(* $Id$ *)
theory Proof
imports Main
begin
chapter {* Proofs *}
text {*
Proof commands perform transitions of Isar/VM machine
configurations, which are block-structured, consisting of a stack of
nodes with three main components: logical proof context, current
facts, and open goals. Isar/VM transitions are \emph{typed}
according to the following three different modes of operation:
\begin{descr}
\item [@{text "proof(prove)"}] means that a new goal has just been
stated that is now to be \emph{proven}; the next command may refine
it by some proof method, and enter a sub-proof to establish the
actual result.
\item [@{text "proof(state)"}] is like a nested theory mode: the
context may be augmented by \emph{stating} additional assumptions,
intermediate results etc.
\item [@{text "proof(chain)"}] is intermediate between @{text
"proof(state)"} and @{text "proof(prove)"}: existing facts (i.e.\
the contents of the special ``@{fact_ref this}'' register) have been
just picked up in order to be used when refining the goal claimed
next.
\end{descr}
The proof mode indicator may be read as a verb telling the writer
what kind of operation may be performed next. The corresponding
typings of proof commands restricts the shape of well-formed proof
texts to particular command sequences. So dynamic arrangements of
commands eventually turn out as static texts of a certain structure.
\Appref{ap:refcard} gives a simplified grammar of the overall
(extensible) language emerging that way.
*}
section {* Context elements \label{sec:proof-context} *}
text {*
\begin{matharray}{rcl}
@{command_def "fix"} & : & \isartrans{proof(state)}{proof(state)} \\
@{command_def "assume"} & : & \isartrans{proof(state)}{proof(state)} \\
@{command_def "presume"} & : & \isartrans{proof(state)}{proof(state)} \\
@{command_def "def"} & : & \isartrans{proof(state)}{proof(state)} \\
\end{matharray}
The logical proof context consists of fixed variables and
assumptions. The former closely correspond to Skolem constants, or
meta-level universal quantification as provided by the Isabelle/Pure
logical framework. Introducing some \emph{arbitrary, but fixed}
variable via ``@{command "fix"}~@{text x}'' results in a local value
that may be used in the subsequent proof as any other variable or
constant. Furthermore, any result @{text "\<turnstile> \<phi>[x]"} exported from
the context will be universally closed wrt.\ @{text x} at the
outermost level: @{text "\<turnstile> \<And>x. \<phi>[x]"} (this is expressed in normal
form using Isabelle's meta-variables).
Similarly, introducing some assumption @{text \<chi>} has two effects.
On the one hand, a local theorem is created that may be used as a
fact in subsequent proof steps. On the other hand, any result
@{text "\<chi> \<turnstile> \<phi>"} exported from the context becomes conditional wrt.\
the assumption: @{text "\<turnstile> \<chi> \<Longrightarrow> \<phi>"}. Thus, solving an enclosing goal
using such a result would basically introduce a new subgoal stemming
from the assumption. How this situation is handled depends on the
version of assumption command used: while @{command "assume"}
insists on solving the subgoal by unification with some premise of
the goal, @{command "presume"} leaves the subgoal unchanged in order
to be proved later by the user.
Local definitions, introduced by ``@{command "def"}~@{text "x \<equiv>
t"}'', are achieved by combining ``@{command "fix"}~@{text x}'' with
another version of assumption that causes any hypothetical equation
@{text "x \<equiv> t"} to be eliminated by the reflexivity rule. Thus,
exporting some result @{text "x \<equiv> t \<turnstile> \<phi>[x]"} yields @{text "\<turnstile>
\<phi>[t]"}.
\begin{rail}
'fix' (vars + 'and')
;
('assume' | 'presume') (props + 'and')
;
'def' (def + 'and')
;
def: thmdecl? \\ name ('==' | equiv) term termpat?
;
\end{rail}
\begin{descr}
\item [@{command "fix"}~@{text x}] introduces a local variable
@{text x} that is \emph{arbitrary, but fixed.}
\item [@{command "assume"}~@{text "a: \<phi>"} and @{command
"presume"}~@{text "a: \<phi>"}] introduce a local fact @{text "\<phi> \<turnstile> \<phi>"} by
assumption. Subsequent results applied to an enclosing goal (e.g.\
by @{command_ref "show"}) are handled as follows: @{command
"assume"} expects to be able to unify with existing premises in the
goal, while @{command "presume"} leaves @{text \<phi>} as new subgoals.
Several lists of assumptions may be given (separated by
@{keyword_ref "and"}; the resulting list of current facts consists
of all of these concatenated.
\item [@{command "def"}~@{text "x \<equiv> t"}] introduces a local
(non-polymorphic) definition. In results exported from the context,
@{text x} is replaced by @{text t}. Basically, ``@{command
"def"}~@{text "x \<equiv> t"}'' abbreviates ``@{command "fix"}~@{text
x}~@{command "assume"}~@{text "x \<equiv> t"}'', with the resulting
hypothetical equation solved by reflexivity.
The default name for the definitional equation is @{text x_def}.
Several simultaneous definitions may be given at the same time.
\end{descr}
The special name @{fact_ref prems} refers to all assumptions of the
current context as a list of theorems. This feature should be used
with great care! It is better avoided in final proof texts.
