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Untersuchungsergebnis.rst Download desHaskell {Haskell[232] Fortran[423] Ada[431]}zum Wurzelverzeichnis wechseln
.. _ltac:
The tactic language
===================
This chapter documents the tactic language |Ltac|.
We start by giving the syntax, and next, we present the informal
semantics. To learn more about the language and
especially about its foundations, please refer to :cite:`Del00`.
.. example:: Basic tactic macros
Here are some examples of simple tactic macros that the
language lets you write.
.. coqdoc::
Ltac reduce_and_try_to_solve := simpl; intros; auto.
Ltac destruct_bool_and_rewrite b H1 H2 :=
destruct b; [ rewrite H1; eauto | rewrite H2; eauto ].
See Section :ref:`ltac-examples` for more advanced examples.
.. _ltac-syntax:
Syntax
------
The syntax of the tactic language is given below. See Chapter
:ref:`gallinaspecificationlanguage` for a description of the BNF metasyntax used
in these grammar rules. Various already defined entries will be used in this
chapter: entries :token:`natural`, :token:`integer`, :token:`ident`,
:token:`qualid`, :token:`term`, :token:`cpattern` and :token:`tactic`
represent respectively the natural and integer numbers, the authorized
identificators and qualified names, Coq terms and patterns and all the atomic
tactics described in Chapter :ref:`tactics`.
The syntax of :production:`cpattern` is
the same as that of terms, but it is extended with pattern matching
metavariables. In :token:`cpattern`, a pattern matching metavariable is
represented with the syntax :n:`?@ident`. The
notation :g:`_` can also be used to denote metavariable whose instance is
irrelevant. In the notation :n:`?@ident`, the identifier allows us to keep
instantiations and to make constraints whereas :g:`_` shows that we are not
interested in what will be matched. On the right hand side of pattern matching
clauses, the named metavariables are used without the question mark prefix. There
is also a special notation for second-order pattern matching problems: in an
applicative pattern of the form :n:`%@?@ident @ident__1 … @ident__n`,
the variable :token:`ident` matches any complex expression with (possible)
dependencies in the variables :n:`@ident__i` and returns a functional term
of the form :n:`fun @ident__1 … ident__n => @term`.
The main entry of the grammar is :n:`@ltac_expr`. This language is used in proof
mode but it can also be used in toplevel definitions as shown below.
.. note::
- The infix tacticals ``… || …`` , ``… + …`` , and ``… ; …`` are associative.
.. example::
If you want that :n:`@tactic__2; @tactic__3` be fully run on the first
subgoal generated by :n:`@tactic__1`, before running on the other
subgoals, then you should not write
:n:`@tactic__1; (@tactic__2; @tactic__3)` but rather
:n:`@tactic__1; [> @tactic__2; @tactic__3 .. ]`.
- In :token:`tacarg`, there is an overlap between :token:`qualid` as a
direct tactic argument and :token:`qualid` as a particular case of
:token:`term`. The resolution is done by first looking for a reference
of the tactic language and if it fails, for a reference to a term.
To force the resolution as a reference of the tactic language, use the
form :n:`ltac:(@qualid)`. To force the resolution as a reference to a
term, use the syntax :n:`(@qualid)`.
- As shown by the figure, tactical ``… || …`` binds more than the prefix
tacticals :tacn:`try`, :tacn:`repeat`, :tacn:`do` and :tacn:`abstract`
which themselves bind more than the postfix tactical ``… ;[ … ]``
which binds at the same level as ``… ; …`` .
.. example::
:n:`try repeat @tactic__1 || @tactic__2; @tactic__3; [ {+| @tactic } ]; @tactic__4`
is understood as:
:n:`((try (repeat (@tactic__1 || @tactic__2)); @tactic__3); [ {+| @tactic } ]); @tactic__4`
.. productionlist:: coq
ltac_expr : `ltac_expr` ; `ltac_expr`
: [> `ltac_expr` | ... | `ltac_expr` ]
: `ltac_expr` ; [ `ltac_expr` | ... | `ltac_expr` ]
: `ltac_expr3`
ltac_expr3 : do (`natural` | `ident`) `ltac_expr3`
: progress `ltac_expr3`
: repeat `ltac_expr3`
: try `ltac_expr3`
: once `ltac_expr3`
: exactly_once `ltac_expr3`
: timeout (`natural` | `ident`) `ltac_expr3`
: time [`string`] `ltac_expr3`
: only `selector`: `ltac_expr3`
: `ltac_expr2`
ltac_expr2 : `ltac_expr1` || `ltac_expr3`
: `ltac_expr1` + `ltac_expr3`
: tryif `ltac_expr1` then `ltac_expr1` else `ltac_expr1`
: `ltac_expr1`
ltac_expr1 : fun `name` ... `name` => `atom`
: let [rec] `let_clause` with ... with `let_clause` in `atom`
: match goal with `context_rule` | ... | `context_rule` end
: match reverse goal with `context_rule` | ... | `context_rule` end
: match `ltac_expr` with `match_rule` | ... | `match_rule` end
: lazymatch goal with `context_rule` | ... | `context_rule` end
: lazymatch reverse goal with `context_rule` | ... | `context_rule` end
: lazymatch `ltac_expr` with `match_rule` | ... | `match_rule` end
: multimatch goal with `context_rule` | ... | `context_rule` end
: multimatch reverse goal with `context_rule` | ... | `context_rule` end
: multimatch `ltac_expr` with `match_rule` | ... | `match_rule` end
: abstract `atom`
: abstract `atom` using `ident`
: first [ `ltac_expr` | ... | `ltac_expr` ]
: solve [ `ltac_expr` | ... | `ltac_expr` ]
: idtac [ `message_token` ... `message_token`]
: fail [`natural`] [`message_token` ... `message_token`]
: gfail [`natural`] [`message_token` ... `message_token`]
: fresh [ `component` … `component` ]
: context `ident` [`term`]
: eval `redexpr` in `term`
: type of `term`
: constr : `term`
: uconstr : `term`
: type_term `term`
: numgoals
: guard `test`
: assert_fails `ltac_expr3`
: assert_succeeds `ltac_expr3`
: `tactic`
: `qualid` `tacarg` ... `tacarg`
: `atom`
atom : `qualid`
: ()
: `integer`
: ( `ltac_expr` )
component : `string` | `qualid`
message_token : `string` | `ident` | `integer`
tacarg : `qualid`
: ()
: ltac : `atom`
: `term`
let_clause : `ident` [`name` ... `name`] := `ltac_expr`
context_rule : `context_hyp`, ..., `context_hyp` |- `cpattern` => `ltac_expr`
: `cpattern` => `ltac_expr`
: |- `cpattern` => `ltac_expr`
: _ => `ltac_expr`
context_hyp : `name` : `cpattern`
: `name` := `cpattern` [: `cpattern`]
match_rule : `cpattern` => `ltac_expr`
: context [`ident`] [ `cpattern` ] => `ltac_expr`
: _ => `ltac_expr`
test : `integer` = `integer`
: `integer` (< | <= | > | >=) `integer`
selector : [`ident`]
: `integer`
: (`integer` | `integer` - `integer`), ..., (`integer` | `integer` - `integer`)
toplevel_selector : `selector`
: all
: par
: !
