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author | oheimb |

Mon, 06 Sep 1999 18:17:48 +0200 | |

changeset 7490 | 9a74b57740d1 |

parent 7489 | 77d654ea31a9 |

child 7491 | 95a4af0e10a7 |

added smp_tac

doc-src/HOL/HOL.tex | file | annotate | diff | comparison | revisions | |

src/HOL/HOL_lemmas.ML | file | annotate | diff | comparison | revisions |

--- a/doc-src/HOL/HOL.tex Mon Sep 06 17:03:19 1999 +0200 +++ b/doc-src/HOL/HOL.tex Mon Sep 06 18:17:48 1999 +0200 @@ -10,7 +10,7 @@ Church-style higher-order logic. Experience with the {\sc hol} system has demonstrated that higher-order logic is widely applicable in many areas of mathematics and computer science, not just hardware -verification, {\sc hol}'s original \textit{raison d'\^etre\/}. It is +verification, {\sc hol}'s original \textit{raison d'{\^e}tre\/}. It is weaker than {\ZF} set theory but for most applications this does not matter. If you prefer {\ML} to Lisp, you will probably prefer \HOL\ to~{\ZF}. @@ -46,7 +46,7 @@ \begin{constants} \it name &\it meta-type & \it description \\ \cdx{Trueprop}& $bool\To prop$ & coercion to $prop$\\ - \cdx{Not} & $bool\To bool$ & negation ($\neg$) \\ + \cdx{Not} & $bool\To bool$ & negation ($\lnot$) \\ \cdx{True} & $bool$ & tautology ($\top$) \\ \cdx{False} & $bool$ & absurdity ($\bot$) \\ \cdx{If} & $[bool,\alpha,\alpha]\To\alpha$ & conditional \\ @@ -132,13 +132,13 @@ Figure~\ref{hol-constants} lists the constants (including infixes and binders), while Fig.\ts\ref{hol-grammar} presents the grammar of higher-order logic. Note that $a$\verb|~=|$b$ is translated to -$\neg(a=b)$. +$\lnot(a=b)$. \begin{warn} \HOL\ has no if-and-only-if connective; logical equivalence is expressed using equality. But equality has a high priority, as befitting a relation, while if-and-only-if typically has the lowest priority. Thus, - $\neg\neg P=P$ abbreviates $\neg\neg (P=P)$ and not $(\neg\neg P)=P$. + $\lnot\lnot P=P$ abbreviates $\lnot\lnot (P=P)$ and not $(\lnot\lnot P)=P$. When using $=$ to mean logical equivalence, enclose both operands in parentheses. \end{warn} @@ -157,14 +157,14 @@ functions. \HOL\ offers various methods for introducing new types. -See~\S\ref{sec:HOL:Types} and~\S\ref{sec:HOL:datatype}. +See~{\S}\ref{sec:HOL:Types} and~{\S}\ref{sec:HOL:datatype}. Theory \thydx{Ord} defines the syntactic class \cldx{ord} of order signatures; the relations $<$ and $\leq$ are polymorphic over this class, as are the functions \cdx{mono}, \cdx{min} and \cdx{max}, and the \cdx{LEAST} operator. \thydx{Ord} also defines a subclass \cldx{order} of \cldx{ord} which axiomatizes partially ordered types -(w.r.t.\ $\le$). +(w.r.t.\ $\leq$). Three other syntactic type classes --- \cldx{plus}, \cldx{minus} and \cldx{times} --- permit overloading of the operators {\tt+},\index{*"+ @@ -175,10 +175,10 @@ If you state a goal containing overloaded functions, you may need to include type constraints. Type inference may otherwise make the goal more polymorphic than you intended, with confusing results. For example, the -variables $i$, $j$ and $k$ in the goal $i \le j \Imp i \le j+k$ have type +variables $i$, $j$ and $k$ in the goal $i \leq j \Imp i \leq j+k$ have type $\alpha::\{ord,plus\}$, although you may have expected them to have some numeric type, e.g. $nat$. Instead you should have stated the goal as -$(i::nat) \le j \Imp i \le j+k$, which causes all three variables to have +$(i::nat) \leq j \Imp i \leq j+k$, which causes all three variables to have type $nat$. \begin{warn} @@ -231,12 +231,12 @@ If $\tau$ is a type of class \cldx{ord}, $P$ a formula and $x$ a variable of type $\tau$, then the term \cdx{LEAST}~$x. P[x]$ is defined -to be the least (w.r.t.