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doc-src/Intro/foundations.tex

changeset 331 | 13660d5f6a77 |

parent 312 | 7ceea59b4748 |

child 1865 | 484956c42436 |

--- a/doc-src/Intro/foundations.tex Fri Apr 22 12:43:53 1994 +0200 +++ b/doc-src/Intro/foundations.tex Fri Apr 22 18:08:57 1994 +0200 @@ -50,21 +50,24 @@ \end{figure} \section{Formalizing logical syntax in Isabelle}\label{sec:logical-syntax} +\index{first-order logic} + Figure~\ref{fol-fig} presents intuitionistic first-order logic, including equality. Let us see how to formalize this logic in Isabelle, illustrating the main features of Isabelle's polymorphic meta-logic. +\index{lambda calc@$\lambda$-calculus} Isabelle represents syntax using the simply typed $\lambda$-calculus. We declare a type for each syntactic category of the logic. We declare a constant for each symbol of the logic, giving each $n$-place operation an $n$-argument curried function type. Most importantly, $\lambda$-abstraction represents variable binding in quantifiers. -\index{lambda calc@$\lambda$-calculus} \index{lambda - abs@$\lambda$-abstractions} -\index{types!syntax of}\index{types!function}\index{*fun type} -Isabelle has \ML-style type constructors such as~$(\alpha)list$, where +\index{types!syntax of}\index{types!function}\index{*fun type} +\index{type constructors} +Isabelle has \ML-style polymorphic types such as~$(\alpha)list$, where +$list$ is a type constructor and $\alpha$ is a type variable; for example, $(bool)list$ is the type of lists of booleans. Function types have the form $(\sigma,\tau)fun$ or $\sigma\To\tau$, where $\sigma$ and $\tau$ are types. Curried function types may be abbreviated: @@ -116,8 +119,9 @@ precedences, so that we write $P\conj Q\disj R$ instead of $\disj(\conj(P,Q), R)$. -\S\ref{sec:defining-theories} below describes the syntax of Isabelle theory -files and illustrates it by extending our logic with mathematical induction. +Section~\ref{sec:defining-theories} below describes the syntax of Isabelle +theory files and illustrates it by extending our logic with mathematical +induction. \subsection{Polymorphic types and constants} \label{polymorphic} @@ -164,7 +168,7 @@ We give~$o$ and function types the class $logic$ rather than~$term$, since they are not legal types for terms. We may introduce new types of class $term$ --- for instance, type $string$ or $real$ --- at any time. We can -even declare type constructors such as $(\alpha)list$, and state that type +even declare type constructors such as~$list$, and state that type $(\tau)list$ belongs to class~$term$ provided $\tau$ does; equality applies to lists of natural numbers but not to lists of formulae. We may summarize this paragraph by a set of {\bf arity declarations} for type @@ -176,7 +180,7 @@ list & :: & (term)term \end{eqnarray*} (Recall that $fun$ is the type constructor for function types.) -In higher-order logic, equality does apply to truth values and +In \rmindex{higher-order logic}, equality does apply to truth values and functions; this requires the arity declarations ${o::term}$ and ${fun::(term,term)term}$. The class system can also handle overloading.\index{overloading|bold} We could declare $arith$ to be the @@ -186,7 +190,7 @@ {+},{-},{\times},{/} & :: & [\alpha{::}arith,\alpha]\To \alpha \end{eqnarray*} If we declare new types $real$ and $complex$ of class $arith$, then we -effectively have three sets of operators: +in effect have three sets of operators: \begin{eqnarray*} {+},{-},{\times},{/} & :: & [nat,nat]\To nat \\ {+},{-},{\times},{/} & :: & [real,real]\To real \\ @@ -216,7 +220,10 @@ Even with overloading, each term has a unique, most general type. For this to be possible, the class and type declarations must satisfy certain -technical constraints~\cite{nipkow-prehofer}. +technical constraints; see +\iflabelundefined{sec:ref-defining-theories}% + {Sect.