(* $Id$ *)
theory "proof" imports base begin
chapter {* Structured proofs *}
section {* Variables \label{sec:variables} *}
text {*
Any variable that is not explicitly bound by @{text "\<lambda>"}-abstraction
is considered as ``free''. Logically, free variables act like
outermost universal quantification at the sequent level: @{text
"A\<^isub>1(x), \<dots>, A\<^isub>n(x) \<turnstile> B(x)"} means that the result
holds \emph{for all} values of @{text "x"}. Free variables for
terms (not types) can be fully internalized into the logic: @{text
"\<turnstile> B(x)"} and @{text "\<turnstile> \<And>x. B(x)"} are interchangeable, provided
that @{text "x"} does not occur elsewhere in the context.
Inspecting @{text "\<turnstile> \<And>x. B(x)"} more closely, we see that inside the
quantifier, @{text "x"} is essentially ``arbitrary, but fixed'',
while from outside it appears as a place-holder for instantiation
(thanks to @{text "\<And>"} 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.\ @{text "x"}) vs.\
\emph{schematic variables} (e.g.\ @{text "?x"}). Incidently, a
universal result @{text "\<turnstile> \<And>x. B(x)"} has the HHF normal form @{text
"\<turnstile> B(?x)"}, which represents its generality nicely without requiring
an explicit quantifier. The same principle works for type
variables: @{text "\<turnstile> B(?\<alpha>)"} represents the idea of ``@{text "\<turnstile>
\<forall>\<alpha>. B(\<alpha>)"}'' 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:
@{text "\<Gamma> = \<alpha>: type, x: \<alpha>, a: A(x\<^isub>\<alpha>)"}.
We allow a slightly less formalistic mode of operation: term
variables @{text "x"} are fixed without specifying a type yet
(essentially \emph{all} potential occurrences of some instance
@{text "x\<^isub>\<tau>"} are fixed); the first occurrence of @{text "x"}
within a specific term assigns its most general type, which is then
maintained consistently in the context. The above example becomes
@{text "\<Gamma> = x: term, \<alpha>: type, A(x\<^isub>\<alpha>)"}, where type @{text
"\<alpha>"} is fixed \emph{after} term @{text "x"}, and the constraint
@{text "x :: \<alpha>"} is an implicit consequence of the occurrence of
@{text "x\<^isub>\<alpha>"} in the subsequent proposition.
This twist of dependencies is also accommodated by the reverse
operation of exporting results from a context: a type variable
@{text "\<alpha>"} is considered fixed as long as it occurs in some fixed
term variable of the context. For example, exporting @{text "x:
term, \<alpha>: type \<turnstile> x\<^isub>\<alpha> = x\<^isub>\<alpha>"} produces in the first step
@{text "x: term \<turnstile> x\<^isub>\<alpha> = x\<^isub>\<alpha>"} for fixed @{text "\<alpha>"},
and only in the second step @{text "\<turnstile> ?x\<^isub>?\<^isub>\<alpha> =
?x\<^isub>?\<^isub>\<alpha>"} for schematic @{text "?x"} and @{text "?\<alpha>"}.
\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 @{text "add_fixes"} operation explictly declares fixed
variables; the @{text "declare_term"} operation absorbs a term into
a context by fixing new type variables and adding syntactic
constraints.
The @{text "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 @{text "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 @{text "export"} later,
potentially with an extended context; the result is equivalent to
the original modulo renaming of schematic variables.
The @{text "focus"} operation provides a variant of @{text "import"}
for nested propositions (with explicit quantification): @{text
"\<And>x\<^isub>1 \<dots> x\<^isub>n. B(x\<^isub>1, \<dots>, x\<^isub>n)"} is
decomposed by inventing fixed variables @{text "x\<^isub>1, \<dots>,
x\<^isub>n"} for the body.
