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Ltac¶

This chapter documents the tactic language L_tac.

We start by giving the syntax, and next, we present the informal semantics. To learn more about the language and especially about its foundations, please refer to [Del00].

Example: Basic tactic macros

Here are some examples of simple tactic macros that the language lets you write.

Ltac reduce_and_try_to_solve := simpl; intros; auto. Ltac destruct_bool_and_rewrite b H1 H2 := destruct b; [ rewrite H1; eauto | rewrite H2; eauto ].

See Section Examples of using Ltac for more advanced examples.

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 num, int, ident qualid, term, cpattern and tactic represent respectively natural and integer numbers, identifiers, qualified names, Coq terms, patterns and 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 ?ident. The notation _ can also be used to denote metavariable whose instance is irrelevant. In the notation ?ident, 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 %@?ident ident₁ … ident_n, the variable ident matches any complex expression with (possible) dependencies in the variables ident_i and returns a functional term of the form fun ident₁ … ident_n => term.

The main entry of the grammar is ltac_expr. This language is used in proof mode but it can also be used in toplevel definitions as shown below.

Note

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

Example

If you want that 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

ltac_expr         ::=  ltac_expr ; ltac_expr
                       [> ltac_expr | ... | ltac_expr ]
                       ltac_expr ; [ ltac_expr | ... | ltac_expr ]
                       ltac_expr3
ltac_expr3        ::=  do (natural | ident) ltac_expr3
                       progress ltac_expr3
                       repeat ltac_expr3
                       try ltac_expr3
                       once ltac_expr3
                       exactly_once ltac_expr3
                       timeout (natural | ident) ltac_expr3
                       time [string] ltac_expr3
                       only selector: ltac_expr3
                       ltac_expr2
ltac_expr2        ::=  ltac_expr1 || ltac_expr3
                       ltac_expr1 + ltac_expr3
                       tryif ltac_expr1 then ltac_expr1 else ltac_expr1
                       ltac_expr1
ltac_expr1        ::=  fun name ... name => atom
                       let [rec] let_clause with ... with let_clause in atom
                       match goal with context_rule | ... | context_rule end
                       match reverse goal with context_rule | ... | context_rule end
                       match ltac_expr with match_rule | ... | match_rule end
                       lazymatch goal with context_rule | ... | context_rule end
                       lazymatch reverse goal with context_rule | ... | context_rule end
                       lazymatch ltac_expr with match_rule | ... | match_rule end
                       multimatch goal with context_rule | ... | context_rule end
                       multimatch reverse goal with context_rule | ... | context_rule end
                       multimatch ltac_expr with match_rule | ... | match_rule end
                       abstract atom
                       abstract atom using ident
                       first [ ltac_expr | ... | ltac_expr ]
                       solve [ ltac_expr | ... | ltac_expr ]
                       idtac [ message_token ... message_token]
                       fail [natural] [message_token ... message_token]
                       gfail [natural] [message_token ... message_token]
                       fresh [ component … component ]
                       context ident [term]
                       eval redexpr in term
                       type of term
                       constr : term
                       uconstr : term
                       type_term term
                       numgoals
                       guard test
                       assert_fails ltac_expr3
                       assert_succeeds ltac_expr3
                       tactic
                       qualid tacarg ... tacarg
                       atom
atom              ::=  qualid
                       ()
                       int
                       ( ltac_expr )
component         ::=  string | qualid
message_token     ::=  string | ident | int
tacarg            ::=  qualid
                       ()
                       ltac : atom
                       term
let_clause        ::=  ident [name ... name] := ltac_expr
context_rule      ::=  context_hyp, ..., context_hyp |- cpattern => ltac_expr
                       cpattern => ltac_expr
                       |- cpattern => ltac_expr
                       _ => ltac_expr
context_hyp       ::=  name : cpattern
                       name := cpattern [: cpattern]
match_rule        ::=  cpattern => ltac_expr
                       context [ident] [ cpattern ] => ltac_expr
                       _ => ltac_expr
test              ::=  int = int
                       int (< | <= | > | >=) int
selector          ::=  [ident]
                       int
                       (int | int - int), ..., (int | int - int)
toplevel_selector ::=  selector
                       all
                       par
                       !

top      ::=  [Local] Ltac ltac_def with ... with ltac_def
ltac_def ::=  ident [ident ... ident] := ltac_expr
              qualid [ident ... ident] ::= ltac_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:

ltac_expr₁ ; ltac_expr₂¶: The expression ltac_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 ltac_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:

[> ltac_expr*|]¶

The expressions ltac_expr_i are evaluated to v_i, for i = 1, ..., n and all have to be tactics. The v_i is applied to the i-th goal, for i = 1, ..., n. It fails if the number of focused goals is not exactly n.

