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doc-src/Logics/ZF.tex

changeset 114 | 96c627d2815e |

parent 111 | 1b3cddf41b2d |

child 131 | bb0caac13eff |

--- a/doc-src/Logics/ZF.tex Thu Nov 11 13:24:47 1993 +0100 +++ b/doc-src/Logics/ZF.tex Fri Nov 12 10:41:13 1993 +0100 @@ -753,11 +753,10 @@ \ttindexbold{Pair_inject1} and \ttindexbold{Pair_inject2}, and equivalently as the elimination rule \ttindexbold{Pair_inject}. -Note the rule \ttindexbold{Pair_neq_0}, which asserts -$\pair{a,b}\neq\emptyset$. This is no arbitrary property of -$\{\{a\},\{a,b\}\}$, but one that we can reasonably expect to hold for any -encoding of ordered pairs. It turns out to be useful for constructing -Lisp-style S-expressions in set theory. +The rule \ttindexbold{Pair_neq_0} asserts $\pair{a,b}\neq\emptyset$. This +is a property of $\{\{a\},\{a,b\}\}$, and need not hold for other +encoding of ordered pairs. The non-standard ordered pairs mentioned below +satisfy $\pair{\emptyset;\emptyset}=\emptyset$. The natural deduction rules \ttindexbold{SigmaI} and \ttindexbold{SigmaE} assert that \ttindex{Sigma}$(A,B)$ consists of all pairs of the form @@ -1193,14 +1192,13 @@ \idx{Fin_0I} 0 : Fin(A) \idx{Fin_consI} [| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A) -\idx{Fin_mono} A<=B ==> Fin(A) <= Fin(B) - \idx{Fin_induct} [| b: Fin(A); P(0); !!x y. [| x: A; y: Fin(A); x~:y; P(y) |] ==> P(cons(x,y)) |] ==> P(b) +\idx{Fin_mono} A<=B ==> Fin(A) <= Fin(B) \idx{Fin_UnI} [| b: Fin(A); c: Fin(A) |] ==> b Un c : Fin(A) \idx{Fin_UnionI} C : Fin(Fin(A)) ==> Union(C) : Fin(A) \idx{Fin_subset} [| c<=b; b: Fin(A) |] ==> c: Fin(A) @@ -1254,6 +1252,10 @@ some of the further constants and infixes that are declared in the various theory extensions. +Figure~\ref{zf-equalities} presents commutative, associative, distributive, +and idempotency laws of union and intersection, along with other equations. +See file \ttindexbold{ZF/equalities.ML}. + Figure~\ref{zf-sum} defines $\{0,1\}$ as a set of booleans, with a conditional operator. It also defines the disjoint union of two sets, with injections and a case analysis operator. See files @@ -1265,16 +1267,13 @@ \ttindexbold{qconverse}, and the summation operator \ttindexbold{QSigma}. These are completely analogous to the corresponding versions for standard ordered pairs. The theory goes on to define a non-standard notion of -disjoint sum using non-standard pairs. See file \ttindexbold{qpair.thy}. +disjoint sum using non-standard pairs. This will support co-inductive +definitions, for example of infinite lists. See file \ttindexbold{qpair.thy}. Monotonicity properties of most of the set-forming operations are proved. These are useful for applying the Knaster-Tarski Fixedpoint Theorem. See file \ttindexbold{ZF/mono.ML}. -Figure~\ref{zf-equalities} presents commutative, associative, distributive, -and idempotency laws of union and intersection, along with other equations. -See file \ttindexbold{ZF/equalities.ML}. - Figure~\ref{zf-fixedpt} presents the Knaster-Tarski Fixedpoint Theorem, proved for the lattice of subsets of a set. The theory defines least and greatest fixedpoint operators with corresponding induction and co-induction rules. @@ -1302,7 +1301,7 @@ where $\alpha$ is any ordinal. The file \ttindexbold{ZF/quniv.ML} defines a ``universe'' ${\tt quniv}(A)$, -for constructing co-datatypes such as streams. It is similar to ${\tt +for constructing co-datatypes such as streams. It is analogous to ${\tt univ}(A)$ but is closed under the non-standard product and sum. Figure~\ref{zf-fin} presents the finite set operator; ${\tt Fin}(A)$ is the @@ -1392,39 +1391,68 @@ ``Composition of homomorphisms'' challenge~\cite{boyer86}. \item[\ttindexbold{ZF/ex/ramsey.ML}] -proves the finite exponent 2 version of Ramsey's Theorem. +proves the finite exponent 2 version of Ramsey's Theorem, following Basin +and Kaufmann's presentation~\cite{basin91}. + +\item[\ttindexbold{ZF/ex/equiv.ML}] +develops a ZF theory