\[\begin{split}\newcommand{\alors}{\textsf{then}} \newcommand{\alter}{\textsf{alter}} \newcommand{\as}{\kw{as}} \newcommand{\Assum}[3]{\kw{Assum}(#1)(#2:#3)} \newcommand{\bool}{\textsf{bool}} \newcommand{\case}{\kw{case}} \newcommand{\conc}{\textsf{conc}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\conshl}{\textsf{cons\_hl}} \newcommand{\Def}[4]{\kw{Def}(#1)(#2:=#3:#4)} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\EqSt}{\textsf{EqSt}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\false}{\textsf{false}} \newcommand{\filter}{\textsf{filter}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\from}{\textsf{from}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\haslength}{\textsf{has\_length}} \newcommand{\hd}{\textsf{hd}} \newcommand{\ident}{\textsf{ident}} \newcommand{\In}{\kw{in}} \newcommand{\Ind}[4]{\kw{Ind}[#2](#3:=#4)} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[5]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)} \newcommand{\Indpstr}[6]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)/{#6}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\lb}{\lambda} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\Nat}{\mathbb{N}} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\Prod}{\textsf{prod}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\si}{\textsf{if}} \newcommand{\sinon}{\textsf{else}} \newcommand{\Sort}{\cal S} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\true}{\textsf{true}} \newcommand{\Type}{\textsf{Type}} \newcommand{\unfold}{\textsf{unfold}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\cal W\!F}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\cal W\!F}(#2)} \newcommand{\WFTWOLINES}[2]{{\cal W\!F}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \newcommand{\zeros}{\textsf{zeros}} \end{split}\]

# 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 [Del00]. Chapter Detailed examples of tactics is devoted to giving small but nontrivial use examples of this language.

## Syntax

The syntax of the tactic language is given below. See Chapter The Gallina specification language for a description of the BNF metasyntax used in these grammar rules. Various already defined entries will be used in this chapter: entries `natural`, `integer`, `ident`, `qualid`, `term`, `cpattern` and `atomic_tactic` represent respectively the natural and integer numbers, the authorized identificators and qualified names, Coq terms and patterns and all the atomic tactics described in Chapter Tactics. The syntax of `cpattern` is the same as that of terms, but it is extended with pattern matching metavariables. In `cpattern`, a pattern matching metavariable is represented with the syntax `?id` where `id` is an `ident`. The notation `_` can also be used to denote metavariable whose instance is irrelevant. In the notation `?id`, the identifier allows us to keep instantiations and to make constraints whereas `_` shows that we are not interested in what will be matched. On the right hand side of pattern matching clauses, the named metavariables are used without the question mark prefix. There is also a special notation for second-order pattern matching problems: in an applicative pattern of the form `@?id id1 … idn`, the variable id matches any complex expression with (possible) dependencies in the variables `id1 … idn` and returns a functional term of the form `fun id1 … idn => term`.

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

Note

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

- In `tacarg`, there is an overlap between qualid as a direct tactic argument and `qualid` as a particular case of term. The resolution is done by first looking for a reference of the tactic language and if it fails, for a reference to a term. To force the resolution as a reference of the tactic language, use the form `ltac:(@qualid)`. To force the resolution as a reference to a term, use the syntax `(@qualid)`.

- As shown by the figure, tactical `\|\|` binds more than the prefix tacticals try, repeat, do and abstract which themselves bind more than the postfix tactical “… ;[ … ]” which binds more than “… ; …”.

  For instance

  try repeat tac1 || tac2; tac3; [tac31 | ... | tac3n]; tac4.
    Toplevel input, characters 0-3: > try repeat tac1 || tac2; tac3; [tac31 | ... > ^^^ Error: Syntax error: illegal begin of vernac. Toplevel input, characters 44-45: > try repeat tac1 || tac2; tac3; [tac31 | ... | tac3n]; tac4. > ^ Error: Syntax error: illegal begin of vernac.

  is understood as

  try (repeat (tac1 || tac2)); ((tac3; [tac31 | ... | tac3n]); tac4).
    Toplevel input, characters 0-3: > try (repeat (tac1 || tac2)); ((tac3; [tac31 | ... > ^^^ Error: Syntax error: illegal begin of vernac. Toplevel input, characters 52-53: > try (repeat (tac1 || tac2)); ((tac3; [tac31 | ... | tac3n]); tac4). > ^ Error: Syntax error: illegal begin of vernac.

