doc-src/IsarImplementation/Thy/logic.thy
 author wenzelm Fri Sep 08 19:44:43 2006 +0200 (2006-09-08) changeset 20494 99ad217b6974 parent 20493 48fea5e99505 child 20498 825a8d2335ce permissions -rw-r--r--
tuned;
2 (* $Id$ *)
4 theory logic imports base begin
6 chapter {* Primitive logic \label{ch:logic} *}
8 text {*
9   The logical foundations of Isabelle/Isar are that of the Pure logic,
10   which has been introduced as a natural-deduction framework in
11   \cite{paulson700}.  This is essentially the same logic as @{text
12   "\<lambda>HOL"}'' in the more abstract setting of Pure Type Systems (PTS)
13   \cite{Barendregt-Geuvers:2001}, although there are some key
14   differences in the specific treatment of simple types in
15   Isabelle/Pure.
17   Following type-theoretic parlance, the Pure logic consists of three
18   levels of @{text "\<lambda>"}-calculus with corresponding arrows: @{text
19   "\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text
20   "\<And>"} for universal quantification (proofs depending on terms), and
21   @{text "\<Longrightarrow>"} for implication (proofs depending on proofs).
23   Pure derivations are relative to a logical theory, which declares
24   type constructors, term constants, and axioms.  Theory declarations
25   support schematic polymorphism, which is strictly speaking outside
26   the logic.\footnote{Incidently, this is the main logical reason, why
27   the theory context @{text "\<Theta>"} is separate from the context @{text
28   "\<Gamma>"} of the core calculus.}
29 *}
32 section {* Types \label{sec:types} *}
34 text {*
35   The language of types is an uninterpreted order-sorted first-order
36   algebra; types are qualified by ordered type classes.
38   \medskip A \emph{type class} is an abstract syntactic entity
39   declared in the theory context.  The \emph{subclass relation} @{text
40   "c\<^isub>1 \<subseteq> c\<^isub>2"} is specified by stating an acyclic
41   generating relation; the transitive closure is maintained
42   internally.  The resulting relation is an ordering: reflexive,
43   transitive, and antisymmetric.
45   A \emph{sort} is a list of type classes written as @{text
46   "{c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
47   intersection.  Notationally, the curly braces are omitted for
48   singleton intersections, i.e.\ any class @{text "c"} may be read as
49   a sort @{text "{c}"}.  The ordering on type classes is extended to
50   sorts according to the meaning of intersections: @{text
51   "{c\<^isub>1, \<dots> c\<^isub>m} \<subseteq> {d\<^isub>1, \<dots>, d\<^isub>n}"} iff
52   @{text "\<forall>j. \<exists>i. c\<^isub>i \<subseteq> d\<^isub>j"}.  The empty intersection
53   @{text "{}"} refers to the universal sort, which is the largest
54   element wrt.\ the sort order.  The intersections of all (finitely
55   many) classes declared in the current theory are the minimal
56   elements wrt.\ the sort order.
58   \medskip A \emph{fixed type variable} is a pair of a basic name
59   (starting with a @{text "'"} character) and a sort constraint.  For
60   example, @{text "('a, s)"} which is usually printed as @{text
61   "\<alpha>\<^isub>s"}.  A \emph{schematic type variable} is a pair of an
62   indexname and a sort constraint.  For example, @{text "(('a, 0),
63   s)"} which is usually printed as @{text "?\<alpha>\<^isub>s"}.
65   Note that \emph{all} syntactic components contribute to the identity
66   of type variables, including the sort constraint.  The core logic
67   handles type variables with the same name but different sorts as
68   different, although some outer layers of the system make it hard to
69   produce anything like this.
71   A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator
72   on types declared in the theory.  Type constructor application is
73   usually written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}.
74   For @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text
75   "prop"} instead of @{text "()prop"}.  For @{text "k = 1"} the
76   parentheses are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text
77   "(\<alpha>)list"}.  Further notation is provided for specific constructors,
78   notably the right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of
79   @{text "(\<alpha>, \<beta>)fun"}.
81   A \emph{type} is defined inductively over type variables and type
82   constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s |
83   (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}.
85   A \emph{type abbreviation} is a syntactic abbreviation @{text
86   "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over
87   variables @{text "\<^vec>\<alpha>"}.  Type abbreviations looks like type
88   constructors at the surface, but are fully expanded before entering
89   the logical core.
91   A \emph{type arity} declares the image behavior of a type
92   constructor wrt.\ the algebra of sorts: @{text "\<kappa> :: (s\<^isub>1, \<dots>,
93   s\<^isub>k)s"} means that @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>k)\<kappa>"} is
94   of sort @{text "s"} if every argument type @{text "\<tau>\<^isub>i"} is
95   of sort @{text "s\<^isub>i"}.  Arity declarations are implicitly
96   completed, i.e.\ @{text "\<kappa> :: (\<^vec>s)c"} entails @{text "\<kappa> ::
97   (\<^vec>s)c'"} for any @{text "c' \<supseteq> c"}.