.. productionlist:: coq
top : [Local] Ltac `ltac_def` with ... with `ltac_def`
ltac_def : `ident` [`ident` ... `ident`] := `ltac_expr`
: `qualid` [`ident` ... `ident`] ::= `ltac_expr`
.. _ltac-semantics:
Semantics
---------
Tactic expressions can only be applied in the context of a proof. The
evaluation yields either a term, an integer or a tactic. Intermediate
results can be terms or integers but the final result must be a tactic
which is then applied to the focused goals.
There is a special case for ``match goal`` expressions of which the clauses
evaluate to tactics. Such expressions can only be used as end result of
a tactic expression (never as argument of a non-recursive local
definition or of an application).
The rest of this section explains the semantics of every construction of
|Ltac|.
Sequence
~~~~~~~~
A sequence is an expression of the following form:
.. tacn:: @ltac_expr__1 ; @ltac_expr__2
:name: ltac-seq
The expression :n:`@ltac_expr__1` is evaluated to :n:`v__1`, which must be
a tactic value. The tactic :n:`v__1` is applied to the current goal,
possibly producing more goals. Then :n:`@ltac_expr__2` is evaluated to
produce :n:`v__2`, which must be a tactic value. The tactic
:n:`v__2` is applied to all the goals produced by the prior
application. Sequence is associative.
Local application of tactics
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Different tactics can be applied to the different goals using the
following form:
.. tacn:: [> {*| @ltac_expr }]
:name: [> ... | ... | ... ] (dispatch)
The expressions :n:`@ltac_expr__i` are evaluated to :n:`v__i`, for
i = 1, ..., n and all have to be tactics. The :n:`v__i` is applied to the
i-th goal, for i = 1, ..., n. It fails if the number of focused goals is not
exactly n.
.. note::
If no tactic is given for the i-th goal, it behaves as if the tactic idtac
were given. For instance, ``[> | auto]`` is a shortcut for ``[> idtac | auto
]``.
.. tacv:: [> {*| @ltac_expr__i} | @ltac_expr .. | {*| @ltac_expr__j}]
In this variant, :n:`@ltac_expr` is used for each goal coming after those
covered by the list of :n:`@ltac_expr__i` but before those covered by the
list of :n:`@ltac_expr__j`.
.. tacv:: [> {*| @ltac_expr} | .. | {*| @ltac_expr}]
In this variant, idtac is used for the goals not covered by the two lists of
:n:`@ltac_expr`.
.. tacv:: [> @ltac_expr .. ]
In this variant, the tactic :n:`@ltac_expr` is applied independently to each of
the goals, rather than globally. In particular, if there are no goals, the
tactic is not run at all. A tactic which expects multiple goals, such as
``swap``, would act as if a single goal is focused.
.. tacv:: @ltac_expr__0 ; [{*| @ltac_expr__i}]
This variant of local tactic application is paired with a sequence. In this
variant, there must be as many :n:`@ltac_expr__i` as goals generated
by the application of :n:`@ltac_expr__0` to each of the individual goals
independently. All the above variants work in this form too.
Formally, :n:`@ltac_expr ; [ ... ]` is equivalent to :n:`[> @ltac_expr ; [> ... ] .. ]`.
.. _goal-selectors:
Goal selectors
~~~~~~~~~~~~~~
We can restrict the application of a tactic to a subset of the currently
focused goals with:
.. tacn:: @toplevel_selector : @ltac_expr
:name: ... : ... (goal selector)
We can also use selectors as a tactical, which allows to use them nested
in a tactic expression, by using the keyword ``only``:
.. tacv:: only @selector : @ltac_expr
:name: only ... : ...
When selecting several goals, the tactic :token:`ltac_expr` is applied globally to all
selected goals.
.. tacv:: [@ident] : @ltac_expr
In this variant, :token:`ltac_expr` is applied locally to a goal previously named
by the user (see :ref:`existential-variables`).
.. tacv:: @num : @ltac_expr
In this variant, :token:`ltac_expr` is applied locally to the :token:`num`-th goal.
.. tacv:: {+, @num-@num} : @ltac_expr
In this variant, :n:`@ltac_expr` is applied globally to the subset of goals
described by the given ranges. You can write a single ``n`` as a shortcut
for ``n-n`` when specifying multiple ranges.
.. tacv:: all: @ltac_expr
:name: all: ...
In this variant, :token:`ltac_expr` is applied to all focused goals. ``all:`` can only
be used at the toplevel of a tactic expression.
.. tacv:: !: @ltac_expr
In this variant, if exactly one goal is focused, :token:`ltac_expr` is
applied to it. Otherwise the tactic fails. ``!:`` can only be
used at the toplevel of a tactic expression.
.. tacv:: par: @ltac_expr
:name: par: ...
In this variant, :n:`@ltac_expr` is applied to all focused goals in parallel.
The number of workers can be controlled via the command line option
``-async-proofs-tac-j`` taking as argument the desired number of workers.
Limitations: ``par:`` only works on goals containing no existential
variables and :n:`@ltac_expr` must either solve the goal completely or do
nothing (i.e. it cannot make some progress). ``par:`` can only be used at
the toplevel of a tactic expression.
.. exn:: No such goal.
:name: No such goal. (Goal selector)
:undocumented:
.. TODO change error message index entry
For loop
~~~~~~~~
There is a for loop that repeats a tactic :token:`num` times:
.. tacn:: do @num @ltac_expr
:name: do
:n:`@ltac_expr` is evaluated to ``v`` which must be a tactic value. This tactic
value ``v`` is applied :token:`num` times. Supposing :token:`num` > 1, after the
first application of ``v``, ``v`` is applied, at least once, to the generated
subgoals and so on. It fails if the application of ``v`` fails before the num
applications have been completed.
Repeat loop
~~~~~~~~~~~
We have a repeat loop with:
.. tacn:: repeat @ltac_expr
:name: repeat
:n:`@ltac_expr` is evaluated to ``v``. If ``v`` denotes a tactic, this tactic is
applied to each focused goal independently. If the application succeeds, the
tactic is applied recursively to all the generated subgoals until it eventually
fails. The recursion stops in a subgoal when the tactic has failed *to make
progress*. The tactic :n:`repeat @ltac_expr` itself never fails.
Error catching
~~~~~~~~~~~~~~
We can catch the tactic errors with:
.. tacn:: try @ltac_expr
:name: try
:n:`@ltac_expr` is evaluated to ``v`` which must be a tactic value. The tactic
value ``v`` is applied to each focused goal independently. If the application of
``v`` fails in a goal, it catches the error and leaves the goal unchanged. If the
level of the exception is positive, then the exception is re-raised with its
level decremented.
Detecting progress
~~~~~~~~~~~~~~~~~~
We can check if a tactic made progress with:
.. tacn:: progress @ltac_expr
:name: progress
:n:`@ltac_expr` is evaluated to v which must be a tactic value. The tactic value ``v``
is applied to each focused subgoal independently. If the application of ``v``
to one of the focused subgoal produced subgoals equal to the initial
goals (up to syntactical equality), then an error of level 0 is raised.
.. exn:: Failed to progress.
:undocumented:
Backtracking branching
~~~~~~~~~~~~~~~~~~~~~~
We can branch with the following structure:
.. tacn:: @ltac_expr__1 + @ltac_expr__2
:name: + (backtracking branching)
:n:`@ltac_expr__1` and :n:`@ltac_expr__2` are evaluated respectively to :n:`v__1` and
:n:`v__2` which must be tactic values. The tactic value :n:`v__1` is applied to
each focused goal independently and if it fails or a later tactic fails, then
the proof backtracks to the current goal and :n:`v__2` is applied.
