Chapter 9 The tactic language
This chapter gives a compact documentation of Ltac, 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 . Chapter 10 is devoted to giving examples of use of this language on small but also with non-trivial problems.
The syntax of the tactic language is given Figures 9.1 and 9.2. See Chapter 1 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’s terms and patterns and all the atomic tactics described in Chapter 8. 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 metavariable 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 @?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 in Figure 9.3.
- The infix tacticals “… || …”, “… + …”, and “… ; …” are associative.
- In tacarg, there is an overlap between qualid as a direct tactic argument and qualid 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 form ltac : 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 “… ;
try repeat tactic1 || tactic2;tactic3;[tactic31|…|tactic3n];tactic4.
is understood as
(try (repeat (tactic1 || tactic2)));
expr ::= expr ; expr | [> expr | … | expr ] | expr ; [ expr | … | expr ] | tacexpr3 tacexpr3 ::= do (natural | ident) tacexpr3 | progress tacexpr3 | repeat tacexpr3 | try tacexpr3 | once tacexpr3 | exactly_once tacexpr3 | timeout (natural | ident) tacexpr3 | time [string] tacexpr3 | only selector : tacexpr3 | tacexpr2 tacexpr2 ::= tacexpr1 || tacexpr3 | tacexpr1 + tacexpr3 | tryif tacexpr1 then tacexpr1 else tacexpr1 | tacexpr1 tacexpr1 ::= 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 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 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 expr with match_rule | … | match_rule end | abstract atom | abstract atom using ident | first [ expr | … | expr ] | solve [ expr | … | expr ] | idtac [message_token … message_token] | fail [natural] [message_token … message_token] | gfail [natural] [message_token … message_token] | fresh | fresh string| fresh qualid | context ident [ term ] | eval redexpr in term | type of term | external string string tacarg … tacarg | constr : term | uconstr : term | type_term term | numgoals | guard test | atomic_tactic | qualid tacarg … tacarg | atom
atom ::= qualid | () | integer | ( expr ) 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 | _ => expr context_hyp ::= name : cpattern | name := cpattern [: cpattern] match_rule ::= cpattern => expr | context [ident] [ cpattern ] => expr | appcontext [ident] [ cpattern ] => expr | _ => expr test ::= integer = integer | integer < integer | integer <= integer | integer > integer | integer >= integer selector ::= [ident] | integer | (integer | integer - integer) , … , (integer | integer - integer) toplevel_selector ::= selector | all | par
top ::= [Local] Ltac ltac_def with … with ltac_def ltac_def ::= ident [ident … ident] := expr | qualid [ident … ident] ::=expr
Tactic expressions can only be applied in the context of a proof. The evaluation yields either a term, an integer or a tactic. Intermediary 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.
A sequence is an expression of the following form:
expr1 ; expr2
The expression expr1 is evaluated to v1, which must be a tactic value. The tactic v1 is applied to the current goal, possibly producing more goals. Then expr2 is evaluated to produce v2, which must be a tactic value. The tactic v2 is applied to all the goals produced by the prior application. Sequence is associative.
Different tactics can be applied to the different goals using the following form:
[ > expr1 | ... | exprn ]
The expressions expri are evaluated to vi, for i=0,...,n and all have to be tactics. The vi is applied to the i-th goal, for =1,...,n. It fails if the number of focused goals is not exactly n.
- 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 ].
- [ > expr1 | ... |
expri | expr .. |
expri+1+j | ... | exprn ]
In this variant, expr is used for each goal numbered from i+1 to i+j (assuming n is the number of goals).
Note that .. is part of the syntax, while ... is the meta-symbol used to describe a list of expr of arbitrary length. goals numbered from i+1 to i+j.
- [ > expr1 | ... |
expri | .. | expri+1+j |
... | exprn ]
In this variant, idtac is used for the goals numbered from i+1 to i+j.
- [ > expr .. ]
In this variant, the tactic expr is applied independently to each of the goals, rather than globally. In particular, if there are no goal, 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.
