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author | wenzelm |

Fri, 09 May 1997 19:41:46 +0200 | |

changeset 3149 | 434b33c5f827 |

parent 3148 | f081757ce757 |

child 3150 | a8faa68c68b5 |

minor tuning;

--- a/doc-src/Logics/ZF.tex Fri May 09 19:41:17 1997 +0200 +++ b/doc-src/Logics/ZF.tex Fri May 09 19:41:46 1997 +0200 @@ -19,13 +19,14 @@ 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. The default -simpset and claset are usually satisfactory. Named simpsets include -\ttindexbold{ZF_ss} (basic set theory rules) and \ttindexbold{rank_ss} (for -proving termination of well-founded recursion). Named clasets sets include -\ttindexbold{ZF_cs} (basic set theory) and \ttindexbold{le_cs} (useful for -reasoning about the relations $<$ and $\le$). +Because {\ZF} is an extension of {\FOL}, it provides the same +packages, namely {\tt hyp_subst_tac}, the simplifier, and the +classical reasoner. The default simpset and claset are usually +satisfactory. Named simpsets include \ttindexbold{ZF_ss} (basic set +theory rules) and \ttindexbold{rank_ss} (for proving termination of +well-founded recursion). Named clasets include \ttindexbold{ZF_cs} +(basic set theory) and \ttindexbold{le_cs} (useful for reasoning about +the relations $<$ and $\le$). {\tt ZF} has a flexible package for handling inductive definitions, such as inference systems, and datatype definitions, such as lists and @@ -110,8 +111,8 @@ \tt `` & $[i,i]\To i$ & Left 90 & image \\ \tt -`` & $[i,i]\To i$ & Left 90 & inverse image \\ \tt ` & $[i,i]\To i$ & Left 90 & application \\ - \sdx{Int} & $[i,i]\To i$ & Left 70 & intersection ($\inter$) \\ - \sdx{Un} & $[i,i]\To i$ & Left 65 & union ($\union$) \\ + \sdx{Int} & $[i,i]\To i$ & Left 70 & intersection ($\int$) \\ + \sdx{Un} & $[i,i]\To i$ & Left 65 & union ($\un$) \\ \tt - & $[i,i]\To i$ & Left 65 & set difference ($-$) \\[1ex] \tt: & $[i,i]\To o$ & Left 50 & membership ($\in$) \\ \tt <= & $[i,i]\To o$ & Left 50 & subset ($\subseteq$) @@ -146,8 +147,8 @@ {\ZF} have type~\tydx{i}, which is the type of individuals and has class~{\tt term}. The type of first-order formulae, remember, is~{\tt o}. -Infix operators include binary union and intersection ($A\union B$ and -$A\inter B$), set difference ($A-B$), and the subset and membership +Infix operators include binary union and intersection ($A\un B$ and +$A\int B$), set difference ($A-B$), and the subset and membership relations. Note that $a$\verb|~:|$b$ is translated to $\neg(a\in b)$. The union and intersection operators ($\bigcup A$ and $\bigcap A$) form the union or intersection of a set of sets; $\bigcup A$ means the same as @@ -160,9 +161,9 @@ \cdx{cons}: \begin{eqnarray*} A\cup B & \equiv & \bigcup({\tt Upair}(A,B)) \\ - {\tt cons}(a,B) & \equiv & {\tt Upair}(a,a) \union B + {\tt cons}(a,B) & \equiv & {\tt Upair}(a,a) \un B \end{eqnarray*} -The $\{\ldots\}$ notation abbreviates finite sets constructed in the +The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the obvious manner using~{\tt cons} and~$\emptyset$ (the empty set): \begin{eqnarray*} \{a,b,c\} & \equiv & {\tt cons}(a,{\tt cons}(b,{\tt cons}(c,\emptyset))) @@ -189,7 +190,7 @@ \begin{tabular}{rrr} \it external & \it internal & \it description \\ $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm negated membership\\ - \ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace & cons($a@1$,$\cdots$,cons($a@n$,0)) & + \ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace & cons($a@1$,$\ldots$,cons($a@n$,0)) & \rm finite set \\ <$a@1$, $\ldots$, $a@{n-1}$, $a@n$> & Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots) & @@ -526,12 +527,12 @@ \section{From basic lemmas to function spaces} Faced with so many definitions, it is essential to prove lemmas. Even -trivial theorems like $A\inter B=B\inter A$ would be difficult to prove -from the definitions alone. Isabelle's set theory derives many rules using -a natural deduction style. Ideally, a natural deduction rule should -introduce or eliminate just one operator, but this is not always practical. -For most operators, we may forget its definition and use its derived rules -instead. +trivial theorems like $A \int B = B \int A$ would be difficult to +prove from the definitions alone. Isabelle's set theory derives many +rules using a natural deduction style. Ideally, a natural deduction +rule should introduce or eliminate just one operator, but this is not +always practical. For most operators, we may forget its definition +and use its derived rules instead. \subsection{Fundamental lemmas} Figure~\ref{zf-lemmas1} presents the derived rules for the most basic @@ -811,7 +812,7 @@ \begin{warn} The translation between patterns and {\tt split} is performed automatically by the parser and printer. Thus the internal and external form of a term - may differ, whichs affects proofs. For example the term {\tt + may differ, which affects proofs. For example the term {\tt (\%<x,y>.<y,x>)<a,b>} requires the theorem {\tt split} to rewrite to {\tt<b,a>}. \end{warn} @@ -1517,8 +1518,8 @@ \begin{figure} \begin{eqnarray*} a\in \emptyset & \bimp & \bot\\ - a \in A \union B & \bimp & a\in A \disj a\in B\\ - a \in A \inter B & \bimp & a\in A \conj a\in B\\ + a \in A \un B & \bimp & a\in A \disj a\in B\\ + a \in A \int B & \bimp & a\in A \conj a\in B\\ a \in A-B & \bimp & a\in A \conj \neg (a\in B)\\ \pair{a,b}\in {\tt Sigma}(A,B) & \bimp & a\in A \conj b\in B(a)\\ @@ -1625,7 +1626,7 @@ {\out 2. Pow(A Int B) <= Pow(B)} {\out 3. Pow(A) Int Pow(B) <= Pow(A Int B)} \end{ttbox} -Subgoal~1 follows by applying the monotonicity of {\tt Pow} to $A\inter +Subgoal~1 follows by applying the monotonicity of {\tt Pow} to $A\int B\subseteq A$; subgoal~2 follows similarly: \index{*Int_lower1 theorem}\index{*Int_lower2 theorem} \begin{ttbox} @@ -1675,7 +1676,7 @@ {\out 1. !!x. [| x <= A; x <= B |] ==> x <= A Int B} \end{ttbox} The assumptions are that $x$ is a lower bound of both $A$ and~$B$, but -$A\inter B$ is the greatest lower bound:\index{*Int_greatest theorem} +$A\int B$ is the greatest lower bound:\index{*Int_greatest theorem} \begin{ttbox} by (resolve_tac [Int_greatest] 1); {\out Level 9} @@ -1868,7 +1869,7 @@ {\out (f Un g) ` a = f ` a} {\out 1. f Un g : (PROD x:?A. ?B(x))} \end{ttbox} -To construct functions of the form $f\union g$, we apply +To construct functions of the form $f\un g$, we apply \tdx{fun_disjoint_Un}: \begin{ttbox} by (resolve_tac [fun_disjoint_Un] 1);