Ltac — Coq 8.14.1 documentation

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# Ltac

This chapter documents the tactic language `L`tac.

We start by giving the syntax followed by the informal semantics. To learn more about the language and especially about its foundations, please refer to [Del00]. (Note the examples in the paper won't work as-is; Coq has evolved since the paper was written.)

Example: Basic tactic macros

Here are some examples of simple tactic macros you can create with `L`tac:

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

See Section Examples of using Ltac for more advanced examples.

## Syntax

The syntax of the tactic language is given below.

The main entry of the grammar is `ltac_expr`, which is used in proof mode as well as to define new tactics with the `Ltac` command.

The grammar uses multiple `ltac_expr*` nonterminals to express how subexpressions are grouped when they're not fully parenthesized. For example, in many programming languages, `a*b+c` is interpreted as `(a*b)+c` because `*` has higher precedence than `+`. Usually `a/b/c` is given the left associative interpretation `(a/b)/c` rather than the right associative interpretation `a/(b/c)`.

In Coq, the expression `try repeat tactic1 || tactic2; tactic3; tactic4` is interpreted as `(try (repeat (tactic1 || tactic2)); tactic3); tactic4` because `||` is part of `ltac_expr2`, which has higher precedence than `try` and `repeat` (at the level of `ltac_expr3`), which in turn have higher precedence than `;`, which is part of `ltac_expr4`. (A lower number in the nonterminal name means higher precedence in this grammar.)

The constructs in `ltac_expr` are left associative.

ltac_expr`::=`ltac_expr4​binder_tactic`` ltac_expr4`::=`ltac_expr3 ; ltac_expr3​binder_tactic`````|`ltac_expr3 ; [ for_each_goal ]`````|`ltac_expr3`` ltac_expr3`::=`l3_tactic`````|`ltac_expr2`` ltac_expr2`::=`ltac_expr1 + ltac_expr2​binder_tactic`````|`ltac_expr1 || ltac_expr2​binder_tactic`````|`l2_tactic`````|`ltac_expr1`` ltac_expr1`::=`tactic_value`````|`qualid tactic_arg+`````|`l1_tactic`````|`ltac_expr0`` tactic_value`::=`value_tactic​syn_value`` tactic_arg`::=`tactic_value`````|`term`````|`()``ltac_expr0`::=`( ltac_expr )`````|`[> for_each_goal ]`````|`tactic_atom`` tactic_atom`::=`integer`````|`qualid`````|`()``

Note

Tactics described in other chapters of the documentation are `simple_tactic`s, which only modify the proof state. `L`tac provides additional constructs that can generally be used wherever a `simple_tactic` can appear, even though they don't modify the proof state and that syntactically they're at varying levels in `ltac_expr`. For simplicity of presentation, the `L`tac constructs are documented as tactics. Tactics are grouped as follows:

- `binder_tactic`s are: `fun` and `let`

- `l3_tactic`s include `L`tac tactics: `try`, `do`, `repeat`, `timeout`, `time`, `progress`, `once`, `exactly_once`, `only` and `abstract`

- `l2_tactic`s are: `tryif`

- `l1_tactic`s are the `simple_tactic`s, `first`, `solve`, `idtac`, `fail` and `gfail` as well as `match`, `match goal` and their `lazymatch` and `multimatch` variants.

- `value_tactic`s, which return values rather than change the proof state. They are: `eval`, `context`, `numgoals`, `fresh`, `type of` and `type_term`.

The documentation for these `L`tac constructs mentions which group they belong to.

The difference is only relevant in some compound tactics where extra parentheses may be needed. For example, parenthesees are required in `idtac + (once idtac)` because `once` is an `l3_tactic`, which the production `ltac_expr2 ::= ltac_expr1 + ltac_expr2​binder_tactic` doesn't accept after the `+`.

Note

- The grammar reserves the token `||`.

## Semantics

### Types of values

An `L`tac value can be a tactic, integer, string, unit (written as "`()`" ) or syntactic value. Syntactic values correspond to certain nonterminal symbols in the grammar, each of which is a distinct type of value. Most commonly, the value of an `L`tac expression is a tactic that can be executed.

While there are a number of constructs that let you combine multiple tactics into compound tactics, there are no operations for combining most other types of values. For example, there's no function to add two integers. Syntactic values are entered with the `syn_value` construct. Values of all types can be assigned to toplevel symbols with the `Ltac` command or to local symbols with the `let` tactic. `L`tac `functions` can return values of any type.

### Syntactic values

syn_value`::=`ident : ( nonterminal )``

Provides a way to use the syntax and semantics of a grammar nonterminal as a value in an `ltac_expr`. The table below describes the most useful of these. You can see the others by running "`Print Grammar` `tactic`" and examining the part at the end under "Entry tactic:tactic_value".

> `ident`

>   name of a grammar nonterminal listed in the table

> `nonterminal`

>   represents syntax described by `nonterminal`.

> Specified `ident` | Parsed as | Interpreted as | as in tactic
> --- | --- | --- | ---
> `ident` | `ident` | a user-specified name | `intro`
> `string` | `string` | a string |
> `integer` | `integer` | an integer |
> `reference` | `qualid` | a qualified identifier |
> `uconstr` | `term` | an untyped term | `refine`
> `constr` | `term` | a term | `exact`
> `ltac` | `ltac_expr` | a tactic |

`ltac:(ltac_expr)` can be used to indicate that the parenthesized item should be interpreted as a tactic and not as a term. The constructs can also be used to pass parameters to tactics written in OCaml. (While all of the `syn_value`s can appear at the beginning of an `ltac_expr`, the others are not useful because they will not evaluate to tactics.)

`uconstr:(term)` can be used to build untyped terms. Terms built in `L`tac are well-typed by default. Building large terms in recursive `L`tac functions may give very slow behavior because terms must be fully type checked at each step. In this case, using an untyped term may avoid most of the repetitive type checking for the term, improving performance.

Untyped terms built using `uconstr:(…)` can be used as arguments to the `refine` tactic, for example. 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.

### Tactics in terms

term_ltac`::=`ltac : ( ltac_expr )``

Allows including an `ltac_expr` within a term. Semantically, it's the same as the `syn_value` for `ltac`, but these are distinct in the grammar.