    expr              ::=  expr ; expr
                           | [> expr | ... | expr ]
                           | expr ; [ expr | ... | expr ]
                           | tacexpr3
    tacexpr3          ::=  do (natural | ident) tacexpr3
                           | progress tacexpr3
                           | repeat tacexpr3
                           | try tacexpr3
                           | once tacexpr3
                           | exactly_once tacexpr3
                           | timeout (natural | ident) tacexpr3
                           | time [string] tacexpr3
                           | only selector: tacexpr3
                           | tacexpr2
    tacexpr2          ::=  tacexpr1 || tacexpr3
                           | tacexpr1 + tacexpr3
                           | tryif tacexpr1 then tacexpr1 else tacexpr1
                           | tacexpr1
    tacexpr1          ::=  fun name ... name => atom
                           | let [rec] let_clause with ... with let_clause in atom
                           | match goal with context_rule | ... | context_rule end
                           | match reverse goal with context_rule | ... | context_rule end
                           | match expr with match_rule | ... | match_rule end
                           | lazymatch goal with context_rule | ... | context_rule end
                           | lazymatch reverse goal with context_rule | ... | context_rule end
                           | lazymatch expr with match_rule | ... | match_rule end
                           | multimatch goal with context_rule | ... | context_rule end
                           | multimatch reverse goal with context_rule | ... | context_rule end
                           | multimatch expr with match_rule | ... | match_rule end
                           | abstract atom
                           | abstract atom using ident
                           | first [ expr | ... | expr ]
                           | solve [ expr | ... | expr ]
                           | idtac [ message_token ... message_token]
                           | fail [natural] [message_token ... message_token]
                           | fresh [ component … component ]
                           | context ident [term]
                           | eval redexpr in term
                           | type of term
                           | constr : term
                           | uconstr : term
                           | type_term term
                           | numgoals
                           | guard test
                           | assert_fails tacexpr3
                           | assert_succeeds tacexpr3
                           | atomic_tactic
                           | qualid tacarg ... tacarg
                           | atom
    atom              ::=  qualid
                           | ()
                           | integer
                           | ( expr )
    component         ::=  string | qualid
    message_token     ::=  string | ident | integer
    tacarg            ::=  qualid
                           | ()
                           | ltac : atom
                           | term
    let_clause        ::=  ident [name ... name] := expr
    context_rule      ::=  context_hyp, ..., context_hyp |- cpattern => expr
                           | cpattern => expr
                           | |- cpattern => expr
                           | _ => expr
    context_hyp       ::=  name : cpattern
                           | name := cpattern [: cpattern]
    match_rule        ::=  cpattern => expr
                           | context [ident] [ cpattern ] => expr
                           | _ => expr
    test              ::=  integer = integer
                           | integer (< | <= | > | >=) integer
    selector          ::=  [ident]
                           | integer
                           (integer | integer - integer), ..., (integer | integer - integer)
    toplevel_selector ::=  selector
                           | all
                           | par

    top      ::=  [Local] Ltac ltac_def with ... with ltac_def
    ltac_def ::=  ident [ident ... ident] := expr
                  | qualid [ident ... ident] ::= expr

## Semantics

Tactic expressions can only be applied in the context of a proof. The evaluation yields either a term, an integer or a tactic. Intermediate results can be terms or integers but the final result must be a tactic which is then applied to the focused goals.

There is a special case for `match goal` expressions of which the clauses evaluate to tactics. Such expressions can only be used as end result of a tactic expression (never as argument of a non-recursive local definition or of an application).

The rest of this section explains the semantics of every construction of Ltac.

### Sequence

A sequence is an expression of the following form:

`expr1 ; expr2`

  The expression `expr1` is evaluated to `v1`, which must be a tactic value. The tactic `v1` is applied to the current goal, possibly producing more goals. Then `expr2` is evaluated to produce `v2`, which must be a tactic value. The tactic `v2` is applied to all the goals produced by the prior application. Sequence is associative.

### Local application of tactics

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

`[> expr*|]`

  The expressions `expri` are evaluated to `vi`, for i = 0, ..., n and all have to be tactics. The `vi` is applied to the i-th goal, for i = 1, ..., n. It fails if the number of focused goals is not exactly n.