*}
section {* Facts and forward chaining *}
text {*
\begin{matharray}{rcl}
@{command_def "note"} & : & \isartrans{proof(state)}{proof(state)} \\
@{command_def "then"} & : & \isartrans{proof(state)}{proof(chain)} \\
@{command_def "from"} & : & \isartrans{proof(state)}{proof(chain)} \\
@{command_def "with"} & : & \isartrans{proof(state)}{proof(chain)} \\
@{command_def "using"} & : & \isartrans{proof(prove)}{proof(prove)} \\
@{command_def "unfolding"} & : & \isartrans{proof(prove)}{proof(prove)} \\
\end{matharray}
New facts are established either by assumption or proof of local
statements. Any fact will usually be involved in further proofs,
either as explicit arguments of proof methods, or when forward
chaining towards the next goal via @{command "then"} (and variants);
@{command "from"} and @{command "with"} are composite forms
involving @{command "note"}. The @{command "using"} elements
augments the collection of used facts \emph{after} a goal has been
stated. Note that the special theorem name @{fact_ref this} refers
to the most recently established facts, but only \emph{before}
issuing a follow-up claim.
\begin{rail}
'note' (thmdef? thmrefs + 'and')
;
('from' | 'with' | 'using' | 'unfolding') (thmrefs + 'and')
;
\end{rail}
\begin{descr}
\item [@{command "note"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}]
recalls existing facts @{text "b\<^sub>1, \<dots>, b\<^sub>n"}, binding
the result as @{text a}. Note that attributes may be involved as
well, both on the left and right hand sides.
\item [@{command "then"}] indicates forward chaining by the current
facts in order to establish the goal to be claimed next. The
initial proof method invoked to refine that will be offered the
facts to do ``anything appropriate'' (see also
\secref{sec:proof-steps}). For example, method @{method_ref rule}
(see \secref{sec:pure-meth-att}) would typically do an elimination
rather than an introduction. Automatic methods usually insert the
facts into the goal state before operation. This provides a simple
scheme to control relevance of facts in automated proof search.
\item [@{command "from"}~@{text b}] abbreviates ``@{command
"note"}~@{text b}~@{command "then"}''; thus @{command "then"} is
equivalent to ``@{command "from"}~@{text this}''.
\item [@{command "with"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"}]
abbreviates ``@{command "from"}~@{text "b\<^sub>1 \<dots> b\<^sub>n \<AND>
this"}''; thus the forward chaining is from earlier facts together
with the current ones.
\item [@{command "using"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"}] augments
the facts being currently indicated for use by a subsequent
refinement step (such as @{command_ref "apply"} or @{command_ref
"proof"}).
\item [@{command "unfolding"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"}] is
structurally similar to @{command "using"}, but unfolds definitional
equations @{text "b\<^sub>1, \<dots> b\<^sub>n"} throughout the goal state
and facts.
\end{descr}
Forward chaining with an empty list of theorems is the same as not
chaining at all. Thus ``@{command "from"}~@{text nothing}'' has no
effect apart from entering @{text "prove(chain)"} mode, since
@{fact_ref nothing} is bound to the empty list of theorems.
Basic proof methods (such as @{method_ref rule}) expect multiple
facts to be given in their proper order, corresponding to a prefix
of the premises of the rule involved. Note that positions may be
easily skipped using something like @{command "from"}~@{text "_
\<AND> a \<AND> b"}, for example. This involves the trivial rule
@{text "PROP \<psi> \<Longrightarrow> PROP \<psi>"}, which is bound in Isabelle/Pure as
``@{fact_ref "_"}'' (underscore).
Automated methods (such as @{method simp} or @{method auto}) just
insert any given facts before their usual operation. Depending on
the kind of procedure involved, the order of facts is less
significant here.
*}
section {* Goal statements \label{sec:goals} *}
text {*
\begin{matharray}{rcl}
@{command_def "lemma"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
@{command_def "theorem"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
@{command_def "corollary"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
@{command_def "have"} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\
@{command_def "show"} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\
@{command_def "hence"} & : & \isartrans{proof(state)}{proof(prove)} \\
@{command_def "thus"} & : & \isartrans{proof(state)}{proof(prove)} \\
@{command_def "print_statement"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
\end{matharray}
From a theory context, proof mode is entered by an initial goal
command such as @{command "lemma"}, @{command "theorem"}, or
@{command "corollary"}. Within a proof, new claims may be
introduced locally as well; four variants are available here to
indicate whether forward chaining of facts should be performed
initially (via @{command_ref "then"}), and whether the final result
is meant to solve some pending goal.
Goals may consist of multiple statements, resulting in a list of
facts eventually. A pending multi-goal is internally represented as
a meta-level conjunction (printed as @{text "&&"}), which is usually
split into the corresponding number of sub-goals prior to an initial
method application, via @{command_ref "proof"}
(\secref{sec:proof-steps}) or @{command_ref "apply"}
(\secref{sec:tactic-commands}). The @{method_ref induct} method
covered in \secref{sec:cases-induct} acts on multiple claims
simultaneously.
