‹
Interactive theorem proving is traditionally associated with ``proof
scripts'', but Isabelle/Isar is centered around structured 🪙‹proof
documents› instead (see also \chref{ch:proofs}).
Nonetheless, it is possible to emulate proof scripts by sequential
refinements of a proof state in backwards mode, notably with the @{command
apply} command (see \secref{sec:tactic-commands}).
There are also various proof methods that allow to refer to implicit goal
state information that is not accessible to structured Isar proofs (see \secref{sec:tactics}). Note that the @{command subgoal}
(\secref{sec:subgoal}) command usually eliminates the need for implicit goal
state references. ›
‹Commands for step-wise refinement \label{sec:tactic-commands}›
🪙
is similar to @{command "note"}, but it operates in backwards mode and does
not have any impact on chained facts.
🪙 @{command "apply"}~‹m› applies proof method ‹m› in initial position, but
unlike @{command "proof"} it retains ``‹proof(prove)›
consecutive method applications may be given just as in tactic scripts.
Facts are passed to ‹m› as indicated by the goal's forward-chain mode, and
are 🪙‹consumed› afterwards. Thus any further @{command "apply"} command
would always work in a purely backward manner.
🪙 @{command "apply_end"}~‹m› applies proof method ‹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 ‹m› here. Furthermore, the static context is that of
ommand "qe}).Thus thet proofmethod
may not refer to any assumptions introduced in the current body, for
example.
🪙 @{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.
🪙 @{command "defer"}~‹n› and @{command "prefer"}~‹n› shuffle the list of
pending goals: @{command "defer"} puts off sub-goal ‹
list (‹n = 1› by default), while @{command "prefer"} brings sub-goal ‹n› to
the front.
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
latest proof command. Any proof command may return multiple results, and
this command explores the possibilities step-by-step. It is mainly useful
for experimentation and interactive exploration, and should be avoided in
finished proofs. ›
🪙 @{command "subgoal"} allows to impose some structure on backward
refinements, to avoid proof scripts degenerating into long of @{command
apply} sequences.
The current goal state, which is essentially a hidden part of the Isar/VM
configuration, is turned into a proof context and remaining conclusion.
This corresponds to @{command fix}~/ @{command assume}~/ @{command show} in
structured proofs, but the text of the parameters, premises and conclusion
is not given explicitly.
Goal parameters may be specified separately, in order to allow referring to
them in the proof body: ``@{command subgoal}~@{keyword "for"}~‹x y z›''
names a 🪙‹prefix›, and ``@{command subgoal}~@{keyword "for"}~‹
names a 🪙‹suffix› of goal parameters. The latter uses a literal 🍋‹…› symbol
as notation. Parameter positions may be skipped via dummies (underscore).
Unspecified names remain internal, and thus inaccessible in the proof text.
``@{command subgoal}~@{keyword "premises"}~‹prems›'' indicates that goal
premises should be turned into assumptions of the context (otherwise the
remaining conclusion is a Pure implication). The fact name and attributes
are optional; the particular name ``‹prems›'' is a common convention for the
premises of an arbitrary goal context in proof scripts.
``@{command subgoal}~‹
proven subgoal. Thus it may be re-used in further reasoning, similar to the
result of @{command show} in structured Isar proofs.
Here are some abstract examples: ›
"∧x y z. A x ==> B y ==> C z"
and "∧u v. X u ==> Y v"
subgoal 🍋
subgoal 🍋
done
"∧x y z. A x ==> B y ==> C z"
and "∧C of str |
subgoal for x y z 🍋
subgoal for u v 🍋
done
"∧x y z. A x ==> B y ==> C z"
and "∧u v. X u ==> Y v"
subgoal premises for x y z
using ‹A x›‹string | 🍋
subgoal premises for u v
using ‹X u› 🍋
done
"∧x y z. A x ==> B y ==> C z"
and "∧u v. X u ==> Y v"
subgoal r premises prems for x y z
proof -
have "A x" by (fact prems)
moreover have "B y" by (fact prems)
ultimately show ?thesis 🍋
qed
subgoal premises prems for u v
proof -
have "∧x y z. A x ==> B y ==> C z" by (fact r)
moreover
have "X u" by (fact prems)
ultimately show ?thesis 🍋
qed
done
"∧x y z. A x ==> B y ==> C z"
subgoal premises prems for … z
proof -
from prems show "C z" 🍋
qed
done
‹
The following improper proof methods emulate traditional tactics. These
admit direct access to the goal state, which is normally considered harmful!
In particular, this may involve both numbered goal addressing (default 1),
and dynamic instantiation within the scope of some subgoal.
\begin{warn}
Dynamic instantiations refer to universally quantified parameters of a
subgoal (the d context) rath than fix variab and term
abbreviations of a (static) Isar context. \end{warn}
Tactic emulation methods, unlike their ML counterparts, admit simultaneous
instantiation from both dynamic and static contexts. If names occur in both
contexts goal parameters hide locally fixed variables. Likewise, schematic
variables refer to term abbreviations, if present in the static context.
Otherwise the schematic variable is interpreted as a schematic variable and
left to be solved by unification with certain parts of the subgoal.
Note that the tactic emulation proof methods in Isabelle/Isar are
consistently named ‹foo_tac› unota that resembl
left hand sides of instantiations must be preceded by a question mark if
they coincide with a keyword or contain dots. This is consistent with the
attribute @{attribute "where"} (see \secref{sec:pure-meth-att}).
🪙 @{method rule_tac} etc. do resolution of rules with explicit
instantiation. This works the same way as the ML tactics \<^ML ›
Multiple rules may be only given if there is no instantiation; then @{method
rule_tac} is the same as 🪙‹resolve_tac› in ML (see cite‹"isabelle-implementation"›).
🪙 @{method cut_tac} inserts facts into the proof state as assumption of a
subgoal; instantiations may be given as well. Note that the scope of
schematic variables is spread over the main goal statement and rule premises
are turned into new subgoals. This is in contrast to the regular method
@{method insert} which inserts closed rule statements.
🪙 @{method thin_tac}~‹φ› deletes the specified premise from a subgoal. Note
that ‹φ› may contain schematic variables, to abbreviate the intended
proposition; the first matching subgoal premise will be deleted. Removing
useless premises from a subgoal increases its readability and can make
search tactics run faster.
🪙 @{method subgoal_tac}~‹φ1… φn› adds the propositions ‹φ1… φn› as
local premises to a subgoal, and poses the same as new subgoals (in the
original context).
🪙 @{method rename_tac}~‹x1… xn› renames parameters of a goal according to
the list ‹x1, …, xn›
🪙 @{method rotate_tac}~‹n› rotates the premises of a subgoal by ‹n›
positions: from right to left if ‹n› is positive, and from left to right if ‹n› is negative; the default value is 1.
🪙cert 🪙‹tactic›. Apart from the usual ML environment and the current proof
context, the ML code may refer to the locally bound values 🪙‹facts›,
which indicates any current facts used for forward-chaining.
🪙 @{method raw_tactic} is similar to @{method tactic}, but presents the goal
state in its raw internal form, where simultaneous subgoals appear as
conjunction of the logical framework instead of the usual split into several
subgoals. While feature this is useful for debugging of complex method
definitions, it sres.. Thus 🍋 ›
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