Note

If no tactic is given for the i-th goal, it behaves as if the tactic idtac were given. For instance, [> | auto] is a shortcut for [> idtac | auto ].

Variant [> ltac_expr_i*| | ltac_expr .. | ltac_expr_j*|]: In this variant, ltac_expr is used for each goal coming after those covered by the list of ltac_expr_i but before those covered by the list of ltac_expr_j.

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

Variant [> ltac_expr .. ]: In this variant, the tactic ltac_expr is applied independently to each of the goals, rather than globally. In particular, if there are no goals, the tactic is not run at all. A tactic which expects multiple goals, such as swap, would act as if a single goal is focused.

Variant ltac_expr₀ ; [ltac_expr_i*|]: This variant of local tactic application is paired with a sequence. In this variant, there must be as many ltac_expr_i as goals generated by the application of ltac_expr₀ to each of the individual goals independently. All the above variants work in this form too. Formally, ltac_expr ; [ ... ] is equivalent to [> ltac_expr ; [> ... ] .. ].

Goal selectors¶

We can restrict the application of a tactic to a subset of the currently focused goals with:

toplevel_selector : ltac_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 : ltac_expr¶: When selecting several goals, the tactic ltac_expr is applied globally to all selected goals.

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

Variant num : ltac_expr: In this variant, ltac_expr is applied locally to the num-th goal.

Variant num-num+, : ltac_expr: In this variant, ltac_expr is applied globally to the subset of goals described by the given ranges. You can write a single n as a shortcut for n-n when specifying multiple ranges.

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

Variant !: ltac_expr: In this variant, if exactly one goal is focused, ltac_expr is applied to it. Otherwise the tactic fails. !: can only be used at the toplevel of a tactic expression.

Variant par: ltac_expr¶: In this variant, ltac_expr is applied to all focused goals in parallel. The number of workers can be controlled via the command line option -async-proofs-tac-j taking as argument the desired number of workers. Limitations: par: only works on goals containing no existential variables and ltac_expr must either solve the goal completely or do nothing (i.e. it cannot make some progress). par: can only be used at the toplevel of a tactic expression.

Error No such goal.¶

For loop¶

There is a for loop that repeats a tactic num times:

do num ltac_expr¶: ltac_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 ltac_expr¶: ltac_expr is evaluated to v. If v denotes a tactic, this tactic is applied to each focused goal independently. If the application succeeds, the tactic is applied recursively to all the generated subgoals until it eventually fails. The recursion stops in a subgoal when the tactic has failed to make progress. The tactic repeat ltac_expr itself never fails.

Error catching¶

We can catch the tactic errors with:

try ltac_expr¶: ltac_expr is evaluated to v which must be a tactic value. The tactic value v is applied to each focused goal independently. If the application of v fails in a goal, it catches the error and leaves the goal unchanged. If the level of the exception is positive, then the exception is re-raised with its level decremented.

Detecting progress¶

We can check if a tactic made progress with:

progress ltac_expr¶

ltac_expr is evaluated to v which must be a tactic value. The tactic value v is applied to each focused subgoal independently. If the application of v to one of the focused subgoal produced subgoals equal to the initial goals (up to syntactical equality), then an error of level 0 is raised.

