doc-src/IsarImplementation/Thy/logic.thy
 author wenzelm Mon Sep 11 14:35:25 2006 +0200 (2006-09-11) changeset 20501 de0b523b0d62 parent 20498 825a8d2335ce child 20514 5ede702cd2ca permissions -rw-r--r--
more rules;
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.
204   $205 \infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{} 206 \qquad 207 \infer{@{text "(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>"}}{@{text "t :: \<sigma>"}} 208 \qquad 209 \infer{@{text "t u :: \<sigma>"}}{@{text "t :: \<tau> \<Rightarrow> \<sigma>"} & @{text "u :: \<tau>"}} 210$
212 *}
215 text {*
217 FIXME
219 \glossary{Schematic polymorphism}{FIXME}
221 \glossary{Type variable}{FIXME}
223 *}
226 section {* Theorems \label{sec:thms} *}
228 text {*
229   \glossary{Proposition}{A \seeglossary{term} of \seeglossary{type}
230   @{text "prop"}.  Internally, there is nothing special about
231   propositions apart from their type, but the concrete syntax enforces
232   a clear distinction.  Propositions are structured via implication
233   @{text "A \<Longrightarrow> B"} or universal quantification @{text "\<And>x. B x"} ---
234   anything else is considered atomic.  The canonical form for
235   propositions is that of a \seeglossary{Hereditary Harrop Formula}. FIXME}
237   \glossary{Theorem}{A proven proposition within a certain theory and
238   proof context, formally @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}; both contexts are
239   rarely spelled out explicitly.  Theorems are usually normalized
240   according to the \seeglossary{HHF} format. FIXME}
242   \glossary{Fact}{Sometimes used interchangably for
243   \seeglossary{theorem}.  Strictly speaking, a list of theorems,
244   essentially an extra-logical conjunction.  Facts emerge either as
245   local assumptions, or as results of local goal statements --- both
246   may be simultaneous, hence the list representation. FIXME}
248   \glossary{Schematic variable}{FIXME}
250   \glossary{Fixed variable}{A variable that is bound within a certain
251   proof context; an arbitrary-but-fixed entity within a portion of
252   proof text. FIXME}
254   \glossary{Free variable}{Synonymous for \seeglossary{fixed
255   variable}. FIXME}
257   \glossary{Bound variable}{FIXME}
259   \glossary{Variable}{See \seeglossary{schematic variable},
260   \seeglossary{fixed variable}, \seeglossary{bound variable}, or
261   \seeglossary{type variable}.  The distinguishing feature of
262   different variables is their binding scope. FIXME}
264   A \emph{proposition} is a well-formed term of type @{text "prop"}.
265   The connectives of minimal logic are declared as constants of the
266   basic theory:
268   \smallskip
269   \begin{tabular}{ll}
270   @{text "all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop"} & universal quantification (binder @{text "\<And>"}) \\
271   @{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\
272   \end{tabular}
274   \medskip A \emph{theorem} is a proven proposition, depending on a
275   collection of assumptions, and axioms from the theory context.  The
276   judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is defined
277   inductively by the primitive inferences given in
278   \figref{fig:prim-rules}; there is a global syntactic restriction
279   that the hypotheses may not contain schematic variables.
281   \begin{figure}[htb]
282   \begin{center}
283   $284 \infer[@{text "(axiom)"}]{@{text "\<turnstile> A"}}{@{text "A \<in> \<Theta>"}} 285 \qquad 286 \infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{} 287$
288   $289 \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}{@{text "\<Gamma> \<turnstile> b x"} & @{text "x \<notin> \<Gamma>"}} 290 \qquad 291 \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b a"}}{@{text "\<Gamma> \<turnstile> \<And>x. b x"}} 292$
293   $294 \infer[@{text "(\<Longrightarrow>_intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}} 295 \qquad 296 \infer[@{text "(\<Longrightarrow>_elim)"}]{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}} 297$
298   \caption{Primitive inferences of the Pure logic}\label{fig:prim-rules}
299   \end{center}
300   \end{figure}
302   The introduction and elimination rules for @{text "\<And>"} and @{text
303   "\<Longrightarrow>"} are analogous to formation of (dependently typed) @{text
304   "\<lambda>"}-terms representing the underlying proof objects.  Proof terms
305   are \emph{irrelevant} in the Pure logic, they may never occur within
306   propositions, i.e.\ the @{text "\<Longrightarrow>"} arrow of the framework is a
307   non-dependent one.
