author | wenzelm |
Mon, 16 Feb 2009 21:39:52 +0100 | |
changeset 29762 | e5324b8b4df5 |
parent 29756 | df70c0291579 |
child 30121 | 5c7bcb296600 |
permissions | -rw-r--r-- |
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\begin{isabellebody}% |
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\def\isabellecontext{Proof}% |
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\isadelimtheory |
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\endisadelimtheory |
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\isatagtheory |
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\isacommand{theory}\isamarkupfalse% |
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\ Proof\isanewline |
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\isakeyword{imports}\ Base\isanewline |
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\isakeyword{begin}% |
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\endisatagtheory |
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{\isafoldtheory}% |
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\isadelimtheory |
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\endisadelimtheory |
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\isamarkupchapter{Structured proofs% |
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} |
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\isamarkuptrue% |
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\isamarkupsection{Variables \label{sec:variables}% |
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} |
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\isamarkuptrue% |
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\begin{isamarkuptext}% |
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Any variable that is not explicitly bound by \isa{{\isasymlambda}}-abstraction |
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is considered as ``free''. Logically, free variables act like |
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outermost universal quantification at the sequent level: \isa{A\isactrlisub {\isadigit{1}}{\isacharparenleft}x{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n{\isacharparenleft}x{\isacharparenright}\ {\isasymturnstile}\ B{\isacharparenleft}x{\isacharparenright}} means that the result |
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holds \emph{for all} values of \isa{x}. Free variables for |
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terms (not types) can be fully internalized into the logic: \isa{{\isasymturnstile}\ B{\isacharparenleft}x{\isacharparenright}} and \isa{{\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ B{\isacharparenleft}x{\isacharparenright}} are interchangeable, provided |
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that \isa{x} does not occur elsewhere in the context. |
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Inspecting \isa{{\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ B{\isacharparenleft}x{\isacharparenright}} more closely, we see that inside the |
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quantifier, \isa{x} is essentially ``arbitrary, but fixed'', |
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while from outside it appears as a place-holder for instantiation |
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(thanks to \isa{{\isasymAnd}} elimination). |
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The Pure logic represents the idea of variables being either inside |
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or outside the current scope by providing separate syntactic |
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categories for \emph{fixed variables} (e.g.\ \isa{x}) vs.\ |
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\emph{schematic variables} (e.g.\ \isa{{\isacharquery}x}). Incidently, a |
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universal result \isa{{\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ B{\isacharparenleft}x{\isacharparenright}} has the HHF normal form \isa{{\isasymturnstile}\ B{\isacharparenleft}{\isacharquery}x{\isacharparenright}}, which represents its generality nicely without requiring |
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an explicit quantifier. The same principle works for type |
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variables: \isa{{\isasymturnstile}\ B{\isacharparenleft}{\isacharquery}{\isasymalpha}{\isacharparenright}} represents the idea of ``\isa{{\isasymturnstile}\ {\isasymforall}{\isasymalpha}{\isachardot}\ B{\isacharparenleft}{\isasymalpha}{\isacharparenright}}'' without demanding a truly polymorphic framework. |
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\medskip Additional care is required to treat type variables in a |
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way that facilitates type-inference. In principle, term variables |
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depend on type variables, which means that type variables would have |
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to be declared first. For example, a raw type-theoretic framework |
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would demand the context to be constructed in stages as follows: |
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\isa{{\isasymGamma}\ {\isacharequal}\ {\isasymalpha}{\isacharcolon}\ type{\isacharcomma}\ x{\isacharcolon}\ {\isasymalpha}{\isacharcomma}\ a{\isacharcolon}\ A{\isacharparenleft}x\isactrlisub {\isasymalpha}{\isacharparenright}}. |
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We allow a slightly less formalistic mode of operation: term |
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variables \isa{x} are fixed without specifying a type yet |
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(essentially \emph{all} potential occurrences of some instance |
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\isa{x\isactrlisub {\isasymtau}} are fixed); the first occurrence of \isa{x} |
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within a specific term assigns its most general type, which is then |
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maintained consistently in the context. The above example becomes |
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\isa{{\isasymGamma}\ {\isacharequal}\ x{\isacharcolon}\ term{\isacharcomma}\ {\isasymalpha}{\isacharcolon}\ type{\isacharcomma}\ A{\isacharparenleft}x\isactrlisub {\isasymalpha}{\isacharparenright}}, where type \isa{{\isasymalpha}} is fixed \emph{after} term \isa{x}, and the constraint |
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\isa{x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}} is an implicit consequence of the occurrence of |
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\isa{x\isactrlisub {\isasymalpha}} in the subsequent proposition. |
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This twist of dependencies is also accommodated by the reverse |
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operation of exporting results from a context: a type variable |
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\isa{{\isasymalpha}} is considered fixed as long as it occurs in some fixed |
