
diff -r a42c773ec8ab -r 841f7b9c0a6a thys/notes.tex
--- a/thys/notes.tex	Tue May 17 14:28:22 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,442 +0,0 @@
-\documentclass[11pt]{article}
-\usepackage[left]{lineno}
-\usepackage{amsmath}
-\usepackage{stmaryrd}
-
-\begin{document}
-%%%\linenumbers
-
-\noindent 
-We already proved that
-
-\[
-\text{If}\;nullable(r)\;\text{then}\;POSIX\;(mkeps\; r)\;r
-\]
-
-\noindent 
-holds. This is essentially the ``base case'' for the
-correctness proof of the algorithm. For the ``induction
-case'' we need the following main theorem, which we are 
-currently after:
-
-\begin{center}
-\begin{tabular}{lll}
-If & (*) & $POSIX\;v\;(der\;c\;r)$ and $\vdash v : der\;c\;r$\\
-then & & $POSIX\;(inj\;r\;c\;v)\;r$
-\end{tabular}
-\end{center}
-
-\noindent 
-That means a POSIX value $v$ is still $POSIX$ after injection.
-I am not sure whether this theorem is actually true in this
-full generality. Maybe it requires some restrictions.
-
-If we unfold the $POSIX$ definition in the then-part, we 
-arrive at
-
-\[
-\forall v'.\;
-\text{if}\;\vdash v' : r\; \text{and} \;|inj\;r\;c\;v| = |v'|\;
-\text{then}\; |inj\;r\;c\;v| \succ_r v' 
-\]
-
-\noindent 
-which is what we need to prove assuming the if-part (*) in the
-theorem above. Since this is a universally quantified formula,
-we just need to fix a $v'$. We can then prove the implication
-by assuming
-
-\[
-\text{(a)}\;\;\vdash v' : r\;\; \text{and} \;\;
-\text{(b)}\;\;inj\;r\;c\;v = |v'|
-\]
-
-\noindent 
-and our goal is
-
-\[
-(goal)\;\;inj\;r\;c\;v \succ_r v'
-\]
-
-\noindent 
-There are already two lemmas proved that can transform 
-the assumptions (a) and (b) into
-
-\[
-\text{(a*)}\;\;\vdash proj\;r\;c\;v' : der\;c\;r\;\; \text{and} \;\;
-\text{(b*)}\;\;c\,\#\,|v| = |v'|
-\]
-
-\noindent 
-Another lemma shows that
-
-\[
-|v'| = c\,\#\,|proj\;r\;c\;v|
-\]
-
-\noindent 
-Using (b*) we can therefore infer 
-
-\[
-\text{(b**)}\;\;|v| = |proj\;r\;c\;v|
-\]
-
-\noindent 
-The main idea of the proof is now a simple instantiation
-of the assumption $POSIX\;v\;(der\;c\;r)$. If we unfold 
-the $POSIX$ definition, we get
-
-\[
-\forall v'.\;
-\text{if}\;\vdash v' : der\;c\;r\; \text{and} \;|v| = |v'|\;
-\text{then}\; v \succ_{der\;c\;r}\; v' 
-\]
-
-\noindent 
-We can instantiate this $v'$ with $proj\;r\;c\;v'$ and can use 
-(a*) and (b**) in order to infer
-
-\[
-v \succ_{der\;c\;r}\; proj\;r\;c\;v'
-\]
-
-\noindent 
-The point of the side-lemma below is that we can ``add'' an
-$inj$ to both sides to obtain
-
-\[
-inj\;r\;c\;v \succ_r\; inj\;r\;c\;(proj\;r\;c\;v')
-\]
-
-\noindent Finally there is already a lemma proved that shows
-that an injection and projection is the identity, meaning
-
-\[
-inj\;r\;c\;(proj\;r\;c\;v') = v'
-\]
-
-\noindent 
-With this we have shown our goal (pending a proof of the side-lemma 
-next).