\ $\le$) $x$ such that $P~x$ holds (see +to be the least (w.r.t.\ $\leq$) $x$ such that $P~x$ holds (see Fig.~\ref{hol-defs}). The definition uses Hilbert's $\varepsilon$ choice operator, so \texttt{Least} is always meaningful, but may yield nothing useful in case there is not a unique least element satisfying $P$.\footnote{Class $ord$ does not require much of its instances, so - $\le$ need not be a well-ordering, not even an order at all!} + $\leq$ need not be a well-ordering, not even an order at all!} \medskip All these binders have priority 10. @@ -260,7 +260,7 @@ logical meaning. By declaring translations, you can cause instances of the \texttt{case} construct to denote applications of particular case operators. This is what happens automatically for each \texttt{datatype} definition -(see~\S\ref{sec:HOL:datatype}). +(see~{\S}\ref{sec:HOL:datatype}). \begin{warn} Both \texttt{if} and \texttt{case} constructs have as low a priority as @@ -335,7 +335,7 @@ The definitions above should never be expanded and are shown for completeness only. Instead users should reason in terms of the derived rules shown below or, better still, using high-level tactics -(see~\S\ref{sec:HOL:generic-packages}). +(see~{\S}\ref{sec:HOL:generic-packages}). \end{warn} Some of the rules mention type variables; for example, \texttt{refl} @@ -441,7 +441,14 @@ from subgoal $i$. \item[\ttindexbold{case_tac} {\tt"}$P${\tt"} $i$] performs case distinction on $P$ for subgoal $i$: the latter is replaced by two identical subgoals - with the added assumptions $P$ and $\neg P$, respectively. + with the added assumptions $P$ and $\lnot P$, respectively. +\item[\ttindexbold{smp_tac} $j$ $i$] applies $j$ times \texttt{spec} and then + \texttt{mp} in subgoal $i$, which is typically useful when forward-chaining + from an induction hypothesis. As a generalization of \texttt{mp_tac}, + if there are assumptions $\forall \vec{x}. P \vec{x} \imp Q \vec{x}$ and + $P \vec{a}$, ($\vec{x}$ being a vector of $j$ variables) + then it replaces the universally quantified implication by $Q \vec{a}$. + It may instantiate unknowns. It fails if it can do nothing. \end{ttdescription} @@ -592,7 +599,7 @@ constructs. Infix operators include union and intersection ($A\un B$ and $A\int B$), the subset and membership relations, and the image operator~{\tt``}\@. Note that $a$\verb|~:|$b$ is translated to -$\neg(a\in b)$. +$\lnot(a\in b)$. The $\{a@1,\ldots\}$ notation abbreviates finite sets constructed in the obvious manner using~\texttt{insert} and~$\{\}$: @@ -955,7 +962,7 @@ expressed by the theorem \tdx{split_if}: $$ \Var{P}(\mbox{\tt if}~\Var{b}~{\tt then}~\Var{x}~\mbox{\tt else}~\Var{y})~=~ -((\Var{b} \to \Var{P}(\Var{x})) \land (\neg \Var{b} \to \Var{P}(\Var{y}))) +((\Var{b} \to \Var{P}(\Var{x})) \land (\lnot \Var{b} \to \Var{P}(\Var{y}))) \eqno{(*)} $$ For example, a simple instance of $(*)$ is @@ -992,8 +999,8 @@ where $c$ is a constant and $rhs$ is arbitrary. Note that $(*)$ is of the right form because internally the left-hand side is $\Var{P}(\mathtt{If}~\Var{b}~\Var{x}~~\Var{y})$. Important further