Error Failed to progress.¶

Backtracking branching¶

We can branch with the following structure:

ltac_expr₁ + ltac_expr₂¶

ltac_expr₁ and ltac_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. ltac_expr₁ + ltac_expr₂ has all the successes of v₁ followed by all the successes of v₂. Algebraically, (ltac_expr₁ + ltac_expr₂); ltac_expr₃ = (ltac_expr₁; ltac_expr₃) + (ltac_expr₂; ltac_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 [ltac_expr*|]¶

The ltac_expr_i are evaluated to v_i and v_i must be tactic values for i = 1, ..., n. Supposing n > 1, first [ltac_expr₁ | ... | ltac_expr_n] 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 [ltac_expr₁ | ... | ltac_expr_n] behaves, in each goal, as the first v_i to have at least one success.

Error No applicable tactic.¶

Variant first ltac_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:

ltac_expr₁ || ltac_expr₂¶: ltac_expr₁ and ltac_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. ltac_expr₁ || ltac_expr₂ is equivalent to first [ progress ltac_expr₁ | ltac_expr₂ ] (except that if it fails, it fails like v₂). Branching is left-associative.

Generalized biased branching¶

The tactic

tryif ltac_expr₁ then ltac_expr₂ else ltac_expr₃¶: is a generalization of the biased-branching tactics above. The expression ltac_expr₁ is evaluated to v₁, which is then applied to each subgoal independently. For each goal where v₁ succeeds at least once, ltac_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 ltac_expr₁ then ltac_expr₂ else ltac_expr₃ is equivalent to v₁; v₂. In each of the goals where v₁ does not succeed at least once, ltac_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 ltac_expr¶: ltac_expr is evaluated to v which must be a tactic value. The tactic value v is applied but only its first success is used. If v fails, once ltac_expr fails like v. If v has at least one success, once ltac_expr succeeds once, but cannot produce more successes.

Checking the successes¶

Coq provides an experimental way to check that a tactic has exactly one success:

exactly_once ltac_expr¶

ltac_expr is evaluated to v which must be a tactic value. The tactic value v is applied if it has at most one success. If v fails, exactly_once ltac_expr fails like v. If v has a exactly one success, exactly_once ltac_expr succeeds like v. If v has two or more successes, exactly_once ltac_expr fails.

Warning

The experimental status of this tactic pertains to the fact if v performs side effects, they may occur in an unpredictable way. Indeed, normally v would only be executed up to the first success until backtracking is needed, however exactly_once needs to look ahead to see whether a second success exists, and may run further effects immediately.

Error This tactic has more than one success.¶

Checking the failure¶

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

assert_fails ltac_expr¶: This behaves like idtac if ltac_expr fails, and behaves like fail 0 ltac_expr "succeeds" if ltac_expr has at least one success.

Checking the success¶

Coq provides a derived tactic to check that a tactic has at least one success:

assert_succeeds ltac_expr¶: This behaves like tryif (assert_fails ltac_expr) then fail 0 ltac_expr "fails" else idtac.

Solving¶

We may consider the first to solve (i.e. which generates no subgoal) among a panel of tactics:

solve [ltac_expr*|]¶

The ltac_expr_i are evaluated to v_i and v_i must be tactic values, for i = 1, ..., n. Supposing n > 1, solve [ltac_expr₁ | ... | ltac_expr_n] 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 ltac_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.
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.
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 ltac_expr¶: ltac_expr is evaluated to v which must be a tactic value. The tactic value v is applied normally, except that it is interrupted after 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 ltac_expr¶: evaluates ltac_expr and displays the running time of the tactic expression, whether it fails or succeeds. In case of several successes, the time for each successive run is displayed. Time is in seconds and is machine-dependent. The 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 ltac_expr¶

which evaluates ltac_expr () and displays the time the tactic expression evaluated, assuming successful evaluation. Time is in seconds and is machine-dependent.

This tactic currently does not support nesting, and will report times based on the innermost execution. This is due to the fact that it is implemented using the following internal tactics:

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 achieve 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.u,0.s) 1 subgoal n := 100 : nat ============================ True

Local definitions¶

Local definitions can be done as follows:

let ident₁ := ltac_expr₁ with ident_i := ltac_expr_i* in ltac_expr¶

each ltac_expr_i is evaluated to v_i, then, ltac_expr is evaluated by substituting v_i to each occurrence of ident_i, for i = 1, ..., n. There are no dependencies between the ltac_expr_i and the ident_i.

