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author | lcp |

Fri, 15 Apr 1994 11:35:44 +0200 | |

changeset 310 | 66fc71f3a347 |

parent 309 | 3751567696bf |

child 311 | 3fb8cdb32e10 |

penultimate Springer draft

--- a/doc-src/Intro/advanced.tex Wed Apr 06 16:36:34 1994 +0200 +++ b/doc-src/Intro/advanced.tex Fri Apr 15 11:35:44 1994 +0200 @@ -28,30 +28,30 @@ A mathematical development goes through a progression of stages. Each stage defines some concepts and derives rules about them. We shall see how to derive rules, perhaps involving definitions, using Isabelle. The -following section will explain how to declare types, constants, axioms and +following section will explain how to declare types, constants, rules and definitions. \subsection{Deriving a rule using tactics and meta-level assumptions} \label{deriving-example} -\index{examples!of deriving rules} +\index{examples!of deriving rules}\index{assumptions!of main goal} The subgoal module supports the derivation of rules, as discussed in \S\ref{deriving}. The \ttindex{goal} command, when supplied a goal of the form $\List{\theta@1; \ldots; \theta@k} \Imp \phi$, creates $\phi\Imp\phi$ as the initial proof state and returns a list consisting of the theorems ${\theta@i\;[\theta@i]}$, for $i=1$, \ldots,~$k$. These meta-assumptions -are also recorded internally, allowing \ttindex{result} to discharge them +are also recorded internally, allowing {\tt result} to discharge them in the original order. Let us derive $\conj$ elimination using Isabelle. -Until now, calling \ttindex{goal} has returned an empty list, which we have +Until now, calling {\tt goal} has returned an empty list, which we have thrown away. In this example, the list contains the two premises of the rule. We bind them to the \ML\ identifiers {\tt major} and {\tt minor}:\footnote{Some ML compilers will print a message such as {\em binding not exhaustive}. This warns that {\tt goal} must return a 2-element list. Otherwise, the pattern-match will fail; ML will -raise exception \ttindex{Match}.} +raise exception \xdx{Match}.} \begin{ttbox} val [major,minor] = goal FOL.thy "[| P&Q; [| P; Q |] ==> R |] ==> R"; @@ -104,9 +104,8 @@ \subsection{Definitions and derived rules} \label{definitions} -\index{rules!derived} -\index{Isabelle!definitions in} -\index{definitions!reasoning about|bold} +\index{rules!derived}\index{definitions!and derived rules|(} + Definitions are expressed as meta-level equalities. Let us define negation and the if-and-only-if connective: \begin{eqnarray*} @@ -114,8 +113,7 @@ \Var{P}\bimp \Var{Q} & \equiv & (\Var{P}\imp \Var{Q}) \conj (\Var{Q}\imp \Var{P}) \end{eqnarray*} -\index{rewriting!meta-level|bold} -\index{unfolding|bold}\index{folding|bold} +\index{meta-rewriting} Isabelle permits {\bf meta-level rewriting} using definitions such as these. {\bf Unfolding} replaces every instance of $\neg \Var{P}$ by the corresponding instance of $\Var{P}\imp\bot$. For @@ -147,7 +145,7 @@ \subsection{Deriving the $\neg$ introduction rule} -To derive $(\neg I)$, we may call \ttindex{goal} with the appropriate +To derive $(\neg I)$, we may call {\tt goal} with the appropriate formula. Again, {\tt goal} returns a list consisting of the rule's premises. We bind this one-element list to the \ML\ identifier {\tt prems}. @@ -158,7 +156,7 @@ {\out 1. ~P} {\out val prems = ["P ==> False [P ==> False]"] : thm list} \end{ttbox} -Calling \ttindex{rewrite_goals_tac} with \ttindex{not_def}, which is the +Calling \ttindex{rewrite_goals_tac} with \tdx{not_def}, which is the definition of negation, unfolds that definition in the subgoals. It leaves the main goal alone. \begin{ttbox} @@ -169,7 +167,7 @@ {\out ~P} {\out 1. P --> False} \end{ttbox} -Using \ttindex{impI} and the premise, we reduce subgoal~1 to a triviality: +Using \tdx{impI} and the premise, we reduce subgoal~1 to a triviality: \begin{ttbox} by (resolve_tac [impI] 1); {\out Level 2} @@ -191,10 +189,10 @@ val