- expr ; [ expr1 | ... | exprn ]
This variant of local tactic application is paired with a sequence. In this variant, n must be the number of goals generated by the application of expr to each of the individual goals independently. All the above variants work in this form too. Formally, expr ; [ ... ] is equivalent to
[ > expr ; [ > ... ] .. ]
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:
only selector : expr
When selecting several goals, the tactic expr is applied globally to all selected goals.
[ident] : expr
In this variant, expr is applied locally to a goal previously named by the user (see 2.11).
- num : expr
In this variant, expr is applied locally to the num-th goal.
- n1-m1, …, nk-mk : expr
In this variant, 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.
- all: expr
In this variant, expr is applied to all focused goals. all: can only be used at the toplevel of a tactic expression.
- 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 option -async-proofs-tac-j taking as argument the desired number of workers. Limitations: par: only works on goals containing no existential variables and 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.
There is a for loop that repeats a tactic num times:
do num expr
expr is evaluated to v which must be a tactic value. This tactic value v is applied num times. Supposing 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.
We have a repeat loop with:
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 repeat expr itself never fails.
We can catch the tactic errors with:
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.
We can check if a tactic made progress with:
expr is evaluated to v which must be a tactic value. The tactic value v is applied to each focued 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.
We can branch with the following structure:
expr1 + expr2
expr1 and expr2 are evaluated to v1 and v2 which must be tactic values. The tactic value v1 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 v2 is applied.
Tactics can be seen as having several successes. When a tactic fails it asks for more successes of the prior tactics. expr1 + expr2 has all the successes of v1 followed by all the successes of v2. Algebraically, (expr1 + expr2);expr3 = (expr1;expr3) + (expr2;expr3).
Branching is left-associative.
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 [ expr1 | ... | exprn ]
expri are evaluated to vi and vi must be tactic values, for i=1,...,n. Supposing n>1, it applies, in each focused goal independently, v1, if it works, it stops otherwise it tries to apply v2 and so on. It fails when there is no applicable tactic. In other words, first [ expr1 | ... | exprn ] behaves, in each goal, as the the first vi to have at least one success.
Variant: first expr
This is an Ltac alias that gives a primitive access to the first tactical as a 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.
Tactic Notation "foo" tactic_list(tacs) := first tacs.
Yet another way of branching without backtracking is the following structure:
expr1 || expr2
expr1 and expr2 are evaluated to v1 and v2 which must be tactic values. The tactic value v1 is applied in each subgoal independently and if it fails to progress then v2 is applied. expr1 || expr2 is equivalent to first [ progress expr1 | expr2 ] (except that if it fails, it fails like v2). Branching is left-associative.
tryif expr1 then expr2 else expr3
is a generalization of the biased-branching tactics above. The expression expr1 is evaluated to v1, which is then applied to each subgoal independently. For each goal where v1 succeeds at least once, expr2 is evaluated to v2 which is then applied collectively to the generated subgoals. The v2 tactic can trigger backtracking points in v1: where v1 succeeds at least once, tryif expr1 then expr2 else expr3 is equivalent to v1;v2. In each of the goals where v1 does not succeed at least once, expr3 is evaluated in v3 which is is then applied to the goal.
Another way of restricting backtracking is to restrict a tactic to a single success a posteriori:
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, once expr fails like v. If v has a least one success, once expr succeeds once, but cannot produce more successes.
Coq provides an experimental way to check that a tactic has exactly one success:
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, exactly_once expr fails like v. If v has a exactly one success, exactly_once expr succeeds like v. If v has two or more successes, exactly_once expr fails.
The experimental status of this tactic pertains to the fact if v performs side effects, they may occur in a 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.
We may consider the first to solve (i.e. which generates no subgoal) among a panel of tactics:
solve [ expr1 | ... | exprn ]
expri are evaluated to vi and vi must be tactic values, for i=1,...,n. Supposing n>1, it applies v1 to each goal independently, if it doesn’t solve the goal then it tries to apply v2 and so on. It fails if there is no solving tactic.
Variant: solve expr
This is an Ltac alias that gives a primitive access to the solve tactical. See the first tactical for more information.
