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author | lcp |

Wed, 01 Dec 1993 13:00:04 +0100 | |

changeset 179 | ceb948cefb93 |

parent 178 | afbb13cb34ca |

child 180 | 8962c2b0dc2b |

minor corrections

--- a/doc-src/ind-defs.tex Wed Dec 01 12:48:47 1993 +0100 +++ b/doc-src/ind-defs.tex Wed Dec 01 13:00:04 1993 +0100 @@ -1,4 +1,5 @@ \documentstyle[12pt,a4,proof,lcp,alltt,amssymbols]{article} +\hyphenation{data-type} %CADE version should use 11pt and the Springer style \newif\ifCADE \CADEfalse @@ -97,7 +98,6 @@ \newcommand\Con{{\tt Con}} \newcommand\data{{\tt data}} -\sloppy \binperiod %%%treat . like a binary operator \begin{document} @@ -106,19 +106,18 @@ \maketitle \begin{abstract} Several theorem provers provide commands for formalizing recursive - datatypes and/or inductively defined sets. This paper presents a new - approach, based on fixedpoint definitions. It is unusually general: - it admits all monotone inductive definitions. It is conceptually simple, - which has allowed the easy implementation of mutual recursion and other - conveniences. It also handles coinductive definitions: simply replace - the least fixedpoint by a greatest fixedpoint. This represents the first - automated support for coinductive definitions. + data\-types or inductively defined sets. This paper presents a new + approach, based on fixedpoint definitions. It is unusually general: it + admits all provably monotone inductive definitions. It is conceptually + simple, which has allowed the easy implementation of mutual recursion and + other conveniences. It also handles coinductive definitions: simply + replace the least fixedpoint by a greatest fixedpoint. This represents + the first automated support for coinductive definitions. The method has been implemented in Isabelle's formalization of ZF set - theory. It should - be applicable to any logic in which the Knaster-Tarski Theorem can be - proved. The paper briefly describes a method of formalizing - non-well-founded data structures in standard ZF set theory. + theory. It should be applicable to any logic in which the Knaster-Tarski + Theorem can be proved. The paper briefly describes a method of + formalizing non-well-founded data structures in standard ZF set theory. Examples include lists of $n$ elements, the accessible part of a relation and the set of primitive recursive functions. One example of a @@ -229,8 +228,8 @@ This fixedpoint theory is simple. The Knaster-Tarski theorem is easy to prove. Showing monotonicity of~$h$ is trivial, in typical cases. We must also exhibit a bounding set~$D$ for~$h$. Sometimes this is trivial, as -when a set of ``theorems'' is (co)inductively defined over some previously -existing set of ``formulae.'' But defining the bounding set for +when a set of `theorems' is (co)inductively defined over some previously +existing set of `formulae.' But defining the bounding set for (co)datatype definitions requires some effort; see~\S\ref{univ-sec} below. @@ -534,10 +533,10 @@ as {\tt Fin.induct}, and so forth. \subsection{Lists of $n$ elements}\label{listn-sec} -This has become a standard example in the -literature. Following Paulin-Mohring~\cite{paulin92}, we could attempt to -define a new datatype $\listn(A,n)$, for lists of length~$n$, as an $n$-indexed -family of sets. But her introduction rules +This has become a standard example of an inductive definition. Following +Paulin-Mohring~\cite{paulin92}, we could attempt to define a new datatype +$\listn(A,n)$, for lists of length~$n$, as an $n$-indexed family of sets. +But her introduction rules \[ {\tt Niln}\in\listn(A,0) \qquad \infer{{\tt Consn}(n,a,l)\in\listn(A,\succ(n))} {n\in\nat & a\in A & l\in\listn(A,n)} @@ -628,11 +627,11 @@ \pair{n,l}\in\listn(A) \end{array} \right]_{a,l,n}}} \] -The ML function {\tt ListN.mk\_cases} generates simplified instances of this -rule. It works by freeness reasoning on the list constructors: $\Cons$ is -injective and $\Cons(a,l)\not=\Nil$. If $x$ is $\pair{i,\Nil}$ or -$\pair{i,\Cons(a,l)}$ then {\tt ListN.mk\_cases} deduces the corresponding -form of~$i$. For example, +The ML function {\tt ListN.mk\_cases} generates simplified instances of +this rule. It works by freeness reasoning on the list constructors: +$\Cons(a,l)$ is injective in its two arguments and differs from~$\Nil$. If +$x$ is $\pair{i,\Nil}$ or $\pair{i,\Cons(a,l)}$ then {\tt ListN.mk\_cases} +deduces the corresponding form of~$i$. For example, \begin{ttbox} ListN.mk_cases List.con_defs "<i,Cons(a,l)> : listn(A)" \end{ttbox} @@ -947,9 +946,10 @@ \subsection{The case analysis operator} The (co)datatype package automatically defines a case analysis operator, -called {\tt$R$\_case}. A mutually recursive definition still has only -one operator, called {\tt$R_1$\_\ldots\_$R_n$\_case}. The case operator is -analogous to those for products and sums. +called {\tt$R$\_case}. A mutually recursive definition still has only one +operator, whose name combines those of the recursive sets: it is called +{\tt$R_1$\_\ldots\_$R_n$\_case}. The case operator is analogous to those +for products and sums. Datatype definitions employ standard products and sums, whose operators are $\split$ and $\case$ and satisfy the equations @@ -1519,7 +1519,7 @@ The result structure also inherits everything from the underlying (co)inductive definition, such as the introduction rules, elimination rule, -and induction/coinduction rule. +and (co)induction rule. \begin{figure} @@ -1576,16 +1576,16 @@ $\quniv(A_1\un\cdots\un A_k)$. The {\tt sintrs} specify the introduction rules, which govern the recursive -structure of the datatype. Introduction rules may involve monotone operators -and side-conditions to express things that go beyond the usual notion of -datatype. The theorem lists {\tt monos}, {\tt type\_intrs} and {\tt -type\_elims} should contain precisely what is needed for the underlying -(co)inductive definition. Isabelle/ZF defines theorem lists that can be -defined for the latter two components: +structure of the datatype. Introduction rules may involve monotone +operators and side-conditions to express things that go beyond the usual +notion of datatype. The theorem lists {\tt monos}, {\tt type\_intrs} and +{\tt type\_elims} should contain precisely what is needed for the +underlying (co)inductive definition. Isabelle/ZF defines lists of +type-checking rules that can be supplied for the latter two components: \begin{itemize} -\item {\tt datatype\_intrs} and {\tt datatype\_elims} are type-checking rules +\item {\tt datatype\_intrs} and {\tt datatype\_elims} are rules for $\univ(A)$. -\item {\tt co\_datatype\_intrs} and {\tt co\_datatype\_elims} are type-checking +\item {\tt co\_datatype\_intrs} and {\tt co\_datatype\_elims} are rules for $\quniv(A)$. \end{itemize} In typical definitions, these theorem lists need not be supplemented with