Tactics can be seen as having several successes. When a tactic fails it
asks for more successes of the prior tactics.
:n:`@ltac_expr__1 + @ltac_expr__2` has all the successes of :n:`v__1` followed by all the
successes of :n:`v__2`. Algebraically,
:n:`(@ltac_expr__1 + @ltac_expr__2); @ltac_expr__3 = (@ltac_expr__1; @ltac_expr__3) + (@ltac_expr__2; @ltac_expr__3)`.
Branching is left-associative.
First tactic to work
~~~~~~~~~~~~~~~~~~~~
Backtracking branching may be too expensive. In this case we may
restrict to a local, left biased, branching and consider the first
tactic to work (i.e. which does not fail) among a panel of tactics:
.. tacn:: first [{*| @ltac_expr}]
:name: first
The :n:`@ltac_expr__i` are evaluated to :n:`v__i` and :n:`v__i` must be
tactic values for i = 1, ..., n. Supposing n > 1,
:n:`first [@ltac_expr__1 | ... | @ltac_expr__n]` applies :n:`v__1` in each
focused goal independently and stops if it succeeds; otherwise it
tries to apply :n:`v__2` and so on. It fails when there is no
applicable tactic. In other words,
:n:`first [@ltac_expr__1 | ... | @ltac_expr__n]` behaves, in each goal, as the first
:n:`v__i` to have *at least* one success.
.. exn:: No applicable tactic.
:undocumented:
.. tacv:: first @ltac_expr
This is an |Ltac| alias that gives a primitive access to the first
tactical as an |Ltac| definition without going through a parsing rule. It
expects to be given a list of tactics through a ``Tactic Notation``,
allowing to write notations of the following form:
.. example::
.. coqtop:: in
Tactic Notation "foo" tactic_list(tacs) := first tacs.
Left-biased branching
~~~~~~~~~~~~~~~~~~~~~
Yet another way of branching without backtracking is the following
structure:
.. tacn:: @ltac_expr__1 || @ltac_expr__2
:name: || (left-biased branching)
:n:`@ltac_expr__1` and :n:`@ltac_expr__2` are evaluated respectively to :n:`v__1` and
:n:`v__2` which must be tactic values. The tactic value :n:`v__1` is
applied in each subgoal independently and if it fails *to progress* then
:n:`v__2` is applied. :n:`@ltac_expr__1 || @ltac_expr__2` is
equivalent to :n:`first [ progress @ltac_expr__1 | @ltac_expr__2 ]` (except that
if it fails, it fails like :n:`v__2`). Branching is left-associative.
Generalized biased branching
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The tactic
.. tacn:: tryif @ltac_expr__1 then @ltac_expr__2 else @ltac_expr__3
:name: tryif
is a generalization of the biased-branching tactics above. The
expression :n:`@ltac_expr__1` is evaluated to :n:`v__1`, which is then
applied to each subgoal independently. For each goal where :n:`v__1`
succeeds at least once, :n:`@ltac_expr__2` is evaluated to :n:`v__2` which
is then applied collectively to the generated subgoals. The :n:`v__2`
tactic can trigger backtracking points in :n:`v__1`: where :n:`v__1`
succeeds at least once,
:n:`tryif @ltac_expr__1 then @ltac_expr__2 else @ltac_expr__3` is equivalent to
:n:`v__1; v__2`. In each of the goals where :n:`v__1` does not succeed at least
once, :n:`@ltac_expr__3` is evaluated in :n:`v__3` which is is then applied to the
goal.
Soft cut
~~~~~~~~
Another way of restricting backtracking is to restrict a tactic to a
single success *a posteriori*:
.. tacn:: once @ltac_expr
:name: once
:n:`@ltac_expr` is evaluated to ``v`` which must be a tactic value. The tactic value
``v`` is applied but only its first success is used. If ``v`` fails,
:n:`once @ltac_expr` fails like ``v``. If ``v`` has at least one success,
:n:`once @ltac_expr` succeeds once, but cannot produce more successes.
Checking the successes
~~~~~~~~~~~~~~~~~~~~~~
Coq provides an experimental way to check that a tactic has *exactly
one* success:
.. tacn:: exactly_once @ltac_expr
:name: exactly_once
:n:`@ltac_expr` is evaluated to ``v`` which must be a tactic value. The tactic value
``v`` is applied if it has at most one success. If ``v`` fails,
:n:`exactly_once @ltac_expr` fails like ``v``. If ``v`` has a exactly one success,
:n:`exactly_once @ltac_expr` succeeds like ``v``. If ``v`` has two or more
successes, :n:`exactly_once @ltac_expr` fails.
.. warning::
The experimental status of this tactic pertains to the fact if ``v``
performs side effects, they may occur in an unpredictable way. Indeed,
normally ``v`` would only be executed up to the first success until
backtracking is needed, however exactly_once needs to look ahead to see
whether a second success exists, and may run further effects
immediately.
.. exn:: This tactic has more than one success.
:undocumented:
Checking the failure
~~~~~~~~~~~~~~~~~~~~
Coq provides a derived tactic to check that a tactic *fails*:
.. tacn:: assert_fails @ltac_expr
:name: assert_fails
This behaves like :n:`tryif @ltac_expr then fail 0 tac "succeeds" else idtac`.
Checking the success
~~~~~~~~~~~~~~~~~~~~
Coq provides a derived tactic to check that a tactic has *at least one*
success:
.. tacn:: assert_succeeds @ltac_expr
:name: assert_succeeds
This behaves like
:n:`tryif (assert_fails tac) then fail 0 tac "fails" else idtac`.
Solving
~~~~~~~
We may consider the first to solve (i.e. which generates no subgoal)
among a panel of tactics:
.. tacn:: solve [{*| @ltac_expr}]
:name: solve
The :n:`@ltac_expr__i` are evaluated to :n:`v__i` and :n:`v__i` must be
tactic values, for i = 1, ..., n. Supposing n > 1,
:n:`solve [@ltac_expr__1 | ... | @ltac_expr__n]` applies :n:`v__1` to
each goal independently and stops if it succeeds; otherwise it tries to
apply :n:`v__2` and so on. It fails if there is no solving tactic.
.. exn:: Cannot solve the goal.
:undocumented:
.. tacv:: solve @ltac_expr
This is an |Ltac| alias that gives a primitive access to the :n:`solve:`
tactical. See the :n:`first` tactical for more information.
Identity
~~~~~~~~
The constant :n:`idtac` is the identity tactic: it leaves any goal unchanged but
it appears in the proof script.
.. tacn:: idtac {* @message_token}
:name: idtac
This prints the given tokens. Strings and integers are printed
literally. If a (term) variable is given, its contents are printed.
Failing
~~~~~~~
.. tacn:: fail
:name: fail
This is the always-failing tactic: it does not solve any
goal. It is useful for defining other tacticals since it can be caught by
:tacn:`try`, :tacn:`repeat`, :tacn:`match goal`, or the branching tacticals.
.. tacv:: fail @num
The number is the failure level. If no level is specified, it defaults to 0.
The level is used by :tacn:`try`, :tacn:`repeat`, :tacn:`match goal` and the branching
tacticals. If 0, it makes :tacn:`match goal` consider the next clause
(backtracking). If nonzero, the current :tacn:`match goal` block, :tacn:`try`,
:tacn:`repeat`, or branching command is aborted and the level is decremented. In
the case of :n:`+`, a nonzero level skips the first backtrack point, even if
the call to :n:`fail @num` is not enclosed in a :n:`+` command,
respecting the algebraic identity.