### Substitution

`name`s within `L`tac expressions are used to represent both terms and `L`tac variables. If the `name` corresponds to an `L`tac variable or tactic name, `L`tac substitutes the value before applying the expression. Generally it's best to choose distinctive names for `L`tac variables that won't clash with term names. You can use `ltac:(name)` or `(name)` to control whether a `name` is interpreted as, respectively, an `L`tac variable or a term.

Note that values from toplevel symbols, unlike locally-defined symbols, are substituted only when they appear at the beginning of an `ltac_expr` or as a `tactic_arg`. Local symbols are also substituted into tactics:

Example: Substitution of global and local symbols

Goal True.
  1 goal ============================ True

Ltac n := 1.
  n is defined
let n2 := n in idtac n2.
  1
Fail idtac n.
  The command has indeed failed with message: n not found.

### Sequence: ;

A sequence is an expression of the following form:

Tactic ltac_expr31 ; ltac_expr32​binder_tactic

  The expression `ltac_expr31` is evaluated to `v1`, which must be a tactic value. The tactic `v1` is applied to the current goals, possibly producing more goals. Then the right-hand side 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.

  Note

  - If you want `tactic2; tactic3` to be fully applied to the first subgoal generated by `tactic1` before applying it to the other subgoals, then you should write:

    - `tactic1; [> tactic2; tactic3 .. ]` rather than

    - `tactic1; (tactic2; tactic3)`.

### Local application of tactics: [> ... ]

Tactic [> for_each_goal ]
  for_each_goal`::=`goal_tactics`````|`goal_tactics |? ltac_expr? .. | goal_tactics?``goal_tactics`::=`ltac_expr?*|``

  Applies a different `ltac_expr?` to each of the focused goals. In the first form of `for_each_goal` (without `..`), the construct fails if the number of specified `ltac_expr?` is not the same as the number of focused goals. Omitting an `ltac_expr` leaves the corresponding goal unchanged.

  In the second form (with `ltac_expr? ..`), the left and right `goal_tactics` are applied respectively to a prefix or suffix of the list of focused goals. The `ltac_expr?` before the `..` is applied to any focused goals in the middle (possibly none) that are not covered by the `goal_tactics`. The number of `ltac_expr?` in the `goal_tactics` must be no more than the number of focused goals.

  In particular:

  `goal_tactics | .. | goal_tactics`

    The goals not covered by the two `goal_tactics` are left unchanged.

  `[> ltac_expr .. ]`

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

  Note that `ltac_expr3 ; [ ltac_expr*| ]` is a convenient idiom to process the goals generated by applying `ltac_expr3`.

Tactic ltac_expr3 ; [ for_each_goal ]

  `ltac_expr3 ; [ ... ]` is equivalent to `[> ltac_expr3 ; [> ... ] .. ]`.

### Goal selectors

By default, tactic expressions are applied only to the first goal. Goal selectors provide a way to apply a tactic expression to another goal or multiple goals. (The `Default Goal Selector` option can be used to change the default behavior.)

Tactic toplevel_selector : ltac_expr
  toplevel_selector`::=`selector`````|`all`````|`!`````|`par``

  Reorders the goals and applies `ltac_expr` to the selected goals. It can only be used at the top level of a tactic expression; it cannot be used within a tactic expression. The selected goals are reordered so they appear after the lowest-numbered selected goal, ordered by goal number. Example. If the selector applies to a single goal or to all goals, the reordering will not be apparent. The order of the goals in the `selector` is irrelevant. (This may not be what you expect; see #8481.)

  `all`

    Selects all focused goals.

  `!`

    If exactly one goal is in focus, apply `ltac_expr` to it. Otherwise the tactic fails.

  `par`

    Applies `ltac_expr` to all focused goals in parallel. The number of workers can be controlled via the command line option `-async-proofs-tac-j natural` to specify the desired number of workers. Limitations: `par:` only works on goals that don't contain existential variables. `ltac_expr` must either solve the goal completely or do nothing (i.e. it cannot make some progress).

Selectors can also be used nested within a tactic expression with the `only` tactic:

Tactic only selector : ltac_expr3
  selector`::=`range_selector+,`````|`[ ident ]``range_selector`::=`natural - natural`````|`natural``

  Applies `ltac_expr3` to the selected goals.

  `only` is an `l3_tactic`.

  `range_selector+,`

    The selected goals are the union of the specified `range_selector`s.

  `[ ident ]`

    Limits the application of `ltac_expr3` to the goal previously named `ident` by the user (see Existential variables).

  `natural1 - natural2`

    Selects the goals `natural1` through `natural2`, inclusive.

  `natural`

    Selects a single goal.

Error No such goal.

Example: Selector reordering goals

Goal 1=0 /\ 2=0 /\ 3=0.
  1 goal ============================ 1 = 0 /\ 2 = 0 /\ 3 = 0

repeat split.
  3 goals ============================ 1 = 0 goal 2 is: 2 = 0 goal 3 is: 3 = 0
1,3: idtac.
  3 goals ============================ 1 = 0 goal 2 is: 3 = 0 goal 3 is: 2 = 0

### Processing multiple goals

When presented with multiple focused goals, most `L`tac constructs process each goal separately. They succeed only if there is a success for each goal. For example:

Example: Multiple focused goals

This tactic fails because there no match for the second goal (`False`).

Goal True /\ False.
  1 goal ============================ True /\ False

split.
  2 goals ============================ True goal 2 is: False

Fail all: let n := numgoals in idtac "numgoals =" n; match goal with | |- True => idtac end.
  numgoals = 2 The command has indeed failed with message: No matching clauses for match.

### Do loop

Tactic do nat_or_var ltac_expr3

  The do loop repeats a tactic `nat_or_var` times:

  `ltac_expr` is evaluated to `v`, which must be a tactic value. This tactic value `v` is applied `nat_or_var` times. If `nat_or_var` > 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 `nat_or_var` applications have been completed.

  `do` is an `l3_tactic`.

### Repeat loop

Tactic repeat ltac_expr3

  The repeat loop repeats a tactic until it fails.

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

  `repeat` is an `l3_tactic`.

### Catching errors: try

We can catch the tactic errors with:

Tactic try ltac_expr3

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

  `try` is an `l3_tactic`.