  Note

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

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

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

  Variant `[> expr*| | .. | expr*|]`

    In this variant, idtac is used for the goals not covered by the two lists of `expr`.

  Variant `[> expr .. ]`

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

  Variant `expr0 ; [expri*|]`

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

### Goal selectors

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

`toplevel_selector : expr`

  We can also use selectors as a tactical, which allows to use them nested in a tactic expression, by using the keyword `only`:

  Variant `only selector : expr`

    When selecting several goals, the tactic `expr` is applied globally to all selected goals.

  Variant `[ident] : expr`

    In this variant, `expr` is applied locally to a goal previously named by the user (see Existential variables).

  Variant `num : expr`

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

  Variant `num-num+, : expr`

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

  Variant `all: expr`

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

  Variant `par: expr`

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

  Error `No such goal.`

### For loop

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

`do num expr`

  `expr` is evaluated to `v` which must be a tactic value. This tactic value `v` is applied `num` times. Supposing `num` > 1, after the first application of `v`, `v` is applied, at least once, to the generated subgoals and so on. It fails if the application of `v` fails before the num applications have been completed.

### Repeat loop

We have a repeat loop with:

`repeat expr`

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

### Error catching

We can catch the tactic errors with:

`try expr`

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

### Detecting progress

We can check if a tactic made progress with:

`progress expr`

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

  Error `Failed to progress.`

### Backtracking branching

We can branch with the following structure:

`expr1 + expr2`

  `expr1` and `expr2` are evaluated respectively to `v1` and `v2` which must be tactic values. The tactic value `v1` is applied to each focused goal independently and if it fails or a later tactic fails, then the proof backtracks to the current goal and `v2` is applied.

  Tactics can be seen as having several successes. When a tactic fails it asks for more successes of the prior tactics. `expr1 + expr2` has all the successes of `v1` followed by all the successes of `v2`. Algebraically, `(expr1 + expr2); expr3 = (expr1; expr3) + (expr2; expr3)`.

  Branching is left-associative.

### First tactic to work

Backtracking branching may be too expensive. In this case we may restrict to a local, left biased, branching and consider the first tactic to work (i.e. which does not fail) among a panel of tactics:

`first [expr*|]`

  The `expri` are evaluated to `vi` and `vi` must be tactic values for i = 1, ..., n. Supposing n > 1, `first [expr1 | ... | exprn]` applies `v1` in each focused goal independently and stops if it succeeds; otherwise it tries to apply `v2` and so on. It fails when there is no applicable tactic. In other words, `first [expr1 | ... | exprn]` behaves, in each goal, as the first `vi` to have at least one success.

  Error `No applicable tactic.`
  Variant `first expr`

    This is an Ltac alias that gives a primitive access to the first tactical as an 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

    Tactic Notation "foo" tactic_list(tacs) := first tacs.

### Left-biased branching

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

`expr1 || expr2`

  `expr1` and `expr2` are evaluated respectively to `v1` and `v2` which must be tactic values. The tactic value `v1` is applied in each subgoal independently and if it fails to progress then `v2` is applied. `expr1 || expr2` is equivalent to `first [ progress expr1 | expr2 ]` (except that if it fails, it fails like `v2`). Branching is left-associative.

### Generalized biased branching

The tactic

`tryif expr1 then expr2 else expr3`

  is a generalization of the biased-branching tactics above. The expression `expr1` is evaluated to `v1`, which is then applied to each subgoal independently. For each goal where `v1` succeeds at least once, `expr2` is evaluated to `v2` which is then applied collectively to the generated subgoals. The `v2` tactic can trigger backtracking points in `v1`: where `v1` succeeds at least once, `tryif expr1 then expr2 else expr3` is equivalent to `v1; v2`. In each of the goals where `v1` does not succeed at least once, `expr3` is evaluated in `v3` which is is then applied to the goal.

### Soft cut

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

`once expr`

  `expr` is evaluated to `v` which must be a tactic value. The tactic value `v` is applied but only its first success is used. If `v` fails, `once expr` fails like `v`. If `v` has at least one success, `once expr` succeeds once, but cannot produce more successes.

