The tactic language¶
This chapter gives a compact documentation of L_{tac}, the tactic language available in Coq. We start by giving the syntax, and next, we present the informal semantics. If you want to know more regarding this language and especially about its foundations, you can refer to [Del00]. Chapter Detailed examples of tactics is devoted to giving small but nontrivial use examples of this language.
Syntax¶
The syntax of the tactic language is given below. See Chapter
仕様記述言語 Gallina for a description of the BNF metasyntax used
in these grammar rules. Various already defined entries will be used in this
chapter: entries natural
, integer
, ident
,
qualid
, term
, cpattern
and atomic_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 Tactics. The syntax of cpattern
is
the same as that of terms, but it is extended with pattern matching
metavariables. In cpattern
, a pattern matching metavariable is
represented with the syntax ?id
where id
is an ident
. The
notation _
can also be used to denote metavariable whose instance is
irrelevant. In the notation ?id
, the identifier allows us to keep
instantiations and to make constraints whereas _
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 secondorder pattern matching problems: in an
applicative pattern of the form @?id id1 … idn
, the variable id matches any
complex expression with (possible) dependencies in the variables id1 … idn
and returns a functional term of the form fun id1 … idn => term
.
The main entry of the grammar is expr
. This language is used in proof
mode but it can also be used in toplevel definitions as shown below.
注釈
The infix tacticals “…  …”, “… + …”, and “… ; …” are associative.
In
tacarg
, there is an overlap between qualid as a direct tactic argument andqualid
as a particular case of 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 formltac:(@qualid)
. To force the resolution as a reference to a term, use the syntax(@qualid)
.As shown by the figure, tactical
\\
binds more than the prefix tacticals try, repeat, do and abstract which themselves bind more than the postfix tactical “… ;[ … ]” which binds more than “… ; …”.For instance
 try repeat tac1  tac2; tac3; [tac31  ...  tac3n]; tac4.
 Toplevel input, characters 03: > try repeat tac1  tac2; tac3; [tac31  ... > ^^^ Error: Syntax error: illegal begin of vernac. Toplevel input, characters 4445: > try repeat tac1  tac2; tac3; [tac31  ...  tac3n]; tac4. > ^ Error: Syntax error: illegal begin of vernac.
is understood as
 try (repeat (tac1  tac2)); ((tac3; [tac31  ...  tac3n]); tac4).
 Toplevel input, characters 03: > try (repeat (tac1  tac2)); ((tac3; [tac31  ... > ^^^ Error: Syntax error: illegal begin of vernac. Toplevel input, characters 5253: > try (repeat (tac1  tac2)); ((tac3; [tac31  ...  tac3n]); tac4). > ^ Error: Syntax error: illegal begin of vernac.
expr ::=expr
;expr
 [>expr
 ... expr
] expr
; [expr
 ... expr
] tacexpr3
tacexpr3 ::= do (natural
ident
) tacexpr3  progresstacexpr3
 repeattacexpr3
 trytacexpr3
 oncetacexpr3
 exactly_oncetacexpr3
 timeout (natural
ident
)tacexpr3
 time [string
]tacexpr3
 onlyselector
:tacexpr3
tacexpr2
tacexpr2 ::=tacexpr1
tacexpr3
tacexpr1
+tacexpr3
 tryiftacexpr1
thentacexpr1
elsetacexpr1
tacexpr1
tacexpr1 ::= funname
...name
=>atom
 let [rec]let_clause
with ... withlet_clause
inatom
 match goal withcontext_rule
 ... context_rule
end  match reverse goal withcontext_rule
 ... context_rule
end  matchexpr
withmatch_rule
 ... match_rule
end  lazymatch goal withcontext_rule
 ... context_rule
end  lazymatch reverse goal withcontext_rule
 ... context_rule
end  lazymatchexpr
withmatch_rule
 ... match_rule
end  multimatch goal withcontext_rule
 ... context_rule
end  multimatch reverse goal withcontext_rule
 ... context_rule
end  multimatchexpr
withmatch_rule
 ... match_rule
end  abstractatom
 abstractatom
usingident
 first [expr
 ... expr
]  solve [expr
 ... expr
]  idtac [message_token
...message_token
]  fail [natural
] [message_token
...message_token
]  fresh [component
…component
]  contextident
[term
]  evalredexpr
interm
 type ofterm
 constr :term
 uconstr :term
 type_termterm
 numgoals  guardtest
 assert_failstacexpr3
 assert_succeedstacexpr3
atomic_tactic
qualid
tacarg
...tacarg
atom
atom ::=qualid
 () integer
 (expr
) component ::=string
qualid
message_token ::=string
ident
integer
tacarg ::=qualid
 ()  ltac :atom
term
let_clause ::=ident
[name
...name
] :=expr
context_rule ::=context_hyp
, ...,context_hyp
cpattern
=>expr
cpattern
=>expr
 cpattern
=>expr
 _ =>expr
context_hyp ::=name
:cpattern
name
:=cpattern
[:cpattern
] match_rule ::=cpattern
=>expr
 context [ident] [cpattern
] =>expr
 _ =>expr
test ::=integer
=integer
integer
(<  <=  >  >=)integer
selector ::= [ident
] integer
(integer
integer
integer
), ..., (integer
integer
integer
) toplevel_selector ::=selector
all
par
top ::= [Local] Ltacltac_def
with ... withltac_def
ltac_def ::=ident
[ident
...ident
] :=expr
qualid
[ident
...ident
] ::=expr
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 nonrecursive local
definition or of an application).
The rest of this section explains the semantics of every construction of L_{tac}.
Sequence¶
A sequence is an expression of the following form:

expr_{1} ; expr_{2}
¶ The expression
expr_{1}
is evaluated tov_{1}
, which must be a tactic value. The tacticv_{1}
is applied to the current goal, possibly producing more goals. Thenexpr_{2}
is evaluated to producev_{2}
, which must be a tactic value. The tacticv_{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:

[> expr*]
¶ The expressions
expr_{i}
are evaluated tov_{i}
, for i = 0, ..., n and all have to be tactics. Thev_{i}
is applied to the ith goal, for i = 1, ..., n. It fails if the number of focused goals is not exactly n.注釈
If no tactic is given for the ith goal, it behaves as if the tactic idtac were given. For instance,
[>  auto]
is a shortcut for[> idtac  auto ]
.
Variant
[> expr_{i}*  expr ..  expr_{j}*]
In this variant,
expr
is used for each goal coming after those covered by the list ofexpr_{i}
but before those covered by the list ofexpr_{j}
.

Variant
[> expr*  ..  expr*]
In this variant, idtac is used for the goals not covered by the two lists of
expr
.

Variant
[> expr .. ]
In this variant, the tactic
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 asswap
, would act as if a single goal is focused.

Variant
expr_{0} ; [expr_{i}*]
This variant of local tactic application is paired with a sequence. In this variant, there must be as many
expr_{i}
as goals generated by the application ofexpr_{0}
to each of the individual goals independently. All the above variants work in this form too. Formally,expr ; [ ... ]
is equivalent to[> expr ; [> ... ] .. ]
.

Variant
Goal selectors¶
We can restrict the application of a tactic to a subset of the currently focused goals with:

toplevel_selector : expr
¶ We can also use selectors as a tactical, which allows to use them nested in a tactic expression, by using the keyword
only
:
Variant
only selector : expr
¶ When selecting several goals, the tactic
expr
is applied globally to all selected goals.

Variant
[ident] : expr
In this variant,
expr
is applied locally to a goal previously named by the user (see Existential variables).

Variant
numnum+, : expr
In this variant,
expr
is applied globally to the subset of goals described by the given ranges. You can write a singlen
as a shortcut fornn
when specifying multiple ranges.

Variant
all: expr
¶ In this variant,
expr
is applied to all focused goals.all:
can only be used at the toplevel of a tactic expression.

Variant
par: expr
¶ In this variant,
expr
is applied to all focused goals in parallel. The number of workers can be controlled via the command line optionasyncproofstacj
taking as argument the desired number of workers. Limitations:par:
only works on goals containing no existential variables andexpr
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.