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term variable of the context. For example, exporting \isa{x{\isacharcolon}\ term{\isacharcomma}\ {\isasymalpha}{\isacharcolon}\ type\ {\isasymturnstile}\ x\isactrlisub {\isasymalpha}\ {\isacharequal}\ x\isactrlisub {\isasymalpha}} produces in the first step |
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\isa{x{\isacharcolon}\ term\ {\isasymturnstile}\ x\isactrlisub {\isasymalpha}\ {\isacharequal}\ x\isactrlisub {\isasymalpha}} for fixed \isa{{\isasymalpha}}, |
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and only in the second step \isa{{\isasymturnstile}\ {\isacharquery}x\isactrlisub {\isacharquery}\isactrlisub {\isasymalpha}\ {\isacharequal}\ {\isacharquery}x\isactrlisub {\isacharquery}\isactrlisub {\isasymalpha}} for schematic \isa{{\isacharquery}x} and \isa{{\isacharquery}{\isasymalpha}}. |
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\medskip The Isabelle/Isar proof context manages the gory details of |
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term vs.\ type variables, with high-level principles for moving the |
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frontier between fixed and schematic variables. |
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The \isa{add{\isacharunderscore}fixes} operation explictly declares fixed |
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variables; the \isa{declare{\isacharunderscore}term} operation absorbs a term into |
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a context by fixing new type variables and adding syntactic |
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constraints. |
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The \isa{export} operation is able to perform the main work of |
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generalizing term and type variables as sketched above, assuming |
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that fixing variables and terms have been declared properly. |
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There \isa{import} operation makes a generalized fact a genuine |
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part of the context, by inventing fixed variables for the schematic |
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ones. The effect can be reversed by using \isa{export} later, |
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potentially with an extended context; the result is equivalent to |
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the original modulo renaming of schematic variables. |
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The \isa{focus} operation provides a variant of \isa{import} |
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for nested propositions (with explicit quantification): \isa{{\isasymAnd}x\isactrlisub {\isadigit{1}}\ {\isasymdots}\ x\isactrlisub n{\isachardot}\ B{\isacharparenleft}x\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlisub n{\isacharparenright}} is |
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decomposed by inventing fixed variables \isa{x\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlisub n} for the body.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\isadelimmlref |
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\endisadelimmlref |
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\isatagmlref |
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\begin{isamarkuptext}% |
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\begin{mldecls} |
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\indexml{Variable.add\_fixes}\verb|Variable.add_fixes: |\isasep\isanewline% |
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\verb| string list -> Proof.context -> string list * Proof.context| \\ |
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\indexml{Variable.variant\_fixes}\verb|Variable.variant_fixes: |\isasep\isanewline% |
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\verb| string list -> Proof.context -> string list * Proof.context| \\ |
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\indexml{Variable.declare\_term}\verb|Variable.declare_term: term -> Proof.context -> Proof.context| \\ |
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\indexml{Variable.declare\_constraints}\verb|Variable.declare_constraints: term -> Proof.context -> Proof.context| \\ |
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\indexml{Variable.export}\verb|Variable.export: Proof.context -> Proof.context -> thm list -> thm list| \\ |
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\indexml{Variable.polymorphic}\verb|Variable.polymorphic: Proof.context -> term list -> term list| \\ |
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\indexml{Variable.import\_thms}\verb|Variable.import_thms: bool -> thm list -> Proof.context ->|\isasep\isanewline% |
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\verb| ((ctyp list * cterm list) * thm list) * Proof.context| \\ |
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\indexml{Variable.focus}\verb|Variable.focus: cterm -> Proof.context -> (cterm list * cterm) * Proof.context| \\ |
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\end{mldecls} |
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\begin{description} |
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\item \verb|Variable.add_fixes|~\isa{xs\ ctxt} fixes term |
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variables \isa{xs}, returning the resulting internal names. By |
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default, the internal representation coincides with the external |
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one, which also means that the given variables must not be fixed |
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already. There is a different policy within a local proof body: the |
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given names are just hints for newly invented Skolem variables. |
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\item \verb|Variable.variant_fixes| is similar to \verb|Variable.add_fixes|, but always produces fresh variants of the given |
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names. |
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\item \verb|Variable.declare_term|~\isa{t\ ctxt} declares term |
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\isa{t} to belong to the context. This automatically fixes new |
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type variables, but not term variables. Syntactic constraints for |
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type and term variables are declared uniformly, though. |
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\item \verb|Variable.declare_constraints|~\isa{t\ ctxt} declares |
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syntactic constraints from term \isa{t}, without making it part |
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of the context yet. |
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\item \verb|Variable.export|~\isa{inner\ outer\ thms} generalizes |
