.. include:: ../preamble.rst .. include:: ../replaces.rst .. _ltac: The tactic language =================== This chapter gives a compact documentation of |Ltac|, the tactic language available in |Coq|. We start by giving the syntax, and next, we present the informal semantics. If you want to know more regarding this language and especially about its foundations, you can refer to :cite:`Del00`. Chapter :ref:`detailedexamplesoftactics` is devoted to giving examples of use of this language on small but also with non-trivial problems. .. _ltac-syntax: Syntax ------ The syntax of the tactic language is given below. See Chapter :ref:`gallinaspecificationlanguage` for a description of the BNF metasyntax used in these grammar rules. Various already defined entries will be used in this chapter: entries :token:`natural`, :token:`integer`, :token:`ident`, :token:`qualid`, :token:`term`, :token:`cpattern` and :token:`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 :ref:`tactics`. The syntax of :token:`cpattern` is the same as that of terms, but it is extended with pattern matching metavariables. In :token:`cpattern`, a pattern-matching metavariable is represented with the syntax :g:`?id` where :g:`id` is an :token:`ident`. The notation :g:`_` can also be used to denote metavariable whose instance is irrelevant. In the notation :g:`?id`, the identifier allows us to keep instantiations and to make constraints whereas :g:`_` shows that we are not interested in what will be matched. On the right hand side of pattern-matching clauses, the named metavariable are used without the question mark prefix. There is also a special notation for second-order pattern-matching problems: in an applicative pattern of the form :g:`@?id id1 … idn`, the variable id matches any complex expression with (possible) dependencies in the variables :g:`id1 … idn` and returns a functional term of the form :g:`fun id1 … idn => term`. The main entry of the grammar is :n:`@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. - In :token:`tacarg`, there is an overlap between qualid as a direct tactic argument and :token:`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 :g:`ltac:(@qualid)`. To force the resolution as a reference to a term, use the syntax :g:`(@qualid)`. - As shown by the figure, tactical ``\|\|`` binds more than the prefix tacticals try, repeat, do and abstract which themselves bind more than the postfix tactical “… ;[ … ]” which binds more than “… ; …”. For instance .. coqtop:: in try repeat tac1 || tac2; tac3; [tac31 | ... | tac3n]; tac4. is understood as .. coqtop:: in try (repeat (tac1 || tac2)); ((tac3; [tac31 | ... | tac3n]); tac4). .. productionlist:: coq 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 | fresh `string` | fresh `qualid` : | 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` ) 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` .. productionlist:: coq top : [Local] Ltac `ltac_def` with ... with `ltac_def` ltac_def : `ident` [`ident` ... `ident`] := `expr` : | `qualid` [`ident` ... `ident`] ::= `expr` .. _ltac-semantics: Semantics --------- Tactic expressions can only be applied in the context of a proof. The evaluation yields either a term, an integer or a tactic. Intermediary results can be terms or integers but the final result must be a tactic which is then applied to the focused goals. There is a special case for ``match goal`` expressions of which the clauses evaluate to tactics. Such expressions can only be used as end result of a tactic expression (never as argument of a non recursive local definition or of an application). The rest of this section explains the semantics of every construction of |Ltac|. Sequence ~~~~~~~~ A sequence is an expression of the following form: .. tacn:: @expr ; @expr :name: ltac-seq The expression :n:`@expr__1` is evaluated to :n:`v__1`, which must be a tactic value. The tactic :n:`v__1` is applied to the current goal, possibly producing more goals. Then :n:`@expr__2` is evaluated to produce :n:`v__2`, which must be a tactic value. The tactic :n:`v__2` is applied to all the goals produced by the prior application. Sequence is associative. Local application of tactics ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Different tactics can be applied to the different goals using the following form: .. tacn:: [> {*| @expr }] :name: [> ... | ... | ... ] (dispatch) The expressions :n:`@expr__i` are evaluated to :n:`v__i`, for i=0,...,n and all have to be tactics. The :n:`v__i` is applied to the i-th goal, for =1,...,n. It fails if the number of focused goals is not exactly n. .. note:: If no tactic is given for the i-th goal, it behaves as if the tactic idtac were given. For instance, ``[> | auto]`` is a shortcut for ``[> idtac | auto ]``. .. tacv:: [> {*| @expr} | @expr .. | {*| @expr}] In this variant, token:`expr` is used for each goal coming after those covered