Error
No such goal.
¶

Variant
For loop¶
There is a for loop that repeats a tactic num
times:

do num expr
¶ expr
is evaluated tov
which must be a tactic value. This tactic valuev
is appliednum
times. Supposingnum
> 1, after the first application ofv
,v
is applied, at least once, to the generated subgoals and so on. It fails if the application ofv
fails before the num applications have been completed.
Repeat loop¶
We have a repeat loop with:

repeat expr
¶ expr
is evaluated tov
. Ifv
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 tacticrepeat expr
itself never fails.
Error catching¶
We can catch the tactic errors with:

try expr
¶ expr
is evaluated tov
which must be a tactic value. The tactic valuev
is applied to each focused goal independently. If the application ofv
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 reraised with its level decremented.
Detecting progress¶
We can check if a tactic made progress with:

progress expr
¶ expr
is evaluated to v which must be a tactic value. The tactic valuev
is applied to each focued subgoal independently. If the application ofv
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.
Error
Failed to progress.
¶

Error
Backtracking branching¶
We can branch with the following structure:

expr_{1} + expr_{2}
¶ expr_{1}
andexpr_{2}
are evaluated respectively tov_{1}
andv_{2}
which must be tactic values. The tactic valuev_{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 andv_{2}
is applied.Tactics can be seen as having several successes. When a tactic fails it asks for more successes of the prior tactics.
expr_{1} + expr_{2}
has all the successes ofv_{1}
followed by all the successes ofv_{2}
. Algebraically,(expr_{1} + expr_{2}); expr_{3} = (expr_{1}; expr_{3}) + (expr_{2}; expr_{3})
.Branching is leftassociative.
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:

first [expr*]
¶ The
expr_{i}
are evaluated tov_{i}
andv_{i}
must be tactic values for i = 1, ..., n. Supposing n > 1,first [expr_{1}  ...  expr_{n}]
appliesv_{1}
in each focused goal independently and stops if it succeeds; otherwise it tries to applyv_{2}
and so on. It fails when there is no applicable tactic. In other words,first [expr_{1}  ...  expr_{n}]
behaves, in each goal, as the firstv_{i}
to have at least one success.
Error
No applicable tactic.
¶

Variant
first expr
This is an L_{tac} alias that gives a primitive access to the first tactical as an L_{tac} 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
 Tactic Notation "foo" tactic_list(tacs) := first tacs.

Error
Leftbiased branching¶
Yet another way of branching without backtracking is the following structure:

expr_{1}  expr_{2}
¶ expr_{1}
andexpr_{2}
are evaluated respectively tov_{1}
andv_{2}
which must be tactic values. The tactic valuev_{1}
is applied in each subgoal independently and if it fails to progress thenv_{2}
is applied.expr_{1}  expr_{2}
is equivalent tofirst [ progress expr_{1}  expr_{2} ]
(except that if it fails, it fails likev_{2}
). Branching is leftassociative.
Generalized biased branching¶
The tactic

tryif expr_{1} then expr_{2} else expr_{3}
¶ is a generalization of the biasedbranching tactics above. The expression
expr_{1}
is evaluated tov_{1}
, which is then applied to each subgoal independently. For each goal wherev_{1}
succeeds at least once,expr_{2}
is evaluated tov_{2}
which is then applied collectively to the generated subgoals. Thev_{2}
tactic can trigger backtracking points inv_{1}
: wherev_{1}
succeeds at least once,tryif expr_{1} then expr_{2} else expr_{3}
is equivalent tov_{1}; v_{2}
. In each of the goals wherev_{1}
does not succeed at least once,expr_{3}
is evaluated inv_{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:
Checking the successes¶
Coq provides an experimental way to check that a tactic has exactly one success:

exactly_once expr
¶ expr
is evaluated tov
which must be a tactic value. The tactic valuev
is applied if it has at most one success. Ifv
fails,exactly_once expr
fails likev
. Ifv
has a exactly one success,exactly_once expr
succeeds likev
. Ifv
has two or more successes, exactly_once expr fails.警告
The experimental status of this tactic pertains to the fact if
v
performs side effects, they may occur in an unpredictable way. Indeed, normallyv
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.
Error
This tactic has more than one success.
¶

Error
Checking the failure¶
Coq provides a derived tactic to check that a tactic fails:
Checking the success¶
Coq provides a derived tactic to check that a tactic has at least one success:
Solving¶
We may consider the first to solve (i.e. which generates no subgoal) among a panel of tactics:

solve [expr*]
¶ The
expr_{i}
are evaluated tov_{i}
andv_{i}
must be tactic values, for i = 1, ..., n. Supposing n > 1,solve [expr_{1}  ...  expr_{n}]
appliesv_{1}
to each goal independently and stops if it succeeds; otherwise it tries to applyv_{2}
and so on. It fails if there is no solving tactic.
Error
Cannot solve the goal.
¶

Variant
solve expr
This is an L_{tac} alias that gives a primitive access to the
solve:
tactical. See thefirst
tactical for more information.