notI = result(); {\out val notI = "(?P ==> False) ==> ~?P" : thm} \end{ttbox} -\indexbold{*notI} +\indexbold{*notI theorem} There is a simpler way of conducting this proof. The \ttindex{goalw} -command starts a backward proof, as does \ttindex{goal}, but it also +command starts a backward proof, as does {\tt goal}, but it also unfolds definitions. Thus there is no need to call \ttindex{rewrite_goals_tac}: \begin{ttbox} @@ -211,8 +209,8 @@ Let us derive the rule $(\neg E)$. The proof follows that of~{\tt conjE} above, with an additional step to unfold negation in the major premise. Although the {\tt goalw} command is best for this, let us -try~\ttindex{goal} to see another way of unfolding definitions. After -binding the premises to \ML\ identifiers, we apply \ttindex{FalseE}: +try~{\tt goal} to see another way of unfolding definitions. After +binding the premises to \ML\ identifiers, we apply \tdx{FalseE}: \begin{ttbox} val [major,minor] = goal FOL.thy "[| ~P; P |] ==> R"; {\out Level 0} @@ -257,7 +255,7 @@ val notE = result(); {\out val notE = "[| ~?P; ?P |] ==> ?R" : thm} \end{ttbox} -\indexbold{*notE} +\indexbold{*notE theorem} \medskip Again, there is a simpler way of conducting this proof. Recall that @@ -282,8 +280,9 @@ {\out R} {\out No subgoals!} \end{ttbox} +\index{definitions!and derived rules|)} -\goodbreak\medskip +\goodbreak\medskip\index{*"!"! symbol!in main goal} Finally, here is a trick that is sometimes useful. If the goal has an outermost meta-quantifier, then \ttindex{goal} and \ttindex{goalw} do not return the rule's premises in the list of theorems; instead, the @@ -297,8 +296,8 @@ val it = [] : thm list \end{ttbox} The proof continues as before. But instead of referring to \ML\ -identifiers, we refer to assumptions using \ttindex{eresolve_tac} or -\ttindex{assume_tac}: +identifiers, we refer to assumptions using {\tt eresolve_tac} or +{\tt assume_tac}: \begin{ttbox} by (resolve_tac [FalseE] 1); {\out Level 1} @@ -329,6 +328,7 @@ \section{Defining theories}\label{sec:defining-theories} \index{theories!defining|(} + Isabelle makes no distinction between simple extensions of a logic --- like defining a type~$bool$ with constants~$true$ and~$false$ --- and defining an entire logic. A theory definition has the form @@ -350,7 +350,8 @@ default sort for type variables. A constant declaration can specify an associated concrete syntax. The translations section specifies rewrite rules on abstract syntax trees, for defining notations and abbreviations. -The {\ML} section contains code to perform arbitrary syntactic +\index{*ML section} +The {\tt ML} section contains code to perform arbitrary syntactic transformations. The main declaration forms are discussed below. The full syntax can be found in \iflabelundefined{app:TheorySyntax}{the appendix of the {\it Reference Manual}}{App.\ts\ref{app:TheorySyntax}}. @@ -358,7 +359,7 @@ All the declaration parts can be omitted. In the simplest case, $T$ is just the union of $S@1$,~\ldots,~$S@n$. New theories always extend one or more other theories, inheriting their types, constants, syntax, etc. -The theory \ttindexbold{Pure} contains nothing but Isabelle's meta-logic. +The theory \thydx{Pure} contains nothing but Isabelle's meta-logic. Each theory definition must reside in a separate file, whose name is determined as follows: the theory name, say {\tt ListFn}, is converted to @@ -388,7 +389,8 @@ \subsection{Declaring constants and rules} -\indexbold{constants!declaring}\indexbold{rules!declaring} +\indexbold{constants!declaring}\index{rules!declaring} + Most theories simply declare constants and rules. The {\bf constant declaration part} has the form \begin{ttbox} @@ -414,6 +416,7 @@ $rule@n$ are expressions of type~$prop$. Each rule {\em must\/} be enclosed in quotation marks. +\indexbold{definitions} {\bf Definitions} are rules of the form $t\equiv u$. Normally definitions should be conservative, serving only as abbreviations. As