examples -are splitting rules for \texttt{case} expressions (see~\S\ref{subsec:list} -and~\S\ref{subsec:datatype:basics}). +are splitting rules for \texttt{case} expressions (see~{\S}\ref{subsec:list} +and~{\S}\ref{subsec:datatype:basics}). Analogous to \texttt{Addsimps} and \texttt{Delsimps}, there are also imperative versions of \texttt{addsplits} and \texttt{delsplits} @@ -1117,7 +1124,7 @@ \beta$, $\alpha + \beta$, $nat$ and $\alpha \; list$) and ways for introducing new types in general. The most important type construction, the \texttt{datatype}, is treated separately in -\S\ref{sec:HOL:datatype}. +{\S}\ref{sec:HOL:datatype}. \subsection{Product and sum types}\index{*"* type}\index{*"+ type} @@ -1225,7 +1232,7 @@ shown. The constructions are fairly standard and can be found in the respective theory files. Although the sum and product types are constructed manually for foundational reasons, they are represented as -actual datatypes later (see \S\ref{subsec:datatype:rep_datatype}). +actual datatypes later (see {\S}\ref{subsec:datatype:rep_datatype}). Therefore, the theory \texttt{Datatype} should be used instead of \texttt{Sum} or \texttt{Prod}. @@ -1335,7 +1342,7 @@ \texttt{nat} are part of the default simpset. Functions on \tydx{nat} can be defined by primitive or well-founded recursion; -see \S\ref{sec:HOL:recursive}. A simple example is addition. +see {\S}\ref{sec:HOL:recursive}. A simple example is addition. Here, \texttt{op +} is the name of the infix operator~\texttt{+}, following the standard convention. \begin{ttbox} @@ -1352,7 +1359,7 @@ the order of the two cases. Both \texttt{primrec} and \texttt{case} are realized by a recursion operator \cdx{nat_rec}, which is available because \textit{nat} is represented as -a datatype (see \S\ref{subsec:datatype:rep_datatype}). +a datatype (see {\S}\ref{subsec:datatype:rep_datatype}). %The predecessor relation, \cdx{pred_nat}, is shown to be well-founded. %Recursion along this relation resembles primitive recursion, but is @@ -1513,7 +1520,7 @@ \begin{center}\tt case $e$ of [] => $a$ | \(x\)\#\(xs\) => b \end{center} -is defined by translation. For details see~\S\ref{sec:HOL:datatype}. There +is defined by translation. For details see~{\S}\ref{sec:HOL:datatype}. There is also a case splitting rule \tdx{split_list_case} \[ \begin{array}{l} @@ -1524,10 +1531,10 @@ \end{array} \] which can be fed to \ttindex{addsplits} just like -\texttt{split_if} (see~\S\ref{subsec:HOL:case:splitting}). +\texttt{split_if} (see~{\S}\ref{subsec:HOL:case:splitting}). \texttt{List} provides a basic library of list processing functions defined by -primitive recursion (see~\S\ref{sec:HOL:primrec}). The recursion equations +primitive recursion (see~{\S}\ref{sec:HOL:primrec}). The recursion equations are shown in Fig.\ts\ref{fig:HOL:list-simps}. \index{list@{\textit{list}} type|)} @@ -1543,7 +1550,7 @@ \begin{warn} Types in \HOL\ must be non-empty; otherwise the quantifier rules would be - unsound, because $\exists x. x=x$ is a theorem \cite[\S7]{paulson-COLOG}. + unsound, because $\exists x. x=x$ is a theorem \cite[{\S}7]{paulson-COLOG}. \end{warn} A \bfindex{type definition} identifies the new type with a subset of an existing type. More precisely, the new type is defined by @@ -1699,11 +1706,11 @@ In Isabelle/HOL record types have to be defined explicitly, fixing their field names and types, and their (optional) parent