Error
Identity¶
The constant idtac
is the identity tactic: it leaves any goal unchanged but
it appears in the proof script.

idtac message_token*
¶ This prints the given tokens. Strings and integers are printed literally. If a (term) variable is given, its contents are printed.
Failing¶

fail
¶ This is the alwaysfailing tactic: it does not solve any goal. It is useful for defining other tacticals since it can be caught by
try
,repeat
,match goal
, or the branching tacticals.
Variant
fail num
The number is the failure level. If no level is specified, it defaults to 0. The level is used by
try
,repeat
,match goal
and the branching tacticals. If 0, it makesmatch goal
consider the next clause (backtracking). If nonzero, the currentmatch goal
block,try
,repeat
, or branching command is aborted and the level is decremented. In the case of+
, a nonzero level skips the first backtrack point, even if the call tofail num
is not enclosed in a+
command, respecting the algebraic identity.

Variant
fail message_token*
The given tokens are used for printing the failure message.

Variant
fail num message_token*
This is a combination of the previous variants.

Variant
gfail
¶ This variant fails even when used after
;
and there are no goals left. Similarly,gfail
fails even when used afterall:
and there are no goals left. See the example for clarification.

Variant
gfail message_token*

Variant
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
let x := H in fail 0 x
will succeed.

Error
No such goal.
¶
Example
 Goal True.
 1 subgoal ============================ True
 Proof.
 fail.
 Toplevel input, characters 05: > fail. > ^^^^^ Error: Tactic failure.
 Abort.
 Goal True.
 1 subgoal ============================ True
 Proof.
 trivial; fail.
 No more subgoals.
 Qed.
 Unnamed_thm is defined
 Goal True.
 1 subgoal ============================ True
 Proof.
 trivial.
 No more subgoals.
 fail.
 Toplevel input, characters 05: > fail. > ^^^^^ Error: No such goal.
 Abort.
 Goal True.
 1 subgoal ============================ True
 Proof.
 trivial.
 No more subgoals.
 all: fail.
 Qed.
 Unnamed_thm0 is defined
 Goal True.
 1 subgoal ============================ True
 Proof.
 gfail.
 Toplevel input, characters 06: > gfail. > ^^^^^^ Error: Tactic failure.
 Abort.
 Goal True.
 1 subgoal ============================ True
 Proof.
 trivial; gfail.
 Toplevel input, characters 015: > trivial; gfail. > ^^^^^^^^^^^^^^^ Error: Tactic failure.
 Abort.
 Goal True.
 1 subgoal ============================ True
 Proof.
 trivial.
 No more subgoals.
 gfail.
 Toplevel input, characters 06: > gfail. > ^^^^^^ Error: No such goal.
 Abort.
 Goal True.
 1 subgoal ============================ True
 Proof.
 trivial.
 No more subgoals.
 all: gfail.
 Toplevel input, characters 011: > all: gfail. > ^^^^^^^^^^^ Error: Tactic failure.
 Abort.

Variant
Timeout¶
We can force a tactic to stop if it has not finished after a certain amount of time:

timeout num expr
¶ expr
is evaluated tov
which must be a tactic value. The tactic valuev
is applied normally, except that it is interrupted afternum
seconds if it is still running. In this case the outcome is a failure.警告
For the moment, timeout is based on elapsed time in seconds, which is very machinedependent: 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:

time string expr
¶ evaluates
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 machinedependent. Thestring
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

time_constr expr
¶ which evaluates
expr ()
and displays the time the tactic expression evaluated, assuming successful evaluation. Time is in seconds and is machinedependent.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 tactics
and
which (re)set and display an optionally named timer, respectively. The parenthesized string argument to
finish_timing
is also optional, and determines the label associated with the timer for printing.By copying the definition of
time_constr
from the standard library, users can achive support for a fixed pattern of nesting by passing differentstring
parameters torestart_timer
andfinish_timing
at each level of nesting.Example
 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.
 time_constr1 is defined
 Goal True.
 1 subgoal ============================ 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.
 Tactic evaluation (depth 1) ran for 0. secs (0.u,0.s) Tactic evaluation (depth 1) ran for 0. secs (0.u,0.s) Tactic evaluation ran for 0. secs (0.u,0.s) 1 subgoal n := 100 : nat ============================ True
 Abort.
Local definitions¶
Local definitions can be done as follows:

let ident_{1} := expr_{1} with ident_{i} := expr_{i}* in expr
¶ each
expr_{i}
is evaluated tov_{i}
, then,expr
is evaluated by substitutingv_{i}
to each occurrence ofident_{i}
, for i = 1, ..., n. There are no dependencies between theexpr_{i}
and theident_{i}
.Local definitions can be made recursive by using
let rec
instead oflet
. In this latter case, the definitions are evaluated lazily so that the rec keyword can be used also in nonrecursive cases so as to avoid the eager evaluation of local definitions.
Application¶
An application is an expression of the following form:
Function construction¶
A parameterized tactic can be built anonymously (without resorting to local definitions) with:
Pattern matching on terms¶
We can carry out pattern matching on terms with:

match expr with cpattern_{i} => expr_{i}+ end
The expression
expr
is evaluated and should yield a term which is matched againstcpattern_{1}
. The matching is nonlinear: if a metavariable occurs more than once, it should match the same expression every time. It is firstorder except on the variables of the form@?id
that occur in head position of an application. For these variables, the matching is secondorder and returns a functional term.Alternatively, when a metavariable of the form
?id
occurs under binders, sayx_{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 variablesx_{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
cpattern_{1}
succeeds, thenexpr_{1}
is evaluated into some value by substituting the pattern matching instantiations to the metavariables. Ifexpr_{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 alet in
), then this tactic is applied. If the tactic succeeds, the list of resulting subgoals is the result of the match expression. Ifexpr_{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 ofexpr_{1}
is the result of the match expression.If the matching with
cpattern_{1}
fails, or if it succeeds but the evaluation ofexpr_{1}
fails, or if the evaluation ofexpr_{1}
succeeds but returns a tactic in execution position whose execution fails, thencpattern_{2}
is used and so on. The pattern_
matches any term and shadows all remaining patterns if any. If all clauses fail (in particular, there is no pattern_
) then a nomatchingclause error is raised.Failures in subsequent tactics do not cause backtracking to select new branches or inside the righthand side of the selected branch even if it has backtracking points.

Error
No matching clauses for match.
¶ No pattern can be used and, in particular, there is no
_
pattern.

Error
Argument of match does not evaluate to a term.
¶ This happens when
expr
does not denote a term.

Variant
multimatch expr with cpattern_{i} => expr_{i}+ end
Using multimatch instead of match will allow subsequent tactics to backtrack into a righthand side tactic which has backtracking points left and trigger the selection of a new matching branch when all the backtracking points of the righthand side have been consumed.
The syntax
match …
is, in fact, a shorthand foronce multimatch …
.

Variant
lazymatch expr with cpattern_{i} => 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 righthand side fails. If the righthand side of the selected branch is a tactic with backtracking points, then subsequent failures cause this tactic to backtrack.

Variant
context ident [cpattern]
This special form of patterns matches any term with a subterm matching cpattern. If there is a match, the optional
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 righthandside 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 topbottom and from left to right (with respect to the raw printing obtained by setting option
Printing All
).
Example
 Ltac f x := match x with context f [S ?X] => idtac X; assert (p := eq_refl 1 : X=1); let x:= context f[O] in assert (x=O) end.
 f is defined
 Goal True.
 1 subgoal ============================ True
 f (3+4).
 2 1 2 subgoals p : 1 = 1 ============================ 1 + 4 = 0 subgoal 2 is: True

Error
Pattern matching on goals¶
We can perform pattern matching on goals using the following expression:

match goal with hyp+  cpattern => expr+  _ => expr end
¶ If each hypothesis pattern
hyp
_{1,i}, with i = 1, ..., m_{1} is matched (nonlinear firstorder unification) by a hypothesis of the goal and ifcpattern_1
is matched by the conclusion of the goal, thenexpr_{1}
is evaluated tov_{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. Ifv_{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 lastexpr
is evaluated tov
andv
is applied. Note also that matching against subterms (using thecontext 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 righthand side of the selected branch even if it has backtracking points.

Error
No matching clauses for match goal.
¶ No clause succeeds, i.e. all matching patterns, if any, fail at the application of the righthandside.
注釈
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_{i,m}_{i`} before hyp_{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
match reverse goal with
variant.
Variant
multimatch goal with hyp+  cpattern => expr+  _ => expr end
Using
multimatch
instead ofmatch
will allow subsequent tactics to backtrack into a righthand 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 righthand side have been consumed.The syntax
match [reverse] goal …
is, in fact, a shorthand foronce multimatch [reverse] goal …
.