Claims at the theory level may be either in short or long form. A
short goal merely consists of several simultaneous propositions
(often just one). A long goal includes an explicit context
specification for the subsequent conclusion, involving local
parameters and assumptions. Here the role of each part of the
statement is explicitly marked by separate keywords (see also
\secref{sec:locale}); the local assumptions being introduced here
are available as @{fact_ref assms} in the proof. Moreover, there
are two kinds of conclusions: @{element_def "shows"} states several
simultaneous propositions (essentially a big conjunction), while
@{element_def "obtains"} claims several simultaneous simultaneous
contexts of (essentially a big disjunction of eliminated parameters
and assumptions, cf.\ \secref{sec:obtain}).
\begin{rail}
('lemma' | 'theorem' | 'corollary') target? (goal | longgoal)
;
('have' | 'show' | 'hence' | 'thus') goal
;
'print\_statement' modes? thmrefs
;
goal: (props + 'and')
;
longgoal: thmdecl? (contextelem *) conclusion
;
conclusion: 'shows' goal | 'obtains' (parname? case + '|')
;
case: (vars + 'and') 'where' (props + 'and')
;
\end{rail}
\begin{descr}
\item [@{command "lemma"}~@{text "a: \<phi>"}] enters proof mode with
@{text \<phi>} as main goal, eventually resulting in some fact @{text "\<turnstile>
\<phi>"} to be put back into the target context. An additional
\railnonterm{context} specification may build up an initial proof
context for the subsequent claim; this includes local definitions
and syntax as well, see the definition of @{syntax contextelem} in
\secref{sec:locale}.
\item [@{command "theorem"}~@{text "a: \<phi>"} and @{command
"corollary"}~@{text "a: \<phi>"}] are essentially the same as @{command
"lemma"}~@{text "a: \<phi>"}, but the facts are internally marked as
being of a different kind. This discrimination acts like a formal
comment.
\item [@{command "have"}~@{text "a: \<phi>"}] claims a local goal,
eventually resulting in a fact within the current logical context.
This operation is completely independent of any pending sub-goals of
an enclosing goal statements, so @{command "have"} may be freely
used for experimental exploration of potential results within a
proof body.
\item [@{command "show"}~@{text "a: \<phi>"}] is like @{command
"have"}~@{text "a: \<phi>"} plus a second stage to refine some pending
sub-goal for each one of the finished result, after having been
exported into the corresponding context (at the head of the
sub-proof of this @{command "show"} command).
To accommodate interactive debugging, resulting rules are printed
before being applied internally. Even more, interactive execution
of @{command "show"} predicts potential failure and displays the
resulting error as a warning beforehand. Watch out for the
following message:
%FIXME proper antiquitation
\begin{ttbox}
Problem! Local statement will fail to solve any pending goal
\end{ttbox}
\item [@{command "hence"}] abbreviates ``@{command "then"}~@{command
"have"}'', i.e.\ claims a local goal to be proven by forward
chaining the current facts. Note that @{command "hence"} is also
equivalent to ``@{command "from"}~@{text this}~@{command "have"}''.
\item [@{command "thus"}] abbreviates ``@{command "then"}~@{command
"show"}''. Note that @{command "thus"} is also equivalent to
``@{command "from"}~@{text this}~@{command "show"}''.
\item [@{command "print_statement"}~@{text a}] prints facts from the
current theory or proof context in long statement form, according to
the syntax for @{command "lemma"} given above.
\end{descr}
Any goal statement causes some term abbreviations (such as
@{variable_ref "?thesis"}) to be bound automatically, see also
\secref{sec:term-abbrev}. Furthermore, the local context of a
(non-atomic) goal is provided via the @{case_ref rule_context} case.
The optional case names of @{element_ref "obtains"} have a twofold
meaning: (1) during the of this claim they refer to the the local
context introductions, (2) the resulting rule is annotated
accordingly to support symbolic case splits when used with the
@{method_ref cases} method (cf. \secref{sec:cases-induct}).
\medskip
\begin{warn}
Isabelle/Isar suffers theory-level goal statements to contain
\emph{unbound schematic variables}, although this does not conform
to the aim of human-readable proof documents! The main problem
with schematic goals is that the actual outcome is usually hard to
predict, depending on the behavior of the proof methods applied
during the course of reasoning. Note that most semi-automated
methods heavily depend on several kinds of implicit rule
declarations within the current theory context. As this would
also result in non-compositional checking of sub-proofs,
\emph{local goals} are not allowed to be schematic at all.
Nevertheless, schematic goals do have their use in Prolog-style
interactive synthesis of proven results, usually by stepwise
refinement via emulation of traditional Isabelle tactic scripts
(see also \secref{sec:tactic-commands}). In any case, users
should know what they are doing.
\end{warn}
*}
section {* Initial and terminal proof steps \label{sec:proof-steps} *}
text {*
\begin{matharray}{rcl}
@{command_def "proof"} & : & \isartrans{proof(prove)}{proof(state)} \\
@{command_def "qed"} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\
@{command_def "by"} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
@{command_def ".."} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
@{command_def "."} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
@{command_def "sorry"} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
\end{matharray}
Arbitrary goal refinement via tactics is considered harmful.
Structured proof composition in Isar admits proof methods to be
invoked in two places only.
\begin{enumerate}
\item An \emph{initial} refinement step @{command_ref
"proof"}~@{text "m\<^sub>1"} reduces a newly stated goal to a number
of sub-goals that are to be solved later. Facts are passed to
@{text "m\<^sub>1"} for forward chaining, if so indicated by @{text
"proof(chain)"} mode.
\item A \emph{terminal} conclusion step @{command_ref "qed"}~@{text
"m\<^sub>2"} is intended to solve remaining goals. No facts are
passed to @{text "m\<^sub>2"}.
\end{enumerate}
The only other (proper) way to affect pending goals in a proof body
is by @{command_ref "show"}, which involves an explicit statement of
what is to be solved eventually. Thus we avoid the fundamental
problem of unstructured tactic scripts that consist of numerous
consecutive goal transformations, with invisible effects.
\medskip As a general rule of thumb for good proof style, initial
proof methods should either solve the goal completely, or constitute
some well-understood reduction to new sub-goals. Arbitrary
automatic proof tools that are prone leave a large number of badly
structured sub-goals are no help in continuing the proof document in
an intelligible manner.