by the first list of :n:`@expr` but before those coevered by the last list of :n:`@expr`. .. tacv:: [> {*| @expr} | .. | {*| @expr}] In this variant, idtac is used for the goals not covered by the two lists of :n:`@expr`. .. tacv:: [> @expr .. ] In this variant, the tactic :n:`@expr` is applied independently to each of the goals, rather than globally. In particular, if there are no goal, the tactic is not run at all. A tactic which expects multiple goals, such as ``swap``, would act as if a single goal is focused. .. tacv:: expr ; [{*| @expr}] This variant of local tactic application is paired with a sequence. In this variant, there must be as many :n:`@expr` in the list as goals generated by the application of the first :n:`@expr` to each of the individual goals independently. All the above variants work in this form too. Formally, :n:`@expr ; [ ... ]` is equivalent to :n:`[> @expr ; [> ... ] .. ]`. .. _goal-selectors: Goal selectors ~~~~~~~~~~~~~~ We can restrict the application of a tactic to a subset of the currently focused goals with: .. tacn:: @toplevel_selector : @expr :name: ... : ... (goal selector) We can also use selectors as a tactical, which allows to use them nested in a tactic expression, by using the keyword ``only``: .. tacv:: only selector : expr :name: only ... : ... When selecting several goals, the tactic expr is applied globally to all selected goals. .. tacv:: [@ident] : @expr In this variant, :n:`@expr` is applied locally to a goal previously named by the user (see :ref:`existential-variables`). .. tacv:: @num : @expr In this variant, :n:`@expr` is applied locally to the :token:`num`-th goal. .. tacv:: {+, @num-@num} : @expr In this variant, :n:`@expr` is applied globally to the subset of goals described by the given ranges. You can write a single ``n`` as a shortcut for ``n-n`` when specifying multiple ranges. .. tacv:: all: @expr :name: all: ... In this variant, :n:`@expr` is applied to all focused goals. ``all:`` can only be used at the toplevel of a tactic expression. .. tacv:: !: @expr In this variant, if exactly one goal is focused :n:`expr` is applied to it. Otherwise the tactical fails. ``!:`` can only be used at the toplevel of a tactic expression. .. tacv:: par: @expr :name: par: ... In this variant, :n:`@expr` is applied to all focused goals in parallel. The number of workers can be controlled via the command line option ``-async-proofs-tac-j`` taking as argument the desired number of workers. Limitations: ``par:`` only works on goals containing no existential variables and :n:`@expr` must either solve the goal completely or do nothing (i.e. it cannot make some progress). ``par:`` can only be used at the toplevel of a tactic expression. .. exn:: No such goal. :name: No such goal. (Goal selector) .. TODO change error message index entry For loop ~~~~~~~~ There is a for loop that repeats a tactic :token:`num` times: .. tacn:: do @num @expr :name: do :n:`@expr` is evaluated to ``v`` which must be a tactic value. This tactic value ``v`` is applied :token:`num` times. Supposing :token:`num` > 1, after the first application of ``v``, ``v`` is applied, at least once, to the generated subgoals and so on. It fails if the application of ``v`` fails before the num applications have been completed. Repeat loop ~~~~~~~~~~~ We have a repeat loop with: .. tacn:: repeat @expr :name: repeat :n:`@expr` is evaluated to ``v``. If ``v`` denotes a tactic, this tactic is applied to each focused goal independently. If the application succeeds, the tactic is applied recursively to all the generated subgoals until it eventually fails. The recursion stops in a subgoal when the tactic has failed *to make progress*. The tactic :n:`repeat @expr` itself never fails. Error catching ~~~~~~~~~~~~~~ We can catch the tactic errors with: .. tacn:: try @expr :name: try :n:`@expr` is evaluated to ``v`` which must be a tactic value. The tactic value ``v`` is applied to each focused goal independently. If the application of ``v`` fails in a goal, it catches the error and leaves the goal unchanged. If the level of the exception is positive, then the exception is re-raised with its level decremented. Detecting progress ~~~~~~~~~~~~~~~~~~ We can check if a tactic made progress with: .. tacn:: progress expr :name: progress :n:`@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. .. exn:: Failed to progress. Backtracking branching ~~~~~~~~~~~~~~~~~~~~~~ We can branch with the following structure: .. tacn:: @expr__1 + @expr__2 :name: + (backtracking branching) :n:`@expr__1` and :n:`@expr__2` are evaluated respectively to :n:`v__1` and :n:`v__2` which must be tactic values. The tactic value :n:`v__1` is applied to each focused goal independently and if it fails or a later tactic fails, then the proof backtracks to the current goal and :n:`v__2` is applied. Tactics can be seen as having several successes. When a tactic