Variant
lazymatch goal with hyp+  cpattern => expr+  _ => 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 righthand side fails. If the righthand side of the selected branch is a tactic with backtracking points, then subsequent failures cause this tactic to backtrack.

Error
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:
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:

fresh component*
It evaluates to an identifier unbound in the goal. This fresh identifier is obtained by concatenating the value of the
component
s (each of them is, either aqualid
which has to refer to a (unqualified) name, or directly a name denoted by astring
).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:
Manipulating untyped terms¶

uconstr : term
The terms built in L_{tac} are welltyped by default. It may not be appropriate for building large terms using a recursive L_{tac} 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 L_{tac} with the
uconstr : term
syntax.

type_term term
An untyped term, in L_{tac}, can contain references to hypotheses or to L_{tac} 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 welltyped term which can be used in tactics.
Untyped terms built using uconstr :
can also be used as arguments to the
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¶

numgoals
The number of goals under focus can be recovered using the
numgoals
function. Combined with the guard command below, it can be used to branch over the number of goals produced by previous tactics.Example
 Ltac pr_numgoals := let n := numgoals in idtac "There are" n "goals".
 pr_numgoals is defined
 Goal True /\ True /\ True.
 pr_numgoals is defined 1 subgoal ============================ True /\ True /\ True
 split;[split].
 3 subgoals ============================ True subgoal 2 is: True subgoal 3 is: True
 all:pr_numgoals.
 There are 3 goals
Testing boolean expressions¶

guard test
¶ The
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
 Goal True /\ True /\ True.
 1 subgoal ============================ True /\ True /\ True
 split;[split].
 3 subgoals ============================ True subgoal 2 is: True subgoal 3 is: True
 all:let n:= numgoals in guard n<4.
 Fail all:let n:= numgoals in guard n=2.
 The command has indeed failed with message: Ltac call to "guard (test)" failed. Condition not satisfied: 3=2

Error
Condition not satisfied.
¶
Proving a subgoal as a separate lemma¶

abstract expr
¶ From the outside,
abstract expr
is the same assolve expr
. Internally it saves an auxiliary lemma calledident_subproofn
whereident
is the name of the current goal andn
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
omega
ordiscriminate
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.

Variant
abstract expr using ident
Give explicitly the name of the auxiliary lemma.
警告
Use this feature at your own risk; explicitly named and reused subterms don’t play well with asynchronous proofs.

Variant
transparent_abstract expr
¶ Save the subproof in a transparent lemma rather than an opaque one.
警告
Use this feature at your own risk; building computationally relevant terms with tactics is fragile.

Variant
transparent_abstract expr using ident
Give explicitly the name of the auxiliary transparent lemma.
警告
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.

Error
Proof is not complete.
¶

Variant
Tactic toplevel definitions¶
Defining L_{tac} functions¶
Basically, L_{tac} toplevel definitions are made as follows:

Command
Ltac ident ident* := expr
¶ This defines a new L_{tac} function that can be used in any tactic script or new L_{tac} toplevel definition.
Recursive and mutual recursive function definitions are also possible with the syntax:

Variant
Ltac ident ident* with ident ident** := expr
It is also possible to redefine an existing userdefined tactic using the syntax:

Variant
Ltac qualid ident* ::= expr
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.
If preceded by the keyword Local the tactic definition will not be exported outside the current module.

Variant
Debugging L_{tac} tactics¶
Info trace¶

Command
Info num expr
¶ This command can be used to print the trace of the path eventually taken by an L_{tac} script. That is, the list of executed tactics, discarding all the branches which have failed. To that end the
Info
command can be used with the following syntax.The number
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
 Ltac t x := exists x; reflexivity.
 t is defined
 Goal exists n, n=0.
 1 subgoal ============================ exists n : nat, n = 0
 Info 0 t 1t 0.
 t <constr:(0)> No more subgoals.
 Undo.
 1 subgoal ============================ exists n : nat, n = 0
 Info 1 t 1t 0.
 exists with 0;reflexivity No more subgoals.
The trace produced by
Info
tries its best to be a reparsable L_{tac} 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
Info
contains (in comments) the messages produced by theidtac
tactical at the right position in the script. In particular, the calls to idtac in branches which failed are not printed.
Interactive debugger¶

Flag
Ltac Debug
¶ This option governs the stepbystep debugger that comes with the L_{tac} interpreter
When the debugger is activated, it stops at every step of the evaluation of the current L_{tac} 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” 

Error
Debug mode not available in the IDE
¶
A noninteractive 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 L_{tac} tactics¶
It is possible to measure the time spent in invocations of primitive tactics as well as tactics defined in L_{tac} 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 performence degradation can be intricate, like a slowly performing L_{tac} match or a subtactic 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.