Unless given explicitly by the user, the default initial method is
``@{method_ref rule}'', which applies a single standard elimination
or introduction rule according to the topmost symbol involved.
There is no separate default terminal method. Any remaining goals
are always solved by assumption in the very last step.
\begin{rail}
'proof' method?
;
'qed' method?
;
'by' method method?
;
('.' | '..' | 'sorry')
;
\end{rail}
\begin{descr}
\item [@{command "proof"}~@{text "m\<^sub>1"}] refines the goal by
proof method @{text "m\<^sub>1"}; facts for forward chaining are
passed if so indicated by @{text "proof(chain)"} mode.
\item [@{command "qed"}~@{text "m\<^sub>2"}] refines any remaining
goals by proof method @{text "m\<^sub>2"} and concludes the
sub-proof by assumption. If the goal had been @{text "show"} (or
@{text "thus"}), some pending sub-goal is solved as well by the rule
resulting from the result \emph{exported} into the enclosing goal
context. Thus @{text "qed"} may fail for two reasons: either @{text
"m\<^sub>2"} fails, or the resulting rule does not fit to any
pending goal\footnote{This includes any additional ``strong''
assumptions as introduced by @{command "assume"}.} of the enclosing
context. Debugging such a situation might involve temporarily
changing @{command "show"} into @{command "have"}, or weakening the
local context by replacing occurrences of @{command "assume"} by
@{command "presume"}.
\item [@{command "by"}~@{text "m\<^sub>1 m\<^sub>2"}] is a
\emph{terminal proof}\index{proof!terminal}; it abbreviates
@{command "proof"}~@{text "m\<^sub>1"}~@{text "qed"}~@{text
"m\<^sub>2"}, but with backtracking across both methods. Debugging
an unsuccessful @{command "by"}~@{text "m\<^sub>1 m\<^sub>2"}
command can be done by expanding its definition; in many cases
@{command "proof"}~@{text "m\<^sub>1"} (or even @{text
"apply"}~@{text "m\<^sub>1"}) is already sufficient to see the
problem.
\item [``@{command ".."}''] is a \emph{default
proof}\index{proof!default}; it abbreviates @{command "by"}~@{text
"rule"}.
\item [``@{command "."}''] is a \emph{trivial
proof}\index{proof!trivial}; it abbreviates @{command "by"}~@{text
"this"}.
\item [@{command "sorry"}] is a \emph{fake proof}\index{proof!fake}
pretending to solve the pending claim without further ado. This
only works in interactive development, or if the @{ML
quick_and_dirty} flag is enabled (in ML). Facts emerging from fake
proofs are not the real thing. Internally, each theorem container
is tainted by an oracle invocation, which is indicated as ``@{text
"[!]"}'' in the printed result.
The most important application of @{command "sorry"} is to support
experimentation and top-down proof development.
\end{descr}
*}
section {* Fundamental methods and attributes \label{sec:pure-meth-att} *}
text {*
The following proof methods and attributes refer to basic logical
operations of Isar. Further methods and attributes are provided by
several generic and object-logic specific tools and packages (see
\chref{ch:gen-tools} and \chref{ch:hol}).
\begin{matharray}{rcl}
@{method_def "-"} & : & \isarmeth \\
@{method_def "fact"} & : & \isarmeth \\
@{method_def "assumption"} & : & \isarmeth \\
@{method_def "this"} & : & \isarmeth \\
@{method_def "rule"} & : & \isarmeth \\
@{method_def "iprover"} & : & \isarmeth \\[0.5ex]
@{attribute_def (Pure) "intro"} & : & \isaratt \\
@{attribute_def (Pure) "elim"} & : & \isaratt \\
@{attribute_def (Pure) "dest"} & : & \isaratt \\
@{attribute_def "rule"} & : & \isaratt \\[0.5ex]
@{attribute_def "OF"} & : & \isaratt \\
@{attribute_def "of"} & : & \isaratt \\
@{attribute_def "where"} & : & \isaratt \\
\end{matharray}
\begin{rail}
'fact' thmrefs?
;
'rule' thmrefs?
;
'iprover' ('!' ?) (rulemod *)
;
rulemod: ('intro' | 'elim' | 'dest') ((('!' | () | '?') nat?) | 'del') ':' thmrefs
;
('intro' | 'elim' | 'dest') ('!' | () | '?') nat?
;
'rule' 'del'
;
'OF' thmrefs
;
'of' insts ('concl' ':' insts)?
;
'where' ((name | var | typefree | typevar) '=' (type | term) * 'and')
;
\end{rail}
\begin{descr}
\item [``@{method "-"}'' (minus)] does nothing but insert the
forward chaining facts as premises into the goal. Note that command
@{command_ref "proof"} without any method actually performs a single
reduction step using the @{method_ref rule} method; thus a plain
\emph{do-nothing} proof step would be ``@{command "proof"}~@{text
"-"}'' rather than @{command "proof"} alone.
\item [@{method "fact"}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}] composes
some fact from @{text "a\<^sub>1, \<dots>, a\<^sub>n"} (or implicitly from
the current proof context) modulo unification of schematic type and
term variables. The rule structure is not taken into account, i.e.\
meta-level implication is considered atomic. This is the same
principle underlying literal facts (cf.\ \secref{sec:syn-att}):
``@{command "have"}~@{text "\<phi>"}~@{command "by"}~@{text fact}'' is
equivalent to ``@{command "note"}~@{verbatim "`"}@{text \<phi>}@{verbatim
"`"}'' provided that @{text "\<turnstile> \<phi>"} is an instance of some known
@{text "\<turnstile> \<phi>"} in the proof context.