of equivalence classes, not using the Axiom of Choice. + +\item[\ttindexbold{ZF/ex/integ.ML}] +develops a theory of the integers as equivalence classes of pairs of +natural numbers. + +\item[\ttindexbold{ZF/ex/bin.ML}] +defines a datatype for two's complement binary integers. File +\ttindexbold{ZF/ex/binfn.ML} then develops rewrite rules for binary +arithmetic. For instance, $1359\times {-}2468 = {-}3354012$ takes under +14 seconds. \item[\ttindexbold{ZF/ex/bt.ML}] defines the recursive data structure ${\tt bt}(A)$, labelled binary trees. -\item[\ttindexbold{ZF/ex/sexp.ML}] -defines the set of Lisp $S$-expressions over~$A$, ${\tt sexp}(A)$. These -are unlabelled binary trees whose leaves contain elements of $(A)$. - -\item[\ttindexbold{ZF/ex/term.ML}] -defines a recursive data structure for terms and term lists. +\item[\ttindexbold{ZF/ex/term.ML}] + and \ttindexbold{ZF/ex/termfn.ML} define a recursive data structure for + terms and term lists. These are simply finite branching trees. \item[\ttindexbold{ZF/ex/tf.ML}] -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. + and \ttindexbold{ZF/ex/tf_fn.ML} define 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[\ttindexbold{ZF/ex/prop.ML}] + and \ttindexbold{ZF/ex/proplog.ML} proves soundness and completeness of + propositional logic. This illustrates datatype definitions, inductive + definitions, structural induction and rule induction. + +\item[\ttindexbold{ZF/ex/listn.ML}] +presents the inductive definition of the lists of $n$ elements~\cite{paulin92}. -\item[\ttindexbold{ZF/ex/finite.ML}] -inductively defines a finite powerset operator. +\item[\ttindexbold{ZF/ex/acc.ML}] +presents the inductive definition of the accessible part of a +relation~\cite{paulin92}. -\item[\ttindexbold{ZF/ex/proplog.ML}] -proves soundness and completeness of propositional logic. This illustrates -the main forms of induction. +\item[\ttindexbold{ZF/ex/comb.ML}] + presents the datatype definition of combinators. File + \ttindexbold{ZF/ex/contract0.ML} defines contraction, while file + \ttindexbold{ZF/ex/parcontract.ML} defines parallel contraction and + proves the Church-Rosser Theorem. This case study follows Camilleri and + Melham~\cite{camilleri92}. + +\item[\ttindexbold{ZF/ex/llist.ML}] + and \ttindexbold{ZF/ex/llist_eq.ML} develop lazy lists in ZF and a notion + of co-induction for proving equations between them. \end{description} \section{A proof about powersets} -To demonstrate high-level reasoning about subsets, let us prove the equation -${Pow(A)\cap Pow(B)}= Pow(A\cap B)$. Compared with first-order logic, set -theory involves a maze of rules, and theorems have many different proofs. -Attempting other proofs of the theorem might be instructive. This proof -exploits the lattice properties of intersection. It also uses the -monotonicity of the powerset operation, from {\tt ZF/mono.ML}: +To demonstrate high-level reasoning about subsets, let us prove the +equation ${{\tt Pow}(A)\cap {\tt Pow}(B)}= {\tt Pow}(A\cap B)$. Compared +with first-order logic, set theory involves a maze of rules, and theorems +have many different proofs. Attempting other proofs of the theorem might +be instructive. This proof exploits the lattice properties of +intersection. It also uses the monotonicity of the powerset operation, +from {\tt ZF/mono.ML}: \begin{ttbox} \idx{Pow_mono} A<=B ==> Pow(A) <= Pow(B) \end{ttbox} @@ -1536,6 +1564,8 @@ {\out Level 0} {\out Union(C) <= Union(D)} {\out 1. Union(C) <= Union(D)} +{\out val prem = "C <= D [C <= D]" : thm} +\ttbreak by (resolve_tac [subsetI] 1); {\out Level 1} {\out Union(C) <= Union(D)} @@ -1613,6 +1643,11 @@ {\out Level 0} {\out (f Un g) ` a = f ` a} {\out 1. (f Un g) ` a = f ` a} +\ttbreak +{\out val prems = ["a : A [a : A]",} +{\out "f : A -> B [f : A -> B]",} +{\out "g : C -> D [g : C -> D]",} +{\out "A Int C = 0 [A Int C = 0]"] : thm list} \end{ttbox} Using \ttindex{apply_equality}, we reduce the equality to reasoning about ordered pairs.