### Checking the successes

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

`exactly_once expr`

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

  Warning

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

  Error `This tactic has more than one success.`

### Checking the failure

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

`assert_fails expr`

  This behaves like `tryif expr then fail 0 tac "succeeds" else idtac`.

### Checking the success

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

`assert_succeeds expr`

  This behaves like `tryif (assert_fails tac) then fail 0 tac "fails" else idtac`.

### Solving

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

`solve [expr*|]`

  The `expri` are evaluated to `vi` and `vi` must be tactic values, for i = 1, ..., n. Supposing n > 1, `solve [expr1 | ... | exprn]` applies `v1` to each goal independently and stops if it succeeds; otherwise it tries to apply `v2` and so on. It fails if there is no solving tactic.

  Error `Cannot solve the goal.`
  Variant `solve expr`

    This is an Ltac alias that gives a primitive access to the `solve:` tactical. See the `first` tactical for more information.

### Identity

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

`idtac message_token*`

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

### Failing

`fail`

  This is the always-failing tactic: it does not solve any goal. It is useful for defining other tacticals since it can be caught by `try`, `repeat`, `match goal`, or the branching tacticals.

  Variant `fail num`

    The number is the failure level. If no level is specified, it defaults to 0. The level is used by `try`, `repeat`, `match goal` and the branching tacticals. If 0, it makes `match goal` consider the next clause (backtracking). If nonzero, the current `match goal` block, `try`, `repeat`, or branching command is aborted and the level is decremented. In the case of `+`, a nonzero level skips the first backtrack point, even if the call to `fail num` is not enclosed in a `+` command, respecting the algebraic identity.

  Variant `fail message_token*`

    The given tokens are used for printing the failure message.

  Variant `fail num message_token*`

    This is a combination of the previous variants.

  Variant `gfail`

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

  Variant `gfail message_token*`
  Variant `gfail num message_token*`

    These variants fail with an error message or an error level even if there are no goals left. Be careful however if Coq terms have to be printed as part of the failure: term construction always forces the tactic into the goals, meaning that if there are no goals when it is evaluated, a tactic call like `let x := H in fail 0 x` will succeed.

  Error `Tactic Failure message (level num).`
  Error `No such goal.`

  Example

  Goal True.
    1 subgoal ============================ True
  Proof.
  fail.
    Toplevel input, characters 0-5: > fail. > ^^^^^ Error: Tactic failure.
  Abort.
  Goal True.
    1 subgoal ============================ True
  Proof.
  trivial; fail.
    No more subgoals.
  Qed.
    Unnamed_thm is defined
  Goal True.
    1 subgoal ============================ True
  Proof.
  trivial.
    No more subgoals.
  fail.
    Toplevel input, characters 0-5: > fail. > ^^^^^ Error: No such goal.
  Abort.
  Goal True.
    1 subgoal ============================ True
  Proof.
  trivial.
    No more subgoals.
  all: fail.
  Qed.
    Unnamed_thm0 is defined
  Goal True.
    1 subgoal ============================ True
  Proof.
  gfail.
    Toplevel input, characters 0-6: > gfail. > ^^^^^^ Error: Tactic failure.
  Abort.
  Goal True.
    1 subgoal ============================ True
  Proof.
  trivial; gfail.
    Toplevel input, characters 0-15: > trivial; gfail. > ^^^^^^^^^^^^^^^ Error: Tactic failure.
  Abort.
  Goal True.
    1 subgoal ============================ True
  Proof.
  trivial.
    No more subgoals.
  gfail.
    Toplevel input, characters 0-6: > gfail. > ^^^^^^ Error: No such goal.
  Abort.
  Goal True.
    1 subgoal ============================ True
  Proof.
  trivial.
    No more subgoals.
  all: gfail.
    Toplevel input, characters 0-11: > all: gfail. > ^^^^^^^^^^^ Error: Tactic failure.
  Abort.

### Timeout

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

`timeout num expr`

  `expr` is evaluated to `v` which must be a tactic value. The tactic value `v` is applied normally, except that it is interrupted after `num` seconds if it is still running. In this case the outcome is a failure.

  Warning

  For the moment, timeout is based on elapsed time in seconds, which is very machine-dependent: a script that works on a quick machine may fail on a slow one. The converse is even possible if you combine a timeout with some other tacticals. This tactical is hence proposed only for convenience during debugging or other development phases, we strongly advise you to not leave any timeout in final scripts. Note also that this tactical isn’t available on the native Windows port of Coq.