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 ltac_expr_i are evaluated to v_i, for i = 1, ..., n.

Function construction¶

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

fun ident+ => ltac_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 ltac_expr with cpattern_i => ltac_expr_i+| end

The expression ltac_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, …, x_n and the expression matches, the metavariable is instantiated by a term which can then be used in any context which also binds the variables x₁, …, 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 cpattern₁ succeeds, then ltac_expr₁ is evaluated into some value by substituting the pattern matching instantiations to the metavariables. If ltac_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 ltac_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 ltac_expr₁ is the result of the match expression.

If the matching with cpattern₁ fails, or if it succeeds but the evaluation of ltac_expr₁ fails, or if the evaluation of ltac_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 ltac_expr does not denote a term.

Variant multimatch ltac_expr with cpattern_i => ltac_expr_i+| end

Using multimatch instead of match will allow subsequent tactics to backtrack into a right-hand side tactic which has backtracking points left and trigger the selection of a new matching branch when all the backtracking points of the right-hand side have been consumed.

The syntax match … is, in fact, a shorthand for once multimatch ….

Variant lazymatch ltac_expr with cpattern_i => ltac_expr_i+| end: Using lazymatch instead of match will perform the same pattern matching procedure but will commit to the first matching branch rather than trying a new matching if the right-hand side fails. If the right-hand side of the selected branch is a tactic with backtracking points, then subsequent failures cause this tactic to backtrack.

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 the Printing All flag).

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

Pattern matching on goals¶

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

match goal with context_hyp+, |- cpattern => ltac_expr+| | _ => ltac_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 ltac_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 ltac_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,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 context_hyp+, |- cpattern => ltac_expr+| | _ => ltac_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 context_hyp+, |- cpattern => ltac_expr+| | _ => ltac_expr end: Using lazymatch instead of match will perform the same pattern matching procedure but will commit to the first matching branch with the first matching combination of hypotheses rather than trying a new matching if the right-hand side fails. If the right-hand side of the selected branch is a tactic with backtracking points, then subsequent failures cause this tactic to backtrack.

Filling a term context¶

The following expression is not a tactic in the sense that it does not produce subgoals but generates a term to be used in tactic expressions:

context ident [ltac_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 ltac_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

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: Condition not satisfied: 3=2

Error Condition not satisfied.¶

Proving a subgoal as a separate lemma¶

abstract ltac_expr¶

From the outside, abstract ltac_expr is the same as solve ltac_expr. Internally it saves an auxiliary lemma called ident_subproofn where ident is the name of the current goal and n is chosen so that this is a fresh name. Such an auxiliary lemma is inlined in the final proof term.

This tactical is useful with tactics such as 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.

Warning

The abstract tactic, while very useful, still has some known limitations, see https://github.com/coq/coq/issues/9146 for more details. Thus we recommend using it caution in some "non-standard" contexts. In particular, abstract won't properly work when used inside quotations ltac:(...), or if used as part of typeclass resolution, it may produce wrong terms when in universe polymorphic mode.

Variant abstract ltac_expr using ident: Give explicitly the name of the auxiliary lemma.

Warning

Use this feature at your own risk; explicitly named and reused subterms don’t play well with asynchronous proofs.

Variant transparent_abstract ltac_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 ltac_expr using ident: Give explicitly the name of the auxiliary transparent lemma.

Warning

Use this feature at your own risk; building computationally relevant terms with tactics is fragile, and explicitly named and reused subterms don’t play well with asynchronous proofs.

Error Proof is not complete.¶

Tactic toplevel definitions¶

Defining `L`_tac functions¶

Basically, L_tac toplevel definitions are made as follows:

Command Local? Ltac ident ident* := ltac_expr¶

This defines a new L_tac function that can be used in any tactic script or new L_tac toplevel definition.

If preceded by the keyword Local, the tactic definition will not be exported outside the current module.

Note

The preceding definition can equivalently be written:

Ltac ident := fun ident+ => ltac_expr

Variant Ltac ident ident* with ident ident** := ltac_expr: This syntax allows recursive and mutual recursive function definitions.

Variant Ltac qualid ident* ::= ltac_expr

This syntax redefines an existing user-defined tactic.

