
# Ltac2¶

The Ltac tactic language is probably one of the ingredients of the success of Coq, yet it is at the same time its Achilles' heel. Indeed, Ltac:

• has often unclear semantics

• is very non-uniform due to organic growth

• lacks expressivity (data structures, combinators, types, ...)

• is slow

• is error-prone and fragile

• has an intricate implementation

Following the need of users who are developing huge projects relying critically on Ltac, we believe that we should offer a proper modern language that features at least the following:

• at least informal, predictable semantics

• a type system

• standard programming facilities (e.g., datatypes)

This new language, called Ltac2, is described in this chapter. It is still experimental but we nonetheless encourage users to start testing it, especially wherever an advanced tactic language is needed. The previous implementation of Ltac, described in the previous chapter, will be referred to as Ltac1.

Current limitations include:

• There are a number of tactics that are not yet supported in Ltac2 because the interface OCaml and/or Ltac2 notations haven't been written. See Defining tactics.

• Missing usability features such as:

• Printing functions are limited and awkward to use. Only a few data types are printable.

• Deep pattern matching and matching on tuples don't work.

• A convenient way to build terms with casts through the low-level API. Because the cast type is opaque, building terms with casts currently requires an awkward construction like the following, which also incurs extra overhead to repeat typechecking for each call to get_vm_cast:

Constr.Unsafe.make (Constr.Unsafe.Cast 'I (get_vm_cast ()) 'True)

with:

From Ltac2 Require Import Ltac2.
Ltac2 get_vm_cast () :=   match Constr.Unsafe.kind '(I <: True) with   | Constr.Unsafe.Cast _ cst _ => cst   | _ => Control.throw Not_found   end.
• Missing low-level primitives that are convenient for writing automation, such as:

• An easy way to get the number of constructors of an inductive type. Currently only way to do this is to destruct a variable of the inductive type and count the number of goals that result.

• The deprecated attribute is not supported for Ltac2 definitions.

• Error messages may be cryptic.

## General design¶

There are various alternatives to Ltac1, such as Mtac or Rtac for instance. While those alternatives can be quite different from Ltac1, we designed Ltac2 to be as close as reasonably possible to Ltac1, while fixing the aforementioned defects.

In particular, Ltac2 is:

• a member of the ML family of languages, i.e.

• a call-by-value functional language

• with effects

• together with the Hindley-Milner type system

• a language featuring meta-programming facilities for the manipulation of Coq-side terms

• a language featuring notation facilities to help write palatable scripts

We describe these in more detail in the remainder of this document.

## ML component¶

### Overview¶

Ltac2 is a member of the ML family of languages, in the sense that it is an effectful call-by-value functional language, with static typing à la Hindley-Milner (see [DM82]). It is commonly accepted that ML constitutes a sweet spot in PL design, as it is relatively expressive while not being either too lax (unlike dynamic typing) nor too strict (unlike, say, dependent types).

The main goal of Ltac2 is to serve as a meta-language for Coq. As such, it naturally fits in the ML lineage, just as the historical ML was designed as the tactic language for the LCF prover. It can also be seen as a general-purpose language, by simply forgetting about the Coq-specific features.

Sticking to a standard ML type system can be considered somewhat weak for a meta-language designed to manipulate Coq terms. In particular, there is no way to statically guarantee that a Coq term resulting from an Ltac2 computation will be well-typed. This is actually a design choice, motivated by backward compatibility with Ltac1. Instead, well-typedness is deferred to dynamic checks, allowing many primitive functions to fail whenever they are provided with an ill-typed term.

The language is naturally effectful as it manipulates the global state of the proof engine. This allows to think of proof-modifying primitives as effects in a straightforward way. Semantically, proof manipulation lives in a monad, which allows to ensure that Ltac2 satisfies the same equations as a generic ML with unspecified effects would do, e.g. function reduction is substitution by a value.

Use the following command to import Ltac2:

From Ltac2 Require Import Ltac2.

### Type Syntax¶

At the level of terms, we simply elaborate on Ltac1 syntax, which is quite close to OCaml. Types follow the simply-typed syntax of OCaml.

::=|ltac2_type2::=|ltac2_type1::=|ltac2_type0::=( ) |ltac2_typevar|_|qualid::=

The set of base types can be extended thanks to the usual ML type declarations such as algebraic datatypes and records.

