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author | lcp |

Fri, 15 Apr 1994 12:13:37 +0200 | |

changeset 312 | 7ceea59b4748 |

parent 311 | 3fb8cdb32e10 |

child 313 | a45ae7b38672 |

penultimate Springer draft

--- a/doc-src/Intro/foundations.tex Fri Apr 15 11:48:23 1994 +0200 +++ b/doc-src/Intro/foundations.tex Fri Apr 15 12:13:37 1994 +0200 @@ -50,29 +50,32 @@ \end{figure} \section{Formalizing logical syntax in Isabelle}\label{sec:logical-syntax} -\index{Isabelle!formalizing syntax|bold} 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. -Isabelle represents syntax using the 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. +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{$\To$|bold}\index{types} +\index{types!syntax of}\index{types!function}\index{*fun type} Isabelle has \ML-style type constructors such as~$(\alpha)list$, where $(bool)list$ is the type of lists of booleans. Function types have the -form $\sigma\To\tau$, where $\sigma$ and $\tau$ are types. Curried -function types may be abbreviated: +form $(\sigma,\tau)fun$ or $\sigma\To\tau$, where $\sigma$ and $\tau$ are +types. Curried function types may be abbreviated: \[ \sigma@1\To (\cdots \sigma@n\To \tau\cdots) \quad \hbox{as} \quad - [\sigma@1, \ldots, \sigma@n] \To \tau $$ + [\sigma@1, \ldots, \sigma@n] \To \tau \] +\index{terms!syntax of} The syntax for terms is summarised below. Note that function application is written $t(u)$ rather than the usual $t\,u$. \[ +\index{lambda abs@$\lambda$-abstractions}\index{function applications} \begin{array}{ll} t :: \tau & \hbox{type constraint, on a term or bound variable} \\ \lambda x.t & \hbox{abstraction} \\ @@ -92,6 +95,7 @@ let us declare a type~$nat$ of natural numbers. Later, we shall see how to admit terms of other types. +\index{constants}\index{*nat type}\index{*o type} After declaring the types~$o$ and~$nat$, we may declare constants for the symbols of our logic. Since $\bot$ denotes a truth value (falsity) and 0 denotes a number, we put \begin{eqnarray*} @@ -119,10 +123,11 @@ \subsection{Polymorphic types and constants} \label{polymorphic} \index{types!polymorphic|bold} \index{equality!polymorphic} +\index{constants!polymorphic} Which type should we assign to the equality symbol? If we tried $[nat,nat]\To o$, then equality would be restricted to the natural -numbers; we would have to declare different equality symbols for each +numbers; we should have to declare different equality symbols for each type. Isabelle's type system is polymorphic, so we could declare \begin{eqnarray*} {=} & :: & [\alpha,\alpha]\To o, @@ -140,11 +145,13 @@ types. They closely resemble the classes of the functional language Haskell~\cite{haskell-tutorial,haskell-report}. +\index{*logic class}\index{*term class} Isabelle provides the built-in class $logic$, which consists of the logical types: the ones we want to reason about. Let us declare a class $term$, to consist of all legal types of terms in our logic. The subclass structure is now $term\le logic$. +\index{*nat type} We put $nat$ in class $term$ by declaring $nat{::}term$. We declare the equality constant by \begin{eqnarray*} @@ -154,23 +161,24 @@ $term$. Such type variables resemble Standard~\ML's equality type variables. -We give function types and~$o$ the class $logic$ rather than~$term$, since +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 $(\tau)list$ 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 -constructors: \index{$\To$|bold}\index{arities!declaring} -\begin{eqnarray*} - o & :: & logic \\ - {\To} & :: & (logic,logic)logic \\ +constructors:\index{arities!declaring} +\begin{eqnarray*}\index{*o type}\index{*fun type} + o & :: & logic \\ + fun & :: & (logic,logic)logic \\ nat, string, real & :: & term \\ - list & :: & (term)term + 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 functions; this requires the arity declarations ${o::term}$ -and ${{\To}::(term,term)term}$. The class system can also handle +and ${fun::(term,term)term}$. The class system can also handle overloading.