\item [@{method assumption}] solves some goal by a single assumption
step. All given facts are guaranteed to participate in the
refinement; this means there may be only 0 or 1 in the first place.
Recall that @{command "qed"} (\secref{sec:proof-steps}) already
concludes any remaining sub-goals by assumption, so structured
proofs usually need not quote the @{method assumption} method at
all.
\item [@{method this}] applies all of the current facts directly as
rules. Recall that ``@{command "."}'' (dot) abbreviates ``@{command
"by"}~@{text this}''.
\item [@{method rule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}] applies some
rule given as argument in backward manner; facts are used to reduce
the rule before applying it to the goal. Thus @{method rule}
without facts is plain introduction, while with facts it becomes
elimination.
When no arguments are given, the @{method rule} method tries to pick
appropriate rules automatically, as declared in the current context
using the @{attribute (Pure) intro}, @{attribute (Pure) elim},
@{attribute (Pure) dest} attributes (see below). This is the
default behavior of @{command "proof"} and ``@{command ".."}''
(double-dot) steps (see \secref{sec:proof-steps}).
\item [@{method iprover}] performs intuitionistic proof search,
depending on specifically declared rules from the context, or given
as explicit arguments. Chained facts are inserted into the goal
before commencing proof search; ``@{method iprover}@{text "!"}''
means to include the current @{fact prems} as well.
Rules need to be classified as @{attribute (Pure) intro},
@{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
``@{text "!"}'' indicator refers to ``safe'' rules, which may be
applied aggressively (without considering back-tracking later).
Rules declared with ``@{text "?"}'' are ignored in proof search (the
single-step @{method rule} method still observes these). An
explicit weight annotation may be given as well; otherwise the
number of rule premises will be taken into account here.
\item [@{attribute (Pure) intro}, @{attribute (Pure) elim}, and
@{attribute (Pure) dest}] declare introduction, elimination, and
destruct rules, to be used with the @{method rule} and @{method
iprover} methods. Note that the latter will ignore rules declared
with ``@{text "?"}'', while ``@{text "!"}'' are used most
aggressively.
The classical reasoner (see \secref{sec:classical}) introduces its
own variants of these attributes; use qualified names to access the
present versions of Isabelle/Pure, i.e.\ @{attribute (Pure)
"Pure.intro"}.
\item [@{attribute rule}~@{text del}] undeclares introduction,
elimination, or destruct rules.
\item [@{attribute OF}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}] applies some
theorem to all of the given rules @{text "a\<^sub>1, \<dots>, a\<^sub>n"}
(in parallel). This corresponds to the @{ML "op MRS"} operation in
ML, but note the reversed order. Positions may be effectively
skipped by including ``@{text _}'' (underscore) as argument.
\item [@{attribute of}~@{text "t\<^sub>1 \<dots> t\<^sub>n"}] performs
positional instantiation of term variables. The terms @{text
"t\<^sub>1, \<dots>, t\<^sub>n"} are substituted for any schematic
variables occurring in a theorem from left to right; ``@{text _}''
(underscore) indicates to skip a position. Arguments following a
``@{text "concl:"}'' specification refer to positions of the
conclusion of a rule.
\item [@{attribute "where"}~@{text "x\<^sub>1 = t\<^sub>1 \<AND> \<dots>
x\<^sub>n = t\<^sub>n"}] performs named instantiation of schematic
type and term variables occurring in a theorem. Schematic variables
have to be specified on the left-hand side (e.g.\ @{text "?x1.3"}).
The question mark may be omitted if the variable name is a plain
identifier without index. As type instantiations are inferred from
term instantiations, explicit type instantiations are seldom
necessary.
\end{descr}
*}
section {* Term abbreviations \label{sec:term-abbrev} *}
text {*
\begin{matharray}{rcl}
@{command_def "let"} & : & \isartrans{proof(state)}{proof(state)} \\
@{keyword_def "is"} & : & syntax \\
\end{matharray}
Abbreviations may be either bound by explicit @{command
"let"}~@{text "p \<equiv> t"} statements, or by annotating assumptions or
goal statements with a list of patterns ``@{text "(\<IS> p\<^sub>1 \<dots>
p\<^sub>n)"}''. In both cases, higher-order matching is invoked to
bind extra-logical term variables, which may be either named
schematic variables of the form @{text ?x}, or nameless dummies
``@{variable _}'' (underscore). Note that in the @{command "let"}
form the patterns occur on the left-hand side, while the @{keyword
"is"} patterns are in postfix position.
Polymorphism of term bindings is handled in Hindley-Milner style,
similar to ML. Type variables referring to local assumptions or
open goal statements are \emph{fixed}, while those of finished
results or bound by @{command "let"} may occur in \emph{arbitrary}
instances later. Even though actual polymorphism should be rarely
used in practice, this mechanism is essential to achieve proper
incremental type-inference, as the user proceeds to build up the
Isar proof text from left to right.
