_of_nl_app_Suc) +done + +declare tape_of_nat[simp del] + +fun wcode_double_case_inv :: "nat \ bin_inv_t" + where + "wcode_double_case_inv st ires rs (l, r) = + (if st = Suc 0 then wcode_on_left_moving_1 ires rs (l, r) + else if st = Suc (Suc 0) then wcode_on_checking_1 ires rs (l, r) + else if st = 3 then wcode_erase1 ires rs (l, r) + else if st = 4 then wcode_on_right_moving_1 ires rs (l, r) + else if st = 5 then wcode_goon_right_moving_1 ires rs (l, r) + else if st = 6 then wcode_backto_standard_pos ires rs (l, r) + else if st = 13 then wcode_before_double ires rs (l, r) + else False)" + +declare wcode_double_case_inv.simps[simp del] + +fun wcode_double_case_state :: "t_conf \ nat" + where + "wcode_double_case_state (st, l, r) = + 13 - st" + +fun wcode_double_case_step :: "t_conf \ nat" + where + "wcode_double_case_step (st, l, r) = + (if st = Suc 0 then (length l) + else if st = Suc (Suc 0) then (length r) + else if st = 3 then + if hd r = Oc then 1 else 0 + else if st = 4 then (length r) + else if st = 5 then (length r) + else if st = 6 then (length l) + else 0)" + +fun wcode_double_case_measure :: "t_conf \ nat \ nat" + where + "wcode_double_case_measure (st, l, r) = + (wcode_double_case_state (st, l, r), + wcode_double_case_step (st, l, r))" + +definition wcode_double_case_le :: "(t_conf \ t_conf) set" + where "wcode_double_case_le \ (inv_image lex_pair wcode_double_case_measure)" + +lemma [intro]: "wf lex_pair" +by(auto intro:wf_lex_prod simp:lex_pair_def) + +lemma wf_wcode_double_case_le[intro]: "wf wcode_double_case_le" +by(auto intro:wf_inv_image simp: wcode_double_case_le_def ) +term fetch + +lemma [simp]: "fetch t_wcode_main (Suc 0) Bk = (L, Suc 0)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main (Suc 0) Oc = (L, Suc (Suc 0))" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Oc = (R, 3)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Bk = (R, 4)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Oc = (W0, 3)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 4 Bk = (R, 4)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 4 Oc = (R, 5)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 5 Oc = (R, 5)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 5 Bk = (W1, 6)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 6 Bk = (R, 13)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 6 Oc = (L, 6)" +apply(simp add: t_wcode_main_def t_wcode_main_first_part_def + fetch.simps nth_of.simps) +done +lemma [elim]: "Bk\<^bsup>mr\<^esup> = [] \ mr = 0" +apply(case_tac mr, auto simp: exponent_def) +done + +lemma [simp]: "wcode_on_left_moving_1 ires rs (b, []) = False" +apply(simp add: wcode_on_left_moving_1.simps wcode_on_left_moving_1_B.simps + wcode_on_left_moving_1_O.simps, auto) +done + + +declare wcode_on_checking_1.simps[simp del] + +lemmas wcode_double_case_inv_simps = + wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps + wcode_on_left_moving_1_B.simps wcode_on_checking_1.simps + wcode_erase1.simps wcode_on_right_moving_1.simps + wcode_goon_right_moving_1.simps wcode_backto_standard_pos.simps + + +lemma [simp]: "wcode_on_left_moving_1 ires rs (b, r) \ b \ []" +apply(simp add: wcode_double_case_inv_simps, auto) +done + + +lemma [elim]: "\wcode_on_left_moving_1 ires rs (b, Bk # list); + tl b = aa \ hd b # Bk # list = ba\ \ + wcode_on_left_moving_1 ires rs (aa, ba)" +apply(simp only: wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps + wcode_on_left_moving_1_B.simps) +apply(erule_tac disjE) +apply(erule_tac exE)+ +apply(case_tac ml, simp) +apply(rule_tac x = "mr - Suc (Suc 0)" in exI, rule_tac x = rn in exI) +apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind) +apply(rule_tac disjI1) +apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI, + simp add: exp_ind_def) +apply(erule_tac exE)+ +apply(simp) +done + + +lemma [elim]: + "\wcode_on_left_moving_1 ires rs (b, Oc # list); tl b = aa \ hd b # Oc # list = ba\ + \ wcode_on_checking_1 ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps) +apply(erule_tac disjE) +apply(erule_tac [!] exE)+ +apply(case_tac mr, simp, simp add: exp_ind_def) +apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp) +done + + +lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False" +apply(auto simp: wcode_double_case_inv_simps) +done + +lemma [simp]: "wcode_on_checking_1 ires rs (b, Bk # list) = False" +apply(auto simp: wcode_double_case_inv_simps) +done + +lemma [elim]: "\wcode_on_checking_1 ires rs (b, Oc # ba);Oc # b = aa \ list = ba\ + \ wcode_erase1 ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps) +apply(erule_tac exE)+ +apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp) +done + + +lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False" +apply(simp add: wcode_double_case_inv_simps) +done + +lemma [simp]: "wcode_on_checking_1 ires rs ([], Bk # list) = False" +apply(simp add: wcode_double_case_inv_simps) +done + +lemma [simp]: "wcode_erase1 ires rs (b, []) = False" +apply(simp add: wcode_double_case_inv_simps) +done + +lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False" +apply(simp add: wcode_double_case_inv_simps exp_ind_def) +done + +lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False" +apply(simp add: wcode_double_case_inv_simps exp_ind_def) +done + +lemma [elim]: "\wcode_on_right_moving_1 ires rs (b, Bk # ba); Bk # b = aa \ list = b\ \ + wcode_on_right_moving_1 ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps) +apply(erule_tac exE)+ +apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI, + rule_tac x = rn in exI) +apply(simp add: exp_ind_def) +apply(case_tac mr, simp, simp add: exp_ind_def) +done + +lemma [elim]: + "\wcode_on_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \ list = ba\ + \ wcode_goon_right_moving_1 ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps) +apply(erule_tac exE)+ +apply(rule_tac x = "Suc 0" in exI, rule_tac x = "rs" in exI, + rule_tac x = "ml - Suc (Suc 0)" in exI, rule_tac x = rn in exI) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(case_tac ml, simp, case_tac nat, simp, simp) +apply(simp add: exp_ind_def) +done + +lemma [simp]: + "wcode_on_right_moving_1 ires rs (b, []) \ False" +apply(simp add: wcode_double_case_inv_simps exponent_def) +done + +lemma [elim]: "\wcode_erase1 ires rs (b, Bk # ba); Bk # b = aa \ list = ba; c = Bk # ba\ + \ wcode_on_right_moving_1 ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps) +apply(erule_tac exE)+ +apply(rule_tac x = "Suc 0" in exI, rule_tac x = "Suc (Suc ln)" in exI, + rule_tac x = rn in exI, simp add: exp_ind) +done + +lemma [elim]: "\wcode_erase1 ires rs (aa, Oc # list); b = aa \ Bk # list = ba\ \ + wcode_erase1 ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps) +apply(erule_tac exE)+ +apply(rule_tac x = ln in exI, rule_tac x = rn in exI, auto) +done + +lemma [elim]: "\wcode_goon_right_moving_1 ires rs (aa, []); b = aa \ [Oc] = ba\ + \ wcode_backto_standard_pos ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps) +apply(erule_tac exE)+ +apply(rule_tac disjI2) +apply(simp only:wcode_backto_standard_pos_O.simps) +apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI, + rule_tac x = rn in exI, simp) +apply(case_tac mr, simp_all add: exponent_def) +done + +lemma [elim]: + "\wcode_goon_right_moving_1 ires rs (aa, Bk # list); b = aa \ Oc # list = ba\ + \ wcode_backto_standard_pos ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps) +apply(erule_tac exE)+ +apply(rule_tac disjI2) +apply(simp only:wcode_backto_standard_pos_O.simps) +apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI, + rule_tac x = "rn - Suc 0" in exI, simp) +apply(case_tac mr, simp, case_tac rn, simp, simp_all add: exp_ind_def) +done + +lemma [elim]: "\wcode_goon_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \ list = ba\ + \ wcode_goon_right_moving_1 ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps) +apply(erule_tac exE)+ +apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI, + rule_tac x = ln in exI, rule_tac x = rn in exI) +apply(simp add: exp_ind_def) +apply(case_tac mr, simp, case_tac rn, simp_all add: exp_ind_def) +done + +lemma [elim]: "\wcode_backto_standard_pos ires rs (b, []); Bk # b = aa\ \ False" +apply(auto simp: wcode_double_case_inv_simps wcode_backto_standard_pos_O.simps + wcode_backto_standard_pos_B.simps) +apply(case_tac mr, simp_all add: exp_ind_def) +done + +lemma [elim]: "\wcode_backto_standard_pos ires rs (b, Bk # ba); Bk # b = aa \ list = ba\ + \ wcode_before_double ires rs (aa, ba)" +apply(simp only: wcode_double_case_inv_simps wcode_backto_standard_pos_B.simps + wcode_backto_standard_pos_O.simps wcode_before_double.simps) +apply(erule_tac disjE) +apply(erule_tac exE)+ +apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp) +apply(auto) +apply(case_tac [!] mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wcode_backto_standard_pos ires rs ([], Oc # list) = False" +apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps + wcode_backto_standard_pos_O.simps) +done + +lemma [simp]: "wcode_backto_standard_pos ires rs (b, []) = False" +apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps + wcode_backto_standard_pos_O.simps) +apply(case_tac mr, simp, simp add: exp_ind_def) +done + +lemma [elim]: "\wcode_backto_standard_pos ires rs (b, Oc # list); tl b = aa; hd b # Oc # list = ba\ + \ wcode_backto_standard_pos ires rs (aa, ba)" +apply(simp only: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps + wcode_backto_standard_pos_O.simps) +apply(erule_tac disjE) +apply(simp) +apply(erule_tac exE)+ +apply(case_tac ml, simp) +apply(rule_tac disjI1, rule_tac conjI) +apply(rule_tac x = ln in exI, simp, rule_tac x = rn in exI, simp) +apply(rule_tac disjI2) +apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = ln in exI, + rule_tac x = rn in exI, simp) +apply(simp add: exp_ind_def) +done + +declare new_tape.simps[simp del] nth_of.simps[simp del] fetch.simps[simp del] +lemma wcode_double_case_first_correctness: + "let P = (\ (st, l, r). st = 13) in + let Q = (\ (st, l, r). wcode_double_case_inv st ires rs (l, r)) in + let f = (\ stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in + \ n .P (f n) \ Q (f (n::nat))" +proof - + let ?P = "(\ (st, l, r). st = 13)" + let ?Q = "(\ (st, l, r). wcode_double_case_inv st ires rs (l, r))" + let ?f = "(\ stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)" + have "\ n. ?P (?f n) \ ?Q (?f (n::nat))" + proof(rule_tac halt_lemma2) + show "wf wcode_double_case_le" + by auto + next + show "\ na. \ ?P (?f na) \ ?Q (?f na) \ + ?Q (?f (Suc na)) \ (?f (Suc na), ?f na) \ wcode_double_case_le" + proof(rule_tac allI, case_tac "?f na", simp add: tstep_red) + fix na a b c + show "a \ 13 \ wcode_double_case_inv a ires rs (b, c) \ + (case tstep (a, b, c) t_wcode_main of (st, x) \ + wcode_double_case_inv st ires rs x) \ + (tstep (a, b, c) t_wcode_main, a, b, c) \ wcode_double_case_le" + apply(rule_tac impI, simp add: wcode_double_case_inv.simps) + apply(auto split: if_splits simp: tstep.simps, + case_tac [!] c, simp_all, case_tac [!] "(c::block list)!0") + apply(simp_all add: new_tape.simps wcode_double_case_inv.simps wcode_double_case_le_def + lex_pair_def) + apply(auto split: if_splits) + done + qed + next + show "?Q (?f 0)" + apply(simp add: steps.simps wcode_double_case_inv.simps + wcode_on_left_moving_1.simps + wcode_on_left_moving_1_B.simps) + apply(rule_tac disjI1) + apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def) + apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def) + apply(auto) + done + next + show "\ ?P (?f 0)" + apply(simp add: steps.simps) + done + qed + thus "let P = \(st, l, r). st = 13; + Q = \(st, l, r). wcode_double_case_inv st ires rs (l, r); + f = steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main + in \n. P (f n) \ Q (f n)" + apply(simp add: Let_def) + done +qed + +lemma [elim]: "t_ncorrect tp + \ t_ncorrect (tshift tp a)" +apply(simp add: t_ncorrect.simps shift_length) +done + +lemma tshift_fetch: "\ fetch tp a b = (aa, st'); 0 < st'\ + \ fetch (tshift tp (length tp1 div 2)) a b + = (aa, st' + length tp1 div 2)" +apply(subgoal_tac "a > 0") +apply(auto simp: fetch.simps nth_of.simps shift_length nth_map + tshift.simps split: block.splits if_splits) +done + +lemma t_steps_steps_eq: "\steps (st, l, r) tp stp = (st', l', r'); + 0 < st'; + 0 < st \ st \ length tp div 2; + t_ncorrect tp1; + t_ncorrect tp\ + \ t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2), + length tp1 div 2) stp + = (st' + length tp1 div 2, l', r')" +apply(induct stp arbitrary: st' l' r', simp add: steps.simps t_steps.simps, + simp add: tstep_red stepn) +apply(case_tac "(steps (st, l, r) tp stp)", simp) +proof - + fix stp st' l' r' a b c + assume ind: "\st' l' r'. + \a = st' \ b = l' \ c = r'; 0 < st'\ + \ t_steps (st + length tp1 div 2, l, r) + (tshift tp (length tp1 div 2), length tp1 div 2) stp = + (st' + length tp1 div 2, l', r')" + and h: "tstep (a, b, c) tp = (st', l', r')" "0 < st'" "t_ncorrect tp1" "t_ncorrect tp" + have k: "t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2), + length tp1 div 2) stp = (a + length tp1 div 2, b, c)" + apply(rule_tac ind, simp) + using h + apply(case_tac a, simp_all add: tstep.simps fetch.simps) + done + from h and this show "t_step (t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2), length tp1 div 2) stp) + (tshift tp (length tp1 div 2), length tp1 div 2) = + (st' + length tp1 div 2, l', r')" + apply(simp add: k) + apply(simp add: tstep.simps t_step.simps) + apply(case_tac "fetch tp a (case c of [] \ Bk | x # xs \ x)", simp) + apply(subgoal_tac "fetch (tshift tp (length tp1 div 2)) a + (case c of [] \ Bk | x # xs \ x) = (aa, st' + length tp1 div 2)", simp) + apply(simp add: tshift_fetch) + done +qed + +lemma t_tshift_lemma: "\ steps (st, l, r) tp stp = (st', l', r'); + st' \ 0; + stp > 0; + 0 < st \ st \ length tp div 2; + t_ncorrect tp1; + t_ncorrect tp; + t_ncorrect tp2 + \ + \ \ stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp + = (st' + length tp1 div 2, l', r')" +proof - + assume h: "steps (st, l, r) tp stp = (st', l', r')" + "st' \ 0" "stp > 0" + "0 < st \ st \ length tp div 2" + "t_ncorrect tp1" + "t_ncorrect tp" + "t_ncorrect tp2" + from h have + "\stp>0. t_steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2, 0) stp = + (st' + length tp1 div 2, l', r')" + apply(rule_tac stp = stp in turing_shift, simp_all add: shift_length) + apply(simp add: t_steps_steps_eq) + apply(simp add: t_ncorrect.simps shift_length) + done + thus "\ stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp + = (st' + length tp1 div 2, l', r')" + apply(erule_tac exE) + apply(rule_tac x = stp in exI, simp) + apply(subgoal_tac "length (tp1 @ tshift tp (length tp1 div 2) @ tp2) mod 2 = 0") + apply(simp only: steps_eq) + using h + apply(auto simp: t_ncorrect.simps shift_length) + apply arith + done +qed + + +lemma t_twice_len_ge: "Suc 0 \ length t_twice div 2" +apply(simp add: t_twice_def tMp.simps shift_length) +done + +lemma [intro]: "rec_calc_rel (recf.id (Suc 0) 0) [rs] rs" + apply(rule_tac calc_id, simp_all) + done + +lemma [intro]: "rec_calc_rel (constn 2) [rs] 2" +using prime_rel_exec_eq[of "constn 2" "[rs]" 2] +apply(subgoal_tac "primerec (constn 2) 1", auto) +done + +lemma [intro]: "rec_calc_rel rec_mult [rs, 2] (2 * rs)" +using prime_rel_exec_eq[of "rec_mult" "[rs, 2]" "2*rs"] +apply(subgoal_tac "primerec rec_mult (Suc (Suc 0))", auto) +done +lemma t_twice_correct: "\stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp = + (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof(case_tac "rec_ci rec_twice") + fix a b c + assume h: "rec_ci rec_twice = (a, b, c)" + have "\stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_twice @ tMp (Suc 0) + (start_of twice_ly (length abc_twice) - 1)) stp = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2*rs)\<^esup> @ Bk\<^bsup>l\<^esup>)" + proof(rule_tac t_compiled_by_rec) + show "rec_ci rec_twice = (a, b, c)" by (simp add: h) + next + show "rec_calc_rel rec_twice [rs] (2 * rs)" + apply(simp add: rec_twice_def) + apply(rule_tac rs = "[rs, 2]" in calc_cn, simp_all) + apply(rule_tac allI, case_tac k, auto) + done + next + show "length [rs] = Suc 0" by simp + next + show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))" + by simp + next + show "start_of twice_ly (length abc_twice) = + start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))" + using h + apply(simp add: twice_ly_def abc_twice_def) + done + next + show "tm_of abc_twice = tm_of (a [+] dummy_abc (Suc 0))" + using h + apply(simp add: abc_twice_def) + done + qed + thus "\stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp = + (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) + done +qed + +lemma change_termi_state_fetch: "\fetch ap a b = (aa, st); st > 0\ + \ fetch (change_termi_state ap) a b = (aa, st)" +apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map + split: if_splits block.splits) +done + +lemma change_termi_state_exec_in_range: + "\steps (st, l, r) ap stp = (st', l', r'); st' \ 0\ + \ steps (st, l, r) (change_termi_state ap) stp = (st', l', r')" +proof(induct stp arbitrary: st l r st' l' r', simp add: steps.simps) + fix stp st l r st' l' r' + assume ind: "\st l r st' l' r'. + \steps (st, l, r) ap stp = (st', l', r'); st' \ 0\ \ + steps (st, l, r) (change_termi_state ap) stp = (st', l', r')" + and h: "steps (st, l, r) ap (Suc stp) = (st', l', r')" "st' \ 0" + from h show "steps (st, l, r) (change_termi_state ap) (Suc stp) = (st', l', r')" + proof(simp add: tstep_red, case_tac "steps (st, l, r) ap stp", simp) + fix a b c + assume g: "steps (st, l, r) ap stp = (a, b, c)" + "tstep (a, b, c) ap = (st', l', r')" "0 < st'" + hence "steps (st, l, r) (change_termi_state ap) stp = (a, b, c)" + apply(rule_tac ind, simp) + apply(case_tac a, simp_all add: tstep_0) + done + from g and this show "tstep (steps (st, l, r) (change_termi_state ap) stp) + (change_termi_state ap) = (st', l', r')" + apply(simp add: tstep.simps) + apply(case_tac "fetch ap a (case c of [] \ Bk | x # xs \ x)", simp) + apply(subgoal_tac "fetch (change_termi_state ap) a (case c of [] \ Bk | x # xs \ x) + = (aa, st')", simp) + apply(simp add: change_termi_state_fetch) + done + qed +qed + +lemma change_termi_state_fetch0: + "\0 < a; a \ length ap div 2; t_correct ap; fetch ap a b = (aa, 0)\ + \ fetch (change_termi_state ap) a b = (aa, Suc (length ap div 2))" +apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map + split: if_splits block.splits) +done + +lemma turing_change_termi_state: + "\steps (Suc 0, l, r) ap stp = (0, l', r'); t_correct ap\ + \ \ stp. steps (Suc 0, l, r) (change_termi_state ap) stp = + (Suc (length ap div 2), l', r')" +apply(drule first_halt_point) +apply(erule_tac exE) +apply(rule_tac x = "Suc stp" in exI, simp add: tstep_red) +apply(case_tac "steps (Suc 0, l, r) ap stp") +apply(simp add: isS0_def change_termi_state_exec_in_range) +apply(subgoal_tac "steps (Suc 0, l, r) (change_termi_state ap) stp = (a, b, c)", simp) +apply(simp add: tstep.simps) +apply(case_tac "fetch ap a (case c of [] \ Bk | x # xs \ x)", simp) +apply(subgoal_tac "fetch (change_termi_state ap) a + (case c of [] \ Bk | x # xs \ x) = (aa, Suc (length ap div 2))", simp) +apply(rule_tac ap = ap in change_termi_state_fetch0, simp_all) +apply(rule_tac tp = "(l, r)" and l = b and r = c and stp = stp and A = ap in s_keep, simp_all) +apply(simp add: change_termi_state_exec_in_range) +done + +lemma t_twice_change_term_state: + "\ stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp + = (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +using t_twice_correct[of ires rs n] +apply(erule_tac exE) +apply(erule_tac exE) +apply(erule_tac exE) +proof(drule_tac turing_change_termi_state) + fix stp ln rn + show "t_correct (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0))" + apply(rule_tac t_compiled_correct, simp_all) + apply(simp add: twice_ly_def) + done +next + fix stp ln rn + show "\stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + (change_termi_state (tm_of abc_twice @ tMp (Suc 0) + (start_of twice_ly (length abc_twice) - Suc 0))) stp = + (Suc (length (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) div 2), + Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \ + \stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp = + (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(erule_tac exE) + apply(simp add: t_twice_len_def t_twice_def) + apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp) + done +qed + +lemma t_twice_append_pre: + "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp + = (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) + \ \ stp>0. steps (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + (t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @ + ([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp + = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof(rule_tac t_tshift_lemma, simp_all add: t_twice_len_ge) + assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp = + (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + thus "0 < stp" + apply(case_tac stp, simp add: steps.simps t_twice_len_ge t_twice_len_def) + using t_twice_len_ge + apply(simp, simp) + done +next + show "t_ncorrect t_wcode_main_first_part" + apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def) + done +next + show "t_ncorrect t_twice" + using length_tm_even[of abc_twice] + apply(auto simp: t_ncorrect.simps t_twice_def) + apply(arith) + done +next + show "t_ncorrect ((L, Suc 0) # (L, Suc 0) # + tshift t_fourtimes (t_twice_len + 13) @ [(L, Suc 0), (L, Suc 0)])" + using length_tm_even[of abc_fourtimes] + apply(simp add: t_ncorrect.simps shift_length t_fourtimes_def) + apply arith + done +qed + +lemma t_twice_append: + "\ stp ln rn. steps (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + (t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @ + ([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp + = (Suc (t_twice_len) + length t_wcode_main_first_part div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using t_twice_change_term_state[of ires rs n] + apply(erule_tac exE) + apply(erule_tac exE) + apply(erule_tac exE) + apply(drule_tac t_twice_append_pre) + apply(erule_tac exE) + apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI) + apply(simp) + done + +lemma [simp]: "fetch t_wcode_main (Suc (t_twice_len + length t_wcode_main_first_part div 2)) Oc + = (L, Suc 0)" +apply(subgoal_tac "length (t_twice) mod 2 = 0") +apply(simp add: t_wcode_main_def nth_append fetch.simps t_wcode_main_first_part_def + nth_of.simps shift_length t_twice_len_def, auto) +apply(simp add: t_twice_def) +apply(subgoal_tac "length (tm_of abc_twice) mod 2 = 0") +apply arith +apply(rule_tac tm_even) +done + +lemma wcode_jump1: + "\ stp ln rn. steps (Suc (t_twice_len) + length t_wcode_main_first_part div 2, + Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>n\<^esup>) + t_wcode_main stp + = (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +apply(rule_tac x = "Suc 0" in exI, rule_tac x = "m" in exI, rule_tac x = n in exI) +apply(simp add: steps.simps tstep.simps exp_ind_def new_tape.simps) +apply(case_tac m, simp, simp add: exp_ind_def) +apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym]) +done + +lemma wcode_main_first_part_len: + "length t_wcode_main_first_part = 24" + apply(simp add: t_wcode_main_first_part_def) + done + +lemma wcode_double_case: + shows "\stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof - + have "\stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (13, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using wcode_double_case_first_correctness[of ires rs m n] + apply(simp) + apply(erule_tac exE) + apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na", + auto simp: wcode_double_case_inv.simps + wcode_before_double.simps) + apply(rule_tac x = na in exI, rule_tac x = ln in exI, rule_tac x = rn in exI) + apply(simp) + done + from this obtain stpa lna rna where stp1: + "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa = + (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast + have "\ stp ln rn. steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp = + (13 + t_twice_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using t_twice_append[of "Bk\<^bsup>lna\<^esup> @ Oc # ires" "Suc rs" rna] + apply(erule_tac exE) + apply(erule_tac exE) + apply(erule_tac exE) + apply(simp add: wcode_main_first_part_len) + apply(rule_tac x = stp in exI, rule_tac x = "ln + lna" in exI, + rule_tac x = rn in exI) + apply(simp add: t_wcode_main_def) + apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym]) + done + from this obtain stpb lnb rnb where stp2: + "steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb = + (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>)" by blast + have "\stp ln rn. steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, + Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp = + (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using wcode_jump1[of lnb "Oc # ires" "Suc rs" rnb] + apply(erule_tac exE) + apply(erule_tac exE) + apply(erule_tac exE) + apply(rule_tac x = stp in exI, + rule_tac x = ln in exI, + rule_tac x = rn in exI, simp add:wcode_main_first_part_len t_wcode_main_def) + apply(subgoal_tac "Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc # ires = Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires", simp) + apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym]) + apply(simp) + apply(case_tac lnb, simp, simp add: exp_ind_def[THEN sym] exp_ind) + done + from this obtain stpc lnc rnc where stp3: + "steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, + Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stpc = + (Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnc\<^esup>)" + by blast + from stp1 stp2 stp3 show "?thesis" + apply(rule_tac x = "stpa + stpb + stpc" in exI, rule_tac x = lnc in exI, + rule_tac x = rnc in exI) + apply(simp add: steps_add) + done +qed + + +(* Begin: fourtime_case*) +fun wcode_on_left_moving_2_B :: "bin_inv_t" + where + "wcode_on_left_moving_2_B ires rs (l, r) = + (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Oc # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr > Suc 0 \ mr > 0)" + +fun wcode_on_left_moving_2_O :: "bin_inv_t" + where + "wcode_on_left_moving_2_O ires rs (l, r) = + (\ ln rn. l = Bk # Oc # ires \ + r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_on_left_moving_2 :: "bin_inv_t" + where + "wcode_on_left_moving_2 ires rs (l, r) = + (wcode_on_left_moving_2_B ires rs (l, r) \ + wcode_on_left_moving_2_O ires rs (l, r))" + +fun wcode_on_checking_2 :: "bin_inv_t" + where + "wcode_on_checking_2 ires rs (l, r) = + (\ ln rn. l = Oc#ires \ + r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_goon_checking :: "bin_inv_t" + where + "wcode_goon_checking ires rs (l, r) = + (\ ln rn. l = ires \ + r = Oc # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_right_move :: "bin_inv_t" + where + "wcode_right_move ires rs (l, r) = + (\ ln rn. l = Oc # ires \ + r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_erase2 :: "bin_inv_t" + where + "wcode_erase2 ires rs (l, r) = + (\ ln rn. l = Bk # Oc # ires \ + tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_on_right_moving_2 :: "bin_inv_t" + where + "wcode_on_right_moving_2 ires rs (l, r) = + (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr > Suc 0)" + +fun wcode_goon_right_moving_2 :: "bin_inv_t" + where + "wcode_goon_right_moving_2 ires rs (l, r) = + (\ ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr = Suc rs)" + +fun wcode_backto_standard_pos_2_B :: "bin_inv_t" + where + "wcode_backto_standard_pos_2_B ires rs (l, r) = + (\ ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_backto_standard_pos_2_O :: "bin_inv_t" + where + "wcode_backto_standard_pos_2_O ires rs (l, r) = + (\ ml mr ln rn. l = Oc\<^bsup>ml \<^esup>@ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = (Suc (Suc rs)) \ mr > 0)" + +fun wcode_backto_standard_pos_2 :: "bin_inv_t" + where + "wcode_backto_standard_pos_2 ires rs (l, r) = + (wcode_backto_standard_pos_2_O ires rs (l, r) \ + wcode_backto_standard_pos_2_B ires rs (l, r))" + +fun wcode_before_fourtimes :: "bin_inv_t" + where + "wcode_before_fourtimes ires rs (l, r) = + (\ ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \ + r = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +declare wcode_on_left_moving_2_B.simps[simp del] wcode_on_left_moving_2.simps[simp del] + wcode_on_left_moving_2_O.simps[simp del] wcode_on_checking_2.simps[simp del] + wcode_goon_checking.simps[simp del] wcode_right_move.simps[simp del] + wcode_erase2.simps[simp del] + wcode_on_right_moving_2.simps[simp del] wcode_goon_right_moving_2.simps[simp del] + wcode_backto_standard_pos_2_B.simps[simp del] wcode_backto_standard_pos_2_O.simps[simp del] + wcode_backto_standard_pos_2.simps[simp del] + +lemmas wcode_fourtimes_invs = + wcode_on_left_moving_2_B.simps wcode_on_left_moving_2.simps + wcode_on_left_moving_2_O.simps wcode_on_checking_2.simps + wcode_goon_checking.simps wcode_right_move.simps + wcode_erase2.simps + wcode_on_right_moving_2.simps wcode_goon_right_moving_2.simps + wcode_backto_standard_pos_2_B.simps wcode_backto_standard_pos_2_O.simps + wcode_backto_standard_pos_2.simps + +fun wcode_fourtimes_case_inv :: "nat \ bin_inv_t" + where + "wcode_fourtimes_case_inv st ires rs (l, r) = + (if st = Suc 0 then wcode_on_left_moving_2 ires rs (l, r) + else if st = Suc (Suc 0) then wcode_on_checking_2 ires rs (l, r) + else if st = 7 then wcode_goon_checking ires rs (l, r) + else if st = 8 then wcode_right_move ires rs (l, r) + else if st = 9 then wcode_erase2 ires rs (l, r) + else if st = 10 then wcode_on_right_moving_2 ires rs (l, r) + else if st = 11 then wcode_goon_right_moving_2 ires rs (l, r) + else if st = 12 then wcode_backto_standard_pos_2 ires rs (l, r) + else if st = t_twice_len + 14 then wcode_before_fourtimes ires rs (l, r) + else False)" + +declare wcode_fourtimes_case_inv.simps[simp del] + +fun wcode_fourtimes_case_state :: "t_conf \ nat" + where + "wcode_fourtimes_case_state (st, l, r) = 13 - st" + +fun wcode_fourtimes_case_step :: "t_conf \ nat" + where + "wcode_fourtimes_case_step (st, l, r) = + (if st = Suc 0 then length l + else if st = 9 then + (if hd r = Oc then 1 + else 0) + else if st = 10 then length r + else if st = 11 then length r + else if st = 12 then length l + else 0)" + +fun wcode_fourtimes_case_measure :: "t_conf \ nat \ nat" + where + "wcode_fourtimes_case_measure (st, l, r) = + (wcode_fourtimes_case_state (st, l, r), + wcode_fourtimes_case_step (st, l, r))" + +definition wcode_fourtimes_case_le :: "(t_conf \ t_conf) set" + where "wcode_fourtimes_case_le \ (inv_image lex_pair wcode_fourtimes_case_measure)" + +lemma wf_wcode_fourtimes_case_le[intro]: "wf wcode_fourtimes_case_le" +by(auto intro:wf_inv_image simp: wcode_fourtimes_case_le_def) + +lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Bk = (L, 7)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 7 Oc = (R, 8)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 8 Bk = (R, 9)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 9 Bk = (R, 10)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 9 Oc = (W0, 9)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 10 Bk = (R, 10)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 10 Oc = (R, 11)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 11 Bk = (W1, 12)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 11 Oc = (R, 11)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 12 Oc = (L, 12)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_main 12 Bk = (R, t_twice_len + 14)" +apply(simp add: t_wcode_main_def fetch.simps + t_wcode_main_first_part_def nth_of.simps) +done + + +lemma [simp]: "wcode_on_left_moving_2 ires rs (b, []) = False" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_on_checking_2 ires rs (b, []) = False" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_goon_checking ires rs (b, []) = False" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_right_move ires rs (b, []) = False" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_erase2 ires rs (b, []) = False" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_on_right_moving_2 ires rs (b, []) = False" +apply(auto simp: wcode_fourtimes_invs exponent_def) +done + +lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, []) = False" +apply(auto simp: wcode_fourtimes_invs exponent_def) +done + +lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \ b \ []" +apply(simp add: wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \ wcode_on_left_moving_2 ires rs (tl b, hd b # Bk # list)" +apply(simp only: wcode_fourtimes_invs) +apply(erule_tac disjE) +apply(erule_tac exE)+ +apply(case_tac ml, simp) +apply(rule_tac x = "mr - (Suc (Suc 0))" in exI, rule_tac x = rn in exI, simp) +apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind) +apply(rule_tac disjI1) +apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI, + simp add: exp_ind_def) +apply(simp) +done + +lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list) \ b \ []" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list) + \ wcode_goon_checking ires rs (tl b, hd b # Bk # list)" +apply(simp only: wcode_fourtimes_invs) +apply(auto) +done + +lemma [simp]: "wcode_goon_checking ires rs (b, Bk # list) = False" +apply(simp add: wcode_fourtimes_invs) +done + +lemma [simp]: " wcode_right_move ires rs (b, Bk # list) \ b\ []" +apply(simp add: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_right_move ires rs (b, Bk # list) \ wcode_erase2 ires rs (Bk # b, list)" +apply(auto simp:wcode_fourtimes_invs ) +apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp) +done + +lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \ b \ []" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \ wcode_on_right_moving_2 ires rs (Bk # b, list)" +apply(auto simp:wcode_fourtimes_invs ) +apply(rule_tac x = "Suc (Suc 0)" in exI, simp add: exp_ind) +apply(rule_tac x = "Suc (Suc ln)" in exI, simp add: exp_ind, auto) +done + +lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) \ b \ []" +apply(auto simp:wcode_fourtimes_invs ) +done + +lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) + \ wcode_on_right_moving_2 ires rs (Bk # b, list)" +apply(auto simp: wcode_fourtimes_invs) +apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def) +apply(rule_tac x = "mr - 1" in exI, case_tac mr, auto simp: exp_ind_def) +done + +lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \ b \ []" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \ + wcode_backto_standard_pos_2 ires rs (b, Oc # list)" +apply(simp add: wcode_fourtimes_invs, auto) +apply(rule_tac x = ml in exI, auto) +apply(rule_tac x = "Suc 0" in exI, simp) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(rule_tac x = "rn - 1" in exI, simp) +apply(case_tac rn, simp, simp add: exp_ind_def) +done + +lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) \ b \ []" +apply(simp add: wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \ b \ []" +apply(simp add: wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \ + wcode_on_checking_2 ires rs (tl b, hd b # Oc # list)" +apply(auto simp: wcode_fourtimes_invs) +apply(case_tac [!] mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \ b \ []" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \ + wcode_backto_standard_pos_2 ires rs (b, [Oc])" +apply(simp only: wcode_fourtimes_invs) +apply(erule_tac exE)+ +apply(rule_tac disjI1) +apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, + rule_tac x = ln in exI, rule_tac x = rn in exI, simp) +apply(case_tac mr, simp, simp add: exp_ind_def) +done + +lemma "wcode_backto_standard_pos_2 ires rs (b, Bk # list) + \ (\ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \ (\rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +apply(auto simp: wcode_fourtimes_invs) +apply(case_tac [!] mr, auto simp: exp_ind_def) +done + + +lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) \ False" +apply(simp add: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_goon_checking ires rs (b, Oc # list) \ + (b = [] \ wcode_right_move ires rs ([Oc], list)) \ + (b \ [] \ wcode_right_move ires rs (Oc # b, list))" +apply(simp only: wcode_fourtimes_invs) +apply(erule_tac exE)+ +apply(auto) +done + +lemma [simp]: "wcode_right_move ires rs (b, Oc # list) = False" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: " wcode_erase2 ires rs (b, Oc # list) \ b \ []" +apply(simp add: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_erase2 ires rs (b, Oc # list) + \ wcode_erase2 ires rs (b, Bk # list)" +apply(auto simp: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list) \ b \ []" +apply(simp only: wcode_fourtimes_invs) +apply(auto) +done + +lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list) + \ wcode_goon_right_moving_2 ires rs (Oc # b, list)" +apply(auto simp: wcode_fourtimes_invs) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(rule_tac x = "Suc 0" in exI, auto) +apply(rule_tac x = "ml - 2" in exI) +apply(case_tac ml, simp, case_tac nat, simp_all add: exp_ind_def) +done + +lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \ b \ []" +apply(simp only:wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) + \ (\ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \ (\rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +apply(simp add: wcode_fourtimes_invs, auto) +apply(case_tac [!] mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) = False" +apply(simp add: wcode_fourtimes_invs) +done + +lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \ + wcode_goon_right_moving_2 ires rs (Oc # b, list)" +apply(simp only:wcode_fourtimes_invs, auto) +apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def) +apply(rule_tac x = "mr - 1" in exI) +apply(case_tac mr, case_tac rn, auto simp: exp_ind_def) +done + +lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) \ b \ []" +apply(simp only: wcode_fourtimes_invs, auto) +done + +lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) + \ wcode_backto_standard_pos_2 ires rs (tl b, hd b # Oc # list)" +apply(simp only: wcode_fourtimes_invs) +apply(erule_tac disjE) +apply(erule_tac exE)+ +apply(case_tac ml, simp) +apply(rule_tac disjI2) +apply(rule_tac conjI, rule_tac x = ln in exI, simp) +apply(rule_tac x = rn in exI, simp) +apply(rule_tac disjI1) +apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, + rule_tac x = ln in exI, rule_tac x = rn in exI, simp add: exp_ind_def) +apply(simp) +done + +lemma wcode_fourtimes_case_first_correctness: + shows "let P = (\ (st, l, r). st = t_twice_len + 14) in + let Q = (\ (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r)) in + let f = (\ stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in + \ n .P (f n) \ Q (f (n::nat))" +proof - + let ?P = "(\ (st, l, r). st = t_twice_len + 14)" + let ?Q = "(\ (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r))" + let ?f = "(\ stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)" + have "\ n . ?P (?f n) \ ?Q (?f (n::nat))" + proof(rule_tac halt_lemma2) + show "wf wcode_fourtimes_case_le" + by auto + next + show "\ na. \ ?P (?f na) \ ?Q (?f na) \ + ?Q (?f (Suc na)) \ (?f (Suc na), ?f na) \ wcode_fourtimes_case_le" + apply(rule_tac allI, + case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na", simp, + rule_tac impI) + apply(simp add: tstep_red tstep.simps, case_tac c, simp, case_tac [2] aa, simp_all) + + apply(simp_all add: wcode_fourtimes_case_inv.simps new_tape.simps + wcode_fourtimes_case_le_def lex_pair_def split: if_splits) + done + next + show "?Q (?f 0)" + apply(simp add: steps.simps wcode_fourtimes_case_inv.simps) + apply(simp add: wcode_on_left_moving_2.simps wcode_on_left_moving_2_B.simps + wcode_on_left_moving_2_O.simps) + apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def) + apply(rule_tac x ="Suc 0" in exI, auto) + done + next + show "\ ?P (?f 0)" + apply(simp add: steps.simps) + done + qed + thus "?thesis" + apply(erule_tac exE, simp) + done +qed + +definition t_fourtimes_len :: "nat" + where + "t_fourtimes_len = (length t_fourtimes div 2)" + +lemma t_fourtimes_len_gr: "t_fourtimes_len > 0" +apply(simp add: t_fourtimes_len_def t_fourtimes_def) +done + +lemma t_fourtimes_correct: + "\stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp = + (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof(case_tac "rec_ci rec_fourtimes") + fix a b c + assume h: "rec_ci rec_fourtimes = (a, b, c)" + have "\stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_fourtimes @ tMp (Suc 0) + (start_of fourtimes_ly (length abc_fourtimes) - 1)) stp = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4*rs)\<^esup> @ Bk\<^bsup>l\<^esup>)" + proof(rule_tac t_compiled_by_rec) + show "rec_ci rec_fourtimes = (a, b, c)" by (simp add: h) + next + show "rec_calc_rel rec_fourtimes [rs] (4 * rs)" + using prime_rel_exec_eq [of rec_fourtimes "[rs]" "4 * rs"] + apply(subgoal_tac "primerec rec_fourtimes (length [rs])") + apply(simp_all add: rec_fourtimes_def rec_exec.simps) + apply(auto) + apply(simp only: Nat.One_nat_def[THEN sym], auto) + done + next + show "length [rs] = Suc 0" by simp + next + show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))" + by simp + next + show "start_of fourtimes_ly (length abc_fourtimes) = + start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))" + using h + apply(simp add: fourtimes_ly_def abc_fourtimes_def) + done + next + show "tm_of abc_fourtimes = tm_of (a [+] dummy_abc (Suc 0))" + using h + apply(simp add: abc_fourtimes_def) + done + qed + thus "\stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp = + (0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) + done +qed + +lemma t_fourtimes_change_term_state: + "\ stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp + = (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +using t_fourtimes_correct[of ires rs n] +apply(erule_tac exE) +apply(erule_tac exE) +apply(erule_tac exE) +proof(drule_tac turing_change_termi_state) + fix stp ln rn + show "t_correct (tm_of abc_fourtimes @ tMp (Suc 0) + (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))" + apply(rule_tac t_compiled_correct, auto simp: fourtimes_ly_def) + done +next + fix stp ln rn + show "\stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) + (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) stp = + (Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly + (length abc_fourtimes) - Suc 0)) div 2), Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \ + \stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp = + (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(erule_tac exE) + apply(simp add: t_fourtimes_len_def t_fourtimes_def) + apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp) + done +qed + +lemma t_fourtimes_append_pre: + "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp + = (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) + \ \ stp>0. steps (Suc 0 + length (t_wcode_main_first_part @ + tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, + Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + ((t_wcode_main_first_part @ + tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @ + tshift t_fourtimes (length (t_wcode_main_first_part @ + tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2) @ ([(L, 1), (L, 1)])) stp + = (Suc t_fourtimes_len + length (t_wcode_main_first_part @ + tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, + Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof(rule_tac t_tshift_lemma, auto) + assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp = + (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + thus "0 < stp" + using t_fourtimes_len_gr + apply(case_tac stp, simp_all add: steps.simps) + done +next + show "Suc 0 \ length t_fourtimes div 2" + apply(simp add: t_fourtimes_def shift_length tMp.simps) + done +next + show "t_ncorrect (t_wcode_main_first_part @ + tshift t_twice (length t_wcode_main_first_part div 2) @ + [(L, Suc 0), (L, Suc 0)])" + apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def shift_length + t_twice_def) + using tm_even[of abc_twice] + by arith +next + show "t_ncorrect t_fourtimes" + apply(simp add: t_fourtimes_def steps.simps t_ncorrect.simps) + using tm_even[of abc_fourtimes] + by arith +next + show "t_ncorrect [(L, Suc 0), (L, Suc 0)]" + apply(simp add: t_ncorrect.simps) + done +qed + +lemma [simp]: "length t_wcode_main_first_part = 24" +apply(simp add: t_wcode_main_first_part_def) +done + +lemma [simp]: "(26 + length t_twice) div 2 = (length t_twice) div 2 + 13" +using tm_even[of abc_twice] +apply(simp add: t_twice_def) +done + +lemma [simp]: "((26 + length (tshift t_twice 12)) div 2) + = (length (tshift t_twice 12) div 2 + 13)" +using tm_even[of abc_twice] +apply(simp add: t_twice_def) +done + +lemma [simp]: "t_twice_len + 14 = 14 + length (tshift t_twice 12) div 2" +using tm_even[of abc_twice] +apply(simp add: t_twice_def t_twice_len_def shift_length) +done + +lemma t_fourtimes_append: + "\ stp ln rn. + steps (Suc 0 + length (t_wcode_main_first_part @ tshift t_twice + (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, + Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + ((t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @ + [(L, 1), (L, 1)]) @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp + = (Suc t_fourtimes_len + length (t_wcode_main_first_part @ tshift t_twice + (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, + Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using t_fourtimes_change_term_state[of ires rs n] + apply(erule_tac exE) + apply(erule_tac exE) + apply(erule_tac exE) + apply(drule_tac t_fourtimes_append_pre) + apply(erule_tac exE) + apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI) + apply(simp add: t_twice_len_def shift_length) + done + +lemma t_wcode_main_len: "length t_wcode_main = length t_twice + length t_fourtimes + 28" +apply(simp add: t_wcode_main_def shift_length) +done + +lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) b + = (L, Suc 0)" +using tm_even[of "abc_twice"] tm_even[of "abc_fourtimes"] +apply(case_tac b) +apply(simp_all only: fetch.simps) +apply(auto simp: nth_of.simps t_wcode_main_len t_twice_len_def + t_fourtimes_def t_twice_def t_fourtimes_def t_fourtimes_len_def) +apply(auto simp: t_wcode_main_def t_wcode_main_first_part_def shift_length t_twice_def nth_append + t_fourtimes_def) +done + +lemma wcode_jump2: + "\ stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len + , Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp = + (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +apply(rule_tac x = "Suc 0" in exI) +apply(simp add: steps.simps shift_length) +apply(rule_tac x = lnb in exI, rule_tac x = rnb in exI) +apply(simp add: tstep.simps new_tape.simps) +done + +lemma wcode_fourtimes_case: + shows "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof - + have "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (t_twice_len + 14, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using wcode_fourtimes_case_first_correctness[of ires rs m n] + apply(simp add: wcode_fourtimes_case_inv.simps, auto) + apply(rule_tac x = na in exI, rule_tac x = ln in exI, + rule_tac x = rn in exI) + apply(simp) + done + from this obtain stpa lna rna where stp1: + "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Oc # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa = + (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast + have "\stp ln rn. steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>) + t_wcode_main stp = + (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using t_fourtimes_append[of " Bk\<^bsup>lna\<^esup> @ Oc # ires" "rs + 1" rna] + apply(erule_tac exE) + apply(erule_tac exE) + apply(erule_tac exE) + apply(simp add: t_wcode_main_def) + apply(rule_tac x = stp in exI, + rule_tac x = "ln + lna" in exI, + rule_tac x = rn in exI, simp) + apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym]) + done + from this obtain stpb lnb rnb where stp2: + "steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>) + t_wcode_main stpb = + (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>)" + by blast + have "\stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len, + Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>) + t_wcode_main stp = + (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(rule wcode_jump2) + done + from this obtain stpc lnc rnc where stp3: + "steps (t_twice_len + 14 + t_fourtimes_len, + Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>) + t_wcode_main stpc = + (Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnc\<^esup>)" + by blast + from stp1 stp2 stp3 show "?thesis" + apply(rule_tac x = "stpa + stpb + stpc" in exI, + rule_tac x = lnc in exI, rule_tac x = rnc in exI) + apply(simp add: steps_add) + done +qed + +(**********************************************************) + +fun wcode_on_left_moving_3_B :: "bin_inv_t" + where + "wcode_on_left_moving_3_B ires rs (l, r) = + (\ ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Bk # ires \ + r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr > Suc 0 \ mr > 0 )" + +fun wcode_on_left_moving_3_O :: "bin_inv_t" + where + "wcode_on_left_moving_3_O ires rs (l, r) = + (\ ln rn. l = Bk # Bk # ires \ + r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_on_left_moving_3 :: "bin_inv_t" + where + "wcode_on_left_moving_3 ires rs (l, r) = + (wcode_on_left_moving_3_B ires rs (l, r) \ + wcode_on_left_moving_3_O ires rs (l, r))" + +fun wcode_on_checking_3 :: "bin_inv_t" + where + "wcode_on_checking_3 ires rs (l, r) = + (\ ln rn. l = Bk # ires \ + r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_goon_checking_3 :: "bin_inv_t" + where + "wcode_goon_checking_3 ires rs (l, r) = + (\ ln rn. l = ires \ + r = Bk # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_stop :: "bin_inv_t" + where + "wcode_stop ires rs (l, r) = + (\ ln rn. l = Bk # ires \ + r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wcode_halt_case_inv :: "nat \ bin_inv_t" + where + "wcode_halt_case_inv st ires rs (l, r) = + (if st = 0 then wcode_stop ires rs (l, r) + else if st = Suc 0 then wcode_on_left_moving_3 ires rs (l, r) + else if st = Suc (Suc 0) then wcode_on_checking_3 ires rs (l, r) + else if st = 7 then wcode_goon_checking_3 ires rs (l, r) + else False)" + +fun wcode_halt_case_state :: "t_conf \ nat" + where + "wcode_halt_case_state (st, l, r) = + (if st = 1 then 5 + else if st = Suc (Suc 0) then 4 + else if st = 7 then 3 + else 0)" + +fun wcode_halt_case_step :: "t_conf \ nat" + where + "wcode_halt_case_step (st, l, r) = + (if st = 1 then length l + else 0)" + +fun wcode_halt_case_measure :: "t_conf \ nat \ nat" + where + "wcode_halt_case_measure (st, l, r) = + (wcode_halt_case_state (st, l, r), + wcode_halt_case_step (st, l, r))" + +definition wcode_halt_case_le :: "(t_conf \ t_conf) set" + where "wcode_halt_case_le \ (inv_image lex_pair wcode_halt_case_measure)" + +lemma wf_wcode_halt_case_le[intro]: "wf wcode_halt_case_le" +by(auto intro:wf_inv_image simp: wcode_halt_case_le_def) + +declare wcode_on_left_moving_3_B.simps[simp del] wcode_on_left_moving_3_O.simps[simp del] + wcode_on_checking_3.simps[simp del] wcode_goon_checking_3.simps[simp del] + wcode_on_left_moving_3.simps[simp del] wcode_stop.simps[simp del] + +lemmas wcode_halt_invs = + wcode_on_left_moving_3_B.simps wcode_on_left_moving_3_O.simps + wcode_on_checking_3.simps wcode_goon_checking_3.simps + wcode_on_left_moving_3.simps wcode_stop.simps + +lemma [simp]: "fetch t_wcode_main 7 Bk = (R, 0)" +apply(simp add: fetch.simps t_wcode_main_def nth_append nth_of.simps + t_wcode_main_first_part_def) +done + +lemma [simp]: "wcode_on_left_moving_3 ires rs (b, []) = False" +apply(simp only: wcode_halt_invs) +apply(simp add: exp_ind_def) +done + +lemma [simp]: "wcode_on_checking_3 ires rs (b, []) = False" +apply(simp add: wcode_halt_invs) +done + +lemma [simp]: "wcode_goon_checking_3 ires rs (b, []) = False" +apply(simp add: wcode_halt_invs) +done + +lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list) + \ wcode_on_left_moving_3 ires rs (tl b, hd b # Bk # list)" +apply(simp only: wcode_halt_invs) +apply(erule_tac disjE) +apply(erule_tac exE)+ +apply(case_tac ml, simp) +apply(rule_tac x = "mr - 2" in exI, rule_tac x = rn in exI) +apply(case_tac mr, simp, simp add: exp_ind, simp add: exp_ind[THEN sym]) +apply(rule_tac disjI1) +apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, + rule_tac x = rn in exI, simp add: exp_ind_def) +apply(simp) +done + +lemma [simp]: "wcode_goon_checking_3 ires rs (b, Bk # list) \ + (b = [] \ wcode_stop ires rs ([Bk], list)) \ + (b \ [] \ wcode_stop ires rs (Bk # b, list))" +apply(auto simp: wcode_halt_invs) +done + +lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \ b \ []" +apply(auto simp: wcode_halt_invs) +done + +lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \ + wcode_on_checking_3 ires rs (tl b, hd b # Oc # list)" +apply(simp add:wcode_halt_invs, auto) +apply(case_tac [!] mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wcode_on_checking_3 ires rs (b, Oc # list) = False" +apply(auto simp: wcode_halt_invs) +done + +lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list) \ b \ []" +apply(simp add: wcode_halt_invs, auto) +done + + +lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \ b \ []" +apply(auto simp: wcode_halt_invs) +done + +lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \ + wcode_goon_checking_3 ires rs (tl b, hd b # Bk # list)" +apply(auto simp: wcode_halt_invs) +done + +lemma [simp]: "wcode_goon_checking_3 ires rs (b, Oc # list) = False" +apply(simp add: wcode_goon_checking_3.simps) +done + +lemma t_halt_case_correctness: +shows "let P = (\ (st, l, r). st = 0) in + let Q = (\ (st, l, r). wcode_halt_case_inv st ires rs (l, r)) in + let f = (\ stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp) in + \ n .P (f n) \ Q (f (n::nat))" +proof - + let ?P = "(\ (st, l, r). st = 0)" + let ?Q = "(\ (st, l, r). wcode_halt_case_inv st ires rs (l, r))" + let ?f = "(\ stp. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp)" + have "\ n. ?P (?f n) \ ?Q (?f (n::nat))" + proof(rule_tac halt_lemma2) + show "wf wcode_halt_case_le" by auto + next + show "\ na. \ ?P (?f na) \ ?Q (?f na) \ + ?Q (?f (Suc na)) \ (?f (Suc na), ?f na) \ wcode_halt_case_le" + apply(rule_tac allI, rule_tac impI, case_tac "?f na") + apply(simp add: tstep_red tstep.simps) + apply(case_tac c, simp, case_tac [2] aa) + apply(simp_all split: if_splits add: new_tape.simps wcode_halt_case_le_def lex_pair_def) + done + next + show "?Q (?f 0)" + apply(simp add: steps.simps wcode_halt_invs) + apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def) + apply(rule_tac x = "Suc 0" in exI, auto) + done + next + show "\ ?P (?f 0)" + apply(simp add: steps.simps) + done + qed + thus "?thesis" + apply(auto) + done +qed + +declare wcode_halt_case_inv.simps[simp del] +lemma [intro]: "\ xs. ( :: block list) = Oc # xs" +apply(case_tac "rev list", simp) +apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def) +apply(case_tac list, simp, simp) +done + +lemma wcode_halt_case: + "\stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + t_wcode_main stp = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using t_halt_case_correctness[of ires rs m n] +apply(simp) +apply(erule_tac exE) +apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na") +apply(auto simp: wcode_halt_case_inv.simps wcode_stop.simps) +apply(rule_tac x = na in exI, rule_tac x = ln in exI, + rule_tac x = rn in exI, simp) +done + +lemma bl_bin_one: "bl_bin [Oc] = Suc 0" +apply(simp add: bl_bin.simps) +done + +lemma t_wcode_main_lemma_pre: + "\args \ []; lm = \ \ + \ stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main + stp + = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2^(length lm - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof(induct "length args" arbitrary: args lm rs m n, simp) + fix x args lm rs m n + assume ind: + "\args lm rs m n. + \x = length args; (args::nat list) \ []; lm = \ + \ \stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + + and h: "Suc x = length args" "(args::nat list) \ []" "lm = " + from h have "\ (a::nat) xs. args = xs @ [a]" + apply(rule_tac x = "last args" in exI) + apply(rule_tac x = "butlast args" in exI, auto) + done + from this obtain a xs where "args = xs @ [a]" by blast + from h and this show + "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof(case_tac "xs::nat list", simp) + show "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin () + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof(induct "a" arbitrary: m n rs ires, simp) + fix m n rs ires + show "\stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) + t_wcode_main stp = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin [Oc] + rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(simp add: bl_bin_one) + apply(rule_tac wcode_halt_case) + done + next + fix a m n rs ires + assume ind2: + "\m n rs ires. + \stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin () + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" + show "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin () + rs * 2 ^ Suc a\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof - + have "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(simp add: tape_of_nat) + using wcode_double_case[of m "Oc\<^bsup>a\<^esup> @ Bk # Bk # ires" rs n] + apply(simp add: exp_ind_def) + done + from this obtain stpa lna rna where stp1: + "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa = + (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast + moreover have + "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin () + (2*rs + 2) * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using ind2[of lna ires "2*rs + 2" rna] by simp + from this obtain stpb lnb rnb where stp2: + "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev () @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin () + (2*rs + 2) * 2 ^ a\<^esup> @ Bk\<^bsup>rnb\<^esup>)" + by blast + from stp1 and stp2 show "?thesis" + apply(rule_tac x = "stpa + stpb" in exI, + rule_tac x = lnb in exI, rule_tac x = rnb in exI, simp) + apply(simp add: steps_add bl_bin_nat_Suc exponent_def) + done + qed + qed + next + fix aa list + assume g: "Suc x = length args" "args \ []" "lm = " "args = xs @ [a::nat]" "xs = (aa::nat) # list" + thus "\stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev lm @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin lm + rs * 2 ^ (length lm - 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof(induct a arbitrary: m n rs args lm, simp_all add: tape_of_nl_rev, + simp only: tape_of_nl_cons_app1, simp) + fix m n rs args lm + have "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<(aa::nat) # list>) @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev () @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof(simp add: tape_of_nl_rev) + have "\ xs. () = Oc # xs" by auto + from this obtain xs where "() = Oc # xs" .. + thus "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ @ Bk # Bk # ires, Bk # Oc\<^bsup>5 + 4 * rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(simp) + using wcode_fourtimes_case[of m "xs @ Bk # Bk # ires" rs n] + apply(simp) + done + qed + from this obtain stpa lna rna where stp1: + "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev () @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa = + (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev () @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast + from g have + "\ stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp = (0, Bk # ires, + Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()+ (4*rs + 4) * 2^(length () - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(rule_tac args = "(aa::nat)#list" in ind, simp_all) + done + from this obtain stpb lnb rnb where stp2: + "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb = (0, Bk # ires, + Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()+ (4*rs + 4) * 2^(length () - 1) \<^esup> @ Bk\<^bsup>rnb\<^esup>)" + by blast + from stp1 and stp2 and h + show "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # + Bk # Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # ) + rs * (2 * 2 ^ (aa + length ()))\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI, + rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_rev) + done + next + fix ab m n rs args lm + assume ind2: + "\ m n rs args lm. + \lm = ; args = aa # list @ [ab]\ + \ \stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # + Bk # Oc\<^bsup>bl_bin () + rs * 2 ^ (length () - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + and k: "args = aa # list @ [Suc ab]" "lm = " + show "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires,Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # + Bk # Oc\<^bsup>bl_bin () + rs * 2 ^ (length () - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + proof(simp add: tape_of_nl_cons_app1) + have "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # @ Bk # Bk # ires, + Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp + = (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + using wcode_double_case[of m "Oc\<^bsup>ab\<^esup> @ Bk # @ Bk # Bk # ires" + rs n] + apply(simp add: exp_ind_def) + done + from this obtain stpa lna rna where stp1: + "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # @ Bk # Bk # ires, + Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa + = (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast + from k have + "\ stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp + = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # + Bk # Oc\<^bsup>bl_bin ( ) + (2*rs + 2)* 2^(length () - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(rule_tac ind2, simp_all) + done + from this obtain stpb lnb rnb where stp2: + "steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb + = (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # + Bk # Oc\<^bsup>bl_bin ( ) + (2*rs + 2)* 2^(length () - Suc 0)\<^esup> @ Bk\<^bsup>rnb\<^esup>)" + by blast + from stp1 and stp2 show + "\stp ln rn. + steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # @ Bk # Bk # ires, + Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = + (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # + Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # ) + rs * (2 * 2 ^ (aa + length ()))\<^esup> + @ Bk\<^bsup>rn\<^esup>)" + apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI, + rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_cons_app1 exp_ind_def) + done + qed + qed + qed + qed + + + +(* turing_shift can be used*) +term t_wcode_main +definition t_wcode_prepare :: "tprog" + where + "t_wcode_prepare \ + [(W1, 2), (L, 1), (L, 3), (R, 2), (R, 4), (W0, 3), + (R, 4), (R, 5), (R, 6), (R, 5), (R, 7), (R, 5), + (W1, 7), (L, 0)]" + +fun wprepare_add_one :: "nat \ nat list \ tape \ bool" + where + "wprepare_add_one m lm (l, r) = + (\ rn. l = [] \ + (r = @ Bk\<^bsup>rn\<^esup> \ + r = Bk # @ Bk\<^bsup>rn\<^esup>))" + +fun wprepare_goto_first_end :: "nat \ nat list \ tape \ bool" + where + "wprepare_goto_first_end m lm (l, r) = + (\ ml mr rn. l = Oc\<^bsup>ml\<^esup> \ + r = Oc\<^bsup>mr\<^esup> @ Bk # @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc m))" + +fun wprepare_erase :: "nat \ nat list \ tape \ bool" + where + "wprepare_erase m lm (l, r) = + (\ rn. l = Oc\<^bsup>Suc m\<^esup> \ + tl r = Bk # @ Bk\<^bsup>rn\<^esup>)" + +fun wprepare_goto_start_pos_B :: "nat \ nat list \ tape \ bool" + where + "wprepare_goto_start_pos_B m lm (l, r) = + (\ rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk # @ Bk\<^bsup>rn\<^esup>)" + +fun wprepare_goto_start_pos_O :: "nat \ nat list \ tape \ bool" + where + "wprepare_goto_start_pos_O m lm (l, r) = + (\ rn. l = Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + r = @ Bk\<^bsup>rn\<^esup>)" + +fun wprepare_goto_start_pos :: "nat \ nat list \ tape \ bool" + where + "wprepare_goto_start_pos m lm (l, r) = + (wprepare_goto_start_pos_B m lm (l, r) \ + wprepare_goto_start_pos_O m lm (l, r))" + +fun wprepare_loop_start_on_rightmost :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_start_on_rightmost m lm (l, r) = + (\ rn mr. rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # @ Bk\<^bsup>rn\<^esup> \ l \ [] \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wprepare_loop_start_in_middle :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_start_in_middle m lm (l, r) = + (\ rn (mr:: nat) (lm1::nat list). + rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # @ Bk\<^bsup>rn\<^esup> \ l \ [] \ + r = Oc\<^bsup>mr\<^esup> @ Bk # @ Bk\<^bsup>rn\<^esup> \ lm1 \ [])" + +fun wprepare_loop_start :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_start m lm (l, r) = (wprepare_loop_start_on_rightmost m lm (l, r) \ + wprepare_loop_start_in_middle m lm (l, r))" + +fun wprepare_loop_goon_on_rightmost :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_goon_on_rightmost m lm (l, r) = + (\ rn. l = Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>rn\<^esup>)" + +fun wprepare_loop_goon_in_middle :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_goon_in_middle m lm (l, r) = + (\ rn (mr:: nat) (lm1::nat list). + rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # @ Bk\<^bsup>rn\<^esup> \ l \ [] \ + (if lm1 = [] then r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> + else r = Oc\<^bsup>mr\<^esup> @ Bk # @ Bk\<^bsup>rn\<^esup>) \ mr > 0)" + +fun wprepare_loop_goon :: "nat \ nat list \ tape \ bool" + where + "wprepare_loop_goon m lm (l, r) = + (wprepare_loop_goon_in_middle m lm (l, r) \ + wprepare_loop_goon_on_rightmost m lm (l, r))" + +fun wprepare_add_one2 :: "nat \ nat list \ tape \ bool" + where + "wprepare_add_one2 m lm (l, r) = + (\ rn. l = Bk # Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + (r = [] \ tl r = Bk\<^bsup>rn\<^esup>))" + +fun wprepare_stop :: "nat \ nat list \ tape \ bool" + where + "wprepare_stop m lm (l, r) = + (\ rn. l = Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk # Oc # Bk\<^bsup>rn\<^esup>)" + +fun wprepare_inv :: "nat \ nat \ nat list \ tape \ bool" + where + "wprepare_inv st m lm (l, r) = + (if st = 0 then wprepare_stop m lm (l, r) + else if st = Suc 0 then wprepare_add_one m lm (l, r) + else if st = Suc (Suc 0) then wprepare_goto_first_end m lm (l, r) + else if st = Suc (Suc (Suc 0)) then wprepare_erase m lm (l, r) + else if st = 4 then wprepare_goto_start_pos m lm (l, r) + else if st = 5 then wprepare_loop_start m lm (l, r) + else if st = 6 then wprepare_loop_goon m lm (l, r) + else if st = 7 then wprepare_add_one2 m lm (l, r) + else False)" + +fun wprepare_stage :: "t_conf \ nat" + where + "wprepare_stage (st, l, r) = + (if st \ 1 \ st \ 4 then 3 + else if st = 5 \ st = 6 then 2 + else 1)" + +fun wprepare_state :: "t_conf \ nat" + where + "wprepare_state (st, l, r) = + (if st = 1 then 4 + else if st = Suc (Suc 0) then 3 + else if st = Suc (Suc (Suc 0)) then 2 + else if st = 4 then 1 + else if st = 7 then 2 + else 0)" + +fun wprepare_step :: "t_conf \ nat" + where + "wprepare_step (st, l, r) = + (if st = 1 then (if hd r = Oc then Suc (length l) + else 0) + else if st = Suc (Suc 0) then length r + else if st = Suc (Suc (Suc 0)) then (if hd r = Oc then 1 + else 0) + else if st = 4 then length r + else if st = 5 then Suc (length r) + else if st = 6 then (if r = [] then 0 else Suc (length r)) + else if st = 7 then (if (r \ [] \ hd r = Oc) then 0 + else 1) + else 0)" + +fun wcode_prepare_measure :: "t_conf \ nat \ nat \ nat" + where + "wcode_prepare_measure (st, l, r) = + (wprepare_stage (st, l, r), + wprepare_state (st, l, r), + wprepare_step (st, l, r))" + +definition wcode_prepare_le :: "(t_conf \ t_conf) set" + where "wcode_prepare_le \ (inv_image lex_triple wcode_prepare_measure)" + +lemma [intro]: "wf lex_pair" +by(auto intro:wf_lex_prod simp:lex_pair_def) + +lemma wf_wcode_prepare_le[intro]: "wf wcode_prepare_le" +by(auto intro:wf_inv_image simp: wcode_prepare_le_def + recursive.lex_triple_def) + +declare wprepare_add_one.simps[simp del] wprepare_goto_first_end.simps[simp del] + wprepare_erase.simps[simp del] wprepare_goto_start_pos.simps[simp del] + wprepare_loop_start.simps[simp del] wprepare_loop_goon.simps[simp del] + wprepare_add_one2.simps[simp del] + +lemmas wprepare_invs = wprepare_add_one.simps wprepare_goto_first_end.simps + wprepare_erase.simps wprepare_goto_start_pos.simps + wprepare_loop_start.simps wprepare_loop_goon.simps + wprepare_add_one2.simps + +declare wprepare_inv.simps[simp del] +lemma [simp]: "fetch t_wcode_prepare (Suc 0) Bk = (W1, 2)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare (Suc 0) Oc = (L, 1)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Bk = (L, 3)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Oc = (R, 2)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Bk = (R, 4)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Oc = (W0, 3)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare 4 Bk = (R, 4)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare 4 Oc = (R, 5)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare 5 Oc = (R, 5)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare 5 Bk = (R, 6)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare 6 Oc = (R, 5)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare 6 Bk = (R, 7)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare 7 Oc = (L, 0)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_prepare 7 Bk = (W1, 7)" +apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) +done + +lemma tape_of_nl_not_null: "lm \ [] \ \ []" +apply(case_tac lm, auto) +apply(case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) +done + +lemma [simp]: "lm \ [] \ wprepare_add_one m lm (b, []) = False" +apply(simp add: wprepare_invs) +apply(simp add: tape_of_nl_not_null) +done + +lemma [simp]: "lm \ [] \ wprepare_goto_first_end m lm (b, []) = False" +apply(simp add: wprepare_invs) +done + +lemma [simp]: "lm \ [] \ wprepare_erase m lm (b, []) = False" +apply(simp add: wprepare_invs) +done + + + +lemma [simp]: "lm \ [] \ wprepare_goto_start_pos m lm (b, []) = False" +apply(simp add: wprepare_invs tape_of_nl_not_null) +done + +lemma [simp]: "\lm \ []; wprepare_loop_start m lm (b, [])\ \ b \ []" +apply(simp add: wprepare_invs tape_of_nl_not_null, auto) +done + +lemma [simp]: "\lm \ []; wprepare_loop_start m lm (b, [])\ \ + wprepare_loop_goon m lm (Bk # b, [])" +apply(simp only: wprepare_invs tape_of_nl_not_null) +apply(erule_tac disjE) +apply(rule_tac disjI2) +apply(simp add: wprepare_loop_start_on_rightmost.simps + wprepare_loop_goon_on_rightmost.simps, auto) +apply(rule_tac rev_eq, simp add: tape_of_nl_rev) +done + +lemma [simp]: "\lm \ []; wprepare_loop_goon m lm (b, [])\ \ b \ []" +apply(simp only: wprepare_invs tape_of_nl_not_null, auto) +done + +lemma [simp]:"\lm \ []; wprepare_loop_goon m lm (b, [])\ \ + wprepare_add_one2 m lm (Bk # b, [])" +apply(simp only: wprepare_invs tape_of_nl_not_null, auto split: if_splits) +apply(case_tac mr, simp, simp add: exp_ind_def) +done + +lemma [simp]: "wprepare_add_one2 m lm (b, []) \ b \ []" +apply(simp only: wprepare_invs tape_of_nl_not_null, auto) +done + +lemma [simp]: "wprepare_add_one2 m lm (b, []) \ wprepare_add_one2 m lm (b, [Oc])" +apply(simp only: wprepare_invs tape_of_nl_not_null, auto) +done + +lemma [simp]: "Bk # list = <(m::nat) # lm> @ ys = False" +apply(case_tac lm, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) +done + +lemma [simp]: "\lm \ []; wprepare_add_one m lm (b, Bk # list)\ + \ (b = [] \ wprepare_goto_first_end m lm ([], Oc # list)) \ + (b \ [] \ wprepare_goto_first_end m lm (b, Oc # list))" +apply(simp only: wprepare_invs, auto) +apply(rule_tac x = 0 in exI, simp add: exp_ind_def) +apply(case_tac lm, simp, simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) +apply(rule_tac x = rn in exI, simp) +done + +lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \ b \ []" +apply(simp only: wprepare_invs tape_of_nl_not_null, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \ + wprepare_erase m lm (tl b, hd b # Bk # list)" +apply(simp only: wprepare_invs tape_of_nl_not_null, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(case_tac mr, auto simp: exp_ind_def) +done + +lemma [simp]: "wprepare_erase m lm (b, Bk # list) \ b \ []" +apply(simp only: wprepare_invs exp_ind_def, auto) +done + +lemma [simp]: "wprepare_erase m lm (b, Bk # list) \ + wprepare_goto_start_pos m lm (Bk # b, list)" +apply(simp only: wprepare_invs, auto) +done + +lemma [simp]: "\wprepare_add_one m lm (b, Bk # list)\ \ list \ []" +apply(simp only: wprepare_invs) +apply(case_tac lm, simp_all add: tape_of_nl_abv + tape_of_nat_list.simps exp_ind_def, auto) +done + +lemma [simp]: "\lm \ []; wprepare_goto_first_end m lm (b, Bk # list)\ \ list \ []" +apply(simp only: wprepare_invs, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(simp add: tape_of_nl_not_null) +done + +lemma [simp]: "\lm \ []; wprepare_goto_first_end m lm (b, Bk # list)\ \ b \ []" +apply(simp only: wprepare_invs, auto) +apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null) +done + +lemma [simp]: "\lm \ []; wprepare_erase m lm (b, Bk # list)\ \ list \ []" +apply(simp only: wprepare_invs, auto) +done + +lemma [simp]: "\lm \ []; wprepare_erase m lm (b, Bk # list)\ \ b \ []" +apply(simp only: wprepare_invs, auto simp: exp_ind_def) +done + +lemma [simp]: "\lm \ []; wprepare_goto_start_pos m lm (b, Bk # list)\ \ list \ []" +apply(simp only: wprepare_invs, auto) +apply(simp add: tape_of_nl_not_null) +apply(case_tac lm, simp, case_tac list) +apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) +done + +lemma [simp]: "\lm \ []; wprepare_goto_start_pos m lm (b, Bk # list)\ \ b \ []" +apply(simp only: wprepare_invs) +apply(auto) +done + +lemma [simp]: "\lm \ []; wprepare_loop_goon m lm (b, Bk # list)\ \ b \ []" +apply(simp only: wprepare_invs, auto) +done + +lemma [simp]: "\lm \ []; wprepare_loop_goon m lm (b, Bk # list)\ \ + (list = [] \ wprepare_add_one2 m lm (Bk # b, [])) \ + (list \ [] \ wprepare_add_one2 m lm (Bk # b, list))" +apply(simp only: wprepare_invs, simp) +apply(case_tac list, simp_all split: if_splits, auto) +apply(case_tac [1-3] mr, simp_all add: exp_ind_def) +apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null) +apply(case_tac [1-2] mr, simp_all add: exp_ind_def) +apply(case_tac rn, simp, case_tac nat, auto simp: exp_ind_def) +done + +lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \ b \ []" +apply(simp only: wprepare_invs, simp) +done + +lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \ + (list = [] \ wprepare_add_one2 m lm (b, [Oc])) \ + (list \ [] \ wprepare_add_one2 m lm (b, Oc # list))" +apply(simp only: wprepare_invs, auto) +done + +lemma [simp]: "wprepare_goto_first_end m lm (b, Oc # list) + \ (b = [] \ wprepare_goto_first_end m lm ([Oc], list)) \ + (b \ [] \ wprepare_goto_first_end m lm (Oc # b, list))" +apply(simp only: wprepare_invs, auto) +apply(rule_tac x = 1 in exI, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(case_tac ml, simp_all add: exp_ind_def) +apply(rule_tac x = rn in exI, simp) +apply(rule_tac x = "Suc ml" in exI, simp_all add: exp_ind_def) +apply(rule_tac x = "mr - 1" in exI, simp) +apply(case_tac mr, simp_all add: exp_ind_def, auto) +done + +lemma [simp]: "wprepare_erase m lm (b, Oc # list) \ b \ []" +apply(simp only: wprepare_invs, auto simp: exp_ind_def) +done + +lemma [simp]: "wprepare_erase m lm (b, Oc # list) + \ wprepare_erase m lm (b, Bk # list)" +apply(simp only:wprepare_invs, auto simp: exp_ind_def) +done + +lemma [simp]: "\lm \ []; wprepare_goto_start_pos m lm (b, Bk # list)\ + \ wprepare_goto_start_pos m lm (Bk # b, list)" +apply(simp