### Timing a tactic

A tactic execution can be timed:

`time string expr`

  evaluates `expr` and displays the running time of the tactic expression, whether it fails or succeeds. In case of several successes, the time for each successive run is displayed. Time is in seconds and is machine-dependent. The `string` argument is optional. When provided, it is used to identify this particular occurrence of time.

### Timing a tactic that evaluates to a term

Tactic expressions that produce terms can be timed with the experimental tactic

`time_constr expr`

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

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

  `restart_timer string`

  and

  `finish_timing string? string`

  which (re)set and display an optionally named timer, respectively. The parenthesized string argument to `finish_timing` is also optional, and determines the label associated with the timer for printing.

  By copying the definition of `time_constr` from the standard library, users can achive support for a fixed pattern of nesting by passing 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. secs (0.u,0.s) 1 subgoal n := 100 : nat ============================ True
  Abort.

### Local definitions

Local definitions can be done as follows:

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

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

  Local definitions can be made recursive by using `let rec` instead of `let`. In this latter case, the definitions are evaluated lazily so that the rec keyword can be used also in non-recursive cases so as to avoid the eager evaluation of local definitions.

### Application

An application is an expression of the following form:

`qualid tacarg+`

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

### Function construction

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

`fun ident+ => expr`

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

### Pattern matching on terms

We can carry out pattern matching on terms with:

`match expr with cpatterni => expri+| end`

  The expression `expr` is evaluated and should yield a term which is matched against `cpattern1`. The matching is non-linear: if a metavariable occurs more than once, it should match the same expression every time. It is first-order except on the variables of the form `@?id` that occur in head position of an application. For these variables, the matching is second-order and returns a functional term.

  Alternatively, when a metavariable of the form `?id` occurs under binders, say `x1, …, xn` and the expression matches, the metavariable is instantiated by a term which can then be used in any context which also binds the variables `x1, …, xn` with same types. This provides with a primitive form of matching under context which does not require manipulating a functional term.

  If the matching with `cpattern1` succeeds, then `expr1` is evaluated into some value by substituting the pattern matching instantiations to the metavariables. If `expr1` evaluates to a tactic and the match expression is in position to be applied to a goal (e.g. it is not bound to a variable by a `let in`), then this tactic is applied. If the tactic succeeds, the list of resulting subgoals is the result of the match expression. If `expr1` does not evaluate to a tactic or if the match expression is not in position to be applied to a goal, then the result of the evaluation of `expr1` is the result of the match expression.

  If the matching with `cpattern1` fails, or if it succeeds but the evaluation of `expr1` fails, or if the evaluation of `expr1` succeeds but returns a tactic in execution position whose execution fails, then `cpattern2` is used and so on. The pattern `_` matches any term and shadows all remaining patterns if any. If all clauses fail (in particular, there is no pattern `_`) then a no-matching-clause error is raised.

  Failures in subsequent tactics do not cause backtracking to select new branches or inside the right-hand side of the selected branch even if it has backtracking points.

  Error `No matching clauses for match.`

    No pattern can be used and, in particular, there is no `_` pattern.

  Error `Argument of match does not evaluate to a term.`

    This happens when `expr` does not denote a term.

  Variant `multimatch expr with cpatterni => expri+| end`

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

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

  Variant `lazymatch expr with cpatterni => expri+| end`

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

  Variant `context ident [cpattern]`

    This special form of patterns matches any term with a subterm matching cpattern. If there is a match, the optional `ident` is assigned the "matched context", i.e. the initial term where the matched subterm is replaced by a hole. The example below will show how to use such term contexts.

    If the evaluation of the right-hand-side of a valid match fails, the next matching subterm is tried. If no further subterm matches, the next clause is tried. Matching subterms are considered top-bottom and from left to right (with respect to the raw printing obtained by setting option `Printing All`).