\medskip Term abbreviations are quite different from local
definitions as introduced via @{command "def"} (see
\secref{sec:proof-context}). The latter are visible within the
logic as actual equations, while abbreviations disappear during the
input process just after type checking. Also note that @{command
"def"} does not support polymorphism.
\begin{rail}
'let' ((term + 'and') '=' term + 'and')
;
\end{rail}
The syntax of @{keyword "is"} patterns follows \railnonterm{termpat}
or \railnonterm{proppat} (see \secref{sec:term-decls}).
\begin{descr}
\item [@{command "let"}~@{text "p\<^sub>1 = t\<^sub>1 \<AND> \<dots>
p\<^sub>n = t\<^sub>n"}] binds any text variables in patterns @{text
"p\<^sub>1, \<dots>, p\<^sub>n"} by simultaneous higher-order matching
against terms @{text "t\<^sub>1, \<dots>, t\<^sub>n"}.
\item [@{text "(\<IS> p\<^sub>1 \<dots> p\<^sub>n)"}] resembles @{command
"let"}, but matches @{text "p\<^sub>1, \<dots>, p\<^sub>n"} against the
preceding statement. Also note that @{keyword "is"} is not a
separate command, but part of others (such as @{command "assume"},
@{command "have"} etc.).
\end{descr}
Some \emph{implicit} term abbreviations\index{term abbreviations}
for goals and facts are available as well. For any open goal,
@{variable_ref thesis} refers to its object-level statement,
abstracted over any meta-level parameters (if present). Likewise,
@{variable_ref this} is bound for fact statements resulting from
assumptions or finished goals. In case @{variable this} refers to
an object-logic statement that is an application @{text "f t"}, then
@{text t} is bound to the special text variable ``@{variable "\<dots>"}''
(three dots). The canonical application of this convenience are
calculational proofs (see \secref{sec:calculation}).
*}
section {* Block structure *}
text {*
\begin{matharray}{rcl}
@{command_def "next"} & : & \isartrans{proof(state)}{proof(state)} \\
@{command_def "{"} & : & \isartrans{proof(state)}{proof(state)} \\
@{command_def "}"} & : & \isartrans{proof(state)}{proof(state)} \\
\end{matharray}
While Isar is inherently block-structured, opening and closing
blocks is mostly handled rather casually, with little explicit
user-intervention. Any local goal statement automatically opens
\emph{two} internal blocks, which are closed again when concluding
the sub-proof (by @{command "qed"} etc.). Sections of different
context within a sub-proof may be switched via @{command "next"},
which is just a single block-close followed by block-open again.
The effect of @{command "next"} is to reset the local proof context;
there is no goal focus involved here!
For slightly more advanced applications, there are explicit block
parentheses as well. These typically achieve a stronger forward
style of reasoning.
\begin{descr}
\item [@{command "next"}] switches to a fresh block within a
sub-proof, resetting the local context to the initial one.
\item [@{command "{"} and @{command "}"}] explicitly open and close
blocks. Any current facts pass through ``@{command "{"}''
unchanged, while ``@{command "}"}'' causes any result to be
\emph{exported} into the enclosing context. Thus fixed variables
are generalized, assumptions discharged, and local definitions
unfolded (cf.\ \secref{sec:proof-context}). There is no difference
of @{command "assume"} and @{command "presume"} in this mode of
forward reasoning --- in contrast to plain backward reasoning with
the result exported at @{command "show"} time.
\end{descr}
*}
section {* Emulating tactic scripts \label{sec:tactic-commands} *}
text {*
The Isar provides separate commands to accommodate tactic-style
proof scripts within the same system. While being outside the
orthodox Isar proof language, these might come in handy for
interactive exploration and debugging, or even actual tactical proof
within new-style theories (to benefit from document preparation, for
example). See also \secref{sec:tactics} for actual tactics, that
have been encapsulated as proof methods. Proper proof methods may
be used in scripts, too.
\begin{matharray}{rcl}
@{command_def "apply"}@{text "\<^sup>*"} & : & \isartrans{proof(prove)}{proof(prove)} \\
@{command_def "apply_end"}@{text "\<^sup>*"} & : & \isartrans{proof(state)}{proof(state)} \\
@{command_def "done"}@{text "\<^sup>*"} & : & \isartrans{proof(prove)}{proof(state)} \\
@{command_def "defer"}@{text "\<^sup>*"} & : & \isartrans{proof}{proof} \\
@{command_def "prefer"}@{text "\<^sup>*"} & : & \isartrans{proof}{proof} \\
@{command_def "back"}@{text "\<^sup>*"} & : & \isartrans{proof}{proof} \\
\end{matharray}
\begin{rail}
( 'apply' | 'apply\_end' ) method
;
'defer' nat?
;
'prefer' nat
;
\end{rail}
\begin{descr}
\item [@{command "apply"}~@{text m}] applies proof method @{text m}
in initial position, but unlike @{command "proof"} it retains
``@{text "proof(prove)"}'' mode. Thus consecutive method
applications may be given just as in tactic scripts.
Facts are passed to @{text m} as indicated by the goal's
forward-chain mode, and are \emph{consumed} afterwards. Thus any
further @{command "apply"} command would always work in a purely
backward manner.
\item [@{command "apply_end"}~@{text "m"}] applies proof method
@{text m} as if in terminal position. Basically, this simulates a
multi-step tactic script for @{command "qed"}, but may be given
anywhere within the proof body.
No facts are passed to @{text m} here. Furthermore, the static
context is that of the enclosing goal (as for actual @{command
"qed"}). Thus the proof method may not refer to any assumptions
introduced in the current body, for example.
\item [@{command "done"}] completes a proof script, provided that
the current goal state is solved completely. Note that actual
structured proof commands (e.g.\ ``@{command "."}'' or @{command
"sorry"}) may be used to conclude proof scripts as well.