### Detecting progress

We can check if a tactic made progress with:

Tactic progress ltac_expr3

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

  `progress` is an `l3_tactic`.

  Error Failed to progress.

### Branching and backtracking

`L`tac provides several branching tactics that permit trying multiple alternative tactics for a proof step. For example, `first`, which tries several alternatives and selects the first that succeeds, or `tryif`, which tests whether a given tactic would succeed or fail if it was applied and then, depending on the result, applies one of two alternative tactics. There are also looping constructs `do` and `repeat`. The order in which the subparts of these tactics are evaluated is generally similar to structured programming constructs in many languages.

The `+`, `multimatch` and `multimatch goal` tactics provide more complex capability. Rather than applying a single successful tactic, these tactics generate a series of successful tactic alternatives that are tried sequentially when subsequent tactics outside these constructs fail. For example:

> Example: Backtracking

> Fail multimatch True with | True => idtac "branch 1" | _ => idtac "branch 2" end ; idtac "branch A"; fail.
>   branch 1 branch A branch 2 branch A The command has indeed failed with message: Tactic failure.

These constructs are evaluated using backtracking. Each creates a backtracking point. When a subsequent tactic fails, evaluation continues from the nearest prior backtracking point with the next successful alternative and repeats the tactics after the backtracking point. When a backtracking point has no more successful alternatives, evaluation continues from the next prior backtracking point. If there are no more prior backtracking points, the overall tactic fails.

Thus, backtracking tactics can have multiple successes. Non-backtracking constructs that appear after a backtracking point are reprocessed after backtracking, as in the example above, in which the `;` construct is reprocessed after backtracking. When a backtracking construct is within a non-backtracking construct, the latter uses the first success. Backtracking to a point within a non-backtracking construct won't change the branch that was selected by the non-backtracking construct.

The `once` tactic stops further backtracking to backtracking points within that tactic.

### Branching with backtracking: +

We can branch with backtracking with the following structure:

Tactic ltac_expr1 + ltac_expr2​binder_tactic

  Evaluates and applies `ltac_expr1` to each focused goal independently. If it fails (i.e. there is no initial success), then evaluates and applies the right-hand side. If the right-hand side fails, the construct fails.

  If `ltac_expr1` has an initial success and a subsequent tactic (outside the `+` construct) fails, `L`tac backtracks and selects the next success for `ltac_expr1`. If there are no more successes, then `+` similarly evaluates and applies (and backtracks in) the right-hand side. To prevent evaluation of further alternatives after an initial success for a tactic, use `first` instead.

  `+` is left-associative.

  In all cases, `(ltac_expr1 + ltac_expr2); ltac_expr3` is equivalent to `(ltac_expr1; ltac_expr3) + (ltac_expr2; ltac_expr3)`.

  Additionally, in most cases, `(ltac_expr1 + ltac_expr2) + ltac_expr3` is equivalent to `ltac_expr1 + (ltac_expr2 + ltac_expr3)`. Here's an example where the behavior differs slightly:

  > Goal True.
  >   1 goal ============================ True
  > Fail (fail 2 + idtac) + idtac.
  >   The command has indeed failed with message: Tactic failure.
  > Fail fail 2 + (idtac + idtac).
  >   The command has indeed failed with message: Tactic failure (level 1).

  Example: Backtracking branching with +

  In the first tactic, `idtac "2"` is not executed. In the second, the subsequent `fail` causes backtracking and the execution of `idtac "B"`.

  Goal True.
    1 goal ============================ True
  idtac "1" + idtac "2".
    1
  assert_fails ((idtac "A" + idtac "B"); fail).
    A B

### First tactic to succeed

In some cases backtracking may be too expensive.

Tactic first [ ltac_expr*| ]

  For each focused goal, independently apply the first `ltac_expr` that succeeds. The `ltac_expr`s must evaluate to tactic values. Failures in tactics after the `first` won't cause backtracking. (To allow backtracking, use the `+` construct above instead.)

  If the `first` contains a tactic that can backtrack, "success" means the first success of that tactic. Consider the following:

  Example: Backtracking inside a non-backtracking construct

  Goal True.
    1 goal ============================ True

  The `fail` doesn't trigger the second `idtac`:

  assert_fails (first [ idtac "1" | idtac "2" ]; fail).
    1

  This backtracks within `(idtac "1A" + idtac "1B" + fail)` but `first` won't consider the `idtac "2"` alternative:

  assert_fails (first [ (idtac "1A" + idtac "1B" + fail) | idtac "2" ]; fail).
    1A 1B

  `first` is an `l1_tactic`.

  Error No applicable tactic.
  Variant first ltac_expr

    This is an `L`tac alias that gives a primitive access to the first tactical as an `L`tac definition without going through a parsing rule. It expects to be given a list of tactics through a `Tactic Notation` command, permitting notations with the following form to be written:

    Example

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

### Solving

Selects and applies the first tactic that solves each goal (i.e. leaves no subgoal) in a series of alternative tactics:

Tactic solve [ ltac_expri*| ]

  For each current subgoal: evaluates and applies each `ltac_expr` in order until one is found that solves the subgoal.

  If any of the subgoals are not solved, then the overall `solve` fails.

  Note

  In `solve` and `first`, `ltac_expr`s that don't evaluate to tactic values are ignored. So `solve` `[ () | 1 |` `constructor` `]` is equivalent to `solve` `[` `constructor` `]`. This may make it harder to debug scripts that inadvertently include non-tactic values.

  `solve` is an `l1_tactic`.

  Variant solve ltac_expr

    This is an `L`tac alias that gives a primitive access to the `solve` tactic. See the `first` tactic for more information.

### First tactic to make progress: ||

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

Tactic ltac_expr1 || ltac_expr2​binder_tactic

  `ltac_expr1 || ltac_expr2` is equivalent to `first [ progress ltac_expr1 | ltac_expr2 ]`, except that if it fails, it fails like `ltac_expr2. `||` is left-associative.

  `ltac_expr`s that don't evaluate to tactic values are ignored. See the note at `solve`.

### Conditional branching: tryif

Tactic tryif ltac_exprtest then ltac_exprthen else ltac_expr2else

  For each focused goal, independently: Evaluate and apply `ltac_exprtest`. If `ltac_exprtest` succeeds at least once, evaluate and apply `ltac_exprthen` to all the subgoals generated by `ltac_exprtest`. Otherwise, evaluate and apply `ltac_expr2else` to all the subgoals generated by `ltac_exprtest`.

