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\begin{isabellebody}%
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\def\isabellecontext{proof}%
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\isadelimtheory
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\isanewline
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\isanewline
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\isanewline
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\endisadelimtheory
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\isatagtheory
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\isacommand{theory}\isamarkupfalse%
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\ {\isachardoublequoteopen}proof{\isachardoublequoteclose}\ \isakeyword{imports}\ base\ \isakeyword{begin}%
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\endisatagtheory
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{\isafoldtheory}%
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\isadelimtheory
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\isamarkupchapter{Structured proofs%
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}
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\isamarkuptrue%
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%
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\isamarkupsection{Variables%
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}
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\isamarkuptrue%
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%
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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 that
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\isa{x} does not occur elsewhere in the context. Inspecting
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\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 notion of variables being either
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inside 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 canonical 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 variables
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as well: \isa{{\isasymturnstile}\ B{\isacharparenleft}{\isacharquery}{\isasymalpha}{\isacharparenright}} expresses 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}} will be fixed); the first occurrence of \isa{x} within a specific term assigns its most general type, which is
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then maintained consistently in the context. The above example
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becomes \isa{{\isasymGamma}\ {\isacharequal}\ x{\isacharcolon}\ term{\isacharcomma}\ {\isasymalpha}{\isacharcolon}\ type{\isacharcomma}\ A{\isacharparenleft}x\isactrlisub {\isasymalpha}{\isacharparenright}}, where type
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\isa{{\isasymalpha}} is fixed \emph{after} term \isa{x}, and the
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constraint \isa{x{\isacharcolon}\ {\isasymalpha}} is an implicit consequence of the
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occurrence of \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 \isa{x{\isacharcolon}\ term\ {\isasymturnstile}\ x\isactrlisub {\isasymalpha}\ {\isacharequal}\ x\isactrlisub {\isasymalpha}} for fixed \isa{{\isasymalpha}} in the first step,
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and \isa{{\isasymturnstile}\ {\isacharquery}x\isactrlisub {\isacharquery}\isactrlisub {\isasymalpha}\ {\isacharequal}\ {\isacharquery}x\isactrlisub {\isacharquery}\isactrlisub {\isasymalpha}} for
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schematic \isa{{\isacharquery}x} and \isa{{\isacharquery}{\isasymalpha}} only in the second step.
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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. By observing a
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simple discipline of fixing variables and declaring terms
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explicitly, the fine points are treated by the \isa{export}
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operation.
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There is also a separate \isa{import} operation makes a
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generalized fact a genuine part of the context, by inventing fixed
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variables for the schematic ones. The effect can be reversed by
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using \isa{export} later, with a potentially extended context,
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but the result will be only equivalent modulo renaming of schematic
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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{\isachardot}\ B{\isacharparenleft}x{\isacharparenright}} is decomposed by inventing a fixed variable \isa{x}
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and for the body \isa{B{\isacharparenleft}x{\isacharparenright}}.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\endisadelimmlref
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\begin{isamarkuptext}%
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\begin{mldecls}
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\indexml{Variable.add-fixes}\verb|Variable.add_fixes: string list -> Proof.context -> string list * Proof.context| \\
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\indexml{Variable.invent-fixes}\verb|Variable.invent_fixes: 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}\verb|Variable.import: bool ->|\isasep\isanewline%
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\verb| thm list -> Proof.context -> ((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 have been
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fixed already. Within a local proof body, the given names are just
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hints for newly invented Skolem variables.
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\item \verb|Variable.invent_fixes| is similar to \verb|Variable.add_fixes|, but always produces fresh variants of the given
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hints.
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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.
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\item \verb|Variable.declare_constraints|~\isa{t\ ctxt} derives
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type-inference information from term \isa{t}, without making it
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part 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; this is essentially reversed
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with \verb|Variable.polymorphic|, the given terms are detached from
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the context as far as possible.
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\item \verb|Variable.import|~\isa{open\ thms\ ctxt} augments the
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context by new fixes for the schematic type and term variables
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occurring in \isa{thms}. The \isa{open} flag indicates
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whether the fixed names should be accessible to the user, otherwise
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internal names are chosen.
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\verb|Variable.export| essentially reverses the effect of \verb|Variable.import|, modulo renaming of schematic variables.
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\item \verb|Variable.focus|~\isa{{\isasymAnd}x\isactrlisub {\isadigit{1}}\ {\isasymdots}\ x\isactrlisub n{\isachardot}\ B{\isacharparenleft}x\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlisub n{\isacharparenright}} invents fixed variables
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for \isa{x\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlisub n} and replaces these in the
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body.
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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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{\isafoldmlref}%
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%
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\begin{isamarkuptext}%
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FIXME%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Assumptions%
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}
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\isamarkuptrue%
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%
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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, all
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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 arbitrary \isa{{\isacharquery}x}
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of the fixed type \isa{{\isasymalpha}}. Local derivations accumulate more
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and 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 (cf.\
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\secref{sec:isar-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 builtin goal refinement scheme of Isar 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 assumptions may perform arbitrary
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transformations on export, as long as a particular 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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\[
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\infer{\isa{{\isasymGamma}\ {\isacharbackslash}\ x\ {\isasymequiv}\ t\ {\isasymturnstile}\ B\ t}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B\ x}}
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\]
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\medskip The general concept supports block-structured reasoning
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nicely, with arbitrary mechanisms for introducing local assumptions.
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The common reasoning pattern is as follows:
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\medskip
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\begin{tabular}{l}
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\isa{add{\isacharunderscore}assms\ e\isactrlisub {\isadigit{1}}\ A\isactrlisub {\isadigit{1}}} \\
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\isa{{\isasymdots}} \\
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\isa{add{\isacharunderscore}assms\ e\isactrlisub n\ A\isactrlisub n} \\
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\isa{export} \\
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\end{tabular}
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\medskip
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\noindent The final \isa{export} will turn any fact \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n\ {\isasymturnstile}\ B} into some \isa{{\isasymturnstile}\ B{\isacharprime}}, by
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applying the export rules \isa{e\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ e\isactrlisub n}
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inside-out.%
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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{e\ As} augments the context
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by assumptions \isa{As} with export rule \isa{e}. The
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resulting facts are hypothetical theorems as produced by \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\ th}
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exports result \isa{th} 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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%
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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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\isamarkupsection{Conclusions%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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FIXME%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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|
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\isamarkupsection{Proof states \label{sec:isar-proof-state}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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FIXME
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\glossary{Proof state}{The whole configuration of a structured proof,
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consisting of a \seeglossary{proof context} and an optional
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\seeglossary{structured goal}. Internally, an Isar proof state is
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organized as a stack to accomodate block structure of proof texts.
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For historical reasons, a low-level \seeglossary{tactical goal} is
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occasionally called ``proof state'' as well.}
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\glossary{Structured goal}{FIXME}
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\glossary{Goal}{See \seeglossary{tactical goal} or \seeglossary{structured goal}. \norefpage}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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|
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\isamarkupsection{Proof methods%
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327 |
}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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FIXME%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Attributes%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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|
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FIXME ?!%
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18537
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\end{isamarkuptext}%
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\isamarkuptrue%
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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:
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