\item [@{command "defer"}~@{text n} and @{command "prefer"}~@{text
n}] shuffle the list of pending goals: @{command "defer"} puts off
sub-goal @{text n} to the end of the list (@{text "n = 1"} by
default), while @{command "prefer"} brings sub-goal @{text n} to the
front.
\item [@{command "back"}] does back-tracking over the result
sequence of the latest proof command. Basically, any proof command
may return multiple results.
\end{descr}
Any proper Isar proof method may be used with tactic script commands
such as @{command "apply"}. A few additional emulations of actual
tactics are provided as well; these would be never used in actual
structured proofs, of course.
*}
section {* Omitting proofs *}
text {*
\begin{matharray}{rcl}
@{command_def "oops"} & : & \isartrans{proof}{theory} \\
\end{matharray}
The @{command "oops"} command discontinues the current proof
attempt, while considering the partial proof text as properly
processed. This is conceptually quite different from ``faking''
actual proofs via @{command_ref "sorry"} (see
\secref{sec:proof-steps}): @{command "oops"} does not observe the
proof structure at all, but goes back right to the theory level.
Furthermore, @{command "oops"} does not produce any result theorem
--- there is no intended claim to be able to complete the proof
anyhow.
A typical application of @{command "oops"} is to explain Isar proofs
\emph{within} the system itself, in conjunction with the document
preparation tools of Isabelle described in \cite{isabelle-sys}.
Thus partial or even wrong proof attempts can be discussed in a
logically sound manner. Note that the Isabelle {\LaTeX} macros can
be easily adapted to print something like ``@{text "\<dots>"}'' instead of
the keyword ``@{command "oops"}''.
\medskip The @{command "oops"} command is undo-able, unlike
@{command_ref "kill"} (see \secref{sec:history}). The effect is to
get back to the theory just before the opening of the proof.
*}
section {* Generalized elimination \label{sec:obtain} *}
text {*
\begin{matharray}{rcl}
@{command_def "obtain"} & : & \isartrans{proof(state)}{proof(prove)} \\
@{command_def "guess"}@{text "\<^sup>*"} & : & \isartrans{proof(state)}{proof(prove)} \\
\end{matharray}
Generalized elimination means that additional elements with certain
properties may be introduced in the current context, by virtue of a
locally proven ``soundness statement''. Technically speaking, the
@{command "obtain"} language element is like a declaration of
@{command "fix"} and @{command "assume"} (see also see
\secref{sec:proof-context}), together with a soundness proof of its
additional claim. According to the nature of existential reasoning,
assumptions get eliminated from any result exported from the context
later, provided that the corresponding parameters do \emph{not}
occur in the conclusion.
\begin{rail}
'obtain' parname? (vars + 'and') 'where' (props + 'and')
;
'guess' (vars + 'and')
;
\end{rail}
The derived Isar command @{command "obtain"} is defined as follows
(where @{text "b\<^sub>1, \<dots>, b\<^sub>k"} shall refer to (optional)
facts indicated for forward chaining).
\begin{matharray}{l}
@{text "\<langle>using b\<^sub>1 \<dots> b\<^sub>k\<rangle>"}~~@{command "obtain"}~@{text "x\<^sub>1 \<dots> x\<^sub>m \<WHERE> a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n \<langle>proof\<rangle> \<equiv>"} \\[1ex]
\quad @{command "have"}~@{text "\<And>thesis. (\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots> \<phi>\<^sub>n \<Longrightarrow> thesis) \<Longrightarrow> thesis"} \\
\quad @{command "proof"}~@{text succeed} \\
\qquad @{command "fix"}~@{text thesis} \\
\qquad @{command "assume"}~@{text "that [Pure.intro?]: \<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots> \<phi>\<^sub>n \<Longrightarrow> thesis"} \\
\qquad @{command "then"}~@{command "show"}~@{text thesis} \\
\quad\qquad @{command "apply"}~@{text -} \\
\quad\qquad @{command "using"}~@{text "b\<^sub>1 \<dots> b\<^sub>k \<langle>proof\<rangle>"} \\
\quad @{command "qed"} \\
\quad @{command "fix"}~@{text "x\<^sub>1 \<dots> x\<^sub>m"}~@{command "assume"}@{text "\<^sup>* a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"} \\
\end{matharray}
Typically, the soundness proof is relatively straight-forward, often
just by canonical automated tools such as ``@{command "by"}~@{text
simp}'' or ``@{command "by"}~@{text blast}''. Accordingly, the
``@{text that}'' reduction above is declared as simplification and
introduction rule.
In a sense, @{command "obtain"} represents at the level of Isar
proofs what would be meta-logical existential quantifiers and
conjunctions. This concept has a broad range of useful
applications, ranging from plain elimination (or introduction) of
object-level existential and conjunctions, to elimination over
results of symbolic evaluation of recursive definitions, for
example. Also note that @{command "obtain"} without parameters acts
much like @{command "have"}, where the result is treated as a
genuine assumption.
An alternative name to be used instead of ``@{text that}'' above may
be given in parentheses.
\medskip The improper variant @{command "guess"} is similar to
@{command "obtain"}, but derives the obtained statement from the
course of reasoning! The proof starts with a fixed goal @{text
thesis}. The subsequent proof may refine this to anything of the
form like @{text "\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots>
\<phi>\<^sub>n \<Longrightarrow> thesis"}, but must not introduce new subgoals. The
final goal state is then used as reduction rule for the obtain
scheme described above. Obtained parameters @{text "x\<^sub>1, \<dots>,
x\<^sub>m"} are marked as internal by default, which prevents the
proof context from being polluted by ad-hoc variables. The variable
names and type constraints given as arguments for @{command "guess"}
specify a prefix of obtained parameters explicitly in the text.