A previous definition of qualid must exist in the environment. The new definition will always be used instead of the old one and it goes across module boundaries.

Printing `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.

Examples of using `L`_tac¶

Proof that the natural numbers have at least two elements¶

Example: Proof that the natural numbers have at least two elements

The first example shows how to use pattern matching over the proof context to prove that natural numbers have at least two elements. This can be done as follows:

Lemma card_nat : ~ exists x y : nat, forall z:nat, x = z \/ y = z.: 1 subgoal ============================ ~ (exists x y : nat, forall z : nat, x = z \/ y = z)
Proof.
intros (x & y & Hz).: 1 subgoal x, y : nat Hz : forall z : nat, x = z \/ y = z ============================ False
destruct (Hz 0), (Hz 1), (Hz 2).: 8 subgoals x, y : nat Hz : forall z : nat, x = z \/ y = z H : x = 0 H0 : x = 1 H1 : x = 2 ============================ False subgoal 2 is: False subgoal 3 is: False subgoal 4 is: False subgoal 5 is: False subgoal 6 is: False subgoal 7 is: False subgoal 8 is: False

At this point, the congruence tactic would finish the job:

all: congruence.: No more subgoals.

But for the purpose of the example, let's craft our own custom tactic to solve this:

Lemma card_nat : ~ exists x y : nat, forall z:nat, x = z \/ y = z.: 1 subgoal ============================ ~ (exists x y : nat, forall z : nat, x = z \/ y = z)
Proof.
intros (x & y & Hz).: 1 subgoal x, y : nat Hz : forall z : nat, x = z \/ y = z ============================ False
destruct (Hz 0), (Hz 1), (Hz 2).: 8 subgoals x, y : nat Hz : forall z : nat, x = z \/ y = z H : x = 0 H0 : x = 1 H1 : x = 2 ============================ False subgoal 2 is: False subgoal 3 is: False subgoal 4 is: False subgoal 5 is: False subgoal 6 is: False subgoal 7 is: False subgoal 8 is: False

all: match goal with | _ : ?a = ?b, _ : ?a = ?c |- _ => assert (b = c) by now transitivity a end.: 8 subgoals x, y : nat Hz : forall z : nat, x = z \/ y = z H : x = 0 H0 : x = 1 H1 : x = 2 H2 : 1 = 2 ============================ False subgoal 2 is: False subgoal 3 is: False subgoal 4 is: False subgoal 5 is: False subgoal 6 is: False subgoal 7 is: False subgoal 8 is: False
all: discriminate.: No more subgoals.

Notice that all the (very similar) cases coming from the three eliminations (with three distinct natural numbers) are successfully solved by a match goal structure and, in particular, with only one pattern (use of non-linear matching).

Proving that a list is a permutation of a second list¶

Example: Proving that a list is a permutation of a second list

Let's first define the permutation predicate:

Section Sort.
Variable A : Set.: A is declared
Inductive perm : list A -> list A -> Prop := | perm_refl : forall l, perm l l | perm_cons : forall a l0 l1, perm l0 l1 -> perm (a :: l0) (a :: l1) | perm_append : forall a l, perm (a :: l) (l ++ a :: nil) | perm_trans : forall l0 l1 l2, perm l0 l1 -> perm l1 l2 -> perm l0 l2.: perm is defined perm_ind is defined perm_sind is defined
End Sort.

Require Import List.

Next we define an auxiliary tactic perm_aux which takes an argument used to control the recursion depth. This tactic works as follows: If the lists are identical (i.e. convertible), it completes the proof. Otherwise, if the lists have identical heads, it looks at their tails. Finally, if the lists have different heads, it rotates the first list by putting its head at the end.

Every time we perform a rotation, we decrement n. When n drops down to 1, we stop performing rotations and we fail. The idea is to give the length of the list as the initial value of n. This way of counting the number of rotations will avoid going back to a head that had been considered before.

From Section Syntax we know that Ltac has a primitive notion of integers, but they are only used as arguments for primitive tactics and we cannot make computations with them. Thus, instead, we use Coq's natural number type nat.

