author | Christian Urban <christian dot urban at kcl dot ac dot uk> |
Thu, 06 Dec 2012 16:32:03 +0000 | |
changeset 373 | 0679a84b11ad |
parent 371 | 48b231495281 |
permissions | -rw-r--r-- |
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theory UTM |
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imports Main uncomputable recursive abacus UF GCD |
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begin |
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section {* Wang coding of input arguments *} |
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text {* |
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The direct compilation of the universal function @{text "rec_F"} can not give us UTM, because @{text "rec_F"} is of arity 2, |
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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. |
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(Notice, left number is always @{text "0"} at the very beginning). However, UTM needs to simulate the execution of any TM which may |
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very well take many input arguments. Therefore, a initialization TM needs to run before the TM compiled from @{text "rec_F"}, and the sequential |
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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 |
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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 |
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argument. |
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However, this initialization TM (named @{text "t_wcode"}) can not be constructed by compiling from any resurve function, because every recursive |
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function takes a fixed number of input arguments, while @{text "t_wcode"} needs to take varying number of arguments and tranform them into |
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Wang's coding. Therefore, this section give a direct construction of @{text "t_wcode"} with just some parts being obtained from recursive functions. |
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\newlength{\basewidth} |
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\settowidth{\basewidth}{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx} |
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\newlength{\baseheight} |
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\settoheight{\baseheight}{$B:R$} |
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\newcommand{\vsep}{5\baseheight} |
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The TM used to generate the Wang's code of input arguments is divided into three TMs |
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executed sequentially, namely $prepare$, $mainwork$ and $adjust$¡£According to the |
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convention, start state of ever TM is fixed to state $1$ while the final state is |
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fixed to $0$. |
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The input and output of $prepare$ are illustrated respectively by Figure |
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\ref{prepare_input} and \ref{prepare_output}. |
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\begin{figure}[h!] |
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\centering |
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\scalebox{1.2}{ |
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\begin{tikzpicture} |
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[tbox/.style = {draw, thick, inner sep = 5pt}] |
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\node (0) {}; |
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\node (1) [tbox, text height = 3.5pt, right = -0.9pt of 0] {\wuhao $m$}; |
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\node (2) [tbox, right = -0.9pt of 1] {\wuhao $0$}; |
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\node (3) [tbox, right = -0.9pt of 2] {\wuhao $a_1$}; |
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\node (4) [tbox, right = -0.9pt of 3] {\wuhao $0$}; |
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\node (5) [tbox, right = -0.9pt of 4] {\wuhao $a_2$}; |
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\node (6) [right = -0.9pt of 5] {\ldots \ldots}; |
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\node (7) [tbox, right = -0.9pt of 6] {\wuhao $a_n$}; |
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\draw [->, >=latex, thick] (1)+(0, -4\baseheight) -- (1); |
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\end{tikzpicture}} |
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\caption{The input of TM $prepare$} \label{prepare_input} |
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\end{figure} |
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\begin{figure}[h!] |
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\centering |
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\scalebox{1.2}{ |
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\begin{tikzpicture} |
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\node (0) {}; |
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\node (1) [draw, text height = 3.5pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; |
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\node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; |
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\node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; |
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\node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; |
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\node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; |
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\node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; |
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\node (7) [right = -0.9pt of 6] {\ldots \ldots}; |
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\node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_n$}; |
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\node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $0$}; |
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\node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$}; |
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\node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $1$}; |
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\draw [->, >=latex, thick] (10)+(0, -4\baseheight) -- (10); |
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\end{tikzpicture}} |
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\caption{The output of TM $prepare$} \label{prepare_output} |
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\end{figure} |
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As shown in Figure \ref{prepare_input}, the input of $prepare$ is the same as the the input |
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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}, |
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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, |
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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}. |
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\begin{figure}[h!] |
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\centering |
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\scalebox{0.9}{ |
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\begin{tikzpicture} |
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\node[circle,draw] (1) {$1$}; |
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\node[circle,draw] (2) at ($(1)+(0.3\basewidth, 0)$) {$2$}; |
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\node[circle,draw] (3) at ($(2)+(0.3\basewidth, 0)$) {$3$}; |
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\node[circle,draw] (4) at ($(3)+(0.3\basewidth, 0)$) {$4$}; |
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\node[circle,draw] (5) at ($(4)+(0.3\basewidth, 0)$) {$5$}; |
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\node[circle,draw] (6) at ($(5)+(0.3\basewidth, 0)$) {$6$}; |
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\node[circle,draw] (7) at ($(6)+(0.3\basewidth, 0)$) {$7$}; |
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\node[circle,draw] (8) at ($(7)+(0.3\basewidth, 0)$) {$0$}; |
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\draw [->, >=latex] (1) edge [loop above] node[above] {$S_1:L$} (1) |
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; |
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\draw [->, >=latex] (1) -- node[above] {$S_0:S_1$} (2) |
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; |
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\draw [->, >=latex] (2) edge [loop above] node[above] {$S_1:R$} (2) |
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; |
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\draw [->, >=latex] (2) -- node[above] {$S_0:L$} (3) |
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; |
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\draw [->, >=latex] (3) edge[loop above] node[above] {$S_1:S_0$} (3) |
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; |
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\draw [->, >=latex] (3) -- node[above] {$S_0:R$} (4) |
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; |
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\draw [->, >=latex] (4) edge[loop above] node[above] {$S_0:R$} (4) |
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; |
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\draw [->, >=latex] (4) -- node[above] {$S_0:R$} (5) |
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; |
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\draw [->, >=latex] (5) edge[loop above] node[above] {$S_1:R$} (5) |
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; |
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\draw [->, >=latex] (5) -- node[above] {$S_0:R$} (6) |
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\draw [->, >=latex] (6) edge[bend left = 50] node[below] {$S_1:R$} (5) |
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; |
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\draw [->, >=latex] (6) -- node[above] {$S_0:R$} (7) |
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; |
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\draw [->, >=latex] (7) edge[loop above] node[above] {$S_0:S_1$} (7) |
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; |
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\draw [->, >=latex] (7) -- node[above] {$S_1:L$} (8) |
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; |
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\end{tikzpicture}} |
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\caption{The diagram of TM $prepare$} \label{prepare_diag} |
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\end{figure} |
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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. |
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In order to detect the termination condition when the left most bit of $a_1$ is reached, |
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|
128 |
TM $mainwork$ needs to look ahead and consider three different situations at the start of |
48b231495281
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zhang
parents:
370
diff
changeset
|
129 |
every iteration: |
48b231495281
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zhang
parents:
370
diff
changeset
|
130 |
\begin{enumerate} |
48b231495281
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zhang
parents:
370
diff
changeset
|
131 |
\item The TM configuration for the first situation is shown in Figure \ref{mainwork_case_one_input}, |
48b231495281
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zhang
parents:
370
diff
changeset
|
132 |
where the accumulator is stored in $r$, both of the next two bits |
48b231495281
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zhang
parents:
370
diff
changeset
|
133 |
to be encoded are $1$. The configuration at the end of the iteration |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
134 |
is shown in Figure \ref{mainwork_case_one_output}, where the first 1-bit has been |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
135 |
encoded and cleared. Notice that the accumulator has been changed to |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
136 |
$(r+1) \times 2$ to reflect the encoded bit. |
48b231495281
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zhang
parents:
370
diff
changeset
|
137 |
\item The TM configuration for the second situation is shown in Figure |
48b231495281
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zhang
parents:
370
diff
changeset
|
138 |
\ref{mainwork_case_two_input}, |
48b231495281
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zhang
parents:
370
diff
changeset
|
139 |
where the accumulator is stored in $r$, the next two bits |
48b231495281
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zhang
parents:
370
diff
changeset
|
140 |
to be encoded are $1$ and $0$. After the first |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
141 |
$1$-bit was encoded and cleared, the second $0$-bit is difficult to detect |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
142 |
and process. To solve this problem, these two consecutive bits are |
48b231495281
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zhang
parents:
370
diff
changeset
|
143 |
encoded in one iteration. In this situation, only the first $1$-bit needs |
48b231495281
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zhang
parents:
370
diff
changeset
|
144 |
to be cleared since the second one is cleared by definition. |
48b231495281
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zhang
parents:
370
diff
changeset
|
145 |
The configuration at the end of the iteration |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
146 |
is shown in Figure \ref{mainwork_case_two_output}. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
147 |
Notice that the accumulator has been changed to |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
148 |
$(r+1) \times 4$ to reflect the two encoded bits. |
48b231495281
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zhang
parents:
370
diff
changeset
|
149 |
\item The third situation corresponds to the case when the last bit of $a_1$ is reached. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
150 |
The TM configurations at the start and end of the iteration are shown in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
151 |
Figure \ref{mainwork_case_three_input} and \ref{mainwork_case_three_output} |
48b231495281
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zhang
parents:
370
diff
changeset
|
152 |
respectively. For this situation, only the read write head needs to be moved to |
48b231495281
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zhang
parents:
370
diff
changeset
|
153 |
the left to prepare a initial configuration for TM $adjust$ to start with. |
48b231495281
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zhang
parents:
370
diff
changeset
|
154 |
\end{enumerate} |
48b231495281
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zhang
parents:
370
diff
changeset
|
155 |
The diagram of $mainwork$ is given in Figure \ref{mainwork_diag}. The two rectangular nodes |
48b231495281
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zhang
parents:
370
diff
changeset
|
156 |
labeled with $2 \times x$ and $4 \times x$ are two TMs compiling from recursive functions |
48b231495281
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zhang
parents:
370
diff
changeset
|
157 |
so that we do not have to design and verify two quite complicated TMs. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
158 |
|
48b231495281
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zhang
parents:
370
diff
changeset
|
159 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
160 |
\begin{figure}[h!] |
48b231495281
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zhang
parents:
370
diff
changeset
|
161 |
\centering |
48b231495281
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zhang
parents:
370
diff
changeset
|
162 |
\scalebox{1.2}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
163 |
\begin{tikzpicture} |
48b231495281
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zhang
parents:
370
diff
changeset
|
164 |
\node (0) {}; |
48b231495281
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zhang
parents:
370
diff
changeset
|
165 |
\node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; |
48b231495281
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zhang
parents:
370
diff
changeset
|
166 |
\node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
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zhang
parents:
370
diff
changeset
|
167 |
\node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
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zhang
parents:
370
diff
changeset
|
168 |
\node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; |
48b231495281
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zhang
parents:
370
diff
changeset
|
169 |
\node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
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zhang
parents:
370
diff
changeset
|
170 |
\node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; |
48b231495281
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zhang
parents:
370
diff
changeset
|
171 |
\node (7) [right = -0.9pt of 6] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
172 |
\node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
173 |
\node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
174 |
\node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
175 |
\node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
176 |
\node (12) [right = -0.9pt of 11] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
177 |
\node (13) [draw, right = -0.9pt of 12, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
178 |
\node (14) [draw, text height = 3.9pt, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $r$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
179 |
\draw [->, >=latex, thick] (13)+(0, -4\baseheight) -- (13); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
180 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
181 |
\caption{The first situation for TM $mainwork$ to consider} \label{mainwork_case_one_input} |
48b231495281
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zhang
parents:
370
diff
changeset
|
182 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
183 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
184 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
185 |
\begin{figure}[h!] |
48b231495281
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zhang
parents:
370
diff
changeset
|
186 |
\centering |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
187 |
\scalebox{1.2}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
188 |
\begin{tikzpicture} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
189 |
\node (0) {}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
190 |
\node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
191 |
\node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
192 |
\node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
193 |
\node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; |
48b231495281
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zhang
parents:
370
diff
changeset
|
194 |
\node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
195 |
\node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
196 |
\node (7) [right = -0.9pt of 6] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
197 |
\node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$}; |
48b231495281
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zhang
parents:
370
diff
changeset
|
198 |
\node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
199 |
\node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
200 |
\node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
201 |
\node (12) [right = -0.9pt of 11] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
202 |
\node (13) [draw, right = -0.9pt of 12, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
203 |
\node (14) [draw, text height = 2.7pt, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $(r+1) \times 2$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
204 |
\draw [->, >=latex, thick] (13)+(0, -4\baseheight) -- (13); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
205 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
206 |
\caption{The output for the first case of TM $mainwork$'s processing} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
207 |
\label{mainwork_case_one_output} |
48b231495281
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zhang
parents:
370
diff
changeset
|
208 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
209 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
210 |
\begin{figure}[h!] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
211 |
\centering |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
212 |
\scalebox{1.2}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
213 |
\begin{tikzpicture} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
214 |
\node (0) {}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
215 |
\node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
216 |
\node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
217 |
\node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
218 |
\node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
219 |
\node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
220 |
\node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
221 |
\node (7) [right = -0.9pt of 6] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
222 |
\node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
223 |
\node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
224 |
\node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
225 |
\node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
226 |
\node (12) [draw, right = -0.9pt of 11, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
227 |
\node (13) [right = -0.9pt of 12] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
228 |
\node (14) [draw, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
229 |
\node (15) [draw, text height = 3.9pt, right = -0.9pt of 14, thick, inner sep = 5pt] {\wuhao $r$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
230 |
\draw [->, >=latex, thick] (14)+(0, -4\baseheight) -- (14); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
231 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
232 |
\caption{The second situation for TM $mainwork$ to consider} \label{mainwork_case_two_input} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
233 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
234 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
235 |
\begin{figure}[h!] |
48b231495281
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zhang
parents:
370
diff
changeset
|
236 |
\centering |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
237 |
\scalebox{1.2}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
238 |
\begin{tikzpicture} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
239 |
\node (0) {}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
240 |
\node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
241 |
\node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
242 |
\node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
243 |
\node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $a_1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
244 |
\node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
245 |
\node (6) [draw, right = -0.9pt of 5, thick, inner sep = 5pt] {\wuhao $a_2$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
246 |
\node (7) [right = -0.9pt of 6] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
247 |
\node (8) [draw, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $a_i$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
248 |
\node (9) [draw, right = -0.9pt of 8, thick, inner sep = 5pt] {\wuhao $1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
249 |
\node (10) [draw, right = -0.9pt of 9, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
250 |
\node (11) [draw, right = -0.9pt of 10, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
251 |
\node (12) [draw, right = -0.9pt of 11, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
252 |
\node (13) [right = -0.9pt of 12] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
253 |
\node (14) [draw, right = -0.9pt of 13, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
254 |
\node (15) [draw, text height = 2.7pt, right = -0.9pt of 14, thick, inner sep = 5pt] {\wuhao $(r+1) \times 4$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
255 |
\draw [->, >=latex, thick] (14)+(0, -4\baseheight) -- (14); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
256 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
257 |
\caption{The output for the second case of TM $mainwork$'s processing} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
258 |
\label{mainwork_case_two_output} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
259 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
260 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
261 |
\begin{figure}[h!] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
262 |
\centering |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
263 |
\scalebox{1.2}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
264 |
\begin{tikzpicture} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
265 |
\node (0) {}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
266 |
\node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
267 |
\node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
268 |
\node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
269 |
\node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
270 |
\node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
271 |
\node (6) [right = -0.9pt of 5] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
272 |
\node (7) [draw, right = -0.9pt of 6, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
273 |
\node (8) [draw, text height = 3.9pt, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $r$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
274 |
\draw [->, >=latex, thick] (7)+(0, -4\baseheight) -- (7); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
275 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
276 |
\caption{The third situation for TM $mainwork$ to consider} \label{mainwork_case_three_input} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
277 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
278 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
279 |
\begin{figure}[h!] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
280 |
\centering |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
281 |
\scalebox{1.2}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
282 |
\begin{tikzpicture} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
283 |
\node (0) {}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
284 |
\node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
285 |
\node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
286 |
\node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
287 |
\node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
288 |
\node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
289 |
\node (6) [right = -0.9pt of 5] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
290 |
\node (7) [draw, right = -0.9pt of 6, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
291 |
\node (8) [draw, text height = 3.9pt, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $r$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
292 |
\draw [->, >=latex, thick] (3)+(0, -4\baseheight) -- (3); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
293 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
294 |
\caption{The output for the third case of TM $mainwork$'s processing} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
295 |
\label{mainwork_case_three_output} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
296 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
297 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
298 |
\begin{figure}[h!] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
299 |
\centering |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
300 |
\scalebox{0.9}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
301 |
\begin{tikzpicture} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
302 |
\node[circle,draw] (1) {$1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
303 |
\node[circle,draw] (2) at ($(1)+(0.3\basewidth, 0)$) {$2$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
304 |
\node[circle,draw] (3) at ($(2)+(0.3\basewidth, 0)$) {$3$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
305 |
\node[circle,draw] (4) at ($(3)+(0.3\basewidth, 0)$) {$4$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
306 |
\node[circle,draw] (5) at ($(4)+(0.3\basewidth, 0)$) {$5$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
307 |
\node[circle,draw] (6) at ($(5)+(0.3\basewidth, 0)$) {$6$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
308 |
\node[circle,draw] (7) at ($(2)+(0, -7\baseheight)$) {$7$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
309 |
\node[circle,draw] (8) at ($(7)+(0, -7\baseheight)$) {$8$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
310 |
\node[circle,draw] (9) at ($(8)+(0.3\basewidth, 0)$) {$9$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
311 |
\node[circle,draw] (10) at ($(9)+(0.3\basewidth, 0)$) {$10$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
312 |
\node[circle,draw] (11) at ($(10)+(0.3\basewidth, 0)$) {$11$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
313 |
\node[circle,draw] (12) at ($(11)+(0.3\basewidth, 0)$) {$12$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
314 |
\node[draw] (13) at ($(6)+(0.3\basewidth, 0)$) {$2 \times x$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
315 |
\node[circle,draw] (14) at ($(13)+(0.3\basewidth, 0)$) {$j_1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
316 |
\node[draw] (15) at ($(12)+(0.3\basewidth, 0)$) {$4 \times x$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
317 |
\node[draw] (16) at ($(15)+(0.3\basewidth, 0)$) {$j_2$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
318 |
\node[draw] (17) at ($(7)+(0.3\basewidth, 0)$) {$0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
319 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
320 |
\draw [->, >=latex] (1) edge[loop left] node[above] {$S_0:L$} (1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
321 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
322 |
\draw [->, >=latex] (1) -- node[above] {$S_1:L$} (2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
323 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
324 |
\draw [->, >=latex] (2) -- node[above] {$S_1:R$} (3) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
325 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
326 |
\draw [->, >=latex] (2) -- node[left] {$S_1:L$} (7) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
327 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
328 |
\draw [->, >=latex] (3) edge[loop above] node[above] {$S_1:S_0$} (3) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
329 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
330 |
\draw [->, >=latex] (3) -- node[above] {$S_0:R$} (4) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
331 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
332 |
\draw [->, >=latex] (4) edge[loop above] node[above] {$S_0:R$} (4) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
333 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
334 |
\draw [->, >=latex] (4) -- node[above] {$S_1:R$} (5) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
335 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
336 |
\draw [->, >=latex] (5) edge[loop above] node[above] {$S_1:R$} (5) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
337 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
338 |
\draw [->, >=latex] (5) -- node[above] {$S_0:S_1$} (6) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
339 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
340 |
\draw [->, >=latex] (6) edge[loop above] node[above] {$S_1:L$} (6) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
341 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
342 |
\draw [->, >=latex] (6) -- node[above] {$S_0:R$} (13) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
343 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
344 |
\draw [->, >=latex] (13) -- (14) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
345 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
346 |
\draw (14) -- ($(14)+(0, 6\baseheight)$) -- ($(1) + (0, 6\baseheight)$) node [above,midway] {$S_1:L$} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
347 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
348 |
\draw [->, >=latex] ($(1) + (0, 6\baseheight)$) -- (1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
349 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
350 |
\draw [->, >=latex] (7) -- node[above] {$S_0:R$} (17) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
351 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
352 |
\draw [->, >=latex] (7) -- node[left] {$S_1:R$} (8) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
353 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
354 |
\draw [->, >=latex] (8) -- node[above] {$S_0:R$} (9) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
355 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
356 |
\draw [->, >=latex] (9) -- node[above] {$S_0:R$} (10) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
357 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
358 |
\draw [->, >=latex] (10) -- node[above] {$S_1:R$} (11) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
359 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
360 |
\draw [->, >=latex] (10) edge[loop above] node[above] {$S_0:R$} (10) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
361 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
362 |
\draw [->, >=latex] (11) edge[loop above] node[above] {$S_1:R$} (11) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
363 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
364 |
\draw [->, >=latex] (11) -- node[above] {$S_0:S_1$} (12) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
365 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
366 |
\draw [->, >=latex] (12) -- node[above] {$S_0:R$} (15) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
367 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
368 |
\draw [->, >=latex] (12) edge[loop above] node[above] {$S_1:L$} (12) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
369 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
370 |
\draw [->, >=latex] (15) -- (16) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
371 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
372 |
\draw (16) -- ($(16)+(0, -4\baseheight)$) -- ($(1) + (0, -18\baseheight)$) node [below,midway] {$S_1:L$} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
373 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
374 |
\draw [->, >=latex] ($(1) + (0, -18\baseheight)$) -- (1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
375 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
376 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
377 |
\caption{The diagram of TM $mainwork$} \label{mainwork_diag} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
378 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
379 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
380 |
The purpose of TM $adjust$ is to encode the last bit of $a_1$. The initial and final configuration |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
381 |
of this TM are shown in Figure \ref{adjust_initial} and \ref{adjust_final} respectively. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
382 |
The diagram of TM $adjust$ is shown in Figure \ref{adjust_diag}. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
383 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
384 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
385 |
\begin{figure}[h!] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
386 |
\centering |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
387 |
\scalebox{1.2}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
388 |
\begin{tikzpicture} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
389 |
\node (0) {}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
390 |
\node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
391 |
\node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
392 |
\node (3) [draw, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
393 |
\node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
394 |
\node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
395 |
\node (6) [right = -0.9pt of 5] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
396 |
\node (7) [draw, right = -0.9pt of 6, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
397 |
\node (8) [draw, text height = 3.9pt, right = -0.9pt of 7, thick, inner sep = 5pt] {\wuhao $r$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
398 |
\draw [->, >=latex, thick] (3)+(0, -4\baseheight) -- (3); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
399 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
400 |
\caption{Initial configuration of TM $adjust$} \label{adjust_initial} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
401 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
402 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
403 |
\begin{figure}[h!] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
404 |
\centering |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
405 |
\scalebox{1.2}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
406 |
\begin{tikzpicture} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
407 |
\node (0) {}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
408 |
\node (1) [draw, text height = 3.9pt, right = -0.9pt of 0, thick, inner sep = 5pt] {\wuhao $m$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
409 |
\node (2) [draw, right = -0.9pt of 1, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
410 |
\node (3) [draw, text height = 2.9pt, right = -0.9pt of 2, thick, inner sep = 5pt] {\wuhao $r+1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
411 |
\node (4) [draw, right = -0.9pt of 3, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
412 |
\node (5) [draw, right = -0.9pt of 4, thick, inner sep = 5pt] {\wuhao $0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
413 |
\node (6) [right = -0.9pt of 5] {\ldots \ldots}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
414 |
\draw [->, >=latex, thick] (1)+(0, -4\baseheight) -- (1); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
415 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
416 |
\caption{Final configuration of TM $adjust$} \label{adjust_final} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
417 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
418 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
419 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
420 |
\begin{figure}[h!] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
421 |
\centering |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
422 |
\scalebox{0.9}{ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
423 |
\begin{tikzpicture} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
424 |
\node[circle,draw] (1) {$1$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
425 |
\node[circle,draw] (2) at ($(1)+(0.3\basewidth, 0)$) {$2$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
426 |
\node[circle,draw] (3) at ($(2)+(0.3\basewidth, 0)$) {$3$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
427 |
\node[circle,draw] (4) at ($(3)+(0.3\basewidth, 0)$) {$4$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
428 |
\node[circle,draw] (5) at ($(4)+(0.3\basewidth, 0)$) {$5$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
429 |
\node[circle,draw] (6) at ($(5)+(0.3\basewidth, 0)$) {$6$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
430 |
\node[circle,draw] (7) at ($(6)+(0.3\basewidth, 0)$) {$7$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
431 |
\node[circle,draw] (8) at ($(4)+(0, -7\baseheight)$) {$8$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
432 |
\node[circle,draw] (9) at ($(8)+(0.3\basewidth, 0)$) {$9$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
433 |
\node[circle,draw] (10) at ($(9)+(0.3\basewidth, 0)$) {$10$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
434 |
\node[circle,draw] (11) at ($(10)+(0.3\basewidth, 0)$) {$11$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
435 |
\node[circle,draw] (12) at ($(11)+(0.3\basewidth, 0)$) {$0$}; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
436 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
437 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
438 |
\draw [->, >=latex] (1) -- node[above] {$S_1:R$} (2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
439 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
440 |
\draw [->, >=latex] (1) edge[loop above] node[above] {$S_0:S_1$} (1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
441 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
442 |
\draw [->, >=latex] (2) -- node[above] {$S_1:R$} (3) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
443 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
444 |
\draw [->, >=latex] (3) edge[loop above] node[above] {$S_0:R$} (3) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
445 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
446 |
\draw [->, >=latex] (3) -- node[above] {$S_1:R$} (4) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
447 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
448 |
\draw [->, >=latex] (4) -- node[above] {$S_1:L$} (5) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
449 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
450 |
\draw [->, >=latex] (4) -- node[right] {$S_0:L$} (8) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
451 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
452 |
\draw [->, >=latex] (5) -- node[above] {$S_0:L$} (6) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
453 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
454 |
\draw [->, >=latex] (5) edge[loop above] node[above] {$S_1:S_0$} (5) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
455 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
456 |
\draw [->, >=latex] (6) -- node[above] {$S_1:R$} (7) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
457 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
458 |
\draw [->, >=latex] (6) edge[loop above] node[above] {$S_0:L$} (6) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
459 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
460 |
\draw (7) -- ($(7)+(0, 6\baseheight)$) -- ($(2) + (0, 6\baseheight)$) node [above,midway] {$S_0:S_1$} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
461 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
462 |
\draw [->, >=latex] ($(2) + (0, 6\baseheight)$) -- (2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
463 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
464 |
\draw [->, >=latex] (8) edge[loop left] node[left] {$S_1:S_0$} (8) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
465 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
466 |
\draw [->, >=latex] (8) -- node[above] {$S_0:L$} (9) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
467 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
468 |
\draw [->, >=latex] (9) edge[loop above] node[above] {$S_0:L$} (9) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
469 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
470 |
\draw [->, >=latex] (9) -- node[above] {$S_1:L$} (10) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
471 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
472 |
\draw [->, >=latex] (10) edge[loop above] node[above] {$S_0:L$} (10) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
473 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
474 |
\draw [->, >=latex] (10) -- node[above] {$S_0:L$} (11) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
475 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
476 |
\draw [->, >=latex] (11) edge[loop above] node[above] {$S_1:L$} (11) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
477 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
478 |
\draw [->, >=latex] (11) -- node[above] {$S_0:R$} (12) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
479 |
; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
480 |
\end{tikzpicture}} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
481 |
\caption{Diagram of TM $adjust$} \label{adjust_diag} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
482 |
\end{figure} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
483 |
*} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
484 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
485 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
486 |
definition rec_twice :: "recf" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
487 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
488 |
"rec_twice = Cn 1 rec_mult [id 1 0, constn 2]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
489 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
490 |
definition rec_fourtimes :: "recf" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
491 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
492 |
"rec_fourtimes = Cn 1 rec_mult [id 1 0, constn 4]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
493 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
494 |
definition abc_twice :: "abc_prog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
495 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
496 |
"abc_twice = (let (aprog, ary, fp) = rec_ci rec_twice in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
497 |
aprog [+] dummy_abc ((Suc 0)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
498 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
499 |
definition abc_fourtimes :: "abc_prog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
500 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
501 |
"abc_fourtimes = (let (aprog, ary, fp) = rec_ci rec_fourtimes in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
502 |
aprog [+] dummy_abc ((Suc 0)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
503 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
504 |
definition twice_ly :: "nat list" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
505 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
506 |
"twice_ly = layout_of abc_twice" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
507 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
508 |
definition fourtimes_ly :: "nat list" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
509 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
510 |
"fourtimes_ly = layout_of abc_fourtimes" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
511 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
512 |
definition t_twice :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
513 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
514 |
"t_twice = change_termi_state (tm_of (abc_twice) @ (tMp 1 (start_of twice_ly (length abc_twice) - Suc 0)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
515 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
516 |
definition t_fourtimes :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
517 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
518 |
"t_fourtimes = change_termi_state (tm_of (abc_fourtimes) @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
519 |
(tMp 1 (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
520 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
521 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
522 |
definition t_twice_len :: "nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
523 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
524 |
"t_twice_len = length t_twice div 2" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
525 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
526 |
definition t_wcode_main_first_part:: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
527 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
528 |
"t_wcode_main_first_part \<equiv> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
529 |
[(L, 1), (L, 2), (L, 7), (R, 3), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
530 |
(R, 4), (W0, 3), (R, 4), (R, 5), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
531 |
(W1, 6), (R, 5), (R, 13), (L, 6), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
532 |
(R, 0), (R, 8), (R, 9), (Nop, 8), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
533 |
(R, 10), (W0, 9), (R, 10), (R, 11), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
534 |
(W1, 12), (R, 11), (R, t_twice_len + 14), (L, 12)]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
535 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
536 |
definition t_wcode_main :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
537 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
538 |
"t_wcode_main = (t_wcode_main_first_part @ tshift t_twice 12 @ [(L, 1), (L, 1)] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
539 |
@ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
540 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
541 |
fun bl_bin :: "block list \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
542 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
543 |
"bl_bin [] = 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
544 |
| "bl_bin (Bk # xs) = 2 * bl_bin xs" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
545 |
| "bl_bin (Oc # xs) = Suc (2 * bl_bin xs)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
546 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
547 |
declare bl_bin.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
548 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
549 |
type_synonym bin_inv_t = "block list \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
550 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
551 |
fun wcode_before_double :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
552 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
553 |
"wcode_before_double ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
554 |
(\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
555 |
r = Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
556 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
557 |
declare wcode_before_double.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
558 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
559 |
fun wcode_after_double :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
560 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
561 |
"wcode_after_double ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
562 |
(\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
563 |
r = Oc\<^bsup>Suc (Suc (Suc 2*rs))\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
564 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
565 |
declare wcode_after_double.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
566 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
567 |
fun wcode_on_left_moving_1_B :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
568 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
569 |
"wcode_on_left_moving_1_B ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
570 |
(\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
571 |
r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
572 |
ml + mr > Suc 0 \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
573 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
574 |
declare wcode_on_left_moving_1_B.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
575 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
576 |
fun wcode_on_left_moving_1_O :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
577 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
578 |
"wcode_on_left_moving_1_O ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
579 |
(\<exists> ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
580 |
l = Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
581 |
r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
582 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
583 |
declare wcode_on_left_moving_1_O.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
584 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
585 |
fun wcode_on_left_moving_1 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
586 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
587 |
"wcode_on_left_moving_1 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
588 |
(wcode_on_left_moving_1_B ires rs (l, r) \<or> wcode_on_left_moving_1_O ires rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
589 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
590 |
declare wcode_on_left_moving_1.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
591 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
592 |
fun wcode_on_checking_1 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
593 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
594 |
"wcode_on_checking_1 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
595 |
(\<exists> ln rn. l = ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
596 |
r = Oc # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
597 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
598 |
fun wcode_erase1 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