\ Defining Theories in the {\em Reference Manual}}% + {\S\ref{sec:ref-defining-theories}}. \subsection{Higher types and quantifiers} @@ -338,10 +345,10 @@ \noindent Next, consider the disjunction rules. The discharge of assumption in $(\disj E)$ is expressed using $\Imp$: +\index{assumptions!discharge of}% $$ P \Imp P\disj Q \qquad Q \Imp P\disj Q \eqno(\disj I1,2) $$ $$ \List{P\disj Q; P\Imp R; Q\Imp R} \Imp R \eqno(\disj E) $$ - -\noindent\index{assumptions!discharge of} +% To understand this treatment of assumptions in natural deduction, look at implication. The rule $({\imp}I)$ is the classic example of natural deduction: to prove that $P\imp Q$ is true, assume $P$ @@ -421,9 +428,8 @@ A {\bf signature} contains the information necessary for type checking, parsing and pretty printing a term. It specifies classes and their -relationships; types, with their arities, and constants, with -their types. It also contains syntax rules, specified using mixfix -declarations. +relationships, types and their arities, constants and their types, etc. It +also contains syntax rules, specified using mixfix declarations. Two signatures can be merged provided their specifications are compatible --- they must not, for example, assign different types to the same constant. @@ -457,13 +463,13 @@ {} & {} &\hbox{Pure}& {} & {} \end{array} \] -\end{tt} +\end{tt}% Each Isabelle proof typically works within a single theory, which is associated with the proof state. However, many different theories may coexist at the same time, and you may work in each of these during a single session. -\begin{warn}\index{constants!clashes with variables} +\begin{warn}\index{constants!clashes with variables}% Confusing problems arise if you work in the wrong theory. Each theory defines its own syntax. An identifier may be regarded in one theory as a constant and in another as a variable. @@ -481,7 +487,7 @@ properties must be demonstrated separately for each object-logic. The meta-logic is defined by a collection of inference rules, including -equational rules for the $\lambda$-calculus, and logical rules. The rules +equational rules for the $\lambda$-calculus and logical rules. The rules for~$\Imp$ and~$\Forall$ resemble those for~$\imp$ and~$\forall$ in Fig.\ts\ref{fol-fig}. Proofs performed using the primitive meta-rules would be lengthy; Isabelle proofs normally use certain derived rules. @@ -562,7 +568,7 @@ equation \[ t(\Var{h@1}(t),\ldots,\Var{h@m}(t)) \qeq g(u@1,\ldots,u@k). $$ -\begin{warn}\index{unification!incompleteness of} +\begin{warn}\index{unification!incompleteness of}% Huet's unification procedure is complete. Isabelle's polymorphic version, which solves for type unknowns as well as for term unknowns, is incomplete. The problem is that projection requires type information. In @@ -637,7 +643,7 @@ \] When resolving two rules, the unknowns in the first rule are renamed, by subscripting, to make them distinct from the unknowns in the second rule. To -resolve $({\imp}E)$ with itself, the first copy of the rule would become +resolve $({\imp}E)$ with itself, the first copy of the rule becomes \[ \List{\Var{P@1}\imp \Var{Q@1}; \Var{P@1}} \Imp \Var{Q@1}. \] Resolving this with $({\imp}E)$ in the first premise, unifying $\Var{Q@1}$ with $\Var{P}\imp \Var{Q}$, is the meta-level inference @@ -647,9 +653,8 @@ \List{\Var{P}\imp \Var{Q}; \Var{P}} \Imp \Var{Q}} \] Renaming the unknowns in the resolvent, we have derived the -object-level rule +object-level rule\index{rules!derived} \[ \infer{Q.}{R\imp (P\imp Q) & R & P} \] -\index{rules!derived} Joining rules in this fashion is a simple way of proving theorems. The derived rules are conservative extensions of the object-logic, and may permit simpler proofs. Let us consider another example. Suppose we have the axiom @@ -797,9 +802,10 @@ To refine subgoal~$i$ of a proof state by a rule, perform the following resolution: \[ \infer{\hbox{new proof state}}{\hbox{rule} & & \hbox{proof state}} \] -If the rule is $\List{\psi'@1; \ldots; \psi'@m} \Imp \psi'$ after lifting -over subgoal~$i$, and the proof state is $\List{\phi@1; \ldots; \phi@n} -\Imp \phi$, then the new proof state is (for~$1\leq i\leq n$) +Suppose the rule is $\List{\psi'@1; \ldots; \psi'@m} \Imp \psi'$ after +lifting over subgoal~$i$'s assumptions and parameters. If the proof state +is $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, then the new proof state is +(for~$1\leq i\leq n$) \[ (\List{\phi@1; \ldots; \phi@{i-1}; \psi'@1; \ldots; \psi'@m; \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s. \] Substitution~$s$ unifies $\psi'$ with~$\phi@i$. In the proof state, @@ -974,9 +980,8 @@ tries~$tac1$ and returns the result if non-empty; otherwise, it uses~$tac2$. \item[REPEAT $tac$] is a repetition tactic. Applied to a state, it -returns all states reachable by applying~$tac$ as long as possible (until -it would fail). $REPEAT$ can be expressed in a few lines of \ML{} using -{\tt THEN} and~{\tt ORELSE}. +returns all states reachable by applying~$tac$ as long as possible --- until +it would fail. \end{ttdescription} For instance, this tactic repeatedly