Ltac perm_aux n := match goal with | |- (perm _ ?l ?l) => apply perm_refl | |- (perm _ (?a :: ?l1) (?a :: ?l2)) => let newn := eval compute in (length l1) in (apply perm_cons; perm_aux newn) | |- (perm ?A (?a :: ?l1) ?l2) => match eval compute in n with | 1 => fail | _ => let l1' := constr:(l1 ++ a :: nil) in (apply (perm_trans A (a :: l1) l1' l2); [ apply perm_append | compute; perm_aux (pred n) ]) end end.: perm_aux is defined

The main tactic is solve_perm. It computes the lengths of the two lists and uses them as arguments to call perm_aux if the lengths are equal. (If they aren't, the lists cannot be permutations of each other.)

Ltac solve_perm := match goal with | |- (perm _ ?l1 ?l2) => match eval compute in (length l1 = length l2) with | (?n = ?n) => perm_aux n end end.: solve_perm is defined

And now, here is how we can use the tactic solve_perm:

Goal perm nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil).: 1 subgoal ============================ perm nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil)

solve_perm.: No more subgoals.

Goal perm nat (0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil) (0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil).: 1 subgoal ============================ perm nat (0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil) (0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil)

solve_perm.: No more subgoals.

Deciding intuitionistic propositional logic¶

Pattern matching on goals allows powerful backtracking when returning tactic values. An interesting application is the problem of deciding intuitionistic propositional logic. Considering the contraction-free sequent calculi LJT* of Roy Dyckhoff [Dyc92], it is quite natural to code such a tactic using the tactic language as shown below.

Ltac basic := match goal with | |- True => trivial | _ : False |- _ => contradiction | _ : ?A |- ?A => assumption end.: basic is defined

Ltac simplify := repeat (intros; match goal with | H : ~ _ |- _ => red in H | H : _ /\ _ |- _ => elim H; do 2 intro; clear H | H : _ \/ _ |- _ => elim H; intro; clear H | H : ?A /\ ?B -> ?C |- _ => cut (A -> B -> C); [ intro | intros; apply H; split; assumption ] | H: ?A \/ ?B -> ?C |- _ => cut (B -> C); [ cut (A -> C); [ intros; clear H | intro; apply H; left; assumption ] | intro; apply H; right; assumption ] | H0 : ?A -> ?B, H1 : ?A |- _ => cut B; [ intro; clear H0 | apply H0; assumption ] | |- _ /\ _ => split | |- ~ _ => red end).: simplify is defined

Ltac my_tauto := simplify; basic || match goal with | H : (?A -> ?B) -> ?C |- _ => cut (B -> C); [ intro; cut (A -> B); [ intro; cut C; [ intro; clear H | apply H; assumption ] | clear H ] | intro; apply H; intro; assumption ]; my_tauto | H : ~ ?A -> ?B |- _ => cut (False -> B); [ intro; cut (A -> False); [ intro; cut B; [ intro; clear H | apply H; assumption ] | clear H ] | intro; apply H; red; intro; assumption ]; my_tauto | |- _ \/ _ => (left; my_tauto) || (right; my_tauto) end.: my_tauto is defined

The tactic basic tries to reason using simple rules involving truth, falsity and available assumptions. The tactic simplify applies all the reversible rules of Dyckhoff’s system. Finally, the tactic my_tauto (the main tactic to be called) simplifies with simplify, tries to conclude with basic and tries several paths using the backtracking rules (one of the four Dyckhoff’s rules for the left implication to get rid of the contraction and the right or).

Having defined my_tauto, we can prove tautologies like these:

Lemma my_tauto_ex1 : forall A B : Prop, A /\ B -> A \/ B.: 1 subgoal ============================ forall A B : Prop, A /\ B -> A \/ B
Proof.
my_tauto.: No more subgoals.
Qed.

Lemma my_tauto_ex2 : forall A B : Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B.: 1 subgoal ============================ forall A B : Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B
Proof.
my_tauto.: No more subgoals.
Qed.

Deciding type isomorphisms¶

A trickier problem is to decide equalities between types modulo isomorphisms. Here, we choose to use the isomorphisms of the simply typed λ-calculus with Cartesian product and unit type (see, for example, [dC95]). The axioms of this λ-calculus are given below.