Built-in types include:

• int, machine integers (size not specified, in practice inherited from OCaml)

• string, mutable strings

• 'a array, mutable arrays

• exn, exceptions

• constr, kernel-side terms

• pattern, term patterns

• ident, well-formed identifiers

### Type declarations¶

One can define new types with the following commands.

Command Ltac2 Type rec? tac2typ_def with tac2typ_def*
::= qualid :=::= tac2typ_knd?::=ltac2_typevar|( )::=ltac2_type|[ ? ]|[ .. ]|{ ;?? }::=ident|ident ( )::=mutable? ident : ltac2_type
:=

Defines a type with with an explicit set of constructors

::=

Extends an existing open variant type, a special kind of variant type whose constructors are not statically defined, but can instead be extended dynamically. A typical example is the standard exn type for exceptions. Pattern matching on open variants must always include a catch-all clause. They can be extended with this form, in which case tac2typ_knd should be in the form [ |? tac2alg_constructor+|? ].

Without :=​::=

Defines an abstract type for use representing data from OCaml. Not for end users.

with tac2typ_def

Permits definition of mutually recursive type definitions.

Each production of tac2typ_knd defines one of four possible kinds of definitions, respectively: alias, variant, open variant and record types.

Aliases are names for a given type expression and are transparently unfoldable to that expression. They cannot be recursive.

Variants are sum types defined by constructors and eliminated by pattern-matching. They can be recursive, but the rec flag must be explicitly set. Pattern matching must be exhaustive.

Open variants can be extended with additional constructors using the ::= form.

Records are product types with named fields and eliminated by projection. Likewise they can be recursive if the rec flag is set.

Command Ltac2 @ external ident : ltac2_type := string string

Declares abstract terms. Frequently, these declare OCaml functions defined in Coq and give their type information. They can also declare data structures from OCaml. This command has no use for the end user.

### APIs¶

Ltac2 provides over 150 API functions that provide various capabilities. These are declared with Ltac2 external in lib/coq/user-contrib/Ltac2/*.v. For example, Message.print defined in Message.v is used to print messages:

Goal True.
1 subgoal ============================ True
Message.print (Message.of_string "fully qualified calls").
fully qualified calls
From Ltac2 Require Import Message.
print (of_string "unqualified calls").
unqualified calls

### Term Syntax¶

The syntax of the functional fragment is very close to that of Ltac1, except that it adds a true pattern-matching feature, as well as a few standard constructs from ML.

In practice, there is some additional syntactic sugar that allows the user to bind a variable and match on it at the same time, in the usual ML style.

There is dedicated syntax for list and array literals.

::=|ltac2_expr5::=fun => ltac2_expr|let rec? ltac2_let_clause with ltac2_let_clause* in ltac2_expr|ltac2_expr3::=::=::=|ltac2_expr1::=|||ltac2_expr0::=::=||()|[ ]|{ ? }|ltac2_tactic_atom::=integer|string|qualid|||' term|ltac2_quotations

The non-terminal lident designates identifiers starting with a lowercase letter.

'term is equivalent to open_constr:(term).

### Ltac2 Definitions¶

Command Ltac2 mutable? rec? tac2def_body with tac2def_body*
::=

This command defines a new global Ltac2 value. If one or more tac2pat0 are specified, the new value is a function. This is a shortcut for one of the ltac2_expr5 productions. For example: Ltac2 foo a b := … is equivalent to Ltac2 foo := fun a b => ….

The body of an Ltac2 definition is required to be a syntactical value that is, a function, a constant, a pure constructor recursively applied to values or a (non-recursive) let binding of a value in a value.

If rec is set, the tactic is expanded into a recursive binding.

If mutable is set, the definition can be redefined at a later stage (see below).

Command Ltac2 Set qualid as ident? := ltac2_expr

This command redefines a previous mutable definition. Mutable definitions act like dynamic binding, i.e. at runtime, the last defined value for this entry is chosen. This is useful for global flags and the like. The previous value of the binding can be optionally accessed using the as binding syntax.

Example: Dynamic nature of mutable cells

Ltac2 mutable x := true.
Ltac2 y := x.
Ltac2 Eval y.
- : bool = true
Ltac2 Set x := false.
Ltac2 Eval y.
- : bool = false

Example: Interaction with recursive calls

Ltac2 mutable rec f b := if b then 0 else f true.
Ltac2 Set f := fun b => if b then 1 else f true.
Ltac2 Eval (f false).
- : int = 1
Ltac2 Set f as oldf := fun b => if b then 2 else oldf false.
Ltac2 Eval (f false).
- : int = 2

In the definition, the f in the body is resolved statically because the definition is marked recursive. In the first re-definition, the f in the body is resolved dynamically. This is witnessed by the second re-definition.