\index{overloading|bold} We could declare $arith$ to be the subclass of $term$ consisting of the `arithmetic' types, such as~$nat$. Then we could declare the operators @@ -212,7 +220,7 @@ \subsection{Higher types and quantifiers} -\index{types!higher|bold} +\index{types!higher|bold}\index{quantifiers} Quantifiers are regarded as operations upon functions. Ignoring polymorphism for the moment, consider the formula $\forall x. P(x)$, where $x$ ranges over type~$nat$. This is true if $P(x)$ is true for all~$x$. Abstracting @@ -244,12 +252,9 @@ \section{Formalizing logical rules in Isabelle} -\index{meta-logic|bold} -\index{Isabelle!formalizing rules|bold} -\index{$\Imp$|bold}\index{$\Forall$|bold}\index{$\equiv$|bold} -\index{implication!meta-level|bold} -\index{quantifiers!meta-level|bold} -\index{equality!meta-level|bold} +\index{meta-implication|bold} +\index{meta-quantifiers|bold} +\index{meta-equality|bold} Object-logics are formalized by extending Isabelle's meta-logic~\cite{paulson89}, which is intuitionistic higher-order logic. @@ -263,26 +268,27 @@ all $x$', and expresses {\bf generality} in rules and axiom schemes. \item The equality \(a\equiv b\) means `$a$ equals $b$', for expressing - \bfindex{definitions} (see~\S\ref{definitions}). Equalities left over - from the unification process, so called \bfindex{flex-flex equations}, - are written $a\qeq b$. The two equality symbols have the same logical - meaning. + {\bf definitions} (see~\S\ref{definitions}).\index{definitions} + Equalities left over from the unification process, so called {\bf + flex-flex constraints},\index{flex-flex constraints} are written $a\qeq + b$. The two equality symbols have the same logical meaning. \end{itemize} -The syntax of the meta-logic is formalized in precisely in the same manner -as object-logics, using the typed $\lambda$-calculus. Analogous to +The syntax of the meta-logic is formalized in the same manner +as object-logics, using the simply typed $\lambda$-calculus. Analogous to type~$o$ above, there is a built-in type $prop$ of meta-level truth values. Meta-level formulae will have this type. Type $prop$ belongs to class~$logic$; also, $\sigma\To\tau$ belongs to $logic$ provided $\sigma$ and $\tau$ do. Here are the types of the built-in connectives: -\begin{eqnarray*} +\begin{eqnarray*}\index{*prop type}\index{*logic class} \Imp & :: & [prop,prop]\To prop \\ \Forall & :: & (\alpha{::}logic\To prop) \To prop \\ {\equiv} & :: & [\alpha{::}\{\},\alpha]\To prop \\ - \qeq & :: & [\alpha{::}\{\},\alpha]\To prop c + \qeq & :: & [\alpha{::}\{\},\alpha]\To prop \end{eqnarray*} -The restricted polymorphism in $\Forall$ excludes certain types, those used -just for parsing. +The polymorphism in $\Forall$ is restricted to class~$logic$ to exclude +certain types, those used just for parsing. The type variable +$\alpha{::}\{\}$ ranges over the universal sort. In our formalization of first-order logic, we declared a type~$o$ of object-level truth values, rather than using~$prop$ for this purpose. If @@ -292,7 +298,7 @@ the distinction between the meta-level and the object-level. To formalize the inference rules, we shall need to relate the two levels; accordingly, we declare the constant -\index{Trueprop@{$Trueprop$}} +\index{*Trueprop constant} \begin{eqnarray*} Trueprop & :: & o\To prop. \end{eqnarray*} @@ -302,7 +308,7 @@ \subsection{Expressing propositional rules} -\index{rules!propositional|bold} +\index{rules!propositional} We shall illustrate the use of the meta-logic by formalizing the rules of Fig.\ts\ref{fol-fig}. Each object-level rule is expressed as a meta-level axiom. @@ -315,10 +321,11 @@ reading is correct because the meta-logic has simple models, where types denote sets and $\Forall$ really means `for all.' -\index{Trueprop@{$Trueprop$}} +\index{*Trueprop constant} Isabelle adopts notational conventions to ease the writing of rules. We may hide the occurrences of $Trueprop$ by making it an implicit coercion. Outer universal quantifiers may be dropped. Finally, the nested implication +\index{meta-implication} \[ \phi@1\Imp(\cdots \phi@n\Imp\psi\cdots) \] may be abbreviated as $\List{\phi@1; \ldots; \phi@n} \Imp \psi$, which formalizes a rule of $n$~premises. @@ -334,8 +341,8 @@ $$ P \Imp P\disj Q \qquad Q \Imp P\disj Q \eqno(\disj I1,2) $$ $$ \List{P\disj Q; P\Imp R; Q\Imp R} \Imp