  Example

  Ltac f x := match x with context f [S ?X] => idtac X; assert (p := eq_refl 1 : X=1); let x:= context f[O] in assert (x=O) end.
    f is defined
  Goal True.
    1 subgoal ============================ True
  f (3+4).
    2 1 2 subgoals p : 1 = 1 ============================ 1 + 4 = 0 subgoal 2 is: True

### Pattern matching on goals

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

`match goal with hyp+ |- cpattern => expr+| | _ => expr end`

  If each hypothesis pattern `hyp`1,i, with i = 1, ..., m1 is matched (non-linear first-order unification) by a hypothesis of the goal and if `cpattern_1` is matched by the conclusion of the goal, then `expr1` is evaluated to `v1` by substituting the pattern matching to the metavariables and the real hypothesis names bound to the possible hypothesis names occurring in the hypothesis patterns. If `v1` is a tactic value, then it is applied to the goal. If this application fails, then another combination of hypotheses is tried with the same proof context pattern. If there is no other combination of hypotheses then the second proof context pattern is tried and so on. If the next to last proof context pattern fails then the last `expr` is evaluated to `v` and `v` is applied. Note also that matching against subterms (using the `context ident [ cpattern ]`) is available and is also subject to yielding several matchings.

  Failures in subsequent tactics do not cause backtracking to select new branches or combinations of hypotheses, or inside the right-hand side of the selected branch even if it has backtracking points.

  Error `No matching clauses for match goal.`

    No clause succeeds, i.e. all matching patterns, if any, fail at the application of the right-hand-side.

  Note

  It is important to know that each hypothesis of the goal can be matched by at most one hypothesis pattern. The order of matching is the following: hypothesis patterns are examined from right to left (i.e. hypi,mi` before hypi,1). For each hypothesis pattern, the goal hypotheses are matched in order (newest first), but it possible to reverse this order (oldest first) with the `match reverse goal with` variant.

  Variant `multimatch goal with hyp+ |- cpattern => expr+| | _ => expr end`

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

    The syntax `match [reverse] goal …` is, in fact, a shorthand for `once multimatch [reverse] goal …`.

  Variant `lazymatch goal with hyp+ |- cpattern => expr+| | _ => expr end`

    Using lazymatch instead of match will perform the same pattern matching procedure but will commit to the first matching branch with the first matching combination of hypotheses rather than trying a new matching if the right-hand side fails. If the right-hand side of the selected branch is a tactic with backtracking points, then subsequent failures cause this tactic to backtrack.

### Filling a term context

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

`context ident [expr]`

  `ident` must denote a context variable bound by a context pattern of a match expression. This expression evaluates replaces the hole of the value of `ident` by the value of `expr`.

  Error `Not a context variable.`
  Error `Unbound context identifier ident.`

### Generating fresh hypothesis names

Tactics sometimes have to generate new names for hypothesis. Letting the system decide a name with the intro tactic is not so good since it is very awkward to retrieve the name the system gave. The following expression returns an identifier:

`fresh component*`

  It evaluates to an identifier unbound in the goal. This fresh identifier is obtained by concatenating the value of the `component`s (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

The following returns the type of term:

`type of term`

### Manipulating untyped terms

`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 `uconstr : term` syntax.

`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 `uconstr :` can also be used as arguments to the `refine` tactic. In that case the untyped term is type checked against the conclusion of the goal, and the holes which are not solved by the typing procedure are turned into new subgoals.

### Counting the goals

`numgoals`

  The number of goals under focus can be recovered using the `numgoals` function. Combined with the guard command below, it can be used to branch over the number of goals produced by previous tactics.

  Example

  Ltac pr_numgoals := let n := numgoals in idtac "There are" n "goals".
    pr_numgoals is defined
  Goal True /\ True /\ True.
    pr_numgoals is defined 1 subgoal ============================ True /\ True /\ True
  split;[|split].
    3 subgoals ============================ True subgoal 2 is: True subgoal 3 is: True
  all:pr_numgoals.
    There are 3 goals

### Testing boolean expressions

`guard test`

  The `guard` tactic tests a boolean expression, and fails if the expression evaluates to false. If the expression evaluates to true, it succeeds without affecting the proof.

  The accepted tests are simple integer comparisons.

  Example

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

### Proving a subgoal as a separate lemma

`abstract expr`

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

  This tactical is useful with tactics such as `omega` or `discriminate` that generate huge proof terms. With that tool the user can avoid the explosion at time of the Save command without having to cut manually the proof in smaller lemmas.