99   \medskip The sort algebra is always maintained as \emph{coregular},
100   which means that type arities are consistent with the subclass
101   relation: for each type constructor @{text "\<kappa>"} and classes @{text
102   "c\<^isub>1 \<subseteq> c\<^isub>2"}, any arity @{text "\<kappa> ::
103   (\<^vec>s\<^isub>1)c\<^isub>1"} has a corresponding arity @{text "\<kappa>
104   :: (\<^vec>s\<^isub>2)c\<^isub>2"} where @{text "\<^vec>s\<^isub>1 \<subseteq>
105   \<^vec>s\<^isub>2"} holds componentwise.
107   The key property of a coregular order-sorted algebra is that sort
108   constraints may be always solved in a most general fashion: for each
109   type constructor @{text "\<kappa>"} and sort @{text "s"} there is a most
110   general vector of argument sorts @{text "(s\<^isub>1, \<dots>,
111   s\<^isub>k)"} such that a type scheme @{text
112   "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is
113   of sort @{text "s"}.  Consequently, the unification problem on the
114   algebra of types has most general solutions (modulo renaming and
115   equivalence of sorts).  Moreover, the usual type-inference algorithm
116   will produce primary types as expected \cite{nipkow-prehofer}.
117 *}
119 text %mlref {*
120   \begin{mldecls}
121   @{index_ML_type class} \\
122   @{index_ML_type sort} \\
123   @{index_ML_type arity} \\
124   @{index_ML_type typ} \\
125   @{index_ML fold_atyps: "(typ -> 'a -> 'a) -> typ -> 'a -> 'a"} \\
126   @{index_ML Sign.subsort: "theory -> sort * sort -> bool"} \\
127   @{index_ML Sign.of_sort: "theory -> typ * sort -> bool"} \\
128   @{index_ML Sign.add_types: "(bstring * int * mixfix) list -> theory -> theory"} \\
129   @{index_ML Sign.add_tyabbrs_i: "
130   (bstring * string list * typ * mixfix) list -> theory -> theory"} \\
131   @{index_ML Sign.primitive_class: "string * class list -> theory -> theory"} \\
132   @{index_ML Sign.primitive_classrel: "class * class -> theory -> theory"} \\
133   @{index_ML Sign.primitive_arity: "arity -> theory -> theory"} \\
134   \end{mldecls}
136   \begin{description}
138   \item @{ML_type class} represents type classes; this is an alias for
139   @{ML_type string}.
141   \item @{ML_type sort} represents sorts; this is an alias for
142   @{ML_type "class list"}.
144   \item @{ML_type arity} represents type arities; this is an alias for
145   triples of the form @{text "(\<kappa>, \<^vec>s, s)"} for @{text "\<kappa> ::
146   (\<^vec>s)s"} described above.
148   \item @{ML_type typ} represents types; this is a datatype with
149   constructors @{ML TFree}, @{ML TVar}, @{ML Type}.
151   \item @{ML fold_atyps}~@{text "f \<tau>"} iterates function @{text "f"}
152   over all occurrences of atoms (@{ML TFree} or @{ML TVar}) of @{text
153   "\<tau>"}; the type structure is traversed from left to right.
155   \item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"}
156   tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}.
158   \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether a type
159   is of a given sort.
161   \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares new
162   type constructors @{text "\<kappa>"} with @{text "k"} arguments and
163   optional mixfix syntax.
165   \item @{ML Sign.add_tyabbrs_i}~@{text "[(\<kappa>, \<^vec>\<alpha>, \<tau>, mx), \<dots>]"}
166   defines a new type abbreviation @{text "(\<^vec>\<alpha>)\<kappa> = \<tau>"} with
167   optional mixfix syntax.
169   \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
170   c\<^isub>n])"} declares new class @{text "c"}, together with class
171   relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
173   \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
174   c\<^isub>2)"} declares class relation @{text "c\<^isub>1 \<subseteq>
175   c\<^isub>2"}.
177   \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
178   arity @{text "\<kappa> :: (\<^vec>s)s"}.
180   \end{description}
181 *}
185 section {* Terms \label{sec:terms} *}
187 text {*
188   \glossary{Term}{FIXME}
190   The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
191   with de-Bruijn indices for bound variables, and named free
192   variables, and constants.  Terms with loose bound variables are
193   usually considered malformed.  The types of variables and constants
194   is stored explicitly at each occurrence in the term (which is a
195   known performance issue).
197   FIXME de-Bruijn representation of lambda terms
199   Term syntax provides explicit abstraction @{text "\<lambda>x :: \<alpha>. b(x)"}
200   and application @{text "t u"}, while types are usually implicit
201   thanks to type-inference.
203   Terms of type @{text "prop"} are called
204   propositions.  Logical statements are composed via @{text "\<And>x ::
205   \<alpha>. B(x)"} and @{text "A \<Longrightarrow> B"}.