R \eqno(\disj E) $$ -\noindent -To understand this treatment of assumptions\index{assumptions} in natural +\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 $P$ is true and show that $Q$ must then be true. More concisely, if $P$ @@ -354,12 +361,13 @@ \begin{warn} Earlier versions of Isabelle, and certain papers~\cite{paulson89,paulson700}, use $\List{P}$ to mean $Trueprop(P)$. -\index{Trueprop@{$Trueprop$}} \end{warn} \subsection{Quantifier rules and substitution} -\index{rules!quantifier|bold}\index{substitution|bold} -\index{variables!bound} +\index{quantifiers}\index{rules!quantifier}\index{substitution|bold} +\index{variables!bound}\index{lambda abs@$\lambda$-abstractions} +\index{function applications} + Isabelle expresses variable binding using $\lambda$-abstraction; for instance, $\forall x.P$ is formalized as $\forall(\lambda x.P)$. Recall that $F(t)$ is Isabelle's syntax for application of the function~$F$ to the argument~$t$; @@ -375,8 +383,8 @@ \ldots {\it (some set of formulae)},' where $x$ is a variable (called a {\bf parameter} or {\bf eigenvariable}) in some premise. Isabelle treats provisos using~$\Forall$, its inbuilt notion of `for all'. +\index{meta-quantifiers} -\index{$\Forall$} The purpose of the proviso `$x$ not free in \ldots' is to ensure that the premise may not make assumptions about the value of~$x$, and therefore holds for all~$x$. We formalize $(\forall I)$ by @@ -394,7 +402,7 @@ $$ P(t) \Imp \exists x.P(x) \eqno(\exists I)$$ $$ \List{\exists x.P(x);\; \Forall x. P(x)\Imp Q} \Imp Q \eqno(\exists E) $$ Let us verify $(\exists E)$ semantically. Suppose that the premises -hold; since $\exists x.P(x)$ is true, we may choose $a$ such that $P(a)$ is +hold; since $\exists x.P(x)$ is true, we may choose an~$a$ such that $P(a)$ is true. Instantiating $\Forall x. P(x)\Imp Q$ with $a$ yields $P(a)\Imp Q$, and we obtain the desired conclusion, $Q$. @@ -402,16 +410,17 @@ \[ \infer{P[u/t]}{t=u & P} \] would be hard to formalize in Isabelle. It calls for replacing~$t$ by $u$ throughout~$P$, which cannot be expressed using $\beta$-conversion. Our -rule~$(subst)$ uses the occurrences of~$x$ in~$P$ as a template for -substitution, inferring $P[u/x]$ from~$P[t/x]$. When we formalize this as -an axiom, the template becomes a function variable: +rule~$(subst)$ uses~$P$ as a template for substitution, inferring $P[u/x]$ +from~$P[t/x]$. When we formalize this as an axiom, the template becomes a +function variable: $$ \List{t=u; P(t)} \Imp P(u). \eqno(subst)$$ \subsection{Signatures and theories} -\index{signatures|bold}\index{theories|bold} +\index{signatures|bold} + A {\bf signature} contains the information necessary for type checking, -parsing and pretty printing. It specifies classes and their +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. @@ -421,6 +430,7 @@ Under similar conditions, a signature can be extended. Signatures are managed internally by Isabelle; users seldom encounter them. +\index{theories|bold} A {\bf theory} consists of a signature plus a collection of axioms. The {\bf pure} theory contains only the meta-logic. Theories can be combined provided their signatures are compatible. A theory definition extends an @@ -453,15 +463,13 @@ coexist at the same time, and you may work in each of these during a single session. -\begin{warn} +\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. \end{warn} \section{Proof construction in Isabelle} -\index{Isabelle!proof construction in|bold} - I have elsewhere described the meta-logic and demonstrated it by formalizing first-order logic~\cite{paulson89}. There is a one-to-one correspondence between meta-level proofs and object-level proofs. To each @@ -487,23 +495,24 @@ its inputs, we need not state how many clock ticks will be required. Such quantities often emerge from the proof. -Isabelle provides {\bf schematic variables}, or \bfindex{unknowns}, for -unification. Logically, unknowns are free variables. But while ordinary -variables remain fixed, unification may instantiate unknowns. Unknowns are -written with a ?