599 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
600 |
"wcode_erase1 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
601 |
(\<exists> ln rn. l = Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
602 |
tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
603 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
604 |
declare wcode_erase1.simps [simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
605 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
606 |
fun wcode_on_right_moving_1 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
607 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
608 |
"wcode_on_right_moving_1 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
609 |
(\<exists> ml mr rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
610 |
l = Bk\<^bsup>ml\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
611 |
r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
612 |
ml + mr > Suc 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
613 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
614 |
declare wcode_on_right_moving_1.simps [simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
615 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
616 |
declare wcode_on_right_moving_1.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
617 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
618 |
fun wcode_goon_right_moving_1 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
619 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
620 |
"wcode_goon_right_moving_1 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
621 |
(\<exists> ml mr ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
622 |
l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
623 |
r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
624 |
ml + mr = Suc rs)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
625 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
626 |
declare wcode_goon_right_moving_1.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
627 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
628 |
fun wcode_backto_standard_pos_B :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
629 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
630 |
"wcode_backto_standard_pos_B ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
631 |
(\<exists> ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
632 |
r = Bk # Oc\<^bsup>(Suc (Suc rs))\<^esup> @ Bk\<^bsup>rn \<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
633 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
634 |
declare wcode_backto_standard_pos_B.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
635 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
636 |
fun wcode_backto_standard_pos_O :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
637 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
638 |
"wcode_backto_standard_pos_O ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
639 |
(\<exists> ml mr ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
640 |
l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
641 |
r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
642 |
ml + mr = Suc (Suc rs) \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
643 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
644 |
declare wcode_backto_standard_pos_O.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
645 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
646 |
fun wcode_backto_standard_pos :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
647 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
648 |
"wcode_backto_standard_pos ires rs (l, r) = (wcode_backto_standard_pos_B ires rs (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
649 |
wcode_backto_standard_pos_O ires rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
650 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
651 |
declare wcode_backto_standard_pos.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
652 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
653 |
lemma [simp]: "<0::nat> = [Oc]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
654 |
apply(simp add: tape_of_nat_abv exponent_def tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
655 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
656 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
657 |
lemma tape_of_Suc_nat: "<Suc (a ::nat)> = replicate a Oc @ [Oc, Oc]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
658 |
apply(simp add: tape_of_nat_abv exp_ind tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
659 |
apply(simp only: exp_ind_def[THEN sym]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
660 |
apply(simp only: exp_ind, simp, simp add: exponent_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
661 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
662 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
663 |
lemma [simp]: "length (<a::nat>) = Suc a" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
664 |
apply(simp add: tape_of_nat_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
665 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
666 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
667 |
lemma [simp]: "<[a::nat]> = <a>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
668 |
apply(simp add: tape_of_nat_abv tape_of_nl_abv exponent_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
669 |
tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
670 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
671 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
672 |
lemma bin_wc_eq: "bl_bin xs = bl2wc xs" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
673 |
proof(induct xs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
674 |
show " bl_bin [] = bl2wc []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
675 |
apply(simp add: bl_bin.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
676 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
677 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
678 |
fix a xs |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
679 |
assume "bl_bin xs = bl2wc xs" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
680 |
thus " bl_bin (a # xs) = bl2wc (a # xs)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
681 |
apply(case_tac a, simp_all add: bl_bin.simps bl2wc.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
682 |
apply(simp_all add: bl2nat.simps bl2nat_double) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
683 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
684 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
685 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
686 |
declare exp_def[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
687 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
688 |
lemma bl_bin_nat_Suc: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
689 |
"bl_bin (<Suc a>) = bl_bin (<a>) + 2^(Suc a)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
690 |
apply(simp add: tape_of_nat_abv bin_wc_eq) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
691 |
apply(simp add: bl2wc.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
692 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
693 |
lemma [simp]: " rev (a\<^bsup>aa\<^esup>) = a\<^bsup>aa\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
694 |
apply(simp add: exponent_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
695 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
696 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
697 |
declare tape_of_nl_abv_cons[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
698 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
699 |
lemma tape_of_nl_rev: "rev (<lm::nat list>) = (<rev lm>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
700 |
apply(induct lm rule: list_tl_induct, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
701 |
apply(case_tac "list = []", simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
702 |
apply(simp add: tape_of_nat_list_butlast_last tape_of_nl_abv_cons) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
703 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
704 |
lemma [simp]: "a\<^bsup>Suc 0\<^esup> = [a]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
705 |
by(simp add: exp_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
706 |
lemma tape_of_nl_cons_app1: "(<a # xs @ [b]>) = (Oc\<^bsup>Suc a\<^esup> @ Bk # (<xs@ [b]>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
707 |
apply(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
708 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
709 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
710 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
711 |
lemma bl_bin_bk_oc[simp]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
712 |
"bl_bin (xs @ [Bk, Oc]) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
713 |
bl_bin xs + 2*2^(length xs)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
714 |
apply(simp add: bin_wc_eq) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
715 |
using bl2nat_cons_oc[of "xs @ [Bk]"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
716 |
apply(simp add: bl2nat_cons_bk bl2wc.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
717 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
718 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
719 |
lemma tape_of_nat[simp]: "(<a::nat>) = Oc\<^bsup>Suc a\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
720 |
apply(simp add: tape_of_nat_abv) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
721 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
722 |
lemma tape_of_nl_cons_app2: "(<c # xs @ [b]>) = (<c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
723 |
proof(induct "length xs" arbitrary: xs c, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
724 |
simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
725 |
fix x xs c |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
726 |
assume ind: "\<And>xs c. x = length xs \<Longrightarrow> <c # xs @ [b]> = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
727 |
<c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
728 |
and h: "Suc x = length (xs::nat list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
729 |
show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
730 |
proof(case_tac xs, simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
731 |
fix a list |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
732 |
assume g: "xs = a # list" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
733 |
hence k: "<a # list @ [b]> = <a # list> @ Bk # Oc\<^bsup>Suc b\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
734 |
apply(rule_tac ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
735 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
736 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
737 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
738 |
from g and k show "<c # xs @ [b]> = <c # xs> @ Bk # Oc\<^bsup>Suc b\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
739 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
740 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
741 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
742 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
743 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
744 |
lemma [simp]: "length (<aa # a # list>) = Suc (Suc aa) + length (<a # list>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
745 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
746 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
747 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
748 |
lemma [simp]: "bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista) @ [Bk, Oc]) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
749 |
bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)) + |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
750 |
2* 2^(length (Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
751 |
using bl_bin_bk_oc[of "Oc\<^bsup>Suc aa\<^esup> @ Bk # tape_of_nat_list (a # lista)"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
752 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
753 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
754 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
755 |
lemma [simp]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
756 |
"bl_bin (<aa # list>) + (4 * rs + 4) * 2 ^ (length (<aa # list>) - Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
757 |
= bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
758 |
apply(case_tac "list", simp add: add_mult_distrib, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
759 |
apply(simp add: tape_of_nl_cons_app2 add_mult_distrib) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
760 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
761 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
762 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
763 |
lemma tape_of_nl_app_Suc: "((<list @ [Suc ab]>)) = (<list @ [ab]>) @ [Oc]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
764 |
apply(induct list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
765 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
766 |
apply(case_tac list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
767 |
apply(simp_all add:tape_of_nl_abv tape_of_nat_list.simps exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
768 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
769 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
770 |
lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]> @ [Oc]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
771 |
= bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) + |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
772 |
2^(length (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
773 |
apply(simp add: bin_wc_eq) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
774 |
apply(simp add: bl2nat_cons_oc bl2wc.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
775 |
using bl2nat_cons_oc[of "Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
776 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
777 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
778 |
lemma [simp]: "bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [ab]>) + (4 * 2 ^ (aa + length (<list @ [ab]>)) + |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
779 |
4 * (rs * 2 ^ (aa + length (<list @ [ab]>)))) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
780 |
bl_bin (Oc # Oc\<^bsup>aa\<^esup> @ Bk # <list @ [Suc ab]>) + |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
781 |
rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
782 |
apply(simp add: tape_of_nl_app_Suc) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
783 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
784 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
785 |
declare tape_of_nat[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
786 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
787 |
fun wcode_double_case_inv :: "nat \<Rightarrow> bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
788 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
789 |
"wcode_double_case_inv st ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
790 |
(if st = Suc 0 then wcode_on_left_moving_1 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
791 |
else if st = Suc (Suc 0) then wcode_on_checking_1 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
792 |
else if st = 3 then wcode_erase1 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
793 |
else if st = 4 then wcode_on_right_moving_1 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
794 |
else if st = 5 then wcode_goon_right_moving_1 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
795 |
else if st = 6 then wcode_backto_standard_pos ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
796 |
else if st = 13 then wcode_before_double ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
797 |
else False)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
798 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
799 |
declare wcode_double_case_inv.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
800 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
801 |
fun wcode_double_case_state :: "t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
802 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
803 |
"wcode_double_case_state (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
804 |
13 - st" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
805 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
806 |
fun wcode_double_case_step :: "t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
807 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
808 |
"wcode_double_case_step (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
809 |
(if st = Suc 0 then (length l) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
810 |
else if st = Suc (Suc 0) then (length r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
811 |
else if st = 3 then |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
812 |
if hd r = Oc then 1 else 0 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
813 |
else if st = 4 then (length r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
814 |
else if st = 5 then (length r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
815 |
else if st = 6 then (length l) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
816 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
817 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
818 |
fun wcode_double_case_measure :: "t_conf \<Rightarrow> nat \<times> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
819 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
820 |
"wcode_double_case_measure (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
821 |
(wcode_double_case_state (st, l, r), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
822 |
wcode_double_case_step (st, l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
823 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
824 |
definition wcode_double_case_le :: "(t_conf \<times> t_conf) set" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
825 |
where "wcode_double_case_le \<equiv> (inv_image lex_pair wcode_double_case_measure)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
826 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
827 |
lemma [intro]: "wf lex_pair" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
828 |
by(auto intro:wf_lex_prod simp:lex_pair_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
829 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
830 |
lemma wf_wcode_double_case_le[intro]: "wf wcode_double_case_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
831 |
by(auto intro:wf_inv_image simp: wcode_double_case_le_def ) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
832 |
term fetch |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
833 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
834 |
lemma [simp]: "fetch t_wcode_main (Suc 0) Bk = (L, Suc 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
835 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
836 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
837 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
838 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
839 |
lemma [simp]: "fetch t_wcode_main (Suc 0) Oc = (L, Suc (Suc 0))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
840 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
841 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
842 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
843 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
844 |
lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Oc = (R, 3)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
845 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
846 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
847 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
848 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
849 |
lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Bk = (R, 4)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
850 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
851 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
852 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
853 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
854 |
lemma [simp]: "fetch t_wcode_main (Suc (Suc (Suc 0))) Oc = (W0, 3)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
855 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
856 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
857 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
858 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
859 |
lemma [simp]: "fetch t_wcode_main 4 Bk = (R, 4)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
860 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
861 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
862 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
863 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
864 |
lemma [simp]: "fetch t_wcode_main 4 Oc = (R, 5)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
865 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
866 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
867 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
868 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
869 |
lemma [simp]: "fetch t_wcode_main 5 Oc = (R, 5)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
870 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
871 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
872 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
873 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
874 |
lemma [simp]: "fetch t_wcode_main 5 Bk = (W1, 6)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
875 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
876 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
877 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
878 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
879 |
lemma [simp]: "fetch t_wcode_main 6 Bk = (R, 13)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
880 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
881 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
882 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
883 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
884 |
lemma [simp]: "fetch t_wcode_main 6 Oc = (L, 6)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
885 |
apply(simp add: t_wcode_main_def t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
886 |
fetch.simps nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
887 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
888 |
lemma [elim]: "Bk\<^bsup>mr\<^esup> = [] \<Longrightarrow> mr = 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
889 |
apply(case_tac mr, auto simp: exponent_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
890 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
891 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
892 |
lemma [simp]: "wcode_on_left_moving_1 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
893 |
apply(simp add: wcode_on_left_moving_1.simps wcode_on_left_moving_1_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
894 |
wcode_on_left_moving_1_O.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
895 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
896 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
897 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
898 |
declare wcode_on_checking_1.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
899 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
900 |
lemmas wcode_double_case_inv_simps = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
901 |
wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
902 |
wcode_on_left_moving_1_B.simps wcode_on_checking_1.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
903 |
wcode_erase1.simps wcode_on_right_moving_1.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
904 |
wcode_goon_right_moving_1.simps wcode_backto_standard_pos.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
905 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
906 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
907 |
lemma [simp]: "wcode_on_left_moving_1 ires rs (b, r) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
908 |
apply(simp add: wcode_double_case_inv_simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
909 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
910 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
911 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
912 |
lemma [elim]: "\<lbrakk>wcode_on_left_moving_1 ires rs (b, Bk # list); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
913 |
tl b = aa \<and> hd b # Bk # list = ba\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
914 |
wcode_on_left_moving_1 ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
915 |
apply(simp only: wcode_on_left_moving_1.simps wcode_on_left_moving_1_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
916 |
wcode_on_left_moving_1_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
917 |
apply(erule_tac disjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
918 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
919 |
apply(case_tac ml, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
920 |
apply(rule_tac x = "mr - Suc (Suc 0)" in exI, rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
921 |
apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
922 |
apply(rule_tac disjI1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
923 |
apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
924 |
simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
925 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
926 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
927 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
928 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
929 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
930 |
lemma [elim]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
931 |
"\<lbrakk>wcode_on_left_moving_1 ires rs (b, Oc # list); tl b = aa \<and> hd b # Oc # list = ba\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
932 |
\<Longrightarrow> wcode_on_checking_1 ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
933 |
apply(simp only: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
934 |
apply(erule_tac disjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
935 |
apply(erule_tac [!] exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
936 |
apply(case_tac mr, simp, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
937 |
apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
938 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
939 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
940 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
941 |
lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
942 |
apply(auto simp: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
943 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
944 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
945 |
lemma [simp]: "wcode_on_checking_1 ires rs (b, Bk # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
946 |
apply(auto simp: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
947 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
948 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
949 |
lemma [elim]: "\<lbrakk>wcode_on_checking_1 ires rs (b, Oc # ba);Oc # b = aa \<and> list = ba\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
950 |
\<Longrightarrow> wcode_erase1 ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
951 |
apply(simp only: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
952 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
953 |
apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
954 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
955 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
956 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
957 |
lemma [simp]: "wcode_on_checking_1 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
958 |
apply(simp add: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
959 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
960 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
961 |
lemma [simp]: "wcode_on_checking_1 ires rs ([], Bk # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
962 |
apply(simp add: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
963 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
964 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
965 |
lemma [simp]: "wcode_erase1 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
966 |
apply(simp add: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
967 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
968 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
969 |
lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
970 |
apply(simp add: wcode_double_case_inv_simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
971 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
972 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
973 |
lemma [simp]: "wcode_on_right_moving_1 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
974 |
apply(simp add: wcode_double_case_inv_simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
975 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
976 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
977 |
lemma [elim]: "\<lbrakk>wcode_on_right_moving_1 ires rs (b, Bk # ba); Bk # b = aa \<and> list = b\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
978 |
wcode_on_right_moving_1 ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
979 |
apply(simp only: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
980 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
981 |
apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
982 |
rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
983 |
apply(simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
984 |
apply(case_tac mr, simp, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
985 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
986 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
987 |
lemma [elim]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
988 |
"\<lbrakk>wcode_on_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \<and> list = ba\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
989 |
\<Longrightarrow> wcode_goon_right_moving_1 ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
990 |
apply(simp only: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
991 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
992 |
apply(rule_tac x = "Suc 0" in exI, rule_tac x = "rs" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
993 |
rule_tac x = "ml - Suc (Suc 0)" in exI, rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
994 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
995 |
apply(case_tac ml, simp, case_tac nat, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
996 |
apply(simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
997 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
998 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
999 |
lemma [simp]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1000 |
"wcode_on_right_moving_1 ires rs (b, []) \<Longrightarrow> False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1001 |
apply(simp add: wcode_double_case_inv_simps exponent_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1002 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1003 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1004 |
lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba; c = Bk # ba\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1005 |
\<Longrightarrow> wcode_on_right_moving_1 ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1006 |
apply(simp only: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1007 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1008 |
apply(rule_tac x = "Suc 0" in exI, rule_tac x = "Suc (Suc ln)" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1009 |
rule_tac x = rn in exI, simp add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1010 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1011 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1012 |
lemma [elim]: "\<lbrakk>wcode_erase1 ires rs (aa, Oc # list); b = aa \<and> Bk # list = ba\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1013 |
wcode_erase1 ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1014 |
apply(simp only: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1015 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1016 |
apply(rule_tac x = ln in exI, rule_tac x = rn in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1017 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1018 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1019 |
lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (aa, []); b = aa \<and> [Oc] = ba\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1020 |
\<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1021 |
apply(simp only: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1022 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1023 |
apply(rule_tac disjI2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1024 |
apply(simp only:wcode_backto_standard_pos_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1025 |
apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1026 |
rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1027 |
apply(case_tac mr, simp_all add: exponent_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1028 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1029 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1030 |
lemma [elim]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1031 |
"\<lbrakk>wcode_goon_right_moving_1 ires rs (aa, Bk # list); b = aa \<and> Oc # list = ba\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1032 |
\<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1033 |
apply(simp only: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1034 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1035 |
apply(rule_tac disjI2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1036 |
apply(simp only:wcode_backto_standard_pos_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1037 |
apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, rule_tac x = ln in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1038 |
rule_tac x = "rn - Suc 0" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1039 |
apply(case_tac mr, simp, case_tac rn, simp, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1040 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1041 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1042 |
lemma [elim]: "\<lbrakk>wcode_goon_right_moving_1 ires rs (b, Oc # ba); Oc # b = aa \<and> list = ba\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1043 |
\<Longrightarrow> wcode_goon_right_moving_1 ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1044 |
apply(simp only: wcode_double_case_inv_simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1045 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1046 |
apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - Suc 0" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1047 |
rule_tac x = ln in exI, rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1048 |
apply(simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1049 |
apply(case_tac mr, simp, case_tac rn, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1050 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1051 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1052 |
lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, []); Bk # b = aa\<rbrakk> \<Longrightarrow> False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1053 |
apply(auto simp: wcode_double_case_inv_simps wcode_backto_standard_pos_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1054 |
wcode_backto_standard_pos_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1055 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1056 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1057 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1058 |
lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, Bk # ba); Bk # b = aa \<and> list = ba\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1059 |
\<Longrightarrow> wcode_before_double ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1060 |
apply(simp only: wcode_double_case_inv_simps wcode_backto_standard_pos_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1061 |
wcode_backto_standard_pos_O.simps wcode_before_double.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1062 |
apply(erule_tac disjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1063 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1064 |
apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1065 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1066 |
apply(case_tac [!] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1067 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1068 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1069 |
lemma [simp]: "wcode_backto_standard_pos ires rs ([], Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1070 |
apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1071 |
wcode_backto_standard_pos_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1072 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1073 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1074 |
lemma [simp]: "wcode_backto_standard_pos ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1075 |
apply(auto simp: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1076 |
wcode_backto_standard_pos_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1077 |
apply(case_tac mr, simp, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1078 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1079 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1080 |
lemma [elim]: "\<lbrakk>wcode_backto_standard_pos ires rs (b, Oc # list); tl b = aa; hd b # Oc # list = ba\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1081 |
\<Longrightarrow> wcode_backto_standard_pos ires rs (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1082 |
apply(simp only: wcode_backto_standard_pos.simps wcode_backto_standard_pos_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1083 |
wcode_backto_standard_pos_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1084 |
apply(erule_tac disjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1085 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1086 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1087 |
apply(case_tac ml, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1088 |
apply(rule_tac disjI1, rule_tac conjI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1089 |
apply(rule_tac x = ln in exI, simp, rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1090 |
apply(rule_tac disjI2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1091 |
apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = ln in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1092 |
rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1093 |
apply(simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1094 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1095 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1096 |
declare new_tape.simps[simp del] nth_of.simps[simp del] fetch.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1097 |
lemma wcode_double_case_first_correctness: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1098 |
"let P = (\<lambda> (st, l, r). st = 13) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1099 |
let Q = (\<lambda> (st, l, r). wcode_double_case_inv st ires rs (l, r)) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1100 |
let f = (\<lambda> 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 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1101 |
\<exists> n .P (f n) \<and> Q (f (n::nat))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1102 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1103 |
let ?P = "(\<lambda> (st, l, r). st = 13)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1104 |
let ?Q = "(\<lambda> (st, l, r). wcode_double_case_inv st ires rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1105 |
let ?f = "(\<lambda> 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)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1106 |
have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1107 |
proof(rule_tac halt_lemma2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1108 |
show "wf wcode_double_case_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1109 |
by auto |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1110 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1111 |
show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1112 |
?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_double_case_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1113 |
proof(rule_tac allI, case_tac "?f na", simp add: tstep_red) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1114 |
fix na a b c |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1115 |
show "a \<noteq> 13 \<and> wcode_double_case_inv a ires rs (b, c) \<longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1116 |
(case tstep (a, b, c) t_wcode_main of (st, x) \<Rightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1117 |
wcode_double_case_inv st ires rs x) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1118 |
(tstep (a, b, c) t_wcode_main, a, b, c) \<in> wcode_double_case_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1119 |
apply(rule_tac impI, simp add: wcode_double_case_inv.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1120 |
apply(auto split: if_splits simp: tstep.simps, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1121 |
case_tac [!] c, simp_all, case_tac [!] "(c::block list)!0") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1122 |
apply(simp_all add: new_tape.simps wcode_double_case_inv.simps wcode_double_case_le_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1123 |
lex_pair_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1124 |
apply(auto split: if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1125 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1126 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1127 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1128 |
show "?Q (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1129 |
apply(simp add: steps.simps wcode_double_case_inv.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1130 |
wcode_on_left_moving_1.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1131 |
wcode_on_left_moving_1_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1132 |
apply(rule_tac disjI1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1133 |
apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1134 |
apply(rule_tac x = "Suc 0" in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1135 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1136 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1137 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1138 |
show "\<not> ?P (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1139 |
apply(simp add: steps.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1140 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1141 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1142 |
thus "let P = \<lambda>(st, l, r). st = 13; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1143 |
Q = \<lambda>(st, l, r). wcode_double_case_inv st ires rs (l, r); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1144 |
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 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1145 |
in \<exists>n. P (f n) \<and> Q (f n)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1146 |
apply(simp add: Let_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1147 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1148 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1149 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1150 |
lemma [elim]: "t_ncorrect tp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1151 |
\<Longrightarrow> t_ncorrect (abacus.tshift tp a)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1152 |
apply(simp add: t_ncorrect.simps shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1153 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1154 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1155 |
lemma tshift_fetch: "\<lbrakk> fetch tp a b = (aa, st'); 0 < st'\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1156 |
\<Longrightarrow> fetch (abacus.tshift tp (length tp1 div 2)) a b |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1157 |
= (aa, st' + length tp1 div 2)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1158 |
apply(subgoal_tac "a > 0") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1159 |
apply(auto simp: fetch.simps nth_of.simps shift_length nth_map |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1160 |
tshift.simps split: block.splits if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1161 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1162 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1163 |
lemma t_steps_steps_eq: "\<lbrakk>steps (st, l, r) tp stp = (st', l', r'); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1164 |
0 < st'; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1165 |
0 < st \<and> st \<le> length tp div 2; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1166 |
t_ncorrect tp1; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1167 |
t_ncorrect tp\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1168 |
\<Longrightarrow> t_steps (st + length tp1 div 2, l, r) (tshift tp (length tp1 div 2), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1169 |
length tp1 div 2) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1170 |
= (st' + length tp1 div 2, l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1171 |
apply(induct stp arbitrary: st' l' r', simp add: steps.simps t_steps.simps, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1172 |
simp add: tstep_red stepn) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1173 |
apply(case_tac "(steps (st, l, r) tp stp)", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1174 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1175 |
fix stp st' l' r' a b c |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1176 |
assume ind: "\<And>st' l' r'. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1177 |
\<lbrakk>a = st' \<and> b = l' \<and> c = r'; 0 < st'\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1178 |
\<Longrightarrow> t_steps (st + length tp1 div 2, l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1179 |
(abacus.tshift tp (length tp1 div 2), length tp1 div 2) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1180 |
(st' + length tp1 div 2, l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1181 |
and h: "tstep (a, b, c) tp = (st', l', r')" "0 < st'" "t_ncorrect tp1" "t_ncorrect tp" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1182 |
have k: "t_steps (st + length tp1 div 2, l, r) (abacus.tshift tp (length tp1 div 2), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1183 |
length tp1 div 2) stp = (a + length tp1 div 2, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1184 |
apply(rule_tac ind, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1185 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1186 |
apply(case_tac a, simp_all add: tstep.simps fetch.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1187 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1188 |
from h and this show "t_step (t_steps (st + length tp1 div 2, l, r) (abacus.tshift tp (length tp1 div 2), length tp1 div 2) stp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1189 |
(abacus.tshift tp (length tp1 div 2), length tp1 div 2) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1190 |
(st' + length tp1 div 2, l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1191 |
apply(simp add: k) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1192 |
apply(simp add: tstep.simps t_step.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1193 |
apply(case_tac "fetch tp a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1194 |
apply(subgoal_tac "fetch (abacus.tshift tp (length tp1 div 2)) a |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1195 |
(case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, st' + length tp1 div 2)", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1196 |
apply(simp add: tshift_fetch) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1197 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1198 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1199 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1200 |
lemma t_tshift_lemma: "\<lbrakk> steps (st, l, r) tp stp = (st', l', r'); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1201 |
st' \<noteq> 0; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1202 |
stp > 0; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1203 |
0 < st \<and> st \<le> length tp div 2; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1204 |
t_ncorrect tp1; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1205 |
t_ncorrect tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1206 |
t_ncorrect tp2 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1207 |
\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1208 |
\<Longrightarrow> \<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1209 |
= (st' + length tp1 div 2, l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1210 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1211 |
assume h: "steps (st, l, r) tp stp = (st', l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1212 |
"st' \<noteq> 0" "stp > 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1213 |
"0 < st \<and> st \<le> length tp div 2" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1214 |
"t_ncorrect tp1" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1215 |
"t_ncorrect tp" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1216 |
"t_ncorrect tp2" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1217 |
from h have |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1218 |
"\<exists>stp>0. t_steps (st + length tp1 div 2, l, r) (tp1 @ abacus.tshift tp (length tp1 div 2) @ tp2, 0) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1219 |
(st' + length tp1 div 2, l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1220 |
apply(rule_tac stp = stp in turing_shift, simp_all add: shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1221 |
apply(simp add: t_steps_steps_eq) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1222 |
apply(simp add: t_ncorrect.simps shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1223 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1224 |
thus "\<exists> stp>0. steps (st + length tp1 div 2, l, r) (tp1 @ tshift tp (length tp1 div 2) @ tp2) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1225 |
= (st' + length tp1 div 2, l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1226 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1227 |
apply(rule_tac x = stp in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1228 |
apply(subgoal_tac "length (tp1 @ abacus.tshift tp (length tp1 div 2) @ tp2) mod 2 = 0") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1229 |
apply(simp only: steps_eq) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1230 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1231 |
apply(auto simp: t_ncorrect.simps shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1232 |
apply arith |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1233 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1234 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1235 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1236 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1237 |
lemma t_twice_len_ge: "Suc 0 \<le> length t_twice div 2" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1238 |
apply(simp add: t_twice_def tMp.simps shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1239 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1240 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1241 |
lemma [intro]: "rec_calc_rel (recf.id (Suc 0) 0) [rs] rs" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1242 |
apply(rule_tac calc_id, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1243 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1244 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1245 |
lemma [intro]: "rec_calc_rel (constn 2) [rs] 2" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1246 |
using prime_rel_exec_eq[of "constn 2" "[rs]" 2] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1247 |
apply(subgoal_tac "primerec (constn 2) 1", auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1248 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1249 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1250 |
lemma [intro]: "rec_calc_rel rec_mult [rs, 2] (2 * rs)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1251 |
using prime_rel_exec_eq[of "rec_mult" "[rs, 2]" "2*rs"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1252 |
apply(subgoal_tac "primerec rec_mult (Suc (Suc 0))", auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1253 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1254 |
lemma t_twice_correct: "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1255 |
(tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1256 |
(0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1257 |
proof(case_tac "rec_ci rec_twice") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1258 |
fix a b c |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1259 |
assume h: "rec_ci rec_twice = (a, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1260 |
have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_twice @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1261 |