Open Scope type_scope.

Section Iso_axioms.

Variables A B C : Set.: A is declared B is declared C is declared

Axiom Com : A * B = B * A.: Com is declared
Axiom Ass : A * (B * C) = A * B * C.: Ass is declared
Axiom Cur : (A * B -> C) = (A -> B -> C).: Cur is declared
Axiom Dis : (A -> B * C) = (A -> B) * (A -> C).: Dis is declared
Axiom P_unit : A * unit = A.: P_unit is declared
Axiom AR_unit : (A -> unit) = unit.: AR_unit is declared
Axiom AL_unit : (unit -> A) = A.: AL_unit is declared

Lemma Cons : B = C -> A * B = A * C.: 1 subgoal A, B, C : Set ============================ B = C -> A * B = A * C
Proof.
intro Heq; rewrite Heq; reflexivity.: No more subgoals.
Qed.

End Iso_axioms.

Ltac simplify_type ty := match ty with | ?A * ?B * ?C => rewrite <- (Ass A B C); try simplify_type_eq | ?A * ?B -> ?C => rewrite (Cur A B C); try simplify_type_eq | ?A -> ?B * ?C => rewrite (Dis A B C); try simplify_type_eq | ?A * unit => rewrite (P_unit A); try simplify_type_eq | unit * ?B => rewrite (Com unit B); try simplify_type_eq | ?A -> unit => rewrite (AR_unit A); try simplify_type_eq | unit -> ?B => rewrite (AL_unit B); try simplify_type_eq | ?A * ?B => (simplify_type A; try simplify_type_eq) || (simplify_type B; try simplify_type_eq) | ?A -> ?B => (simplify_type A; try simplify_type_eq) || (simplify_type B; try simplify_type_eq) end with simplify_type_eq := match goal with | |- ?A = ?B => try simplify_type A; try simplify_type B end.: simplify_type is defined simplify_type_eq is defined

Ltac len trm := match trm with | _ * ?B => let succ := len B in constr:(S succ) | _ => constr:(1) end.: len is defined

Ltac assoc := repeat rewrite <- Ass.: assoc is defined

Ltac solve_type_eq n := match goal with | |- ?A = ?A => reflexivity | |- ?A * ?B = ?A * ?C => apply Cons; let newn := len B in solve_type_eq newn | |- ?A * ?B = ?C => match eval compute in n with | 1 => fail | _ => pattern (A * B) at 1; rewrite Com; assoc; solve_type_eq (pred n) end end.: solve_type_eq is defined

Ltac compare_structure := match goal with | |- ?A = ?B => let l1 := len A with l2 := len B in match eval compute in (l1 = l2) with | ?n = ?n => solve_type_eq n end end.: compare_structure is defined

Ltac solve_iso := simplify_type_eq; compare_structure.: solve_iso is defined

The tactic to judge equalities modulo this axiomatization is shown above. The algorithm is quite simple. First types are simplified using axioms that can be oriented (this is done by simplify_type and simplify_type_eq). The normal forms are sequences of Cartesian products without a Cartesian product in the left component. These normal forms are then compared modulo permutation of the components by the tactic compare_structure. If they have the same length, the tactic solve_type_eq attempts to prove that the types are equal. The main tactic that puts all these components together is solve_iso.

Here are examples of what can be solved by solve_iso.

Lemma solve_iso_ex1 : forall A B : Set, A * unit * B = B * (unit * A).: 1 subgoal ============================ forall A B : Set, A * unit * B = B * (unit * A)
Proof.
intros; solve_iso.: No more subgoals.
Qed.

Lemma solve_iso_ex2 : forall A B C : Set, (A * unit -> B * (C * unit)) = (A * unit -> (C -> unit) * C) * (unit -> A -> B).: 1 subgoal ============================ forall A B C : Set, (A * unit -> B * (C * unit)) = (A * unit -> (C -> unit) * C) * (unit -> A -> B)
Proof.
intros; solve_iso.: No more subgoals.
Qed.