of this writing, Isabelle does not provide a separate declaration part for definitions; it @@ -544,9 +547,11 @@ negate :: "signal => signal" \end{ttbox} -\subsection{Infixes and Mixfixes} -\indexbold{infix operators}\index{examples!of theories} -The constant declaration part of the theory +\subsection{Infix and Mixfix operators} +\index{infixes}\index{examples!of theories} + +Infix or mixfix syntax may be attached to constants. Consider the +following theory: \begin{ttbox} Gate2 = FOL + consts "~&" :: "[o,o] => o" (infixl 35) @@ -555,55 +560,63 @@ xor_def "P # Q == P & ~Q | ~P & Q" end \end{ttbox} -declares two left-associating infix operators: $\nand$ of precedence~35 and -$\xor$ of precedence~30. Hence $P \xor Q \xor R$ is parsed as $(P\xor -Q) \xor R$ and $P \xor Q \nand R$ as $P \xor (Q \nand R)$. Note the -quotation marks in \verb|"~&"| and \verb|"#"|. +The constant declaration part declares two left-associating infix operators +with their priorities, or precedences; they are $\nand$ of priority~35 and +$\xor$ of priority~30. Hence $P \xor Q \xor R$ is parsed as $(P\xor Q) +\xor R$ and $P \xor Q \nand R$ as $P \xor (Q \nand R)$. Note the quotation +marks in \verb|"~&"| and \verb|"#"|. The constants \hbox{\verb|op ~&|} and \hbox{\verb|op #|} are declared automatically, just as in \ML. Hence you may write propositions like \verb|op #(True) == op ~&(True)|, which asserts that the functions $\lambda Q.True \xor Q$ and $\lambda Q.True \nand Q$ are identical. -\indexbold{mixfix operators} -{\bf Mixfix} operators may have arbitrary context-free syntaxes. For example +\bigskip\index{mixfix declarations} +{\bf Mixfix} operators may have arbitrary context-free syntaxes. Let us +add a line to the constant declaration part: \begin{ttbox} If :: "[o,o,o] => o" ("if _ then _ else _") \end{ttbox} -declares a constant $If$ of type $[o,o,o] \To o$ with concrete syntax {\tt +This declares a constant $If$ of type $[o,o,o] \To o$ with concrete syntax {\tt if~$P$ then~$Q$ else~$R$} as well as {\tt If($P$,$Q$,$R$)}. Underscores -denote argument positions. Pretty-printing information can be specified in -order to improve the layout of formulae with mixfix operations. For -details, see {\em Isabelle's Object-Logics}. +denote argument positions. -Mixfix declarations can be annotated with precedences, just like +The declaration above does not allow the {\tt if}-{\tt then}-{\tt else} +construct to be split across several lines, even if it is too long to fit +on one line. Pretty-printing information can be added to specify the +layout of mixfix operators. For details, see +\iflabelundefined{Defining-Logics}% + {the {\it Reference Manual}, chapter `Defining Logics'}% + {Chap.\ts\ref{Defining-Logics}}. + +Mixfix declarations can be annotated with priorities, just like infixes. The example above is just a shorthand for \begin{ttbox} If :: "[o,o,o] => o" ("if _ then _ else _" [0,0,0] 1000) \end{ttbox} -The numeric components determine precedences. The list of integers -defines, for each argument position, the minimal precedence an expression -at that position must have. The final integer is the precedence of the +The numeric components determine priorities. The list of integers +defines, for each argument position, the minimal priority an expression +at that position must have. The final integer is the priority of the construct itself. In the example above, any argument expression is -acceptable because precedences are non-negative, and conditionals may -appear everywhere because 1000 is the highest precedence. On the other -hand, +acceptable because priorities are non-negative, and conditionals may +appear everywhere because 1000 is the highest priority. On the other +hand, the declaration \begin{ttbox} If :: "[o,o,o] => o" ("if _ then _ else _" [100,0,0] 99) \end{ttbox} defines concrete syntax for a conditional whose first argument cannot have -the form {\tt if~$P$ then~$Q$ else~$R$} because it must have a