.. tacv:: fail {* @message_token}
The given tokens are used for printing the failure message.
.. tacv:: fail @num {* @message_token}
This is a combination of the previous variants.
.. tacv:: gfail
:name: gfail
This variant fails even when used after :n:`;` and there are no goals left.
Similarly, ``gfail`` fails even when used after ``all:`` and there are no
goals left. See the example for clarification.
.. tacv:: gfail {* @message_token}
gfail @num {* @message_token}
These variants fail with an error message or an error level even if
there are no goals left. Be careful however if Coq terms have to be
printed as part of the failure: term construction always forces the
tactic into the goals, meaning that if there are no goals when it is
evaluated, a tactic call like :n:`let x := H in fail 0 x` will succeed.
.. exn:: Tactic Failure message (level @num).
:undocumented:
.. exn:: No such goal.
:name: No such goal. (fail)
:undocumented:
.. example::
.. coqtop:: all fail
Goal True.
Proof. fail. Abort.
Goal True.
Proof. trivial; fail. Qed.
Goal True.
Proof. trivial. fail. Abort.
Goal True.
Proof. trivial. all: fail. Qed.
Goal True.
Proof. gfail. Abort.
Goal True.
Proof. trivial; gfail. Abort.
Goal True.
Proof. trivial. gfail. Abort.
Goal True.
Proof. trivial. all: gfail. Abort.
Timeout
~~~~~~~
We can force a tactic to stop if it has not finished after a certain
amount of time:
.. tacn:: timeout @num @ltac_expr
:name: timeout
:n:`@ltac_expr` is evaluated to ``v`` which must be a tactic value. The tactic value
``v`` is applied normally, except that it is interrupted after :n:`@num` seconds
if it is still running. In this case the outcome is a failure.
.. warning::
For the moment, timeout is based on elapsed time in seconds,
which is very machine-dependent: a script that works on a quick machine
may fail on a slow one. The converse is even possible if you combine a
timeout with some other tacticals. This tactical is hence proposed only
for convenience during debugging or other development phases, we strongly
advise you to not leave any timeout in final scripts. Note also that
this tactical isn’t available on the native Windows port of Coq.
Timing a tactic
~~~~~~~~~~~~~~~
A tactic execution can be timed:
.. tacn:: time @string @ltac_expr
:name: time
evaluates :n:`@ltac_expr` and displays the running time of the tactic expression, whether it
fails or succeeds. In case of several successes, the time for each successive
run is displayed. Time is in seconds and is machine-dependent. The :n:`@string`
argument is optional. When provided, it is used to identify this particular
occurrence of time.
Timing a tactic that evaluates to a term
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Tactic expressions that produce terms can be timed with the experimental
tactic
.. tacn:: time_constr @ltac_expr
:name: time_constr
which evaluates :n:`@ltac_expr ()` and displays the time the tactic expression
evaluated, assuming successful evaluation. Time is in seconds and is
machine-dependent.
This tactic currently does not support nesting, and will report times
based on the innermost execution. This is due to the fact that it is
implemented using the following internal tactics:
.. tacn:: restart_timer @string
:name: restart_timer
Reset a timer
.. tacn:: finish_timing {? (@string)} @string
:name: finish_timing
Display an optionally named timer. The parenthesized string argument
is also optional, and determines the label associated with the timer
for printing.
By copying the definition of :tacn:`time_constr` from the standard library,
users can achieve support for a fixed pattern of nesting by passing
different :token:`string` parameters to :tacn:`restart_timer` and
:tacn:`finish_timing` at each level of nesting.
.. example::
.. coqtop:: all abort
Ltac time_constr1 tac :=
let eval_early := match goal with _ => restart_timer "(depth 1)" end in
let ret := tac () in
let eval_early := match goal with _ => finish_timing ( "Tactic evaluation" ) "(depth 1)" end in
ret.
Goal True.
let v := time_constr
ltac:(fun _ =>
let x := time_constr1 ltac:(fun _ => constr:(10 * 10)) in
let y := time_constr1 ltac:(fun _ => eval compute in x) in
y) in
pose v.
Local definitions
~~~~~~~~~~~~~~~~~
Local definitions can be done as follows:
.. tacn:: let @ident__1 := @ltac_expr__1 {* with @ident__i := @ltac_expr__i} in @ltac_expr
:name: let ... := ...
each :n:`@ltac_expr__i` is evaluated to :n:`v__i`, then, :n:`@ltac_expr` is evaluated
by substituting :n:`v__i` to each occurrence of :n:`@ident__i`, for
i = 1, ..., n. There are no dependencies between the :n:`@ltac_expr__i` and the
:n:`@ident__i`.
Local definitions can be made recursive by using :n:`let rec` instead of :n:`let`.
In this latter case, the definitions are evaluated lazily so that the rec
keyword can be used also in non-recursive cases so as to avoid the eager
evaluation of local definitions.
.. but rec changes the binding!!
Application
~~~~~~~~~~~
An application is an expression of the following form:
.. tacn:: @qualid {+ @tacarg}
The reference :n:`@qualid` must be bound to some defined tactic definition
expecting at least as many arguments as the provided :n:`tacarg`. The
expressions :n:`@ltac_expr__i` are evaluated to :n:`v__i`, for i = 1, ..., n.
.. what expressions ??
Function construction
~~~~~~~~~~~~~~~~~~~~~
A parameterized tactic can be built anonymously (without resorting to
local definitions) with:
.. tacn:: fun {+ @ident} => @ltac_expr
Indeed, local definitions of functions are a syntactic sugar for binding
a :n:`fun` tactic to an identifier.
Pattern matching on terms
~~~~~~~~~~~~~~~~~~~~~~~~~
We can carry out pattern matching on terms with:
.. tacn:: match @ltac_expr with {+| @cpattern__i => @ltac_expr__i} end
The expression :n:`@ltac_expr` is evaluated and should yield a term which is
matched against :n:`cpattern__1`. The matching is non-linear: if a
metavariable occurs more than once, it should match the same expression
every time. It is first-order except on the variables of the form :n:`@?id`
that occur in head position of an application. For these variables, the
matching is second-order and returns a functional term.
Alternatively, when a metavariable of the form :n:`?id` occurs under binders,
say :n:`x__1, …, x__n` and the expression matches, the
metavariable is instantiated by a term which can then be used in any
context which also binds the variables :n:`x__1, …, x__n` with
same types. This provides with a primitive form of matching under
context which does not require manipulating a functional term.
If the matching with :n:`@cpattern__1` succeeds, then :n:`@ltac_expr__1` is
evaluated into some value by substituting the pattern matching
instantiations to the metavariables. If :n:`@ltac_expr__1` evaluates to a
tactic and the match expression is in position to be applied to a goal
(e.g. it is not bound to a variable by a :n:`let in`), then this tactic is
applied. If the tactic succeeds, the list of resulting subgoals is the
result of the match expression. If :n:`@ltac_expr__1` does not evaluate to a
tactic or if the match expression is not in position to be applied to a
goal, then the result of the evaluation of :n:`@ltac_expr__1` is the result
of the match expression.