Debugging `L`_tac tactics¶

Backtraces¶

Flag Ltac Backtrace¶: Setting this flag displays a backtrace on Ltac failures that can be useful to find out what went wrong. It is disabled by default for performance reasons.

Info trace¶

Command Info num ltac_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.: exists with 0;<ltac_plugin::reflexivity@0> No more subgoals.

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

Info 1 t 1||t 0.: <ltac_plugin::exists@1> with 0;simple refine ?X12;<unknown> 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 flag 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 flag:

Flag Ltac Batch Debug¶: This flag 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 flag 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 performance 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 flag 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 newring_plugin.cmxs ... done] [Loading ML file zify_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: 2.576s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac ----------------------------------- 0.1% 100.0% 1 2.576s ─<Coq.Init.Tauto.with_uniform_flags> --- 0.0% 75.3% 26 0.188s ─<Coq.Init.Tauto.tauto_gen> ------------ 0.0% 75.2% 26 0.188s ─<Coq.Init.Tauto.tauto_intuitionistic> - 0.0% 75.2% 26 0.188s ─t_tauto_intuit ------------------------ 0.1% 75.1% 26 0.187s ─<Coq.Init.Tauto.simplif> -------------- 47.2% 72.5% 26 0.185s ─omega --------------------------------- 24.4% 24.4% 28 0.265s ─elim id ------------------------------- 11.6% 11.6% 650 0.131s ─<Coq.Init.Tauto.is_conj> -------------- 10.2% 10.2% 28756 0.000s ─<Coq.Init.Tauto.axioms> --------------- 1.6% 2.5% 0 0.013s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac ----------------------------------- 0.1% 100.0% 1 2.576s ├─<Coq.Init.Tauto.with_uniform_flags> - 0.0% 75.3% 26 0.188s │└<Coq.Init.Tauto.tauto_gen> ---------- 0.0% 75.2% 26 0.188s │└<Coq.Init.Tauto.tauto_intuitionistic> 0.0% 75.2% 26 0.188s │└t_tauto_intuit ---------------------- 0.1% 75.1% 26 0.187s │ ├─<Coq.Init.Tauto.simplif> ---------- 47.2% 72.5% 26 0.185s │ │ ├─elim id ------------------------- 11.6% 11.6% 650 0.131s │ │ └─<Coq.Init.Tauto.is_conj> -------- 10.2% 10.2% 28756 0.000s │ └─<Coq.Init.Tauto.axioms> ----------- 1.6% 2.5% 0 0.013s └─omega ------------------------------- 24.4% 24.4% 28 0.265s
Show Ltac Profile "omega".: total time: 2.576s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─omega --------------------------------- 24.4% 24.4% 28 0.265s 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 flag 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.

Ltac¶

Syntax¶

Semantics¶

Sequence¶

Local application of tactics¶

Goal selectors¶

For loop¶

Repeat loop¶

Error catching¶

Detecting progress¶

Backtracking branching¶

First tactic to work¶

Left-biased branching¶

Generalized biased branching¶

Soft cut¶

Checking the successes¶

Checking the failure¶

Checking the success¶

Solving¶

Identity¶

Failing¶

Timeout¶

Timing a tactic¶

Timing a tactic that evaluates to a term¶

Local definitions¶

Application¶

Function construction¶

Pattern matching on terms¶

Pattern matching on goals¶

Filling a term context¶

Generating fresh hypothesis names¶

Computing in a constr¶

Recovering the type of a term¶

Manipulating untyped terms¶

Counting the goals¶

Testing boolean expressions¶

Proving a subgoal as a separate lemma¶

Tactic toplevel definitions¶

Defining Ltac functions¶

Printing Ltac tactics¶

Examples of using Ltac¶

Proof that the natural numbers have at least two elements¶

Proving that a list is a permutation of a second list¶

Deciding intuitionistic propositional logic¶

Deciding type isomorphisms¶

Debugging Ltac tactics¶

Backtraces¶

Info trace¶

Interactive debugger¶

Profiling Ltac tactics¶

Run-time optimization tactic¶

Defining `L`_tac functions¶

Printing `L`_tac tactics¶

Examples of using `L`_tac¶

Debugging `L`_tac tactics¶

Profiling `L`_tac tactics¶