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