fails it asks for more successes of the prior tactics. :n:`@expr__1 + @expr__2` has all the successes of :n:`v__1` followed by all the successes of :n:`v__2`. Algebraically, :n:`(@expr__1 + @expr__2); @expr__3 = (@expr__1; @expr__3) + (@expr__2; @expr__3)`. Branching is left-associative. First tactic to work ~~~~~~~~~~~~~~~~~~~~ Backtracking branching may be too expensive. In this case we may restrict to a local, left biased, branching and consider the first tactic to work (i.e. which does not fail) among a panel of tactics: .. tacn:: first [{*| @expr}] :name: first The :n:`@expr__i` are evaluated to :n:`v__i` and :n:`v__i` must be tactic values, for i=1,...,n. Supposing n>1, it applies, in each focused goal independently, :n:`v__1`, if it works, it stops otherwise it tries to apply :n:`v__2` and so on. It fails when there is no applicable tactic. In other words, :n:`first [:@expr__1 | ... | @expr__n]` behaves, in each goal, as the the first :n:`v__i` to have *at least* one success. .. exn:: No applicable tactic. .. tacv:: first @expr This is an |Ltac| alias that gives a primitive access to the first tactical as a |Ltac| definition without going through a parsing rule. It expects to be given a list of tactics through a ``Tactic Notation``, allowing to write notations of the following form: .. example:: .. coqtop:: in Tactic Notation "foo" tactic_list(tacs) := first tacs. Left-biased branching ~~~~~~~~~~~~~~~~~~~~~ Yet another way of branching without backtracking is the following structure: .. tacn:: @expr__1 || @expr__2 :name: || (left-biased branching) :n:`@expr__1` and :n:`@expr__2` are evaluated respectively to :n:`v__1` and :n:`v__2` which must be tactic values. The tactic value :n:`v__1` is applied in each subgoal independently and if it fails *to progress* then :n:`v__2` is applied. :n:`@expr__1 || @expr__2` is equivalent to :n:`first [ progress @expr__1 | @expr__2 ]` (except that if it fails, it fails like :n:`v__2`). Branching is left-associative. Generalized biased branching ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The tactic .. tacn:: tryif @expr__1 then @expr__2 else @expr__3 :name: tryif is a generalization of the biased-branching tactics above. The expression :n:`@expr__1` is evaluated to :n:`v__1`, which is then applied to each subgoal independently. For each goal where :n:`v__1` succeeds at least once, :n:`@expr__2` is evaluated to :n:`v__2` which is then applied collectively to the generated subgoals. The :n:`v__2` tactic can trigger backtracking points in :n:`v__1`: where :n:`v__1` succeeds at least once, :n:`tryif @expr__1 then @expr__2 else @expr__3` is equivalent to :n:`v__1; v__2`. In each of the goals where :n:`v__1` does not succeed at least once, :n:`@expr__3` is evaluated in :n:`v__3` which is is then applied to the goal. Soft cut ~~~~~~~~ Another way of restricting backtracking is to restrict a tactic to a single success *a posteriori*: .. tacn:: once @expr :name: once :n:`@expr` is evaluated to ``v`` which must be a tactic value. The tactic value ``v`` is applied but only its first success is used. If ``v`` fails, :n:`once @expr` fails like ``v``. If ``v`` has a least one success, :n:`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: .. tacn:: exactly_once @expr :name: exactly_once :n:`@expr` is evaluated to ``v`` which must be a tactic value. The tactic value ``v`` is applied if it has at most one success. If ``v`` fails, :n:`exactly_once @expr` fails like ``v``. If ``v`` has a exactly one success, :n:`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 a unpredictable way. Indeed, normally ``v`` would only be executed up to the first success until backtracking is needed, however exactly_once needs to look ahead to see whether a second success exists, and may run further effects immediately. .. exn:: This tactic has more than one success. Checking the failure ~~~~~~~~~~~~~~~~~~~~ Coq provides a derived tactic to check that a tactic *fails*: .. tacn:: assert_fails @expr :name: assert_fails This behaves like :n:`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: .. tacn:: assert_succeeds @expr :name: assert_succeeds This behaves like :n:`tryif (assert_fails tac) then fail 0 tac "fails" else idtac`. Solving ~~~~~~~ We may consider the first to solve (i.e. which generates no subgoal) among a panel of tactics: .. tacn:: solve [{*| @expr}] :name: solve The :n:`@expr__i` are evaluated to :n:`v__i` and :n:`v__i` must be tactic values, for i=1,...,n. Supposing n>1, it applies :n:`v__1` to each goal independently, if it doesn’t solve the goal then it tries to apply :n:`v__2` and so on. It fails if there is no solving tactic. .. exn:: Cannot solve the goal. .. tacv:: solve @expr This is an |Ltac| alias that gives a primitive access to the :n:`solve:` tactical. See the :n:`first` tactical for more information. Identity ~~~~~~~~ The constant :n:`idtac` is the identity tactic: it leaves any goal unchanged but it appears in the proof script. .. tacn:: idtac {* message_token} :name: idtac This prints the given tokens. Strings and integers are printed literally. If a (term) variable is given, its contents are printed. Failing ~~~~~~~ .. tacn:: fail :name: fail This is the always-failing tactic: it does not solve any goal. It is useful for defining other tacticals since it can be caught by :tacn:`try`, :tacn:`repeat`, :tacn:`match goal`, or the branching tacticals. The :tacn:`fail` tactic will, however, succeed if all the goals have already been solved. .. tacv:: fail @num The number is the failure level. If no level is specified, it defaults to 0. The level is used by :tacn:`try`, :tacn:`repeat`, :tacn:`match goal` and the branching tacticals. If 0, it makes :tacn:`match goal` considering the next clause (backtracking). If non zero, the current :tacn:`match goal` block, :tacn:`try`, :tacn:`repeat`, or branching command is aborted and the level is decremented. In the case of :n:`+`, a non-zero level skips the first backtrack point, even if the call to :n:`fail @num` is not enclosed in a :n:`+` command, respecting the algebraic identity. .. tacv:: fail {* message_token} The given tokens are used for printing the failure message. .. tacv:: fail @num {* message_token} This is a combination of the previous variants. .. tacv:: gfail :name: gfail This variant fails even if there are no goals left. .. tacv:: gfail {* message_token} .. tacv:: gfail @num {* message_token} These variants fail with an error message or an error level even if there are no goals left. Be careful however if Coq terms have to be printed as part of the failure: term construction always forces the tactic into the goals, meaning that if there are no goals when it is evaluated, a tactic call like :n:`let x:=H in fail 0 x` will succeed. .. exn:: Tactic Failure message (level @num). Timeout ~~~~~~~ We can force a tactic to stop if it has not finished after a certain amount of time: .. tacn:: timeout @num @expr :name: timeout :n:`@expr` is evaluated to ``v`` which must be a tactic value. The tactic value ``v`` is applied normally, except that it is interrupted after :n:`@num` seconds if it is still running. In this case the outcome is a failure. .. warning:: For the moment, timeout is based on elapsed time in seconds, which is very machine-dependent: a script that works on a quick machine may fail on a slow one. The converse is even possible if you combine a timeout with some other tacticals. This tactical is hence proposed only for convenience during debug or other development phases, we strongly advise you to not leave any timeout in final scripts. Note also that this tactical isn’t available on the native Windows port of Coq. Timing a tactic ~~~~~~~~~~~~~~~ A tactic execution can be timed: .. tacn:: time @string @expr :name: time evaluates :n:`@expr` and displays the time the tactic expression ran, whether it fails or successes. In case of several successes, the time for each successive runs is displayed. Time is in seconds and is machine-dependent. The :n:`@string` argument is optional. When provided, it is used to identify this particular occurrence of time. Timing a tactic that evaluates to a term ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Tactic expressions that produce terms can be timed with the experimental tactic .. tacn:: time_constr expr :name: time_constr which evaluates :n:`@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 tactics .. tacn:: restart_timer @string :name: restart_timer and .. tacn:: finish_timing {? @string} @string :name: finish_timing which (re)set and display an optionally named timer, respectively. The parenthesized string argument to :n:`finish_timing` is also optional, and determines the label associated with the timer for printing. By copying the definition of :n:`time_constr` from the standard library, users can achive support for a fixed pattern of nesting by passing different :n:`@string` parameters to :n:`restart_timer` and :n:`finish_timing` at each level of nesting. .. example:: .. coqtop:: all Ltac time_constr1 tac := let eval_early := match goal with _ => restart_timer "(depth 1)" end in let ret := tac () in let eval_early := match goal with _ => finish_timing ( "Tactic evaluation" ) "(depth 1)" end in ret. Goal True. let v := time_constr ltac:(fun _ => let x := time_constr1 ltac:(fun _ => constr:(10 * 10)) in let y := time_constr1 ltac:(fun _ => eval compute in x) in y) in pose v. Abort. Local definitions ~~~~~~~~~~~~~~~~~ Local definitions can be done as follows: .. tacn:: let @ident__1 := @expr__1 {* with @ident__i := @expr__i} in @expr each :n:`@expr__i` is evaluated to :n:`v__i`, then, :n:`@expr` is evaluated by substituting :n:`v__i` to each occurrence of :n:`@ident__i`, for i=1,...,n. There is no dependencies between the :n:`@expr__i` and the :n:`@ident__i`. Local