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fixed type and term variables in \isa{thms} according to the |
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difference of the \isa{inner} and \isa{outer} context, |
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following the principles sketched above. |
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\item \verb|Variable.polymorphic|~\isa{ctxt\ ts} generalizes type |
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variables in \isa{ts} as far as possible, even those occurring |
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in fixed term variables. The default policy of type-inference is to |
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fix newly introduced type variables, which is essentially reversed |
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with \verb|Variable.polymorphic|: here the given terms are detached |
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from the context as far as possible. |
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\item \verb|Variable.import_thms|~\isa{open\ thms\ ctxt} invents fixed |
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type and term variables for the schematic ones occurring in \isa{thms}. The \isa{open} flag indicates whether the fixed names |
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should be accessible to the user, otherwise newly introduced names |
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are marked as ``internal'' (\secref{sec:names}). |
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\item \verb|Variable.focus|~\isa{B} decomposes the outermost \isa{{\isasymAnd}} prefix of proposition \isa{B}. |
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\end{description}% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\endisatagmlref |
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\isamarkupsection{Assumptions \label{sec:assumptions}% |
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} |
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\isamarkuptrue% |
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\begin{isamarkuptext}% |
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An \emph{assumption} is a proposition that it is postulated in the |
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current context. Local conclusions may use assumptions as |
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additional facts, but this imposes implicit hypotheses that weaken |
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the overall statement. |
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Assumptions are restricted to fixed non-schematic statements, i.e.\ |
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all generality needs to be expressed by explicit quantifiers. |
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Nevertheless, the result will be in HHF normal form with outermost |
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quantifiers stripped. For example, by assuming \isa{{\isasymAnd}x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}{\isachardot}\ P\ x} we get \isa{{\isasymAnd}x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}{\isachardot}\ P\ x\ {\isasymturnstile}\ P\ {\isacharquery}x} for schematic \isa{{\isacharquery}x} |
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of fixed type \isa{{\isasymalpha}}. Local derivations accumulate more and |
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more explicit references to hypotheses: \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n\ {\isasymturnstile}\ B} where \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n} needs to |
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be covered by the assumptions of the current context. |
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\medskip The \isa{add{\isacharunderscore}assms} operation augments the context by |
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local assumptions, which are parameterized by an arbitrary \isa{export} rule (see below). |
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The \isa{export} operation moves facts from a (larger) inner |
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context into a (smaller) outer context, by discharging the |
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difference of the assumptions as specified by the associated export |
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rules. Note that the discharged portion is determined by the |
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difference contexts, not the facts being exported! There is a |
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separate flag to indicate a goal context, where the result is meant |
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to refine an enclosing sub-goal of a structured proof state. |
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\medskip The most basic export rule discharges assumptions directly |
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by means of the \isa{{\isasymLongrightarrow}} introduction rule: |
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\[ |
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\infer[(\isa{{\isasymLongrightarrow}{\isacharunderscore}intro})]{\isa{{\isasymGamma}\ {\isacharbackslash}\ A\ {\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B}} |
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\] |
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The variant for goal refinements marks the newly introduced |
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premises, which causes the canonical Isar goal refinement scheme to |
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enforce unification with local premises within the goal: |
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\[ |
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\infer[(\isa{{\isacharhash}{\isasymLongrightarrow}{\isacharunderscore}intro})]{\isa{{\isasymGamma}\ {\isacharbackslash}\ A\ {\isasymturnstile}\ {\isacharhash}A\ {\isasymLongrightarrow}\ B}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B}} |
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\] |
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\medskip Alternative versions of assumptions may perform arbitrary |
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transformations on export, as long as the corresponding portion of |
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hypotheses is removed from the given facts. For example, a local |
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definition works by fixing \isa{x} and assuming \isa{x\ {\isasymequiv}\ t}, |
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with the following export rule to reverse the effect: |
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\[ |
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\infer[(\isa{{\isasymequiv}{\isacharminus}expand})]{\isa{{\isasymGamma}\ {\isacharbackslash}\ x\ {\isasymequiv}\ t\ {\isasymturnstile}\ B\ t}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B\ x}} |
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\] |