applies $tac1$ and~$tac2$, giving $tac1$ priority: @@ -999,7 +1004,7 @@ \index{elim-resolution|bold}\index{assumptions!deleting} Consider proving the theorem $((R\disj R)\disj R)\disj R\imp R$. By -$({\imp}I)$, we prove $R$ from the assumption $((R\disj R)\disj R)\disj R$. +$({\imp}I)$, we prove~$R$ from the assumption $((R\disj R)\disj R)\disj R$. Applying $(\disj E)$ to this assumption yields two subgoals, one that assumes~$R$ (and is therefore trivial) and one that assumes $(R\disj R)\disj R$. This subgoal admits another application of $(\disj E)$. Since @@ -1008,9 +1013,8 @@ \[ \List{((R\disj R)\disj R)\disj R; (R\disj R)\disj R; R\disj R; R} \Imp R. \] In general, using $(\disj E)$ on the assumption $P\disj Q$ creates two new subgoals with the additional assumption $P$ or~$Q$. In these subgoals, -$P\disj Q$ is redundant. It wastes space; worse, it could make a theorem -prover repeatedly apply~$(\disj E)$, looping. Other elimination rules pose -similar problems. In first-order logic, only universally quantified +$P\disj Q$ is redundant. Other elimination rules behave +similarly. In first-order logic, only universally quantified assumptions are sometimes needed more than once --- say, to prove $P(f(f(a)))$ from the assumptions $\forall x.P(x)\imp P(f(x))$ and~$P(a)$. @@ -1053,7 +1057,7 @@ \[ \List{\Var{P}\disj \Var{Q};\; \Var{P}\Imp \Var{R};\; \Var{Q}\Imp \Var{R}} \Imp \Var{R} \] and the proof state is $(P\disj P\Imp P)\Imp (P\disj P\imp P)$. The -lifted rule would be +lifted rule is \[ \begin{array}{l@{}l} \lbrakk\;& P\disj P \Imp \Var{P@1}\disj\Var{Q@1}; \\ & \List{P\disj P ;\; \Var{P@1}} \Imp \Var{R@1}; \\ @@ -1061,10 +1065,10 @@ \rbrakk\;& \Imp \Var{R@1} \end{array} \] -Unification would take the simultaneous equations +Unification takes the simultaneous equations $P\disj P \qeq \Var{P@1}\disj\Var{Q@1}$ and $\Var{R@1} \qeq P$, yielding $\Var{P@1}\equiv\Var{Q@1}\equiv\Var{R@1} \equiv P$. The new proof state -would be simply +is simply \[ \List{P \Imp P;\; P \Imp P} \Imp (P\disj P\imp P). \] Elim-resolution's simultaneous unification gives better control @@ -1110,14 +1114,13 @@ \[ \infer[\quad\hbox{to the elimination rule}\quad]{Q}{P@1 & \ldots & P@m} \infer{R.}{P@1 & \ldots & P@m & \infer*{R}{[Q]}} \] -\index{destruct-resolution} -{\bf Destruct-resolution} combines this transformation with -elim-resolution. It applies a destruction rule to some assumption of a -subgoal. Given the rule above, it replaces the assumption~$P@1$ by~$Q$, -with new subgoals of showing instances of $P@2$, \ldots,~$P@m$. -Destruct-resolution works forward from a subgoal's assumptions. Ordinary -resolution performs forward reasoning from theorems, as illustrated -in~\S\ref{joining}. +{\bf Destruct-resolution}\index{destruct-resolution} combines this +transformation with elim-resolution. It applies a destruction rule to some +assumption of a subgoal. Given the rule above, it replaces the +assumption~$P@1$ by~$Q$, with new subgoals of showing instances of $P@2$, +\ldots,~$P@m$. Destruct-resolution works forward from a subgoal's +assumptions. Ordinary resolution performs forward reasoning from theorems, +as illustrated in~\S\ref{joining}. \subsection{Deriving rules by resolution} \label{deriving} @@ -1149,10 +1152,14 @@ state \[ \phantom{\List{P\conj Q;\; P\conj Q}} \llap{$\List{P;Q}$}\Imp R \quad [\,\List{P;Q}\Imp R\,]. \] -Resolving subgoals~1 and~2 with $({\conj}E1)$ and $({\conj}E2)$, +Resolving subgoals~1 and~2 with~$({\conj}E1)$ and~$({\conj}E2)$, respectively, yields the state -\[ \List{P\conj Q;\; P\conj Q}\Imp R \quad [\,\List{P;Q}\Imp R\,]. \] -Resolving both subgoals with the assumption $P\conj Q$ yields +\[ \List{P\conj \Var{Q@1};\; \Var{P@2}\conj Q}\Imp R + \quad [\,\List{P;Q}\Imp R\,]. +\] +The unknowns $\Var{Q@1}$ and~$\Var{P@2}$ arise from unconstrained +subformulae in the premises of~$({\conj}E1)$ and~$({\conj}E2)$. Resolving +both subgoals with the assumption $P\conj Q$ instantiates the unknowns to yield \[ R \quad [\, \List{P;Q}\Imp R, P\conj Q \,]. \] The proof may use the meta-assumptions in any order, and as often as necessary; when finished, we discharge them in the correct order to @@ -1163,7 +1170,7 @@ schematic variables. \begin{warn} -Schematic variables are not allowed in meta-assumptions because they would -complicate lifting. Meta-assumptions remain fixed throughout a proof. + Schematic variables are not allowed in meta-assumptions, for a variety of + reasons. Meta-assumptions remain fixed throughout a proof. \end{warn}