definitions can be recursive by using :n:`let rec` instead of :n:`let`. In this latter case, the definitions are evaluated lazily so that the rec keyword can be used also in non recursive cases so as to avoid the eager evaluation of local definitions. .. but rec changes the binding!! Application ~~~~~~~~~~~ An application is an expression of the following form: .. tacn:: @qualid {+ @tacarg} The reference :n:`@qualid` must be bound to some defined tactic definition expecting at least as many arguments as the provided :n:`tacarg`. The expressions :n:`@expr__i` are evaluated to :n:`v__i`, for i=1,...,n. .. what expressions ?? Function construction ~~~~~~~~~~~~~~~~~~~~~ A parameterized tactic can be built anonymously (without resorting to local definitions) with: .. tacn:: fun {+ @ident} => @expr Indeed, local definitions of functions are a syntactic sugar for binding a :n:`fun` tactic to an identifier. Pattern matching on terms ~~~~~~~~~~~~~~~~~~~~~~~~~ We can carry out pattern matching on terms with: .. tacn:: match @expr with {+| @cpattern__i => @expr__i} end The expression :n:`@expr` is evaluated and should yield a term which is matched against :n:`cpattern__1`. The matching is non-linear: if a metavariable occurs more than once, it should match the same expression every time. It is first-order except on the variables of the form :n:`@?id` that occur in head position of an application. For these variables, the matching is second-order and returns a functional term. Alternatively, when a metavariable of the form :n:`?id` occurs under binders, say :n:`x__1, …, x__n` and the expression matches, the metavariable is instantiated by a term which can then be used in any context which also binds the variables :n:`x__1, …, x__n` with same types. This provides with a primitive form of matching under context which does not require manipulating a functional term. If the matching with :n:`@cpattern__1` succeeds, then :n:`@expr__1` is evaluated into some value by substituting the pattern matching instantiations to the metavariables. If :n:`@expr__1` evaluates to a tactic and the match expression is in position to be applied to a goal (e.g. it is not bound to a variable by a :n:`let in`), then this tactic is applied. If the tactic succeeds, the list of resulting subgoals is the result of the match expression. If :n:`@expr__1` does not evaluate to a tactic or if the match expression is not in position to be applied to a goal, then the result of the evaluation of :n:`@expr__1` is the result of the match expression. If the matching with :n:`@cpattern__1` fails, or if it succeeds but the evaluation of :n:`@expr__1` fails, or if the evaluation of :n:`@expr__1` succeeds but returns a tactic in execution position whose execution fails, then :n:`cpattern__2` is used and so on. The pattern :n:`_` matches any term and shunts all remaining patterns if any. If all clauses fail (in particular, there is no pattern :n:`_`) then a no-matching-clause error is raised. Failures in subsequent tactics do not cause backtracking to select new branches or inside the right-hand side of the selected branch even if it has backtracking points. .. exn:: No matching clauses for match. No pattern can be used and, in particular, there is no :n:`_` pattern. .. exn:: Argument of match does not evaluate to a term. This happens when :n:`@expr` does not denote a term. .. tacv:: multimatch @expr with {+| @cpattern__i => @expr__i} end Using multimatch instead of match will allow subsequent tactics to backtrack into a right-hand side tactic which has backtracking points left and trigger the selection of a new matching branch when all the backtracking points of the right-hand side have been consumed. The syntax :n:`match …` is, in fact, a shorthand for :n:`once multimatch …`. .. tacv:: lazymatch @expr with {+| @cpattern__i => @expr__i} end Using lazymatch instead of match will perform the same pattern matching procedure but will commit to the first matching branch rather than trying a new matching if the right-hand side fails. If the right-hand side of the selected branch is a tactic with backtracking points, then subsequent failures cause this tactic to backtrack. .. tacv:: context @ident [@cpattern] This special form of patterns matches any term with a subterm matching cpattern. If there is a match, the optional :n:`@ident` is assigned the "matched context", i.e. the initial term where the matched subterm is replaced by a hole. The example below will show how to use such term contexts. If the evaluation of the right-hand-side of a valid match fails, the next matching subterm is tried. If no further subterm matches, the next clause is tried. Matching subterms are considered top-bottom and from left to right (with respect to the raw printing obtained by setting option :opt:`Printing All`). .. example:: .. coqtop:: all Ltac f x := match x with context f [S ?X] => idtac X; (* To display the evaluation order *) assert (p := eq_refl 1 : X=1); (* To filter the case X=1 *) let x:= context f[O] in assert (x=O) (* To observe the context *) end. Goal True. f (3+4). .. _ltac-match-goal: Pattern matching on goals ~~~~~~~~~~~~~~~~~~~~~~~~~ We can make pattern matching on goals using the following expression: .. we should provide the full grammar here .. tacn:: match goal with {+| {+ hyp} |- @cpattern => @expr } | _ => @expr end :name: match goal If each hypothesis pattern :n:`hyp`\ :sub:`1,i`, with i=1,...,m\ :sub:`1` is matched (non-linear first-order unification) by an hypothesis of the goal and if :n:`cpattern_1` is matched by the conclusion of the goal, then :n:`@expr__1` is evaluated to :n:`v__1` by substituting the pattern matching to the metavariables and the real hypothesis names bound to the possible hypothesis names occurring in the hypothesis patterns. If :n:`v__1` is a tactic value, then it is applied to the goal. If this application fails, then another combination of hypotheses is tried with the same proof context pattern. If there is no other combination of hypotheses then the second proof context pattern is tried and so on. If the next to last proof context pattern fails then the last :n:`@expr` is evaluated to :n:`v` and :n:`v` is applied. Note also that matching against subterms (using the :n:`context @ident [ @cpattern ]`) is available and is also subject to yielding several matchings. Failures in subsequent tactics do not cause backtracking to select new branches or combinations of hypotheses, or inside the right-hand side of the selected branch even if it has backtracking points. .. exn:: No matching clauses for match goal. No clause succeeds, i.e. all matching patterns, if any, fail at the application of the right-hand-side. .. note:: It is important to know that each hypothesis of the goal can be matched by at most one hypothesis pattern. The order of matching is the following: hypothesis patterns are examined from the right to the left (i.e. hyp\ :sub:`i,m`\ :sub:`i`` before hyp\ :sub:`i,1`). For each hypothesis pattern, the goal hypothesis are matched in order (fresher hypothesis first), but it possible to reverse this order (older first) with the :n:`match reverse goal with` variant. .. tacv:: multimatch goal with {+| {+ hyp} |- @cpattern => @expr } | _ => @expr end Using :n:`multimatch` instead of :n:`match` will allow subsequent tactics to backtrack into a right-hand side tactic which has backtracking points left and trigger the selection of a new matching branch or combination of hypotheses when all the backtracking points of the right-hand side have been consumed. The syntax :n:`match [reverse] goal …` is, in fact, a shorthand for :n:`once multimatch [reverse] goal …`. .. tacv:: lazymatch goal with {+| {+ 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: .. tacn:: context @ident [@expr] :n:`@ident` must denote a context variable bound by a context pattern of a match expression. This expression evaluates replaces the hole of the value of :n:`@ident` by the value of :n:`@expr`. .. exn:: Not a context variable. Generating fresh hypothesis names ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Tactics sometimes have to generate new names for hypothesis. Letting the system decide a name with the intro tactic is not so good since it is very awkward to retrieve the name the system gave. The following expression returns an identifier: .. tacn:: fresh {* component} It evaluates to an identifier unbound in the goal. This fresh identifier is obtained by concatenating the value of the :n:`@component`s (each of them is, either a :n:`@qualid` which has to refer to a (unqualified) name, or directly a name denoted by a :n:`@string`). .. I don't understand this component thing. Couldn't we give the grammar? If the resulting name is already used, it is padded with a number so that it becomes fresh. If no component is given, the name is a fresh derivative of the name ``H``. Computing in a constr ~~~~~~~~~~~~~~~~~~~~~ Evaluation of a term can be performed with: .. tacn:: eval @redexpr in @term where :n:`@redexpr` is a reduction tactic among :tacn:`red`, :tacn:`hnf`, :tacn:`compute`, :tacn:`simpl`, :tacn:`cbv`, :tacn:`lazy`, :tacn:`unfold`, :tacn:`fold`, :tacn:`pattern`. Recovering the type of a term ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The following returns the type of term: .. tacn:: type of @term Manipulating untyped terms ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. tacn:: uconstr : @term The terms built in |Ltac| are well-typed by default. It may not be appropriate for building large terms using a recursive |Ltac| function: the term has to be entirely type checked at each step, resulting in potentially very slow behavior. It is possible to build untyped terms using |Ltac| with the :n:`uconstr : @term` syntax. .. tacn:: type_term @term