Command
Show Ltac Profile
¶ Prints the profile

Command
Reset Ltac Profile
¶ Resets the profile, that is, deletes all accumulated information.
警告
Backtracking across a
Reset Ltac Profile
will not restore the information.
 Require Import Coq.omega.Omega.
 [Loading ML file z_syntax_plugin.cmxs ... done] [Loading ML file quote_plugin.cmxs ... done] [Loading ML file newring_plugin.cmxs ... done] [Loading ML file omega_plugin.cmxs ... done]
 Ltac mytauto := tauto.
 mytauto is defined
 Ltac tac := intros; repeat split; omega  mytauto.
 tac is defined
 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).
 1 subgoal ============================ forall (x y z : nat) (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 : Prop), x + (y + (z  y)  x) = x + (y  x) + (z  (x + (y  x))) /\ x + (y + (z  y)  x) = x + (y  x) + (z  (x + (y  x))) /\ (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.
 Set Ltac Profiling.
 tac.
 No more subgoals.
 Show Ltac Profile.
 total time: 1.994s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac  0.1% 100.0% 1 1.994s ─<Coq.Init.Tauto.with_uniform_flags>  0.0% 73.9% 26 0.096s ─<Coq.Init.Tauto.tauto_gen>  0.0% 73.8% 26 0.096s ─<Coq.Init.Tauto.tauto_intuitionistic>  0.0% 73.8% 26 0.096s ─t_tauto_intuit  0.1% 73.7% 26 0.096s ─<Coq.Init.Tauto.simplif>  51.2% 71.3% 26 0.094s ─omega  25.7% 25.7% 28 0.218s ─<Coq.Init.Tauto.is_conj>  11.2% 11.2% 28756 0.000s ─elim id  6.0% 6.0% 650 0.037s ─<Coq.Init.Tauto.axioms>  1.8% 2.3% 0 0.003s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac  0.1% 100.0% 1 1.994s ├─<Coq.Init.Tauto.with_uniform_flags>  0.0% 73.9% 26 0.096s │└<Coq.Init.Tauto.tauto_gen>  0.0% 73.8% 26 0.096s │└<Coq.Init.Tauto.tauto_intuitionistic> 0.0% 73.8% 26 0.096s │└t_tauto_intuit  0.1% 73.7% 26 0.096s │ ├─<Coq.Init.Tauto.simplif>  51.2% 71.3% 26 0.094s │ │ ├─<Coq.Init.Tauto.is_conj>  11.2% 11.2% 28756 0.000s │ │ └─elim id  6.0% 6.0% 650 0.037s │ └─<Coq.Init.Tauto.axioms>  1.8% 2.3% 0 0.003s └─omega  25.7% 25.7% 28 0.218s
 Show Ltac Profile "omega".
 total time: 1.994s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─omega  25.7% 25.7% 28 0.218s tactic local total calls max
 Abort.
 Unset Ltac Profiling.

stop ltac profiling
¶ Similarly to
start ltac profiling
, this tactic behaves likeidtac
. Together, they allow you to exclude parts of a proof script from profiling.

reset ltac profile
¶ This tactic behaves like the corresponding vernacular command and allow displaying and resetting the profile from tactic scripts for benchmarking purposes.

show ltac profile
¶ This tactic behaves like the corresponding vernacular command and allow displaying and resetting the profile from tactic scripts for benchmarking purposes.

show ltac profile string
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 profileltac
command line option to coqc
, which
turns the Ltac Profiling
option on at the beginning of each document,
and performs a Show Ltac Profile
at the end.
警告
Note that the profiler currently does not handle backtracking into multisuccess tactics, and issues a warning to this effect in many cases when such backtracking occurs.
Runtime optimization tactic¶

optimize_heap
¶
This tactic behaves like idtac
, except that running it compacts the
heap in the OCaml runtime system. It is analogous to the Vernacular
command Optimize Heap
.