  `tryif` is an `l2_tactic`.

### Soft cut: once

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

Tactic once ltac_expr3

  `ltac_expr3` is evaluated to `v` which must be a tactic value. The tactic value `v` is applied but only its first success is used. If `v` fails, `once` `ltac_expr3` fails like `v`. If `v` has at least one success, `once` `ltac_expr3` succeeds once, but cannot produce more successes.

  `once` is an `l3_tactic`.

### Checking for a single success: exactly_once

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

Tactic exactly_once ltac_expr3

  `ltac_expr3` is evaluated to `v` which must be a tactic value. The tactic value `v` is applied if it has at most one success. If `v` fails, `exactly_once` `ltac_expr3` fails like `v`. If `v` has a exactly one success, `exactly_once` `ltac_expr3` succeeds like `v`. If `v` has two or more successes, `exactly_once` `ltac_expr3` fails.

  `exactly_once` is an `l3_tactic`.

  Warning

  The experimental status of this tactic pertains to the fact if `v` has 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 for failure: assert_fails

Coq defines an `L`tac tactic in `Init.Tactics` to check that a tactic fails:

Tactic assert_fails ltac_expr3

  If `ltac_expr3` fails, the proof state is unchanged and no message is printed. If `ltac_expr3` unexpectedly has at least one success, the tactic performs a `gfail` `0`, printing the following message:

  Error Tactic failure: <tactic closure> succeeds.

  Note

  `assert_fails` and `assert_succeeds` work as described when `ltac_expr3` is a `simple_tactic`. In some more complex expressions, they may report an error from within `ltac_expr3` when they shouldn't. This is due to the order in which parts of the `ltac_expr3` are evaluated and executed. For example:

  Goal True.
    1 goal ============================ True
  assert_fails match True with _ => fail end.
    Toplevel input, characters 0-43: > assert_fails match True with _ => fail end. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Error: Tactic failure.

  should not show any message. The issue is that `assert_fails` is an `L`tac-defined tactic. That makes it a function that's processed in the evaluation phase, causing the `match` to find its first success earlier. One workaround is to prefix `ltac_expr3` with "`idtac;`".

  assert_fails (idtac; match True with _ => fail end).

  Alternatively, substituting the `match` into the definition of `assert_fails` works as expected:

  tryif (once match True with _ => fail end) then gfail 0 (* tac *) "succeeds" else idtac.

### Checking for success: assert_succeeds

Coq defines an `L`tac tactic in `Init.Tactics` to check that a tactic has at least one success:

Tactic assert_succeeds ltac_expr3

  If `ltac_expr3` has at least one success, the proof state is unchanged and no message is printed. If `ltac_expr3` fails, the tactic performs a `gfail` `0`, printing the following message:

  Error Tactic failure: <tactic closure> fails.

### Print/identity tactic: idtac

Tactic idtac ident​string​natural*

  Leaves the proof unchanged and prints the given tokens. `String`s and `natural`s are printed literally. If `ident` is an `L`tac variable, its contents are printed; if not, it is an error.

  `idtac` is an `l1_tactic`.

### Failing

Tactic fail​gfail nat_or_var? ident​string​natural*

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

  `gfail` 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.

  `fail` and `gfail` are `l1_tactic`s.

  See the example for a comparison of the two constructs.

  Note that 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.

  `nat_or_var`

    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` `natural` is not enclosed in a `+` construct, respecting the algebraic identity.

  `ident​string​natural*`

    The given tokens are used for printing the failure message. If `ident` is an `L`tac variable, its contents are printed; if not, it is an error.

  Error Tactic failure.
  Error Tactic failure (level natural).
  Error No such goal.

  Example

  Goal True.
    1 goal ============================ True
  Proof. fail. Abort.
    Toplevel input, characters 7-12: > Proof. fail. > ^^^^^ Error: Tactic failure.
  Goal True.
    1 goal ============================ True
  Proof. trivial; fail. Qed.
    No more goals.
  Goal True.
    1 goal ============================ True
  Proof. trivial. fail. Abort.
    No more goals. Toplevel input, characters 16-21: > Proof. trivial. fail. > ^^^^^ Error: No such goal.
  Goal True.
    1 goal ============================ True
  Proof. trivial. all: fail. Qed.
    No more goals.
  Goal True.
    1 goal ============================ True
  Proof. gfail. Abort.
    Toplevel input, characters 7-13: > Proof. gfail. > ^^^^^^ Error: Tactic failure.
  Goal True.
    1 goal ============================ True
  Proof. trivial; gfail. Abort.
    Toplevel input, characters 7-22: > Proof. trivial; gfail. > ^^^^^^^^^^^^^^^ Error: Tactic failure.
  Goal True.
    1 goal ============================ True
  Proof. trivial. gfail. Abort.
    No more goals. Toplevel input, characters 16-22: > Proof. trivial. gfail. > ^^^^^^ Error: No such goal.
  Goal True.
    1 goal ============================ True
  Proof. trivial. all: gfail. Abort.
    No more goals. Toplevel input, characters 16-27: > Proof. trivial. all: gfail. > ^^^^^^^^^^^ Error: Tactic failure.

### Timeout

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

Tactic timeout nat_or_var ltac_expr3

  `ltac_expr3` is evaluated to `v` which must be a tactic value. The tactic value `v` is applied normally, except that it is interrupted after `nat_or_var` seconds if it is still running. In this case the outcome is a failure.

  `timeout` is an `l3_tactic`.

  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:

Tactic time string? ltac_expr3

  evaluates `ltac_expr3` 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`.

  `time` is an `l3_tactic`.

### Timing a tactic that evaluates to a term: time_constr

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

Tactic time_constr ltac_expr

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

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

Tactic restart_timer string?

  Reset a timer

Tactic finish_timing ( string )? string?

  Display an optionally named timer. The parenthesized string argument is also optional, and determines the label associated with the timer for printing.

By copying the definition of `time_constr` from the standard library, users can achieve support for a fixed pattern of nesting by passing different `string` parameters to `restart_timer` and `finish_timing` at each level of nesting.

