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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 The Gallina specification language 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 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 below.

Note

• The infix tacticals  … || … ,  … + … , and  … ; …  are associative.

  Example

  If you want that tactic2; tactic3 be fully run on the first subgoal generated by tactic1, before running on the other subgoals, then you should not write tactic1; (tactic2; tactic3) but rather tactic1; [> tactic2; tactic3 .. ].

• 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 at the same level as  … ; … .

  Example

  try repeat tactic1 || tactic2; tactic3; [ tactic+| ]; tactic4

  is understood as:

  ((try (repeat (tactic1 || tactic2)); tactic3); [ tactic+| ]); tactic4

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]
                       | 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 tacexpr3
                       | assert_succeeds tacexpr3
                       | 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] Ltac ltac_def with ... with ltac_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 non-recursive 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:

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.

Local application of tactics

Different tactics can be applied to the different goals using the following form:

[> expr*|]

The expressions expri are evaluated to vi, for i = 1, ..., n and all have to be tactics. The vi 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 ].

Variant [> expri*| | expr .. | exprj*|]

In this variant, expr is used for each goal coming after those covered by the list of expri but before those covered by the list of exprj.

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 as swap, would act as if a single goal is focused.

Variant expr0 ; [expri*|]

This variant of local tactic application is paired with a sequence. In this variant, there must be as many expri as goals generated by the application of expr0 to each of the individual goals independently. All the above variants work in this form too. Formally, expr ; [ ... ] is equivalent to [> expr ; [> ... ] .. ].

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 num : expr

In this variant, expr is applied locally to the num-th goal.

Variant num-num+, : 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.

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 !: expr

In this variant, if exactly one goal is focused, expr is applied to it. Otherwise the tactic fails. !: 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 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.

Error No such goal.

For loop

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.

Repeat loop

We have a repeat loop with:

repeat expr

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.

Error catching

We can catch the tactic errors with:

try expr

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:

progress expr

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.

Error Failed to progress.

Backtracking branching

We can branch with the following structure:

expr1 + expr2

expr1 and expr2 are evaluated respectively 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.

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 expri are evaluated to vi and vi must be tactic values for i = 1, ..., n. Supposing n > 1, first [expr1 | ... | exprn] applies v1 in each focused goal independently and stops if it succeeds; 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 first vi 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.

Left-biased branching

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

expr1 || expr2

expr1 and expr2 are evaluated respectively 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.

Generalized biased branching

The tactic

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.

Soft cut

Another way of restricting backtracking is to restrict a tactic to a single success a posteriori:

once expr

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 at least one success, once 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:

exactly_once expr

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.

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.

Error This tactic has more than one success.

Checking the failure

Coq provides a derived tactic to check that a tactic fails:

assert_fails expr

This behaves like tryif 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:

assert_succeeds expr

This behaves like 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:

solve [expr*|]

The expri are evaluated to vi and vi must be tactic values, for i = 1, ..., n. Supposing n > 1, solve [expr1 | ... | exprn] applies v1 to each goal independently and stops if it succeeds; otherwise it tries to apply v2 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 the first tactical for more information.

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 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.

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 makes match goal consider the next clause (backtracking). If nonzero, the current match 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 to fail 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 after all: 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 Tactic Failure message (level num).

Error No such goal.

Example

Goal True.

1 subgoal ============================ True

Proof.

fail.

Toplevel input, characters 0-5: > 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 0-5: > 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 0-6: > gfail. > ^^^^^^ Error: Tactic failure.

Abort.

Goal True.

1 subgoal ============================ True

Proof.

trivial; gfail.

Toplevel input, characters 0-15: > trivial; gfail. > ^^^^^^^^^^^^^^^ Error: Tactic failure.

Abort.

Goal True.

1 subgoal ============================ True

Proof.

trivial.

No more subgoals.

gfail.

Toplevel input, characters 0-6: > gfail. > ^^^^^^ Error: No such goal.

Abort.

Goal True.

1 subgoal ============================ True

Proof.

trivial.

No more subgoals.

all: gfail.

Toplevel input, characters 0-11: > all: gfail. > ^^^^^^^^^^^ Error: Tactic failure.

Abort.

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 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 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 machine-dependent. The 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

time_constr expr

which evaluates 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:

restart_timer string

Reset a timer

finish_timing (string)? string

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 time_constr from the standard library, users can achive support for a fixed pattern of nesting by passing different string parameters to restart_timer and finish_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 0.001 secs (0.001u,0.s) 1 subgoal n := 100 : nat ============================ True

Abort.

Local definitions

Local definitions can be done as follows:

let ident1 := expr1 with identi := expri* in expr

each expri is evaluated to vi, then, expr is evaluated by substituting vi to each occurrence of identi, for i = 1, ..., n. There are no dependencies between the expri and the identi.