-
-
-\subsection*{Side-Lemma}
-
-A side-lemma needed for the theorem above which might be true, but can also be false, is as follows:
-
-\begin{center}
-\begin{tabular}{lll}
-If   & (1) & $v_1 \succ_{der\;c\;r} v_2$,\\
-     & (2) & $\vdash v_1 : der\;c\;r$, and\\ 
-     & (3) & $\vdash v_2 : der\;c\;r$ holds,\\
-then &     & $inj\;r\;c\;v_1 \succ_r inj\;r\;c\;v_2$ also holds.  
-\end{tabular}
-\end{center}
-
-\noindent It essentially states that if one value $v_1$ is 
-bigger than $v_2$ then this ordering is preserved under 
-injections. This is proved by induction (on the definition of 
-$der$\ldots this is very similar to an induction on $r$).
-\bigskip
-
-\noindent
-The case that is still unproved is the sequence case where we 
-assume $r = r_1\cdot r_2$ and also $r_1$ being nullable.
-The derivative $der\;c\;r$ is then
-
-\begin{center}
-$der\;c\;r = ((der\;c\;r_1) \cdot r_2) + (der\;c\;r_2)$
-\end{center}
-
-\noindent 
-or without the parentheses
-
-\begin{center}
-$der\;c\;r = (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$
-\end{center}
-
-\noindent 
-In this case the assumptions are
-
-\begin{center}
-\begin{tabular}{ll}
-(a) & $v_1 \succ_{(der\;c\;r_1) \cdot r_2 + der\;c\;r_2} v_2$\\
-(b) & $\vdash v_1 : (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$\\
-(c) & $\vdash v_2 : (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$\\
-(d) & $nullable(r_1)$
-\end{tabular}
-\end{center}
-
-\noindent 
-The induction hypotheses are
-
-\begin{center}
-\begin{tabular}{ll}
-(IH1) & $\forall v_1 v_2.\;v_1 \succ_{der\;c\;r_1} v_2
-\;\wedge\; \vdash v_1 : der\;c\;r_1 \;\wedge\; 
-\vdash v_2 : der\;c\;r_1\qquad$\\
-      & $\hfill\longrightarrow
-         inj\;r_1\;c\;v_1 \succ{r_1} \;inj\;r_1\;c\;v_2$\smallskip\\
-(IH2) & $\forall v_1 v_2.\;v_1 \succ_{der\;c\;r_2} v_2
-\;\wedge\; \vdash v_2 : der\;c\;r_2 \;\wedge\; 
-\vdash v_2 : der\;c\;r_2\qquad$\\
-      & $\hfill\longrightarrow
-         inj\;r_2\;c\;v_1 \succ{r_2} \;inj\;r_2\;c\;v_2$\\
-\end{tabular}
-\end{center}
-
-\noindent 
-The goal is
-
-\[
-(goal)\qquad
-inj\; (r_1 \cdot r_2)\;c\;v_1 \succ_{r_1 \cdot r_2} 
-inj\; (r_1 \cdot r_2)\;c\;v_2
-\]
-
-\noindent 
-If we analyse how (a) could have arisen (that is make a case
-distinction), then we will find four cases:
-
-\begin{center}
-\begin{tabular}{ll}
-LL & $v_1 = Left(w_1)$, $v_2 = Left(w_2)$\\
-LR & $v_1 = Left(w_1)$, $v_2 = Right(w_2)$\\
-RL & $v_1 = Right(w_1)$, $v_2 = Left(w_2)$\\
-RR & $v_1 = Right(w_1)$, $v_2 = Right(w_2)$\\
-\end{tabular}
-\end{center}
-
-
-\noindent 
-We have to establish our goal in all four cases. 