309   Also note that fixed parameters as in @{text "\<And>_intro"} need not be
310   recorded in the context @{text "\<Gamma>"}, since syntactic types are
311   always inhabitable.  An assumption'' @{text "x :: \<tau>"} is logically
312   vacuous, because @{text "\<tau>"} is always non-empty.  This is the deeper
313   reason why @{text "\<Gamma>"} only consists of hypothetical proofs, but no
314   hypothetical terms.
316   The corresponding proof terms are left implicit in the classic
317   LCF-approach'', although they could be exploited separately
318   \cite{Berghofer-Nipkow:2000}.  The implementation provides a runtime
319   option to control the generation of full proof terms.
321   \medskip The axiomatization of a theory is implicitly closed by
322   forming all instances of type and term variables: @{text "\<turnstile> A\<theta>"} for
323   any substirution instance of axiom @{text "\<turnstile> A"}.  By pushing
324   substitution through derivations inductively, we get admissible
325   substitution rules for theorems shown in \figref{fig:subst-rules}.
327   \begin{figure}[htb]
328   \begin{center}
329   $330 \infer{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}{@{text "\<Gamma> \<turnstile> B[\<alpha>]"} & @{text "\<alpha> \<notin> \<Gamma>"}} 331 \quad 332 \infer[\quad@{text "(generalize)"}]{@{text "\<Gamma> \<turnstile> B[?x]"}}{@{text "\<Gamma> \<turnstile> B[x]"} & @{text "x \<notin> \<Gamma>"}} 333$
334   $335 \infer{@{text "\<Gamma> \<turnstile> B[\<tau>]"}}{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}} 336 \quad 337 \infer[\quad@{text "(instantiate)"}]{@{text "\<Gamma> \<turnstile> B[t]"}}{@{text "\<Gamma> \<turnstile> B[?x]"}} 338$
339   \caption{Admissible substitution rules}\label{fig:subst-rules}
340   \end{center}
341   \end{figure}
343   Note that @{text "instantiate_term"} could be derived using @{text
344   "\<And>_intro/elim"}, but this is not how it is implemented.  The type
345   instantiation rule is a genuine admissible one, due to the lack of
346   true polymorphism in the logic.
348   Since @{text "\<Gamma>"} may never contain any schematic variables, the
349   @{text "instantiate"} do not require an explicit side-condition.  In
350   principle, variables could be substituted in hypotheses as well, but
351   this could disrupt monotonicity of the basic calculus: derivations
352   could leave the current proof context.
354   \medskip The framework also provides builtin equality @{text "\<equiv>"},
355   which is conceptually axiomatized shown in \figref{fig:equality},
356   although the implementation provides derived rules directly:
358   \begin{figure}[htb]
359   \begin{center}
360   \begin{tabular}{ll}
361   @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
362   @{text "\<turnstile> (\<lambda>x. b x) a \<equiv> b a"} & @{text "\<beta>"}-conversion \\
363   @{text "\<turnstile> x \<equiv> x"} & reflexivity law \\
364   @{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution law \\
365   @{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\
366   @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & coincidence with equivalence \\
367   \end{tabular}
368   \caption{Conceptual axiomatization of equality.}\label{fig:equality}
369   \end{center}
370   \end{figure}
372   Since the basic representation of terms already accounts for @{text
373   "\<alpha>"}-conversion, Pure equality essentially acts like @{text
374   "\<alpha>\<beta>\<eta>"}-equivalence on terms, while coinciding with bi-implication.
377   \medskip Conjunction is defined in Pure as a derived connective, see
378   \figref{fig:conjunction}.  This is occasionally useful to represent
379   simultaneous statements behind the scenes --- framework conjunction
380   is usually not exposed to the user.