The constant idtac is the identity tactic: it leaves any goal unchanged but it appears in the proof script.
Variant: idtac message_token … message_token
This prints the given tokens. Strings and integers are printed literally. If a (term) variable is given, its contents are printed.
The tactic fail is the always-failing 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. The fail tactic will, however, succeed if all the goals have already been solved.
The number n 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 makes match goal considering the next clause (backtracking). If non zero, the current match goal block, try, repeat, or branching command is aborted and the level is decremented. In the case of +, a non-zero level skips the first backtrack point, even if the call to fail n is not enclosed in a + command, respecting the algebraic identity.
- fail message_token … message_token
The given tokens are used for printing the failure message.
- fail n message_token … message_token
This is a combination of the previous variants.
This variant fails even if there are no goals left.
- gfail message_token … message_token
gfail n message_token … 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.
We can force a tactic to stop if it has not finished after a certain amount of time:
timeout num expr
expr is evaluated to v which must be a tactic value. The tactic value v is applied normally, except that it is interrupted after 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 debug 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.
A tactic execution can be timed:
time string expr
evaluates expr and displays the time the tactic expression ran, whether it fails or successes. In case of several successes, the time for each successive runs is displayed. Time is in seconds and is machine-dependent. The string argument is optional. When provided, it is used to identify this particular occurrence of time.
Local definitions can be done as follows:
let ident1 := expr1
with ident2 := expr2
with identn := exprn in
each expri is evaluated to vi, then, expr is evaluated by substituting vi to each occurrence of identi, for i=1,...,n. There is no dependencies between the expri and the identi.
Local definitions can be recursive by using let rec instead of 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.
An application is an expression of the following form:
qualid tacarg1 ... tacargn
The reference qualid must be bound to some defined tactic definition expecting at least n arguments. The expressions expri are evaluated to vi, for i=1,...,n.
A parameterized tactic can be built anonymously (without resorting to local definitions) with:
fun ident1 ... identn => expr
Indeed, local definitions of functions are a syntactic sugar for binding a fun tactic to an identifier.
We can carry out pattern matching on terms with:
match expr with
cpattern1 => expr1
| cpattern2 => expr2
| cpatternn => exprn
| _ => exprn+1
The expression expr is evaluated and should yield a term which is matched against cpattern1. 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 @?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 ?id occurs under binders, say x1, …, xn and the expression matches, the metavariable is instantiated by a term which can then be used in any context which also binds the variables x1, …, xn 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 cpattern1 succeeds, then expr1 is evaluated into some value by substituting the pattern matching instantiations to the metavariables. If expr1 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 let in), then this tactic is applied. If the tactic succeeds, the list of resulting subgoals is the result of the match expression. If expr1 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 expr1 is the result of the match expression.
If the matching with cpattern1 fails, or if it succeeds but the evaluation of expr1 fails, or if the evaluation of expr1 succeeds but returns a tactic in execution position whose execution fails, then cpattern2 is used and so on. The pattern _ matches any term and shunts all remaining patterns if any. If all clauses fail (in particular, there is no pattern _) 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.
- No matching clauses for match
No pattern can be used and, in particular, there is no _ pattern.
- Argument of match does not evaluate to a term
This happens when expr does not denote a term.
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
The syntax match … is, in fact, a shorthand for once multimatch ….
- 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.
There is a special form of patterns to match a subterm against the
context ident [ cpattern ]It 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 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 Printing All, see Section 2.9).Coq < 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 *)
f is defined
Coq < Goal True.
Coq < f (3+4).
p : 1 = 1
1 + 4 = 0
subgoal 2 is:
For historical reasons, context used to consider n-ary applications
such as (f 1 2) as a whole, and not as a sequence of unary
applications ((f 1) 2). Hence context [f ?x] would fail
to find a matching subterm in (f 1 2): if the pattern was a partial
application, the matched subterms would have necessarily been
applications with exactly the same number of arguments.
As a workaround, one could use the following variant of context:
appcontext ident [ cpattern ]This syntax is now deprecated, as context behaves as intended. The former behavior can be retrieved with the Tactic Compat Context flag.