If the matching with :n:`@cpattern__1` fails, or if it succeeds but the
evaluation of :n:`@ltac_expr__1` fails, or if the evaluation of
:n:`@ltac_expr__1` succeeds but returns a tactic in execution position whose
execution fails, then :n:`cpattern__2` is used and so on. The pattern
:n:`_` matches any term and shadows all remaining patterns if any. If all
clauses fail (in particular, there is no pattern :n:`_`) then a
no-matching-clause error is raised.
Failures in subsequent tactics do not cause backtracking to select new
branches or inside the right-hand side of the selected branch even if it
has backtracking points.
.. exn:: No matching clauses for match.
No pattern can be used and, in particular, there is no :n:`_` pattern.
.. exn:: Argument of match does not evaluate to a term.
This happens when :n:`@ltac_expr` does not denote a term.
.. tacv:: multimatch @ltac_expr with {+| @cpattern__i => @ltac_expr__i} end
Using multimatch instead of match will allow subsequent tactics to
backtrack into a right-hand side tactic which has backtracking points
left and trigger the selection of a new matching branch when all the
backtracking points of the right-hand side have been consumed.
The syntax :n:`match …` is, in fact, a shorthand for :n:`once multimatch …`.
.. tacv:: lazymatch @ltac_expr with {+| @cpattern__i => @ltac_expr__i} end
Using lazymatch instead of match will perform the same pattern
matching procedure but will commit to the first matching branch
rather than trying a new matching if the right-hand side fails. If
the right-hand side of the selected branch is a tactic with
backtracking points, then subsequent failures cause this tactic to
backtrack.
.. tacv:: context @ident [@cpattern]
This special form of patterns matches any term with a subterm matching
cpattern. If there is a match, the optional :n:`@ident` is assigned the "matched
context", i.e. the initial term where the matched subterm is replaced by a
hole. The example below will show how to use such term contexts.
If the evaluation of the right-hand-side of a valid match fails, the next
matching subterm is tried. If no further subterm matches, the next clause
is tried. Matching subterms are considered top-bottom and from left to
right (with respect to the raw printing obtained by setting option
:flag:`Printing All`).
.. example::
.. coqtop:: all abort
Ltac f x :=
match x with
context f [S ?X] =>
idtac X; (* To display the evaluation order *)
assert (p := eq_refl 1 : X=1); (* To filter the case X=1 *)
let x:= context f[O] in assert (x=O) (* To observe the context *)
end.
Goal True.
f (3+4).
.. _ltac-match-goal:
Pattern matching on goals
~~~~~~~~~~~~~~~~~~~~~~~~~
We can perform pattern matching on goals using the following expression:
.. we should provide the full grammar here
.. tacn:: match goal with {+| {+, @context_hyp} |- @cpattern => @ltac_expr } | _ => @ltac_expr end
:name: match goal
If each hypothesis pattern :n:`hyp`\ :sub:`1,i`, with i = 1, ..., m\ :sub:`1` is
matched (non-linear first-order unification) by a hypothesis of the
goal and if :n:`cpattern_1` is matched by the conclusion of the goal,
then :n:`@ltac_expr__1` is evaluated to :n:`v__1` by substituting the
pattern matching to the metavariables and the real hypothesis names
bound to the possible hypothesis names occurring in the hypothesis
patterns. If :n:`v__1` is a tactic value, then it is applied to the
goal. If this application fails, then another combination of hypotheses
is tried with the same proof context pattern. If there is no other
combination of hypotheses then the second proof context pattern is tried
and so on. If the next to last proof context pattern fails then
the last :n:`@ltac_expr` is evaluated to :n:`v` and :n:`v` is
applied. Note also that matching against subterms (using the :n:`context
@ident [ @cpattern ]`) is available and is also subject to yielding several
matchings.
Failures in subsequent tactics do not cause backtracking to select new
branches or combinations of hypotheses, or inside the right-hand side of
the selected branch even if it has backtracking points.
.. exn:: No matching clauses for match goal.
No clause succeeds, i.e. all matching patterns, if any, fail at the
application of the right-hand-side.
.. note::
It is important to know that each hypothesis of the goal can be matched
by at most one hypothesis pattern. The order of matching is the
following: hypothesis patterns are examined from right to left
(i.e. hyp\ :sub:`i,m`\ :sub:`i`` before hyp\ :sub:`i,1`). For each
hypothesis pattern, the goal hypotheses are matched in order (newest
first), but it possible to reverse this order (oldest first)
with the :n:`match reverse goal with` variant.
.. tacv:: multimatch goal with {+| {+, @context_hyp} |- @cpattern => @ltac_expr } | _ => @ltac_expr end
Using :n:`multimatch` instead of :n:`match` will allow subsequent tactics
to backtrack into a right-hand side tactic which has backtracking points
left and trigger the selection of a new matching branch or combination of
hypotheses when all the backtracking points of the right-hand side have
been consumed.
The syntax :n:`match [reverse] goal …` is, in fact, a shorthand for
:n:`once multimatch [reverse] goal …`.
.. tacv:: lazymatch goal with {+| {+, @context_hyp} |- @cpattern => @ltac_expr } | _ => @ltac_expr end
Using lazymatch instead of match will perform the same pattern matching
procedure but will commit to the first matching branch with the first
matching combination of hypotheses rather than trying a new matching if
the right-hand side fails. If the right-hand side of the selected branch
is a tactic with backtracking points, then subsequent failures cause
this tactic to backtrack.
Filling a term context
~~~~~~~~~~~~~~~~~~~~~~
The following expression is not a tactic in the sense that it does not
produce subgoals but generates a term to be used in tactic expressions:
.. tacn:: context @ident [@ltac_expr]
:n:`@ident` must denote a context variable bound by a context pattern of a
match expression. This expression evaluates replaces the hole of the
value of :n:`@ident` by the value of :n:`@ltac_expr`.
.. exn:: Not a context variable.
:undocumented:
.. exn:: Unbound context identifier @ident.
:undocumented:
Generating fresh hypothesis names
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Tactics sometimes have to generate new names for hypothesis. Letting the
system decide a name with the intro tactic is not so good since it is
very awkward to retrieve the name the system gave. The following
expression returns an identifier:
.. tacn:: fresh {* @component}
It evaluates to an identifier unbound in the goal. This fresh identifier
is obtained by concatenating the value of the :n:`@component`\ s (each of them
is, either a :n:`@qualid` which has to refer to a (unqualified) name, or
directly a name denoted by a :n:`@string`).
If the resulting name is already used, it is padded with a number so that it
becomes fresh. If no component is given, the name is a fresh derivative of
the name ``H``.
Computing in a constr
~~~~~~~~~~~~~~~~~~~~~
Evaluation of a term can be performed with:
.. tacn:: eval @redexpr in @term
where :n:`@redexpr` is a reduction tactic among :tacn:`red`, :tacn:`hnf`,
:tacn:`compute`, :tacn:`simpl`, :tacn:`cbv`, :tacn:`lazy`, :tacn:`unfold`,
:tacn:`fold`, :tacn:`pattern`.
Recovering the type of a term
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. tacn:: type of @term
This tactic returns the type of :token:`term`.
Manipulating untyped terms
~~~~~~~~~~~~~~~~~~~~~~~~~~
.. tacn:: uconstr : @term
The terms built in |Ltac| are well-typed by default. It may not be
appropriate for building large terms using a recursive |Ltac| function: the
term has to be entirely type checked at each step, resulting in potentially
very slow behavior. It is possible to build untyped terms using |Ltac| with
the :n:`uconstr : @term` syntax.