206 *}
209 text {*
211 FIXME
213 \glossary{Schematic polymorphism}{FIXME}
215 \glossary{Type variable}{FIXME}
217 *}
220 section {* Theorems \label{sec:thms} *}
222 text {*
224   Primitive reasoning operates on judgments of the form @{text "\<Gamma> \<turnstile>
225   \<phi>"}, with standard introduction and elimination rules for @{text
226   "\<And>"} and @{text "\<Longrightarrow>"} that refer to fixed parameters @{text "x"} and
227   hypotheses @{text "A"} from the context @{text "\<Gamma>"}.  The
228   corresponding proof terms are left implicit in the classic
229   LCF-approach'', although they could be exploited separately
230   \cite{Berghofer-Nipkow:2000}.
232   The framework also provides definitional equality @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha>
233   \<Rightarrow> prop"}, with @{text "\<alpha>\<beta>\<eta>"}-conversion rules.  The internal
234   conjunction @{text "& :: prop \<Rightarrow> prop \<Rightarrow> prop"} enables the view of
235   assumptions and conclusions emerging uniformly as simultaneous
236   statements.
240   FIXME
242 \glossary{Proposition}{A \seeglossary{term} of \seeglossary{type}
243 @{text "prop"}.  Internally, there is nothing special about
244 propositions apart from their type, but the concrete syntax enforces a
245 clear distinction.  Propositions are structured via implication @{text
246 "A \<Longrightarrow> B"} or universal quantification @{text "\<And>x. B x"} --- anything
247 else is considered atomic.  The canonical form for propositions is
248 that of a \seeglossary{Hereditary Harrop Formula}.}
250 \glossary{Theorem}{A proven proposition within a certain theory and
251 proof context, formally @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}; both contexts are
252 rarely spelled out explicitly.  Theorems are usually normalized
253 according to the \seeglossary{HHF} format.}
255 \glossary{Fact}{Sometimes used interchangably for
256 \seeglossary{theorem}.  Strictly speaking, a list of theorems,
257 essentially an extra-logical conjunction.  Facts emerge either as
258 local assumptions, or as results of local goal statements --- both may
259 be simultaneous, hence the list representation.}
261 \glossary{Schematic variable}{FIXME}
263 \glossary{Fixed variable}{A variable that is bound within a certain
264 proof context; an arbitrary-but-fixed entity within a portion of proof
265 text.}
267 \glossary{Free variable}{Synonymous for \seeglossary{fixed variable}.}
269 \glossary{Bound variable}{FIXME}
271 \glossary{Variable}{See \seeglossary{schematic variable},
272 \seeglossary{fixed variable}, \seeglossary{bound variable}, or
273 \seeglossary{type variable}.  The distinguishing feature of different
274 variables is their binding scope.}
276 *}
279 section {* Proof terms *}
281 text {*
282   FIXME !?
283 *}
286 section {* Rules \label{sec:rules} *}
288 text {*
290 FIXME
292   A \emph{rule} is any Pure theorem in HHF normal form; there is a
293   separate calculus for rule composition, which is modeled after
294   Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows
295   rules to be nested arbitrarily, similar to \cite{extensions91}.
297   Normally, all theorems accessible to the user are proper rules.
298   Low-level inferences are occasional required internally, but the
299   result should be always presented in canonical form.  The higher
300   interfaces of Isabelle/Isar will always produce proper rules.  It is
301   important to maintain this invariant in add-on applications!
303   There are two main principles of rule composition: @{text
304   "resolution"} (i.e.\ backchaining of rules) and @{text
305   "by-assumption"} (i.e.\ closing a branch); both principles are
306   combined in the variants of @{text "elim-resosultion"} and @{text
307   "dest-resolution"}.  Raw @{text "composition"} is occasionally
308   useful as well, also it is strictly speaking outside of the proper
309   rule calculus.
311   Rules are treated modulo general higher-order unification, which is
312   unification modulo the equational theory of @{text "\<alpha>\<beta>\<eta>"}-conversion
313   on @{text "\<lambda>"}-terms.  Moreover, propositions are understood modulo
314   the (derived) equivalence @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.
316   This means that any operations within the rule calculus may be
317   subject to spontaneous @{text "\<alpha>\<beta>\<eta>"}-HHF conversions.  It is common
318   practice not to contract or expand unnecessarily.  Some mechanisms
319   prefer an one form, others the opposite, so there is a potential
320   danger to produce some oscillation!
322   Only few operations really work \emph{modulo} HHF conversion, but
323   expect a normal form: quantifiers @{text "\<And>"} before implications
324   @{text "\<Longrightarrow>"} at each level of nesting.
326 \glossary{Hereditary Harrop Formula}{The set of propositions in HHF
327 format is defined inductively as @{text "H = (\<And>x\<^sup>*. H\<^sup>* \<Longrightarrow>
328 A)"}, for variables @{text "x"} and atomic propositions @{text "A"}.
329 Any proposition may be put into HHF form by normalizing with the rule
330 @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.  In Isabelle, the outermost
331 quantifier prefix is represented via \seeglossary{schematic
332 variables}, such that the top-level structure is merely that of a
333 \seeglossary{Horn Clause}}.
335 \glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.}
337 *}
339 end