record (see -\S\ref{sec:HOL:record-def}). Afterwards, records may be formed using above +{\S}\ref{sec:HOL:record-def}). Afterwards, records may be formed using above syntax, while obeying the canonical order of fields as given by their declaration. The record package also provides several operations like -selectors and updates (see \S\ref{sec:HOL:record-ops}), together with -characteristic properties (see \S\ref{sec:HOL:record-thms}). +selectors and updates (see {\S}\ref{sec:HOL:record-ops}), together with +characteristic properties (see {\S}\ref{sec:HOL:record-thms}). There is an example theory demonstrating most basic aspects of extensible records (see theory \texttt{HOL/ex/Points} in the Isabelle sources). @@ -1891,7 +1898,7 @@ Inductive datatypes, similar to those of \ML, frequently appear in applications of Isabelle/HOL. In principle, such types could be defined by -hand via \texttt{typedef} (see \S\ref{sec:typedef}), but this would be far too +hand via \texttt{typedef} (see {\S}\ref{sec:typedef}), but this would be far too tedious. The \ttindex{datatype} definition package of Isabelle/HOL (cf.\ \cite{Berghofer-Wenzel:1999:TPHOL}) automates such chores. It generates an appropriate \texttt{typedef} based on a least fixed-point construction, and @@ -1931,7 +1938,7 @@ \item $\tau = (\tau'@1,\ldots,\tau'@{h'})t'$, where $t'$ is the type constructor of an already existing datatype and $\tau'@1,\ldots,\tau'@{h'}$ are admissible types. -\item $\tau = \sigma \rightarrow \tau'$, where $\tau'$ is an admissible +\item $\tau = \sigma \to \tau'$, where $\tau'$ is an admissible type and $\sigma$ is non-recursive (i.e. the occurrences of the newly defined types are {\em strictly positive}) \end{itemize} @@ -1972,7 +1979,7 @@ \medskip Types in HOL must be non-empty. Each of the new datatypes -$(\alpha@1,\ldots,\alpha@h)t@j$ with $1 \le j \le n$ is non-empty iff it has a +$(\alpha@1,\ldots,\alpha@h)t@j$ with $1 \leq j \leq n$ is non-empty iff it has a constructor $C^j@i$ with the following property: for all argument types $\tau^j@{i,i'}$ of the form $(\alpha@1,\ldots,\alpha@h)t@{j'}$ the datatype $(\alpha@1,\ldots,\alpha@h)t@{j'}$ is non-empty. @@ -2014,7 +2021,7 @@ The datatype package also provides structural induction rules. For datatypes without nested recursion, this is of the following form: \[ -\infer{P@1~x@1 \wedge \dots \wedge P@n~x@n} +\infer{P@1~x@1 \land \dots \land P@n~x@n} {\begin{array}{lcl} \Forall x@1 \dots x@{m^1@1}. \List{P@{s^1@{1,1}}~x@{r^1@{1,1}}; \dots; @@ -2044,7 +2051,7 @@ \left\{\left(r^j@{i,1},s^j@{i,1}\right),\ldots, \left(r^j@{i,l^j@i},s^j@{i,l^j@i}\right)\right\} = \\[2ex] && \left\{(i',i'')~\left|~ - 1\leq i' \leq m^j@i \wedge 1 \leq i'' \leq n \wedge + 1\leq i' \leq m^j@i \land 1 \leq i'' \leq n \land \tau^j@{i,i'} = (\alpha@1,\ldots,\alpha@h)t@{i''}\right.\right\} \end{array} \] @@ -2070,7 +2077,7 @@ constructors \texttt{Nil} and \texttt{Cons}. Thus, the structural induction rule for \texttt{term} gets the form \[ -\infer{P@1~x@1 \wedge P@2~x@2} +\infer{P@1~x@1 \land P@2~x@2} {\begin{array}{l} \Forall x.~P@1~(\mathtt{Var}~x) \\ \Forall x@1~x@2.