382   \begin{figure}[htb]
383   \begin{center}
384   \begin{tabular}{ll}
385   @{text "& :: prop \<Rightarrow> prop \<Rightarrow> prop"} & conjunction (hidden) \\
386   @{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\
387   \end{tabular}
388   \caption{Definition of conjunction.}\label{fig:equality}
389   \end{center}
390   \end{figure}
392   The definition allows to derive the usual introduction @{text "\<turnstile> A \<Longrightarrow>
393   B \<Longrightarrow> A & B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B
394   \<Longrightarrow> B"}.
395 *}
398 section {* Rules \label{sec:rules} *}
400 text {*
402 FIXME
404   A \emph{rule} is any Pure theorem in HHF normal form; there is a
405   separate calculus for rule composition, which is modeled after
406   Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows
407   rules to be nested arbitrarily, similar to \cite{extensions91}.
409   Normally, all theorems accessible to the user are proper rules.
410   Low-level inferences are occasional required internally, but the
411   result should be always presented in canonical form.  The higher
412   interfaces of Isabelle/Isar will always produce proper rules.  It is
413   important to maintain this invariant in add-on applications!
415   There are two main principles of rule composition: @{text
416   "resolution"} (i.e.\ backchaining of rules) and @{text
417   "by-assumption"} (i.e.\ closing a branch); both principles are
418   combined in the variants of @{text "elim-resosultion"} and @{text
419   "dest-resolution"}.  Raw @{text "composition"} is occasionally
420   useful as well, also it is strictly speaking outside of the proper
421   rule calculus.
423   Rules are treated modulo general higher-order unification, which is
424   unification modulo the equational theory of @{text "\<alpha>\<beta>\<eta>"}-conversion
425   on @{text "\<lambda>"}-terms.  Moreover, propositions are understood modulo
426   the (derived) equivalence @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.
428   This means that any operations within the rule calculus may be
429   subject to spontaneous @{text "\<alpha>\<beta>\<eta>"}-HHF conversions.  It is common
430   practice not to contract or expand unnecessarily.  Some mechanisms
431   prefer an one form, others the opposite, so there is a potential
432   danger to produce some oscillation!
434   Only few operations really work \emph{modulo} HHF conversion, but
435   expect a normal form: quantifiers @{text "\<And>"} before implications
436   @{text "\<Longrightarrow>"} at each level of nesting.
438 \glossary{Hereditary Harrop Formula}{The set of propositions in HHF
439 format is defined inductively as @{text "H = (\<And>x\<^sup>*. H\<^sup>* \<Longrightarrow>
440 A)"}, for variables @{text "x"} and atomic propositions @{text "A"}.
441 Any proposition may be put into HHF form by normalizing with the rule
442 @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.  In Isabelle, the outermost
443 quantifier prefix is represented via \seeglossary{schematic
444 variables}, such that the top-level structure is merely that of a
445 \seeglossary{Horn Clause}}.
447 \glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.}
450   $451 \infer[@{text "(assumption)"}]{@{text "C\<vartheta>"}} 452 {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C"} & @{text "A\<vartheta> = H\<^sub>i\<vartheta>"}~~\text{(for some~@{text i})}} 453$
456   $457 \infer[@{text "(compose)"}]{@{text "\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>"}} 458 {@{text "\<^vec>A \<Longrightarrow> B"} & @{text "B' \<Longrightarrow> C"} & @{text "B\<vartheta> = B'\<vartheta>"}} 459$
462   $463 \infer[@{text "(\<And>_lift)"}]{@{text "(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))"}}{@{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"}} 464$
465   $466 \infer[@{text "(\<Longrightarrow>_lift)"}]{@{text "(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)"}}{@{text "\<^vec>A \<Longrightarrow> B"}} 467$
469   The @{text resolve} scheme is now acquired from @{text "\<And>_lift"},
470   @{text "\<Longrightarrow>_lift"}, and @{text compose}.
472   $473 \infer[@{text "(resolution)"}] 474 {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}} 475 {\begin{tabular}{l} 476 @{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"} \\ 477 @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\ 478 @{text "(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\ 479 \end{tabular}} 480$
483   FIXME @{text "elim_resolution"}, @{text "dest_resolution"}
484 *}
487 end