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This works, because the assumption \isa{x\ {\isasymequiv}\ t} was introduced in |
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a context with \isa{x} being fresh, so \isa{x} does not |
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occur in \isa{{\isasymGamma}} here.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\isadelimmlref |
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\endisadelimmlref |
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\begin{isamarkuptext}% |
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\begin{mldecls} |
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\indexmltype{Assumption.export}\verb|type Assumption.export| \\ |
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\indexml{Assumption.assume}\verb|Assumption.assume: cterm -> thm| \\ |
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\indexml{Assumption.add\_assms}\verb|Assumption.add_assms: Assumption.export ->|\isasep\isanewline% |
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\verb| cterm list -> Proof.context -> thm list * Proof.context| \\ |
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\indexml{Assumption.add\_assumes}\verb|Assumption.add_assumes: |\isasep\isanewline% |
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\verb| cterm list -> Proof.context -> thm list * Proof.context| \\ |
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\indexml{Assumption.export}\verb|Assumption.export: bool -> Proof.context -> Proof.context -> thm -> thm| \\ |
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\end{mldecls} |
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\begin{description} |
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\item \verb|Assumption.export| represents arbitrary export |
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rules, which is any function of type \verb|bool -> cterm list -> thm -> thm|, |
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where the \verb|bool| indicates goal mode, and the \verb|cterm list| the collection of assumptions to be discharged |
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simultaneously. |
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\item \verb|Assumption.assume|~\isa{A} turns proposition \isa{A} into a raw assumption \isa{A\ {\isasymturnstile}\ A{\isacharprime}}, where the conclusion |
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\isa{A{\isacharprime}} is in HHF normal form. |
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\item \verb|Assumption.add_assms|~\isa{r\ As} augments the context |
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by assumptions \isa{As} with export rule \isa{r}. The |
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resulting facts are hypothetical theorems as produced by the raw |
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\verb|Assumption.assume|. |
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\item \verb|Assumption.add_assumes|~\isa{As} is a special case of |
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\verb|Assumption.add_assms| where the export rule performs \isa{{\isasymLongrightarrow}{\isacharunderscore}intro} or \isa{{\isacharhash}{\isasymLongrightarrow}{\isacharunderscore}intro}, depending on goal mode. |
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\item \verb|Assumption.export|~\isa{is{\isacharunderscore}goal\ inner\ outer\ thm} |
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exports result \isa{thm} from the the \isa{inner} context |
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back into the \isa{outer} one; \isa{is{\isacharunderscore}goal\ {\isacharequal}\ true} means |
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this is a goal context. The result is in HHF normal form. Note |
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that \verb|ProofContext.export| combines \verb|Variable.export| |
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and \verb|Assumption.export| in the canonical way. |
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\end{description}% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\isamarkupsection{Results \label{sec:results}% |
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} |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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Local results are established by monotonic reasoning from facts |
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within a context. This allows common combinations of theorems, |
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e.g.\ via \isa{{\isasymAnd}{\isacharslash}{\isasymLongrightarrow}} elimination, resolution rules, or equational |
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reasoning, see \secref{sec:thms}. Unaccounted context manipulations |
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should be avoided, notably raw \isa{{\isasymAnd}{\isacharslash}{\isasymLongrightarrow}} introduction or ad-hoc |
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references to free variables or assumptions not present in the proof |
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context. |
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\medskip The \isa{SUBPROOF} combinator allows to structure a |
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tactical proof recursively by decomposing a selected sub-goal: |
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\isa{{\isacharparenleft}{\isasymAnd}x{\isachardot}\ A{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ B{\isacharparenleft}x{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}} is turned into \isa{B{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}} |
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after fixing \isa{x} and assuming \isa{A{\isacharparenleft}x{\isacharparenright}}. This means |
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the tactic needs to solve the conclusion, but may use the premise as |
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a local fact, for locally fixed variables. |
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|
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The \isa{prove} operation provides an interface for structured |
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backwards reasoning under program control, with some explicit sanity |
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checks of the result. The goal context can be augmented by |
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additional fixed variables (cf.\ \secref{sec:variables}) and |
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assumptions (cf.\ \secref{sec:assumptions}), which will be available |
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as local facts during the proof and discharged into implications in |
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the result. Type and term variables are generalized as usual, |
|
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according to the context. |
|
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|
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The \isa{obtain} operation produces results by eliminating |
