summary |
shortlog |
changelog |
graph |
tags |
bookmarks |
branches |
files |
changeset |
file |
latest |
revisions |
annotate |
diff |
comparison |
raw |
help

doc-src/Logics/ZF.tex

changeset 595 | 96c87d5bb015 |

parent 498 | 689e2bd78c19 |

child 713 | b470cc6326aa |

--- a/doc-src/Logics/ZF.tex Fri Sep 09 11:45:44 1994 +0200 +++ b/doc-src/Logics/ZF.tex Fri Sep 09 11:52:38 1994 +0200 @@ -9,14 +9,15 @@ of it is based on the work of No\"el~\cite{noel}. A tremendous amount of set theory has been formally developed, including -the basic properties of relations, functions and ordinals. Significant -results have been proved, such as the Schr\"oder-Bernstein Theorem and a -version of Ramsey's Theorem. General methods have been developed for -solving recursion equations over monotonic functors; these have been -applied to yield constructions of lists, trees, infinite lists, etc. The -Recursion Theorem has been proved, admitting recursive definitions of -functions over well-founded relations. Thus, we may even regard set theory -as a computational logic, loosely inspired by Martin-L\"of's Type Theory. +the basic properties of relations, functions, ordinals and cardinals. +Significant results have been proved, such as the Schr\"oder-Bernstein +Theorem, the Wellordering Theorem and a version of Ramsey's Theorem. +General methods have been developed for solving recursion equations over +monotonic functors; these have been applied to yield constructions of +lists, trees, infinite lists, etc. The Recursion Theorem has been proved, +admitting recursive definitions of functions over well-founded relations. +Thus, we may even regard set theory as a computational logic, loosely +inspired by Martin-L\"of's Type Theory. Because {\ZF} is an extension of {\FOL}, it provides the same packages, namely {\tt hyp_subst_tac}, the simplifier, and the classical reasoner. @@ -24,11 +25,13 @@ classical rule sets are defined, including {\tt lemmas_cs}, {\tt upair_cs} and~{\tt ZF_cs}. -{\tt ZF} now has a flexible package for handling inductive definitions, +{\tt ZF} has a flexible package for handling inductive definitions, such as inference systems, and datatype definitions, such as lists and -trees. Moreover it also handles coinductive definitions, such as +trees. Moreover it handles coinductive definitions, such as bisimulation relations, and codatatype definitions, such as streams. A -recent paper describes the package~\cite{paulson-fixedpt}. +recent paper describes the package~\cite{paulson-CADE}, but its examples +use an obsolete declaration syntax. Please consult the version of the +paper distributed with Isabelle. Recent reports~\cite{paulson-set-I,paulson-set-II} describe {\tt ZF} less formally than this chapter. Isabelle employs a novel treatment of @@ -1132,7 +1135,7 @@ fixedpoint operators with corresponding induction and coinduction rules. These are essential to many definitions that follow, including the natural numbers and the transitive closure operator. The (co)inductive definition -package also uses the fixedpoint operators~\cite{paulson-fixedpt}. See +package also uses the fixedpoint operators~\cite{paulson-CADE}. See Davey and Priestley~\cite{davey&priestley} for more on the Knaster-Tarski Theorem and my paper~\cite{paulson-set-II} for discussion of the Isabelle proofs. @@ -1290,16 +1293,17 @@ univ}(A)$ (and is defined in terms of it) but is closed under the non-standard product and sum. -Figure~\ref{zf-fin} presents the finite set operator; ${\tt Fin}(A)$ is the -set of all finite sets over~$A$. The definition employs Isabelle's -inductive definition package~\cite{paulson-fixedpt}, which proves various -rules automatically. The induction rule shown is stronger than the one -proved by the package. See file {\tt ZF/Fin.ML}. +Theory {\tt Finite} (Figure~\ref{zf-fin}) defines the finite set operator; +${\tt Fin}(A)$ is the set of all finite sets over~$A$. The theory employs +Isabelle's inductive definition package, which proves various rules +automatically. The induction rule shown is stronger than the one proved by +the package. The theory also defines the set of all finite functions +between two given sets. \begin{figure} \begin{ttbox} -\tdx{Fin_0I} 0 : Fin(A) -\tdx{Fin_consI} [| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A) +\tdx{Fin.emptyI} 0 : Fin(A) +\tdx{Fin.consI} [| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A) \tdx{Fin_induct} [| b: Fin(A); @@ -1400,6 +1404,58 @@ of the cumulative hierarchy (thus $x\in V@{\alpha+1}$). \end{itemize} +Other important theories lead to a theory of cardinal numbers. They have +not yet been written up anywhere. Here is a summary: +\begin{itemize} +\item Theory {\tt Rel} defines the basic properties of relations, such as + (ir)reflexivity, (a)symmetry, and transitivity. + +\item Theory {\tt EquivClass} develops a theory of equivalence + classes, not using the Axiom of Choice. + +\item Theory {\tt Order} defines partial orderings, total orderings and + wellorderings. + +\item Theory {\tt OrderArith} defines orderings on sum and product sets. + These can be used to define ordinal arithmetic and have applications to + cardinal arithmetic. + +\item Theory {\tt OrderType} defines order types. Every wellordering is + equivalent to a unique ordinal, which is its order type. + +\item Theory {\tt Cardinal} defines equipollence and cardinal numbers. + +\item Theory {\tt CardinalArith} defines cardinal addition and + multiplication, and proves their elementary laws. It proves that there + is no greatest cardinal. It also proves a deep result, namely + $\kappa\otimes\kappa=\kappa$ for every infinite cardinal~$\kappa$; see + Kunen~\cite[page 29]{kunen80}. None of these results assume the Axiom of + Choice, which complicates their proofs