-
-
-\subsubsection*{Case LR}
-
-The corresponding rule (instantiated) is:
-
-\begin{center}
-\begin{tabular}{c}
-$len\,|w_1| \geq len\,|w_2|$\\
-\hline
-$Left(w_1) \succ_{(der\;c\;r_1) \cdot r_2 + der\;c\;r_2} Right(w_2)$
-\end{tabular}
-\end{center}
-
-\noindent 
-This means we can also assume in this case
-
-\[
-(e)\quad len\,|w_1| \geq len\,|w_2|
-\] 
-
-\noindent 
-which is the premise of the rule above.
-Instantiating $v_1$ and $v_2$ in the assumptions (b) and (c)
-gives us
-
-\begin{center}
-\begin{tabular}{ll}
-(b*) & $\vdash Left(w_1) : (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$\\
-(c*) & $\vdash Right(w_2) : (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$\\
-\end{tabular}
-\end{center}
-
-\noindent Since these are assumptions, we can further analyse
-how they could have arisen according to the rules of $\vdash
-\_ : \_\,$. This gives us two new assumptions
-
-\begin{center}
-\begin{tabular}{ll}
-(b**) & $\vdash w_1 : (der\;c\;r_1) \cdot r_2$\\
-(c**) & $\vdash w_2 : der\;c\;r_2$\\
-\end{tabular}
-\end{center}
-
-\noindent 
-Looking at (b**) we can further analyse how this
-judgement could have arisen. This tells us that $w_1$
-must have been a sequence, say $u_1\cdot u_2$, with
-
-\begin{center}
-\begin{tabular}{ll}
-(b***) & $\vdash u_1 : der\;c\;r_1$\\
-       & $\vdash u_2 : r_2$\\
-\end{tabular}
-\end{center}
-
-\noindent 
-Instantiating the goal means we need to prove
-
-\[
-inj\; (r_1 \cdot r_2)\;c\;(Left(u_1\cdot u_2)) \succ_{r_1 \cdot r_2} 
-inj\; (r_1 \cdot r_2)\;c\;(Right(w_2))
-\]
-
-\noindent 
-We can simplify this according to the rules of $inj$:
-
-\[
-(inj\; r_1\;c\;u_1)\cdot u_2 \succ_{r_1 \cdot r_2} 
-(mkeps\;r_1) \cdot (inj\; r_2\;c\;w_2)
-\]
-
-\noindent
-This is what we need to prove. There are only two rules that
-can be used to prove this judgement:
-
-\begin{center}
-\begin{tabular}{cc}
-\begin{tabular}{c}
-$v_1 = v_1'$\qquad $v_2 \succ_{r_2} v_2'$\\
-\hline
-$v_1\cdot v_2 \succ_{r_1\cdot r_2} v_1'\cdot v_2'$
-\end{tabular} &
-\begin{tabular}{c}
-$v_1 \succ_{r_1} v_1'$\\
-\hline
-$v_1\cdot v_2 \succ_{r_1\cdot r_2} v_1'\cdot v_2'$
-\end{tabular}
-\end{tabular}
-\end{center}
-
-\noindent 
-Using the left rule would mean we need to show that
-
-\[
-inj\; r_1\;c\;u_1 = mkeps\;r_1
-\]
-
-\noindent 
-but this can never be the case.\footnote{Actually Isabelle
-found this out after analysing its argument. ;o)} Lets assume
-it would be true, then also if we flat each side, it must hold
-that
-
-\[
-|inj\; r_1\;c\;u_1| = |mkeps\;r_1|
-\]
-
-\noindent 
-But this leads to a contradiction, because the right-hand side
-will be equal to the empty list, or empty string. This is 
-because we assumed $nullable(r_1)$ and there is a lemma
-called \texttt{mkeps\_flat} which shows this. On the other
-side we know by assumption (b***) and lemma \texttt{v4} that 
-the other side needs to be a string starting with $c$ (since
-we inject $c$ into $u_1$). The empty string can never be equal 
-to something starting with $c$\ldots therefore there is a 
-contradiction.