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existing facts by means of a given tactic. This acts like a dual |
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conclusion: the proof demonstrates that the context may be augmented |
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by certain fixed variables and assumptions. See also |
|
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\cite{isabelle-isar-ref} for the user-level \isa{{\isasymOBTAIN}} and |
|
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\isa{{\isasymGUESS}} elements. Final results, which may not refer to |
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the parameters in the conclusion, need to exported explicitly into |
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the original context.% |
|
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
|
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\isadelimmlref |
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% |
|
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\endisadelimmlref |
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% |
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\isatagmlref |
|
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% |
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\begin{isamarkuptext}% |
|
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\begin{mldecls} |
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\indexml{SUBPROOF}\verb|SUBPROOF: ({context: Proof.context, schematics: ctyp list * cterm list,|\isasep\isanewline% |
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\verb| params: cterm list, asms: cterm list, concl: cterm,|\isasep\isanewline% |
|
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\verb| prems: thm list} -> tactic) -> Proof.context -> int -> tactic| \\ |
|
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\end{mldecls} |
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\begin{mldecls} |
|
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\indexml{Goal.prove}\verb|Goal.prove: Proof.context -> string list -> term list -> term ->|\isasep\isanewline% |
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\verb| ({prems: thm list, context: Proof.context} -> tactic) -> thm| \\ |
|
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changeset
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\indexml{Goal.prove\_multi}\verb|Goal.prove_multi: Proof.context -> string list -> term list -> term list ->|\isasep\isanewline% |
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\verb| ({prems: thm list, context: Proof.context} -> tactic) -> thm list| \\ |
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\end{mldecls} |
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\begin{mldecls} |
|
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\indexml{Obtain.result}\verb|Obtain.result: (Proof.context -> tactic) ->|\isasep\isanewline% |
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\verb| thm list -> Proof.context -> (cterm list * thm list) * Proof.context| \\ |
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\end{mldecls} |
341 |
||
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\begin{description} |
|
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|
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\item \verb|SUBPROOF|~\isa{tac\ ctxt\ i} decomposes the structure |
345 |
of the specified sub-goal, producing an extended context and a |
|
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reduced goal, which needs to be solved by the given tactic. All |
|
347 |
schematic parameters of the goal are imported into the context as |
|
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fixed ones, which may not be instantiated in the sub-proof. |
|
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|
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\item \verb|Goal.prove|~\isa{ctxt\ xs\ As\ C\ tac} states goal \isa{C} in the context augmented by fixed variables \isa{xs} and |
351 |
assumptions \isa{As}, and applies tactic \isa{tac} to solve |
|
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it. The latter may depend on the local assumptions being presented |
|
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as facts. The result is in HHF normal form. |
|
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|
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\item \verb|Goal.prove_multi| is simular to \verb|Goal.prove|, but |
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states several conclusions simultaneously. The goal is encoded by |
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means of Pure conjunction; \verb|Goal.conjunction_tac| will turn this |
358 |
into a collection of individual subgoals. |
|
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|
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\item \verb|Obtain.result|~\isa{tac\ thms\ ctxt} eliminates the |
361 |
given facts using a tactic, which results in additional fixed |
|
362 |
variables and assumptions in the context. Final results need to be |
|
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exported explicitly. |
|
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|
365 |
\end{description}% |
|
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\end{isamarkuptext}% |
367 |
\isamarkuptrue% |
|
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% |
|
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\endisatagmlref |
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{\isafoldmlref}% |
|
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% |
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\isadelimmlref |
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% |
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\endisadelimmlref |
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% |
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\isadelimtheory |
|
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% |
|
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\endisadelimtheory |
|
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% |
|
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\isatagtheory |
|
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\isacommand{end}\isamarkupfalse% |
|
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% |
|
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\endisatagtheory |
|
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{\isafoldtheory}% |
|
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% |
|
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\isadelimtheory |
|
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% |
|
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\endisadelimtheory |
|
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\isanewline |
|
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\end{isabellebody}% |
|
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%%% Local Variables: |
|
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%%% mode: latex |
|
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%%% TeX-master: "root" |
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%%% End: |