  It may be useful to generate lemmas minimal w.r.t. the assumptions they depend on. This can be obtained thanks to the option below.

  Variant `abstract expr using ident`

    Give explicitly the name of the auxiliary lemma.

    Warning

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

  Variant `transparent_abstract expr`

    Save the subproof in a transparent lemma rather than an opaque one.

    Warning

    Use this feature at your own risk; building computationally relevant terms with tactics is fragile.

  Variant `transparent_abstract expr using ident`

    Give explicitly the name of the auxiliary transparent lemma.

    Warning

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

  Error `Proof is not complete.`

## Tactic toplevel definitions

### Defining Ltac functions

Basically, Ltac toplevel definitions are made as follows:

Command `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:

  `Ltac ident := fun ident+ => expr`

  Recursive and mutual recursive function definitions are also possible with the syntax:

  Variant `Ltac ident ident* with ident ident** := expr`

    It is also possible to redefine an existing user-defined tactic using the syntax:

  Variant `Ltac qualid ident* ::= expr`

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

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

### Printing Ltac tactics

Command `Print Ltac qualid`

  Defined Ltac functions can be displayed using this command.

Command `Print Ltac Signatures`

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

## Debugging Ltac tactics

### Info trace

Command `Info num expr`

  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 `Info` command can be used with the following syntax.

  The number `num` is the unfolding level of tactics in the trace. At level 0, the trace contains a sequence of tactics in the actual script, at level 1, the trace will be the concatenation of the traces of these tactics, etc…

  Example

  Ltac t x := exists x; reflexivity.
    t is defined
  Goal exists n, n=0.
    1 subgoal ============================ exists n : nat, n = 0
  Info 0 t 1||t 0.
    t <constr:(0)> No more subgoals.
  Undo.
    1 subgoal ============================ exists n : nat, n = 0
  Info 1 t 1||t 0.
    exists with 0;reflexivity No more subgoals.

  The trace produced by `Info` tries its best to be a reparsable Ltac script, but this goal is not achievable in all generality. So some of the output traces will contain oddities.

  As an additional help for debugging, the trace produced by `Info` contains (in comments) the messages produced by the `idtac` tactical at the right position in the script. In particular, the calls to idtac in branches which failed are not printed.

  Option `Info Level num`

    This option is an alternative to the `Info` command.

    This will automatically print the same trace as `Info num` at each tactic call. The unfolding level can be overridden by a call to the `Info` command.

### Interactive debugger

Flag `Ltac Debug`

  This option governs the step-by-step debugger that comes with the Ltac interpreter

When the debugger is activated, it stops at every step of the evaluation of the current Ltac expression and prints information on what it is doing. The debugger stops, prompting for a command which can be one of the following:

simple newline: | go to the next step
h: | get help
x: | exit current evaluation
s: | continue current evaluation without stopping
r n: | advance n steps further
r string: | advance up to the next call to “idtac string”

Error `Debug mode not available in the IDE`

A non-interactive mode for the debugger is available via the option:

Flag `Ltac Batch Debug`

  This option has the effect of presenting a newline at every prompt, when the debugger is on. The debug log thus created, which does not require user input to generate when this option is set, can then be run through external tools such as diff.

### Profiling 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 on its calling context. Thus it allows to locate the part of a tactic definition that contains the performance issue.

Flag `Ltac Profiling`

  This option enables and disables the profiler.

Command `Show Ltac Profile`

  Prints the profile

  Variant `Show Ltac Profile string`

    Prints a profile for all tactics that start with `string`. Append a period (.) to the string if you only want exactly that name.