.. tacn:: type_term @term
An untyped term, in |Ltac|, can contain references to hypotheses or to
|Ltac| variables containing typed or untyped terms. An untyped term can be
type checked using the function type_term whose argument is parsed as an
untyped term and returns a well-typed term which can be used in tactics.
Untyped terms built using :n:`uconstr :` can also be used as arguments to the
:tacn:`refine` tactic. In that case the untyped term is type
checked against the conclusion of the goal, and the holes which are not solved
by the typing procedure are turned into new subgoals.
Counting the goals
~~~~~~~~~~~~~~~~~~
.. tacn:: numgoals
The number of goals under focus can be recovered using the :n:`numgoals`
function. Combined with the guard command below, it can be used to
branch over the number of goals produced by previous tactics.
.. example::
.. coqtop:: in
Ltac pr_numgoals := let n := numgoals in idtac "There are" n "goals".
Goal True /\ True /\ True.
split;[|split].
.. coqtop:: all abort
all:pr_numgoals.
Testing boolean expressions
~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. tacn:: guard @test
:name: guard
The :tacn:`guard` tactic tests a boolean expression, and fails if the expression
evaluates to false. If the expression evaluates to true, it succeeds
without affecting the proof.
The accepted tests are simple integer comparisons.
.. example::
.. coqtop:: in
Goal True /\ True /\ True.
split;[|split].
.. coqtop:: all
all:let n:= numgoals in guard n<4.
Fail all:let n:= numgoals in guard n=2.
.. exn:: Condition not satisfied.
:undocumented:
Proving a subgoal as a separate lemma
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. tacn:: abstract @ltac_expr
:name: abstract
From the outside, :n:`abstract @ltac_expr` is the same as :n:`solve @ltac_expr`.
Internally it saves an auxiliary lemma called ``ident_subproofn`` where
``ident`` is the name of the current goal and ``n`` is chosen so that this is
a fresh name. Such an auxiliary lemma is inlined in the final proof term.
This tactical is useful with tactics such as :tacn:`omega` or
:tacn:`discriminate` that generate huge proof terms. With that tool the user
can avoid the explosion at time of the Save command without having to cut
manually the proof in smaller lemmas.
It may be useful to generate lemmas minimal w.r.t. the assumptions they
depend on. This can be obtained thanks to the option below.
.. warning::
The abstract tactic, while very useful, still has some known
limitations, see https://github.com/coq/coq/issues/9146 for more
details. Thus we recommend using it caution in some
"non-standard" contexts. In particular, ``abstract`` won't
properly work when used inside quotations ``ltac:(...)``, or
if used as part of typeclass resolution, it may produce wrong
terms when in universe polymorphic mode.
.. tacv:: abstract @ltac_expr using @ident
Give explicitly the name of the auxiliary lemma.
.. warning::
Use this feature at your own risk; explicitly named and reused subterms
don’t play well with asynchronous proofs.
.. tacv:: transparent_abstract @ltac_expr
:name: transparent_abstract
Save the subproof in a transparent lemma rather than an opaque one.
.. warning::
Use this feature at your own risk; building computationally relevant
terms with tactics is fragile.
.. tacv:: transparent_abstract @ltac_expr using @ident
Give explicitly the name of the auxiliary transparent lemma.
.. warning::
Use this feature at your own risk; building computationally relevant terms
with tactics is fragile, and explicitly named and reused subterms
don’t play well with asynchronous proofs.
.. exn:: Proof is not complete.
:name: Proof is not complete. (abstract)
:undocumented:
Tactic toplevel definitions
---------------------------
Defining |Ltac| functions
~~~~~~~~~~~~~~~~~~~~~~~~~
Basically, |Ltac| toplevel definitions are made as follows:
.. cmd:: {? Local} Ltac @ident {* @ident} := @ltac_expr
:name: Ltac
This defines a new |Ltac| function that can be used in any tactic
script or new |Ltac| toplevel definition.
If preceded by the keyword ``Local``, the tactic definition will not be
exported outside the current module.
.. note::
The preceding definition can equivalently be written:
:n:`Ltac @ident := fun {+ @ident} => @ltac_expr`
.. cmdv:: Ltac @ident {* @ident} {* with @ident {* @ident}} := @ltac_expr
This syntax allows recursive and mutual recursive function definitions.
.. cmdv:: Ltac @qualid {* @ident} ::= @ltac_expr
This syntax *redefines* an existing user-defined tactic.
A previous definition of qualid must exist in the environment. The new
definition will always be used instead of the old one and it goes across
module boundaries.
Printing |Ltac| tactics
~~~~~~~~~~~~~~~~~~~~~~~
.. cmd:: Print Ltac @qualid
Defined |Ltac| functions can be displayed using this command.
.. cmd:: Print Ltac Signatures
This command displays a list of all user-defined tactics, with their arguments.
.. _ltac-examples:
Examples of using |Ltac|
-------------------------
Proof that the natural numbers have at least two elements
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. example:: Proof that the natural numbers have at least two elements
The first example shows how to use pattern matching over the proof
context to prove that natural numbers have at least two
elements. This can be done as follows:
.. coqtop:: reset all
Lemma card_nat :
~ exists x y : nat, forall z:nat, x = z \/ y = z.
Proof.
intros (x & y & Hz).
destruct (Hz 0), (Hz 1), (Hz 2).
At this point, the :tacn:`congruence` tactic would finish the job:
.. coqtop:: all abort
all: congruence.
But for the purpose of the example, let's craft our own custom
tactic to solve this:
.. coqtop:: none
Lemma card_nat :
~ exists x y : nat, forall z:nat, x = z \/ y = z.
Proof.
intros (x & y & Hz).
destruct (Hz 0), (Hz 1), (Hz 2).
.. coqtop:: all abort
all: match goal with
| _ : ?a = ?b, _ : ?a = ?c |- _ => assert (b = c) by now transitivity a
end.
all: discriminate.
Notice that all the (very similar) cases coming from the three
eliminations (with three distinct natural numbers) are successfully
solved by a ``match goal`` structure and, in particular, with only one
pattern (use of non-linear matching).
Proving that a list is a permutation of a second list
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. example:: Proving that a list is a permutation of a second list
Let's first define the permutation predicate:
.. coqtop:: in reset
Section Sort.
Variable A : Set.
Inductive perm : list A -> list A -> Prop :=
| perm_refl : forall l, perm l l
| perm_cons : forall a l0 l1, perm l0 l1 -> perm (a :: l0) (a :: l1)
| perm_append : forall a l, perm (a :: l) (l ++ a :: nil)
| perm_trans : forall l0 l1 l2, perm l0 l1 -> perm l1 l2 -> perm l0 l2.
End Sort.
.. coqtop:: none
Require Import List.
Next we define an auxiliary tactic :g:`perm_aux` which takes an
argument used to control the recursion depth. This tactic works as
follows: If the lists are identical (i.e. convertible), it
completes the proof. Otherwise, if the lists have identical heads,
it looks at their tails. Finally, if the lists have different
heads, it rotates the first list by putting its head at the end.
Every time we perform a rotation, we decrement :g:`n`. When :g:`n`
drops down to :g:`1`, we stop performing rotations and we fail.
The idea is to give the length of the list as the initial value of
:g:`n`. This way of counting the number of rotations will avoid
going back to a head that had been considered before.
From Section :ref:`ltac-syntax` we know that Ltac has a primitive
notion of integers, but they are only used as arguments for
primitive tactics and we cannot make computations with them. Thus,
instead, we use Coq's natural number type :g:`nat`.