Example

Ltac time_constr1 tac := let eval_early := match goal with _ => restart_timer "(depth 1)" end in let ret := tac () in let eval_early := match goal with _ => finish_timing ( "Tactic evaluation" ) "(depth 1)" end in ret.
  time_constr1 is defined
Goal True.
  1 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.
  Tactic evaluation (depth 1) ran for 0. secs (0.u,0.s) Tactic evaluation (depth 1) ran for 0. secs (0.u,0.s) Tactic evaluation ran for 0.001 secs (0.u,0.s) 1 goal n := 100 : nat ============================ True

### Local definitions: let

Tactic let rec? let_clause with let_clause* in ltac_expr
  let_clause`::=`name := ltac_expr`````|`ident name+ := ltac_expr``

  Binds symbols within `ltac_expr`. `let` evaluates each `let_clause`, substitutes the bound variables into `ltac_expr` and then evaluates `ltac_expr`. There are no dependencies between the `let_clause`s.

  Use `let` `rec` to create recursive or mutually recursive bindings, which causes the definitions to be evaluated lazily.

  `let` is a `binder_tactic`.

### Function construction and application

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

Tactic fun name+ => ltac_expr

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

  `fun` is a `binder_tactic`.

Functions can return values of any type.

A function application is an expression of the form:

Tactic qualid tactic_arg+

  `qualid` must be bound to a `L`tac function with at least as many arguments as the provided `tactic_arg`s. The `tactic_arg`s are evaluated before the function is applied or partially applied.

  Functions may be defined with the `fun` and `let` tactics and with the `Ltac` command.

### Pattern matching on terms: match

Tactic match_key ltac_exprterm with |? match_pattern => ltac_expr+| end
  match_key`::=`lazymatch`````|`match`````|`multimatch`` match_pattern`::=`cpattern`````|`context ident? [ cpattern ]``cpattern`::=`term``

  `lazymatch`, `match` and `multimatch` are `ltac_expr1`s.

  Evaluates `ltac_exprterm`, which must yield a term, and matches it sequentially with the `match_pattern`s, which may have metavariables. When a match is found, metavariable values are substituted into `ltac_expr`, which is then applied.

  Matching may continue depending on whether `lazymatch`, `match` or `multimatch` is specified.

  In the `match_pattern`s, metavariables have the form `?ident`, whereas in the `ltac_expr`s, the question mark is omitted. Choose your metavariable names with care to avoid name conflicts. For example, if you use the metavariable `S`, then the `ltac_expr` can't use `S` to refer to the constructor of `nat` without qualifying the constructor as `Datatypes.S`.

  Matching is non-linear: if a metavariable occurs more than once, each occurrence must match the same expression. Expressions match if they are syntactically equal or are α-convertible. Matching is first-order except on variables of the form `@?ident` that occur in the head position of an application. For these variables, matching is second-order and returns a functional term.

  `lazymatch`

    Causes the match to commit to the first matching branch rather than trying a new match if `ltac_expr` fails. Example.

  `match`

    If `ltac_expr` fails, continue matching with the next branch. Failures in subsequent tactics (after the `match`) will not cause selection of a new branch. Examples here and here.

  `multimatch`

    If `ltac_expr` fails, continue matching with the next branch. When an `ltac_expr` succeeds for a branch, subsequent failures (after the `multimatch`) causing consumption of all the successes of `ltac_expr` trigger selection of a new matching branch. Example.

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

  `cpattern`

    The syntax of `cpattern` is the same as that of `term`s, but it can contain pattern matching metavariables in the form `?ident`. `_` can be used to match irrelevant terms. Example.

    When a metavariable in 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.

    There is also a special notation for second-order pattern matching: in an applicative pattern of the form `@?ident ident1 … identn`, the variable `ident` matches any complex expression with (possible) dependencies in the variables `identi` and returns a functional term of the form `fun ident1 … identn => term`.

  `context ident? [ cpattern ]`

    Matches any term with a subterm matching `cpattern`. If there is a match and `ident` is present, it is assigned the "matched context", i.e. the initial term where the matched subterm is replaced by a hole. Note that `context` (with very similar syntax) appearing after the `=>` is the `context` tactic.

    For `match` and `multimatch`, if the evaluation of the `ltac_expr` fails, the next matching subterm is tried. If no further subterm matches, the next branch is tried. Matching subterms are considered from top to bottom and from left to right (with respect to the raw printing obtained by setting the `Printing All` flag). Example.

  `ltac_expr`

    The tactic to apply if the construct matches. Metavariable values from the pattern match are substituted into `ltac_expr` before it's applied. Note that metavariables are not prefixed with the question mark as they are in `cpattern`.

    If `ltac_expr` evaluates to a tactic, then it is applied. If the tactic succeeds, the result of the match expression is `idtac`. If `ltac_expr` does not evaluate to a tactic, that value is the result of the match expression.

    If `ltac_expr` is a tactic with backtracking points, then subsequent failures after a `lazymatch` or `multimatch` (but not `match`) can cause backtracking into `ltac_expr` to select its next success. (`match` `…` is equivalent to `once` `multimatch` `…`. The `once` prevents backtracking into the `match` after it has succeeded.)

  Note

  Each `L`tac construct is processed in two phases: an evaluation phase and an execution phase. In most cases, tactics that may change the proof state are applied in the second phase. (Tactics that generate integer, string or syntactic values, such as `fresh`, are processed during the evaluation phase.)

  Unlike other tactics, `*match*` tactics get their first success (applying tactics to do so) as part of the evaluation phase. Among other things, this can affect how early failures are processed in `assert_fails`. Please see the note in `assert_fails`.

  Error Expression does not evaluate to a tactic.

    `ltac_expr` must evaluate to a tactic.

  Error No matching clauses for match.

    For at least one of the focused goals, there is no branch that matches its pattern and gets at least one success for `ltac_expr`.

  Error Argument of match does not evaluate to a term.

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

> Example: Comparison of lazymatch and match

> In `lazymatch`, if `ltac_expr` fails, the `lazymatch` fails; it doesn't look for further matches. In `match`, if `ltac_expr` fails in a matching branch, it will try to match on subsequent branches.