precedence -of at least~100. Since expressions put in parentheses have maximal -precedence, we may of course write +the form {\tt if~$P$ then~$Q$ else~$R$} because it must have a priority +of at least~100. We may of course write \begin{quote}\tt if (if $P$ then $Q$ else $R$) then $S$ else $T$ \end{quote} +because expressions in parentheses have maximal priority. Binary type constructors, like products and sums, may also be declared as infixes. The type declaration below introduces a type constructor~$*$ with infix notation $\alpha*\beta$, together with the mixfix notation ${<}\_,\_{>}$ for pairs. -\index{examples!of theories} +\index{examples!of theories}\index{mixfix declarations} \begin{ttbox} Prod = FOL + types ('a,'b) "*" (infixl 20) @@ -643,7 +656,7 @@ types with the constants ${+} :: [\alpha,\alpha]\To \alpha$ and $0,1 {::} \alpha$, for $\alpha{::}arith$. We introduce $arith$ as a subclass of $term$ and add the three polymorphic constants of this class. -\index{examples!of theories} +\index{examples!of theories}\index{constants!overloaded} \begin{ttbox} Arith = FOL + classes arith < term @@ -696,7 +709,7 @@ Section\ts\ref{sec:logical-syntax} has formalized a first-order logic, including a type~$nat$ and the constants $0::nat$ and $Suc::nat\To nat$. Let us introduce the Peano axioms for mathematical induction and the -freeness of $0$ and~$Suc$: +freeness of $0$ and~$Suc$:\index{axioms!Peano} \[ \vcenter{\infer[(induct)]{P[n/x]}{P[0/x] & \infer*{P[Suc(x)/x]}{[P]}}} \qquad \parbox{4.5cm}{provided $x$ is not free in any assumption except~$P$} \] @@ -748,12 +761,12 @@ \subsection{Declaring the theory to Isabelle} \index{examples!of theories} -Let us create the theory \ttindexbold{Nat} starting from theory~\verb$FOL$, +Let us create the theory \thydx{Nat} starting from theory~\verb$FOL$, which contains only classical logic with no natural numbers. We declare the 0-place type constructor $nat$ and the associated constants. Note that the constant~0 requires a mixfix annotation because~0 is not a legal identifier, and could not otherwise be written in terms: -\begin{ttbox} +\begin{ttbox}\index{mixfix declarations} Nat = FOL + types nat arities nat :: term @@ -811,8 +824,9 @@ \section{Refinement with explicit instantiation} -\index{refinement!with instantiation|bold} -\index{instantiation!explicit|bold} +\index{resolution!with instantiation} +\index{instantiation|(} + In order to employ mathematical induction, we need to refine a subgoal by the rule~$(induct)$. The conclusion of this rule is $\Var{P}(\Var{n})$, which is highly ambiguous in higher-order unification. It matches every @@ -825,12 +839,12 @@ a rule. But it also accepts explicit instantiations for the rule's schematic variables. \begin{description} -\item[\ttindexbold{res_inst_tac} {\it insts} {\it thm} {\it i}] +\item[\ttindex{res_inst_tac} {\it insts} {\it thm} {\it i}] instantiates the rule {\it thm} with the instantiations {\it insts}, and then performs resolution on subgoal~$i$. -\item[\ttindexbold{eres_inst_tac}] -and \ttindexbold{dres_inst_tac} are similar, but perform elim-resolution +\item[\ttindex{eres_inst_tac}] +and \ttindex{dres_inst_tac} are similar, but perform elim-resolution and destruct-resolution, respectively. \end{description} The list {\it insts} consists of pairs $[(v@1,e@1), \ldots, (v@n,e@n)]$, @@ -844,7 +858,7 @@ whenever the type of~$e@i$ is an instance of the type of~$\Var{v@i}$. \subsection{A simple proof by induction} -\index{proof!by induction}\index{examples!of induction} +\index{examples!of induction} Let us prove that no natural number~$k$ equals its own successor. To use~$(induct)$, we instantiate~$\Var{n}$ to~$k$; Isabelle finds a good instantiation for~$\Var{P}$. @@ -863,8 +877,8 @@ We should check that Isabelle has correctly applied induction. Subgoal~1 is the base case, with $k$ replaced by~0. Subgoal~2 is the inductive step, with $k$ replaced by~$Suc(x)$ and with an induction hypothesis