.. coqtop:: in
Ltac perm_aux n :=
match goal with
| |- (perm _ ?l ?l) => apply perm_refl
| |- (perm _ (?a :: ?l1) (?a :: ?l2)) =>
let newn := eval compute in (length l1) in
(apply perm_cons; perm_aux newn)
| |- (perm ?A (?a :: ?l1) ?l2) =>
match eval compute in n with
| 1 => fail
| _ =>
let l1' := constr:(l1 ++ a :: nil) in
(apply (perm_trans A (a :: l1) l1' l2);
[ apply perm_append | compute; perm_aux (pred n) ])
end
end.
The main tactic is :g:`solve_perm`. It computes the lengths of the
two lists and uses them as arguments to call :g:`perm_aux` if the
lengths are equal. (If they aren't, the lists cannot be
permutations of each other.)
.. coqtop:: in
Ltac solve_perm :=
match goal with
| |- (perm _ ?l1 ?l2) =>
match eval compute in (length l1 = length l2) with
| (?n = ?n) => perm_aux n
end
end.
And now, here is how we can use the tactic :g:`solve_perm`:
.. coqtop:: out
Goal perm nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil).
.. coqtop:: all abort
solve_perm.
.. coqtop:: out
Goal perm nat
(0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil)
(0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil).
.. coqtop:: all abort
solve_perm.
Deciding intuitionistic propositional logic
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Pattern matching on goals allows powerful backtracking when returning tactic
values. An interesting application is the problem of deciding intuitionistic
propositional logic. Considering the contraction-free sequent calculi LJT* of
Roy Dyckhoff :cite:`Dyc92`, it is quite natural to code such a tactic using the
tactic language as shown below.
.. coqtop:: in reset
Ltac basic :=
match goal with
| |- True => trivial
| _ : False |- _ => contradiction
| _ : ?A |- ?A => assumption
end.
.. coqtop:: in
Ltac simplify :=
repeat (intros;
match goal with
| H : ~ _ |- _ => red in H
| H : _ /\ _ |- _ =>
elim H; do 2 intro; clear H
| H : _ \/ _ |- _ =>
elim H; intro; clear H
| H : ?A /\ ?B -> ?C |- _ =>
cut (A -> B -> C);
[ intro | intros; apply H; split; assumption ]
| H: ?A \/ ?B -> ?C |- _ =>
cut (B -> C);
[ cut (A -> C);
[ intros; clear H
| intro; apply H; left; assumption ]
| intro; apply H; right; assumption ]
| H0 : ?A -> ?B, H1 : ?A |- _ =>
cut B; [ intro; clear H0 | apply H0; assumption ]
| |- _ /\ _ => split
| |- ~ _ => red
end).
.. coqtop:: in
Ltac my_tauto :=
simplify; basic ||
match goal with
| H : (?A -> ?B) -> ?C |- _ =>
cut (B -> C);
[ intro; cut (A -> B);
[ intro; cut C;
[ intro; clear H | apply H; assumption ]
| clear H ]
| intro; apply H; intro; assumption ]; my_tauto
| H : ~ ?A -> ?B |- _ =>
cut (False -> B);
[ intro; cut (A -> False);
[ intro; cut B;
[ intro; clear H | apply H; assumption ]
| clear H ]
| intro; apply H; red; intro; assumption ]; my_tauto
| |- _ \/ _ => (left; my_tauto) || (right; my_tauto)
end.
The tactic ``basic`` tries to reason using simple rules involving truth, falsity
and available assumptions. The tactic ``simplify`` applies all the reversible
rules of Dyckhoff’s system. Finally, the tactic ``my_tauto`` (the main
tactic to be called) simplifies with ``simplify``, tries to conclude with
``basic`` and tries several paths using the backtracking rules (one of the
four Dyckhoff’s rules for the left implication to get rid of the contraction
and the right ``or``).
Having defined ``my_tauto``, we can prove tautologies like these:
.. coqtop:: in
Lemma my_tauto_ex1 :
forall A B : Prop, A /\ B -> A \/ B.
Proof. my_tauto. Qed.
.. coqtop:: in
Lemma my_tauto_ex2 :
forall A B : Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B.
Proof. my_tauto. Qed.
Deciding type isomorphisms
~~~~~~~~~~~~~~~~~~~~~~~~~~
A trickier problem is to decide equalities between types modulo
isomorphisms. Here, we choose to use the isomorphisms of the simply
typed λ-calculus with Cartesian product and unit type (see, for
example, :cite:`RC95`). The axioms of this λ-calculus are given below.
.. coqtop:: in reset
Open Scope type_scope.
.. coqtop:: in
Section Iso_axioms.
.. coqtop:: in
Variables A B C : Set.
.. coqtop:: in
Axiom Com : A * B = B * A.
Axiom Ass : A * (B * C) = A * B * C.
Axiom Cur : (A * B -> C) = (A -> B -> C).
Axiom Dis : (A -> B * C) = (A -> B) * (A -> C).
Axiom P_unit : A * unit = A.
Axiom AR_unit : (A -> unit) = unit.
Axiom AL_unit : (unit -> A) = A.
.. coqtop:: in
Lemma Cons : B = C -> A * B = A * C.
Proof.
intro Heq; rewrite Heq; reflexivity.
Qed.
.. coqtop:: in
End Iso_axioms.
.. coqtop:: in
Ltac simplify_type ty :=
match ty with
| ?A * ?B * ?C =>
rewrite <- (Ass A B C); try simplify_type_eq
| ?A * ?B -> ?C =>
rewrite (Cur A B C); try simplify_type_eq
| ?A -> ?B * ?C =>
rewrite (Dis A B C); try simplify_type_eq
| ?A * unit =>
rewrite (P_unit A); try simplify_type_eq
| unit * ?B =>
rewrite (Com unit B); try simplify_type_eq
| ?A -> unit =>
rewrite (AR_unit A); try simplify_type_eq
| unit -> ?B =>
rewrite (AL_unit B); try simplify_type_eq
| ?A * ?B =>
(simplify_type A; try simplify_type_eq) ||
(simplify_type B; try simplify_type_eq)
| ?A -> ?B =>
(simplify_type A; try simplify_type_eq) ||
(simplify_type B; try simplify_type_eq)
end
with simplify_type_eq :=
match goal with
| |- ?A = ?B => try simplify_type A; try simplify_type B
end.
.. coqtop:: in
Ltac len trm :=
match trm with
| _ * ?B => let succ := len B in constr:(S succ)
| _ => constr:(1)
end.
.. coqtop:: in
Ltac assoc := repeat rewrite <- Ass.
.. coqtop:: in
Ltac solve_type_eq n :=
match goal with
| |- ?A = ?A => reflexivity
| |- ?A * ?B = ?A * ?C =>
apply Cons; let newn := len B in solve_type_eq newn
| |- ?A * ?B = ?C =>
match eval compute in n with
| 1 => fail
| _ =>
pattern (A * B) at 1; rewrite Com; assoc; solve_type_eq (pred n)
end
end.
.. coqtop:: in
Ltac compare_structure :=
match goal with
| |- ?A = ?B =>
let l1 := len A
with l2 := len B in
match eval compute in (l1 = l2) with
| ?n = ?n => solve_type_eq n
end
end.
.. coqtop:: in
Ltac solve_iso := simplify_type_eq; compare_structure.
The tactic to judge equalities modulo this axiomatization is shown above.