~P@2~x@2 \Imp P@1~(\mathtt{App}~x@1~x@2) \\ @@ -2105,7 +2112,7 @@ \end{array} \] where the $x@{i,j}$ are either identifiers or nested tuple patterns as in -\S\ref{subsec:prod-sum}. +{\S}\ref{subsec:prod-sum}. \begin{warn} All constructors must be present, their order is fixed, and nested patterns are not supported (with the exception of tuples). Violating this @@ -2126,7 +2133,7 @@ \] where $t@j$\texttt{_case} is the internal name of the \texttt{case}-construct. This theorem can be added to a simpset via \ttindex{addsplits} -(see~\S\ref{subsec:HOL:case:splitting}). +(see~{\S}\ref{subsec:HOL:case:splitting}). \subsubsection{The function \cdx{size}}\label{sec:HOL:size} @@ -2456,7 +2463,7 @@ \subsubsection{Example: Evaluation of expressions} Using mutual primitive recursion, we can define evaluation functions \texttt{evala} and \texttt{eval_bexp} for the datatypes of arithmetic and boolean expressions mentioned in -\S\ref{subsec:datatype:basics}: +{\S}\ref{subsec:datatype:basics}: \begin{ttbox} consts evala :: "['a => nat, 'a aexp] => nat" @@ -2516,7 +2523,7 @@ \subsubsection{Example: A substitution function for terms} Functions on datatypes with nested recursion, such as the type -\texttt{term} mentioned in \S\ref{subsec:datatype:basics}, are +\texttt{term} mentioned in {\S}\ref{subsec:datatype:basics}, are also defined by mutual primitive recursion. A substitution function \texttt{subst_term} on type \texttt{term}, similar to the functions \texttt{substa} and \texttt{substb} described above, can @@ -2536,7 +2543,7 @@ subst_term f t # subst_term_list f ts" \end{ttbox} The recursion scheme follows the structure of the unfolded definition of type -\texttt{term} shown in \S\ref{subsec:datatype:basics}. To prove properties of +\texttt{term} shown in {\S}\ref{subsec:datatype:basics}. To prove properties of this substitution function, mutual induction is needed: \begin{ttbox} Goal @@ -2625,7 +2632,7 @@ Typically, $f$ takes the recursive function's arguments (as a tuple) and returns a result expressed in terms of the function \texttt{size}. It is called a \textbf{measure function}. Recall that \texttt{size} is overloaded - and is defined on all datatypes (see \S\ref{sec:HOL:size}). + and is defined on all datatypes (see {\S}\ref{sec:HOL:size}). \item $\mathop{\mathtt{inv_image}} f\;R$ is a generalisation of \texttt{measure}. It specifies a relation such that $x\prec y$ iff $f(x)$ @@ -2989,7 +2996,7 @@ procedure. These are mostly taken from Pelletier \cite{pelletier86}. \item File \texttt{set.ML} proves Cantor's Theorem, which is presented in - \S\ref{sec:hol-cantor} below, and the Schr\"oder-Bernstein Theorem. + {\S}\ref{sec:hol-cantor} below, and the Schr{\"o}der-Bernstein Theorem. \item Theory \texttt{MT} contains Jacob Frost's formalization~\cite{frost93} of Milner and Tofte's coinduction example~\cite{milner-coind}. This

--- a/src/HOL/HOL_lemmas.ML Mon Sep 06 17:03:19 1999 +0200 +++ b/src/HOL/HOL_lemmas.ML Mon Sep 06 18:17:48 1999 +0200 @@ -424,9 +424,22 @@ in CHANGED_GOAL (rtac (th' RS ssubst)) end; +(* combination of (spec RS spec RS ...(j times) ... spec RS mp *) +local + fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t + | wrong_prem (Bound _) = true + | wrong_prem _ = false; + fun wrong (Const ("==>", _) $ (Const ("Trueprop", _) $ t) $ _) = wrong_prem t + | wrong _ = true; + val filter_right = filter (fn t => not (wrong (#prop (rep_thm t)))); +in + fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]); + fun smp_tac j = EVERY'[dresolve_tac (smp j), atac] +end; + + fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); - (** strip ! and --> from proved goal while preserving !-bound var names **) local