(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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1262 |
proof(rule_tac t_compiled_by_rec) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1263 |
show "rec_ci rec_twice = (a, b, c)" by (simp add: h) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1264 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1265 |
show "rec_calc_rel rec_twice [rs] (2 * rs)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1266 |
apply(simp add: rec_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1267 |
apply(rule_tac rs = "[rs, 2]" in calc_cn, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1268 |
apply(rule_tac allI, case_tac k, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1269 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1270 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1271 |
show "length [rs] = Suc 0" by simp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1272 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1273 |
show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1274 |
by simp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1275 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1276 |
show "start_of twice_ly (length abc_twice) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1277 |
start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1278 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1279 |
apply(simp add: twice_ly_def abc_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1280 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1281 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1282 |
show "tm_of abc_twice = tm_of (a [+] dummy_abc (Suc 0))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1283 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1284 |
apply(simp add: abc_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1285 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1286 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1287 |
thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1288 |
(tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1289 |
(0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1290 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1291 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1292 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1293 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1294 |
lemma change_termi_state_fetch: "\<lbrakk>fetch ap a b = (aa, st); st > 0\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1295 |
\<Longrightarrow> fetch (change_termi_state ap) a b = (aa, st)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1296 |
apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1297 |
split: if_splits block.splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1298 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1299 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1300 |
lemma change_termi_state_exec_in_range: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1301 |
"\<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1302 |
\<Longrightarrow> steps (st, l, r) (change_termi_state ap) stp = (st', l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1303 |
proof(induct stp arbitrary: st l r st' l' r', simp add: steps.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1304 |
fix stp st l r st' l' r' |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1305 |
assume ind: "\<And>st l r st' l' r'. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1306 |
\<lbrakk>steps (st, l, r) ap stp = (st', l', r'); st' \<noteq> 0\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1307 |
steps (st, l, r) (change_termi_state ap) stp = (st', l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1308 |
and h: "steps (st, l, r) ap (Suc stp) = (st', l', r')" "st' \<noteq> 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1309 |
from h show "steps (st, l, r) (change_termi_state ap) (Suc stp) = (st', l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1310 |
proof(simp add: tstep_red, case_tac "steps (st, l, r) ap stp", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1311 |
fix a b c |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1312 |
assume g: "steps (st, l, r) ap stp = (a, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1313 |
"tstep (a, b, c) ap = (st', l', r')" "0 < st'" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1314 |
hence "steps (st, l, r) (change_termi_state ap) stp = (a, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1315 |
apply(rule_tac ind, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1316 |
apply(case_tac a, simp_all add: tstep_0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1317 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1318 |
from g and this show "tstep (steps (st, l, r) (change_termi_state ap) stp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1319 |
(change_termi_state ap) = (st', l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1320 |
apply(simp add: tstep.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1321 |
apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1322 |
apply(subgoal_tac "fetch (change_termi_state ap) a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1323 |
= (aa, st')", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1324 |
apply(simp add: change_termi_state_fetch) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1325 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1326 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1327 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1328 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1329 |
lemma change_termi_state_fetch0: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1330 |
"\<lbrakk>0 < a; a \<le> length ap div 2; t_correct ap; fetch ap a b = (aa, 0)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1331 |
\<Longrightarrow> fetch (change_termi_state ap) a b = (aa, Suc (length ap div 2))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1332 |
apply(case_tac b, auto simp: fetch.simps nth_of.simps change_termi_state.simps nth_map |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1333 |
split: if_splits block.splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1334 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1335 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1336 |
lemma turing_change_termi_state: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1337 |
"\<lbrakk>steps (Suc 0, l, r) ap stp = (0, l', r'); t_correct ap\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1338 |
\<Longrightarrow> \<exists> stp. steps (Suc 0, l, r) (change_termi_state ap) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1339 |
(Suc (length ap div 2), l', r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1340 |
apply(drule first_halt_point) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1341 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1342 |
apply(rule_tac x = "Suc stp" in exI, simp add: tstep_red) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1343 |
apply(case_tac "steps (Suc 0, l, r) ap stp") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1344 |
apply(simp add: isS0_def change_termi_state_exec_in_range) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1345 |
apply(subgoal_tac "steps (Suc 0, l, r) (change_termi_state ap) stp = (a, b, c)", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1346 |
apply(simp add: tstep.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1347 |
apply(case_tac "fetch ap a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1348 |
apply(subgoal_tac "fetch (change_termi_state ap) a |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1349 |
(case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = (aa, Suc (length ap div 2))", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1350 |
apply(rule_tac ap = ap in change_termi_state_fetch0, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1351 |
apply(rule_tac tp = "(l, r)" and l = b and r = c and stp = stp and A = ap in s_keep, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1352 |
apply(simp add: change_termi_state_exec_in_range) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1353 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1354 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1355 |
lemma t_twice_change_term_state: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1356 |
"\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1357 |
= (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1358 |
using t_twice_correct[of ires rs n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1359 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1360 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1361 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1362 |
proof(drule_tac turing_change_termi_state) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1363 |
fix stp ln rn |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1364 |
show "t_correct (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1365 |
apply(rule_tac t_compiled_correct, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1366 |
apply(simp add: twice_ly_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1367 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1368 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1369 |
fix stp ln rn |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1370 |
show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1371 |
(change_termi_state (tm_of abc_twice @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1372 |
(start_of twice_ly (length abc_twice) - Suc 0))) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1373 |
(Suc (length (tm_of abc_twice @ tMp (Suc 0) (start_of twice_ly (length abc_twice) - Suc 0)) div 2), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1374 |
Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1375 |
\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1376 |
(Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1377 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1378 |
apply(simp add: t_twice_len_def t_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1379 |
apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1380 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1381 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1382 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1383 |
lemma t_twice_append_pre: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1384 |
"steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1385 |
= (Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1386 |
\<Longrightarrow> \<exists> 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>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1387 |
(t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1388 |
([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1389 |
= (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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1390 |
proof(rule_tac t_tshift_lemma, simp_all add: t_twice_len_ge) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1391 |
assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_twice stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1392 |
(Suc t_twice_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1393 |
thus "0 < stp" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1394 |
apply(case_tac stp, simp add: steps.simps t_twice_len_ge t_twice_len_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1395 |
using t_twice_len_ge |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1396 |
apply(simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1397 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1398 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1399 |
show "t_ncorrect t_wcode_main_first_part" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1400 |
apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1401 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1402 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1403 |
show "t_ncorrect t_twice" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1404 |
using length_tm_even[of abc_twice] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1405 |
apply(auto simp: t_ncorrect.simps t_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1406 |
apply(arith) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1407 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1408 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1409 |
show "t_ncorrect ((L, Suc 0) # (L, Suc 0) # |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1410 |
abacus.tshift t_fourtimes (t_twice_len + 13) @ [(L, Suc 0), (L, Suc 0)])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1411 |
using length_tm_even[of abc_fourtimes] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1412 |
apply(simp add: t_ncorrect.simps shift_length t_fourtimes_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1413 |
apply arith |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1414 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1415 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1416 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1417 |
lemma t_twice_append: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1418 |
"\<exists> 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>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1419 |
(t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1420 |
([(L, 1), (L, 1)] @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1421 |
= (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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1422 |
using t_twice_change_term_state[of ires rs n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1423 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1424 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1425 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1426 |
apply(drule_tac t_twice_append_pre) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1427 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1428 |
apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1429 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1430 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1431 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1432 |
lemma [simp]: "fetch t_wcode_main (Suc (t_twice_len + length t_wcode_main_first_part div 2)) Oc |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1433 |
= (L, Suc 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1434 |
apply(subgoal_tac "length (t_twice) mod 2 = 0") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1435 |
apply(simp add: t_wcode_main_def nth_append fetch.simps t_wcode_main_first_part_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1436 |
nth_of.simps shift_length t_twice_len_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1437 |
apply(simp add: t_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1438 |
apply(subgoal_tac "length (tm_of abc_twice) mod 2 = 0") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1439 |
apply arith |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1440 |
apply(rule_tac tm_even) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1441 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1442 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1443 |
lemma wcode_jump1: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1444 |
"\<exists> stp ln rn. steps (Suc (t_twice_len) + length t_wcode_main_first_part div 2, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1445 |
Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>n\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1446 |
t_wcode_main stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1447 |
= (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1448 |
apply(rule_tac x = "Suc 0" in exI, rule_tac x = "m" in exI, rule_tac x = n in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1449 |
apply(simp add: steps.simps tstep.simps exp_ind_def new_tape.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1450 |
apply(case_tac m, simp, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1451 |
apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1452 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1453 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1454 |
lemma wcode_main_first_part_len: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1455 |
"length t_wcode_main_first_part = 24" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1456 |
apply(simp add: t_wcode_main_first_part_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1457 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1458 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1459 |
lemma wcode_double_case: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1460 |
shows "\<exists>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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1461 |
(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1462 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1463 |
have "\<exists>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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1464 |
(13, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1465 |
using wcode_double_case_first_correctness[of ires rs m n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1466 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1467 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1468 |
apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Oc # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1469 |
Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na", |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1470 |
auto simp: wcode_double_case_inv.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1471 |
wcode_before_double.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1472 |
apply(rule_tac x = na in exI, rule_tac x = ln in exI, rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1473 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1474 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1475 |
from this obtain stpa lna rna where stp1: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1476 |
"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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1477 |
(13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1478 |
have "\<exists> 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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1479 |
(13 + t_twice_len, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1480 |
using t_twice_append[of "Bk\<^bsup>lna\<^esup> @ Oc # ires" "Suc rs" rna] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1481 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1482 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1483 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1484 |
apply(simp add: wcode_main_first_part_len) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1485 |
apply(rule_tac x = stp in exI, rule_tac x = "ln + lna" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1486 |
rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1487 |
apply(simp add: t_wcode_main_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1488 |
apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1489 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1490 |
from this obtain stpb lnb rnb where stp2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1491 |
"steps (13, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1492 |
(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 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1493 |
have "\<exists>stp ln rn. steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1494 |
Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1495 |
(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1496 |
using wcode_jump1[of lnb "Oc # ires" "Suc rs" rnb] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1497 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1498 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1499 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1500 |
apply(rule_tac x = stp in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1501 |
rule_tac x = ln in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1502 |
rule_tac x = rn in exI, simp add:wcode_main_first_part_len t_wcode_main_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1503 |
apply(subgoal_tac "Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc # ires = Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1504 |
apply(simp add: exp_ind_def[THEN sym] exp_ind[THEN sym]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1505 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1506 |
apply(case_tac lnb, simp, simp add: exp_ind_def[THEN sym] exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1507 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1508 |
from this obtain stpc lnc rnc where stp3: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1509 |
"steps (13 + t_twice_len, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1510 |
Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stpc = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1511 |
(Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (Suc (Suc (2 *rs)))\<^esup> @ Bk\<^bsup>rnc\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1512 |
by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1513 |
from stp1 stp2 stp3 show "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1514 |
apply(rule_tac x = "stpa + stpb + stpc" in exI, rule_tac x = lnc in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1515 |
rule_tac x = rnc in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1516 |
apply(simp add: steps_add) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1517 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1518 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1519 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1520 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1521 |
(* Begin: fourtime_case*) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1522 |
fun wcode_on_left_moving_2_B :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1523 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1524 |
"wcode_on_left_moving_2_B ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1525 |
(\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1526 |
r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1527 |
ml + mr > Suc 0 \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1528 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1529 |
fun wcode_on_left_moving_2_O :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1530 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1531 |
"wcode_on_left_moving_2_O ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1532 |
(\<exists> ln rn. l = Bk # Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1533 |
r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1534 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1535 |
fun wcode_on_left_moving_2 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1536 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1537 |
"wcode_on_left_moving_2 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1538 |
(wcode_on_left_moving_2_B ires rs (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1539 |
wcode_on_left_moving_2_O ires rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1540 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1541 |
fun wcode_on_checking_2 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1542 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1543 |
"wcode_on_checking_2 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1544 |
(\<exists> ln rn. l = Oc#ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1545 |
r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1546 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1547 |
fun wcode_goon_checking :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1548 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1549 |
"wcode_goon_checking ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1550 |
(\<exists> ln rn. l = ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1551 |
r = Oc # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1552 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1553 |
fun wcode_right_move :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1554 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1555 |
"wcode_right_move ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1556 |
(\<exists> ln rn. l = Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1557 |
r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1558 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1559 |
fun wcode_erase2 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1560 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1561 |
"wcode_erase2 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1562 |
(\<exists> ln rn. l = Bk # Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1563 |
tl r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1564 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1565 |
fun wcode_on_right_moving_2 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1566 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1567 |
"wcode_on_right_moving_2 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1568 |
(\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1569 |
r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr > Suc 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1570 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1571 |
fun wcode_goon_right_moving_2 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1572 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1573 |
"wcode_goon_right_moving_2 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1574 |
(\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1575 |
r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = Suc rs)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1576 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1577 |
fun wcode_backto_standard_pos_2_B :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1578 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1579 |
"wcode_backto_standard_pos_2_B ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1580 |
(\<exists> ln rn. l = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1581 |
r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1582 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1583 |
fun wcode_backto_standard_pos_2_O :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1584 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1585 |
"wcode_backto_standard_pos_2_O ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1586 |
(\<exists> ml mr ln rn. l = Oc\<^bsup>ml \<^esup>@ Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1587 |
r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1588 |
ml + mr = (Suc (Suc rs)) \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1589 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1590 |
fun wcode_backto_standard_pos_2 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1591 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1592 |
"wcode_backto_standard_pos_2 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1593 |
(wcode_backto_standard_pos_2_O ires rs (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1594 |
wcode_backto_standard_pos_2_B ires rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1595 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1596 |
fun wcode_before_fourtimes :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1597 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1598 |
"wcode_before_fourtimes ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1599 |
(\<exists> ln rn. l = Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1600 |
r = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1601 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1602 |
declare wcode_on_left_moving_2_B.simps[simp del] wcode_on_left_moving_2.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1603 |
wcode_on_left_moving_2_O.simps[simp del] wcode_on_checking_2.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1604 |
wcode_goon_checking.simps[simp del] wcode_right_move.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1605 |
wcode_erase2.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1606 |
wcode_on_right_moving_2.simps[simp del] wcode_goon_right_moving_2.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1607 |
wcode_backto_standard_pos_2_B.simps[simp del] wcode_backto_standard_pos_2_O.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1608 |
wcode_backto_standard_pos_2.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1609 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1610 |
lemmas wcode_fourtimes_invs = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1611 |
wcode_on_left_moving_2_B.simps wcode_on_left_moving_2.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1612 |
wcode_on_left_moving_2_O.simps wcode_on_checking_2.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1613 |
wcode_goon_checking.simps wcode_right_move.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1614 |
wcode_erase2.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1615 |
wcode_on_right_moving_2.simps wcode_goon_right_moving_2.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1616 |
wcode_backto_standard_pos_2_B.simps wcode_backto_standard_pos_2_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1617 |
wcode_backto_standard_pos_2.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1618 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1619 |
fun wcode_fourtimes_case_inv :: "nat \<Rightarrow> bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1620 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1621 |
"wcode_fourtimes_case_inv st ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1622 |
(if st = Suc 0 then wcode_on_left_moving_2 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1623 |
else if st = Suc (Suc 0) then wcode_on_checking_2 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1624 |
else if st = 7 then wcode_goon_checking ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1625 |
else if st = 8 then wcode_right_move ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1626 |
else if st = 9 then wcode_erase2 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1627 |
else if st = 10 then wcode_on_right_moving_2 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1628 |
else if st = 11 then wcode_goon_right_moving_2 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1629 |
else if st = 12 then wcode_backto_standard_pos_2 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1630 |
else if st = t_twice_len + 14 then wcode_before_fourtimes ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1631 |
else False)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1632 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1633 |
declare wcode_fourtimes_case_inv.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1634 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1635 |
fun wcode_fourtimes_case_state :: "t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1636 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1637 |
"wcode_fourtimes_case_state (st, l, r) = 13 - st" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1638 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1639 |
fun wcode_fourtimes_case_step :: "t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1640 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1641 |
"wcode_fourtimes_case_step (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1642 |
(if st = Suc 0 then length l |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1643 |
else if st = 9 then |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1644 |
(if hd r = Oc then 1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1645 |
else 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1646 |
else if st = 10 then length r |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1647 |
else if st = 11 then length r |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1648 |
else if st = 12 then length l |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1649 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1650 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1651 |
fun wcode_fourtimes_case_measure :: "t_conf \<Rightarrow> nat \<times> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1652 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1653 |
"wcode_fourtimes_case_measure (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1654 |
(wcode_fourtimes_case_state (st, l, r), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1655 |
wcode_fourtimes_case_step (st, l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1656 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1657 |
definition wcode_fourtimes_case_le :: "(t_conf \<times> t_conf) set" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1658 |
where "wcode_fourtimes_case_le \<equiv> (inv_image lex_pair wcode_fourtimes_case_measure)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1659 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1660 |
lemma wf_wcode_fourtimes_case_le[intro]: "wf wcode_fourtimes_case_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1661 |
by(auto intro:wf_inv_image simp: wcode_fourtimes_case_le_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1662 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1663 |
lemma [simp]: "fetch t_wcode_main (Suc (Suc 0)) Bk = (L, 7)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1664 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1665 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1666 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1667 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1668 |
lemma [simp]: "fetch t_wcode_main 7 Oc = (R, 8)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1669 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1670 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1671 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1672 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1673 |
lemma [simp]: "fetch t_wcode_main 8 Bk = (R, 9)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1674 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1675 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1676 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1677 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1678 |
lemma [simp]: "fetch t_wcode_main 9 Bk = (R, 10)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1679 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1680 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1681 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1682 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1683 |
lemma [simp]: "fetch t_wcode_main 9 Oc = (W0, 9)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1684 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1685 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1686 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1687 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1688 |
lemma [simp]: "fetch t_wcode_main 10 Bk = (R, 10)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1689 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1690 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1691 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1692 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1693 |
lemma [simp]: "fetch t_wcode_main 10 Oc = (R, 11)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1694 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1695 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1696 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1697 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1698 |
lemma [simp]: "fetch t_wcode_main 11 Bk = (W1, 12)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1699 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1700 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1701 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1702 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1703 |
lemma [simp]: "fetch t_wcode_main 11 Oc = (R, 11)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1704 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1705 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1706 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1707 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1708 |
lemma [simp]: "fetch t_wcode_main 12 Oc = (L, 12)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1709 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1710 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1711 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1712 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1713 |
lemma [simp]: "fetch t_wcode_main 12 Bk = (R, t_twice_len + 14)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1714 |
apply(simp add: t_wcode_main_def fetch.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1715 |
t_wcode_main_first_part_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1716 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1717 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1718 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1719 |
lemma [simp]: "wcode_on_left_moving_2 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1720 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1721 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1722 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1723 |
lemma [simp]: "wcode_on_checking_2 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1724 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1725 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1726 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1727 |
lemma [simp]: "wcode_goon_checking ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1728 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1729 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1730 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1731 |
lemma [simp]: "wcode_right_move ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1732 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1733 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1734 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1735 |
lemma [simp]: "wcode_erase2 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1736 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1737 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1738 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1739 |
lemma [simp]: "wcode_on_right_moving_2 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1740 |
apply(auto simp: wcode_fourtimes_invs exponent_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1741 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1742 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1743 |
lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1744 |
apply(auto simp: wcode_fourtimes_invs exponent_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1745 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1746 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1747 |
lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1748 |
apply(simp add: wcode_fourtimes_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1749 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1750 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1751 |
lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Bk # list) \<Longrightarrow> wcode_on_left_moving_2 ires rs (tl b, hd b # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1752 |
apply(simp only: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1753 |
apply(erule_tac disjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1754 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1755 |
apply(case_tac ml, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1756 |
apply(rule_tac x = "mr - (Suc (Suc 0))" in exI, rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1757 |
apply(case_tac mr, simp, case_tac nat, simp, simp add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1758 |
apply(rule_tac disjI1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1759 |
apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1760 |
simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1761 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1762 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1763 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1764 |
lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1765 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1766 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1767 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1768 |
lemma [simp]: "wcode_on_checking_2 ires rs (b, Bk # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1769 |
\<Longrightarrow> wcode_goon_checking ires rs (tl b, hd b # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1770 |
apply(simp only: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1771 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1772 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1773 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1774 |
lemma [simp]: "wcode_goon_checking ires rs (b, Bk # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1775 |
apply(simp add: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1776 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1777 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1778 |
lemma [simp]: " wcode_right_move ires rs (b, Bk # list) \<Longrightarrow> b\<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1779 |
apply(simp add: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1780 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1781 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1782 |
lemma [simp]: "wcode_right_move ires rs (b, Bk # list) \<Longrightarrow> wcode_erase2 ires rs (Bk # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1783 |
apply(auto simp:wcode_fourtimes_invs ) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1784 |
apply(rule_tac x = ln in exI, rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1785 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1786 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1787 |
lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1788 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1789 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1790 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1791 |
lemma [simp]: "wcode_erase2 ires rs (b, Bk # list) \<Longrightarrow> wcode_on_right_moving_2 ires rs (Bk # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1792 |
apply(auto simp:wcode_fourtimes_invs ) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1793 |
apply(rule_tac x = "Suc (Suc 0)" in exI, simp add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1794 |
apply(rule_tac x = "Suc (Suc ln)" in exI, simp add: exp_ind, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1795 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1796 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1797 |
lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1798 |
apply(auto simp:wcode_fourtimes_invs ) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1799 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1800 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1801 |
lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Bk # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1802 |
\<Longrightarrow> wcode_on_right_moving_2 ires rs (Bk # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1803 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1804 |
apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1805 |
apply(rule_tac x = "mr - 1" in exI, case_tac mr, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1806 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1807 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1808 |
lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1809 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1810 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1811 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1812 |
lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1813 |
wcode_backto_standard_pos_2 ires rs (b, Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1814 |
apply(simp add: wcode_fourtimes_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1815 |
apply(rule_tac x = ml in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1816 |
apply(rule_tac x = "Suc 0" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1817 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1818 |
apply(rule_tac x = "rn - 1" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1819 |
apply(case_tac rn, simp, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1820 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1821 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1822 |
lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1823 |
apply(simp add: wcode_fourtimes_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1824 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1825 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1826 |
lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1827 |
apply(simp add: wcode_fourtimes_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1828 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1829 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1830 |
lemma [simp]: "wcode_on_left_moving_2 ires rs (b, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1831 |
wcode_on_checking_2 ires rs (tl b, hd b # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1832 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1833 |
apply(case_tac [!] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1834 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1835 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1836 |
lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1837 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1838 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1839 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1840 |
lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, []) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1841 |
wcode_backto_standard_pos_2 ires rs (b, [Oc])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1842 |
apply(simp only: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1843 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1844 |
apply(rule_tac disjI1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1845 |
apply(rule_tac x = ml in exI, rule_tac x = "Suc 0" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1846 |
rule_tac x = ln in exI, rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1847 |
apply(case_tac mr, simp, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1848 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1849 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1850 |
lemma "wcode_backto_standard_pos_2 ires rs (b, Bk # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1851 |
\<Longrightarrow> (\<exists>ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \<and> (\<exists>rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1852 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1853 |
apply(case_tac [!] mr, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1854 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1855 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1856 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1857 |
lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) \<Longrightarrow> False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1858 |
apply(simp add: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1859 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1860 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1861 |
lemma [simp]: "wcode_goon_checking ires rs (b, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1862 |
(b = [] \<longrightarrow> wcode_right_move ires rs ([Oc], list)) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1863 |
(b \<noteq> [] \<longrightarrow> wcode_right_move ires rs (Oc # b, list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1864 |
apply(simp only: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1865 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1866 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1867 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1868 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1869 |
lemma [simp]: "wcode_right_move ires rs (b, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1870 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1871 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1872 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1873 |
lemma [simp]: " wcode_erase2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1874 |
apply(simp add: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1875 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1876 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1877 |
lemma [simp]: "wcode_erase2 ires rs (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1878 |
\<Longrightarrow> wcode_erase2 ires rs (b, Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1879 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1880 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1881 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1882 |
lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1883 |
apply(simp only: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1884 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1885 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1886 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1887 |
lemma [simp]: "wcode_on_right_moving_2 ires rs (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1888 |
\<Longrightarrow> wcode_goon_right_moving_2 ires rs (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1889 |
apply(auto simp: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1890 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1891 |
apply(rule_tac x = "Suc 0" in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1892 |
apply(rule_tac x = "ml - 2" in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1893 |
apply(case_tac ml, simp, case_tac nat, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1894 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1895 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1896 |
lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1897 |
apply(simp only:wcode_fourtimes_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1898 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1899 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1900 |
lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Bk # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1901 |
\<Longrightarrow> (\<exists>ln. b = Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires) \<and> (\<exists>rn. list = Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1902 |
apply(simp add: wcode_fourtimes_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1903 |
apply(case_tac [!] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1904 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1905 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1906 |
lemma [simp]: "wcode_on_checking_2 ires rs (b, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1907 |
apply(simp add: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1908 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1909 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1910 |
lemma [simp]: "wcode_goon_right_moving_2 ires rs (b, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1911 |
wcode_goon_right_moving_2 ires rs (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1912 |
apply(simp only:wcode_fourtimes_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1913 |
apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1914 |
apply(rule_tac x = "mr - 1" in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1915 |
apply(case_tac mr, case_tac rn, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1916 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1917 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1918 |
lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1919 |
apply(simp only: wcode_fourtimes_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1920 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1921 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1922 |
lemma [simp]: "wcode_backto_standard_pos_2 ires rs (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1923 |
\<Longrightarrow> wcode_backto_standard_pos_2 ires rs (tl b, hd b # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1924 |
apply(simp only: wcode_fourtimes_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1925 |
apply(erule_tac disjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1926 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1927 |
apply(case_tac ml, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1928 |
apply(rule_tac disjI2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1929 |
apply(rule_tac conjI, rule_tac x = ln in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1930 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1931 |
apply(rule_tac disjI1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1932 |
apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1933 |
rule_tac x = ln in exI, rule_tac x = rn in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1934 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1935 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1936 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1937 |
lemma wcode_fourtimes_case_first_correctness: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1938 |
shows "let P = (\<lambda> (st, l, r). st = t_twice_len + 14) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1939 |