*}
text %mlref {*
\begin{mldecls}
@{index_ML Variable.add_fixes: "
string list -> Proof.context -> string list * Proof.context"} \\
@{index_ML Variable.variant_fixes: "
string list -> Proof.context -> string list * Proof.context"} \\
@{index_ML Variable.declare_term: "term -> Proof.context -> Proof.context"} \\
@{index_ML Variable.declare_constraints: "term -> Proof.context -> Proof.context"} \\
@{index_ML Variable.export: "Proof.context -> Proof.context -> thm list -> thm list"} \\
@{index_ML Variable.polymorphic: "Proof.context -> term list -> term list"} \\
@{index_ML Variable.import_thms: "bool -> thm list -> Proof.context ->
((ctyp list * cterm list) * thm list) * Proof.context"} \\
@{index_ML Variable.focus: "cterm -> Proof.context -> (cterm list * cterm) * Proof.context"} \\
\end{mldecls}
\begin{description}
\item @{ML Variable.add_fixes}~@{text "xs ctxt"} fixes term
variables @{text "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 @{ML Variable.variant_fixes} is similar to @{ML
Variable.add_fixes}, but always produces fresh variants of the given
names.
\item @{ML Variable.declare_term}~@{text "t ctxt"} declares term
@{text "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 @{ML Variable.declare_constraints}~@{text "t ctxt"} declares
syntactic constraints from term @{text "t"}, without making it part
of the context yet.
\item @{ML Variable.export}~@{text "inner outer thms"} generalizes
fixed type and term variables in @{text "thms"} according to the
difference of the @{text "inner"} and @{text "outer"} context,
following the principles sketched above.
\item @{ML Variable.polymorphic}~@{text "ctxt ts"} generalizes type
variables in @{text "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 @{ML Variable.polymorphic}: here the given terms are detached
from the context as far as possible.
\item @{ML Variable.import_thms}~@{text "open thms ctxt"} invents fixed
type and term variables for the schematic ones occurring in @{text
"thms"}. The @{text "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 @{ML Variable.focus}~@{text B} decomposes the outermost @{text
"\<And>"} prefix of proposition @{text "B"}.
\end{description}
*}
section {* Assumptions \label{sec:assumptions} *}
text {*
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 @{text "\<And>x :: \<alpha>. P
x"} we get @{text "\<And>x :: \<alpha>. P x \<turnstile> P ?x"} for schematic @{text "?x"}
of fixed type @{text "\<alpha>"}. Local derivations accumulate more and
more explicit references to hypotheses: @{text "A\<^isub>1, \<dots>,
A\<^isub>n \<turnstile> B"} where @{text "A\<^isub>1, \<dots>, A\<^isub>n"} needs to
be covered by the assumptions of the current context.
\medskip The @{text "add_assms"} operation augments the context by
local assumptions, which are parameterized by an arbitrary @{text
"export"} rule (see below).
The @{text "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 @{text "\<Longrightarrow>"} introduction rule:
\[
\infer[(@{text "\<Longrightarrow>_intro"})]{@{text "\<Gamma> \\ A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
\]
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:
\[
\infer[(@{text "#\<Longrightarrow>_intro"})]{@{text "\<Gamma> \\ A \<turnstile> #A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
\]
\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 @{text "x"} and assuming @{text "x \<equiv> t"},
with the following export rule to reverse the effect:
\[
\infer[(@{text "\<equiv>-expand"})]{@{text "\<Gamma> \\ x \<equiv> t \<turnstile> B t"}}{@{text "\<Gamma> \<turnstile> B x"}}
\]
This works, because the assumption @{text "x \<equiv> t"} was introduced in
a context with @{text "x"} being fresh, so @{text "x"} does not
occur in @{text "\<Gamma>"} here.
*}
text %mlref {*
\begin{mldecls}
@{index_ML_type Assumption.export} \\
@{index_ML Assumption.assume: "cterm -> thm"} \\
@{index_ML Assumption.add_assms:
"Assumption.export ->
cterm list -> Proof.context -> thm list * Proof.context"} \\
@{index_ML Assumption.add_assumes: "
cterm list -> Proof.context -> thm list * Proof.context"} \\
@{index_ML Assumption.export: "bool -> Proof.context -> Proof.context -> thm -> thm"} \\
\end{mldecls}
\begin{description}
\item @{ML_type Assumption.export} represents arbitrary export
rules, which is any function of type @{ML_type "bool -> cterm list -> thm -> thm"},
where the @{ML_type "bool"} indicates goal mode, and the @{ML_type
"cterm list"} the collection of assumptions to be discharged
simultaneously.