> Goal True.
>   1 goal ============================ True
> Fail lazymatch True with | True => idtac "branch 1"; fail | _ => idtac "branch 2" end.
>   branch 1 The command has indeed failed with message: Tactic failure.
> match True with | True => idtac "branch 1"; fail | _ => idtac "branch 2" end.
>   branch 1 branch 2

> Example: Comparison of match and multimatch

> `match` tactics are only evaluated once, whereas `multimatch` tactics may be evaluated more than once if the following constructs trigger backtracking:

> Fail match True with | True => idtac "branch 1" | _ => idtac "branch 2" end ; idtac "branch A"; fail.
>   branch 1 branch A The command has indeed failed with message: Tactic failure.
> Fail multimatch True with | True => idtac "branch 1" | _ => idtac "branch 2" end ; idtac "branch A"; fail.
>   branch 1 branch A branch 2 branch A The command has indeed failed with message: Tactic failure.

> Example: Matching a pattern with holes

> Notice the `idtac` prints `(z + 1)` while the `pose` substitutes `(x + 1)`.

> Goal True.
>   1 goal ============================ True
> match constr:(fun x => (x + 1) * 3) with | fun z => ?y * 3 => idtac "y =" y; pose (fun z: nat => y * 5) end.
>   y = (z + 1) 1 goal n := fun x : nat => (x + 1) * 5 : nat -> nat ============================ True

> Example: Multiple matches for a "context" pattern.

> Internally "x <> y" is represented as "(~ (x = y))", which produces the first match.

> Ltac f t := match t with | context [ (~ ?t) ] => idtac "?t = " t; fail | _ => idtac end.
>   f is defined
> Goal True.
>   1 goal ============================ True
> f ((~ True) <> (~ False)).
>   ?t = ((~ True) = (~ False)) ?t = True ?t = False

### Pattern matching on goals and hypotheses: match goal

Tactic match_key reverse? goal with |? goal_pattern => ltac_expr+| end
  goal_pattern`::=`match_hyp*, |- match_pattern`````|`[ match_hyp*, |- match_pattern ]`````|`_`` match_hyp`::=`name : match_pattern`````|`name := match_pattern`````|`name := [ match_pattern ] : match_pattern``

  `lazymatch goal`, `match goal` and `multimatch goal` are `l1_tactic`s.

  Use this form to match hypotheses and/or goals in the local context. These patterns have zero or more subpatterns to match hypotheses followed by a subpattern to match the conclusion. Except for the differences noted below, this works the same as the corresponding `match_key ltac_expr` construct (see `match`). Each current goal is processed independently.

  Matching is non-linear: if a metavariable occurs more than once, each occurrence must match the same expression. Within a single term, expressions match if they are syntactically equal or α-convertible. When a metavariable is used across multiple hypotheses or across a hypothesis and the current goal, the expressions match if they are convertible.

  `match_hyp*,`

    Patterns to match with hypotheses. Each pattern must match a distinct hypothesis in order for the branch to match.

    Hypotheses have the form `name := termbinder? : type`. Patterns bind each of these nonterminals separately:

    Pattern syntax | Example pattern
    --- | ---
    `name : match_patterntype` | `n : ?t`
    `name := match_patternbinder` | `n := ?b`
    `name := termbinder : type` | `n := ?b : ?t`
    `name := [ match_patternbinder ] : match_patterntype` | `n := [ ?b ] : ?t`

    > `name` can't have a `?`. Note that the last two forms are equivalent except that:

    > - if the `:` in the third form has been bound to something else in a notation, you must use the fourth form. Note that cmd:`Require Import` `ssreflect` loads a notation that does this.

    > - a `termbinder` such as `[ ?l ]` (e.g., denoting a singleton list after `Import` `ListNotations`) must be parenthesized or, for the fourth form, use double brackets: `[ [ ?l ] ]`.

    > `termbinder`s in the form `[?x ; ?y]` for a list are not parsed correctly. The workaround is to add parentheses or to use the underlying term instead of the notation, i.e. `(cons ?x ?y)`.

    If there are multiple `match_hyp`s in a branch, there may be multiple ways to match them to hypotheses. For `match goal` and `multimatch goal`, if the evaluation of the `ltac_expr` fails, matching will continue with the next hypothesis combination. When those are exhausted, the next alternative from any `context` constructs in the `match_pattern`s is tried and then, when the context alternatives are exhausted, the next branch is tried. Example.

  `reverse`

    Hypothesis matching for `match_hyp`s normally begins by matching them from left to right, to hypotheses, last to first. Specifying `reverse` begins matching in the reverse order, from first to last. Normal and reverse examples.

  `|- match_pattern`

    A pattern to match with the current goal

  `goal_pattern with [ ... ]`

    The square brackets don't affect the semantics. They are permitted for aesthetics.

  Error No matching clauses for match goal.

    No clause succeeds, i.e. all matching patterns, if any, fail at the application of the `ltac_expr`.

Examples:

> Example: Matching hypotheses

> Hypotheses are matched from the last hypothesis (which is by default the newest hypothesis) to the first until the `apply` succeeds.

> Goal forall A B : Prop, A -> B -> (A->B).
>   1 goal ============================ forall A B : Prop, A -> B -> A -> B
> intros.
>   1 goal A, B : Prop H : A H0 : B H1 : A ============================ B
> match goal with | H : _ |- _ => idtac "apply " H; apply H end.
>   apply H1 apply H0 No more goals.

> Example: Matching hypotheses with reverse

> Hypotheses are matched from the first hypothesis to the last until the `apply` succeeds.

> Goal forall A B : Prop, A -> B -> (A->B).
>   1 goal ============================ forall A B : Prop, A -> B -> A -> B
> intros.
>   1 goal A, B : Prop H : A H0 : B H1 : A ============================ B
> match reverse goal with | H : _ |- _ => idtac "apply " H; apply H end.
>   apply A apply B apply H apply H0 No more goals.

> Example: Multiple ways to match hypotheses

> Every possible match for the hypotheses is evaluated until the right-hand side succeeds. Note that `H1` and `H2` are never matched to the same hypothesis. Observe that the number of permutations can grow as the factorial of the number of hypotheses and hypothesis patterns.