let Q = (\<lambda> (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r)) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1940 |
let f = (\<lambda> 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 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1941 |
\<exists> n .P (f n) \<and> Q (f (n::nat))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1942 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1943 |
let ?P = "(\<lambda> (st, l, r). st = t_twice_len + 14)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1944 |
let ?Q = "(\<lambda> (st, l, r). wcode_fourtimes_case_inv st ires rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1945 |
let ?f = "(\<lambda> 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)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1946 |
have "\<exists> n . ?P (?f n) \<and> ?Q (?f (n::nat))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1947 |
proof(rule_tac halt_lemma2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1948 |
show "wf wcode_fourtimes_case_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1949 |
by auto |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1950 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1951 |
show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1952 |
?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_fourtimes_case_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1953 |
apply(rule_tac allI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1954 |
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, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1955 |
rule_tac impI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1956 |
apply(simp add: tstep_red tstep.simps, case_tac c, simp, case_tac [2] aa, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1957 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1958 |
apply(simp_all add: wcode_fourtimes_case_inv.simps new_tape.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1959 |
wcode_fourtimes_case_le_def lex_pair_def split: if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1960 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1961 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1962 |
show "?Q (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1963 |
apply(simp add: steps.simps wcode_fourtimes_case_inv.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1964 |
apply(simp add: wcode_on_left_moving_2.simps wcode_on_left_moving_2_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1965 |
wcode_on_left_moving_2_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1966 |
apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1967 |
apply(rule_tac x ="Suc 0" in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1968 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1969 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1970 |
show "\<not> ?P (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1971 |
apply(simp add: steps.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1972 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1973 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1974 |
thus "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1975 |
apply(erule_tac exE, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1976 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1977 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1978 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1979 |
definition t_fourtimes_len :: "nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1980 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1981 |
"t_fourtimes_len = (length t_fourtimes div 2)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1982 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1983 |
lemma t_fourtimes_len_gr: "t_fourtimes_len > 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1984 |
apply(simp add: t_fourtimes_len_def t_fourtimes_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1985 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1986 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1987 |
lemma t_fourtimes_correct: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1988 |
"\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1989 |
(tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1990 |
(0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1991 |
proof(case_tac "rec_ci rec_fourtimes") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1992 |
fix a b c |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1993 |
assume h: "rec_ci rec_fourtimes = (a, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1994 |
have "\<exists>stp m l. steps (Suc 0, Bk # Bk # ires, <[rs]> @ Bk\<^bsup>n\<^esup>) (tm_of abc_fourtimes @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1995 |
(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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1996 |
proof(rule_tac t_compiled_by_rec) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1997 |
show "rec_ci rec_fourtimes = (a, b, c)" by (simp add: h) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1998 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
1999 |
show "rec_calc_rel rec_fourtimes [rs] (4 * rs)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2000 |
using prime_rel_exec_eq [of rec_fourtimes "[rs]" "4 * rs"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2001 |
apply(subgoal_tac "primerec rec_fourtimes (length [rs])") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2002 |
apply(simp_all add: rec_fourtimes_def rec_exec.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2003 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2004 |
apply(simp only: Nat.One_nat_def[THEN sym], auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2005 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2006 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2007 |
show "length [rs] = Suc 0" by simp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2008 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2009 |
show "layout_of (a [+] dummy_abc (Suc 0)) = layout_of (a [+] dummy_abc (Suc 0))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2010 |
by simp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2011 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2012 |
show "start_of fourtimes_ly (length abc_fourtimes) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2013 |
start_of (layout_of (a [+] dummy_abc (Suc 0))) (length (a [+] dummy_abc (Suc 0)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2014 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2015 |
apply(simp add: fourtimes_ly_def abc_fourtimes_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2016 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2017 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2018 |
show "tm_of abc_fourtimes = tm_of (a [+] dummy_abc (Suc 0))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2019 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2020 |
apply(simp add: abc_fourtimes_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2021 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2022 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2023 |
thus "\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2024 |
(tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2025 |
(0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2026 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2027 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2028 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2029 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2030 |
lemma t_fourtimes_change_term_state: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2031 |
"\<exists> stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2032 |
= (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2033 |
using t_fourtimes_correct[of ires rs n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2034 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2035 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2036 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2037 |
proof(drule_tac turing_change_termi_state) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2038 |
fix stp ln rn |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2039 |
show "t_correct (tm_of abc_fourtimes @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2040 |
(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2041 |
apply(rule_tac t_compiled_correct, auto simp: fourtimes_ly_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2042 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2043 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2044 |
fix stp ln rn |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2045 |
show "\<exists>stp. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2046 |
(change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2047 |
(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2048 |
(Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) (start_of fourtimes_ly |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2049 |
(length abc_fourtimes) - Suc 0)) div 2), Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2050 |
\<exists>stp ln rn. steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2051 |
(Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2052 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2053 |
apply(simp add: t_fourtimes_len_def t_fourtimes_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2054 |
apply(rule_tac x = stp in exI, rule_tac x = ln in exI, rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2055 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2056 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2057 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2058 |
lemma t_fourtimes_append_pre: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2059 |
"steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2060 |
= (Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2061 |
\<Longrightarrow> \<exists> stp>0. steps (Suc 0 + length (t_wcode_main_first_part @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2062 |
tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2063 |
Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2064 |
((t_wcode_main_first_part @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2065 |
tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2066 |
tshift t_fourtimes (length (t_wcode_main_first_part @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2067 |
tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2) @ ([(L, 1), (L, 1)])) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2068 |
= (Suc t_fourtimes_len + length (t_wcode_main_first_part @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2069 |
tshift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2070 |
Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2071 |
proof(rule_tac t_tshift_lemma, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2072 |
assume "steps (Suc 0, Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_fourtimes stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2073 |
(Suc t_fourtimes_len, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2074 |
thus "0 < stp" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2075 |
using t_fourtimes_len_gr |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2076 |
apply(case_tac stp, simp_all add: steps.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2077 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2078 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2079 |
show "Suc 0 \<le> length t_fourtimes div 2" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2080 |
apply(simp add: t_fourtimes_def shift_length tMp.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2081 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2082 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2083 |
show "t_ncorrect (t_wcode_main_first_part @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2084 |
abacus.tshift t_twice (length t_wcode_main_first_part div 2) @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2085 |
[(L, Suc 0), (L, Suc 0)])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2086 |
apply(simp add: t_ncorrect.simps t_wcode_main_first_part_def shift_length |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2087 |
t_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2088 |
using tm_even[of abc_twice] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2089 |
by arith |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2090 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2091 |
show "t_ncorrect t_fourtimes" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2092 |
apply(simp add: t_fourtimes_def steps.simps t_ncorrect.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2093 |
using tm_even[of abc_fourtimes] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2094 |
by arith |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2095 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2096 |
show "t_ncorrect [(L, Suc 0), (L, Suc 0)]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2097 |
apply(simp add: t_ncorrect.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2098 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2099 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2100 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2101 |
lemma [simp]: "length t_wcode_main_first_part = 24" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2102 |
apply(simp add: t_wcode_main_first_part_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2103 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2104 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2105 |
lemma [simp]: "(26 + length t_twice) div 2 = (length t_twice) div 2 + 13" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2106 |
using tm_even[of abc_twice] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2107 |
apply(simp add: t_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2108 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2109 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2110 |
lemma [simp]: "((26 + length (abacus.tshift t_twice 12)) div 2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2111 |
= (length (abacus.tshift t_twice 12) div 2 + 13)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2112 |
using tm_even[of abc_twice] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2113 |
apply(simp add: t_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2114 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2115 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2116 |
lemma [simp]: "t_twice_len + 14 = 14 + length (abacus.tshift t_twice 12) div 2" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2117 |
using tm_even[of abc_twice] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2118 |
apply(simp add: t_twice_def t_twice_len_def shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2119 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2120 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2121 |
lemma t_fourtimes_append: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2122 |
"\<exists> stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2123 |
steps (Suc 0 + length (t_wcode_main_first_part @ tshift t_twice |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2124 |
(length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2125 |
Bk # Bk # ires, Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2126 |
((t_wcode_main_first_part @ tshift t_twice (length t_wcode_main_first_part div 2) @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2127 |
[(L, 1), (L, 1)]) @ tshift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2128 |
= (Suc t_fourtimes_len + length (t_wcode_main_first_part @ tshift t_twice |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2129 |
(length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2130 |
Oc\<^bsup>Suc (4 * rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2131 |
using t_fourtimes_change_term_state[of ires rs n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2132 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2133 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2134 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2135 |
apply(drule_tac t_fourtimes_append_pre) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2136 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2137 |
apply(rule_tac x = stpa in exI, rule_tac x = ln in exI, rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2138 |
apply(simp add: t_twice_len_def shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2139 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2140 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2141 |
lemma t_wcode_main_len: "length t_wcode_main = length t_twice + length t_fourtimes + 28" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2142 |
apply(simp add: t_wcode_main_def shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2143 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2144 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2145 |
lemma [simp]: "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) b |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2146 |
= (L, Suc 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2147 |
using tm_even[of "abc_twice"] tm_even[of "abc_fourtimes"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2148 |
apply(case_tac b) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2149 |
apply(simp_all only: fetch.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2150 |
apply(auto simp: nth_of.simps t_wcode_main_len t_twice_len_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2151 |
t_fourtimes_def t_twice_def t_fourtimes_def t_fourtimes_len_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2152 |
apply(auto simp: t_wcode_main_def t_wcode_main_first_part_def shift_length t_twice_def nth_append |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2153 |
t_fourtimes_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2154 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2155 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2156 |
lemma wcode_jump2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2157 |
"\<exists> stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2158 |
, Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2159 |
(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4 * rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2160 |
apply(rule_tac x = "Suc 0" in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2161 |
apply(simp add: steps.simps shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2162 |
apply(rule_tac x = lnb in exI, rule_tac x = rnb in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2163 |
apply(simp add: tstep.simps new_tape.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2164 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2165 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2166 |
lemma wcode_fourtimes_case: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2167 |
shows "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2168 |
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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2169 |
(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2170 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2171 |
have "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2172 |
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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2173 |
(t_twice_len + 14, Bk # Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2174 |
using wcode_fourtimes_case_first_correctness[of ires rs m n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2175 |
apply(simp add: wcode_fourtimes_case_inv.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2176 |
apply(rule_tac x = na in exI, rule_tac x = ln in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2177 |
rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2178 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2179 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2180 |
from this obtain stpa lna rna where stp1: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2181 |
"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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2182 |
(t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2183 |
have "\<exists>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>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2184 |
t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2185 |
(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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2186 |
using t_fourtimes_append[of " Bk\<^bsup>lna\<^esup> @ Oc # ires" "rs + 1" rna] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2187 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2188 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2189 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2190 |
apply(simp add: t_wcode_main_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2191 |
apply(rule_tac x = stp in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2192 |
rule_tac x = "ln + lna" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2193 |
rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2194 |
apply(simp add: exp_ind_def[THEN sym] exp_add[THEN sym]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2195 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2196 |
from this obtain stpb lnb rnb where stp2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2197 |
"steps (t_twice_len + 14, Bk # Bk # Bk\<^bsup>lna\<^esup> @ Oc # ires, Oc\<^bsup>Suc (rs + 1)\<^esup> @ Bk\<^bsup>rna\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2198 |
t_wcode_main stpb = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2199 |
(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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2200 |
by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2201 |
have "\<exists>stp ln rn. steps (t_twice_len + 14 + t_fourtimes_len, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2202 |
Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2203 |
t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2204 |
(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2205 |
apply(rule wcode_jump2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2206 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2207 |
from this obtain stpc lnc rnc where stp3: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2208 |
"steps (t_twice_len + 14 + t_fourtimes_len, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2209 |
Bk # Bk # Bk\<^bsup>lnb\<^esup> @ Oc # ires, Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnb\<^esup>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2210 |
t_wcode_main stpc = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2211 |
(Suc 0, Bk # Bk\<^bsup>lnc\<^esup> @ Oc # ires, Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rnc\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2212 |
by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2213 |
from stp1 stp2 stp3 show "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2214 |
apply(rule_tac x = "stpa + stpb + stpc" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2215 |
rule_tac x = lnc in exI, rule_tac x = rnc in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2216 |
apply(simp add: steps_add) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2217 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2218 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2219 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2220 |
(**********************************************************) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2221 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2222 |
fun wcode_on_left_moving_3_B :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2223 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2224 |
"wcode_on_left_moving_3_B ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2225 |
(\<exists> ml mr rn. l = Bk\<^bsup>ml\<^esup> @ Oc # Bk # Bk # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2226 |
r = Bk\<^bsup>mr\<^esup> @ Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2227 |
ml + mr > Suc 0 \<and> mr > 0 )" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2228 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2229 |
fun wcode_on_left_moving_3_O :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2230 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2231 |
"wcode_on_left_moving_3_O ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2232 |
(\<exists> ln rn. l = Bk # Bk # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2233 |
r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2234 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2235 |
fun wcode_on_left_moving_3 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2236 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2237 |
"wcode_on_left_moving_3 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2238 |
(wcode_on_left_moving_3_B ires rs (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2239 |
wcode_on_left_moving_3_O ires rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2240 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2241 |
fun wcode_on_checking_3 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2242 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2243 |
"wcode_on_checking_3 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2244 |
(\<exists> ln rn. l = Bk # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2245 |
r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2246 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2247 |
fun wcode_goon_checking_3 :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2248 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2249 |
"wcode_goon_checking_3 ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2250 |
(\<exists> ln rn. l = ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2251 |
r = Bk # Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2252 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2253 |
fun wcode_stop :: "bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2254 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2255 |
"wcode_stop ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2256 |
(\<exists> ln rn. l = Bk # ires \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2257 |
r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2258 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2259 |
fun wcode_halt_case_inv :: "nat \<Rightarrow> bin_inv_t" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2260 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2261 |
"wcode_halt_case_inv st ires rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2262 |
(if st = 0 then wcode_stop ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2263 |
else if st = Suc 0 then wcode_on_left_moving_3 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2264 |
else if st = Suc (Suc 0) then wcode_on_checking_3 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2265 |
else if st = 7 then wcode_goon_checking_3 ires rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2266 |
else False)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2267 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2268 |
fun wcode_halt_case_state :: "t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2269 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2270 |
"wcode_halt_case_state (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2271 |
(if st = 1 then 5 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2272 |
else if st = Suc (Suc 0) then 4 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2273 |
else if st = 7 then 3 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2274 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2275 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2276 |
fun wcode_halt_case_step :: "t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2277 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2278 |
"wcode_halt_case_step (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2279 |
(if st = 1 then length l |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2280 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2281 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2282 |
fun wcode_halt_case_measure :: "t_conf \<Rightarrow> nat \<times> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2283 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2284 |
"wcode_halt_case_measure (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2285 |
(wcode_halt_case_state (st, l, r), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2286 |
wcode_halt_case_step (st, l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2287 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2288 |
definition wcode_halt_case_le :: "(t_conf \<times> t_conf) set" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2289 |
where "wcode_halt_case_le \<equiv> (inv_image lex_pair wcode_halt_case_measure)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2290 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2291 |
lemma wf_wcode_halt_case_le[intro]: "wf wcode_halt_case_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2292 |
by(auto intro:wf_inv_image simp: wcode_halt_case_le_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2293 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2294 |
declare wcode_on_left_moving_3_B.simps[simp del] wcode_on_left_moving_3_O.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2295 |
wcode_on_checking_3.simps[simp del] wcode_goon_checking_3.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2296 |
wcode_on_left_moving_3.simps[simp del] wcode_stop.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2297 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2298 |
lemmas wcode_halt_invs = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2299 |
wcode_on_left_moving_3_B.simps wcode_on_left_moving_3_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2300 |
wcode_on_checking_3.simps wcode_goon_checking_3.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2301 |
wcode_on_left_moving_3.simps wcode_stop.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2302 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2303 |
lemma [simp]: "fetch t_wcode_main 7 Bk = (R, 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2304 |
apply(simp add: fetch.simps t_wcode_main_def nth_append nth_of.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2305 |
t_wcode_main_first_part_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2306 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2307 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2308 |
lemma [simp]: "wcode_on_left_moving_3 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2309 |
apply(simp only: wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2310 |
apply(simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2311 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2312 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2313 |
lemma [simp]: "wcode_on_checking_3 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2314 |
apply(simp add: wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2315 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2316 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2317 |
lemma [simp]: "wcode_goon_checking_3 ires rs (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2318 |
apply(simp add: wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2319 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2320 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2321 |
lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2322 |
\<Longrightarrow> wcode_on_left_moving_3 ires rs (tl b, hd b # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2323 |
apply(simp only: wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2324 |
apply(erule_tac disjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2325 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2326 |
apply(case_tac ml, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2327 |
apply(rule_tac x = "mr - 2" in exI, rule_tac x = rn in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2328 |
apply(case_tac mr, simp, simp add: exp_ind, simp add: exp_ind[THEN sym]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2329 |
apply(rule_tac disjI1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2330 |
apply(rule_tac x = nat in exI, rule_tac x = "Suc mr" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2331 |
rule_tac x = rn in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2332 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2333 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2334 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2335 |
lemma [simp]: "wcode_goon_checking_3 ires rs (b, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2336 |
(b = [] \<longrightarrow> wcode_stop ires rs ([Bk], list)) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2337 |
(b \<noteq> [] \<longrightarrow> wcode_stop ires rs (Bk # b, list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2338 |
apply(auto simp: wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2339 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2340 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2341 |
lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2342 |
apply(auto simp: wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2343 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2344 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2345 |
lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2346 |
wcode_on_checking_3 ires rs (tl b, hd b # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2347 |
apply(simp add:wcode_halt_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2348 |
apply(case_tac [!] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2349 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2350 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2351 |
lemma [simp]: "wcode_on_checking_3 ires rs (b, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2352 |
apply(auto simp: wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2353 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2354 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2355 |
lemma [simp]: "wcode_on_left_moving_3 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2356 |
apply(simp add: wcode_halt_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2357 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2358 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2359 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2360 |
lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2361 |
apply(auto simp: wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2362 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2363 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2364 |
lemma [simp]: "wcode_on_checking_3 ires rs (b, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2365 |
wcode_goon_checking_3 ires rs (tl b, hd b # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2366 |
apply(auto simp: wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2367 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2368 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2369 |
lemma [simp]: "wcode_goon_checking_3 ires rs (b, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2370 |
apply(simp add: wcode_goon_checking_3.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2371 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2372 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2373 |
lemma t_halt_case_correctness: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2374 |
shows "let P = (\<lambda> (st, l, r). st = 0) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2375 |
let Q = (\<lambda> (st, l, r). wcode_halt_case_inv st ires rs (l, r)) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2376 |
let f = (\<lambda> 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 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2377 |
\<exists> n .P (f n) \<and> Q (f (n::nat))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2378 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2379 |
let ?P = "(\<lambda> (st, l, r). st = 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2380 |
let ?Q = "(\<lambda> (st, l, r). wcode_halt_case_inv st ires rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2381 |
let ?f = "(\<lambda> 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)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2382 |
have "\<exists> n. ?P (?f n) \<and> ?Q (?f (n::nat))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2383 |
proof(rule_tac halt_lemma2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2384 |
show "wf wcode_halt_case_le" by auto |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2385 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2386 |
show "\<forall> na. \<not> ?P (?f na) \<and> ?Q (?f na) \<longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2387 |
?Q (?f (Suc na)) \<and> (?f (Suc na), ?f na) \<in> wcode_halt_case_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2388 |
apply(rule_tac allI, rule_tac impI, case_tac "?f na") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2389 |
apply(simp add: tstep_red tstep.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2390 |
apply(case_tac c, simp, case_tac [2] aa) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2391 |
apply(simp_all split: if_splits add: new_tape.simps wcode_halt_case_le_def lex_pair_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2392 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2393 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2394 |
show "?Q (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2395 |
apply(simp add: steps.simps wcode_halt_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2396 |
apply(rule_tac x = "Suc m" in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2397 |
apply(rule_tac x = "Suc 0" in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2398 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2399 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2400 |
show "\<not> ?P (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2401 |
apply(simp add: steps.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2402 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2403 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2404 |
thus "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2405 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2406 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2407 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2408 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2409 |
declare wcode_halt_case_inv.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2410 |
lemma [intro]: "\<exists> xs. (<rev list @ [aa::nat]> :: block list) = Oc # xs" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2411 |
apply(case_tac "rev list", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2412 |
apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2413 |
apply(case_tac list, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2414 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2415 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2416 |
lemma wcode_halt_case: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2417 |
"\<exists>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>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2418 |
t_wcode_main stp = (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2419 |
using t_halt_case_correctness[of ires rs m n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2420 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2421 |
apply(erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2422 |
apply(case_tac "steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2423 |
Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main na") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2424 |
apply(auto simp: wcode_halt_case_inv.simps wcode_stop.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2425 |
apply(rule_tac x = na in exI, rule_tac x = ln in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2426 |
rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2427 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2428 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2429 |
lemma bl_bin_one: "bl_bin [Oc] = Suc 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2430 |
apply(simp add: bl_bin.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2431 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2432 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2433 |
lemma t_wcode_main_lemma_pre: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2434 |
"\<lbrakk>args \<noteq> []; lm = <args::nat list>\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2435 |
\<exists> 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 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2436 |
stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2437 |
= (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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2438 |
proof(induct "length args" arbitrary: args lm rs m n, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2439 |
fix x args lm rs m n |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2440 |
assume ind: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2441 |
"\<And>args lm rs m n. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2442 |
\<lbrakk>x = length args; (args::nat list) \<noteq> []; lm = <args>\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2443 |
\<Longrightarrow> \<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2444 |
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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2445 |
(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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2446 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2447 |
and h: "Suc x = length args" "(args::nat list) \<noteq> []" "lm = <args>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2448 |
from h have "\<exists> (a::nat) xs. args = xs @ [a]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2449 |
apply(rule_tac x = "last args" in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2450 |
apply(rule_tac x = "butlast args" in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2451 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2452 |
from this obtain a xs where "args = xs @ [a]" by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2453 |
from h and this show |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2454 |
"\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2455 |
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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2456 |
(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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2457 |
proof(case_tac "xs::nat list", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2458 |
show "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2459 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2460 |
(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2461 |
proof(induct "a" arbitrary: m n rs ires, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2462 |
fix m n rs ires |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2463 |
show "\<exists>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>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2464 |
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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2465 |
apply(simp add: bl_bin_one) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2466 |
apply(rule_tac wcode_halt_case) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2467 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2468 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2469 |
fix a m n rs ires |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2470 |
assume ind2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2471 |
"\<And>m n rs ires. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2472 |
\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2473 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2474 |
(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + rs * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2475 |
show "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2476 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2477 |
(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<Suc a>) + rs * 2 ^ Suc a\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2478 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2479 |
have "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2480 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2481 |
(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2482 |
apply(simp add: tape_of_nat) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2483 |
using wcode_double_case[of m "Oc\<^bsup>a\<^esup> @ Bk # Bk # ires" rs n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2484 |
apply(simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2485 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2486 |
from this obtain stpa lna rna where stp1: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2487 |
"steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ rev (<Suc a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2488 |
(Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2489 |
moreover have |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2490 |
"\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2491 |
steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2492 |
(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2) * 2 ^ a\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2493 |
using ind2[of lna ires "2*rs + 2" rna] by simp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2494 |
from this obtain stpb lnb rnb where stp2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2495 |
"steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<a>) @ Bk # Bk # ires, Bk # Oc\<^bsup>Suc (2 * rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2496 |
(0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<a>) + (2*rs + 2) * 2 ^ a\<^esup> @ Bk\<^bsup>rnb\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2497 |
by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2498 |
from stp1 and stp2 show "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2499 |
apply(rule_tac x = "stpa + stpb" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2500 |
rule_tac x = lnb in exI, rule_tac x = rnb in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2501 |
apply(simp add: steps_add bl_bin_nat_Suc exponent_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2502 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2503 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2504 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2505 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2506 |
fix aa list |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2507 |
assume g: "Suc x = length args" "args \<noteq> []" "lm = <args>" "args = xs @ [a::nat]" "xs = (aa::nat) # list" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2508 |
thus "\<exists>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 = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2509 |
(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>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2510 |
proof(induct a arbitrary: m n rs args lm, simp_all add: tape_of_nl_rev, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2511 |
simp only: tape_of_nl_cons_app1, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2512 |
fix m n rs args lm |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2513 |
have "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2514 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<(aa::nat) # list>) @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2515 |
Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2516 |
(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2517 |
Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2518 |
proof(simp add: tape_of_nl_rev) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2519 |
have "\<exists> xs. (<rev list @ [aa]>) = Oc # xs" by auto |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2520 |
from this obtain xs where "(<rev list @ [aa]>) = Oc # xs" .. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2521 |
thus "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2522 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2523 |
Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2524 |
(Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ <rev list @ [aa]> @ Bk # Bk # ires, Bk # Oc\<^bsup>5 + 4 * rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2525 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2526 |
using wcode_fourtimes_case[of m "xs @ Bk # Bk # ires" rs n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2527 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2528 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2529 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2530 |
from this obtain stpa lna rna where stp1: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2531 |
"steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # rev (<aa # list>) @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2532 |
Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2533 |
(Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<aa # list>) @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2534 |
Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2535 |
from g have |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2536 |
"\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2537 |
Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp = (0, Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2538 |
Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) \<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2539 |
apply(rule_tac args = "(aa::nat)#list" in ind, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2540 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2541 |
from this obtain stpb lnb rnb where stp2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2542 |
"steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ rev (<(aa::nat) # list>) @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2543 |
Bk # Oc\<^bsup>Suc (4*rs + 4)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb = (0, Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2544 |
Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<aa#list>)+ (4*rs + 4) * 2^(length (<aa#list>) - 1) \<^esup> @ Bk\<^bsup>rnb\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2545 |
by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2546 |
from stp1 and stp2 and h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2547 |
show "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2548 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc # Bk # <rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2549 |
Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2550 |
(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2551 |
Bk # Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [0]>) + rs * (2 * 2 ^ (aa + length (<list @ [0]>)))\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2552 |
apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2553 |
rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_rev) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2554 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2555 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2556 |
fix ab m n rs args lm |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2557 |
assume ind2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2558 |
"\<And> m n rs args lm. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2559 |
\<lbrakk>lm = <aa # list @ [ab]>; args = aa # list @ [ab]\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2560 |
\<Longrightarrow> \<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2561 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2562 |
Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2563 |
(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2564 |
Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]>) + rs * 2 ^ (length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2565 |
and k: "args = aa # list @ [Suc ab]" "lm = <aa # list @ [Suc ab]>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2566 |
show "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2567 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ <Suc ab # rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2568 |
Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2569 |
(0, Bk # ires,Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2570 |
Bk # Oc\<^bsup>bl_bin (<aa # list @ [Suc ab]>) + rs * 2 ^ (length (<aa # list @ [Suc ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2571 |
proof(simp add: tape_of_nl_cons_app1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2572 |
have "\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2573 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2574 |
Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2575 |
= (Suc 0, Bk # Bk\<^bsup>ln\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2576 |
Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2577 |
using wcode_double_case[of m "Oc\<^bsup>ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2578 |
rs n] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2579 |
apply(simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2580 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2581 |
from this obtain stpa lna rna where stp1: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2582 |
"steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2583 |
Bk # Oc # Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stpa |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2584 |