for~$x$. -The rest of the proof demonstrates~\ttindex{notI}, \ttindex{notE} and the -other rules of~\ttindex{Nat.thy}. The base case holds by~\ttindex{Suc_neq_0}: +The rest of the proof demonstrates~\tdx{notI}, \tdx{notE} and the +other rules of theory {\tt Nat}. The base case holds by~\ttindex{Suc_neq_0}: \begin{ttbox} by (resolve_tac [notI] 1); {\out Level 2} @@ -949,23 +963,26 @@ \end{ttbox} Inspecting subgoal~1 reveals that induction has been applied to just the second occurrence of~$n$. This perfectly legitimate induction is useless -here. The main goal admits fourteen different applications of induction. -The number is exponential in the size of the formula. +here. + +The main goal admits fourteen different applications of induction. The +number is exponential in the size of the formula. \subsection{Proving that addition is associative} -Let us invoke the induction rule properly properly, -using~\ttindex{res_inst_tac}. At the same time, we shall have a glimpse at -Isabelle's rewriting tactics, which are described in the {\em Reference - Manual}. - -\index{rewriting!object-level}\index{examples!of rewriting} +Let us invoke the induction rule properly properly, using~{\tt + res_inst_tac}. At the same time, we shall have a glimpse at Isabelle's +simplification tactics, which are described in +\iflabelundefined{simp-chap}% + {the {\em Reference Manual}}{Chap.\ts\ref{simp-chap}}. -Isabelle's rewriting tactics repeatedly applies equations to a subgoal, -simplifying or proving it. For efficiency, the rewriting rules must be -packaged into a \bfindex{simplification set}, or {\bf simpset}. We take -the standard simpset for first-order logic and insert the equations -for~{\tt add} proved in the previous section, namely $0+n=n$ and ${\tt - Suc}(m)+n={\tt Suc}(m+n)$: +\index{simplification}\index{examples!of simplification} + +Isabelle's simplification tactics repeatedly apply equations to a subgoal, +perhaps proving it. For efficiency, the rewrite rules must be +packaged into a {\bf simplification set},\index{simplification sets} +or {\bf simpset}. We take the standard simpset for first-order logic and +insert the equations for~{\tt add} proved in the previous section, namely +$0+n=n$ and ${\tt Suc}(m)+n={\tt Suc}(m+n)$: \begin{ttbox} val add_ss = FOL_ss addrews [add_0, add_Suc]; \end{ttbox} @@ -1004,6 +1021,7 @@ {\out k + m + n = k + (m + n)} {\out No subgoals!} \end{ttbox} +\index{instantiation|)} \section{A Prolog interpreter} @@ -1037,9 +1055,9 @@ \] \index{examples!of theories} -Theory \ttindex{Prolog} extends first-order logic in order to make use +Theory \thydx{Prolog} extends first-order logic in order to make use of the class~$term$ and the type~$o$. The interpreter does not use the -rules of~\ttindex{FOL}. +rules of~{\tt FOL}. \begin{ttbox} Prolog = FOL + types 'a list @@ -1103,7 +1121,7 @@ \end{ttbox} -\subsection{Backtracking}\index{backtracking} +\subsection{Backtracking}\index{backtracking!Prolog style} Prolog backtracking can answer questions that have multiple solutions. Which lists $x$ and $y$ can be appended to form the list $[a,b,c,d]$? This question has five solutions. Using \ttindex{REPEAT} to apply the rules, we @@ -1196,11 +1214,12 @@ {\out rev(c : b : a : Nil, a : b : c : Nil)} {\out No subgoals!} \end{ttbox} -\ttindex{REPEAT} stops when it cannot continue, regardless of which state -is reached. The tactical \ttindex{DEPTH_FIRST} searches for a satisfactory -state, as specified by an \ML{} predicate. Below, +\ttindex{REPEAT} goes wrong because it is only a repetition tactical, not a +search tactical. {\tt REPEAT} stops when it cannot continue, regardless of +which state is reached. The tactical \ttindex{DEPTH_FIRST} searches for a +satisfactory state, as specified by an \ML{} predicate. Below, \ttindex{has_fewer_prems} specifies that the proof state should have no -subgoals. +subgoals. \begin{ttbox} val prolog_tac = DEPTH_FIRST (has_fewer_prems 1) (resolve_tac rules 1);