> Goal forall A B : Prop, A -> B -> (A->B).
>   1 goal ============================ forall A B : Prop, A -> B -> A -> B
> intros A B H.
>   1 goal A, B : Prop H : A ============================ B -> A -> B
> match goal with | H1 : _, H2 : _ |- _ => idtac "match " H1 H2; fail | _ => idtac end.
>   match B H match A H match H B match A B match H A match B A

### 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:

Tactic context ident [ term ]

  Returns the term matched with the `context` pattern (described here) substituting `term` for the hole created by the pattern.

  `context` is a `value_tactic`.

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

  Example: Substituting a matched context

  Goal True /\ True.
    1 goal ============================ True /\ True
  match goal with | |- context G [True] => let x := context G [False] in idtac x end.
    (False /\ True)

### Generating fresh hypothesis names

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

Tactic fresh string​qualid*

  Returns a fresh identifier name (i.e. one that is not already used in the local context and not previously returned by `fresh` in the current `ltac_expr`). The fresh identifier is formed by concatenating the final `ident` of each `qualid` (dropping any qualified components) and each specified `string`. If the resulting name is already used, a number is appended to make it fresh. If no arguments are given, the name is a fresh derivative of the name `H`.

  Note

  We recommend generating the fresh identifier immediately before adding it to the local context. Using `fresh` in a local function may not work as you expect:

  Successive calls to `fresh` give distinct names even if the names haven't yet been added to the local context:

  Goal True -> True.
    1 goal ============================ True -> True
  intro x.
    1 goal x : True ============================ True
  let a := fresh "x" in let b := fresh "x" in idtac a b.
    x0 x1

  When applying `fresh` in a function, the name is chosen based on the tactic context at the point where the function was defined:

  let a := fresh "x" in let f := fun _ => fresh "x" in let c := f () in let d := f () in idtac a c d.
    x0 x1 x1

  `fresh` is a `value_tactic`.

### Computing in a term: eval

Evaluation of a term can be performed with:

`eval red_expr in term`

See `eval`. `eval` is a `value_tactic`.

### Getting the type of a term

Tactic type of term

  This tactic returns the type of `term`.

  `type of` is a `value_tactic`.

### Manipulating untyped terms: type_term

The `uconstr : ( term )` construct can be used to build an untyped term. See `syn_value`.

Tactic type_term one_term

  In `L`tac, an untyped term can contain references to hypotheses or to `L`tac variables containing typed or untyped terms. An untyped term can be type checked with `type_term` whose argument is parsed as an untyped term and returns a well-typed term which can be used in tactics.

  `type_term` is a `value_tactic`.

### Counting goals: numgoals

Tactic numgoals

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

  `numgoals` is a `value_tactic`.

  Example

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

### Testing boolean expressions: guard

Tactic guard int_or_var comparison int_or_var
  int_or_var`::=`integer​ident`` comparison`::=`=`````|`<`````|`<=`````|`>`````|`>=``

  Tests a boolean expression. If the expression evaluates to true, it succeeds without affecting the proof. The tactic fails if the expression is false.

  The accepted tests are simple integer comparisons.

  Example

  Goal True /\ True /\ True.
    1 goal ============================ True /\ True /\ True
  split;[|split].
    3 goals ============================ True goal 2 is: True goal 3 is: True
  all: let n:= numgoals in guard n<4.
  Fail all: let n:= numgoals in guard n=2.
    The command has indeed failed with message: Condition not satisfied: 3=2
  Error Condition not satisfied.

### Proving a subgoal as a separate lemma: abstract

Tactic abstract ltac_expr2 using identname?

  Does a `solve` `[ ltac_expr2 ]` and saves the subproof as an auxiliary lemma. if `identname` is specified, the lemma is saved with that name; otherwise the lemma is saved with the name `ident` `_subproof` `natural?` where `ident` is the name of the current goal (e.g. the theorem name) and `natural` is chosen to get a fresh name. If the proof is closed with `Qed`, the auxiliary lemma is inlined in the final proof term.

  This is useful with tactics such as `discriminate` that generate huge proof terms with many intermediate goals. It can significantly reduce peak memory use. In most cases it doesn't have a significant impact on run time. One case in which it can reduce run time is when a tactic `foo` is known to always pass type checking when it succeeds, such as in reflective proofs. In this case, the idiom "`abstract` `exact_no_check` `foo`" will save half the type checking type time compared to "`exact` `foo`".

  `abstract` is an `l3_tactic`.

  Warning

  The abstract tactic, while very useful, still has some known limitations. See #9146 for more details. We recommend caution when using it in some "non-standard" contexts. In particular, `abstract` doesn't work properly when used inside quotations `ltac:(...)`. If used as part of typeclass resolution, it may produce incorrect terms when in polymorphic universe mode.

  Warning

  Provide `identname` at your own risk; explicitly named and reused subterms don’t play well with asynchronous proofs.

Tactic transparent_abstract ltac_expr3 using ident?

  Like `abstract`, but save the subproof in a transparent lemma with a name in the form `ident` `_subterm` `natural?`.

  Warning

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

  Error Proof is not complete.

## Tactic toplevel definitions

### Defining `L`tac symbols

`L`tac toplevel definitions are made as follows:

Command Ltac tacdef_body with tacdef_body*
  tacdef_body`::=`qualid name* :=​::= ltac_expr``

  Defines or redefines an `L`tac symbol.

  If the `local` attribute is specified, the definition will not be exported outside the current module.

  `qualid`

    Name of the symbol being defined or redefined. For definitions, `qualid` must be a simple `ident`.

  `name*`

    If specified, the symbol defines a function with the given parameter names. If no names are specified, `qualid` is assigned the value of `ltac_expr`.

  `:=`

    Defines a user-defined symbol, but gives an error if the symbol has already been defined.

    Error There is already an Ltac named qualid
  `::=`

    Redefines an existing user-defined symbol, but gives an error if the symbol doesn't exist. Note that `Tactic Notation`s do not count as user-defined tactics for `::=`. If `local` is not specified, the redefinition applies across module boundaries.

    Error There is no Ltac named qualid
  `with tacdef_body*`

    Permits definition of mutually recursive tactics.

  Note

  The following definitions are equivalent:

  - `Ltac qualid name+ := ltac_expr`

  - `Ltac qualid := fun name+ => ltac_expr`

### Printing `L`tac tactics

Command Print Ltac qualid

  Defined `L`tac functions can be displayed using this command.