Command `Reset Ltac Profile`

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

  Warning

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

Require Import Coq.omega.Omega.
  [Loading ML file z_syntax_plugin.cmxs ... done] [Loading ML file quote_plugin.cmxs ... done] [Loading ML file newring_plugin.cmxs ... done] [Loading ML file omega_plugin.cmxs ... done]
Ltac mytauto := tauto.
  mytauto is defined
Ltac tac := intros; repeat split; omega || mytauto.
  tac is defined
Notation max x y := (x + (y - x)) (only parsing).
Goal forall x y z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z, max x (max y z) = max (max x y) z /\ max x (max y z) = max (max x y) z /\ (A /\ B /\ C /\ D /\ E /\ F /\ G /\ H /\ I /\ J /\ K /\ L /\ M /\ N /\ O /\ P /\ Q /\ R /\ S /\ T /\ U /\ V /\ W /\ X /\ Y /\ Z -> Z /\ Y /\ X /\ W /\ V /\ U /\ T /\ S /\ R /\ Q /\ P /\ O /\ N /\ M /\ L /\ K /\ J /\ I /\ H /\ G /\ F /\ E /\ D /\ C /\ B /\ A).
  1 subgoal ============================ forall (x y z : nat) (A B C D E F G H I J K L M N O P Q R S T U V W X Y Z : Prop), x + (y + (z - y) - x) = x + (y - x) + (z - (x + (y - x))) /\ x + (y + (z - y) - x) = x + (y - x) + (z - (x + (y - x))) /\ (A /\ B /\ C /\ D /\ E /\ F /\ G /\ H /\ I /\ J /\ K /\ L /\ M /\ N /\ O /\ P /\ Q /\ R /\ S /\ T /\ U /\ V /\ W /\ X /\ Y /\ Z -> Z /\ Y /\ X /\ W /\ V /\ U /\ T /\ S /\ R /\ Q /\ P /\ O /\ N /\ M /\ L /\ K /\ J /\ I /\ H /\ G /\ F /\ E /\ D /\ C /\ B /\ A)
Proof.

Set Ltac Profiling.
tac.
  No more subgoals.
Show Ltac Profile.
  total time: 1.563s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac ----------------------------------- 0.1% 100.0% 1 1.563s ─<Coq.Init.Tauto.with_uniform_flags> --- 0.0% 72.8% 26 0.057s ─<Coq.Init.Tauto.tauto_gen> ------------ 0.0% 72.7% 26 0.057s ─<Coq.Init.Tauto.tauto_intuitionistic> - 0.0% 72.7% 26 0.057s ─t_tauto_intuit ------------------------ 0.1% 72.6% 26 0.057s ─<Coq.Init.Tauto.simplif> -------------- 49.1% 70.1% 26 0.055s ─omega --------------------------------- 26.7% 26.7% 28 0.189s ─<Coq.Init.Tauto.is_conj> -------------- 13.0% 13.0% 28756 0.017s ─elim id ------------------------------- 4.1% 4.1% 650 0.007s ─<Coq.Init.Tauto.axioms> --------------- 1.8% 2.4% 0 0.002s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac ----------------------------------- 0.1% 100.0% 1 1.563s ├─<Coq.Init.Tauto.with_uniform_flags> - 0.0% 72.8% 26 0.057s │└<Coq.Init.Tauto.tauto_gen> ---------- 0.0% 72.7% 26 0.057s │└<Coq.Init.Tauto.tauto_intuitionistic> 0.0% 72.7% 26 0.057s │└t_tauto_intuit ---------------------- 0.1% 72.6% 26 0.057s │ ├─<Coq.Init.Tauto.simplif> ---------- 49.1% 70.1% 26 0.055s │ │ ├─<Coq.Init.Tauto.is_conj> -------- 13.0% 13.0% 28756 0.017s │ │ └─elim id ------------------------- 4.1% 4.1% 650 0.007s │ └─<Coq.Init.Tauto.axioms> ----------- 1.8% 2.4% 0 0.002s └─omega ------------------------------- 26.7% 26.7% 28 0.189s
Show Ltac Profile "omega".
  total time: 1.563s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─omega --------------------------------- 26.7% 26.7% 28 0.189s tactic local total calls max

Abort.
Unset Ltac Profiling.

`start ltac profiling`

  This tactic behaves like `idtac` but enables the profiler.

`stop ltac profiling`

  Similarly to `start ltac profiling`, this tactic behaves like `idtac`. Together, they allow you to exclude parts of a proof script from profiling.

`reset ltac profile`

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

`show ltac profile`

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

`show ltac profile string`

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

You can also pass the `-profile-ltac` command line option to `coqc`, which turns the `Ltac Profiling` option on at the beginning of each document, and performs a `Show Ltac Profile` at the end.

Warning

Note that the profiler currently does not handle backtracking into multi-success tactics, and issues a warning to this effect in many cases when such backtracking occurs.

### Run-time optimization tactic

`optimize_heap`

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