only:wprepare_invs, auto) +apply(case_tac [!] lm, simp, simp_all) +done + +lemma [simp]: "wprepare_loop_start m lm (b, aa) \ b \ []" +apply(simp only:wprepare_invs, auto) +done +lemma [elim]: "Bk # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ \rn. list = Bk\<^bsup>rn\<^esup>" +apply(case_tac mr, simp_all) +apply(case_tac rn, simp_all add: exp_ind_def, auto) +done + +lemma rev_equal_iff: "x = y \ rev x = rev y" +by simp + +lemma tape_of_nl_false1: + "lm \ [] \ rev b @ [Bk] \ Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # " +apply(auto) +apply(drule_tac rev_equal_iff, simp add: tape_of_nl_rev) +apply(case_tac "rev lm") +apply(case_tac [2] list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) +done + +lemma [simp]: "wprepare_loop_start_in_middle m lm (b, [Bk]) = False" +apply(simp add: wprepare_loop_start_in_middle.simps, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(case_tac lm1, simp, simp add: tape_of_nl_not_null) +done + +declare wprepare_loop_start_in_middle.simps[simp del] + +declare wprepare_loop_start_on_rightmost.simps[simp del] + wprepare_loop_goon_in_middle.simps[simp del] + wprepare_loop_goon_on_rightmost.simps[simp del] + +lemma [simp]: "wprepare_loop_goon_in_middle m lm (Bk # b, []) = False" +apply(simp add: wprepare_loop_goon_in_middle.simps, auto) +done + +lemma [simp]: "\lm \ []; wprepare_loop_start m lm (b, [Bk])\ \ + wprepare_loop_goon m lm (Bk # b, [])" +apply(simp only: wprepare_invs, simp) +apply(simp add: wprepare_loop_goon_on_rightmost.simps + wprepare_loop_start_on_rightmost.simps, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(rule_tac rev_eq) +apply(simp add: tape_of_nl_rev) +apply(simp add: exp_ind_def[THEN sym] exp_ind) +done + +lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista) + \ wprepare_loop_goon_in_middle m lm (Bk # b, a # lista) = False" +apply(auto simp: wprepare_loop_start_on_rightmost.simps + wprepare_loop_goon_in_middle.simps) +apply(case_tac [!] mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "\lm \ []; wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)\ + \ wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista)" +apply(simp only: wprepare_loop_start_on_rightmost.simps + wprepare_loop_goon_on_rightmost.simps, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(simp add: tape_of_nl_rev) +apply(simp add: exp_ind_def[THEN sym] exp_ind) +done + +lemma [simp]: "\lm \ []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\ + \ wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista) = False" +apply(simp add: wprepare_loop_start_in_middle.simps + wprepare_loop_goon_on_rightmost.simps, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(case_tac "lm1::nat list", simp_all, case_tac list, simp) +apply(simp add: tape_of_nl_abv tape_of_nat_list.simps tape_of_nat_abv exp_ind_def) +apply(case_tac [!] rna, simp_all add: exp_ind_def) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(case_tac lm1, simp, case_tac list, simp) +apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def tape_of_nat_abv) +done + +lemma [simp]: + "\lm \ []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\ + \ wprepare_loop_goon_in_middle m lm (Bk # b, a # lista)" +apply(simp add: wprepare_loop_start_in_middle.simps + wprepare_loop_goon_in_middle.simps, auto) +apply(rule_tac x = rn in exI, simp) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(case_tac lm1, simp) +apply(rule_tac x = "Suc aa" in exI, simp) +apply(rule_tac x = list in exI) +apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps) +done + +lemma [simp]: "\lm \ []; wprepare_loop_start m lm (b, Bk # a # lista)\ \ + wprepare_loop_goon m lm (Bk # b, a # lista)" +apply(simp add: wprepare_loop_start.simps + wprepare_loop_goon.simps) +apply(erule_tac disjE, simp, auto) +done + +lemma start_2_goon: + "\lm \ []; wprepare_loop_start m lm (b, Bk # list)\ \ + (list = [] \ wprepare_loop_goon m lm (Bk # b, [])) \ + (list \ [] \ wprepare_loop_goon m lm (Bk # b, list))" +apply(case_tac list, auto) +done + +lemma add_one_2_add_one: "wprepare_add_one m lm (b, Oc # list) + \ (hd b = Oc \ (b = [] \ wprepare_add_one m lm ([], Bk # Oc # list)) \ + (b \ [] \ wprepare_add_one m lm (tl b, Oc # Oc # list))) \ + (hd b \ Oc \ (b = [] \ wprepare_add_one m lm ([], Bk # Oc # list)) \ + (b \ [] \ wprepare_add_one m lm (tl b, hd b # Oc # list)))" +apply(simp only: wprepare_add_one.simps, auto) +done + +lemma [simp]: "wprepare_loop_start m lm (b, Oc # list) \ b \ []" +apply(simp) +done + +lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Oc # list) \ + wprepare_loop_start_on_rightmost m lm (Oc # b, list)" +apply(simp add: wprepare_loop_start_on_rightmost.simps, auto) +apply(rule_tac x = rn in exI, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(case_tac rn, auto simp: exp_ind_def) +done + +lemma [simp]: "wprepare_loop_start_in_middle m lm (b, Oc # list) \ + wprepare_loop_start_in_middle m lm (Oc # b, list)" +apply(simp add: wprepare_loop_start_in_middle.simps, auto) +apply(rule_tac x = rn in exI, auto) +apply(case_tac mr, simp, simp add: exp_ind_def) +apply(rule_tac x = nat in exI, simp) +apply(rule_tac x = lm1 in exI, simp) +done + +lemma start_2_start: "wprepare_loop_start m lm (b, Oc # list) \ + wprepare_loop_start m lm (Oc # b, list)" +apply(simp add: wprepare_loop_start.simps) +apply(erule_tac disjE, simp_all ) +done + +lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list) \ b \ []" +apply(simp add: wprepare_loop_goon.simps + wprepare_loop_goon_in_middle.simps + wprepare_loop_goon_on_rightmost.simps) +apply(auto) +done + +lemma [simp]: "wprepare_goto_start_pos m lm (b, Oc # list) \ b \ []" +apply(simp add: wprepare_goto_start_pos.simps) +done + +lemma [simp]: "wprepare_loop_goon_on_rightmost m lm (b, Oc # list) = False" +apply(simp add: wprepare_loop_goon_on_rightmost.simps) +done +lemma wprepare_loop1: "\rev b @ Oc\<^bsup>mr\<^esup> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ; + b \ []; 0 < mr; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>\ + \ wprepare_loop_start_on_rightmost m lm (Oc # b, list)" +apply(simp add: wprepare_loop_start_on_rightmost.simps) +apply(rule_tac x = rn in exI, simp) +apply(case_tac mr, simp, simp add: exp_ind_def, auto) +done + +lemma wprepare_loop2: "\rev b @ Oc\<^bsup>mr\<^esup> @ Bk # = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ; + b \ []; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk # <(a::nat) # lista> @ Bk\<^bsup>rn\<^esup>\ + \ wprepare_loop_start_in_middle m lm (Oc # b, list)" +apply(simp add: wprepare_loop_start_in_middle.simps) +apply(rule_tac x = rn in exI, simp) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(rule_tac x = nat in exI, simp) +apply(rule_tac x = "a#lista" in exI, simp) +done + +lemma [simp]: "wprepare_loop_goon_in_middle m lm (b, Oc # list) \ + wprepare_loop_start_on_rightmost m lm (Oc # b, list) \ + wprepare_loop_start_in_middle m lm (Oc # b, list)" +apply(simp add: wprepare_loop_goon_in_middle.simps split: if_splits) +apply(case_tac lm1, simp_all add: wprepare_loop1 wprepare_loop2) +done + +lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list) + \ wprepare_loop_start m lm (Oc # b, list)" +apply(simp add: wprepare_loop_goon.simps + wprepare_loop_start.simps) +done + +lemma [simp]: "wprepare_add_one m lm (b, Oc # list) + \ b = [] \ wprepare_add_one m lm ([], Bk # Oc # list)" +apply(auto) +apply(simp add: wprepare_add_one.simps) +done + +lemma [simp]: "wprepare_goto_start_pos m [a] (b, Oc # list) + \ wprepare_loop_start_on_rightmost m [a] (Oc # b, list) " +apply(auto simp: wprepare_goto_start_pos.simps + wprepare_loop_start_on_rightmost.simps) +apply(rule_tac x = rn in exI, simp) +apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def, auto) +done + +lemma [simp]: "wprepare_goto_start_pos m (a # aa # listaa) (b, Oc # list) + \wprepare_loop_start_in_middle m (a # aa # listaa) (Oc # b, list)" +apply(auto simp: wprepare_goto_start_pos.simps + wprepare_loop_start_in_middle.simps) +apply(rule_tac x = rn in exI, simp) +apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) +apply(rule_tac x = a in exI, rule_tac x = "aa#listaa" in exI, simp) +done + +lemma [simp]: "\lm \ []; wprepare_goto_start_pos m lm (b, Oc # list)\ + \ wprepare_loop_start m lm (Oc # b, list)" +apply(case_tac lm, simp_all) +apply(case_tac lista, simp_all add: wprepare_loop_start.simps) +done + +lemma [simp]: "wprepare_add_one2 m lm (b, Oc # list) \ b \ []" +apply(auto simp: wprepare_add_one2.simps) +done + +lemma add_one_2_stop: + "wprepare_add_one2 m lm (b, Oc # list) + \ wprepare_stop m lm (tl b, hd b # Oc # list)" +apply(simp add: wprepare_stop.simps wprepare_add_one2.simps) +done + +declare wprepare_stop.simps[simp del] + +lemma wprepare_correctness: + assumes h: "lm \ []" + shows "let P = (\ (st, l, r). st = 0) in + let Q = (\ (st, l, r). wprepare_inv st m lm (l, r)) in + let f = (\ stp. steps (Suc 0, [], ()) t_wcode_prepare stp) in + \ n .P (f n) \ Q (f n)" +proof - + let ?P = "(\ (st, l, r). st = 0)" + let ?Q = "(\ (st, l, r). wprepare_inv st m lm (l, r))" + let ?f = "(\ stp. steps (Suc 0, [], ()) t_wcode_prepare stp)" + have "\ n. ?P (?f n) \ ?Q (?f n)" + proof(rule_tac halt_lemma2) + show "wf wcode_prepare_le" by auto + next + show "\ n. \ ?P (?f n) \ ?Q (?f n) \ + ?Q (?f (Suc n)) \ (?f (Suc n), ?f n) \ wcode_prepare_le" + using h + apply(rule_tac allI, rule_tac impI, case_tac "?f n", + simp add: tstep_red tstep.simps) + apply(case_tac c, simp, case_tac [2] aa) + apply(simp_all add: wprepare_inv.simps wcode_prepare_le_def new_tape.simps + lex_triple_def lex_pair_def + + split: if_splits) + apply(simp_all add: start_2_goon start_2_start + add_one_2_add_one add_one_2_stop) + apply(auto simp: wprepare_add_one2.simps) + done + next + show "?Q (?f 0)" + apply(simp add: steps.simps wprepare_inv.simps wprepare_invs) + done + next + show "\ ?P (?f 0)" + apply(simp add: steps.simps) + done + qed + thus "?thesis" + apply(auto) + done +qed + +lemma [intro]: "t_correct t_wcode_prepare" +apply(simp add: t_correct.simps t_wcode_prepare_def iseven_def) +apply(rule_tac x = 7 in exI, simp) +done + +lemma twice_len_even: "length (tm_of abc_twice) mod 2 = 0" +apply(simp add: tm_even) +done + +lemma fourtimes_len_even: "length (tm_of abc_fourtimes) mod 2 = 0" +apply(simp add: tm_even) +done + +lemma t_correct_termi: "t_correct tp \ + list_all (\(acn, st). (st \ Suc (length tp div 2))) (change_termi_state tp)" +apply(auto simp: t_correct.simps List.list_all_length) +apply(erule_tac x = n in allE, simp) +apply(case_tac "tp!n", auto simp: change_termi_state.simps split: if_splits) +done + + +lemma t_correct_shift: + "list_all (\(acn, st). (st \ y)) tp \ + list_all (\(acn, st). (st \ y + off)) (tshift tp off) " +apply(auto simp: t_correct.simps List.list_all_length) +apply(erule_tac x = n in allE, simp add: shift_length) +apply(case_tac "tp!n", auto simp: tshift.simps) +done + +lemma [intro]: + "t_correct (tm_of abc_twice @ tMp (Suc 0) + (start_of twice_ly (length abc_twice) - Suc 0))" +apply(rule_tac t_compiled_correct, simp_all) +apply(simp add: twice_ly_def) +done + +lemma [intro]: "t_correct (tm_of abc_fourtimes @ tMp (Suc 0) + (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))" +apply(rule_tac t_compiled_correct, simp_all) +apply(simp add: fourtimes_ly_def) +done + + +lemma [intro]: "t_correct t_wcode_main" +apply(auto simp: t_wcode_main_def t_correct.simps shift_length + t_twice_def t_fourtimes_def) +proof - + show "iseven (60 + (length (tm_of abc_twice) + + length (tm_of abc_fourtimes)))" + using twice_len_even fourtimes_len_even + apply(auto simp: iseven_def) + apply(rule_tac x = "30 + q + qa" in exI, simp) + done +next + show " list_all (\(acn, s). s \ (60 + (length (tm_of abc_twice) + + length (tm_of abc_fourtimes))) div 2) t_wcode_main_first_part" + apply(auto simp: t_wcode_main_first_part_def shift_length t_twice_def) + done +next + have "list_all (\(acn, s). s \ Suc (length (tm_of abc_twice @ tMp (Suc 0) + (start_of twice_ly (length abc_twice) - Suc 0)) div 2)) + (change_termi_state (tm_of abc_twice @ tMp (Suc 0) + (start_of twice_ly (length abc_twice) - Suc 0)))" + apply(rule_tac t_correct_termi, auto) + done + hence "list_all (\(acn, s). s \ Suc (length (tm_of abc_twice @ tMp (Suc 0) + (start_of twice_ly (length abc_twice) - Suc 0)) div 2) + 12) + (tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0) + (start_of twice_ly (length abc_twice) - Suc 0))) 12)" + apply(rule_tac t_correct_shift, simp) + done + thus "list_all (\(acn, s). s \ + (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2) + (tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0) + (start_of twice_ly (length abc_twice) - Suc 0))) 12)" + apply(simp) + apply(simp add: list_all_length, auto) + done +next + have "list_all (\(acn, s). s \ Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) + (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2)) + (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) + (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) " + apply(rule_tac t_correct_termi, auto) + done + hence "list_all (\(acn, s). s \ Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) + (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2) + (t_twice_len + 13)) + (tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) + (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))" + apply(rule_tac t_correct_shift, simp) + done + thus "list_all (\(acn, s). s \ (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2) + (tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) + (start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))" + apply(simp add: t_twice_len_def t_twice_def) + using twice_len_even fourtimes_len_even + apply(auto simp: list_all_length) + done +qed + +lemma [intro]: "t_correct (t_wcode_prepare |+| t_wcode_main)" +apply(auto intro: t_correct_add) +done + +lemma prepare_mainpart_lemma: + "args \ [] \ + \ stp ln rn. steps (Suc 0, [], ) (t_wcode_prepare |+| t_wcode_main) stp + = (0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof - + let ?P1 = "\ (l, r). l = [] \ r = " + let ?Q1 = "\ (l, r). wprepare_stop m args (l, r)" + let ?P2 = ?Q1 + let ?Q2 = "\ (l, r). (\ ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" + let ?P3 = "\ tp. False" + assume h: "args \ []" + have "?P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) + (t_wcode_prepare |+| t_wcode_main) stp = (0, tp') \ ?Q2 tp')" + proof(rule_tac turing_merge.t_merge_halt[of t_wcode_prepare t_wcode_main ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], + auto simp: turing_merge_def) + show "\stp. case steps (Suc 0, [], ) t_wcode_prepare stp of (st, tp') + \ st = 0 \ wprepare_stop m args tp'" + using wprepare_correctness[of args m] h + apply(simp, auto) + apply(rule_tac x = n in exI, simp add: wprepare_inv.simps) + done + next + fix a b + assume "wprepare_stop m args (a, b)" + thus "\stp. case steps (Suc 0, a, b) t_wcode_main stp of + (st, tp') \ (st = 0) \ (case tp' of (l, r) \ l = Bk # Oc\<^bsup>Suc m\<^esup> \ + (\ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>))" + proof(simp only: wprepare_stop.simps, erule_tac exE) + fix rn + assume "a = Bk # @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \ + b = Bk # Oc # Bk\<^bsup>rn\<^esup>" + thus "?thesis" + using t_wcode_main_lemma_pre[of "args" "" 0 "Oc\<^bsup>Suc m\<^esup>" 0 rn] h + apply(simp) + apply(erule_tac exE)+ + apply(rule_tac x = stp in exI, simp add: tape_of_nl_rev, auto) + done + qed + next + show "wprepare_stop m args \-> wprepare_stop m args" + by(simp add: t_imply_def) + qed + thus "\ stp ln rn. steps (Suc 0, [], ) (t_wcode_prepare |+| t_wcode_main) stp + = (0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(simp add: t_imply_def) + apply(erule_tac exE)+ + apply(auto) + done +qed + + +lemma [simp]: "tinres r r' \ + fetch t ss (case r of [] \ Bk | x # xs \ x) = + fetch t ss (case r' of [] \ Bk | x # xs \ x)" +apply(simp add: fetch.simps, auto split: if_splits simp: tinres_def) +apply(case_tac [!] r', simp_all) +apply(case_tac [!] n, simp_all add: exp_ind_def) +apply(case_tac [!] r, simp_all) +done + +lemma [intro]: "\ n. (a::block)\<^bsup>n\<^esup> = []" +by auto + +lemma [simp]: "\tinres r r'; r \ []; r' \ []\ \ hd r = hd r'" +apply(auto simp: tinres_def) +done + +lemma [intro]: "hd (Bk\<^bsup>Suc n\<^esup>) = Bk" +apply(simp add: exp_ind_def) +done + +lemma [simp]: "\tinres r []; r \ []\ \ hd r = Bk" +apply(auto simp: tinres_def) +apply(case_tac n, auto) +done + +lemma [simp]: "\tinres [] r'; r' \ []\ \ hd r' = Bk" +apply(auto simp: tinres_def) +done + +lemma [intro]: "\na. tl r = tl (r @ Bk\<^bsup>n\<^esup>) @ Bk\<^bsup>na\<^esup> \ tl (r @ Bk\<^bsup>n\<^esup>) = tl r @ Bk\<^bsup>na\<^esup>" +apply(case_tac r, simp) +apply(case_tac n, simp) +apply(rule_tac x = 0 in exI, simp) +apply(rule_tac x = nat in exI, simp add: exp_ind_def) +apply(simp) +apply(rule_tac x = n in exI, simp) +done + +lemma [simp]: "tinres r r' \ tinres (tl r) (tl r')" +apply(auto simp: tinres_def) +apply(case_tac r', simp_all) +apply(case_tac n, simp_all add: exp_ind_def) +apply(rule_tac x = 0 in exI, simp) +apply(rule_tac x = nat in exI, simp_all) +apply(rule_tac x = n in exI, simp) +done + +lemma [simp]: "\tinres r []; r \ []\ \ tinres (tl r) []" +apply(case_tac r, auto simp: tinres_def) +apply(case_tac n, simp_all add: exp_ind_def) +apply(rule_tac x = nat in exI, simp) +done + +lemma [simp]: "\tinres [] r'\ \ tinres [] (tl r')" +apply(case_tac r', auto simp: tinres_def) +apply(case_tac n, simp_all add: exp_ind_def) +apply(rule_tac x = nat in exI, simp) +done + +lemma [simp]: "tinres r r' \ tinres (b # r) (b # r')" +apply(auto simp: tinres_def) +done + +lemma tinres_step2: + "\tinres r r'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l, r') t = (sb, lb, rb)\ + \ la = lb \ tinres ra rb \ sa = sb" +apply(case_tac "ss = 0", simp add: tstep_0) +apply(simp add: tstep.simps [simp del]) +apply(case_tac "fetch t ss (case r of [] \ Bk | x # xs \ x)", simp) +apply(auto simp: new_tape.simps) +apply(simp_all split: taction.splits if_splits) +apply(auto) +done + + +lemma tinres_steps2: + "\tinres r r'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l, r') t stp = (sb, lb, rb)\ + \ la = lb \ tinres ra rb \ sa = sb" +apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps) +apply(simp add: tstep_red) +apply(case_tac "(steps (ss, l, r) t stp)") +apply(case_tac "(steps (ss, l, r') t stp)") +proof - + fix stp sa la ra sb lb rb a b c aa ba ca + assume ind: "\sa la ra sb lb rb. \steps (ss, l, r) t stp = (sa, la, ra); + steps (ss, l, r') t stp = (sb, lb, rb)\ \ la = lb \ tinres ra rb \ sa = sb" + and h: " tinres r r'" "tstep (steps (ss, l, r) t stp) t = (sa, la, ra)" + "tstep (steps (ss, l, r') t stp) t = (sb, lb, rb)" "steps (ss, l, r) t stp = (a, b, c)" + "steps (ss, l, r') t stp = (aa, ba, ca)" + have "b = ba \ tinres c ca \ a = aa" + apply(rule_tac ind, simp_all add: h) + done + thus "la = lb \ tinres ra rb \ sa = sb" + apply(rule_tac l = b and r = c and ss = a and r' = ca + and t = t in tinres_step2) + using h + apply(simp, simp, simp) + done +qed + +definition t_wcode_adjust :: "tprog" + where + "t_wcode_adjust = [(W1, 1), (R, 2), (Nop, 2), (R, 3), (R, 3), (R, 4), + (L, 8), (L, 5), (L, 6), (W0, 5), (L, 6), (R, 7), + (W1, 2), (Nop, 7), (L, 9), (W0, 8), (L, 9), (L, 10), + (L, 11), (L, 10), (R, 0), (L, 11)]" + +lemma [simp]: "fetch t_wcode_adjust (Suc 0) Bk = (W1, 1)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust (Suc 0) Oc = (R, 2)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust (Suc (Suc 0)) Oc = (R, 3)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc 0))) Oc = (R, 4)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc 0))) Bk = (R, 3)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 4 Bk = (L, 8)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 4 Oc = (L, 5)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 5 Oc = (W0, 5)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 5 Bk = (L, 6)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 6 Oc = (R, 7)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 6 Bk = (L, 6)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 7 Bk = (W1, 2)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 8 Bk = (L, 9)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 8 Oc = (W0, 8)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 9 Oc = (L, 10)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 9 Bk = (L, 9)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 10 Bk = (L, 11)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 10 Oc = (L, 10)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 11 Oc = (L, 11)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +lemma [simp]: "fetch t_wcode_adjust 11 Bk = (R, 0)" +apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) +done + +fun wadjust_start :: "nat \ nat \ tape \ bool" + where + "wadjust_start m rs (l, r) = + (\ ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \ + tl r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wadjust_loop_start :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_start m rs (l, r) = + (\ ln rn ml mr. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc rs) \ mr > 0)" + +fun wadjust_loop_right_move :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_right_move m rs (l, r) = + (\ ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>nr\<^esup> @ Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc rs) \ mr > 0 \ + nl + nr > 0)" + +fun wadjust_loop_check :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_check m rs (l, r) = + (\ ml mr ln rn. l = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr = (Suc rs))" + +fun wadjust_loop_erase :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_erase m rs (l, r) = + (\ ml mr ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + tl r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ ml + mr = (Suc rs) \ mr > 0)" + +fun wadjust_loop_on_left_moving_O :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_on_left_moving_O m rs (l, r) = + (\ ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m \<^esup>\ + r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc rs \ mr > 0)" + +fun wadjust_loop_on_left_moving_B :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_on_left_moving_B m rs (l, r) = + (\ ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>nr\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc rs \ mr > 0)" + +fun wadjust_loop_on_left_moving :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_on_left_moving m rs (l, r) = + (wadjust_loop_on_left_moving_O m rs (l, r) \ + wadjust_loop_on_left_moving_B m rs (l, r))" + +fun wadjust_loop_right_move2 :: "nat \ nat \ tape \ bool" + where + "wadjust_loop_right_move2 m rs (l, r) = + (\ ml mr ln rn. l = Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc rs \ mr > 0)" + +fun wadjust_erase2 :: "nat \ nat \ tape \ bool" + where + "wadjust_erase2 m rs (l, r) = + (\ ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + tl r = Bk\<^bsup>rn\<^esup>)" + +fun wadjust_on_left_moving_O :: "nat \ nat \ tape \ bool" + where + "wadjust_on_left_moving_O m rs (l, r) = + (\ rn. l = Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Oc # Bk\<^bsup>rn\<^esup>)" + +fun wadjust_on_left_moving_B :: "nat \ nat \ tape \ bool" + where + "wadjust_on_left_moving_B m rs (l, r) = + (\ ln rn. l = Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Bk\<^bsup>rn\<^esup>)" + +fun wadjust_on_left_moving :: "nat \ nat \ tape \ bool" + where + "wadjust_on_left_moving m rs (l, r) = + (wadjust_on_left_moving_O m rs (l, r) \ + wadjust_on_left_moving_B m rs (l, r))" + +fun wadjust_goon_left_moving_B :: "nat \ nat \ tape \ bool" + where + "wadjust_goon_left_moving_B m rs (l, r) = + (\ rn. l = Oc\<^bsup>Suc m\<^esup> \ + r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wadjust_goon_left_moving_O :: "nat \ nat \ tape \ bool" + where + "wadjust_goon_left_moving_O m rs (l, r) = + (\ ml mr rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \ + r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc (Suc rs) \ mr > 0)" + +fun wadjust_goon_left_moving :: "nat \ nat \ tape \ bool" + where + "wadjust_goon_left_moving m rs (l, r) = + (wadjust_goon_left_moving_B m rs (l, r) \ + wadjust_goon_left_moving_O m rs (l, r))" + +fun wadjust_backto_standard_pos_B :: "nat \ nat \ tape \ bool" + where + "wadjust_backto_standard_pos_B m rs (l, r) = + (\ rn. l = [] \ + r = Bk # Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +fun wadjust_backto_standard_pos_O :: "nat \ nat \ tape \ bool" + where + "wadjust_backto_standard_pos_O m rs (l, r) = + (\ ml mr rn. l = Oc\<^bsup>ml\<^esup> \ + r = Oc\<^bsup>mr\<^esup> @ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup> \ + ml + mr = Suc m \ mr > 0)" + +fun wadjust_backto_standard_pos :: "nat \ nat \ tape \ bool" + where + "wadjust_backto_standard_pos m rs (l, r) = + (wadjust_backto_standard_pos_B m rs (l, r) \ + wadjust_backto_standard_pos_O m rs (l, r))" + +fun wadjust_stop :: "nat \ nat \ tape \ bool" +where + "wadjust_stop m rs (l, r) = + (\ rn. l = [Bk] \ + r = Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" + +declare wadjust_start.simps[simp del] wadjust_loop_start.simps[simp del] + wadjust_loop_right_move.simps[simp del] wadjust_loop_check.simps[simp del] + wadjust_loop_erase.simps[simp del] wadjust_loop_on_left_moving.simps[simp del] + wadjust_loop_right_move2.simps[simp del] wadjust_erase2.simps[simp del] + wadjust_on_left_moving_O.simps[simp del] wadjust_on_left_moving_B.simps[simp del] + wadjust_on_left_moving.simps[simp del] wadjust_goon_left_moving_B.simps[simp del] + wadjust_goon_left_moving_O.simps[simp del] wadjust_goon_left_moving.simps[simp del] + wadjust_backto_standard_pos.simps[simp del] wadjust_backto_standard_pos_B.simps[simp del] + wadjust_backto_standard_pos_O.simps[simp del] wadjust_stop.simps[simp del] + +fun wadjust_inv :: "nat \ nat \ nat \ tape \ bool" + where + "wadjust_inv st m rs (l, r) = + (if st = Suc 0 then wadjust_start m rs (l, r) + else if st = Suc (Suc 0) then wadjust_loop_start m rs (l, r) + else if st = Suc (Suc (Suc 0)) then wadjust_loop_right_move m rs (l, r) + else if st = 4 then wadjust_loop_check m rs (l, r) + else if st = 5 then wadjust_loop_erase m rs (l, r) + else if st = 6 then wadjust_loop_on_left_moving m rs (l, r) + else if st = 7 then wadjust_loop_right_move2 m rs (l, r) + else if st = 8 then wadjust_erase2 m rs (l, r) + else if st = 9 then wadjust_on_left_moving m rs (l, r) + else if st = 10 then wadjust_goon_left_moving m rs (l, r) + else if st = 11 then wadjust_backto_standard_pos m rs (l, r) + else if st = 0 then wadjust_stop m rs (l, r) + else False +)" + +declare wadjust_inv.simps[simp del] + +fun wadjust_phase :: "nat \ t_conf \ nat" + where + "wadjust_phase rs (st, l, r) = + (if st = 1 then 3 + else if st \ 2 \ st \ 7 then 2 + else if st \ 8 \ st \ 11 then 1 + else 0)" + +thm dropWhile.simps + +fun wadjust_stage :: "nat \ t_conf \ nat" + where + "wadjust_stage rs (st, l, r) = + (if st \ 2 \ st \ 7 then + rs - length (takeWhile (\ a. a = Oc) + (tl (dropWhile (\ a. a = Oc) (rev l @ r)))) + else 0)" + +fun wadjust_state :: "nat \ t_conf \ nat" + where + "wadjust_state rs (st, l, r) = + (if st \ 2 \ st \ 7 then 8 - st + else if st \ 8 \ st \ 11 then 12 - st + else 0)" + +fun wadjust_step :: "nat \ t_conf \ nat" + where + "wadjust_step rs (st, l, r) = + (if st = 1 then (if hd r = Bk then 1 + else 0) + else if st = 3 then length r + else if st = 5 then (if hd r = Oc then 1 + else 0) + else if st = 6 then length l + else if st = 8 then (if hd r = Oc then 1 + else 0) + else if st = 9 then length l + else if st = 10 then length l + else if st = 11 then (if hd r = Bk then 0 + else Suc (length l)) + else 0)" + +fun wadjust_measure :: "(nat \ t_conf) \ nat \ nat \ nat \ nat" + where + "wadjust_measure (rs, (st, l, r)) = + (wadjust_phase rs (st, l, r), + wadjust_stage rs (st, l, r), + wadjust_state rs (st, l, r), + wadjust_step rs (st, l, r))" + +definition wadjust_le :: "((nat \ t_conf) \ nat \ t_conf) set" + where "wadjust_le \ (inv_image lex_square wadjust_measure)" + +lemma [intro]: "wf lex_square" +by(auto intro:wf_lex_prod simp: abacus.lex_pair_def lex_square_def + abacus.lex_triple_def) + +lemma wf_wadjust_le[intro]: "wf wadjust_le" +by(auto intro:wf_inv_image simp: wadjust_le_def + abacus.lex_triple_def abacus.lex_pair_def) + +lemma [simp]: "wadjust_start m rs (c, []) = False" +apply(auto simp: wadjust_start.simps) +done + +lemma [simp]: "wadjust_loop_right_move m rs (c, []) \ c \ []" +apply(auto simp: wadjust_loop_right_move.simps) +done + +lemma [simp]: "wadjust_loop_right_move m rs (c, []) + \ wadjust_loop_check m rs (Bk # c, [])" +apply(simp only: wadjust_loop_right_move.simps wadjust_loop_check.simps) +apply(auto) +apply(case_tac [!] mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wadjust_loop_check m rs (c, []) \ c \ []" +apply(simp only: wadjust_loop_check.simps, auto) +done + +lemma [simp]: "wadjust_loop_start m rs (c, []) = False" +apply(simp add: wadjust_loop_start.simps) +done + +lemma [simp]: "wadjust_loop_right_move m rs (c, []) \ + wadjust_loop_right_move m rs (Bk # c, [])" +apply(simp only: wadjust_loop_right_move.simps) +apply(erule_tac exE)+ +apply(auto) +apply(case_tac [!] mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wadjust_loop_check m rs (c, []) \ wadjust_erase2 m rs (tl c, [hd c])" +apply(simp only: wadjust_loop_check.simps wadjust_erase2.simps, auto) +apply(case_tac mr, simp_all add: exp_ind_def, auto) +done + +lemma [simp]: " wadjust_loop_erase m rs (c, []) + \ (c = [] \ wadjust_loop_on_left_moving m rs ([], [Bk])) \ + (c \ [] \ wadjust_loop_on_left_moving m rs (tl c, [hd c]))" +apply(simp add: wadjust_loop_erase.simps, auto) +apply(case_tac [!] mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wadjust_loop_on_left_moving m rs (c, []) = False" +apply(auto simp: wadjust_loop_on_left_moving.simps) +done + + +lemma [simp]: "wadjust_loop_right_move2 m rs (c, []) = False" +apply(auto simp: wadjust_loop_right_move2.simps) +done + +lemma [simp]: "wadjust_erase2 m rs ([], []) = False" +apply(auto simp: wadjust_erase2.simps) +done + +lemma [simp]: "wadjust_on_left_moving_B m rs + (Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])" +apply(simp add: wadjust_on_left_moving_B.simps, auto) +apply(rule_tac x = 0 in exI, simp add: exp_ind_def) +done + +lemma [simp]: "wadjust_on_left_moving_B m rs + (Bk\<^bsup>n\<^esup> @ Bk # Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])" +apply(simp add: wadjust_on_left_moving_B.simps exp_ind_def, auto) +apply(rule_tac x = "Suc n" in exI, simp add: exp_ind) +done + +lemma [simp]: "\wadjust_erase2 m rs (c, []); c \ []\ \ + wadjust_on_left_moving m rs (tl c, [hd c])" +apply(simp only: wadjust_erase2.simps) +apply(erule_tac exE)+ +apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps) +done + +lemma [simp]: "wadjust_erase2 m rs (c, []) + \ (c = [] \ wadjust_on_left_moving m rs ([], [Bk])) \ + (c \ [] \ wadjust_on_left_moving m rs (tl c, [hd c]))" +apply(auto) +done + +lemma [simp]: "wadjust_on_left_moving m rs ([], []) = False" +apply(simp add: wadjust_on_left_moving.simps + wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps) +done + +lemma [simp]: "wadjust_on_left_moving_O m rs (c, []) = False" +apply(simp add: wadjust_on_left_moving_O.simps) +done + +lemma [simp]: " \wadjust_on_left_moving_B m rs (c, []); c \ []; hd c = Bk\ \ + wadjust_on_left_moving_B m rs (tl c, [Bk])" +apply(simp add: wadjust_on_left_moving_B.simps, auto) +apply(case_tac [!] ln, simp_all add: exp_ind_def, auto) +done + +lemma [simp]: "\wadjust_on_left_moving_B m rs (c, []); c \ []; hd c = Oc\ \ + wadjust_on_left_moving_O m rs (tl c, [Oc])" +apply(simp add: wadjust_on_left_moving_B.simps wadjust_on_left_moving_O.simps, auto) +apply(case_tac [!] ln, simp_all add: exp_ind_def) +done + +lemma [simp]: "\wadjust_on_left_moving m rs (c, []); c \ []\ \ + wadjust_on_left_moving m rs (tl c, [hd c])" +apply(simp add: wadjust_on_left_moving.simps) +apply(case_tac "hd c", simp_all) +done + +lemma [simp]: "wadjust_on_left_moving m rs (c, []) + \ (c = [] \ wadjust_on_left_moving m rs ([], [Bk])) \ + (c \ [] \ wadjust_on_left_moving m rs (tl c, [hd c]))" +apply(auto) +done + +lemma [simp]: "wadjust_goon_left_moving m rs (c, []) = False" +apply(auto simp: wadjust_goon_left_moving.simps wadjust_goon_left_moving_B.simps + wadjust_goon_left_moving_O.simps) +done + +lemma [simp]: "wadjust_backto_standard_pos m rs (c, []) = False" +apply(auto simp: wadjust_backto_standard_pos.simps + wadjust_backto_standard_pos_B.simps wadjust_backto_standard_pos_O.simps) +done + +lemma [simp]: + "wadjust_start m rs (c, Bk # list) \ + (c = [] \ wadjust_start m rs ([], Oc # list)) \ + (c \ [] \ wadjust_start m rs (c, Oc # list))" +apply(auto simp: wadjust_start.simps) +done + +lemma [simp]: "wadjust_loop_start m rs (c, Bk # list) = False" +apply(auto simp: wadjust_loop_start.simps) +done + +lemma [simp]: "wadjust_loop_right_move m rs (c, b) \ c \ []" +apply(simp only: wadjust_loop_right_move.simps, auto) +done + +lemma [simp]: "wadjust_loop_right_move m rs (c, Bk # list) + \ wadjust_loop_right_move m rs (Bk # c, list)" +apply(simp only: wadjust_loop_right_move.simps) +apply(erule_tac exE)+ +apply(rule_tac x = ml in exI, simp) +apply(rule_tac x = mr in exI, simp) +apply(rule_tac x = "Suc nl" in exI, simp add: exp_ind_def) +apply(case_tac nr, simp, case_tac mr, simp_all add: exp_ind_def) +apply(rule_tac x = nat in exI, auto) +done + +lemma [simp]: "wadjust_loop_check m rs (c, b) \ c \ []" +apply(simp only: wadjust_loop_check.simps, auto) +done + +lemma [simp]: "wadjust_loop_check m rs (c, Bk # list) + \ wadjust_erase2 m rs (tl c, hd c # Bk # list)" +apply(auto simp: wadjust_loop_check.simps wadjust_erase2.simps) +apply(case_tac [!] mr, simp_all add: exp_ind_def, auto) +done + +lemma [simp]: "wadjust_loop_erase m rs (c, b) \ c \ []" +apply(simp only: wadjust_loop_erase.simps, auto) +done + +declare wadjust_loop_on_left_moving_O.simps[simp del] + wadjust_loop_on_left_moving_B.simps[simp del] + +lemma [simp]: "\wadjust_loop_erase m rs (c, Bk # list); hd c = Bk\ + \ wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)" +apply(simp only: wadjust_loop_erase.simps + wadjust_loop_on_left_moving_B.simps) +apply(erule_tac exE)+ +apply(rule_tac x = ml in exI, rule_tac x = mr in exI, + rule_tac x = ln in exI, rule_tac x = 0 in exI, simp) +apply(case_tac ln, simp_all add: exp_ind_def, auto) +apply(simp add: exp_ind exp_ind_def[THEN sym]) +done + +lemma [simp]: "\wadjust_loop_erase m rs (c, Bk # list); c \ []; hd c = Oc\ \ + wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)" +apply(simp only: wadjust_loop_erase.simps wadjust_loop_on_left_moving_O.simps, + auto) +apply(case_tac [!] ln, simp_all add: exp_ind_def) +done + +lemma [simp]: "\wadjust_loop_erase m rs (c, Bk # list); c \ []\ \ + wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)" +apply(case_tac "hd c", simp_all add:wadjust_loop_on_left_moving.simps) +done + +lemma [simp]: "wadjust_loop_on_left_moving m rs (c, b) \ c \ []" +apply(simp add: wadjust_loop_on_left_moving.simps +wadjust_loop_on_left_moving_O.simps wadjust_loop_on_left_moving_B.simps, auto) +done + +lemma [simp]: "wadjust_loop_on_left_moving_O m rs (c, Bk # list) = False" +apply(simp add: wadjust_loop_on_left_moving_O.simps) +done + +lemma [simp]: "\wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Bk\ + \ wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)" +apply(simp only: wadjust_loop_on_left_moving_B.simps) +apply(erule_tac exE)+ +apply(rule_tac x = ml in exI, rule_tac x = mr in exI) +apply(case_tac nl, simp_all add: exp_ind_def, auto) +apply(rule_tac x = "Suc nr" in exI, auto simp: exp_ind_def) +done + +lemma [simp]: "\wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Oc\ + \ wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)" +apply(simp only: wadjust_loop_on_left_moving_O.simps + wadjust_loop_on_left_moving_B.simps) +apply(erule_tac exE)+ +apply(rule_tac x = ml in exI, rule_tac x = mr in exI) +apply(case_tac nl, simp_all add: exp_ind_def, auto) +done + +lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Bk # list) + \ wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)" +apply(simp add: wadjust_loop_on_left_moving.simps) +apply(case_tac "hd c", simp_all) +done + +lemma [simp]: "wadjust_loop_right_move2 m rs (c, b) \ c \ []" +apply(simp only: wadjust_loop_right_move2.simps, auto) +done + +lemma [simp]: "wadjust_loop_right_move2 m rs (c, Bk # list) \ wadjust_loop_start m rs (c, Oc # list)" +apply(auto simp: wadjust_loop_right_move2.simps wadjust_loop_start.simps) +apply(case_tac ln, simp_all add: exp_ind_def) +apply(rule_tac x = 0 in exI, simp) +apply(rule_tac x = rn in exI, simp) +apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def, auto) +apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind) +apply(rule_tac x = rn in exI, auto) +apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def) +done + +lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \ c \ []" +apply(auto simp:wadjust_erase2.simps ) +done + +lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \ + wadjust_on_left_moving m rs (tl c, hd c # Bk # list)" +apply(auto simp: wadjust_erase2.simps) +apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps + wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps) +apply(auto) +apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def) +apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind) +apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def) +done + +lemma [simp]: "wadjust_on_left_moving m rs (c,b) \ c \ []" +apply(simp only:wadjust_on_left_moving.simps + wadjust_on_left_moving_O.simps + wadjust_on_left_moving_B.simps + , auto) +done + +lemma [simp]: "wadjust_on_left_moving_O m rs (c, Bk # list) = False" +apply(simp add: wadjust_on_left_moving_O.simps) +done + +lemma [simp]: "\wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Bk\ + \ wadjust_on_left_moving_B m rs (tl c, Bk # Bk # list)" +apply(auto simp: wadjust_on_left_moving_B.simps) +apply(case_tac ln, simp_all add: exp_ind_def, auto) +done + +lemma [simp]: "\wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Oc\ + \ wadjust_on_left_moving_O m rs (tl c, Oc # Bk # list)" +apply(auto simp: wadjust_on_left_moving_O.simps + wadjust_on_left_moving_B.simps) +apply(case_tac ln, simp_all add: exp_ind_def) +done + +lemma [simp]: "wadjust_on_left_moving m rs (c, Bk # list) \ + wadjust_on_left_moving m rs (tl c, hd c # Bk # list)" +apply(simp add: wadjust_on_left_moving.simps) +apply(case_tac "hd c", simp_all) +done + +lemma [simp]: "wadjust_goon_left_moving m rs (c, b) \ c \ []" +apply(simp add: wadjust_goon_left_moving.simps + wadjust_goon_left_moving_B.simps + wadjust_goon_left_moving_O.simps exp_ind_def, auto) +done + +lemma [simp]: "wadjust_goon_left_moving_O m rs (c, Bk # list) = False" +apply(simp add: wadjust_goon_left_moving_O.simps, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "\wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Bk\ + \ wadjust_backto_standard_pos_B m rs (tl c, Bk # Bk # list)" +apply(auto simp: wadjust_goon_left_moving_B.simps + wadjust_backto_standard_pos_B.simps exp_ind_def) +done + +lemma [simp]: "\wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Oc\ + \ wadjust_backto_standard_pos_O m rs (tl c, Oc # Bk # list)" +apply(auto simp: wadjust_goon_left_moving_B.simps + wadjust_backto_standard_pos_O.simps exp_ind_def) +apply(rule_tac x = m in exI, simp, auto) +done + +lemma [simp]: "wadjust_goon_left_moving m rs (c, Bk # list) \ + wadjust_backto_standard_pos m rs (tl c, hd c # Bk # list)" +apply(case_tac "hd c", simp_all add: wadjust_backto_standard_pos.simps + wadjust_goon_left_moving.simps) +done + +lemma [simp]: "wadjust_backto_standard_pos m rs (c, Bk # list) \ + (c = [] \ wadjust_stop m rs ([Bk], list)) \ (c \ [] \ wadjust_stop m rs (Bk # c, list))" +apply(auto simp: wadjust_backto_standard_pos.simps + wadjust_backto_standard_pos_B.simps + wadjust_backto_standard_pos_O.simps wadjust_stop.simps) +apply(case_tac [!] mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wadjust_start m rs (c, Oc # list) + \ (c = [] \ wadjust_loop_start m rs ([Oc], list)) \ + (c \ [] \ wadjust_loop_start m rs (Oc # c, list))" +apply(auto simp:wadjust_loop_start.simps wadjust_start.simps ) +apply(rule_tac x = ln in exI, rule_tac x = rn in exI, + rule_tac x = "Suc 0" in exI, simp) +done + +lemma [simp]: "wadjust_loop_start m rs (c, b) \ c \ []" +apply(simp add: wadjust_loop_start.simps, auto) +done + +lemma [simp]: "wadjust_loop_start m rs (c, Oc # list) + \ wadjust_loop_right_move m rs (Oc # c, list)" +apply(simp add: wadjust_loop_start.simps wadjust_loop_right_move.simps, auto) +apply(rule_tac x = ml in exI, rule_tac x = mr in exI, + rule_tac x = 0 in exI, simp) +apply(rule_tac x = "Suc ln" in exI, simp add: exp_ind, auto) +done + +lemma [simp]: "wadjust_loop_right_move m rs (c, Oc # list) \ + wadjust_loop_check m rs (Oc # c, list)" +apply(simp add: wadjust_loop_right_move.simps + wadjust_loop_check.simps, auto) +apply(rule_tac [!] x = ml in exI, simp_all, auto) +apply(case_tac nl, auto simp: exp_ind_def) +apply(rule_tac x = "mr - 1" in exI, case_tac mr, simp_all add: exp_ind_def) +apply(case_tac [!] nr, simp_all add: exp_ind_def, auto) +done + +lemma [simp]: "wadjust_loop_check m rs (c, Oc # list) \ + wadjust_loop_erase m rs (tl c, hd c # Oc # list)" +apply(simp only: wadjust_loop_check.simps wadjust_loop_erase.simps) +apply(erule_tac exE)+ +apply(rule_tac x = ml in exI, rule_tac x = mr in exI, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +apply(case_tac rn, simp_all add: exp_ind_def) +done + +lemma [simp]: "wadjust_loop_erase m rs (c, Oc # list) \ + wadjust_loop_erase m rs (c, Bk # list)" +apply(auto simp: wadjust_loop_erase.simps) +done + +lemma [simp]: "wadjust_loop_on_left_moving_B m rs (c, Oc # list) = False" +apply(auto simp: wadjust_loop_on_left_moving_B.simps) +apply(case_tac nr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Oc # list) + \ wadjust_loop_right_move2 m rs (Oc # c, list)" +apply(simp add:wadjust_loop_on_left_moving.simps) +apply(auto simp: wadjust_loop_on_left_moving_O.simps + wadjust_loop_right_move2.simps) +done + +lemma [simp]: "wadjust_loop_right_move2 m rs (c, Oc # list) = False" +apply(auto simp: wadjust_loop_right_move2.simps ) +apply(case_tac ln, simp_all add: exp_ind_def) +done + +lemma [simp]: "wadjust_erase2 m rs (c, Oc # list) + \ (c = [] \ wadjust_erase2 m rs ([], Bk # list)) + \ (c \ [] \ wadjust_erase2 m rs (c, Bk # list))" +apply(auto simp: wadjust_erase2.simps ) +done + +lemma [simp]: "wadjust_on_left_moving_B m rs (c, Oc # list) = False" +apply(auto simp: wadjust_on_left_moving_B.simps) +done + +lemma [simp]: "\wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Bk\ \ + wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)" +apply(auto simp: wadjust_on_left_moving_O.simps + wadjust_goon_left_moving_B.simps exp_ind_def) +done + +lemma [simp]: "\wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Oc\ + \ wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)" +apply(auto simp: wadjust_on_left_moving_O.simps + wadjust_goon_left_moving_O.simps exp_ind_def) +apply(rule_tac x = rs in exI, simp) +apply(auto simp: exp_ind_def numeral_2_eq_2) +done + + +lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \ + wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)" +apply(simp add: wadjust_on_left_moving.simps + wadjust_goon_left_moving.simps) +apply(case_tac "hd c", simp_all) +done + +lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \ + wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)" +apply(simp add: wadjust_on_left_moving.simps + wadjust_goon_left_moving.simps) +apply(case_tac "hd c", simp_all) +done + +lemma [simp]: "wadjust_goon_left_moving_B m rs (c, Oc # list) = False" +apply(auto simp: wadjust_goon_left_moving_B.simps) +done + +lemma [simp]: "\wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Bk\ + \ wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)" +apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps) +apply(case_tac [!] ml, auto simp: exp_ind_def) +done + +lemma [simp]: "\wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Oc\ \ + wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)" +apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps) +apply(rule_tac x = "ml - 1" in exI, simp) +apply(case_tac ml, simp_all add: exp_ind_def) +apply(rule_tac x = "Suc mr" in exI, auto simp: exp_ind_def) +done + +lemma [simp]: "wadjust_goon_left_moving m rs (c, Oc # list) \ + wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)" +apply(simp add: wadjust_goon_left_moving.simps) +apply(case_tac "hd c", simp_all) +done + +lemma [simp]: "wadjust_backto_standard_pos_B m rs (c, Oc # list) = False" +apply(simp add: wadjust_backto_standard_pos_B.simps) +done + +lemma [simp]: "wadjust_backto_standard_pos_O m rs (c, Bk # xs) = False" +apply(simp add: wadjust_backto_standard_pos_O.simps, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +done + + + +lemma [simp]: "wadjust_backto_standard_pos_O m rs ([], Oc # list) \ + wadjust_backto_standard_pos_B m rs ([], Bk # Oc # list)" +apply(auto simp: wadjust_backto_standard_pos_O.simps + wadjust_backto_standard_pos_B.simps) +apply(rule_tac x = rn in exI, simp) +apply(case_tac ml, simp_all add: exp_ind_def) +done + + +lemma [simp]: + "\wadjust_backto_standard_pos_O m rs (c, Oc # list); c \ []; hd c = Bk\ + \ wadjust_backto_standard_pos_B m rs (tl c, Bk # Oc # list)" +apply(simp add:wadjust_backto_standard_pos_O.simps + wadjust_backto_standard_pos_B.simps, auto) +apply(case_tac [!] ml, simp_all add: exp_ind_def) +done + +lemma [simp]: "\wadjust_backto_standard_pos_O m rs (c, Oc # list); c \ []; hd c = Oc\ + \ wadjust_backto_standard_pos_O m rs (tl c, Oc # Oc # list)" +apply(simp add: wadjust_backto_standard_pos_O.simps, auto) +apply(case_tac ml, simp_all add: exp_ind_def, auto) +apply(rule_tac x = nat in exI, auto simp: exp_ind_def) +done + +lemma [simp]: "wadjust_backto_standard_pos m rs (c, Oc # list) + \ (c = [] \ wadjust_backto_standard_pos m rs ([], Bk # Oc # list)) \ + (c \ [] \ wadjust_backto_standard_pos m rs (tl c, hd c # Oc # list))" +apply(auto simp: wadjust_backto_standard_pos.simps) +apply(case_tac "hd c", simp_all) +done +thm wadjust_loop_right_move.simps + +lemma [simp]: "wadjust_loop_right_move m rs (c, []) = False" +apply(simp only: wadjust_loop_right_move.simps) +apply(rule_tac iffI) +apply(erule_tac exE)+ +apply(case_tac nr, simp_all add: exp_ind_def) +apply(case_tac mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "wadjust_loop_erase m rs (c, []) = False" +apply(simp only: wadjust_loop_erase.simps, auto) +apply(case_tac mr, simp_all add: exp_ind_def) +done + +lemma [simp]: "\Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Bk # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) \ + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) = + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list))))" +apply(simp only: wadjust_loop_erase.simps) +apply(rule_tac disjI2) +apply(case_tac c, simp, simp) +done + +lemma [simp]: + "\Suc (Suc rs) = a; wadjust_loop_on_left_moving m rs (c, Bk # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) \ + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) = + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list))))" +apply(subgoal_tac "c \ []") +apply(case_tac c, simp_all) +done + +lemma dropWhile_exp1: "dropWhile (\a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = dropWhile (\a. a = Oc) xs" +apply(induct n, simp_all add: exp_ind_def) +done +lemma takeWhile_exp1: "takeWhile (\a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = Oc\<^bsup>n\<^esup> @ takeWhile (\a. a = Oc) xs" +apply(induct n, simp_all add: exp_ind_def) +done + +lemma [simp]: "\Suc (Suc rs) = a; wadjust_loop_right_move2 m rs (c, Bk # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list))))" +apply(simp add: wadjust_loop_right_move2.simps, auto) +apply(simp add: dropWhile_exp1 takeWhile_exp1) +apply(case_tac ln, simp, simp add: exp_ind_def) +done + +lemma [simp]: "wadjust_loop_check m rs ([], b) = False" +apply(simp add: wadjust_loop_check.simps) +done + +lemma [simp]: "\Suc (Suc rs) = a; wadjust_loop_check m rs (c, Oc # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list)))) \ + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) = + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list))))" +apply(case_tac "c", simp_all) +done + +lemma [simp]: + "\Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Oc # list)\ + \ a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) + < a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list)))) \ + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Bk # list)))) = + a - length (takeWhile (\a. a = Oc) (tl (dropWhile (\a. a = Oc) (rev c @ Oc # list))))" +apply(simp add: wadjust_loop_erase.simps) +apply(rule_tac disjI2) +apply(auto) +apply(simp add: dropWhile_exp1 takeWhile_exp1) +done + +declare numeral_2_eq_2[simp del] + +lemma wadjust_correctness: + shows "let P = (\ (len, st, l, r). st = 0) in + let Q = (\ (len, st, l, r). wadjust_inv st m rs (l, r)) in + let f = (\ stp. (Suc (Suc rs), steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, + Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)) in + \ n .P (f n) \ Q (f n)" +proof - + let ?P = "(\ (len, st, l, r). st = 0)" + let ?Q = "\ (len, st, l, r). wadjust_inv st m rs (l, r)" + let ?f = "\ stp. (Suc (Suc rs), steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, + Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)" + have "\ n. ?P (?f n) \ ?Q (?f n)" + proof(rule_tac halt_lemma2) + show "wf wadjust_le" by auto + next + show "\ n. \ ?P (?f n) \ ?Q (?f n) \ + ?Q (?f (Suc n)) \ (?f (Suc n), ?f n) \ wadjust_le" + proof(rule_tac allI, rule_tac impI, case_tac "?f n", + simp add: tstep_red tstep.simps, rule_tac conjI, erule_tac conjE, + erule_tac conjE) + fix n a b c d + assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a" + thus "case case fetch t_wcode_adjust b (case d of [] \ Bk | x # xs \ x) + of (ac, ns) \ (ns, new_tape ac (c, d)) of (st, x) \ wadjust_inv st m rs x" + apply(case_tac d, simp, case_tac [2] aa) + apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps + abacus.lex_triple_def abacus.lex_pair_def lex_square_def + split: if_splits) + done + next + fix n a b c d + assume "0 < b \ wadjust_inv b m rs (c, d)" + "Suc (Suc rs) = a \ steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, + Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust n = (b, c, d)" + thus "((a, case fetch t_wcode_adjust b (case d of [] \ Bk | x # xs \ x) + of (ac, ns) \ (ns, new_tape ac (c, d))), a, b, c, d) \ wadjust_le" + proof(erule_tac conjE, erule_tac conjE, erule_tac conjE) + assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a" + thus "?thesis" + apply(case_tac d, case_tac [2] aa) + apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps + abacus.lex_triple_def abacus.lex_pair_def lex_square_def + split: if_splits) + done + qed + qed + next + show "?Q (?f 0)" + apply(simp add: steps.simps wadjust_inv.simps wadjust_start.simps) + apply(rule_tac x = ln in exI,auto) + done + next + show "\ ?P (?f 0)" + apply(simp add: steps.simps) + done + qed + thus "?thesis" + apply(auto) + done +qed + +lemma [intro]: "t_correct t_wcode_adjust" +apply(auto simp: t_wcode_adjust_def t_correct.simps iseven_def) +apply(rule_tac x = 11 in exI, simp) +done + +lemma wcode_lemma_pre': + "args \ [] \ + \ stp rn. steps (Suc 0, [], ) + ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp + = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>)" +proof - + let ?P1 = "\ (l, r). l = [] \ r = " + let ?Q1 = "\(l, r). l = Bk # Oc\<^bsup>Suc m\<^esup> \ + (\ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>)" + let ?P2 = ?Q1 + let ?Q2 = "\ (l, r). (wadjust_stop m (bl_bin () - 1) (l, r))" + let ?P3 = "\ tp. False" + assume h: "args \ []" + have "?P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) + ((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp = (0, tp') \ ?Q2 tp')" + proof(rule_tac turing_merge.t_merge_halt[of "t_wcode_prepare |+| t_wcode_main" + t_wcode_adjust ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], + auto simp: turing_merge_def) + + show "\stp. case steps (Suc 0, [], ) (t_wcode_prepare |+| t_wcode_main) stp of + (st, tp') \ st = 0 \ (case tp' of (l, r) \ l = Bk # Oc\<^bsup>Suc m\<^esup> \ + (\ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>))" + using h prepare_mainpart_lemma[of args m] + apply(auto) + apply(rule_tac x = stp in exI, simp) + apply(rule_tac x = ln in exI, auto) + done + next + fix ln rn + show "\stp. case steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # + Oc\<^bsup>bl_bin ()\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp of + (st, tp') \ st = 0 \ wadjust_stop m (bl_bin () - Suc 0) tp'" + using wadjust_correctness[of m "bl_bin () - 1" "Suc ln" rn] + apply(subgoal_tac "bl_bin () > 0", auto simp: wadjust_inv.simps) + apply(rule_tac x = n in exI, simp add: exp_ind) + using h + apply(case_tac args, simp_all, case_tac list, + simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def + bl_bin.simps) + done + next + show "?Q1 \-> ?P2" + by(simp add: t_imply_def) + qed + thus "\stp rn. steps (Suc 0, [], ) ((t_wcode_prepare |+| t_wcode_main) |+| + t_wcode_adjust) stp = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>)" + apply(simp add: t_imply_def) + apply(erule_tac exE)+ + apply(subgoal_tac "bl_bin () > 0", auto simp: wadjust_stop.simps) + using h + apply(case_tac args, simp_all, case_tac list, + simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def + bl_bin.simps) + done +qed + +text {* + The initialization TM @{text "t_wcode"}. + *} +definition t_wcode :: "tprog" + where + "t_wcode = (t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust" + + +text {* + The correctness of @{text "t_wcode"}. + *} +lemma wcode_lemma_1: + "args \ [] \ + \ stp ln rn. steps (Suc 0, [], ) (t_wcode) stp = + (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>)" +apply(simp add: wcode_lemma_pre' t_wcode_def) +done + +lemma wcode_lemma: + "args \ [] \ + \ stp ln rn. steps (Suc 0, [], ) (t_wcode) stp = + (0, [Bk], <[m ,bl_bin ()]> @ Bk\<^bsup>rn\<^esup>)" +using wcode_lemma_1[of args m] +apply(simp add: t_wcode_def tape_of_nl_abv tape_of_nat_list.simps) +done + +section {* The universal TM *} + +text {* + This section gives the explicit construction of {\em Universal Turing Machine}, defined as @{text "UTM"} and proves its + correctness. It is pretty easy by composing the partial results we have got so far. + *} + + +definition UTM :: "tprog" + where + "UTM = (let (aprog, rs_pos, a_md) = rec_ci rec_F in + let abc_F = aprog [+] dummy_abc (Suc (Suc 0)) in + (t_wcode |+| (tm_of abc_F @ tMp (Suc (Suc 0)) (start_of (layout_of abc_F) + (length abc_F) - Suc 0))))" + +definition F_aprog :: "abc_prog" + where + "F_aprog \ (let (aprog, rs_pos, a_md) = rec_ci rec_F in + aprog [+] dummy_abc (Suc (Suc 0)))" + +definition F_tprog :: "tprog" + where + "F_tprog = tm_of (F_aprog)" + +definition t_utm :: "tprog" + where + "t_utm \ + (F_tprog) @ tMp (Suc (Suc 0)) (start_of (layout_of (F_aprog)) + (length (F_aprog)) - Suc 0)" + +definition UTM_pre :: "tprog" + where + "UTM_pre = t_wcode |+| t_utm" + +lemma F_abc_halt_eq: + "\turing_basic.t_correct tp; + length lm = k; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>); + rs > 0\ + \ \ stp m. abc_steps_l (0, [code tp, bl2wc ()]) (F_aprog) stp = + (length (F_aprog), code tp # bl2wc () # (rs - 1) # 0\<^bsup>m\<^esup>)" +apply(drule_tac F_t_halt_eq, simp, simp, simp) +apply(case_tac "rec_ci rec_F") +apply(frule_tac abc_append_dummy_complie, simp, simp, erule_tac exE, + erule_tac exE) +apply(rule_tac x = stp in exI, rule_tac x = m in exI) +apply(simp add: F_aprog_def dummy_abc_def) +done + +lemma F_abc_utm_halt_eq: + "\rs > 0; + abc_steps_l (0, [code tp, bl2wc ()]) F_aprog stp = + (length F_aprog, code tp # bl2wc () # (rs - 1) # 0\<^bsup>m\<^esup>)\ + \ \stp m n.(steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stp = + (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>))" + thm abacus_turing_eq_halt + using abacus_turing_eq_halt + [of "layout_of F_aprog" "F_aprog" "F_tprog" "length (F_aprog)" + "[code tp, bl2wc ()]" stp "code tp # bl2wc () # (rs - 1) # 0\<^bsup>m\<^esup>" "Suc (Suc 0)" + "start_of (layout_of (F_aprog)) (length (F_aprog))" "[]" 0] +apply(simp add: F_tprog_def t_utm_def abc_lm_v.simps nth_append) +apply(erule_tac exE)+ +apply(rule_tac x = stpa in exI, rule_tac x = "Suc (Suc ma)" in exI, + rule_tac x = l in exI, simp add: exp_ind) +done + +declare tape_of_nl_abv_cons[simp del] + +lemma t_utm_halt_eq': + "\turing_basic.t_correct tp; + 0 < rs; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\ + \ \stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stp = + (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" +apply(drule_tac l = l in F_abc_halt_eq, simp, simp, simp) +apply(erule_tac exE, erule_tac exE) +apply(rule_tac F_abc_utm_halt_eq, simp_all) +done + +lemma [simp]: "tinres xs (xs @ Bk\<^bsup>i\<^esup>)" +apply(auto simp: tinres_def) +done + +lemma [elim]: "\rs > 0; Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup> = c @ Bk\<^bsup>n\<^esup>\ + \ \n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" +apply(case_tac "na > n") +apply(subgoal_tac "\ d. na = d + n", auto simp: exp_add) +apply(rule_tac x = "na - n" in exI, simp) +apply(subgoal_tac "\ d. n = d + na", auto simp: exp_add) +apply(case_tac rs, simp_all add: exp_ind, case_tac d, + simp_all add: exp_ind) +apply(rule_tac x = "n - na" in exI, simp) +done + + +lemma t_utm_halt_eq'': + "\turing_basic.t_correct tp; + 0 < rs; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\ + \ \stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stp = + (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" +apply(drule_tac t_utm_halt_eq', simp_all) +apply(erule_tac exE)+ +proof - + fix stpa ma na + assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" + and gr: "rs > 0" + thus "\stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" + apply(rule_tac x = stpa in exI, rule_tac x = ma in exI, simp) + proof(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp) + fix a b c + assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" + "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)" + thus " a = 0 \ b = Bk\<^bsup>ma\<^esup> \ (\n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" + using tinres_steps2[of "<[code tp, bl2wc ()]>" "<[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>" + "Suc 0" " [Bk, Bk]" t_utm stpa 0 "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c] + apply(simp) + using gr + apply(simp only: tinres_def, auto) + apply(rule_tac x = "na + n" in exI, simp add: exp_add) + done + qed +qed + +lemma [simp]: "tinres [Bk, Bk] [Bk]" +apply(auto simp: tinres_def) +done + +lemma [elim]: "Bk\<^bsup>ma\<^esup> = b @ Bk\<^bsup>n\<^esup> \ \m. b = Bk\<^bsup>m\<^esup>" +apply(subgoal_tac "ma = length b + n") +apply(rule_tac x = "ma - n" in exI, simp add: exp_add) +apply(drule_tac length_equal) +apply(simp) +done + +lemma t_utm_halt_eq: + "\turing_basic.t_correct tp; + 0 < rs; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\ + \ \stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stp = + (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" +apply(drule_tac i = i in t_utm_halt_eq'', simp_all) +apply(erule_tac exE)+ +proof - + fix stpa ma na + assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" + and gr: "rs > 0" + thus "\stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" + apply(rule_tac x = stpa in exI) + proof(case_tac "steps (Suc 0, [Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp) + fix a b c + assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" + "steps (Suc 0, [Bk], <[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)" + thus "a = 0 \ (\m. b = Bk\<^bsup>m\<^esup>) \ (\n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" + using tinres_steps[of "[Bk, Bk]" "[Bk]" "Suc 0" "<[code tp, bl2wc ()]> @ Bk\<^bsup>i\<^esup>" t_utm stpa 0 + "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c] + apply(simp) + apply(auto simp: tinres_def) + apply(rule_tac x = "ma + n" in exI, simp add: exp_add) + done + qed +qed + +lemma [intro]: "t_correct t_wcode" +apply(simp add: t_wcode_def) +apply(auto) +done + +lemma [intro]: "t_correct t_utm" +apply(simp add: t_utm_def F_tprog_def) +apply(rule_tac t_compiled_correct, auto) +done + +lemma UTM_halt_lemma_pre: + "\turing_basic.t_correct tp; + 0 < rs; + args \ []; + steps (Suc 0, Bk\<^bsup>i\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\ + \ \stp m n. steps (Suc 0, [], ) UTM_pre stp = + (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" +proof - + let ?Q2 = "\ (l, r). (\ ln rn. l = Bk\<^bsup>ln\<^esup> \ r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" + term ?Q2 + let ?P1 = "\ (l, r). l = [] \ r = " + let ?Q1 = "\ (l, r). (l = [Bk] \ + (\ rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>))" + let ?P2 = ?Q1 + let ?P3 = "\ (l, r). False" + assume h: "turing_basic.t_correct tp" "0 < rs" + "args \ []" "steps (Suc 0, Bk\<^bsup>i\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)" + have "?P1 \-> \ tp. (\ stp tp'. steps (Suc 0, tp) + (t_wcode |+| t_utm) stp = (0, tp') \ ?Q2 tp')" + proof(rule_tac turing_merge.t_merge_halt [of "t_wcode" "t_utm" + ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], auto simp: turing_merge_def) + show "\stp. case steps (Suc 0, [], ) t_wcode stp of (st, tp') \ + st = 0 \ (case tp' of (l, r) \ l = [Bk] \ + (\rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>))" + using wcode_lemma_1[of args "code tp"] h + apply(simp, auto) + apply(rule_tac x = stpa in exI, auto) + done + next + fix rn + show "\stp. case steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ + Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp of + (st, tp') \ st = 0 \ (case tp' of (l, r) \ + (\ln. l = Bk\<^bsup>ln\<^esup>) \ (\rn. r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>))" + using t_utm_halt_eq[of tp rs i args stp m k rn] h + apply(auto) + apply(rule_tac x = stpa in exI, simp add: bin_wc_eq + tape_of_nat_list.simps tape_of_nl_abv) + apply(auto) + done + next + show "?Q1 \-> ?P2" + apply(simp add: t_imply_def) + done + qed + thus "?thesis" + apply(simp add: t_imply_def) + apply(auto simp: UTM_pre_def) + done +qed + +text {* + The correctness of @{text "UTM"}, the halt case. +*} +lemma UTM_halt_lemma: + "\turing_basic.t_correct tp; + 0 < rs; + args \ []; + steps (Suc 0, Bk\<^bsup>i\<^esup>, ) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\ + \ \stp m n. steps (Suc 0, [], ) UTM stp = + (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" +using UTM_halt_lemma_pre[of tp rs args i stp m k] +apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def) +apply(case_tac "rec_ci rec_F", simp) +done + +definition TSTD:: "t_conf \ bool" + where + "TSTD c = (let (st, l, r) = c in + st = 0 \ (\ m. l = Bk\<^bsup>m\<^esup>) \ (\ rs n. r = Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>))" + +thm abacus_turing_eq_uhalt + +lemma nstd_case1: "0 < a \ NSTD (trpl_code (a, b, c))" +apply(simp add: NSTD.simps trpl_code.simps) +done + +lemma [simp]: "\m. b \ Bk\<^bsup>m\<^esup> \ 0 < bl2wc b" +apply(rule classical, simp) +apply(induct b, erule_tac x = 0 in allE, simp) +apply(simp add: bl2wc.simps, case_tac a, simp_all + add: bl2nat.simps bl2nat_double) +apply(case_tac "\ m. b = Bk\<^bsup>m\<^esup>", erule exE) +apply(erule_tac x = "Suc m" in allE, simp add: exp_ind_def, simp) +done +lemma nstd_case2: "\m. b \ Bk\<^bsup>m\<^esup> \ NSTD (trpl_code (a, b, c))" +apply(simp add: NSTD.simps trpl_code.simps) +done + +thm lg.simps +thm lgR.simps + +lemma [elim]: "Suc (2 * x) = 2 * y \ RR" +apply(induct x arbitrary: y, simp, simp) +apply(case_tac y, simp, simp) +done + +lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\n. c = Bk\<^bsup>n\<^esup>)" +apply(auto) +apply(induct c, simp add: bl2nat.simps) +apply(rule_tac x = 0 in exI, simp) +apply(case_tac a, auto simp: bl2nat.simps bl2nat_double) +done + +lemma bl2wc_exp_ex: + "\Suc (bl2wc c) = 2 ^ m\ \ \ rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" +apply(induct c arbitrary: m, simp add: bl2wc.simps bl2nat.simps) +apply(case_tac a, auto) +apply(case_tac m, simp_all add: bl2wc.simps, auto) +apply(rule_tac x = 0 in exI, rule_tac x = "Suc n" in exI, + simp add: exp_ind_def) +apply(simp add: bl2wc.simps bl2nat.simps bl2nat_double) +apply(case_tac m, simp, simp) +proof - + fix c m nat + assume ind: + "\m. Suc (bl2nat c 0) = 2 ^ m \ \rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" + and h: + "Suc (Suc (2 * bl2nat c 0)) = 2 * 2 ^ nat" + have "\rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" + apply(rule_tac m = nat in ind) + using h + apply(simp) + done + from this obtain rs n where " c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" by blast + thus "\rs n. Oc # c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" + apply(rule_tac x = "Suc rs" in exI, simp add: exp_ind_def) + apply(rule_tac x = n in exI, simp) + done +qed + +lemma [elim]: + "\\rs n. c \ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>; + bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0\ \ bl2wc c = 0" +apply(subgoal_tac "\ m. Suc (bl2wc c) = 2^m", erule_tac exE) +apply(drule_tac bl2wc_exp_ex, simp, erule_tac exE, erule_tac exE) +apply(case_tac rs, simp, simp, erule_tac x = nat in allE, + erule_tac x = n in allE, simp) +using bl2wc_exp_ex[of c "lg (Suc (bl2wc c)) 2"] +apply(case_tac "(2::nat) ^ lg (Suc (bl2wc c)) 2", + simp, simp, erule_tac exE, erule_tac exE, simp) +apply(simp add: bl2wc.simps) +apply(rule_tac x = rs in exI) +apply(case_tac "(2::nat)^rs", simp, simp) +done + +lemma nstd_case3: + "\rs n. c \ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup> \ NSTD (trpl_code (a, b, c))" +apply(simp add: NSTD.simps trpl_code.simps) +apply(rule_tac impI) +apply(rule_tac disjI2, rule_tac disjI2, auto) +done + +lemma NSTD_1: "\ TSTD (a, b, c) + \ rec_exec rec_NSTD [trpl_code (a, b, c)] = Suc 0" + using NSTD_lemma1[of "trpl_code (a, b, c)"] + NSTD_lemma2[of "trpl_code (a, b, c)"] + apply(simp add: TSTD_def) + apply(erule_tac disjE, erule_tac nstd_case1) + apply(erule_tac disjE, erule_tac nstd_case2) + apply(erule_tac nstd_case3) + done + +lemma nonstop_t_uhalt_eq: + "\turing_basic.t_correct tp; + steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp = (a, b, c); + \ TSTD (a, b, c)\ + \ rec_exec rec_nonstop [code tp, bl2wc (), stp] = Suc 0" +apply(simp add: rec_nonstop_def rec_exec.simps) +apply(subgoal_tac + "rec_exec rec_conf [code tp, bl2wc (), stp] = + trpl_code (a, b, c)", simp) +apply(erule_tac NSTD_1) +using rec_t_eq_steps[of tp l lm stp] +apply(simp) +done + +lemma nonstop_true: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ + \ \y. rec_calc_rel rec_nonstop + ([code tp, bl2wc (), y]) (Suc 0)" +apply(rule_tac allI, erule_tac x = y in allE) +apply(case_tac "steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp y", simp) +apply(rule_tac nonstop_t_uhalt_eq, simp_all) +done + +(* +lemma [simp]: + "\jturing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); + rec_ci rec_F = (F_ap, rs_pos, a_md)\ + \ \ stp. case abc_steps_l (0, [code tp, bl2wc ()] @ 0\<^bsup>a_md - rs_pos \<^esup> + @ suflm) (F_ap) stp of (ss, e) \ ss < length (F_ap)" +apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf + ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])") +apply(simp only: rec_F_def, rule_tac i = 0 and ga = a and gb = b and + gc = c in cn_gi_uhalt, simp, simp, simp, simp, simp, simp, simp) +apply(simp add: ci_cn_para_eq) +apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_conf + ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))") +apply(rule_tac rf = "(Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf + ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])" + and n = "Suc (Suc 0)" and f = rec_right and + gs = "[Cn (Suc (Suc 0)) rec_conf + ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])]" + and i = 0 and ga = aa and gb = ba and gc = ca in + cn_gi_uhalt) +apply(simp, simp, simp, simp, simp, simp, simp, + simp add: ci_cn_para_eq) +apply(case_tac "rec_ci rec_halt") +apply(rule_tac rf = "(Cn (Suc (Suc 0)) rec_conf + ([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))" + and n = "Suc (Suc 0)" and f = "rec_conf" and + gs = "([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])" and + i = "Suc (Suc 0)" and gi = "rec_halt" and ga = ab and gb = bb and + gc = cb in cn_gi_uhalt) +apply(simp, simp, simp, simp, simp add: nth_append, simp, + simp add: nth_append, simp add: rec_halt_def) +apply(simp only: rec_halt_def) +apply(case_tac [!] "rec_ci ((rec_nonstop))") +apply(rule_tac allI, rule_tac impI, simp) +apply(case_tac j, simp) +apply(rule_tac x = "code tp" in exI, rule_tac calc_id, simp, simp, simp, simp) +apply(rule_tac x = "bl2wc ()" in exI, rule_tac calc_id, simp, simp, simp) +apply(rule_tac rf = "Mn (Suc (Suc 0)) (rec_nonstop)" + and f = "(rec_nonstop)" and n = "Suc (Suc 0)" + and aprog' = ac and rs_pos' = bc and a_md' = cc in Mn_unhalt) +apply(simp, simp add: rec_halt_def , simp, simp) +apply(drule_tac nonstop_true, simp_all) +apply(rule_tac allI) +apply(erule_tac x = y in allE)+ +apply(simp) +done + +thm abc_list_crsp_steps + +lemma uabc_uhalt': + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); + rec_ci rec_F = (ap, pos, md)\ + \ \ stp. case abc_steps_l (0, [code tp, bl2wc ()]) ap stp of (ss, e) + \ ss < length ap" +proof(frule_tac F_ap = ap and rs_pos = pos and a_md = md + and suflm = "[]" in F_aprog_uhalt, auto) + fix stp a b + assume h: + "\stp. case abc_steps_l (0, code tp # bl2wc () # 0\<^bsup>md - pos\<^esup>) ap stp of + (ss, e) \ ss < length ap" + "abc_steps_l (0, [code tp, bl2wc ()]) ap stp = (a, b)" + "turing_basic.t_correct tp" + "rec_ci rec_F = (ap, pos, md)" + moreover have "ap \ []" + using h apply(rule_tac rec_ci_not_null, simp) + done + ultimately show "a < length ap" + proof(erule_tac x = stp in allE, + case_tac "abc_steps_l (0, code tp # bl2wc () # 0\<^bsup>md - pos\<^esup>) ap stp", simp) + fix aa ba + assume g: "aa < length ap" + "abc_steps_l (0, code tp # bl2wc () # 0\<^bsup>md - pos\<^esup>) ap stp = (aa, ba)" + "ap \ []" + thus "?thesis" + using abc_list_crsp_steps[of "[code tp, bl2wc ()]" + "md - pos" ap stp aa ba] h + apply(simp) + done + qed +qed + +lemma uabc_uhalt: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ + \ \ stp. case abc_steps_l (0, [code tp, bl2wc ()]) F_aprog + stp of (ss, e) \ ss < length F_aprog" +apply(case_tac "rec_ci rec_F", simp add: F_aprog_def) +thm uabc_uhalt' +apply(drule_tac ap = a and pos = b and md = c in uabc_uhalt', simp_all) +proof - + fix a b c + assume + "\stp. case abc_steps_l (0, [code tp, bl2wc ()]) a stp of (ss, e) + \ ss < length a" + "rec_ci rec_F = (a, b, c)" + thus + "\stp. case abc_steps_l (0, [code tp, bl2wc ()]) + (a [+] dummy_abc (Suc (Suc 0))) stp of (ss, e) \ + ss < Suc (Suc (Suc (length a)))" + using abc_append_uhalt1[of a "[code tp, bl2wc ()]" + "a [+] dummy_abc (Suc (Suc 0))" "[]" "dummy_abc (Suc (Suc 0))"] + apply(simp) + done +qed + +lemma tutm_uhalt': + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ + \ \ stp. \ isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stp)" + using abacus_turing_eq_uhalt[of "layout_of (F_aprog)" + "F_aprog" "F_tprog" "[code tp, bl2wc ()]" + "start_of (layout_of (F_aprog )) (length (F_aprog))" + "Suc (Suc 0)"] +apply(simp add: F_tprog_def) +apply(subgoal_tac "\stp. case abc_steps_l (0, [code tp, bl2wc ()]) + (F_aprog) stp of (as, am) \ as < length (F_aprog)", simp) +thm abacus_turing_eq_uhalt +apply(simp add: t_utm_def F_tprog_def) +apply(rule_tac uabc_uhalt, simp_all) +done + +lemma tinres_commute: "tinres r r' \ tinres r' r" +apply(auto simp: tinres_def) +done + +lemma inres_tape: + "\steps (st, l, r) tp stp = (a, b, c); steps (st, l', r') tp stp = (a', b', c'); + tinres l l'; tinres r r'\ + \ a = a' \ tinres b b' \ tinres c c'" +proof(case_tac "steps (st, l', r) tp stp") + fix aa ba ca + assume h: "steps (st, l, r) tp stp = (a, b, c)" + "steps (st, l', r') tp stp = (a', b', c')" + "tinres l l'" "tinres r r'" + "steps (st, l', r) tp stp = (aa, ba, ca)" + have "tinres b ba \ c = ca \ a = aa" + using h + apply(rule_tac tinres_steps, auto) + done + + thm tinres_steps2 + moreover have "b' = ba \ tinres c' ca \ a' = aa" + using h + apply(rule_tac tinres_steps2, auto intro: tinres_commute) + done + ultimately show "?thesis" + apply(auto intro: tinres_commute) + done +qed + +lemma tape_normalize: "\ stp. \ isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stp) + \ \ stp. \ isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc ()]> @ Bk\<^bsup>n\<^esup>) t_utm stp)" +apply(rule_tac allI, case_tac "(steps (Suc 0, Bk\<^bsup>m\<^esup>, + <[code tp, bl2wc ()]> @ Bk\<^bsup>n\<^esup>) t_utm stp)", simp add: isS0_def) +apply(erule_tac x = stp in allE) +apply(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc ()]>) t_utm stp", simp) +apply(drule_tac inres_tape, auto) +apply(auto simp: tinres_def) +apply(case_tac "m > Suc (Suc 0)") +apply(rule_tac x = "m - Suc (Suc 0)" in exI) +apply(case_tac m, simp_all add: exp_ind_def, case_tac nat, simp_all add: exp_ind_def) +apply(rule_tac x = "2 - m" in exI, simp add: exp_ind_def[THEN sym] exp_add[THEN sym]) +apply(simp only: numeral_2_eq_2, simp add: exp_ind_def) +done + +lemma tutm_uhalt: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp))\ + \ \ stp. \ isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc ()]> @ Bk\<^bsup>n\<^esup>) t_utm stp)" +apply(rule_tac tape_normalize) +apply(rule_tac tutm_uhalt', simp_all) +done + +lemma UTM_uhalt_lemma_pre: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); + args \ []\ + \ \ stp. \ isS0 (steps (Suc 0, [], ) UTM_pre stp)" +proof - + let ?P1 = "\ (l, r). l = [] \ r = " + let ?Q1 = "\ (l, r). (l = [Bk] \ + (\ rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>))" + let ?P4 = ?Q1 + let ?P3 = "\ (l, r). False" + assume h: "turing_basic.t_correct tp" "\stp. \ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)" + "args \ []" + have "?P1 \-> \ tp. \ (\ stp. isS0 (steps (Suc 0, tp) (t_wcode |+| t_utm) stp))" + proof(rule_tac turing_merge.t_merge_uhalt [of "t_wcode" "t_utm" + ?P1 ?P3 ?P3 ?P4 ?Q1 ?P3], auto simp: turing_merge_def) + show "\stp. case steps (Suc 0, [], ) t_wcode stp of (st, tp') \ + st = 0 \ (case tp' of (l, r) \ l = [Bk] \ + (\rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>))" + using wcode_lemma_1[of args "code tp"] h + apply(simp, auto) + apply(rule_tac x = stp in exI, auto) + done + next + fix rn stp + show " isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) + \ False" + using tutm_uhalt[of tp l args "Suc 0" rn] h + apply(simp) + apply(erule_tac x = stp in allE) + apply(simp add: tape_of_nl_abv tape_of_nat_list.simps bin_wc_eq) + done + next + fix rn stp + show "isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) \ + isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin ())\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)" + by simp + next + show "?Q1 \-> ?P4" + apply(simp add: t_imply_def) + done + qed + thus "?thesis" + apply(simp add: t_imply_def UTM_pre_def) + done +qed + +text {* + The correctness of @{text "UTM"}, the unhalt case. + *} + +lemma UTM_uhalt_lemma: + "\turing_basic.t_correct tp; + \ stp. (\ TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, ) tp stp)); + args \ []\ + \ \ stp. \ isS0 (steps (Suc 0, [], ) UTM stp)" +using UTM_uhalt_lemma_pre[of tp l args] +apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def) +apply(case_tac "rec_ci rec_F", simp) +done + +end \ No newline at end of file