An untyped term, in |Ltac|, can contain references to hypotheses or to |Ltac| variables containing typed or untyped terms. An untyped term can be type-checked using the function type_term whose argument is parsed as an untyped term and returns a well-typed term which can be used in tactics. Untyped terms built using :n:`uconstr :` can also be used as arguments to the :tacn:`refine` tactic. In that case the untyped term is type checked against the conclusion of the goal, and the holes which are not solved by the typing procedure are turned into new subgoals. Counting the goals ~~~~~~~~~~~~~~~~~~ .. tacn:: numgoals The number of goals under focus can be recovered using the :n:`numgoals` function. Combined with the guard command below, it can be used to branch over the number of goals produced by previous tactics. .. example:: .. coqtop:: in Ltac pr_numgoals := let n := numgoals in idtac "There are" n "goals". Goal True /\ True /\ True. split;[|split]. .. coqtop:: all all:pr_numgoals. Testing boolean expressions ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. tacn:: guard @test :name: guard The :tacn:`guard` tactic tests a boolean expression, and fails if the expression evaluates to false. If the expression evaluates to true, it succeeds without affecting the proof. The accepted tests are simple integer comparisons. .. example:: .. coqtop:: in Goal True /\ True /\ True. split;[|split]. .. coqtop:: all all:let n:= numgoals in guard n<4. Fail all:let n:= numgoals in guard n=2. .. exn:: Condition not satisfied. Proving a subgoal as a separate lemma ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. tacn:: abstract @expr :name: abstract From the outside, :n:`abstract @expr` is the same as :n:`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 :tacn:`omega` or :tacn:`discriminate` that generate huge proof terms. With that tool the user can avoid the explosion at time of the Save command without having to cut manually the proof in smaller lemmas. It may be useful to generate lemmas minimal w.r.t. the assumptions they depend on. This can be obtained thanks to the option below. .. tacv:: 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. .. tacv:: transparent_abstract @expr :name: transparent_abstract Save the subproof in a transparent lemma rather than an opaque one. .. warning:: Use this feature at your own risk; building computationally relevant terms with tactics is fragile. .. tacv:: transparent_abstract @expr using @ident Give explicitly the name of the auxiliary transparent lemma. .. warning:: Use this feature at your own risk; building computationally relevant terms with tactics is fragile, and explicitly named and reused subterms don’t play well with asynchronous proofs. .. exn:: Proof is not complete. :name: Proof is not complete. (abstract) Tactic toplevel definitions --------------------------- Defining |Ltac| functions ~~~~~~~~~~~~~~~~~~~~~~~~~ Basically, |Ltac| toplevel definitions are made as follows: .. cmd:: Ltac @ident {* @ident} := @expr This defines a new |Ltac| function that can be used in any tactic script or new |Ltac| toplevel definition. .. note:: The preceding definition can equivalently be written: :n:`Ltac @ident := fun {+ @ident} => @expr` Recursive and mutual recursive function definitions are also possible with the syntax: .. cmdv:: Ltac @ident {* @ident} {* with @ident {* @ident}} := @expr It is also possible to *redefine* an existing user-defined tactic using the syntax: .. cmdv:: 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 |Ltac| tactics ~~~~~~~~~~~~~~~~~~~~~~~ .. cmd:: Print Ltac @qualid Defined |Ltac| functions can be displayed using this command. .. cmd:: Print Ltac Signatures This command displays a list of all user-defined tactics, with their arguments. Debugging |Ltac| tactics ------------------------ Info trace ~~~~~~~~~~ .. cmd:: Info @num @expr :name: Info This command can be used to print the trace of the path eventually taken by an |Ltac| script. That is, the list of executed tactics, discarding all the branches which have failed. To that end the :cmd:`Info` command can be used with the following syntax. The number :n:`@num` is the unfolding level of tactics in the trace. At level 0, the trace contains a sequence of tactics in the actual script, at level 1, the trace will be the concatenation of the traces of these tactics, etc… .. example:: .. coqtop:: in reset Ltac t x := exists x; reflexivity. Goal exists n, n=0. .. coqtop:: all Info 0 t 1||t 0. .. coqtop:: in Undo. .. coqtop:: all Info 1 t 1||t 0. The trace produced by :cmd:`Info` tries its best to be a reparsable |Ltac| script, but this goal is not achievable in all generality. So some of the output traces will contain oddities. As an additional help for debugging, the trace produced by :cmd:`Info` contains (in comments) the messages produced by the :tacn:`idtac` tactical at the right position