--- a/CookBook/Package/Ind_General_Scheme.thy	Wed Mar 18 23:52:51 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,87 +0,0 @@
-theory Ind_General_Scheme
-imports Main
-begin
-
-section{* The General Construction Principle \label{sec:ind-general-method} *}
-
-text {*
-  
-  The point of these examples is to get a feeling what the automatic proofs 
-  should do in order to solve all inductive definitions we throw at them. For this 
-  it is instructive to look at the general construction principle 
-  of inductive definitions, which we shall do in the next section.
-
-  Before we start with the implementation, it is useful to describe the general
-  form of inductive definitions that our package should accept.
-  Suppose $R_1,\ldots,R_n$ be mutually inductive predicates and $\vec{p}$ be
-  some fixed parameters. Then the introduction rules for $R_1,\ldots,R_n$ may have
-  the form
-
-  \[
-  \bigwedge\vec{x}_i.~\vec{A}_i \Longrightarrow \left(\bigwedge\vec{y}_{ij}.~\vec{B}_{ij} \Longrightarrow
-  R_{k_{ij}}~\vec{p}~\vec{s}_{ij}\right)_{j=1,\ldots,m_i} \Longrightarrow R_{l_i}~\vec{p}~\vec{t}_i
-  \qquad \mbox{for\ } i=1,\ldots,r
-  \]
-
-  where $\vec{A}_i$ and $\vec{B}_{ij}$ are formulae not containing $R_1,\ldots,R_n$.
-  Note that by disallowing the inductive predicates to occur in $\vec{B}_{ij}$ we make sure
-  that all occurrences of the predicates in the premises of the introduction rules are
-  \emph{strictly positive}. This condition guarantees the existence of predicates
-  that are closed under the introduction rules shown above. Then the definitions of the 
-  inductive predicates $R_1,\ldots,R_n$ is:
-
-  \[
-  \begin{array}{l@ {\qquad}l}
-  R_i \equiv \lambda\vec{p}~\vec{z}_i.~\forall P_1 \ldots P_n.~K_1 \longrightarrow \cdots \longrightarrow K_r \longrightarrow P_i~\vec{z}_i &
-  \mbox{for\ } i=1,\ldots,n \\[1.5ex]
-  \mbox{where} \\
-  K_i \equiv \forall\vec{x}_i.~\vec{A}_i \longrightarrow \left(\forall\vec{y}_{ij}.~\vec{B}_{ij} \longrightarrow
-  P_{k_{ij}}~\vec{s}_{ij}\right)_{j=1,\ldots,m_i} \longrightarrow P_{l_i}~\vec{t}_i &
-  \mbox{for\ } i=1,\ldots,r
-  \end{array}
-  \]
-
-  The induction principles for the inductive predicates $R_1,\ldots,R_n$ are
-
-  \[
-  \begin{array}{l@ {\qquad}l}
-  R_i~\vec{p}~\vec{z}_i \Longrightarrow I_1 \Longrightarrow \cdots \Longrightarrow I_r \Longrightarrow P_i~\vec{z}_i &
-  \mbox{for\ } i=1,\ldots,n \\[1.5ex]
-  \mbox{where} \\
-  I_i \equiv \bigwedge\vec{x}_i.~\vec{A}_i \Longrightarrow \left(\bigwedge\vec{y}_{ij}.~\vec{B}_{ij} \Longrightarrow
-  P_{k_{ij}}~\vec{s}_{ij}\right)_{j=1,\ldots,m_i} \Longrightarrow P_{l_i}~\vec{t}_i &
-  \mbox{for\ } i=1,\ldots,r
-  \end{array}
-  \]
-
-  Since $K_i$ and $I_i$ are equivalent modulo conversion between meta-level and object-level
-  connectives, it is clear that the proof of the induction theorem is straightforward. We will
-  therefore focus on the proof of the introduction rules. When proving the introduction rule
-  shown above, we start by unfolding the definition of $R_1,\ldots,R_n$, which yields
-
-  \[
-  \bigwedge\vec{x}_i.~\vec{A}_i \Longrightarrow \left(\bigwedge\vec{y}_{ij}.~\vec{B}_{ij} \Longrightarrow
-  \forall P_1 \ldots P_n.~\vec{K} \longrightarrow P_{k_{ij}}~\vec{s}_{ij}\right)_{j=1,\ldots,m_i} \Longrightarrow \forall P_1 \ldots P_n.~\vec{K} \longrightarrow P_{l_i}~\vec{t}_i
-  \]
-
-  where $\vec{K}$ abbreviates $K_1,\ldots,K_r$. Applying the introduction rules for
-  $\forall$ and $\longrightarrow$ yields a goal state in which we have to prove $P_{l_i}~\vec{t}_i$
-  from the additional assumptions $\vec{K}$. When using $K_{l_i}$ (converted to the meta-logic format)
-  to prove $P_{l_i}~\vec{t}_i$, we get subgoals $\vec{A}_i$ that are trivially solvable by assumption,
-  as well as subgoals of the form
-
-  \[
-  \bigwedge\vec{y}_{ij}.~\vec{B}_{ij} \Longrightarrow P_{k_{ij}}~\vec{s}_{ij} \qquad \mbox{for\ } j=1,\ldots,m_i
-  \]
-
-  that can be solved using the assumptions
-
-  \[
-  \bigwedge\vec{y}_{ij}.~\vec{B}_{ij} \Longrightarrow
-  \forall P_1 \ldots P_n.~\vec{K} \longrightarrow P_{k_{ij}}~\vec{s}_{ij} \qquad \mbox{and} \qquad \vec{K}
-  \]
-
-  What remains is to implement these proofs generically.
-*}
-
-end