= (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ Oc\<^bsup>Suc ab\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2585 |
Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>)" by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2586 |
from k have |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2587 |
"\<exists> stp ln rn. steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2588 |
Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2589 |
= (0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2590 |
Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) + (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2591 |
apply(rule_tac ind2, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2592 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2593 |
from this obtain stpb lnb rnb where stp2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2594 |
"steps (Suc 0, Bk # Bk\<^bsup>lna\<^esup> @ <ab # rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2595 |
Bk # Oc\<^bsup>Suc (2*rs + 2)\<^esup> @ Bk\<^bsup>rna\<^esup>) t_wcode_main stpb |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2596 |
= (0, Bk # ires, Bk # Oc # Bk\<^bsup>lnb\<^esup> @ Bk # |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2597 |
Bk # Oc\<^bsup>bl_bin (<aa # list @ [ab]> ) + (2*rs + 2)* 2^(length (<aa # list @ [ab]>) - Suc 0)\<^esup> @ Bk\<^bsup>rnb\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2598 |
by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2599 |
from stp1 and stp2 show |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2600 |
"\<exists>stp ln rn. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2601 |
steps (Suc 0, Bk # Bk\<^bsup>m\<^esup> @ Oc\<^bsup>Suc (Suc ab)\<^esup> @ Bk # <rev list @ [aa]> @ Bk # Bk # ires, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2602 |
Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>) t_wcode_main stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2603 |
(0, Bk # ires, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2604 |
Oc\<^bsup>bl_bin (Oc\<^bsup>Suc aa\<^esup> @ Bk # <list @ [Suc ab]>) + rs * (2 * 2 ^ (aa + length (<list @ [Suc ab]>)))\<^esup> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2605 |
@ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2606 |
apply(rule_tac x = "stpa + stpb" in exI, rule_tac x = lnb in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2607 |
rule_tac x = rnb in exI, simp add: steps_add tape_of_nl_cons_app1 exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2608 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2609 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2610 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2611 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2612 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2613 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2614 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2615 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2616 |
(* turing_shift can be used*) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2617 |
term t_wcode_main |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2618 |
definition t_wcode_prepare :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2619 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2620 |
"t_wcode_prepare \<equiv> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2621 |
[(W1, 2), (L, 1), (L, 3), (R, 2), (R, 4), (W0, 3), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2622 |
(R, 4), (R, 5), (R, 6), (R, 5), (R, 7), (R, 5), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2623 |
(W1, 7), (L, 0)]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2624 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2625 |
fun wprepare_add_one :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2626 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2627 |
"wprepare_add_one m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2628 |
(\<exists> rn. l = [] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2629 |
(r = <m # lm> @ Bk\<^bsup>rn\<^esup> \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2630 |
r = Bk # <m # lm> @ Bk\<^bsup>rn\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2631 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2632 |
fun wprepare_goto_first_end :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2633 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2634 |
"wprepare_goto_first_end m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2635 |
(\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2636 |
r = Oc\<^bsup>mr\<^esup> @ Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2637 |
ml + mr = Suc (Suc m))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2638 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2639 |
fun wprepare_erase :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2640 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2641 |
"wprepare_erase m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2642 |
(\<exists> rn. l = Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2643 |
tl r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2644 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2645 |
fun wprepare_goto_start_pos_B :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2646 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2647 |
"wprepare_goto_start_pos_B m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2648 |
(\<exists> rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2649 |
r = Bk # <lm> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2650 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2651 |
fun wprepare_goto_start_pos_O :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2652 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2653 |
"wprepare_goto_start_pos_O m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2654 |
(\<exists> rn. l = Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2655 |
r = <lm> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2656 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2657 |
fun wprepare_goto_start_pos :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2658 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2659 |
"wprepare_goto_start_pos m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2660 |
(wprepare_goto_start_pos_B m lm (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2661 |
wprepare_goto_start_pos_O m lm (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2662 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2663 |
fun wprepare_loop_start_on_rightmost :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2664 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2665 |
"wprepare_loop_start_on_rightmost m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2666 |
(\<exists> rn mr. rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2667 |
r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2668 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2669 |
fun wprepare_loop_start_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2670 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2671 |
"wprepare_loop_start_in_middle m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2672 |
(\<exists> rn (mr:: nat) (lm1::nat list). |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2673 |
rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2674 |
r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup> \<and> lm1 \<noteq> [])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2675 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2676 |
fun wprepare_loop_start :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2677 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2678 |
"wprepare_loop_start m lm (l, r) = (wprepare_loop_start_on_rightmost m lm (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2679 |
wprepare_loop_start_in_middle m lm (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2680 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2681 |
fun wprepare_loop_goon_on_rightmost :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2682 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2683 |
"wprepare_loop_goon_on_rightmost m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2684 |
(\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2685 |
r = Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2686 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2687 |
fun wprepare_loop_goon_in_middle :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2688 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2689 |
"wprepare_loop_goon_in_middle m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2690 |
(\<exists> rn (mr:: nat) (lm1::nat list). |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2691 |
rev l @ r = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm> @ Bk\<^bsup>rn\<^esup> \<and> l \<noteq> [] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2692 |
(if lm1 = [] then r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2693 |
else r = Oc\<^bsup>mr\<^esup> @ Bk # <lm1> @ Bk\<^bsup>rn\<^esup>) \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2694 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2695 |
fun wprepare_loop_goon :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2696 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2697 |
"wprepare_loop_goon m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2698 |
(wprepare_loop_goon_in_middle m lm (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2699 |
wprepare_loop_goon_on_rightmost m lm (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2700 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2701 |
fun wprepare_add_one2 :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2702 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2703 |
"wprepare_add_one2 m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2704 |
(\<exists> rn. l = Bk # Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2705 |
(r = [] \<or> tl r = Bk\<^bsup>rn\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2706 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2707 |
fun wprepare_stop :: "nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2708 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2709 |
"wprepare_stop m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2710 |
(\<exists> rn. l = Bk # <rev lm> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2711 |
r = Bk # Oc # Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2712 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2713 |
fun wprepare_inv :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2714 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2715 |
"wprepare_inv st m lm (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2716 |
(if st = 0 then wprepare_stop m lm (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2717 |
else if st = Suc 0 then wprepare_add_one m lm (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2718 |
else if st = Suc (Suc 0) then wprepare_goto_first_end m lm (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2719 |
else if st = Suc (Suc (Suc 0)) then wprepare_erase m lm (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2720 |
else if st = 4 then wprepare_goto_start_pos m lm (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2721 |
else if st = 5 then wprepare_loop_start m lm (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2722 |
else if st = 6 then wprepare_loop_goon m lm (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2723 |
else if st = 7 then wprepare_add_one2 m lm (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2724 |
else False)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2725 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2726 |
fun wprepare_stage :: "t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2727 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2728 |
"wprepare_stage (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2729 |
(if st \<ge> 1 \<and> st \<le> 4 then 3 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2730 |
else if st = 5 \<or> st = 6 then 2 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2731 |
else 1)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2732 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2733 |
fun wprepare_state :: "t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2734 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2735 |
"wprepare_state (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2736 |
(if st = 1 then 4 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2737 |
else if st = Suc (Suc 0) then 3 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2738 |
else if st = Suc (Suc (Suc 0)) then 2 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2739 |
else if st = 4 then 1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2740 |
else if st = 7 then 2 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2741 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2742 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2743 |
fun wprepare_step :: "t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2744 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2745 |
"wprepare_step (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2746 |
(if st = 1 then (if hd r = Oc then Suc (length l) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2747 |
else 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2748 |
else if st = Suc (Suc 0) then length r |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2749 |
else if st = Suc (Suc (Suc 0)) then (if hd r = Oc then 1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2750 |
else 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2751 |
else if st = 4 then length r |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2752 |
else if st = 5 then Suc (length r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2753 |
else if st = 6 then (if r = [] then 0 else Suc (length r)) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2754 |
else if st = 7 then (if (r \<noteq> [] \<and> hd r = Oc) then 0 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2755 |
else 1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2756 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2757 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2758 |
fun wcode_prepare_measure :: "t_conf \<Rightarrow> nat \<times> nat \<times> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2759 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2760 |
"wcode_prepare_measure (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2761 |
(wprepare_stage (st, l, r), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2762 |
wprepare_state (st, l, r), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2763 |
wprepare_step (st, l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2764 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2765 |
definition wcode_prepare_le :: "(t_conf \<times> t_conf) set" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2766 |
where "wcode_prepare_le \<equiv> (inv_image lex_triple wcode_prepare_measure)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2767 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2768 |
lemma [intro]: "wf lex_pair" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2769 |
by(auto intro:wf_lex_prod simp:lex_pair_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2770 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2771 |
lemma wf_wcode_prepare_le[intro]: "wf wcode_prepare_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2772 |
by(auto intro:wf_inv_image simp: wcode_prepare_le_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2773 |
recursive.lex_triple_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2774 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2775 |
declare wprepare_add_one.simps[simp del] wprepare_goto_first_end.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2776 |
wprepare_erase.simps[simp del] wprepare_goto_start_pos.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2777 |
wprepare_loop_start.simps[simp del] wprepare_loop_goon.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2778 |
wprepare_add_one2.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2779 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2780 |
lemmas wprepare_invs = wprepare_add_one.simps wprepare_goto_first_end.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2781 |
wprepare_erase.simps wprepare_goto_start_pos.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2782 |
wprepare_loop_start.simps wprepare_loop_goon.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2783 |
wprepare_add_one2.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2784 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2785 |
declare wprepare_inv.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2786 |
lemma [simp]: "fetch t_wcode_prepare (Suc 0) Bk = (W1, 2)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2787 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2788 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2789 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2790 |
lemma [simp]: "fetch t_wcode_prepare (Suc 0) Oc = (L, 1)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2791 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2792 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2793 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2794 |
lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Bk = (L, 3)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2795 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2796 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2797 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2798 |
lemma [simp]: "fetch t_wcode_prepare (Suc (Suc 0)) Oc = (R, 2)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2799 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2800 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2801 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2802 |
lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Bk = (R, 4)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2803 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2804 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2805 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2806 |
lemma [simp]: "fetch t_wcode_prepare (Suc (Suc (Suc 0))) Oc = (W0, 3)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2807 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2808 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2809 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2810 |
lemma [simp]: "fetch t_wcode_prepare 4 Bk = (R, 4)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2811 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2812 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2813 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2814 |
lemma [simp]: "fetch t_wcode_prepare 4 Oc = (R, 5)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2815 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2816 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2817 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2818 |
lemma [simp]: "fetch t_wcode_prepare 5 Oc = (R, 5)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2819 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2820 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2821 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2822 |
lemma [simp]: "fetch t_wcode_prepare 5 Bk = (R, 6)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2823 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2824 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2825 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2826 |
lemma [simp]: "fetch t_wcode_prepare 6 Oc = (R, 5)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2827 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2828 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2829 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2830 |
lemma [simp]: "fetch t_wcode_prepare 6 Bk = (R, 7)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2831 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2832 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2833 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2834 |
lemma [simp]: "fetch t_wcode_prepare 7 Oc = (L, 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2835 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2836 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2837 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2838 |
lemma [simp]: "fetch t_wcode_prepare 7 Bk = (W1, 7)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2839 |
apply(simp add: fetch.simps t_wcode_prepare_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2840 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2841 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2842 |
lemma tape_of_nl_not_null: "lm \<noteq> [] \<Longrightarrow> <lm::nat list> \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2843 |
apply(case_tac lm, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2844 |
apply(case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2845 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2846 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2847 |
lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_add_one m lm (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2848 |
apply(simp add: wprepare_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2849 |
apply(simp add: tape_of_nl_not_null) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2850 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2851 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2852 |
lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_goto_first_end m lm (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2853 |
apply(simp add: wprepare_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2854 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2855 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2856 |
lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_erase m lm (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2857 |
apply(simp add: wprepare_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2858 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2859 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2860 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2861 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2862 |
lemma [simp]: "lm \<noteq> [] \<Longrightarrow> wprepare_goto_start_pos m lm (b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2863 |
apply(simp add: wprepare_invs tape_of_nl_not_null) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2864 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2865 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2866 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2867 |
apply(simp add: wprepare_invs tape_of_nl_not_null, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2868 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2869 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2870 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [])\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2871 |
wprepare_loop_goon m lm (Bk # b, [])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2872 |
apply(simp only: wprepare_invs tape_of_nl_not_null) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2873 |
apply(erule_tac disjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2874 |
apply(rule_tac disjI2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2875 |
apply(simp add: wprepare_loop_start_on_rightmost.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2876 |
wprepare_loop_goon_on_rightmost.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2877 |
apply(rule_tac rev_eq, simp add: tape_of_nl_rev) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2878 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2879 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2880 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, [])\<rbrakk> \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2881 |
apply(simp only: wprepare_invs tape_of_nl_not_null, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2882 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2883 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2884 |
lemma [simp]:"\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, [])\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2885 |
wprepare_add_one2 m lm (Bk # b, [])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2886 |
apply(simp only: wprepare_invs tape_of_nl_not_null, auto split: if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2887 |
apply(case_tac mr, simp, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2888 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2889 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2890 |
lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2891 |
apply(simp only: wprepare_invs tape_of_nl_not_null, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2892 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2893 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2894 |
lemma [simp]: "wprepare_add_one2 m lm (b, []) \<Longrightarrow> wprepare_add_one2 m lm (b, [Oc])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2895 |
apply(simp only: wprepare_invs tape_of_nl_not_null, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2896 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2897 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2898 |
lemma [simp]: "Bk # list = <(m::nat) # lm> @ ys = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2899 |
apply(case_tac lm, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2900 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2901 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2902 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_add_one m lm (b, Bk # list)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2903 |
\<Longrightarrow> (b = [] \<longrightarrow> wprepare_goto_first_end m lm ([], Oc # list)) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2904 |
(b \<noteq> [] \<longrightarrow> wprepare_goto_first_end m lm (b, Oc # list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2905 |
apply(simp only: wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2906 |
apply(rule_tac x = 0 in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2907 |
apply(case_tac lm, simp, simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2908 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2909 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2910 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2911 |
lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2912 |
apply(simp only: wprepare_invs tape_of_nl_not_null, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2913 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2914 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2915 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2916 |
lemma [simp]: "wprepare_goto_first_end m lm (b, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2917 |
wprepare_erase m lm (tl b, hd b # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2918 |
apply(simp only: wprepare_invs tape_of_nl_not_null, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2919 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2920 |
apply(case_tac mr, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2921 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2922 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2923 |
lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2924 |
apply(simp only: wprepare_invs exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2925 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2926 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2927 |
lemma [simp]: "wprepare_erase m lm (b, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2928 |
wprepare_goto_start_pos m lm (Bk # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2929 |
apply(simp only: wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2930 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2931 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2932 |
lemma [simp]: "\<lbrakk>wprepare_add_one m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2933 |
apply(simp only: wprepare_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2934 |
apply(case_tac lm, simp_all add: tape_of_nl_abv |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2935 |
tape_of_nat_list.simps exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2936 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2937 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2938 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_first_end m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2939 |
apply(simp only: wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2940 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2941 |
apply(simp add: tape_of_nl_not_null) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2942 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2943 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2944 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_first_end m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2945 |
apply(simp only: wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2946 |
apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2947 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2948 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2949 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_erase m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2950 |
apply(simp only: wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2951 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2952 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2953 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_erase m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2954 |
apply(simp only: wprepare_invs, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2955 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2956 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2957 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> list \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2958 |
apply(simp only: wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2959 |
apply(simp add: tape_of_nl_not_null) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2960 |
apply(case_tac lm, simp, case_tac list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2961 |
apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2962 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2963 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2964 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2965 |
apply(simp only: wprepare_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2966 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2967 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2968 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2969 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2970 |
apply(simp only: wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2971 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2972 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2973 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2974 |
(list = [] \<longrightarrow> wprepare_add_one2 m lm (Bk # b, [])) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2975 |
(list \<noteq> [] \<longrightarrow> wprepare_add_one2 m lm (Bk # b, list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2976 |
apply(simp only: wprepare_invs, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2977 |
apply(case_tac list, simp_all split: if_splits, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2978 |
apply(case_tac [1-3] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2979 |
apply(case_tac mr, simp_all add: exp_ind_def tape_of_nl_not_null) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2980 |
apply(case_tac [1-2] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2981 |
apply(case_tac rn, simp, case_tac nat, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2982 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2983 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2984 |
lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2985 |
apply(simp only: wprepare_invs, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2986 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2987 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2988 |
lemma [simp]: "wprepare_add_one2 m lm (b, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2989 |
(list = [] \<longrightarrow> wprepare_add_one2 m lm (b, [Oc])) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2990 |
(list \<noteq> [] \<longrightarrow> wprepare_add_one2 m lm (b, Oc # list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2991 |
apply(simp only: wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2992 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2993 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2994 |
lemma [simp]: "wprepare_goto_first_end m lm (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2995 |
\<Longrightarrow> (b = [] \<longrightarrow> wprepare_goto_first_end m lm ([Oc], list)) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2996 |
(b \<noteq> [] \<longrightarrow> wprepare_goto_first_end m lm (Oc # b, list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2997 |
apply(simp only: wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2998 |
apply(rule_tac x = 1 in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
2999 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3000 |
apply(case_tac ml, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3001 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3002 |
apply(rule_tac x = "Suc ml" in exI, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3003 |
apply(rule_tac x = "mr - 1" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3004 |
apply(case_tac mr, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3005 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3006 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3007 |
lemma [simp]: "wprepare_erase m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3008 |
apply(simp only: wprepare_invs, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3009 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3010 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3011 |
lemma [simp]: "wprepare_erase m lm (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3012 |
\<Longrightarrow> wprepare_erase m lm (b, Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3013 |
apply(simp only:wprepare_invs, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3014 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3015 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3016 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3017 |
\<Longrightarrow> wprepare_goto_start_pos m lm (Bk # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3018 |
apply(simp only:wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3019 |
apply(case_tac [!] lm, simp, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3020 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3021 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3022 |
lemma [simp]: "wprepare_loop_start m lm (b, aa) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3023 |
apply(simp only:wprepare_invs, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3024 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3025 |
lemma [elim]: "Bk # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<Longrightarrow> \<exists>rn. list = Bk\<^bsup>rn\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3026 |
apply(case_tac mr, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3027 |
apply(case_tac rn, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3028 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3029 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3030 |
lemma rev_equal_iff: "x = y \<Longrightarrow> rev x = rev y" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3031 |
by simp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3032 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3033 |
lemma tape_of_nl_false1: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3034 |
"lm \<noteq> [] \<Longrightarrow> rev b @ [Bk] \<noteq> Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # <lm::nat list>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3035 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3036 |
apply(drule_tac rev_equal_iff, simp add: tape_of_nl_rev) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3037 |
apply(case_tac "rev lm") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3038 |
apply(case_tac [2] list, auto simp: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3039 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3040 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3041 |
lemma [simp]: "wprepare_loop_start_in_middle m lm (b, [Bk]) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3042 |
apply(simp add: wprepare_loop_start_in_middle.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3043 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3044 |
apply(case_tac lm1, simp, simp add: tape_of_nl_not_null) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3045 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3046 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3047 |
declare wprepare_loop_start_in_middle.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3048 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3049 |
declare wprepare_loop_start_on_rightmost.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3050 |
wprepare_loop_goon_in_middle.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3051 |
wprepare_loop_goon_on_rightmost.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3052 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3053 |
lemma [simp]: "wprepare_loop_goon_in_middle m lm (Bk # b, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3054 |
apply(simp add: wprepare_loop_goon_in_middle.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3055 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3056 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3057 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, [Bk])\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3058 |
wprepare_loop_goon m lm (Bk # b, [])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3059 |
apply(simp only: wprepare_invs, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3060 |
apply(simp add: wprepare_loop_goon_on_rightmost.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3061 |
wprepare_loop_start_on_rightmost.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3062 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3063 |
apply(rule_tac rev_eq) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3064 |
apply(simp add: tape_of_nl_rev) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3065 |
apply(simp add: exp_ind_def[THEN sym] exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3066 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3067 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3068 |
lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3069 |
\<Longrightarrow> wprepare_loop_goon_in_middle m lm (Bk # b, a # lista) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3070 |
apply(auto simp: wprepare_loop_start_on_rightmost.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3071 |
wprepare_loop_goon_in_middle.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3072 |
apply(case_tac [!] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3073 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3074 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3075 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3076 |
\<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3077 |
apply(simp only: wprepare_loop_start_on_rightmost.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3078 |
wprepare_loop_goon_on_rightmost.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3079 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3080 |
apply(simp add: tape_of_nl_rev) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3081 |
apply(simp add: exp_ind_def[THEN sym] exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3082 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3083 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3084 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3085 |
\<Longrightarrow> wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3086 |
apply(simp add: wprepare_loop_start_in_middle.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3087 |
wprepare_loop_goon_on_rightmost.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3088 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3089 |
apply(case_tac "lm1::nat list", simp_all, case_tac list, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3090 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps tape_of_nat_abv exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3091 |
apply(case_tac [!] rna, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3092 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3093 |
apply(case_tac lm1, simp, case_tac list, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3094 |
apply(simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def tape_of_nat_abv) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3095 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3096 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3097 |
lemma [simp]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3098 |
"\<lbrakk>lm \<noteq> []; wprepare_loop_start_in_middle m lm (b, Bk # a # lista)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3099 |
\<Longrightarrow> wprepare_loop_goon_in_middle m lm (Bk # b, a # lista)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3100 |
apply(simp add: wprepare_loop_start_in_middle.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3101 |
wprepare_loop_goon_in_middle.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3102 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3103 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3104 |
apply(case_tac lm1, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3105 |
apply(rule_tac x = "Suc aa" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3106 |
apply(rule_tac x = list in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3107 |
apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3108 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3109 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3110 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, Bk # a # lista)\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3111 |
wprepare_loop_goon m lm (Bk # b, a # lista)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3112 |
apply(simp add: wprepare_loop_start.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3113 |
wprepare_loop_goon.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3114 |
apply(erule_tac disjE, simp, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3115 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3116 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3117 |
lemma start_2_goon: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3118 |
"\<lbrakk>lm \<noteq> []; wprepare_loop_start m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3119 |
(list = [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, [])) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3120 |
(list \<noteq> [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3121 |
apply(case_tac list, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3122 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3123 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3124 |
lemma add_one_2_add_one: "wprepare_add_one m lm (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3125 |
\<Longrightarrow> (hd b = Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3126 |
(b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, Oc # Oc # list))) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3127 |
(hd b \<noteq> Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3128 |
(b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, hd b # Oc # list)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3129 |
apply(simp only: wprepare_add_one.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3130 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3131 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3132 |
lemma [simp]: "wprepare_loop_start m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3133 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3134 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3135 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3136 |
lemma [simp]: "wprepare_loop_start_on_rightmost m lm (b, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3137 |
wprepare_loop_start_on_rightmost m lm (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3138 |
apply(simp add: wprepare_loop_start_on_rightmost.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3139 |
apply(rule_tac x = rn in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3140 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3141 |
apply(case_tac rn, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3142 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3143 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3144 |
lemma [simp]: "wprepare_loop_start_in_middle m lm (b, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3145 |
wprepare_loop_start_in_middle m lm (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3146 |
apply(simp add: wprepare_loop_start_in_middle.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3147 |
apply(rule_tac x = rn in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3148 |
apply(case_tac mr, simp, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3149 |
apply(rule_tac x = nat in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3150 |
apply(rule_tac x = lm1 in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3151 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3152 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3153 |
lemma start_2_start: "wprepare_loop_start m lm (b, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3154 |
wprepare_loop_start m lm (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3155 |
apply(simp add: wprepare_loop_start.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3156 |
apply(erule_tac disjE, simp_all ) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3157 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3158 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3159 |
lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3160 |
apply(simp add: wprepare_loop_goon.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3161 |
wprepare_loop_goon_in_middle.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3162 |
wprepare_loop_goon_on_rightmost.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3163 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3164 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3165 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3166 |
lemma [simp]: "wprepare_goto_start_pos m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3167 |
apply(simp add: wprepare_goto_start_pos.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3168 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3169 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3170 |
lemma [simp]: "wprepare_loop_goon_on_rightmost m lm (b, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3171 |
apply(simp add: wprepare_loop_goon_on_rightmost.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3172 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3173 |
lemma wprepare_loop1: "\<lbrakk>rev b @ Oc\<^bsup>mr\<^esup> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm>; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3174 |
b \<noteq> []; 0 < mr; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup>\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3175 |
\<Longrightarrow> wprepare_loop_start_on_rightmost m lm (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3176 |
apply(simp add: wprepare_loop_start_on_rightmost.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3177 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3178 |
apply(case_tac mr, simp, simp add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3179 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3180 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3181 |
lemma wprepare_loop2: "\<lbrakk>rev b @ Oc\<^bsup>mr\<^esup> @ Bk # <a # lista> = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # <lm>; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3182 |
b \<noteq> []; Oc # list = Oc\<^bsup>mr\<^esup> @ Bk # <(a::nat) # lista> @ Bk\<^bsup>rn\<^esup>\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3183 |
\<Longrightarrow> wprepare_loop_start_in_middle m lm (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3184 |
apply(simp add: wprepare_loop_start_in_middle.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3185 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3186 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3187 |
apply(rule_tac x = nat in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3188 |
apply(rule_tac x = "a#lista" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3189 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3190 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3191 |
lemma [simp]: "wprepare_loop_goon_in_middle m lm (b, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3192 |
wprepare_loop_start_on_rightmost m lm (Oc # b, list) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3193 |
wprepare_loop_start_in_middle m lm (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3194 |
apply(simp add: wprepare_loop_goon_in_middle.simps split: if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3195 |
apply(case_tac lm1, simp_all add: wprepare_loop1 wprepare_loop2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3196 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3197 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3198 |
lemma [simp]: "wprepare_loop_goon m lm (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3199 |
\<Longrightarrow> wprepare_loop_start m lm (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3200 |
apply(simp add: wprepare_loop_goon.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3201 |
wprepare_loop_start.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3202 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3203 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3204 |
lemma [simp]: "wprepare_add_one m lm (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3205 |
\<Longrightarrow> b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3206 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3207 |
apply(simp add: wprepare_add_one.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3208 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3209 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3210 |
lemma [simp]: "wprepare_goto_start_pos m [a] (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3211 |
\<Longrightarrow> wprepare_loop_start_on_rightmost m [a] (Oc # b, list) " |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3212 |
apply(auto simp: wprepare_goto_start_pos.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3213 |
wprepare_loop_start_on_rightmost.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3214 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3215 |
apply(simp add: tape_of_nat_abv tape_of_nat_list.simps exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3216 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3217 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3218 |
lemma [simp]: "wprepare_goto_start_pos m (a # aa # listaa) (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3219 |
\<Longrightarrow>wprepare_loop_start_in_middle m (a # aa # listaa) (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3220 |
apply(auto simp: wprepare_goto_start_pos.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3221 |
wprepare_loop_start_in_middle.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3222 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3223 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3224 |
apply(rule_tac x = a in exI, rule_tac x = "aa#listaa" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3225 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3226 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3227 |
lemma [simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Oc # list)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3228 |
\<Longrightarrow> wprepare_loop_start m lm (Oc # b, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3229 |
apply(case_tac lm, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3230 |
apply(case_tac lista, simp_all add: wprepare_loop_start.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3231 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3232 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3233 |
lemma [simp]: "wprepare_add_one2 m lm (b, Oc # list) \<Longrightarrow> b \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3234 |
apply(auto simp: wprepare_add_one2.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3235 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3236 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3237 |
lemma add_one_2_stop: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3238 |
"wprepare_add_one2 m lm (b, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3239 |
\<Longrightarrow> wprepare_stop m lm (tl b, hd b # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3240 |
apply(simp add: wprepare_stop.simps wprepare_add_one2.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3241 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3242 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3243 |