considerably. +\end{itemize} + +The following developments involve the Axiom of Choice (AC): +\begin{itemize} +\item Theory {\tt AC} asserts the Axiom of Choice and proves some simple + equivalent forms. + +\item Theory {\tt Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma + and the Wellordering Theorem, following Abrial and + Laffitte~\cite{abrial93}. + +\item Theory \verb|Cardinal_AC| uses AC to prove simplified theorems about + the cardinals. It also proves a theorem needed to justify + infinitely branching datatype declarations: if $\kappa$ is an infinite + cardinal and $|X(\alpha)| \le \kappa$ for all $\alpha<\kappa$ then + $|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$. + +\item Theory {\tt InfDatatype} proves theorems to justify infinitely + branching datatypes. Arbitrary index sets are allowed, provided their + cardinalities have an upper bound. The theory also justifies some + unusual cases of finite branching, involving the finite powerset operator + and the finite function space operator. +\end{itemize} + + \section{Simplification rules} {\ZF} does not merely inherit simplification from \FOL, but modifies it @@ -1433,69 +1489,63 @@ \end{figure} -\section{The examples directory} +\section{The examples directories} +Directory {\tt HOL/IMP} contains a mechanised version of a semantic +equivalence proof taken from Winskel~\cite{winskel93}. It formalises the +denotational and operational semantics of a simple while-language, then +proves the two equivalent. It contains several datatype and inductive +definitions, and demonstrates their use. + The directory {\tt ZF/ex} contains further developments in {\ZF} set theory. Here is an overview; see the files themselves for more details. I describe much of this material in other -publications~\cite{paulson-fixedpt,paulson-set-I,paulson-set-II}. -\begin{ttdescription} -\item[ZF/ex/misc.ML] contains miscellaneous examples such as - Cantor's Theorem, the Schr\"oder-Bernstein Theorem and the - `Composition of homomorphisms' challenge~\cite{boyer86}. +publications~\cite{paulson-set-I,paulson-set-II,paulson-CADE}. +\begin{itemize} +\item File {\tt misc.ML} contains miscellaneous examples such as + Cantor's Theorem, the Schr\"oder-Bernstein Theorem and the `Composition + of homomorphisms' challenge~\cite{boyer86}. -\item[ZF/ex/Ramsey.ML] -proves the finite exponent 2 version of Ramsey's Theorem, following Basin -and Kaufmann's presentation~\cite{basin91}. - -\item[ZF/ex/Equiv.ML] -develops a theory of equivalence classes, not using the Axiom of Choice. +\item Theory {\tt Ramsey} proves the finite exponent 2 version of + Ramsey's Theorem, following Basin and Kaufmann's + presentation~\cite{basin91}. -\item[ZF/ex/Integ.ML] -develops a theory of the integers as equivalence classes of pairs of -natural numbers. +\item Theory {\tt Integ} develops a theory of the integers as + equivalence classes of pairs of natural numbers. -\item[ZF/ex/Bin.ML] -defines a datatype for two's complement binary integers. File -{\tt BinFn.ML} then develops rewrite rules for binary -arithmetic. For instance, $1359\times {-}2468 = {-}3354012$ takes under -14 seconds. +\item Theory {\tt Bin} defines a datatype for two's complement binary + integers, then proves rewrite rules to perform binary arithmetic. For + instance, $1359\times {-}2468 = {-}3354012$ takes under 14 seconds. -\item[ZF/ex/BT.ML] -defines the recursive data structure ${\tt bt}(A)$, labelled binary trees. +\item Theory {\tt BT} defines the recursive data structure ${\tt + bt}(A)$, labelled binary trees. -\item[ZF/ex/Term.ML] - and {\tt TermFn.ML} define a recursive data structure for - terms and term lists. These are simply finite branching trees. +\item Theory {\tt Term} defines a recursive data structure for terms + and term lists. These are simply finite branching trees. -\item[ZF/ex/TF.ML] - and {\tt TF_Fn.ML} define primitives for solving mutually +\item Theory {\tt TF} defines primitives for solving mutually recursive equations over sets. It constructs sets of trees and forests as an example, including induction and recursion rules that handle the mutual recursion. -\item[ZF/ex/Prop.ML] - and {\tt PropLog.ML} proves soundness and completeness of +\item Theory {\tt Prop} proves soundness and completeness of propositional logic~\cite{paulson-set-II}. This illustrates datatype - definitions, inductive definitions, structural induction and rule induction. + definitions, inductive definitions, structural induction and rule + induction. -\item[ZF/ex/ListN.ML] -presents the inductive definition of the lists of $n$ elements~\cite{paulin92}. - -\item[ZF/ex/Acc.ML] -presents the inductive definition of the accessible part of a -relation~\cite{paulin92}. +\item Theory {\tt ListN} inductively defines the lists of $n$ + elements~\cite{paulin92}. -\item[ZF/ex/Comb.ML] - presents the datatype definition of combinators. The file - {\tt Contract0.ML} defines contraction, while file - {\tt ParContract.ML} defines parallel contraction and - proves the Church-Rosser Theorem. This case study follows Camilleri and - Melham~\cite{camilleri92}. +\item Theory {\tt Acc} inductively defines the accessible part of a + relation~\cite{paulin92}. -\item[ZF/ex/LList.ML] - and {\tt LList_Eq.ML} develop lazy lists and a notion - of coinduction for proving equations between them. -\end{ttdescription} +\item Theory {\tt Comb} defines the datatype of combinators and + inductively defines contraction and parallel contraction. It goes on to + prove the Church-Rosser Theorem. This case study follows Camilleri and + Melham~\cite{camilleri92}. + +\item Theory {\tt LList} defines lazy lists and a coinduction + principle for proving equations between them. +\end{itemize} \section{A proof about powersets}\label{sec:ZF-pow-example}