The algorithm is quite simple. First types are simplified using axioms that
can be oriented (this is done by ``simplify_type`` and ``simplify_type_eq``).
The normal forms are sequences of Cartesian products without a Cartesian product
in the left component. These normal forms are then compared modulo permutation
of the components by the tactic ``compare_structure``. If they have the same
length, the tactic ``solve_type_eq`` attempts to prove that the types are equal.
The main tactic that puts all these components together is ``solve_iso``.
Here are examples of what can be solved by ``solve_iso``.
.. coqtop:: in
Lemma solve_iso_ex1 :
forall A B : Set, A * unit * B = B * (unit * A).
Proof.
intros; solve_iso.
Qed.
.. coqtop:: in
Lemma solve_iso_ex2 :
forall A B C : Set,
(A * unit -> B * (C * unit)) =
(A * unit -> (C -> unit) * C) * (unit -> A -> B).
Proof.
intros; solve_iso.
Qed.
Debugging |Ltac| tactics
------------------------
Backtraces
~~~~~~~~~~
.. flag:: Ltac Backtrace
Setting this flag displays a backtrace on Ltac failures that can be useful
to find out what went wrong. It is disabled by default for performance
reasons.
Info trace
~~~~~~~~~~
.. cmd:: Info @num @ltac_expr
:name: Info
This command can be used to print the trace of the path eventually taken by an
|Ltac| script. That is, the list of executed tactics, discarding
all the branches which have failed. To that end the :cmd:`Info` command can be
used with the following syntax.
The number :n:`@num` is the unfolding level of tactics in the trace. At level
0, the trace contains a sequence of tactics in the actual script, at level 1,
the trace will be the concatenation of the traces of these tactics, etc…
.. example::
.. coqtop:: in reset
Ltac t x := exists x; reflexivity.
Goal exists n, n=0.
.. coqtop:: all
Info 0 t 1||t 0.
.. coqtop:: in
Undo.
.. coqtop:: all
Info 1 t 1||t 0.
The trace produced by :cmd:`Info` tries its best to be a reparsable
|Ltac| script, but this goal is not achievable in all generality.
So some of the output traces will contain oddities.
As an additional help for debugging, the trace produced by :cmd:`Info` contains
(in comments) the messages produced by the :tacn:`idtac` tactical at the right
position in the script. In particular, the calls to idtac in branches which failed are
not printed.
.. opt:: Info Level @num
:name: Info Level
This option is an alternative to the :cmd:`Info` command.
This will automatically print the same trace as :n:`Info @num` at each
tactic call. The unfolding level can be overridden by a call to the
:cmd:`Info` command.
Interactive debugger
~~~~~~~~~~~~~~~~~~~~
.. flag:: Ltac Debug
This option governs the step-by-step debugger that comes with the |Ltac| interpreter.
When the debugger is activated, it stops at every step of the evaluation of
the current |Ltac| expression and prints information on what it is doing.
The debugger stops, prompting for a command which can be one of the
following:
+-----------------+-----------------------------------------------+
| simple newline: | go to the next step |
+-----------------+-----------------------------------------------+
| h: | get help |
+-----------------+-----------------------------------------------+
| x: | exit current evaluation |
+-----------------+-----------------------------------------------+
| s: | continue current evaluation without stopping |
+-----------------+-----------------------------------------------+
| r n: | advance n steps further |
+-----------------+-----------------------------------------------+
| r string: | advance up to the next call to “idtac string” |
+-----------------+-----------------------------------------------+
.. exn:: Debug mode not available in the IDE
:undocumented:
A non-interactive mode for the debugger is available via the option:
.. flag:: Ltac Batch Debug
This option has the effect of presenting a newline at every prompt, when
the debugger is on. The debug log thus created, which does not require
user input to generate when this option is set, can then be run through
external tools such as diff.
Profiling |Ltac| tactics
~~~~~~~~~~~~~~~~~~~~~~~~
It is possible to measure the time spent in invocations of primitive
tactics as well as tactics defined in |Ltac| and their inner
invocations. The primary use is the development of complex tactics,
which can sometimes be so slow as to impede interactive usage. The
reasons for the performance degradation can be intricate, like a slowly
performing |Ltac| match or a sub-tactic whose performance only
degrades in certain situations. The profiler generates a call tree and
indicates the time spent in a tactic depending on its calling context. Thus
it allows to locate the part of a tactic definition that contains the
performance issue.
.. flag:: Ltac Profiling
This option enables and disables the profiler.
.. cmd:: Show Ltac Profile
Prints the profile
.. cmdv:: Show Ltac Profile @string
Prints a profile for all tactics that start with :n:`@string`. Append a period
(.) to the string if you only want exactly that name.
.. cmd:: Reset Ltac Profile
Resets the profile, that is, deletes all accumulated information.
.. warning::
Backtracking across a :cmd:`Reset Ltac Profile` will not restore the information.
.. coqtop:: reset in
Require Import Coq.omega.Omega.
Ltac mytauto := tauto.
Ltac tac := intros; repeat split; omega || mytauto.
Notation max x y := (x + (y - x)) (only parsing).
Goal forall x y z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z,
max x (max y z) = max (max x y) z /\ max x (max y z) = max (max x y) z
/\
(A /\ B /\ C /\ D /\ E /\ F /\ G /\ H /\ I /\ J /\ K /\ L /\ M /\
N /\ O /\ P /\ Q /\ R /\ S /\ T /\ U /\ V /\ W /\ X /\ Y /\ Z
->
Z /\ Y /\ X /\ W /\ V /\ U /\ T /\ S /\ R /\ Q /\ P /\ O /\ N /\
M /\ L /\ K /\ J /\ I /\ H /\ G /\ F /\ E /\ D /\ C /\ B /\ A).
Proof.
.. coqtop:: all
Set Ltac Profiling.
tac.
Show Ltac Profile.
Show Ltac Profile "omega".
.. coqtop:: in
Abort.
Unset Ltac Profiling.
.. tacn:: start ltac profiling
:name: start ltac profiling
This tactic behaves like :tacn:`idtac` but enables the profiler.
.. tacn:: stop ltac profiling
:name: stop ltac profiling
Similarly to :tacn:`start ltac profiling`, this tactic behaves like
:tacn:`idtac`. Together, they allow you to exclude parts of a proof script
from profiling.
.. tacn:: reset ltac profile
:name: reset ltac profile
This tactic behaves like the corresponding vernacular command
and allow displaying and resetting the profile from tactic scripts for
benchmarking purposes.
.. tacn:: show ltac profile
:name: show ltac profile
This tactic behaves like the corresponding vernacular command
and allow displaying and resetting the profile from tactic scripts for
benchmarking purposes.
.. tacn:: show ltac profile @string
:name: show ltac profile
This tactic behaves like the corresponding vernacular command
and allow displaying and resetting the profile from tactic scripts for
benchmarking purposes.
You can also pass the ``-profile-ltac`` command line option to ``coqc``, which
turns the :flag:`Ltac Profiling` option on at the beginning of each document,
and performs a :cmd:`Show Ltac Profile` at the end.
.. warning::
Note that the profiler currently does not handle backtracking into
multi-success tactics, and issues a warning to this effect in many cases
when such backtracking occurs.
Run-time optimization tactic
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. tacn:: optimize_heap
:name: optimize_heap
This tactic behaves like :n:`idtac`, except that running it compacts the
heap in the OCaml run-time system. It is analogous to the Vernacular
command :cmd:`Optimize Heap`.
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