declare wprepare_stop.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3244 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3245 |
lemma wprepare_correctness: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3246 |
assumes h: "lm \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3247 |
shows "let P = (\<lambda> (st, l, r). st = 0) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3248 |
let Q = (\<lambda> (st, l, r). wprepare_inv st m lm (l, r)) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3249 |
let f = (\<lambda> stp. steps (Suc 0, [], (<m # lm>)) t_wcode_prepare stp) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3250 |
\<exists> n .P (f n) \<and> Q (f n)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3251 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3252 |
let ?P = "(\<lambda> (st, l, r). st = 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3253 |
let ?Q = "(\<lambda> (st, l, r). wprepare_inv st m lm (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3254 |
let ?f = "(\<lambda> stp. steps (Suc 0, [], (<m # lm>)) t_wcode_prepare stp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3255 |
have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3256 |
proof(rule_tac halt_lemma2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3257 |
show "wf wcode_prepare_le" by auto |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3258 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3259 |
show "\<forall> n. \<not> ?P (?f n) \<and> ?Q (?f n) \<longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3260 |
?Q (?f (Suc n)) \<and> (?f (Suc n), ?f n) \<in> wcode_prepare_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3261 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3262 |
apply(rule_tac allI, rule_tac impI, case_tac "?f n", |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3263 |
simp add: tstep_red tstep.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3264 |
apply(case_tac c, simp, case_tac [2] aa) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3265 |
apply(simp_all add: wprepare_inv.simps wcode_prepare_le_def new_tape.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3266 |
lex_triple_def lex_pair_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3267 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3268 |
split: if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3269 |
apply(simp_all add: start_2_goon start_2_start |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3270 |
add_one_2_add_one add_one_2_stop) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3271 |
apply(auto simp: wprepare_add_one2.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3272 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3273 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3274 |
show "?Q (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3275 |
apply(simp add: steps.simps wprepare_inv.simps wprepare_invs) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3276 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3277 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3278 |
show "\<not> ?P (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3279 |
apply(simp add: steps.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3280 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3281 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3282 |
thus "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3283 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3284 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3285 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3286 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3287 |
lemma [intro]: "t_correct t_wcode_prepare" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3288 |
apply(simp add: t_correct.simps t_wcode_prepare_def iseven_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3289 |
apply(rule_tac x = 7 in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3290 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3291 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3292 |
lemma twice_len_even: "length (tm_of abc_twice) mod 2 = 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3293 |
apply(simp add: tm_even) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3294 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3295 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3296 |
lemma fourtimes_len_even: "length (tm_of abc_fourtimes) mod 2 = 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3297 |
apply(simp add: tm_even) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3298 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3299 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3300 |
lemma t_correct_termi: "t_correct tp \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3301 |
list_all (\<lambda>(acn, st). (st \<le> Suc (length tp div 2))) (change_termi_state tp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3302 |
apply(auto simp: t_correct.simps List.list_all_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3303 |
apply(erule_tac x = n in allE, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3304 |
apply(case_tac "tp!n", auto simp: change_termi_state.simps split: if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3305 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3306 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3307 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3308 |
lemma t_correct_shift: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3309 |
"list_all (\<lambda>(acn, st). (st \<le> y)) tp \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3310 |
list_all (\<lambda>(acn, st). (st \<le> y + off)) (tshift tp off) " |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3311 |
apply(auto simp: t_correct.simps List.list_all_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3312 |
apply(erule_tac x = n in allE, simp add: shift_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3313 |
apply(case_tac "tp!n", auto simp: tshift.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3314 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3315 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3316 |
lemma [intro]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3317 |
"t_correct (tm_of abc_twice @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3318 |
(start_of twice_ly (length abc_twice) - Suc 0))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3319 |
apply(rule_tac t_compiled_correct, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3320 |
apply(simp add: twice_ly_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3321 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3322 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3323 |
lemma [intro]: "t_correct (tm_of abc_fourtimes @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3324 |
(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3325 |
apply(rule_tac t_compiled_correct, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3326 |
apply(simp add: fourtimes_ly_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3327 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3328 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3329 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3330 |
lemma [intro]: "t_correct t_wcode_main" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3331 |
apply(auto simp: t_wcode_main_def t_correct.simps shift_length |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3332 |
t_twice_def t_fourtimes_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3333 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3334 |
show "iseven (60 + (length (tm_of abc_twice) + |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3335 |
length (tm_of abc_fourtimes)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3336 |
using twice_len_even fourtimes_len_even |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3337 |
apply(auto simp: iseven_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3338 |
apply(rule_tac x = "30 + q + qa" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3339 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3340 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3341 |
show " list_all (\<lambda>(acn, s). s \<le> (60 + (length (tm_of abc_twice) + |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3342 |
length (tm_of abc_fourtimes))) div 2) t_wcode_main_first_part" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3343 |
apply(auto simp: t_wcode_main_first_part_def shift_length t_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3344 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3345 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3346 |
have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_twice @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3347 |
(start_of twice_ly (length abc_twice) - Suc 0)) div 2)) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3348 |
(change_termi_state (tm_of abc_twice @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3349 |
(start_of twice_ly (length abc_twice) - Suc 0)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3350 |
apply(rule_tac t_correct_termi, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3351 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3352 |
hence "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_twice @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3353 |
(start_of twice_ly (length abc_twice) - Suc 0)) div 2) + 12) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3354 |
(abacus.tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3355 |
(start_of twice_ly (length abc_twice) - Suc 0))) 12)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3356 |
apply(rule_tac t_correct_shift, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3357 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3358 |
thus "list_all (\<lambda>(acn, s). s \<le> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3359 |
(60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3360 |
(abacus.tshift (change_termi_state (tm_of abc_twice @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3361 |
(start_of twice_ly (length abc_twice) - Suc 0))) 12)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3362 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3363 |
apply(simp add: list_all_length, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3364 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3365 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3366 |
have "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3367 |
(start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2)) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3368 |
(change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3369 |
(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) " |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3370 |
apply(rule_tac t_correct_termi, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3371 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3372 |
hence "list_all (\<lambda>(acn, s). s \<le> Suc (length (tm_of abc_fourtimes @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3373 |
(start_of fourtimes_ly (length abc_fourtimes) - Suc 0)) div 2) + (t_twice_len + 13)) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3374 |
(abacus.tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3375 |
(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3376 |
apply(rule_tac t_correct_shift, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3377 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3378 |
thus "list_all (\<lambda>(acn, s). s \<le> (60 + (length (tm_of abc_twice) + length (tm_of abc_fourtimes))) div 2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3379 |
(abacus.tshift (change_termi_state (tm_of abc_fourtimes @ tMp (Suc 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3380 |
(start_of fourtimes_ly (length abc_fourtimes) - Suc 0))) (t_twice_len + 13))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3381 |
apply(simp add: t_twice_len_def t_twice_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3382 |
using twice_len_even fourtimes_len_even |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3383 |
apply(auto simp: list_all_length) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3384 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3385 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3386 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3387 |
lemma [intro]: "t_correct (t_wcode_prepare |+| t_wcode_main)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3388 |
apply(auto intro: t_correct_add) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3389 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3390 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3391 |
lemma prepare_mainpart_lemma: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3392 |
"args \<noteq> [] \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3393 |
\<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3394 |
= (0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3395 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3396 |
let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3397 |
let ?Q1 = "\<lambda> (l, r). wprepare_stop m args (l, r)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3398 |
let ?P2 = ?Q1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3399 |
let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3400 |
r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3401 |
let ?P3 = "\<lambda> tp. False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3402 |
assume h: "args \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3403 |
have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3404 |
(t_wcode_prepare |+| t_wcode_main) stp = (0, tp') \<and> ?Q2 tp')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3405 |
proof(rule_tac turing_merge.t_merge_halt[of t_wcode_prepare t_wcode_main ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3406 |
auto simp: turing_merge_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3407 |
show "\<exists>stp. case steps (Suc 0, [], <m # args>) t_wcode_prepare stp of (st, tp') |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3408 |
\<Rightarrow> st = 0 \<and> wprepare_stop m args tp'" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3409 |
using wprepare_correctness[of args m] h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3410 |
apply(simp, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3411 |
apply(rule_tac x = n in exI, simp add: wprepare_inv.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3412 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3413 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3414 |
fix a b |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3415 |
assume "wprepare_stop m args (a, b)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3416 |
thus "\<exists>stp. case steps (Suc 0, a, b) t_wcode_main stp of |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3417 |
(st, tp') \<Rightarrow> (st = 0) \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3418 |
(\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3419 |
proof(simp only: wprepare_stop.simps, erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3420 |
fix rn |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3421 |
assume "a = Bk # <rev args> @ Bk # Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3422 |
b = Bk # Oc # Bk\<^bsup>rn\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3423 |
thus "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3424 |
using t_wcode_main_lemma_pre[of "args" "<args>" 0 "Oc\<^bsup>Suc m\<^esup>" 0 rn] h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3425 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3426 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3427 |
apply(rule_tac x = stp in exI, simp add: tape_of_nl_rev, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3428 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3429 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3430 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3431 |
show "wprepare_stop m args \<turnstile>-> wprepare_stop m args" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3432 |
by(simp add: t_imply_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3433 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3434 |
thus "\<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3435 |
= (0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3436 |
apply(simp add: t_imply_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3437 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3438 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3439 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3440 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3441 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3442 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3443 |
lemma [simp]: "tinres r r' \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3444 |
fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3445 |
fetch t ss (case r' of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3446 |
apply(simp add: fetch.simps, auto split: if_splits simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3447 |
apply(case_tac [!] r', simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3448 |
apply(case_tac [!] n, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3449 |
apply(case_tac [!] r, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3450 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3451 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3452 |
lemma [intro]: "\<exists> n. (a::block)\<^bsup>n\<^esup> = []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3453 |
by auto |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3454 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3455 |
lemma [simp]: "\<lbrakk>tinres r r'; r \<noteq> []; r' \<noteq> []\<rbrakk> \<Longrightarrow> hd r = hd r'" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3456 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3457 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3458 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3459 |
lemma [intro]: "hd (Bk\<^bsup>Suc n\<^esup>) = Bk" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3460 |
apply(simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3461 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3462 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3463 |
lemma [simp]: "\<lbrakk>tinres r []; r \<noteq> []\<rbrakk> \<Longrightarrow> hd r = Bk" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3464 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3465 |
apply(case_tac n, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3466 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3467 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3468 |
lemma [simp]: "\<lbrakk>tinres [] r'; r' \<noteq> []\<rbrakk> \<Longrightarrow> hd r' = Bk" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3469 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3470 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3471 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3472 |
lemma [intro]: "\<exists>na. tl r = tl (r @ Bk\<^bsup>n\<^esup>) @ Bk\<^bsup>na\<^esup> \<or> tl (r @ Bk\<^bsup>n\<^esup>) = tl r @ Bk\<^bsup>na\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3473 |
apply(case_tac r, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3474 |
apply(case_tac n, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3475 |
apply(rule_tac x = 0 in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3476 |
apply(rule_tac x = nat in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3477 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3478 |
apply(rule_tac x = n in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3479 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3480 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3481 |
lemma [simp]: "tinres r r' \<Longrightarrow> tinres (tl r) (tl r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3482 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3483 |
apply(case_tac r', simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3484 |
apply(case_tac n, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3485 |
apply(rule_tac x = 0 in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3486 |
apply(rule_tac x = nat in exI, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3487 |
apply(rule_tac x = n in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3488 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3489 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3490 |
lemma [simp]: "\<lbrakk>tinres r []; r \<noteq> []\<rbrakk> \<Longrightarrow> tinres (tl r) []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3491 |
apply(case_tac r, auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3492 |
apply(case_tac n, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3493 |
apply(rule_tac x = nat in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3494 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3495 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3496 |
lemma [simp]: "\<lbrakk>tinres [] r'\<rbrakk> \<Longrightarrow> tinres [] (tl r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3497 |
apply(case_tac r', auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3498 |
apply(case_tac n, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3499 |
apply(rule_tac x = nat in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3500 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3501 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3502 |
lemma [simp]: "tinres r r' \<Longrightarrow> tinres (b # r) (b # r')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3503 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3504 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3505 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3506 |
lemma tinres_step2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3507 |
"\<lbrakk>tinres r r'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l, r') t = (sb, lb, rb)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3508 |
\<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3509 |
apply(case_tac "ss = 0", simp add: tstep_0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3510 |
apply(simp add: tstep.simps [simp del]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3511 |
apply(case_tac "fetch t ss (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3512 |
apply(auto simp: new_tape.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3513 |
apply(simp_all split: taction.splits if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3514 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3515 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3516 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3517 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3518 |
lemma tinres_steps2: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3519 |
"\<lbrakk>tinres r r'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l, r') t stp = (sb, lb, rb)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3520 |
\<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3521 |
apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3522 |
apply(simp add: tstep_red) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3523 |
apply(case_tac "(steps (ss, l, r) t stp)") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3524 |
apply(case_tac "(steps (ss, l, r') t stp)") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3525 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3526 |
fix stp sa la ra sb lb rb a b c aa ba ca |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3527 |
assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps (ss, l, r) t stp = (sa, la, ra); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3528 |
steps (ss, l, r') t stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> la = lb \<and> tinres ra rb \<and> sa = sb" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3529 |
and h: " tinres r r'" "tstep (steps (ss, l, r) t stp) t = (sa, la, ra)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3530 |
"tstep (steps (ss, l, r') t stp) t = (sb, lb, rb)" "steps (ss, l, r) t stp = (a, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3531 |
"steps (ss, l, r') t stp = (aa, ba, ca)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3532 |
have "b = ba \<and> tinres c ca \<and> a = aa" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3533 |
apply(rule_tac ind, simp_all add: h) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3534 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3535 |
thus "la = lb \<and> tinres ra rb \<and> sa = sb" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3536 |
apply(rule_tac l = b and r = c and ss = a and r' = ca |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3537 |
and t = t in tinres_step2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3538 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3539 |
apply(simp, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3540 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3541 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3542 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3543 |
definition t_wcode_adjust :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3544 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3545 |
"t_wcode_adjust = [(W1, 1), (R, 2), (Nop, 2), (R, 3), (R, 3), (R, 4), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3546 |
(L, 8), (L, 5), (L, 6), (W0, 5), (L, 6), (R, 7), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3547 |
(W1, 2), (Nop, 7), (L, 9), (W0, 8), (L, 9), (L, 10), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3548 |
(L, 11), (L, 10), (R, 0), (L, 11)]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3549 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3550 |
lemma [simp]: "fetch t_wcode_adjust (Suc 0) Bk = (W1, 1)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3551 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3552 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3553 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3554 |
lemma [simp]: "fetch t_wcode_adjust (Suc 0) Oc = (R, 2)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3555 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3556 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3557 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3558 |
lemma [simp]: "fetch t_wcode_adjust (Suc (Suc 0)) Oc = (R, 3)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3559 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3560 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3561 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3562 |
lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc 0))) Oc = (R, 4)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3563 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3564 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3565 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3566 |
lemma [simp]: "fetch t_wcode_adjust (Suc (Suc (Suc 0))) Bk = (R, 3)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3567 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3568 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3569 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3570 |
lemma [simp]: "fetch t_wcode_adjust 4 Bk = (L, 8)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3571 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3572 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3573 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3574 |
lemma [simp]: "fetch t_wcode_adjust 4 Oc = (L, 5)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3575 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3576 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3577 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3578 |
lemma [simp]: "fetch t_wcode_adjust 5 Oc = (W0, 5)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3579 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3580 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3581 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3582 |
lemma [simp]: "fetch t_wcode_adjust 5 Bk = (L, 6)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3583 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3584 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3585 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3586 |
lemma [simp]: "fetch t_wcode_adjust 6 Oc = (R, 7)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3587 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3588 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3589 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3590 |
lemma [simp]: "fetch t_wcode_adjust 6 Bk = (L, 6)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3591 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3592 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3593 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3594 |
lemma [simp]: "fetch t_wcode_adjust 7 Bk = (W1, 2)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3595 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3596 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3597 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3598 |
lemma [simp]: "fetch t_wcode_adjust 8 Bk = (L, 9)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3599 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3600 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3601 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3602 |
lemma [simp]: "fetch t_wcode_adjust 8 Oc = (W0, 8)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3603 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3604 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3605 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3606 |
lemma [simp]: "fetch t_wcode_adjust 9 Oc = (L, 10)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3607 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3608 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3609 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3610 |
lemma [simp]: "fetch t_wcode_adjust 9 Bk = (L, 9)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3611 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3612 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3613 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3614 |
lemma [simp]: "fetch t_wcode_adjust 10 Bk = (L, 11)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3615 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3616 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3617 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3618 |
lemma [simp]: "fetch t_wcode_adjust 10 Oc = (L, 10)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3619 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3620 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3621 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3622 |
lemma [simp]: "fetch t_wcode_adjust 11 Oc = (L, 11)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3623 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3624 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3625 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3626 |
lemma [simp]: "fetch t_wcode_adjust 11 Bk = (R, 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3627 |
apply(simp add: fetch.simps t_wcode_adjust_def nth_of.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3628 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3629 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3630 |
fun wadjust_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3631 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3632 |
"wadjust_start m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3633 |
(\<exists> ln rn. l = Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3634 |
tl r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3635 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3636 |
fun wadjust_loop_start :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3637 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3638 |
"wadjust_loop_start m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3639 |
(\<exists> ln rn ml mr. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3640 |
r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3641 |
ml + mr = Suc (Suc rs) \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3642 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3643 |
fun wadjust_loop_right_move :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3644 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3645 |
"wadjust_loop_right_move m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3646 |
(\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3647 |
r = Bk\<^bsup>nr\<^esup> @ Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3648 |
ml + mr = Suc (Suc rs) \<and> mr > 0 \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3649 |
nl + nr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3650 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3651 |
fun wadjust_loop_check :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3652 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3653 |
"wadjust_loop_check m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3654 |
(\<exists> ml mr ln rn. l = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3655 |
r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3656 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3657 |
fun wadjust_loop_erase :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3658 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3659 |
"wadjust_loop_erase m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3660 |
(\<exists> ml mr ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3661 |
tl r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> ml + mr = (Suc rs) \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3662 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3663 |
fun wadjust_loop_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3664 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3665 |
"wadjust_loop_on_left_moving_O m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3666 |
(\<exists> ml mr ln rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m \<^esup>\<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3667 |
r = Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3668 |
ml + mr = Suc rs \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3669 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3670 |
fun wadjust_loop_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3671 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3672 |
"wadjust_loop_on_left_moving_B m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3673 |
(\<exists> ml mr nl nr rn. l = Bk\<^bsup>nl\<^esup> @ Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3674 |
r = Bk\<^bsup>nr\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3675 |
ml + mr = Suc rs \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3676 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3677 |
fun wadjust_loop_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3678 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3679 |
"wadjust_loop_on_left_moving m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3680 |
(wadjust_loop_on_left_moving_O m rs (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3681 |
wadjust_loop_on_left_moving_B m rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3682 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3683 |
fun wadjust_loop_right_move2 :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3684 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3685 |
"wadjust_loop_right_move2 m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3686 |
(\<exists> ml mr ln rn. l = Oc # Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3687 |
r = Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3688 |
ml + mr = Suc rs \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3689 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3690 |
fun wadjust_erase2 :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3691 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3692 |
"wadjust_erase2 m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3693 |
(\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3694 |
tl r = Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3695 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3696 |
fun wadjust_on_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3697 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3698 |
"wadjust_on_left_moving_O m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3699 |
(\<exists> rn. l = Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3700 |
r = Oc # Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3701 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3702 |
fun wadjust_on_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3703 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3704 |
"wadjust_on_left_moving_B m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3705 |
(\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Oc # Oc\<^bsup>Suc rs\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3706 |
r = Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3707 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3708 |
fun wadjust_on_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3709 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3710 |
"wadjust_on_left_moving m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3711 |
(wadjust_on_left_moving_O m rs (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3712 |
wadjust_on_left_moving_B m rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3713 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3714 |
fun wadjust_goon_left_moving_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3715 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3716 |
"wadjust_goon_left_moving_B m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3717 |
(\<exists> rn. l = Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3718 |
r = Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3719 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3720 |
fun wadjust_goon_left_moving_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3721 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3722 |
"wadjust_goon_left_moving_O m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3723 |
(\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> @ Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3724 |
r = Oc\<^bsup>mr\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3725 |
ml + mr = Suc (Suc rs) \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3726 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3727 |
fun wadjust_goon_left_moving :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3728 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3729 |
"wadjust_goon_left_moving m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3730 |
(wadjust_goon_left_moving_B m rs (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3731 |
wadjust_goon_left_moving_O m rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3732 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3733 |
fun wadjust_backto_standard_pos_B :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3734 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3735 |
"wadjust_backto_standard_pos_B m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3736 |
(\<exists> rn. l = [] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3737 |
r = Bk # Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3738 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3739 |
fun wadjust_backto_standard_pos_O :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3740 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3741 |
"wadjust_backto_standard_pos_O m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3742 |
(\<exists> ml mr rn. l = Oc\<^bsup>ml\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3743 |
r = Oc\<^bsup>mr\<^esup> @ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3744 |
ml + mr = Suc m \<and> mr > 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3745 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3746 |
fun wadjust_backto_standard_pos :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3747 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3748 |
"wadjust_backto_standard_pos m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3749 |
(wadjust_backto_standard_pos_B m rs (l, r) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3750 |
wadjust_backto_standard_pos_O m rs (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3751 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3752 |
fun wadjust_stop :: "nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3753 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3754 |
"wadjust_stop m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3755 |
(\<exists> rn. l = [Bk] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3756 |
r = Oc\<^bsup>Suc m \<^esup>@ Bk # Oc\<^bsup>Suc (Suc rs)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3757 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3758 |
declare wadjust_start.simps[simp del] wadjust_loop_start.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3759 |
wadjust_loop_right_move.simps[simp del] wadjust_loop_check.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3760 |
wadjust_loop_erase.simps[simp del] wadjust_loop_on_left_moving.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3761 |
wadjust_loop_right_move2.simps[simp del] wadjust_erase2.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3762 |
wadjust_on_left_moving_O.simps[simp del] wadjust_on_left_moving_B.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3763 |
wadjust_on_left_moving.simps[simp del] wadjust_goon_left_moving_B.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3764 |
wadjust_goon_left_moving_O.simps[simp del] wadjust_goon_left_moving.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3765 |
wadjust_backto_standard_pos.simps[simp del] wadjust_backto_standard_pos_B.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3766 |
wadjust_backto_standard_pos_O.simps[simp del] wadjust_stop.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3767 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3768 |
fun wadjust_inv :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> tape \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3769 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3770 |
"wadjust_inv st m rs (l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3771 |
(if st = Suc 0 then wadjust_start m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3772 |
else if st = Suc (Suc 0) then wadjust_loop_start m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3773 |
else if st = Suc (Suc (Suc 0)) then wadjust_loop_right_move m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3774 |
else if st = 4 then wadjust_loop_check m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3775 |
else if st = 5 then wadjust_loop_erase m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3776 |
else if st = 6 then wadjust_loop_on_left_moving m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3777 |
else if st = 7 then wadjust_loop_right_move2 m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3778 |
else if st = 8 then wadjust_erase2 m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3779 |
else if st = 9 then wadjust_on_left_moving m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3780 |
else if st = 10 then wadjust_goon_left_moving m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3781 |
else if st = 11 then wadjust_backto_standard_pos m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3782 |
else if st = 0 then wadjust_stop m rs (l, r) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3783 |
else False |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3784 |
)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3785 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3786 |
declare wadjust_inv.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3787 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3788 |
fun wadjust_phase :: "nat \<Rightarrow> t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3789 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3790 |
"wadjust_phase rs (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3791 |
(if st = 1 then 3 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3792 |
else if st \<ge> 2 \<and> st \<le> 7 then 2 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3793 |
else if st \<ge> 8 \<and> st \<le> 11 then 1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3794 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3795 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3796 |
thm dropWhile.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3797 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3798 |
fun wadjust_stage :: "nat \<Rightarrow> t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3799 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3800 |
"wadjust_stage rs (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3801 |
(if st \<ge> 2 \<and> st \<le> 7 then |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3802 |
rs - length (takeWhile (\<lambda> a. a = Oc) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3803 |
(tl (dropWhile (\<lambda> a. a = Oc) (rev l @ r)))) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3804 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3805 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3806 |
fun wadjust_state :: "nat \<Rightarrow> t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3807 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3808 |
"wadjust_state rs (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3809 |
(if st \<ge> 2 \<and> st \<le> 7 then 8 - st |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3810 |
else if st \<ge> 8 \<and> st \<le> 11 then 12 - st |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3811 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3812 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3813 |
fun wadjust_step :: "nat \<Rightarrow> t_conf \<Rightarrow> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3814 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3815 |
"wadjust_step rs (st, l, r) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3816 |
(if st = 1 then (if hd r = Bk then 1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3817 |
else 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3818 |
else if st = 3 then length r |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3819 |
else if st = 5 then (if hd r = Oc then 1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3820 |
else 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3821 |
else if st = 6 then length l |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3822 |
else if st = 8 then (if hd r = Oc then 1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3823 |
else 0) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3824 |
else if st = 9 then length l |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3825 |
else if st = 10 then length l |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3826 |
else if st = 11 then (if hd r = Bk then 0 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3827 |
else Suc (length l)) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3828 |
else 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3829 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3830 |
fun wadjust_measure :: "(nat \<times> t_conf) \<Rightarrow> nat \<times> nat \<times> nat \<times> nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3831 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3832 |
"wadjust_measure (rs, (st, l, r)) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3833 |
(wadjust_phase rs (st, l, r), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3834 |
wadjust_stage rs (st, l, r), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3835 |
wadjust_state rs (st, l, r), |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3836 |
wadjust_step rs (st, l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3837 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3838 |
definition wadjust_le :: "((nat \<times> t_conf) \<times> nat \<times> t_conf) set" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3839 |
where "wadjust_le \<equiv> (inv_image lex_square wadjust_measure)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3840 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3841 |
lemma [intro]: "wf lex_square" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3842 |
by(auto intro:wf_lex_prod simp: abacus.lex_pair_def lex_square_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3843 |
abacus.lex_triple_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3844 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3845 |
lemma wf_wadjust_le[intro]: "wf wadjust_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3846 |
by(auto intro:wf_inv_image simp: wadjust_le_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3847 |
abacus.lex_triple_def abacus.lex_pair_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3848 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3849 |
lemma [simp]: "wadjust_start m rs (c, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3850 |
apply(auto simp: wadjust_start.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3851 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3852 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3853 |
lemma [simp]: "wadjust_loop_right_move m rs (c, []) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3854 |
apply(auto simp: wadjust_loop_right_move.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3855 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3856 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3857 |
lemma [simp]: "wadjust_loop_right_move m rs (c, []) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3858 |
\<Longrightarrow> wadjust_loop_check m rs (Bk # c, [])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3859 |
apply(simp only: wadjust_loop_right_move.simps wadjust_loop_check.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3860 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3861 |
apply(case_tac [!] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3862 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3863 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3864 |
lemma [simp]: "wadjust_loop_check m rs (c, []) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3865 |
apply(simp only: wadjust_loop_check.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3866 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3867 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3868 |
lemma [simp]: "wadjust_loop_start m rs (c, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3869 |
apply(simp add: wadjust_loop_start.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3870 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3871 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3872 |
lemma [simp]: "wadjust_loop_right_move m rs (c, []) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3873 |
wadjust_loop_right_move m rs (Bk # c, [])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3874 |
apply(simp only: wadjust_loop_right_move.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3875 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3876 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3877 |
apply(case_tac [!] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3878 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3879 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3880 |
lemma [simp]: "wadjust_loop_check m rs (c, []) \<Longrightarrow> wadjust_erase2 m rs (tl c, [hd c])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3881 |
apply(simp only: wadjust_loop_check.simps wadjust_erase2.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3882 |
apply(case_tac mr, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3883 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3884 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3885 |
lemma [simp]: " wadjust_loop_erase m rs (c, []) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3886 |
\<Longrightarrow> (c = [] \<longrightarrow> wadjust_loop_on_left_moving m rs ([], [Bk])) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3887 |
(c \<noteq> [] \<longrightarrow> wadjust_loop_on_left_moving m rs (tl c, [hd c]))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3888 |
apply(simp add: wadjust_loop_erase.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3889 |
apply(case_tac [!] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3890 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3891 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3892 |
lemma [simp]: "wadjust_loop_on_left_moving m rs (c, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3893 |
apply(auto simp: wadjust_loop_on_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3894 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3895 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3896 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3897 |
lemma [simp]: "wadjust_loop_right_move2 m rs (c, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3898 |
apply(auto simp: wadjust_loop_right_move2.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3899 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3900 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3901 |
lemma [simp]: "wadjust_erase2 m rs ([], []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3902 |
apply(auto simp: wadjust_erase2.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3903 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3904 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3905 |
lemma [simp]: "wadjust_on_left_moving_B m rs |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3906 |
(Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3907 |
apply(simp add: wadjust_on_left_moving_B.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3908 |
apply(rule_tac x = 0 in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3909 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3910 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3911 |
lemma [simp]: "wadjust_on_left_moving_B m rs |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3912 |
(Bk\<^bsup>n\<^esup> @ Bk # Oc # Oc # Oc\<^bsup>rs\<^esup> @ Bk # Oc # Oc\<^bsup>m\<^esup>, [Bk])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3913 |
apply(simp add: wadjust_on_left_moving_B.simps exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3914 |
apply(rule_tac x = "Suc n" in exI, simp add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3915 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3916 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3917 |
lemma [simp]: "\<lbrakk>wadjust_erase2 m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3918 |
wadjust_on_left_moving m rs (tl c, [hd c])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3919 |
apply(simp only: wadjust_erase2.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3920 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3921 |
apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3922 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3923 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3924 |
lemma [simp]: "wadjust_erase2 m rs (c, []) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3925 |
\<Longrightarrow> (c = [] \<longrightarrow> wadjust_on_left_moving m rs ([], [Bk])) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3926 |
(c \<noteq> [] \<longrightarrow> wadjust_on_left_moving m rs (tl c, [hd c]))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3927 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3928 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3929 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3930 |
lemma [simp]: "wadjust_on_left_moving m rs ([], []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3931 |
apply(simp add: wadjust_on_left_moving.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3932 |
wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3933 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3934 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3935 |
lemma [simp]: "wadjust_on_left_moving_O m rs (c, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3936 |
apply(simp add: wadjust_on_left_moving_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3937 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3938 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3939 |
lemma [simp]: " \<lbrakk>wadjust_on_left_moving_B m rs (c, []); c \<noteq> []; hd c = Bk\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3940 |
wadjust_on_left_moving_B m rs (tl c, [Bk])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3941 |
apply(simp add: wadjust_on_left_moving_B.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3942 |
apply(case_tac [!] ln, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3943 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3944 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3945 |
lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, []); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3946 |
wadjust_on_left_moving_O m rs (tl c, [Oc])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3947 |
apply(simp add: wadjust_on_left_moving_B.simps wadjust_on_left_moving_O.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3948 |
apply(case_tac [!] ln, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3949 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3950 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3951 |
lemma [simp]: "\<lbrakk>wadjust_on_left_moving m rs (c, []); c \<noteq> []\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3952 |
wadjust_on_left_moving m rs (tl c, [hd c])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3953 |
apply(simp add: wadjust_on_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3954 |
apply(case_tac "hd c", simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3955 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3956 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3957 |
lemma [simp]: "wadjust_on_left_moving m rs (c, []) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3958 |
\<Longrightarrow> (c = [] \<longrightarrow> wadjust_on_left_moving m rs ([], [Bk])) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3959 |
(c \<noteq> [] \<longrightarrow> wadjust_on_left_moving m rs (tl c, [hd c]))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3960 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3961 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3962 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3963 |
lemma [simp]: "wadjust_goon_left_moving m rs (c, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3964 |
apply(auto simp: wadjust_goon_left_moving.simps wadjust_goon_left_moving_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3965 |
wadjust_goon_left_moving_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3966 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3967 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3968 |
lemma [simp]: "wadjust_backto_standard_pos m rs (c, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3969 |
apply(auto simp: wadjust_backto_standard_pos.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3970 |
wadjust_backto_standard_pos_B.simps wadjust_backto_standard_pos_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3971 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3972 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3973 |
lemma [simp]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3974 |
"wadjust_start m rs (c, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3975 |
(c = [] \<longrightarrow> wadjust_start m rs ([], Oc # list)) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3976 |
(c \<noteq> [] \<longrightarrow> wadjust_start m rs (c, Oc # list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3977 |
apply(auto simp: wadjust_start.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3978 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3979 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3980 |
lemma [simp]: "wadjust_loop_start m rs (c, Bk # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3981 |
apply(auto simp: wadjust_loop_start.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3982 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3983 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3984 |
lemma [simp]: "wadjust_loop_right_move m rs (c, b) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3985 |
apply(simp only: wadjust_loop_right_move.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3986 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3987 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3988 |
lemma [simp]: "wadjust_loop_right_move m rs (c, Bk # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3989 |
\<Longrightarrow> wadjust_loop_right_move m rs (Bk # c, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3990 |
apply(simp only: wadjust_loop_right_move.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3991 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3992 |
apply(rule_tac x = ml in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3993 |
apply(rule_tac x = mr in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3994 |
apply(rule_tac x = "Suc nl" in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3995 |
apply(case_tac nr, simp, case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3996 |
apply(rule_tac x = nat in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3997 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3998 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
3999 |
lemma [simp]: "wadjust_loop_check m rs (c, b) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4000 |
apply(simp only: wadjust_loop_check.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4001 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4002 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4003 |
lemma [simp]: "wadjust_loop_check m rs (c, Bk # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4004 |
\<Longrightarrow> wadjust_erase2 m rs (tl c, hd c # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4005 |
apply(auto simp: wadjust_loop_check.simps wadjust_erase2.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4006 |
apply(case_tac [!] mr, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4007 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4008 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4009 |
lemma [simp]: "wadjust_loop_erase m rs (c, b) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4010 |
apply(simp only: wadjust_loop_erase.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4011 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4012 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4013 |
declare wadjust_loop_on_left_moving_O.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4014 |
wadjust_loop_on_left_moving_B.simps[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4015 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4016 |
lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); hd c = Bk\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4017 |
\<Longrightarrow> wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4018 |
apply(simp only: wadjust_loop_erase.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4019 |
wadjust_loop_on_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4020 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4021 |
apply(rule_tac x = ml in exI, rule_tac x = mr in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4022 |
rule_tac x = ln in exI, rule_tac x = 0 in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4023 |
apply(case_tac ln, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4024 |
apply(simp add: exp_ind exp_ind_def[THEN sym]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4025 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4026 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4027 |
lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []; hd c = Oc\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4028 |
wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4029 |
apply(simp only: wadjust_loop_erase.simps wadjust_loop_on_left_moving_O.simps, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4030 |
auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4031 |
apply(case_tac [!] ln, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4032 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4033 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4034 |
lemma [simp]: "\<lbrakk>wadjust_loop_erase m rs (c, Bk # list); c \<noteq> []\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4035 |
wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4036 |
apply(case_tac "hd c", simp_all add:wadjust_loop_on_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4037 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4038 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4039 |
lemma [simp]: "wadjust_loop_on_left_moving m rs (c, b) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4040 |
apply(simp add: wadjust_loop_on_left_moving.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4041 |
wadjust_loop_on_left_moving_O.simps wadjust_loop_on_left_moving_B.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4042 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4043 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4044 |
lemma [simp]: "wadjust_loop_on_left_moving_O m rs (c, Bk # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4045 |
apply(simp add: wadjust_loop_on_left_moving_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4046 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4047 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4048 |
lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4049 |
\<Longrightarrow> wadjust_loop_on_left_moving_B m rs (tl c, Bk # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4050 |
apply(simp only: wadjust_loop_on_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4051 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4052 |
apply(rule_tac x = ml in exI, rule_tac x = mr in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4053 |
apply(case_tac nl, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4054 |
apply(rule_tac x = "Suc nr" in exI, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4055 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4056 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4057 |
lemma [simp]: "\<lbrakk>wadjust_loop_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4058 |
\<Longrightarrow> wadjust_loop_on_left_moving_O m rs (tl c, Oc # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4059 |
apply(simp only: wadjust_loop_on_left_moving_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4060 |
wadjust_loop_on_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4061 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4062 |
apply(rule_tac x = ml in exI, rule_tac x = mr in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4063 |
apply(case_tac nl, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4064 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4065 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4066 |
lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Bk # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4067 |
\<Longrightarrow> wadjust_loop_on_left_moving m rs (tl c, hd c # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4068 |
apply(simp add: wadjust_loop_on_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4069 |
apply(case_tac "hd c", simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4070 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4071 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4072 |
lemma [simp]: "wadjust_loop_right_move2 m rs (c, b) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4073 |
apply(simp only: wadjust_loop_right_move2.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4074 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4075 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4076 |
lemma [simp]: "wadjust_loop_right_move2 m rs (c, Bk # list) \<Longrightarrow> wadjust_loop_start m rs (c, Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4077 |
apply(auto simp: wadjust_loop_right_move2.simps wadjust_loop_start.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4078 |
apply(case_tac ln, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4079 |
apply(rule_tac x = 0 in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4080 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4081 |
apply(rule_tac x = "Suc ml" in exI, simp add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4082 |
apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4083 |
apply(rule_tac x = rn in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4084 |
apply(rule_tac x = "Suc ml" in exI, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4085 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4086 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4087 |
lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4088 |
apply(auto simp:wadjust_erase2.simps ) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4089 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4090 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4091 |
lemma [simp]: "wadjust_erase2 m rs (c, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4092 |
wadjust_on_left_moving m rs (tl c, hd c # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4093 |
apply(auto simp: wadjust_erase2.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4094 |
apply(case_tac ln, simp_all add: exp_ind_def wadjust_on_left_moving.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4095 |
wadjust_on_left_moving_O.simps wadjust_on_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4096 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4097 |
apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4098 |
apply(rule_tac x = "Suc nat" in exI, simp add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4099 |
apply(rule_tac x = "(Suc (Suc rn))" in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4100 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4101 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4102 |
lemma [simp]: "wadjust_on_left_moving m rs (c,b) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4103 |
apply(simp only:wadjust_on_left_moving.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4104 |
wadjust_on_left_moving_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4105 |
wadjust_on_left_moving_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4106 |
, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4107 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4108 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4109 |
lemma [simp]: "wadjust_on_left_moving_O m rs (c, Bk # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4110 |
apply(simp add: wadjust_on_left_moving_O.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4111 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4112 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4113 |
lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4114 |
\<Longrightarrow> wadjust_on_left_moving_B m rs (tl c, Bk # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4115 |
apply(auto simp: wadjust_on_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4116 |
apply(case_tac ln, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4117 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4118 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4119 |
lemma [simp]: "\<lbrakk>wadjust_on_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4120 |
\<Longrightarrow> wadjust_on_left_moving_O m rs (tl c, Oc # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4121 |
apply(auto simp: wadjust_on_left_moving_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4122 |
wadjust_on_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4123 |
apply(case_tac ln, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4124 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4125 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4126 |
lemma [simp]: "wadjust_on_left_moving m rs (c, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4127 |
wadjust_on_left_moving m rs (tl c, hd c # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4128 |
apply(simp add: wadjust_on_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4129 |
apply(case_tac "hd c", simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4130 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4131 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4132 |
lemma [simp]: "wadjust_goon_left_moving m rs (c, b) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4133 |
apply(simp add: wadjust_goon_left_moving.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4134 |
wadjust_goon_left_moving_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4135 |
wadjust_goon_left_moving_O.simps exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4136 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4137 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4138 |
lemma [simp]: "wadjust_goon_left_moving_O m rs (c, Bk # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4139 |
apply(simp add: wadjust_goon_left_moving_O.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4140 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4141 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4142 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4143 |
lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Bk\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4144 |
\<Longrightarrow> wadjust_backto_standard_pos_B m rs (tl c, Bk # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4145 |
apply(auto simp: wadjust_goon_left_moving_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4146 |
wadjust_backto_standard_pos_B.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4147 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4148 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4149 |
lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_B m rs (c, Bk # list); hd c = Oc\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4150 |
\<Longrightarrow> wadjust_backto_standard_pos_O m rs (tl c, Oc # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4151 |
apply(auto simp: wadjust_goon_left_moving_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4152 |
wadjust_backto_standard_pos_O.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4153 |
apply(rule_tac x = m in exI, simp, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4154 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4155 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4156 |
lemma [simp]: "wadjust_goon_left_moving m rs (c, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4157 |
wadjust_backto_standard_pos m rs (tl c, hd c # Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4158 |
apply(case_tac "hd c", simp_all add: wadjust_backto_standard_pos.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4159 |
wadjust_goon_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4160 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4161 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4162 |
lemma [simp]: "wadjust_backto_standard_pos m rs (c, Bk # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4163 |
(c = [] \<longrightarrow> wadjust_stop m rs ([Bk], list)) \<and> (c \<noteq> [] \<longrightarrow> wadjust_stop m rs (Bk # c, list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4164 |
apply(auto simp: wadjust_backto_standard_pos.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4165 |
wadjust_backto_standard_pos_B.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4166 |
wadjust_backto_standard_pos_O.simps wadjust_stop.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4167 |
apply(case_tac [!] mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4168 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4169 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4170 |
lemma [simp]: "wadjust_start m rs (c, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4171 |
\<Longrightarrow> (c = [] \<longrightarrow> wadjust_loop_start m rs ([Oc], list)) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4172 |
(c \<noteq> [] \<longrightarrow> wadjust_loop_start m rs (Oc # c, list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4173 |
apply(auto simp:wadjust_loop_start.simps wadjust_start.simps ) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4174 |
apply(rule_tac x = ln in exI, rule_tac x = rn in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4175 |
rule_tac x = "Suc 0" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4176 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4177 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4178 |
lemma [simp]: "wadjust_loop_start m rs (c, b) \<Longrightarrow> c \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4179 |
apply(simp add: wadjust_loop_start.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4180 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4181 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4182 |
lemma [simp]: "wadjust_loop_start m rs (c, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4183 |
\<Longrightarrow> wadjust_loop_right_move m rs (Oc # c, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4184 |
apply(simp add: wadjust_loop_start.simps wadjust_loop_right_move.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4185 |
apply(rule_tac x = ml in exI, rule_tac x = mr in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4186 |
rule_tac x = 0 in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4187 |
apply(rule_tac x = "Suc ln" in exI, simp add: exp_ind, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4188 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4189 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4190 |
lemma [simp]: "wadjust_loop_right_move m rs (c, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4191 |
wadjust_loop_check m rs (Oc # c, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4192 |
apply(simp add: wadjust_loop_right_move.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4193 |
wadjust_loop_check.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4194 |
apply(rule_tac [!] x = ml in exI, simp_all, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4195 |
apply(case_tac nl, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4196 |
apply(rule_tac x = "mr - 1" in exI, case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4197 |
apply(case_tac [!] nr, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4198 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4199 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4200 |
lemma [simp]: "wadjust_loop_check m rs (c, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4201 |
wadjust_loop_erase m rs (tl c, hd c # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4202 |
apply(simp only: wadjust_loop_check.simps wadjust_loop_erase.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4203 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4204 |
apply(rule_tac x = ml in exI, rule_tac x = mr in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4205 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4206 |
apply(case_tac rn, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4207 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4208 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4209 |
lemma [simp]: "wadjust_loop_erase m rs (c, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4210 |
wadjust_loop_erase m rs (c, Bk # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4211 |
apply(auto simp: wadjust_loop_erase.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4212 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4213 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4214 |
lemma [simp]: "wadjust_loop_on_left_moving_B m rs (c, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4215 |
apply(auto simp: wadjust_loop_on_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4216 |
apply(case_tac nr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4217 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4218 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4219 |
lemma [simp]: "wadjust_loop_on_left_moving m rs (c, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4220 |
\<Longrightarrow> wadjust_loop_right_move2 m rs (Oc # c, list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4221 |
apply(simp add:wadjust_loop_on_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4222 |
apply(auto simp: wadjust_loop_on_left_moving_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4223 |
wadjust_loop_right_move2.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4224 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4225 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4226 |
lemma [simp]: "wadjust_loop_right_move2 m rs (c, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4227 |
apply(auto simp: wadjust_loop_right_move2.simps ) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4228 |
apply(case_tac ln, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4229 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4230 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4231 |
lemma [simp]: "wadjust_erase2 m rs (c, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4232 |
\<Longrightarrow> (c = [] \<longrightarrow> wadjust_erase2 m rs ([], Bk # list)) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4233 |
\<and> (c \<noteq> [] \<longrightarrow> wadjust_erase2 m rs (c, Bk # list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4234 |
apply(auto simp: wadjust_erase2.simps ) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4235 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4236 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4237 |
lemma [simp]: "wadjust_on_left_moving_B m rs (c, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4238 |
apply(auto simp: wadjust_on_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4239 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4240 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4241 |
lemma [simp]: "\<lbrakk>wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Bk\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4242 |
wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4243 |
apply(auto simp: wadjust_on_left_moving_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4244 |
wadjust_goon_left_moving_B.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4245 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4246 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4247 |
lemma [simp]: "\<lbrakk>wadjust_on_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4248 |
\<Longrightarrow> wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4249 |
apply(auto simp: wadjust_on_left_moving_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4250 |
wadjust_goon_left_moving_O.simps exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4251 |
apply(rule_tac x = rs in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4252 |
apply(auto simp: exp_ind_def numeral_2_eq_2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4253 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4254 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4255 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4256 |
lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4257 |
wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4258 |
apply(simp add: wadjust_on_left_moving.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4259 |
wadjust_goon_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4260 |
apply(case_tac "hd c", simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4261 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4262 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4263 |
lemma [simp]: "wadjust_on_left_moving m rs (c, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4264 |
wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4265 |
apply(simp add: wadjust_on_left_moving.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4266 |
wadjust_goon_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4267 |
apply(case_tac "hd c", simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4268 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4269 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4270 |
lemma [simp]: "wadjust_goon_left_moving_B m rs (c, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4271 |
apply(auto simp: wadjust_goon_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4272 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4273 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4274 |
lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Bk\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4275 |
\<Longrightarrow> wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4276 |
apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4277 |
apply(case_tac [!] ml, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4278 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4279 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4280 |
lemma [simp]: "\<lbrakk>wadjust_goon_left_moving_O m rs (c, Oc # list); hd c = Oc\<rbrakk> \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4281 |
wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4282 |
apply(auto simp: wadjust_goon_left_moving_O.simps wadjust_goon_left_moving_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4283 |
apply(rule_tac x = "ml - 1" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4284 |
apply(case_tac ml, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4285 |
apply(rule_tac x = "Suc mr" in exI, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4286 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4287 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4288 |
lemma [simp]: "wadjust_goon_left_moving m rs (c, Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4289 |
wadjust_goon_left_moving m rs (tl c, hd c # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4290 |
apply(simp add: wadjust_goon_left_moving.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4291 |
apply(case_tac "hd c", simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4292 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4293 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4294 |
lemma [simp]: "wadjust_backto_standard_pos_B m rs (c, Oc # list) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4295 |
apply(simp add: wadjust_backto_standard_pos_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4296 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4297 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4298 |
lemma [simp]: "wadjust_backto_standard_pos_O m rs (c, Bk # xs) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4299 |
apply(simp add: wadjust_backto_standard_pos_O.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4300 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4301 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4302 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4303 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4304 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4305 |
lemma [simp]: "wadjust_backto_standard_pos_O m rs ([], Oc # list) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4306 |
wadjust_backto_standard_pos_B m rs ([], Bk # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4307 |
apply(auto simp: wadjust_backto_standard_pos_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4308 |
wadjust_backto_standard_pos_B.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4309 |
apply(rule_tac x = rn in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4310 |
apply(case_tac ml, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4311 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4312 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4313 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4314 |
lemma [simp]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4315 |
"\<lbrakk>wadjust_backto_standard_pos_O m rs (c, Oc # list); c \<noteq> []; hd c = Bk\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4316 |
\<Longrightarrow> wadjust_backto_standard_pos_B m rs (tl c, Bk # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4317 |
apply(simp add:wadjust_backto_standard_pos_O.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4318 |
wadjust_backto_standard_pos_B.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4319 |
apply(case_tac [!] ml, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4320 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4321 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4322 |
lemma [simp]: "\<lbrakk>wadjust_backto_standard_pos_O m rs (c, Oc # list); c \<noteq> []; hd c = Oc\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4323 |
\<Longrightarrow> wadjust_backto_standard_pos_O m rs (tl c, Oc # Oc # list)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4324 |
apply(simp add: wadjust_backto_standard_pos_O.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4325 |
apply(case_tac ml, simp_all add: exp_ind_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4326 |
apply(rule_tac x = nat in exI, auto simp: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4327 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4328 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4329 |
lemma [simp]: "wadjust_backto_standard_pos m rs (c, Oc # list) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4330 |
\<Longrightarrow> (c = [] \<longrightarrow> wadjust_backto_standard_pos m rs ([], Bk # Oc # list)) \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4331 |
(c \<noteq> [] \<longrightarrow> wadjust_backto_standard_pos m rs (tl c, hd c # Oc # list))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4332 |
apply(auto simp: wadjust_backto_standard_pos.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4333 |
apply(case_tac "hd c", simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4334 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4335 |
thm wadjust_loop_right_move.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4336 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4337 |
lemma [simp]: "wadjust_loop_right_move m rs (c, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4338 |
apply(simp only: wadjust_loop_right_move.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4339 |
apply(rule_tac iffI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4340 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4341 |
apply(case_tac nr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4342 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4343 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4344 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4345 |
lemma [simp]: "wadjust_loop_erase m rs (c, []) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4346 |
apply(simp only: wadjust_loop_erase.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4347 |
apply(case_tac mr, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4348 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4349 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4350 |
lemma [simp]: "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Bk # list)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4351 |
\<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4352 |
< a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4353 |
a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4354 |
a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4355 |
apply(simp only: wadjust_loop_erase.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4356 |
apply(rule_tac disjI2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4357 |
apply(case_tac c, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4358 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4359 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4360 |
lemma [simp]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4361 |
"\<lbrakk>Suc (Suc rs) = a; wadjust_loop_on_left_moving m rs (c, Bk # list)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4362 |
\<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4363 |
< a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4364 |
a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Bk # list)))) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4365 |
a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4366 |
apply(subgoal_tac "c \<noteq> []") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4367 |
apply(case_tac c, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4368 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4369 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4370 |
lemma dropWhile_exp1: "dropWhile (\<lambda>a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = dropWhile (\<lambda>a. a = Oc) xs" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4371 |
apply(induct n, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4372 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4373 |
lemma takeWhile_exp1: "takeWhile (\<lambda>a. a = Oc) (Oc\<^bsup>n\<^esup> @ xs) = Oc\<^bsup>n\<^esup> @ takeWhile (\<lambda>a. a = Oc) xs" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4374 |
apply(induct n, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4375 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4376 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4377 |
lemma [simp]: "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_right_move2 m rs (c, Bk # list)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4378 |
\<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list)))) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4379 |
< a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list))))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4380 |
apply(simp add: wadjust_loop_right_move2.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4381 |
apply(simp add: dropWhile_exp1 takeWhile_exp1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4382 |
apply(case_tac ln, simp, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4383 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4384 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4385 |
lemma [simp]: "wadjust_loop_check m rs ([], b) = False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4386 |
apply(simp add: wadjust_loop_check.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4387 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4388 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4389 |
lemma [simp]: "\<lbrakk>Suc (Suc rs) = a; wadjust_loop_check m rs (c, Oc # list)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4390 |
\<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4391 |
< a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list)))) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4392 |
a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev (tl c) @ hd c # Oc # list)))) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4393 |
a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4394 |
apply(case_tac "c", simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4395 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4396 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4397 |
lemma [simp]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4398 |
"\<lbrakk>Suc (Suc rs) = a; wadjust_loop_erase m rs (c, Oc # list)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4399 |
\<Longrightarrow> a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4400 |
< a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list)))) \<or> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4401 |
a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Bk # list)))) = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4402 |
a - length (takeWhile (\<lambda>a. a = Oc) (tl (dropWhile (\<lambda>a. a = Oc) (rev c @ Oc # list))))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4403 |
apply(simp add: wadjust_loop_erase.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4404 |
apply(rule_tac disjI2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4405 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4406 |
apply(simp add: dropWhile_exp1 takeWhile_exp1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4407 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4408 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4409 |
declare numeral_2_eq_2[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4410 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4411 |
lemma wadjust_correctness: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4412 |
shows "let P = (\<lambda> (len, st, l, r). st = 0) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4413 |
let Q = (\<lambda> (len, st, l, r). wadjust_inv st m rs (l, r)) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4414 |
let f = (\<lambda> stp. (Suc (Suc rs), steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4415 |
Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4416 |
\<exists> n .P (f n) \<and> Q (f n)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4417 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4418 |
let ?P = "(\<lambda> (len, st, l, r). st = 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4419 |
let ?Q = "\<lambda> (len, st, l, r). wadjust_inv st m rs (l, r)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4420 |
let ?f = "\<lambda> stp. (Suc (Suc rs), steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4421 |
Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4422 |
have "\<exists> n. ?P (?f n) \<and> ?Q (?f n)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4423 |
proof(rule_tac halt_lemma2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4424 |
show "wf wadjust_le" by auto |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4425 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4426 |
show "\<forall> n. \<not> ?P (?f n) \<and> ?Q (?f n) \<longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4427 |
?Q (?f (Suc n)) \<and> (?f (Suc n), ?f n) \<in> wadjust_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4428 |
proof(rule_tac allI, rule_tac impI, case_tac "?f n", |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4429 |
simp add: tstep_red tstep.simps, rule_tac conjI, erule_tac conjE, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4430 |
erule_tac conjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4431 |
fix n a b c d |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4432 |
assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4433 |
thus "case case fetch t_wcode_adjust b (case d of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4434 |
of (ac, ns) \<Rightarrow> (ns, new_tape ac (c, d)) of (st, x) \<Rightarrow> wadjust_inv st m rs x" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4435 |
apply(case_tac d, simp, case_tac [2] aa) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4436 |
apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4437 |
abacus.lex_triple_def abacus.lex_pair_def lex_square_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4438 |
split: if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4439 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4440 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4441 |
fix n a b c d |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4442 |
assume "0 < b \<and> wadjust_inv b m rs (c, d)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4443 |
"Suc (Suc rs) = a \<and> steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4444 |
Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust n = (b, c, d)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4445 |
thus "((a, case fetch t_wcode_adjust b (case d of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4446 |
of (ac, ns) \<Rightarrow> (ns, new_tape ac (c, d))), a, b, c, d) \<in> wadjust_le" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4447 |
proof(erule_tac conjE, erule_tac conjE, erule_tac conjE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4448 |
assume "0 < b" "wadjust_inv b m rs (c, d)" "Suc (Suc rs) = a" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4449 |
thus "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4450 |
apply(case_tac d, case_tac [2] aa) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4451 |
apply(simp_all add: wadjust_inv.simps wadjust_le_def new_tape.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4452 |
abacus.lex_triple_def abacus.lex_pair_def lex_square_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4453 |
split: if_splits) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4454 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4455 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4456 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4457 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4458 |
show "?Q (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4459 |
apply(simp add: steps.simps wadjust_inv.simps wadjust_start.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4460 |
apply(rule_tac x = ln in exI,auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4461 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4462 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4463 |
show "\<not> ?P (?f 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4464 |
apply(simp add: steps.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4465 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4466 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4467 |
thus "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4468 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4469 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4470 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4471 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4472 |
lemma [intro]: "t_correct t_wcode_adjust" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4473 |
apply(auto simp: t_wcode_adjust_def t_correct.simps iseven_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4474 |
apply(rule_tac x = 11 in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4475 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4476 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4477 |
lemma wcode_lemma_pre': |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4478 |
"args \<noteq> [] \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4479 |
\<exists> stp rn. steps (Suc 0, [], <m # args>) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4480 |
((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4481 |
= (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4482 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4483 |
let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <m # args>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4484 |
let ?Q1 = "\<lambda>(l, r). l = Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4485 |
(\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4486 |
let ?P2 = ?Q1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4487 |
let ?Q2 = "\<lambda> (l, r). (wadjust_stop m (bl_bin (<args>) - 1) (l, r))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4488 |
let ?P3 = "\<lambda> tp. False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4489 |
assume h: "args \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4490 |
have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4491 |
((t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust) stp = (0, tp') \<and> ?Q2 tp')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4492 |
proof(rule_tac turing_merge.t_merge_halt[of "t_wcode_prepare |+| t_wcode_main" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4493 |
t_wcode_adjust ?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4494 |
auto simp: turing_merge_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4495 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4496 |
show "\<exists>stp. case steps (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp of |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4497 |
(st, tp') \<Rightarrow> st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = Bk # Oc\<^bsup>Suc m\<^esup> \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4498 |
(\<exists>ln rn. r = Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4499 |
using h prepare_mainpart_lemma[of args m] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4500 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4501 |
apply(rule_tac x = stp in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4502 |
apply(rule_tac x = ln in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4503 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4504 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4505 |
fix ln rn |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4506 |
show "\<exists>stp. case steps (Suc 0, Bk # Oc\<^bsup>Suc m\<^esup>, Bk # Oc # Bk\<^bsup>ln\<^esup> @ Bk # Bk # |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4507 |
Oc\<^bsup>bl_bin (<args>)\<^esup> @ Bk\<^bsup>rn\<^esup>) t_wcode_adjust stp of |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4508 |
(st, tp') \<Rightarrow> st = 0 \<and> wadjust_stop m (bl_bin (<args>) - Suc 0) tp'" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4509 |
using wadjust_correctness[of m "bl_bin (<args>) - 1" "Suc ln" rn] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4510 |
apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_inv.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4511 |
apply(rule_tac x = n in exI, simp add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4512 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4513 |
apply(case_tac args, simp_all, case_tac list, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4514 |
simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4515 |
bl_bin.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4516 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4517 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4518 |
show "?Q1 \<turnstile>-> ?P2" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4519 |
by(simp add: t_imply_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4520 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4521 |
thus "\<exists>stp rn. steps (Suc 0, [], <m # args>) ((t_wcode_prepare |+| t_wcode_main) |+| |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4522 |
t_wcode_adjust) stp = (0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4523 |
apply(simp add: t_imply_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4524 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4525 |
apply(subgoal_tac "bl_bin (<args>) > 0", auto simp: wadjust_stop.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4526 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4527 |
apply(case_tac args, simp_all, case_tac list, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4528 |
simp_all add: tape_of_nl_abv tape_of_nat_list.simps exp_ind_def |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4529 |
bl_bin.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4530 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4531 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4532 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4533 |
text {* |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4534 |
The initialization TM @{text "t_wcode"}. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4535 |
*} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4536 |
definition t_wcode :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4537 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4538 |
"t_wcode = (t_wcode_prepare |+| t_wcode_main) |+| t_wcode_adjust" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4539 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4540 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4541 |