\ prefix and are frequently subscripted: $\Var{a}$, -$\Var{a@1}$, $\Var{a@2}$, \ldots, $\Var{P}$, $\Var{P@1}$, \ldots. +Isabelle provides {\bf schematic variables}, or {\bf + unknowns},\index{unknowns} for unification. Logically, unknowns are free +variables. But while ordinary variables remain fixed, unification may +instantiate unknowns. Unknowns are written with a ?\ prefix and are +frequently subscripted: $\Var{a}$, $\Var{a@1}$, $\Var{a@2}$, \ldots, +$\Var{P}$, $\Var{P@1}$, \ldots. Recall that an inference rule of the form \[ \infer{\phi}{\phi@1 & \ldots & \phi@n} \] is formalized in Isabelle's meta-logic as the axiom -$\List{\phi@1; \ldots; \phi@n} \Imp \phi$. +$\List{\phi@1; \ldots; \phi@n} \Imp \phi$.\index{resolution} Such axioms resemble Prolog's Horn clauses, and can be combined by resolution --- Isabelle's principal proof method. Resolution yields both forward and backward proof. Backward proof works by unifying a goal with the conclusion of a rule, whose premises become new subgoals. Forward proof works by unifying theorems with the premises of a rule, deriving a new theorem. -Isabelle axioms require an extended notion of resolution. +Isabelle formulae require an extended notion of resolution. They differ from Horn clauses in two major respects: \begin{itemize} \item They are written in the typed $\lambda$-calculus, and therefore must be @@ -533,9 +542,9 @@ \Var{f}(t) \qeq g(u@1,\ldots,u@k). \end{equation} Huet's~\cite{huet75} search procedure solves equations by imitation and -projection. {\bf Imitation}\index{imitation|bold} makes~$\Var{f}$ apply -leading symbol (if a constant) of the right-hand side. To solve -equation~(\ref{hou-eqn}), it guesses +projection. {\bf Imitation} makes~$\Var{f}$ apply the leading symbol (if a +constant) of the right-hand side. To solve equation~(\ref{hou-eqn}), it +guesses \[ \Var{f} \equiv \lambda x. g(\Var{h@1}(x),\ldots,\Var{h@k}(x)), \] where $\Var{h@1}$, \ldots, $\Var{h@k}$ are new unknowns. Assuming there are no other occurrences of~$\Var{f}$, equation~(\ref{hou-eqn}) simplifies to the @@ -544,7 +553,6 @@ If the procedure solves these equations, instantiating $\Var{h@1}$, \ldots, $\Var{h@k}$, then it yields an instantiation for~$\Var{f}$. -\index{projection|bold} {\bf Projection} makes $\Var{f}$ apply one of its arguments. To solve equation~(\ref{hou-eqn}), if $t$ expects~$m$ arguments and delivers a result of suitable type, it guesses @@ -554,7 +562,7 @@ equation \[ t(\Var{h@1}(t),\ldots,\Var{h@m}(t)) \qeq g(u@1,\ldots,u@k). $$ -\begin{warn} +\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 @@ -565,7 +573,7 @@ to be a function type. \end{warn} -\index{unknowns!of function type|bold} +\index{unknowns!function|bold} Given $\Var{f}(t@1,\ldots,t@n)\qeq u$, Huet's procedure could make up to $n+1$ guesses. The search tree and set of unifiers may be infinite. But higher-order unification can work effectively, provided you are careful @@ -681,7 +689,7 @@ \] \subsection{Lifting over assumptions} -\index{lifting!over assumptions|bold} +\index{assumptions!lifting over} Lifting over $\theta\Imp{}$ is the following meta-inference rule: \[ \infer{\List{\theta\Imp\phi@1; \ldots; \theta\Imp\phi@n} \Imp (\theta \Imp \phi)} @@ -714,7 +722,7 @@ \[ \infer{P\imp(Q\imp R).}{\infer*{R}{[P,Q]}} \] \subsection{Lifting over parameters} -\index{lifting!over parameters|bold} +\index{parameters!lifting over} An analogous form of lifting handles premises of the form $\Forall x\ldots\,$. Here, lifting prefixes an object-rule's premises and conclusion with $\Forall x$. At the same time, lifting introduces a dependence upon~$x$. It replaces @@ -751,8 +759,7 @@ \section{Backward proof by resolution} -\index{resolution!in backward proof}\index{proof!backward|bold} -\index{tactics}\index{tacticals} +\index{resolution!in backward proof} Resolution is convenient for deriving simple rules and for reasoning forward from facts. It can also support backward proof, where we start @@ -762,19 +769,19 @@ expressing proof searches. {\bf Tactics} refine subgoals while {\bf tacticals} combine tactics. -\index{LCF} +\index{LCF system} Isabelle's tactics and tacticals work differently from {\sc lcf}'s. An Isabelle rule is bidirectional: there is no distinction between inputs and outputs. {\sc lcf} has a separate tactic for each rule; Isabelle