Local definitions can be made 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.

Application

An application is an expression of the following form:

qualid tacarg+

The reference qualid must be bound to some defined tactic definition expecting at least as many arguments as the provided tacarg. The expressions expri are evaluated to vi, for i = 1, ..., n.

Function construction

A parameterized tactic can be built anonymously (without resorting to local definitions) with:

fun ident+ => expr

Indeed, local definitions of functions are a syntactic sugar for binding a fun tactic to an identifier.

Pattern matching on terms

We can carry out pattern matching on terms with:

match expr with cpatterni => expri+| end

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 shadows 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.

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 cpatterni => expri+| 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 match … is, in fact, a shorthand for once multimatch ….

Variant lazymatch expr with cpatterni => expri+| 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.

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 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).

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

Abort.

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, ..., m₁ is matched (non-linear first-order unification) by a hypothesis of the goal and if cpattern_1 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 the last expr is evaluated to v and v 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.

Error 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_i,mi` 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 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 ….

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 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:

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.

Error Not a context variable.

Error Unbound context identifier ident.

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 components (each of them is, either a 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.

Computing in a constr

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

type of term

This tactic returns the type of term.

Manipulating untyped terms

uconstr : term

The terms built in L_tac are well-typed 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 well-typed 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.

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

Abort.

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 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 an auxiliary lemma is inlined in the final proof term.

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.

Variant abstract 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.

Variant transparent_abstract expr

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.

Variant transparent_abstract 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.

Error Proof is not complete.

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.

Note

The preceding definition can equivalently be written:

Ltac ident := fun ident+ => expr

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 user-defined 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.

Printing L_tac tactics

Command Print Ltac qualid

Defined L_tac functions can be displayed using this command.

Command Print Ltac Signatures

This command displays a list of all user-defined tactics, with their arguments.

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 1||t 0.

t <constr:(0)> No more subgoals.

Undo.

1 subgoal ============================ exists n : nat, n = 0

Info 1 t 1||t 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 the idtac tactical at the right position in the script. In particular, the calls to idtac in branches which failed are not printed.

Option Info Level num

This option is an alternative to the Info command.

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.

Interactive debugger

Flag Ltac Debug

This option governs the step-by-step 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 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 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 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.

Command Show Ltac Profile

Prints the profile

Variant 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.

Command Reset Ltac Profile

Resets the profile, that is, deletes all accumulated information.

Warning

Backtracking across a Reset Ltac Profile will not restore the information.

Require Import Coq.omega.Omega.

[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: 3.168s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac ----------------------------------- 0.1% 100.0% 1 3.168s ─<Coq.Init.Tauto.with_uniform_flags> --- 0.0% 72.4% 26 0.133s ─<Coq.Init.Tauto.tauto_gen> ------------ 0.0% 72.3% 26 0.133s ─<Coq.Init.Tauto.tauto_intuitionistic> - 0.0% 72.3% 26 0.132s ─t_tauto_intuit ------------------------ 0.1% 72.2% 26 0.132s ─<Coq.Init.Tauto.simplif> -------------- 51.0% 69.8% 26 0.130s ─omega --------------------------------- 27.2% 27.2% 28 0.380s ─<Coq.Init.Tauto.is_conj> -------------- 9.7% 9.7% 28756 0.000s ─elim id ------------------------------- 6.0% 6.0% 650 0.035s ─<Coq.Init.Tauto.axioms> --------------- 1.9% 2.3% 0 0.004s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac ----------------------------------- 0.1% 100.0% 1 3.168s ├─<Coq.Init.Tauto.with_uniform_flags> - 0.0% 72.4% 26 0.133s │└<Coq.Init.Tauto.tauto_gen> ---------- 0.0% 72.3% 26 0.133s │└<Coq.Init.Tauto.tauto_intuitionistic> 0.0% 72.3% 26 0.132s │└t_tauto_intuit ---------------------- 0.1% 72.2% 26 0.132s │ ├─<Coq.Init.Tauto.simplif> ---------- 51.0% 69.8% 26 0.130s │ │ ├─<Coq.Init.Tauto.is_conj> -------- 9.7% 9.7% 28756 0.000s │ │ └─elim id ------------------------- 6.0% 6.0% 650 0.035s │ └─<Coq.Init.Tauto.axioms> ----------- 1.9% 2.3% 0 0.004s └─omega ------------------------------- 27.2% 27.2% 28 0.380s

Show Ltac Profile "omega".

total time: 3.168s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─omega --------------------------------- 27.2% 27.2% 28 0.380s tactic local total calls max

Abort.

Unset Ltac Profiling.

start ltac profiling

This tactic behaves like idtac but enables the profiler.

stop ltac profiling

Similarly to start ltac profiling, this tactic behaves like idtac. 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 -profile-ltac 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.

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

optimize_heap

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