text {* |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4542 |
The correctness of @{text "t_wcode"}. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4543 |
*} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4544 |
lemma wcode_lemma_1: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4545 |
"args \<noteq> [] \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4546 |
\<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4547 |
(0, [Bk], Oc\<^bsup>Suc m\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4548 |
apply(simp add: wcode_lemma_pre' t_wcode_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4549 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4550 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4551 |
lemma wcode_lemma: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4552 |
"args \<noteq> [] \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4553 |
\<exists> stp ln rn. steps (Suc 0, [], <m # args>) (t_wcode) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4554 |
(0, [Bk], <[m ,bl_bin (<args>)]> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4555 |
using wcode_lemma_1[of args m] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4556 |
apply(simp add: t_wcode_def tape_of_nl_abv tape_of_nat_list.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4557 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4558 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4559 |
section {* The universal TM *} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4560 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4561 |
text {* |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4562 |
This section gives the explicit construction of {\em Universal Turing Machine}, defined as @{text "UTM"} and proves its |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4563 |
correctness. It is pretty easy by composing the partial results we have got so far. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4564 |
*} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4565 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4566 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4567 |
definition UTM :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4568 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4569 |
"UTM = (let (aprog, rs_pos, a_md) = rec_ci rec_F in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4570 |
let abc_F = aprog [+] dummy_abc (Suc (Suc 0)) in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4571 |
(t_wcode |+| (tm_of abc_F @ tMp (Suc (Suc 0)) (start_of (layout_of abc_F) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4572 |
(length abc_F) - Suc 0))))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4573 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4574 |
definition F_aprog :: "abc_prog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4575 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4576 |
"F_aprog \<equiv> (let (aprog, rs_pos, a_md) = rec_ci rec_F in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4577 |
aprog [+] dummy_abc (Suc (Suc 0)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4578 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4579 |
definition F_tprog :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4580 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4581 |
"F_tprog = tm_of (F_aprog)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4582 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4583 |
definition t_utm :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4584 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4585 |
"t_utm \<equiv> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4586 |
(F_tprog) @ tMp (Suc (Suc 0)) (start_of (layout_of (F_aprog)) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4587 |
(length (F_aprog)) - Suc 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4588 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4589 |
definition UTM_pre :: "tprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4590 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4591 |
"UTM_pre = t_wcode |+| t_utm" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4592 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4593 |
lemma F_abc_halt_eq: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4594 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4595 |
length lm = k; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4596 |
steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4597 |
rs > 0\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4598 |
\<Longrightarrow> \<exists> stp m. abc_steps_l (0, [code tp, bl2wc (<lm>)]) (F_aprog) stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4599 |
(length (F_aprog), code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4600 |
apply(drule_tac F_t_halt_eq, simp, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4601 |
apply(case_tac "rec_ci rec_F") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4602 |
apply(frule_tac abc_append_dummy_complie, simp, simp, erule_tac exE, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4603 |
erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4604 |
apply(rule_tac x = stp in exI, rule_tac x = m in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4605 |
apply(simp add: F_aprog_def dummy_abc_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4606 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4607 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4608 |
lemma F_abc_utm_halt_eq: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4609 |
"\<lbrakk>rs > 0; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4610 |
abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4611 |
(length F_aprog, code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4612 |
\<Longrightarrow> \<exists>stp m n.(steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4613 |
(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4614 |
thm abacus_turing_eq_halt |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4615 |
using abacus_turing_eq_halt |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4616 |
[of "layout_of F_aprog" "F_aprog" "F_tprog" "length (F_aprog)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4617 |
"[code tp, bl2wc (<lm>)]" stp "code tp # bl2wc (<lm>) # (rs - 1) # 0\<^bsup>m\<^esup>" "Suc (Suc 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4618 |
"start_of (layout_of (F_aprog)) (length (F_aprog))" "[]" 0] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4619 |
apply(simp add: F_tprog_def t_utm_def abc_lm_v.simps nth_append) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4620 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4621 |
apply(rule_tac x = stpa in exI, rule_tac x = "Suc (Suc ma)" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4622 |
rule_tac x = l in exI, simp add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4623 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4624 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4625 |
declare tape_of_nl_abv_cons[simp del] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4626 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4627 |
lemma t_utm_halt_eq': |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4628 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4629 |
0 < rs; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4630 |
steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4631 |
\<Longrightarrow> \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4632 |
(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4633 |
apply(drule_tac l = l in F_abc_halt_eq, simp, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4634 |
apply(erule_tac exE, erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4635 |
apply(rule_tac F_abc_utm_halt_eq, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4636 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4637 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4638 |
lemma [simp]: "tinres xs (xs @ Bk\<^bsup>i\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4639 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4640 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4641 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4642 |
lemma [elim]: "\<lbrakk>rs > 0; Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup> = c @ Bk\<^bsup>n\<^esup>\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4643 |
\<Longrightarrow> \<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4644 |
apply(case_tac "na > n") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4645 |
apply(subgoal_tac "\<exists> d. na = d + n", auto simp: exp_add) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4646 |
apply(rule_tac x = "na - n" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4647 |
apply(subgoal_tac "\<exists> d. n = d + na", auto simp: exp_add) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4648 |
apply(case_tac rs, simp_all add: exp_ind, case_tac d, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4649 |
simp_all add: exp_ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4650 |
apply(rule_tac x = "n - na" in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4651 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4652 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4653 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4654 |
lemma t_utm_halt_eq'': |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4655 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4656 |
0 < rs; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4657 |
steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4658 |
\<Longrightarrow> \<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4659 |
(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4660 |
apply(drule_tac t_utm_halt_eq', simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4661 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4662 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4663 |
fix stpa ma na |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4664 |
assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4665 |
and gr: "rs > 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4666 |
thus "\<exists>stp m n. steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4667 |
apply(rule_tac x = stpa in exI, rule_tac x = ma in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4668 |
proof(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4669 |
fix a b c |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4670 |
assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4671 |
"steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4672 |
thus " a = 0 \<and> b = Bk\<^bsup>ma\<^esup> \<and> (\<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4673 |
using tinres_steps2[of "<[code tp, bl2wc (<lm>)]>" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4674 |
"Suc 0" " [Bk, Bk]" t_utm stpa 0 "Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4675 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4676 |
using gr |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4677 |
apply(simp only: tinres_def, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4678 |
apply(rule_tac x = "na + n" in exI, simp add: exp_add) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4679 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4680 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4681 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4682 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4683 |
lemma [simp]: "tinres [Bk, Bk] [Bk]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4684 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4685 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4686 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4687 |
lemma [elim]: "Bk\<^bsup>ma\<^esup> = b @ Bk\<^bsup>n\<^esup> \<Longrightarrow> \<exists>m. b = Bk\<^bsup>m\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4688 |
apply(subgoal_tac "ma = length b + n") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4689 |
apply(rule_tac x = "ma - n" in exI, simp add: exp_add) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4690 |
apply(drule_tac length_equal) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4691 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4692 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4693 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4694 |
lemma t_utm_halt_eq: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4695 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4696 |
0 < rs; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4697 |
steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>n\<^esup>)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4698 |
\<Longrightarrow> \<exists>stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4699 |
(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4700 |
apply(drule_tac i = i in t_utm_halt_eq'', simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4701 |
apply(erule_tac exE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4702 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4703 |
fix stpa ma na |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4704 |
assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4705 |
and gr: "rs > 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4706 |
thus "\<exists>stp m n. steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4707 |
apply(rule_tac x = stpa in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4708 |
proof(case_tac "steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4709 |
fix a b c |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4710 |
assume "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (0, Bk\<^bsup>ma\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4711 |
"steps (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>) t_utm stpa = (a, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4712 |
thus "a = 0 \<and> (\<exists>m. b = Bk\<^bsup>m\<^esup>) \<and> (\<exists>n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4713 |
using tinres_steps[of "[Bk, Bk]" "[Bk]" "Suc 0" "<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>i\<^esup>" t_utm stpa 0 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4714 |
"Bk\<^bsup>ma\<^esup>" "Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>na\<^esup>" a b c] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4715 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4716 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4717 |
apply(rule_tac x = "ma + n" in exI, simp add: exp_add) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4718 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4719 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4720 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4721 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4722 |
lemma [intro]: "t_correct t_wcode" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4723 |
apply(simp add: t_wcode_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4724 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4725 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4726 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4727 |
lemma [intro]: "t_correct t_utm" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4728 |
apply(simp add: t_utm_def F_tprog_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4729 |
apply(rule_tac t_compiled_correct, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4730 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4731 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4732 |
lemma UTM_halt_lemma_pre: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4733 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4734 |
0 < rs; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4735 |
args \<noteq> []; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4736 |
steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4737 |
\<Longrightarrow> \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM_pre stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4738 |
(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4739 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4740 |
let ?Q2 = "\<lambda> (l, r). (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> \<and> r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4741 |
term ?Q2 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4742 |
let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4743 |
let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4744 |
(\<exists> rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4745 |
let ?P2 = ?Q1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4746 |
let ?P3 = "\<lambda> (l, r). False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4747 |
assume h: "turing_basic.t_correct tp" "0 < rs" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4748 |
"args \<noteq> []" "steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4749 |
have "?P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4750 |
(t_wcode |+| t_utm) stp = (0, tp') \<and> ?Q2 tp')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4751 |
proof(rule_tac turing_merge.t_merge_halt [of "t_wcode" "t_utm" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4752 |
?P1 ?P2 ?P3 ?P3 ?Q1 ?Q2], auto simp: turing_merge_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4753 |
show "\<exists>stp. case steps (Suc 0, [], <code tp # args>) t_wcode stp of (st, tp') \<Rightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4754 |
st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = [Bk] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4755 |
(\<exists>rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4756 |
using wcode_lemma_1[of args "code tp"] h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4757 |
apply(simp, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4758 |
apply(rule_tac x = stpa in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4759 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4760 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4761 |
fix rn |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4762 |
show "\<exists>stp. case steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4763 |
Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp of |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4764 |
(st, tp') \<Rightarrow> st = 0 \<and> (case tp' of (l, r) \<Rightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4765 |
(\<exists>ln. l = Bk\<^bsup>ln\<^esup>) \<and> (\<exists>rn. r = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>rn\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4766 |
using t_utm_halt_eq[of tp rs i args stp m k rn] h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4767 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4768 |
apply(rule_tac x = stpa in exI, simp add: bin_wc_eq |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4769 |
tape_of_nat_list.simps tape_of_nl_abv) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4770 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4771 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4772 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4773 |
show "?Q1 \<turnstile>-> ?P2" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4774 |
apply(simp add: t_imply_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4775 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4776 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4777 |
thus "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4778 |
apply(simp add: t_imply_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4779 |
apply(auto simp: UTM_pre_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4780 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4781 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4782 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4783 |
text {* |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4784 |
The correctness of @{text "UTM"}, the halt case. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4785 |
*} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4786 |
lemma UTM_halt_lemma: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4787 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4788 |
0 < rs; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4789 |
args \<noteq> []; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4790 |
steps (Suc 0, Bk\<^bsup>i\<^esup>, <args::nat list>) tp stp = (0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup>@Bk\<^bsup>k\<^esup>)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4791 |
\<Longrightarrow> \<exists>stp m n. steps (Suc 0, [], <code tp # args>) UTM stp = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4792 |
(0, Bk\<^bsup>m\<^esup>, Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4793 |
using UTM_halt_lemma_pre[of tp rs args i stp m k] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4794 |
apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4795 |
apply(case_tac "rec_ci rec_F", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4796 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4797 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4798 |
definition TSTD:: "t_conf \<Rightarrow> bool" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4799 |
where |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4800 |
"TSTD c = (let (st, l, r) = c in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4801 |
st = 0 \<and> (\<exists> m. l = Bk\<^bsup>m\<^esup>) \<and> (\<exists> rs n. r = Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4802 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4803 |
thm abacus_turing_eq_uhalt |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4804 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4805 |
lemma nstd_case1: "0 < a \<Longrightarrow> NSTD (trpl_code (a, b, c))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4806 |
apply(simp add: NSTD.simps trpl_code.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4807 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4808 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4809 |
lemma [simp]: "\<forall>m. b \<noteq> Bk\<^bsup>m\<^esup> \<Longrightarrow> 0 < bl2wc b" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4810 |
apply(rule classical, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4811 |
apply(induct b, erule_tac x = 0 in allE, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4812 |
apply(simp add: bl2wc.simps, case_tac a, simp_all |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4813 |
add: bl2nat.simps bl2nat_double) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4814 |
apply(case_tac "\<exists> m. b = Bk\<^bsup>m\<^esup>", erule exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4815 |
apply(erule_tac x = "Suc m" in allE, simp add: exp_ind_def, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4816 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4817 |
lemma nstd_case2: "\<forall>m. b \<noteq> Bk\<^bsup>m\<^esup> \<Longrightarrow> NSTD (trpl_code (a, b, c))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4818 |
apply(simp add: NSTD.simps trpl_code.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4819 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4820 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4821 |
thm lg.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4822 |
thm lgR.simps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4823 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4824 |
lemma [elim]: "Suc (2 * x) = 2 * y \<Longrightarrow> RR" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4825 |
apply(induct x arbitrary: y, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4826 |
apply(case_tac y, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4827 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4828 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4829 |
lemma bl2nat_zero_eq[simp]: "(bl2nat c 0 = 0) = (\<exists>n. c = Bk\<^bsup>n\<^esup>)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4830 |
apply(auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4831 |
apply(induct c, simp add: bl2nat.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4832 |
apply(rule_tac x = 0 in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4833 |
apply(case_tac a, auto simp: bl2nat.simps bl2nat_double) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4834 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4835 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4836 |
lemma bl2wc_exp_ex: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4837 |
"\<lbrakk>Suc (bl2wc c) = 2 ^ m\<rbrakk> \<Longrightarrow> \<exists> rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4838 |
apply(induct c arbitrary: m, simp add: bl2wc.simps bl2nat.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4839 |
apply(case_tac a, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4840 |
apply(case_tac m, simp_all add: bl2wc.simps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4841 |
apply(rule_tac x = 0 in exI, rule_tac x = "Suc n" in exI, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4842 |
simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4843 |
apply(simp add: bl2wc.simps bl2nat.simps bl2nat_double) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4844 |
apply(case_tac m, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4845 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4846 |
fix c m nat |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4847 |
assume ind: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4848 |
"\<And>m. Suc (bl2nat c 0) = 2 ^ m \<Longrightarrow> \<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4849 |
and h: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4850 |
"Suc (Suc (2 * bl2nat c 0)) = 2 * 2 ^ nat" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4851 |
have "\<exists>rs n. c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4852 |
apply(rule_tac m = nat in ind) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4853 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4854 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4855 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4856 |
from this obtain rs n where " c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" by blast |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4857 |
thus "\<exists>rs n. Oc # c = Oc\<^bsup>rs\<^esup> @ Bk\<^bsup>n\<^esup>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4858 |
apply(rule_tac x = "Suc rs" in exI, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4859 |
apply(rule_tac x = n in exI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4860 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4861 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4862 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4863 |
lemma [elim]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4864 |
"\<lbrakk>\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup>; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4865 |
bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0\<rbrakk> \<Longrightarrow> bl2wc c = 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4866 |
apply(subgoal_tac "\<exists> m. Suc (bl2wc c) = 2^m", erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4867 |
apply(drule_tac bl2wc_exp_ex, simp, erule_tac exE, erule_tac exE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4868 |
apply(case_tac rs, simp, simp, erule_tac x = nat in allE, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4869 |
erule_tac x = n in allE, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4870 |
using bl2wc_exp_ex[of c "lg (Suc (bl2wc c)) 2"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4871 |
apply(case_tac "(2::nat) ^ lg (Suc (bl2wc c)) 2", |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4872 |
simp, simp, erule_tac exE, erule_tac exE, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4873 |
apply(simp add: bl2wc.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4874 |
apply(rule_tac x = rs in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4875 |
apply(case_tac "(2::nat)^rs", simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4876 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4877 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4878 |
lemma nstd_case3: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4879 |
"\<forall>rs n. c \<noteq> Oc\<^bsup>Suc rs\<^esup> @ Bk\<^bsup>n\<^esup> \<Longrightarrow> NSTD (trpl_code (a, b, c))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4880 |
apply(simp add: NSTD.simps trpl_code.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4881 |
apply(rule_tac impI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4882 |
apply(rule_tac disjI2, rule_tac disjI2, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4883 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4884 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4885 |
lemma NSTD_1: "\<not> TSTD (a, b, c) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4886 |
\<Longrightarrow> rec_exec rec_NSTD [trpl_code (a, b, c)] = Suc 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4887 |
using NSTD_lemma1[of "trpl_code (a, b, c)"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4888 |
NSTD_lemma2[of "trpl_code (a, b, c)"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4889 |
apply(simp add: TSTD_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4890 |
apply(erule_tac disjE, erule_tac nstd_case1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4891 |
apply(erule_tac disjE, erule_tac nstd_case2) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4892 |
apply(erule_tac nstd_case3) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4893 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4894 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4895 |
lemma nonstop_t_uhalt_eq: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4896 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4897 |
steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp = (a, b, c); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4898 |
\<not> TSTD (a, b, c)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4899 |
\<Longrightarrow> rec_exec rec_nonstop [code tp, bl2wc (<lm>), stp] = Suc 0" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4900 |
apply(simp add: rec_nonstop_def rec_exec.simps) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4901 |
apply(subgoal_tac |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4902 |
"rec_exec rec_conf [code tp, bl2wc (<lm>), stp] = |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4903 |
trpl_code (a, b, c)", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4904 |
apply(erule_tac NSTD_1) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4905 |
using rec_t_eq_steps[of tp l lm stp] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4906 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4907 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4908 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4909 |
lemma nonstop_true: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4910 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4911 |
\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4912 |
\<Longrightarrow> \<forall>y. rec_calc_rel rec_nonstop |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4913 |
([code tp, bl2wc (<lm>), y]) (Suc 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4914 |
apply(rule_tac allI, erule_tac x = y in allE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4915 |
apply(case_tac "steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp y", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4916 |
apply(rule_tac nonstop_t_uhalt_eq, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4917 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4918 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4919 |
(* |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4920 |
lemma [simp]: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4921 |
"\<forall>j<Suc k. Ex (rec_calc_rel (get_fstn_args (Suc k) (Suc k) ! j) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4922 |
(code tp # lm))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4923 |
apply(auto simp: get_fstn_args_nth) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4924 |
apply(rule_tac x = "(code tp # lm) ! j" in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4925 |
apply(rule_tac calc_id, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4926 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4927 |
*) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4928 |
declare ci_cn_para_eq[simp] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4929 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4930 |
lemma F_aprog_uhalt: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4931 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4932 |
\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp)); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4933 |
rec_ci rec_F = (F_ap, rs_pos, a_md)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4934 |
\<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)] @ 0\<^bsup>a_md - rs_pos \<^esup> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4935 |
@ suflm) (F_ap) stp of (ss, e) \<Rightarrow> ss < length (F_ap)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4936 |
apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4937 |
([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4938 |
apply(simp only: rec_F_def, rule_tac i = 0 and ga = a and gb = b and |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4939 |
gc = c in cn_gi_uhalt, simp, simp, simp, simp, simp, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4940 |
apply(simp add: ci_cn_para_eq) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4941 |
apply(case_tac "rec_ci (Cn (Suc (Suc 0)) rec_conf |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4942 |
([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4943 |
apply(rule_tac rf = "(Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4944 |
([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])])" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4945 |
and n = "Suc (Suc 0)" and f = rec_right and |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4946 |
gs = "[Cn (Suc (Suc 0)) rec_conf |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4947 |
([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4948 |
and i = 0 and ga = aa and gb = ba and gc = ca in |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4949 |
cn_gi_uhalt) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4950 |
apply(simp, simp, simp, simp, simp, simp, simp, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4951 |
simp add: ci_cn_para_eq) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4952 |
apply(case_tac "rec_ci rec_halt") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4953 |
apply(rule_tac rf = "(Cn (Suc (Suc 0)) rec_conf |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4954 |
([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt]))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4955 |
and n = "Suc (Suc 0)" and f = "rec_conf" and |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4956 |
gs = "([id (Suc (Suc 0)) 0, id (Suc (Suc 0)) (Suc 0), rec_halt])" and |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4957 |
i = "Suc (Suc 0)" and gi = "rec_halt" and ga = ab and gb = bb and |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4958 |
gc = cb in cn_gi_uhalt) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4959 |
apply(simp, simp, simp, simp, simp add: nth_append, simp, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4960 |
simp add: nth_append, simp add: rec_halt_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4961 |
apply(simp only: rec_halt_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4962 |
apply(case_tac [!] "rec_ci ((rec_nonstop))") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4963 |
apply(rule_tac allI, rule_tac impI, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4964 |
apply(case_tac j, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4965 |
apply(rule_tac x = "code tp" in exI, rule_tac calc_id, simp, simp, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4966 |
apply(rule_tac x = "bl2wc (<lm>)" in exI, rule_tac calc_id, simp, simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4967 |
apply(rule_tac rf = "Mn (Suc (Suc 0)) (rec_nonstop)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4968 |
and f = "(rec_nonstop)" and n = "Suc (Suc 0)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4969 |
and aprog' = ac and rs_pos' = bc and a_md' = cc in Mn_unhalt) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4970 |
apply(simp, simp add: rec_halt_def , simp, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4971 |
apply(drule_tac nonstop_true, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4972 |
apply(rule_tac allI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4973 |
apply(erule_tac x = y in allE)+ |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4974 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4975 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4976 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4977 |
thm abc_list_crsp_steps |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4978 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4979 |
lemma uabc_uhalt': |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4980 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4981 |
\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp)); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4982 |
rec_ci rec_F = (ap, pos, md)\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4983 |
\<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp of (ss, e) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4984 |
\<Rightarrow> ss < length ap" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4985 |
proof(frule_tac F_ap = ap and rs_pos = pos and a_md = md |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4986 |
and suflm = "[]" in F_aprog_uhalt, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4987 |
fix stp a b |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4988 |
assume h: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4989 |
"\<forall>stp. case abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp of |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4990 |
(ss, e) \<Rightarrow> ss < length ap" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4991 |
"abc_steps_l (0, [code tp, bl2wc (<lm>)]) ap stp = (a, b)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4992 |
"turing_basic.t_correct tp" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4993 |
"rec_ci rec_F = (ap, pos, md)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4994 |
moreover have "ap \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4995 |
using h apply(rule_tac rec_ci_not_null, simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4996 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4997 |
ultimately show "a < length ap" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4998 |
proof(erule_tac x = stp in allE, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
4999 |
case_tac "abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5000 |
fix aa ba |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5001 |
assume g: "aa < length ap" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5002 |
"abc_steps_l (0, code tp # bl2wc (<lm>) # 0\<^bsup>md - pos\<^esup>) ap stp = (aa, ba)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5003 |
"ap \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5004 |
thus "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5005 |
using abc_list_crsp_steps[of "[code tp, bl2wc (<lm>)]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5006 |
"md - pos" ap stp aa ba] h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5007 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5008 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5009 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5010 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5011 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5012 |
lemma uabc_uhalt: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5013 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5014 |
\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5015 |
\<Longrightarrow> \<forall> stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) F_aprog |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5016 |
stp of (ss, e) \<Rightarrow> ss < length F_aprog" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5017 |
apply(case_tac "rec_ci rec_F", simp add: F_aprog_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5018 |
thm uabc_uhalt' |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5019 |
apply(drule_tac ap = a and pos = b and md = c in uabc_uhalt', simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5020 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5021 |
fix a b c |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5022 |
assume |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5023 |
"\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) a stp of (ss, e) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5024 |
\<Rightarrow> ss < length a" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5025 |
"rec_ci rec_F = (a, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5026 |
thus |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5027 |
"\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5028 |
(a [+] dummy_abc (Suc (Suc 0))) stp of (ss, e) \<Rightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5029 |
ss < Suc (Suc (Suc (length a)))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5030 |
using abc_append_uhalt1[of a "[code tp, bl2wc (<lm>)]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5031 |
"a [+] dummy_abc (Suc (Suc 0))" "[]" "dummy_abc (Suc (Suc 0))"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5032 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5033 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5034 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5035 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5036 |
lemma tutm_uhalt': |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5037 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5038 |
\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <lm>) tp stp))\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5039 |
\<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5040 |
using abacus_turing_eq_uhalt[of "layout_of (F_aprog)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5041 |
"F_aprog" "F_tprog" "[code tp, bl2wc (<lm>)]" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5042 |
"start_of (layout_of (F_aprog )) (length (F_aprog))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5043 |
"Suc (Suc 0)"] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5044 |
apply(simp add: F_tprog_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5045 |
apply(subgoal_tac "\<forall>stp. case abc_steps_l (0, [code tp, bl2wc (<lm>)]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5046 |
(F_aprog) stp of (as, am) \<Rightarrow> as < length (F_aprog)", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5047 |
thm abacus_turing_eq_uhalt |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5048 |
apply(simp add: t_utm_def F_tprog_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5049 |
apply(rule_tac uabc_uhalt, simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5050 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5051 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5052 |
lemma tinres_commute: "tinres r r' \<Longrightarrow> tinres r' r" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5053 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5054 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5055 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5056 |
lemma inres_tape: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5057 |
"\<lbrakk>steps (st, l, r) tp stp = (a, b, c); steps (st, l', r') tp stp = (a', b', c'); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5058 |
tinres l l'; tinres r r'\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5059 |
\<Longrightarrow> a = a' \<and> tinres b b' \<and> tinres c c'" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5060 |
proof(case_tac "steps (st, l', r) tp stp") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5061 |
fix aa ba ca |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5062 |
assume h: "steps (st, l, r) tp stp = (a, b, c)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5063 |
"steps (st, l', r') tp stp = (a', b', c')" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5064 |
"tinres l l'" "tinres r r'" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5065 |
"steps (st, l', r) tp stp = (aa, ba, ca)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5066 |
have "tinres b ba \<and> c = ca \<and> a = aa" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5067 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5068 |
apply(rule_tac tinres_steps, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5069 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5070 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5071 |
thm tinres_steps2 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5072 |
moreover have "b' = ba \<and> tinres c' ca \<and> a' = aa" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5073 |
using h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5074 |
apply(rule_tac tinres_steps2, auto intro: tinres_commute) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5075 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5076 |
ultimately show "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5077 |
apply(auto intro: tinres_commute) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5078 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5079 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5080 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5081 |
lemma tape_normalize: "\<forall> stp. \<not> isS0 (steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5082 |
\<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5083 |
apply(rule_tac allI, case_tac "(steps (Suc 0, Bk\<^bsup>m\<^esup>, |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5084 |
<[code tp, bl2wc (<lm>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)", simp add: isS0_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5085 |
apply(erule_tac x = stp in allE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5086 |
apply(case_tac "steps (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5087 |
apply(drule_tac inres_tape, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5088 |
apply(auto simp: tinres_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5089 |
apply(case_tac "m > Suc (Suc 0)") |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5090 |
apply(rule_tac x = "m - Suc (Suc 0)" in exI) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5091 |
apply(case_tac m, simp_all add: exp_ind_def, case_tac nat, simp_all add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5092 |
apply(rule_tac x = "2 - m" in exI, simp add: exp_ind_def[THEN sym] exp_add[THEN sym]) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5093 |
apply(simp only: numeral_2_eq_2, simp add: exp_ind_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5094 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5095 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5096 |
lemma tutm_uhalt: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5097 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5098 |
\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp))\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5099 |
\<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, Bk\<^bsup>m\<^esup>, <[code tp, bl2wc (<args>)]> @ Bk\<^bsup>n\<^esup>) t_utm stp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5100 |
apply(rule_tac tape_normalize) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5101 |
apply(rule_tac tutm_uhalt', simp_all) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5102 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5103 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5104 |
lemma UTM_uhalt_lemma_pre: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5105 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5106 |
\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp)); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5107 |
args \<noteq> []\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5108 |
\<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>) UTM_pre stp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5109 |
proof - |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5110 |
let ?P1 = "\<lambda> (l, r). l = [] \<and> r = <code tp # args>" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5111 |
let ?Q1 = "\<lambda> (l, r). (l = [Bk] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5112 |
(\<exists> rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5113 |
let ?P4 = ?Q1 |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5114 |
let ?P3 = "\<lambda> (l, r). False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5115 |
assume h: "turing_basic.t_correct tp" "\<forall>stp. \<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5116 |
"args \<noteq> []" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5117 |
have "?P1 \<turnstile>-> \<lambda> tp. \<not> (\<exists> stp. isS0 (steps (Suc 0, tp) (t_wcode |+| t_utm) stp))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5118 |
proof(rule_tac turing_merge.t_merge_uhalt [of "t_wcode" "t_utm" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5119 |
?P1 ?P3 ?P3 ?P4 ?Q1 ?P3], auto simp: turing_merge_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5120 |
show "\<exists>stp. case steps (Suc 0, [], <code tp # args>) t_wcode stp of (st, tp') \<Rightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5121 |
st = 0 \<and> (case tp' of (l, r) \<Rightarrow> l = [Bk] \<and> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5122 |
(\<exists>rn. r = Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>))" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5123 |
using wcode_lemma_1[of args "code tp"] h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5124 |
apply(simp, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5125 |
apply(rule_tac x = stp in exI, auto) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5126 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5127 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5128 |
fix rn stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5129 |
show " isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5130 |
\<Longrightarrow> False" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5131 |
using tutm_uhalt[of tp l args "Suc 0" rn] h |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5132 |
apply(simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5133 |
apply(erule_tac x = stp in allE) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5134 |
apply(simp add: tape_of_nl_abv tape_of_nat_list.simps bin_wc_eq) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5135 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5136 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5137 |
fix rn stp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5138 |
show "isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp) \<Longrightarrow> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5139 |
isS0 (steps (Suc 0, [Bk], Oc\<^bsup>Suc (code tp)\<^esup> @ Bk # Oc\<^bsup>Suc (bl_bin (<args>))\<^esup> @ Bk\<^bsup>rn\<^esup>) t_utm stp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5140 |
by simp |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5141 |
next |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5142 |
show "?Q1 \<turnstile>-> ?P4" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5143 |
apply(simp add: t_imply_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5144 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5145 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5146 |
thus "?thesis" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5147 |
apply(simp add: t_imply_def UTM_pre_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5148 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5149 |
qed |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5150 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5151 |
text {* |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5152 |
The correctness of @{text "UTM"}, the unhalt case. |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5153 |
*} |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5154 |
|
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5155 |
lemma UTM_uhalt_lemma: |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5156 |
"\<lbrakk>turing_basic.t_correct tp; |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5157 |
\<forall> stp. (\<not> TSTD (steps (Suc 0, Bk\<^bsup>l\<^esup>, <args>) tp stp)); |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5158 |
args \<noteq> []\<rbrakk> |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5159 |
\<Longrightarrow> \<forall> stp. \<not> isS0 (steps (Suc 0, [], <code tp # args>) UTM stp)" |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5160 |
using UTM_uhalt_lemma_pre[of tp l args] |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5161 |
apply(simp add: UTM_pre_def t_utm_def UTM_def F_aprog_def F_tprog_def) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5162 |
apply(case_tac "rec_ci rec_F", simp) |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5163 |
done |
48b231495281
Some illustration added together with more explanations.
zhang
parents:
370
diff
changeset
|
5164 |
|
370
1ce04eb1c8ad
Initial upload of the formal construction of Universal Turing Machine.
zhang
parents:
diff
changeset
|
5165 |
end |