We can make pattern matching on goals using the following expression:
match goal with | hyp1,1,...,hyp1,m1 |-cpattern1=> expr1 | hyp2,1,...,hyp2,m2 |-cpattern2=> expr2 ... | hypn,1,...,hypn,mn |-cpatternn=> exprn |_ => exprn+1 end
If each hypothesis pattern hyp1,i, with i=1,...,m1 is matched (non-linear first-order unification) by an hypothesis of the goal and if cpattern1 is matched by the conclusion of the goal, then expr1 is evaluated to v1 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 v1 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 exprn+1 is evaluated to vn+1 and vn+1 is applied. Note also that matching against subterms (using the 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.
No clause succeeds, i.e. all matching patterns, if any, fail at the application of the right-hand-side.
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 the right to the left (i.e. hypi,mi before hypi,1). For each hypothesis pattern, the goal hypothesis are matched in order (fresher hypothesis first), but it possible to reverse this order (older first) with the match reverse goal with variant.
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 or combination of hypotheses when all the backtracking points of the right-hand side have been consumed.
The syntax match [reverse] goal … is, in fact, a shorthand for once multimatch [reverse] goal ….
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.
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:
context ident [ expr ]
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 ident by the value of expr.
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 … 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 an qualid which has to refer to a (unqualified) name, or directly a name denoted by a 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.
Evaluation of a term can be performed with:
eval redexpr in term
where redexpr is a reduction tactic among red, hnf, compute, simpl, cbv, lazy, unfold, fold, pattern.
Recovering the type of a term
The following returns the type of term:
type of 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 syntax
uconstr : 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 uconstr : can also be used as arguments to the refine tactic 8.2.3. 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.
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.
Coq < Goal True /\ True /\ True.
Coq < split;[|split].
Coq < all:pr_numgoals.
There are 3 goals
subgoal 2 is:
subgoal 3 is:
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.
Coq < split;[|split].
Coq < all:let n:= numgoals in guard n<4.
subgoal 2 is:
subgoal 3 is:
Coq < 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
subgoal 2 is:
subgoal 3 is:
From the outside “abstract expr” is the same as solve 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 auxiliary lemma is inlined in the final proof term unless the proof is ended with “Qed exporting”. In such case the lemma is preserved. The syntax “Qed exporting ident1, ..., identn” is also supported. In such case the system checks that the names given by the user actually exist when the proof is ended.
This tactical is useful with tactics such as omega or 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.
Deprecated since 8.7
When set (default), all lemmas generated through abstract expr and transparent_abstract expr are quantified only over the variables that appear in the term constructed by expr.
abstract expr using ident.
Give explicitly the name of the auxiliary lemma. Use this feature at your own risk; explicitly named and reused subterms don’t play well with asynchronous proofs.
- transparent_abstract expr.
Save the subproof in a transparent lemma rather than an opaque one. Use this feature at your own risk; building computationally relevant terms with tactics is fragile.
- transparent_abstract expr using ident.
Give explicitly the name of the auxiliary transparent lemma. 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.
9.3.1 Defining Ltac functions
Basically, Ltac toplevel definitions are made as follows:
Ltac ident ident1 ... identn := expr
This defines a new Ltac function that can be used in any tactic script or new Ltac toplevel definition.
Remark: The preceding definition can equivalently be written:
Ltac ident := fun ident1 ... identn => expr
Recursive and mutual recursive function definitions are also possible with the syntax:
Ltac ident1 ident1,1 ... ident1,m1 := expr1
with ident2 ident2,1 ... ident2,m2 := expr2
with identn identn,1 ... identn,mn := exprn
It is also possible to redefine an existing user-defined tactic using the syntax:
Ltac qualid ident1 ... identn ::= 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.
Defined Ltac functions can be displayed using the command
Print Ltac qualid.
9.4 Debugging Ltac tactics
It is possible 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 Info command can be used with the following syntax.
Info num expr.
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…
Coq < Goal exists n, n=0.
Coq < Info 0 t 1||t 0.
No more subgoals.
Coq < Undo.
Coq < Info 1 t 1||t 0.