It is important to note that the facts introduced by @{command
"obtain"} and @{command "guess"} may not be polymorphic: any
type-variables occurring here are fixed in the present context!
*}
section {* Calculational reasoning \label{sec:calculation} *}
text {*
\begin{matharray}{rcl}
@{command_def "also"} & : & \isartrans{proof(state)}{proof(state)} \\
@{command_def "finally"} & : & \isartrans{proof(state)}{proof(chain)} \\
@{command_def "moreover"} & : & \isartrans{proof(state)}{proof(state)} \\
@{command_def "ultimately"} & : & \isartrans{proof(state)}{proof(chain)} \\
@{command_def "print_trans_rules"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
@{attribute trans} & : & \isaratt \\
@{attribute sym} & : & \isaratt \\
@{attribute symmetric} & : & \isaratt \\
\end{matharray}
Calculational proof is forward reasoning with implicit application
of transitivity rules (such those of @{text "="}, @{text "\<le>"},
@{text "<"}). Isabelle/Isar maintains an auxiliary fact register
@{fact_ref calculation} for accumulating results obtained by
transitivity composed with the current result. Command @{command
"also"} updates @{fact calculation} involving @{fact this}, while
@{command "finally"} exhibits the final @{fact calculation} by
forward chaining towards the next goal statement. Both commands
require valid current facts, i.e.\ may occur only after commands
that produce theorems such as @{command "assume"}, @{command
"note"}, or some finished proof of @{command "have"}, @{command
"show"} etc. The @{command "moreover"} and @{command "ultimately"}
commands are similar to @{command "also"} and @{command "finally"},
but only collect further results in @{fact calculation} without
applying any rules yet.
Also note that the implicit term abbreviation ``@{text "\<dots>"}'' has
its canonical application with calculational proofs. It refers to
the argument of the preceding statement. (The argument of a curried
infix expression happens to be its right-hand side.)
Isabelle/Isar calculations are implicitly subject to block structure
in the sense that new threads of calculational reasoning are
commenced for any new block (as opened by a local goal, for
example). This means that, apart from being able to nest
calculations, there is no separate \emph{begin-calculation} command
required.
\medskip The Isar calculation proof commands may be defined as
follows:\footnote{We suppress internal bookkeeping such as proper
handling of block-structure.}
\begin{matharray}{rcl}
@{command "also"}@{text "\<^sub>0"} & \equiv & @{command "note"}~@{text "calculation = this"} \\
@{command "also"}@{text "\<^sub>n\<^sub>+\<^sub>1"} & \equiv & @{command "note"}~@{text "calculation = trans [OF calculation this]"} \\[0.5ex]
@{command "finally"} & \equiv & @{command "also"}~@{command "from"}~@{text calculation} \\[0.5ex]
@{command "moreover"} & \equiv & @{command "note"}~@{text "calculation = calculation this"} \\
@{command "ultimately"} & \equiv & @{command "moreover"}~@{command "from"}~@{text calculation} \\
\end{matharray}
\begin{rail}
('also' | 'finally') ('(' thmrefs ')')?
;
'trans' (() | 'add' | 'del')
;
\end{rail}
\begin{descr}
\item [@{command "also"}~@{text "(a\<^sub>1 \<dots> a\<^sub>n)"}]
maintains the auxiliary @{fact calculation} register as follows.
The first occurrence of @{command "also"} in some calculational
thread initializes @{fact calculation} by @{fact this}. Any
subsequent @{command "also"} on the same level of block-structure
updates @{fact calculation} by some transitivity rule applied to
@{fact calculation} and @{fact this} (in that order). Transitivity
rules are picked from the current context, unless alternative rules
are given as explicit arguments.
\item [@{command "finally"}~@{text "(a\<^sub>1 \<dots> a\<^sub>n)"}]
maintaining @{fact calculation} in the same way as @{command
"also"}, and concludes the current calculational thread. The final
result is exhibited as fact for forward chaining towards the next
goal. Basically, @{command "finally"} just abbreviates @{command
"also"}~@{command "from"}~@{fact calculation}. Typical idioms for
concluding calculational proofs are ``@{command "finally"}~@{command
"show"}~@{text ?thesis}~@{command "."}'' and ``@{command
"finally"}~@{command "have"}~@{text \<phi>}~@{command "."}''.
\item [@{command "moreover"} and @{command "ultimately"}] are
analogous to @{command "also"} and @{command "finally"}, but collect
results only, without applying rules.
\item [@{command "print_trans_rules"}] prints the list of
transitivity rules (for calculational commands @{command "also"} and
@{command "finally"}) and symmetry rules (for the @{attribute
symmetric} operation and single step elimination patters) of the
current context.
\item [@{attribute trans}] declares theorems as transitivity rules.
\item [@{attribute sym}] declares symmetry rules, as well as
@{attribute "Pure.elim"}@{text "?"} rules.
\item [@{attribute symmetric}] resolves a theorem with some rule
declared as @{attribute sym} in the current context. For example,
``@{command "assume"}~@{text "[symmetric]: x = y"}'' produces a
swapped fact derived from that assumption.
In structured proof texts it is often more appropriate to use an
explicit single-step elimination proof, such as ``@{command
"assume"}~@{text "x = y"}~@{command "then"}~@{command "have"}~@{text
"y = x"}~@{command ".."}''.
\end{descr}
*}
end