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+ −