\item @{ML Assumption.assume}~@{text "A"} turns proposition @{text
"A"} into a raw assumption @{text "A \<turnstile> A'"}, where the conclusion
@{text "A'"} is in HHF normal form.
\item @{ML Assumption.add_assms}~@{text "r As"} augments the context
by assumptions @{text "As"} with export rule @{text "r"}. The
resulting facts are hypothetical theorems as produced by the raw
@{ML Assumption.assume}.
\item @{ML Assumption.add_assumes}~@{text "As"} is a special case of
@{ML Assumption.add_assms} where the export rule performs @{text
"\<Longrightarrow>_intro"} or @{text "#\<Longrightarrow>_intro"}, depending on goal mode.
\item @{ML Assumption.export}~@{text "is_goal inner outer thm"}
exports result @{text "thm"} from the the @{text "inner"} context
back into the @{text "outer"} one; @{text "is_goal = true"} means
this is a goal context. The result is in HHF normal form. Note
that @{ML "ProofContext.export"} combines @{ML "Variable.export"}
and @{ML "Assumption.export"} in the canonical way.
\end{description}
*}
section {* Results *}
text {*
Local results are established by monotonic reasoning from facts
within a context. This allows common combinations of theorems,
e.g.\ via @{text "\<And>/\<Longrightarrow>"} elimination, resolution rules, or equational
reasoning, see \secref{sec:thms}. Unaccounted context manipulations
should be avoided, notably raw @{text "\<And>/\<Longrightarrow>"} introduction or ad-hoc
references to free variables or assumptions not present in the proof
context.
\medskip The @{text "SUBPROOF"} combinator allows to structure a
tactical proof recursively by decomposing a selected sub-goal:
@{text "(\<And>x. A(x) \<Longrightarrow> B(x)) \<Longrightarrow> \<dots>"} is turned into @{text "B(x) \<Longrightarrow> \<dots>"}
after fixing @{text "x"} and assuming @{text "A(x)"}. This means
the tactic needs to solve the conclusion, but may use the premise as
a local fact, for locally fixed variables.
The @{text "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 @{text "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 @{text "\<OBTAIN>"} and
@{text "\<GUESS>"} elements. Final results, which may not refer to
the parameters in the conclusion, need to exported explicitly into
the original context.
*}
text %mlref {*
\begin{mldecls}
@{index_ML SUBPROOF:
"({context: Proof.context, schematics: ctyp list * cterm list,
params: cterm list, asms: cterm list, concl: cterm,
prems: thm list} -> tactic) -> Proof.context -> int -> tactic"} \\
\end{mldecls}
\begin{mldecls}
@{index_ML Goal.prove: "Proof.context -> string list -> term list -> term ->
({prems: thm list, context: Proof.context} -> tactic) -> thm"} \\
@{index_ML Goal.prove_multi: "Proof.context -> string list -> term list -> term list ->
({prems: thm list, context: Proof.context} -> tactic) -> thm list"} \\
\end{mldecls}
\begin{mldecls}
@{index_ML Obtain.result: "(Proof.context -> tactic) ->
thm list -> Proof.context -> (cterm list * thm list) * Proof.context"} \\
\end{mldecls}
\begin{description}
\item @{ML SUBPROOF}~@{text "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 @{ML Goal.prove}~@{text "ctxt xs As C tac"} states goal @{text
"C"} in the context augmented by fixed variables @{text "xs"} and
assumptions @{text "As"}, and applies tactic @{text "tac"} to solve
it. The latter may depend on the local assumptions being presented
as facts. The result is in HHF normal form.
\item @{ML Goal.prove_multi} is simular to @{ML Goal.prove}, but
states several conclusions simultaneously. The goal is encoded by
means of Pure conjunction; @{ML Goal.conjunction_tac} will turn this
into a collection of individual subgoals.
\item @{ML Obtain.result}~@{text "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}
*}
end