diff -r 0b302c0b449a -r 469c26d19f8e UTM.thy --- a/UTM.thy Wed Feb 06 02:25:00 2013 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,5165 +0,0 @@ -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