--- 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.