in the script. In particular, the calls to idtac in branches which failed are not printed. .. opt:: Info Level @num This option is an alternative to the :cmd:`Info` command. This will automatically print the same trace as :n:`Info @num` at each tactic call. The unfolding level can be overridden by a call to the :cmd:`Info` command. Interactive debugger ~~~~~~~~~~~~~~~~~~~~ .. opt:: Ltac Debug This option governs the step-by-step debugger that comes with the |Ltac| interpreter When the debugger is activated, it stops at every step of the evaluation of the current |Ltac| expression and it prints information on what it is doing. The debugger stops, prompting for a command which can be one of the following: +-----------------+-----------------------------------------------+ | simple newline: | go to the next step | +-----------------+-----------------------------------------------+ | 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” | +-----------------+-----------------------------------------------+ A non-interactive mode for the debugger is available via the option: .. opt:: Ltac Batch Debug This option has the effect of presenting a newline at every prompt, when the debugger is on. The debug log thus created, which does not require user input to generate when this option is set, can then be run through external tools such as diff. Profiling |Ltac| tactics ~~~~~~~~~~~~~~~~~~~~~~~~ It is possible to measure the time spent in invocations of primitive tactics as well as tactics defined in |Ltac| and their inner invocations. The primary use is the development of complex tactics, which can sometimes be so slow as to impede interactive usage. The reasons for the performence degradation can be intricate, like a slowly performing |Ltac| match or a sub-tactic whose performance only degrades in certain situations. The profiler generates a call tree and indicates the time spent in a tactic depending its calling context. Thus it allows to locate the part of a tactic definition that contains the performance bug. .. opt:: Ltac Profiling This option enables and disables the profiler. .. cmd:: Show Ltac Profile Prints the profile .. cmdv:: Show Ltac Profile @string Prints a profile for all tactics that start with :n:`@string`. Append a period (.) to the string if you only want exactly that name. .. cmd:: Reset Ltac Profile Resets the profile, that is, deletes all accumulated information. .. warning:: Backtracking across a :cmd:`Reset Ltac Profile` will not restore the information. .. coqtop:: reset in Require Import Coq.omega.Omega. Ltac mytauto := tauto. Ltac tac := intros; repeat split; omega || mytauto. Notation max x y := (x + (y - x)) (only parsing). Goal forall x y z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z, max x (max y z) = max (max x y) z /\ max x (max y z) = max (max x y) z /\ (A /\ B /\ C /\ D /\ E /\ F /\ G /\ H /\ I /\ J /\ K /\ L /\ M /\ N /\ O /\ P /\ Q /\ R /\ S /\ T /\ U /\ V /\ W /\ X /\ Y /\ Z -> Z /\ Y /\ X /\ W /\ V /\ U /\ T /\ S /\ R /\ Q /\ P /\ O /\ N /\ M /\ L /\ K /\ J /\ I /\ H /\ G /\ F /\ E /\ D /\ C /\ B /\ A). Proof. .. coqtop:: all Set Ltac Profiling. tac. Show Ltac Profile. Show Ltac Profile "omega". .. coqtop:: in Abort. Unset Ltac Profiling. .. tacn:: start ltac profiling :name: start ltac profiling This tactic behaves like :tacn:`idtac` but enables the profiler. .. tacn:: stop ltac profiling :name: stop ltac profiling Similarly to :tacn:`start ltac profiling`, this tactic behaves like :tacn:`idtac`. Together, they allow you to exclude parts of a proof script from profiling. .. tacn:: reset ltac profile :name: reset ltac profile This tactic behaves like the corresponding vernacular command and allow displaying and resetting the profile from tactic scripts for benchmarking purposes. .. tacn:: show ltac profile :name: show ltac profile This tactic behaves like the corresponding vernacular command and allow displaying and resetting the profile from tactic scripts for benchmarking purposes. .. tacn:: show ltac profile @string :name: show ltac profile This tactic behaves like the corresponding vernacular command and allow displaying and resetting the profile from tactic scripts for benchmarking purposes. You can also pass the ``-profile-ltac`` command line option to ``coqc``, which turns the :opt:`Ltac Profiling` option on at the beginning of each document, and performs a :cmd:`Show Ltac Profile` at the end. .. warning:: Note that the profiler currently does not handle backtracking into multi-success tactics, and issues a warning to this effect in many cases when such backtracking occurs. Run-time optimization tactic ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. tacn:: optimize_heap :name: optimize_heap This tactic behaves like :n:`idtac`, except that running it compacts the heap in the OCaml run-time system. It is analogous to the Vernacular command :cmd:`Optimize Heap`.