performs refinement by any rule in a uniform fashion, using resolution. -\index{subgoals|bold}\index{main goal|bold} Isabelle works with meta-level theorems of the form \( \List{\phi@1; \ldots; \phi@n} \Imp \phi \). We have viewed this as the {\bf rule} with premises $\phi@1$,~\ldots,~$\phi@n$ and conclusion~$\phi$. It can also be viewed as -the \bfindex{proof state} with subgoals $\phi@1$,~\ldots,~$\phi@n$ and main +the {\bf proof state}\index{proof state} +with subgoals $\phi@1$,~\ldots,~$\phi@n$ and main goal~$\phi$. To prove the formula~$\phi$, take $\phi\Imp \phi$ as the initial proof @@ -787,7 +794,6 @@ \phi@n} \Imp \phi'$, where $\phi'$ is an instance of~$\phi$. \subsection{Refinement by resolution} -\index{refinement|bold} To refine subgoal~$i$ of a proof state by a rule, perform the following resolution: \[ \infer{\hbox{new proof state}}{\hbox{rule} & & \hbox{proof state}} \] @@ -807,7 +813,7 @@ the proof state. \subsection{Proof by assumption} -\index{proof!by assumption|bold} +\index{assumptions!use of} In the course of a natural deduction proof, parameters $x@1$, \ldots,~$x@l$ and assumptions $\theta@1$, \ldots, $\theta@k$ accumulate, forming a context for each subgoal. Repeated lifting steps can lift a rule into any context. To @@ -958,23 +964,25 @@ best-first search (where a heuristic function selects the best state to explore) return their outcomes as a sequence. Isabelle provides such procedures in the form of tacticals. Simpler procedures can be expressed -directly using the basic tacticals {\it THEN}, {\it ORELSE} and {\it REPEAT}: -\begin{description} -\item[$tac1\;THEN\;tac2$] is a tactic for sequential composition. Applied +directly using the basic tacticals {\tt THEN}, {\tt ORELSE} and {\tt REPEAT}: +\begin{ttdescription} +\item[$tac1$ THEN $tac2$] is a tactic for sequential composition. Applied to a proof state, it returns all states reachable in two steps by applying $tac1$ followed by~$tac2$. -\item[$tac1\;ORELSE\;tac2$] is a choice tactic. Applied to a state, it +\item[$tac1$ ORELSE $tac2$] is a choice tactic. Applied to a state, it 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 +\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 -{\it THEN} and~{\it ORELSE}. -\end{description} +{\tt THEN} and~{\tt ORELSE}. +\end{ttdescription} For instance, this tactic repeatedly applies $tac1$ and~$tac2$, giving $tac1$ priority: -\[ REPEAT(tac1\;ORELSE\;tac2) \] +\begin{center} \tt +REPEAT($tac1$ ORELSE $tac2$) +\end{center} \section{Variations on resolution} @@ -988,7 +996,8 @@ can also serve to derive rules. \subsection{Elim-resolution} -\index{elim-resolution|bold} +\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 $((R\disj R)\disj R)\disj R$. Applying $(\disj E)$ to this assumption yields two subgoals, one that @@ -1075,11 +1084,13 @@ Isabelle. \subsection{Destruction rules} \label{destruct} -\index{destruction rules|bold}\index{proof!forward} +\index{rules!destruction}\index{rules!elimination} +\index{forward proof} + Looking back to Fig.\ts\ref{fol-fig}, notice that there are two kinds of elimination rule. The rules $({\conj}E1)$, $({\conj}E2)$, $({\imp}E)$ and $({\forall}E)$ extract the conclusion from the major premise. In Isabelle -parlance, such rules are called \bfindex{destruction rules}; they are readable +parlance, such rules are called {\bf destruction rules}; they are readable and easy to use in forward proof. The rules $({\disj}E)$, $({\bot}E)$ and $({\exists}E)$ work by discharging assumptions; they support backward proof in a style reminiscent of the sequent calculus. @@ -1099,7 +1110,7 @@ \[ \infer[\quad\hbox{to the elimination rule}\quad]{Q}{P@1 & \ldots & P@m} \infer{R.}{P@1 & \ldots & P@m & \infer*{R}{[Q]}} \] -\index{destruct-resolution|bold} +\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~$Q$, @@ -1110,7 +1121,7 @@ \subsection{Deriving rules by resolution} \label{deriving} -\index{rules!derived|bold} +\index{rules!derived|bold}\index{meta-assumptions!syntax of} The meta-logic, itself a form of the predicate calculus, is defined by a system of natural deduction rules. Each theorem may depend upon meta-assumptions. The theorem that~$\phi$ follows from the assumptions