
diff -r 53460ac408b5 -r 5bb2d29553c2 CookBook/Package/Ind_General_Scheme.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/CookBook/Package/Ind_General_Scheme.thy	Fri Oct 10 17:13:21 2008 +0200
@@ -0,0 +1,71 @@
+theory Ind_General_Scheme
+imports Main
+begin
+
+section{* The general construction principle *}
+
+text {*
+\label{sec:ind-general-method}
+Before we start with the implementation, it is useful to describe the general
+form of inductive definitions that our package should accept. We closely follow
+the notation for inductive definitions introduced by Schwichtenberg
+\cite{Schwichtenberg-MLCF} for the Minlog system.
+Let $R_1,\ldots,R_n$ be mutually inductive predicates and $\vec{p}$ be
+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. The inductive predicates
+$R_1,\ldots,R_n$ can then be defined as follows:
+\[
+\begin{array}{l@ {\qquad}l}
+  R_i \equiv \lambda\vec{p}~\vec{z}_i.~\forall P_1 \ldots P_n.~K_1 \longrightarrow \cdots \longrightarrow K_r \longrightarrow P_i~\vec{z}_i &
+  \mbox{for\ } i=1,\ldots,n \\[1.5ex]
+  \mbox{where} \\
+  K_i \equiv \forall\vec{x}_i.~\vec{A}_i \longrightarrow \left(\forall\vec{y}_{ij}.~\vec{B}_{ij} \longrightarrow
+    P_{k_{ij}}~\vec{s}_{ij}\right)_{j=1,\ldots,m_i} \longrightarrow P_{l_i}~\vec{t}_i &
+  \mbox{for\ } i=1,\ldots,r
+\end{array}
+\]
+The (weak) induction rules 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 proof 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 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
+\[
+\bigwedge\vec{y}_{ij}.~\vec{B}_{ij} \Longrightarrow
+  \forall P_1 \ldots P_n.~\vec{K} \longrightarrow P_{k_{ij}}~\vec{s}_{ij} \qquad \mbox{and} \qquad \vec{K}
+\]
+*}
+
+end