No more subgoals.
The trace produced by 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 Info contains (in comments) the messages produced by the idtac tacticals 9.2 at the right possition in the script. In particular, the calls to idtac in branches which failed are not printed.
An alternative to the Info command is to use the Info Level option as follows:
Set Info Level num.
This will automatically print the same trace as Info num at each tactic call. The unfolding level can be overridden by a call to the Info command. And this option can be turned off with:
Unset Info Level num.
The current value for the Info Level option can be checked using the Test Info Level command.
The Ltac interpreter comes with a step-by-step debugger. The debugger can be activated using the command
Set Ltac Debug.
and deactivated using the command
Unset Ltac Debug.
To know if the debugger is on, use the command Test Ltac Debug. When the debugger is activated, it stops at every step of the evaluation of the current Ltac expression and it 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|
|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”|
A non-interactive mode for the debugger is available via the command
Set 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.
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 performence 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 its calling context. Thus it allows to locate the part of a tactic definition that contains the performance bug.
Set Ltac Profiling.
Enables the profiler
Unset Ltac Profiling.
Disables the profiler
Show Ltac Profile.
Prints the profile
Show Ltac Profile string.
Prints a profile for all tactics that start with string. Append a period (.) to the string if you only want exactly that name.
Reset Ltac Profile.
Resets the profile, that is, deletes all accumulated information. Note that backtracking across a Reset Ltac Profile will not restore the information.
Coq < Ltac mytauto := tauto.
Coq < Ltac tac := intros; repeat split; omega || mytauto.
Coq < Notation max x y := (x + (y - x)) (only parsing).
Coq < 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).
Coq < Proof.
Coq < Set Ltac Profiling.
Coq < tac.
No more subgoals.
total time: 1.160s
tactic local total calls max
─tac ----------------------------------- 0.3% 100.0% 1 1.160s
─t_tauto_intuit ------------------------ 1.4% 60.7% 26 0.072s
─<Coq.Init.Tauto.with_uniform_flags> --- 0.0% 60.7% 26 0.072s
─<Coq.Init.Tauto.tauto_intuitionistic> - 0.0% 60.7% 26 0.072s
─<Coq.Init.Tauto.tauto_gen> ------------ 0.0% 60.7% 26 0.072s
─<Coq.Init.Tauto.simplif> -------------- 41.4% 59.0% 26 0.072s
─omega --------------------------------- 38.6% 38.6% 28 0.188s
─elim id ------------------------------- 8.3% 8.3% 650 0.016s
─<Coq.Init.Tauto.is_conj> -------------- 5.5% 5.5% 28756 0.004s
─clear (var_list) ---------------------- 2.1% 2.1% 650 0.004s
tactic local total calls max
─tac ----------------------------------- 0.3% 100.0% 1 1.160s
├─<Coq.Init.Tauto.with_uniform_flags> - 0.0% 60.7% 26 0.072s
│└<Coq.Init.Tauto.tauto_gen> ---------- 0.0% 60.7% 26 0.072s
│└<Coq.Init.Tauto.tauto_intuitionistic> 0.0% 60.7% 26 0.072s
│└t_tauto_intuit ---------------------- 1.4% 60.7% 26 0.072s
│└<Coq.Init.Tauto.simplif> ------------ 41.4% 59.0% 26 0.072s
│ ├─elim id --------------------------- 8.3% 8.3% 650 0.016s
│ ├─<Coq.Init.Tauto.is_conj> ---------- 5.5% 5.5% 28756 0.004s
│ └─clear (var_list) ------------------ 2.1% 2.1% 650 0.004s
└─omega ------------------------------- 38.6% 38.6% 28 0.188s
Coq < Show Ltac Profile "omega".
total time: 1.160s
tactic local total calls max
─omega --------------------------------- 38.6% 38.6% 28 0.188s
tactic local total calls max
Coq < Unset Ltac Profiling.
start ltac profiling.
stop ltac profiling.
You can also pass the -profile-ltac command line option to coqc, which performs a Set Ltac Profiling at the beginning of each document, and a Show Ltac Profile at the end.
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.