-
-That means we can only use the rule on the right-hand side to 
-prove our goal. This implies we need to prove
-
-\[
-inj\; r_1\;c\;u_1 \succ_{r_1} mkeps\;r_1
-\]
-
-
-\subsubsection*{Case RL}
-
-The corresponding rule (instantiated) is:
-
-\begin{center}
-\begin{tabular}{c}
-$len\,|w_1| > len\,|w_2|$\\
-\hline
-$Right(w_1) \succ_{(der\;c\;r_1) \cdot r_2 + der\;c\;r_2} Left(w_2)$
-\end{tabular}
-\end{center}
-
-\subsection*{Test Proof}
-
-We want to prove that
-
-\[
-nullable(r) \;\text{implies}\; POSIX (mkeps\; r)\; r
-\]
-
-\noindent
-We prove this by induction on $r$. There are 5 subcases, and 
-only the $r_1 + r_2$-case is interesting. In this case we know the 
-induction hypotheses are
-
-\begin{center}
-\begin{tabular}{ll}
-(IMP1) & $nullable(r_1) \;\text{implies}\; 
-          POSIX (mkeps\; r_1)\; r_1$ \\
-(IMP2) & $nullable(r_2) \;\text{implies}\;
-          POSIX (mkeps\; r_2)\; r_2$
-\end{tabular}
-\end{center}
-
-\noindent and know that $nullable(r_1 + r_2)$ holds. From this
-we know that either $nullable(r_1)$ holds or $nullable(r_2)$.
-Let us consider the first case where we know $nullable(r_1)$.
-
-
-\subsection*{Problems in the paper proof}
-
-I cannot verify\ldots
-
-
-
-\newpage
-\section*{Isabelle Cheat-Sheet}
- 
-\begin{itemize} 
-\item The main notion in Isabelle is a \emph{theorem}.
-      Definitions, inductive predicates and recursive
-      functions all have underlying theorems. If a definition
-      is called \texttt{foo}, then the theorem will be called
-      \texttt{foo\_def}. Take a recursive function, say
-      \texttt{bar}, it will have a theorem that is called
-      \texttt{bar.simps} and will be added to the simplifier.
-      That means the simplifier will automatically 
-      Inductive predicates called \texttt{baz} will be called
-      \texttt{baz.intros}. For inductive predicates, there are
-      also theorems \texttt{baz.induct} and
-      \texttt{baz.cases}.    
-
-\item A \emph{goal-state} consists of one or more subgoals. If
-      there are \texttt{No more subgoals!} then the theorem is
-      proved. Each subgoal is of the form
-  
-      \[
-      \llbracket \ldots{}premises\ldots \rrbracket \Longrightarrow 
-      conclusion
-      \]  
-  
-      \noindent 
-      where $premises$ and $conclusion$ are formulas of type 
-      \texttt{bool}.
-      
-\item There are three low-level methods for applying one or
-      more theorem to a subgoal, called \texttt{rule},
-      \texttt{drule} and \texttt{erule}. The first applies a 
-      theorem to a conclusion of a goal. For example
-      
-      \[\texttt{apply}(\texttt{rule}\;thm)
-      \]
- 
-      If the conclusion is of the form $\_ \wedge \_$,
-      $\_ \longrightarrow \_$ and $\forall\,x. \_$ the
-      $thm$ is called
-      
-      \begin{center}
-      \begin{tabular}{lcl}
-      $\_ \wedge \_$          &  $\Rightarrow$ & $conjI$\\
-      $\_ \longrightarrow \_$ &  $\Rightarrow$ & $impI$\\
-      $\forall\,x.\_$        &  $\Rightarrow$ & $allI$
-      \end{tabular}
-      \end{center}
-       
-      Many of such rule are called intro-rules and end with 
-      an ``$I$'', or in case of inductive predicates $\_.intros$.
-      
-   
-\end{itemize}
- 
-
-\end{document}  