Command Print Ltac Signatures

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

## Examples of using `L`tac

### Proof that the natural numbers have at least two elements

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

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

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

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

all: congruence.
  No more goals.

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

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

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

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

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

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

Let's first define the permutation predicate:

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

Require Import List.

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

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

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

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

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

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

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

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

solve_perm.
  No more goals.

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

solve_perm.
  No more goals.

### Deciding intuitionistic propositional logic

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

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

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

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

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

Having defined `my_tauto`, we can prove tautologies like these:

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

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

### Deciding type isomorphisms

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

Open Scope type_scope.

Section Iso_axioms.

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

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

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

End Iso_axioms.

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

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

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

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

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

Ltac solve_iso := simplify_type_eq; compare_structure.
  solve_iso is defined

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

Here are examples of what can be solved by `solve_iso`.

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

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

## Debugging `L`tac tactics

### Backtraces

Flag Ltac Backtrace

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

### Tracing execution

Command Info natural ltac_expr

  Applies `ltac_expr` and prints a trace of the tactics that were successfully applied, discarding branches that failed. `idtac` tactics appear in the trace as comments containing the output.

  This command is valid only in proof mode. It accepts Goal selectors.

  The number `natural` 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 goal ============================ exists n : nat, n = 0
  Info 0 t 1||t 0.
    exists with 0;<ltac_plugin::reflexivity@0> No more goals.
  Undo.
    1 goal ============================ exists n : nat, n = 0
  Info 1 t 1||t 0.
    <ltac_plugin::exists@1> with 0;simple refine ?X11 No more goals.

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

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

  Option Info Level natural

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

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

### Interactive debugger

Flag Ltac Debug

  This flag governs the step-by-step debugger that comes with the `L`tac interpreter.

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

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

Error Debug mode not available in the IDE

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

Flag Ltac Batch Debug

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

### Profiling `L`tac tactics

It is possible to measure the time spent in invocations of primitive tactics as well as tactics defined in `L`tac and their inner invocations. The primary use is the development of complex tactics, which can sometimes be so slow as to impede interactive usage. The reasons for the performance degradation can be intricate, like a slowly performing `L`tac match or a sub-tactic whose performance only degrades in certain situations. The profiler generates a call tree and indicates the time spent in a tactic depending on its calling context. Thus it allows to locate the part of a tactic definition that contains the performance issue.

Flag Ltac Profiling

  This flag enables and disables the profiler.

Command Show Ltac Profile CutOff integer​string?

  Prints the profile.

  `CutOff integer`

    By default, tactics that account for less than 2% of the total time are not displayed. `CutOff` lets you specify a different percentage.

  `string`

  > Limits the profile to 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 Lia.
  [Loading ML file ring_plugin.cmxs ... done] [Loading ML file zify_plugin.cmxs ... done] [Loading ML file micromega_plugin.cmxs ... done]
Ltac mytauto := tauto.
  mytauto is defined
Ltac tac := intros; repeat split; lia || 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 goal ============================ 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 goals.
Show Ltac Profile.
  total time: 8.943s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac ----------------------------------- 0.1% 100.0% 1 8.943s ─<Coq.Init.Tauto.with_uniform_flags> --- 0.0% 96.5% 26 0.584s ─<Coq.Init.Tauto.tauto_gen> ------------ 0.0% 96.5% 26 0.584s ─<Coq.Init.Tauto.tauto_intuitionistic> - 0.0% 96.4% 26 0.584s ─t_tauto_intuit ------------------------ 0.1% 96.4% 26 0.584s ─<Coq.Init.Tauto.simplif> -------------- 68.8% 92.7% 26 0.571s ─<Coq.Init.Tauto.is_conj> -------------- 17.5% 17.5% 28756 0.000s ─elim id ------------------------------- 3.9% 3.9% 650 0.058s ─<Coq.Init.Tauto.axioms> --------------- 2.7% 3.6% 0 0.016s ─lia ----------------------------------- 0.1% 3.2% 28 0.217s ─xlia (tactic) ------------------------- 2.3% 2.5% 28 0.210s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─tac ----------------------------------- 0.1% 100.0% 1 8.943s ├─<Coq.Init.Tauto.with_uniform_flags> - 0.0% 96.5% 26 0.584s │└<Coq.Init.Tauto.tauto_gen> ---------- 0.0% 96.5% 26 0.584s │└<Coq.Init.Tauto.tauto_intuitionistic> 0.0% 96.4% 26 0.584s │└t_tauto_intuit ---------------------- 0.1% 96.4% 26 0.584s │ ├─<Coq.Init.Tauto.simplif> ---------- 68.8% 92.7% 26 0.571s │ │ ├─<Coq.Init.Tauto.is_conj> -------- 17.5% 17.5% 28756 0.000s │ │ └─elim id ------------------------- 3.9% 3.9% 650 0.058s │ └─<Coq.Init.Tauto.axioms> ----------- 2.7% 3.6% 0 0.016s └─lia --------------------------------- 0.1% 3.2% 28 0.217s └xlia (tactic) ----------------------- 2.3% 2.5% 28 0.210s
Show Ltac Profile "lia".
  total time: 8.943s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘ ─lia ----------------------------------- 0.1% 3.2% 28 0.217s tactic local total calls max ────────────────────────────────────────┴──────┴──────┴───────┴─────────┘

Abort.
Unset Ltac Profiling.

Tactic start ltac profiling

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

Tactic 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.

Tactic reset ltac profile

  Equivalent to the `Reset Ltac Profile` command, which allows resetting the profile from tactic scripts for benchmarking purposes.

Tactic show ltac profile cutoff integer​string?

  Equivalent to the `Show Ltac Profile` command, which allows displaying the profile from tactic scripts for benchmarking purposes.

Warning Ltac Profiler encountered an invalid stack (no self node). This can happen if you reset the profile during tactic execution

  Currently, `reset ltac profile` is not very well-supported, as it clears all profiling information about all tactics, including ones above the current tactic. As a result, the profiler has trouble understanding where it is in tactic execution. This mixes especially poorly with backtracking into multi-success tactics. In general, non-top-level calls to `reset ltac profile` should be avoided.

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

### Run-time optimization tactic

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 `Optimize Heap` command.

Tactic infoH ltac_expr3

  Used internally by Proof General. See #12423 for some background.

  `infoH` is an `l3_tactic`.

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