### Reduction¶

We use the usual ML call-by-value reduction, with an otherwise unspecified evaluation order. This is a design choice making it compatible with OCaml, if ever we implement native compilation. The expected equations are as follows:

(fun x => t) V ≡ t{x := V} (βv)

let x := V in t ≡ t{x := V} (let)

match C V₀ ... Vₙ with ... | C x₀ ... xₙ  => t | ... end ≡ t {xᵢ := Vᵢ} (ι)

(t any term, V values, C constructor)


Note that call-by-value reduction is already a departure from Ltac1 which uses heuristics to decide when to evaluate an expression. For instance, the following expressions do not evaluate the same way in Ltac1.

foo (idtac; let x := 0 in bar)

foo (let x := 0 in bar)

Instead of relying on the idtac idiom, we would now require an explicit thunk to not compute the argument, and foo would have e.g. type (unit -> unit) -> unit.

foo (fun () => let x := 0 in bar)

### Typing¶

Typing is strict and follows the Hindley-Milner system. Unlike Ltac1, there are no type casts at runtime, and one has to resort to conversion functions. See notations though to make things more palatable.

In this setting, all the usual argument-free tactics have type unit -> unit, but one can return a value of type t thanks to terms of type unit -> t, or take additional arguments.

### Effects¶

Effects in Ltac2 are straightforward, except that instead of using the standard IO monad as the ambient effectful world, Ltac2 is has a tactic monad.

Note that the order of evaluation of application is not specified and is implementation-dependent, as in OCaml.

We recall that the Proofview.tactic monad is essentially a IO monad together with backtracking state representing the proof state.

Intuitively a thunk of type unit -> 'a can do the following:

• It can perform non-backtracking IO like printing and setting mutable variables

• It can fail in a non-recoverable way

• It can use first-class backtracking. One way to think about this is that thunks are isomorphic to this type: (unit -> 'a) ~ (unit -> exn + ('a * (exn -> 'a))) i.e. thunks can produce a lazy list of results where each tail is waiting for a continuation exception.

• It can access a backtracking proof state, consisting among other things of the current evar assignment and the list of goals under focus.

We now describe more thoroughly the various effects in Ltac2.

#### Standard IO¶

The Ltac2 language features non-backtracking IO, notably mutable data and printing operations.

Mutable fields of records can be modified using the set syntax. Likewise, built-in types like string and array feature imperative assignment. See modules String and Array respectively.

A few printing primitives are provided in the Message module for displaying information to the user.

#### Fatal errors¶

The Ltac2 language provides non-backtracking exceptions, also known as panics, through the following primitive in module Control:

val throw : exn -> 'a


Unlike backtracking exceptions from the next section, this kind of error is never caught by backtracking primitives, that is, throwing an exception destroys the stack. This is codified by the following equation, where E is an evaluation context:

E[throw e] ≡ throw e

(e value)


There is currently no way to catch such an exception, which is a deliberate design choice. Eventually there might be a way to catch it and destroy all backtrack and return values.

#### Backtracking¶

In Ltac2, we have the following backtracking primitives, defined in the Control module:

Ltac2 Type 'a result := [ Val ('a) | Err (exn) ].

val zero : exn -> 'a
val plus : (unit -> 'a) -> (exn -> 'a) -> 'a
val case : (unit -> 'a) -> ('a * (exn -> 'a)) result


If one views thunks as lazy lists, then zero is the empty list and plus is list concatenation, while case is pattern-matching.

The backtracking is first-class, i.e. one can write plus (fun () => "x") (fun _ => "y") : string producing a backtracking string.

These operations are expected to satisfy a few equations, most notably that they form a monoid compatible with sequentialization.:

plus t zero ≡ t ()
plus (fun () => zero e) f ≡ f e
plus (plus t f) g ≡ plus t (fun e => plus (f e) g)

case (fun () => zero e) ≡ Err e
case (fun () => plus (fun () => t) f) ≡ Val (t,f)

let x := zero e in u ≡ zero e
let x := plus t f in u ≡ plus (fun () => let x := t in u) (fun e => let x := f e in u)

(t, u, f, g, e values)


#### Goals¶

A goal is given by the data of its conclusion and hypotheses, i.e. it can be represented as [Γ ⊢ A].

The tactic monad naturally operates over the whole proofview, which may represent several goals, including none. Thus, there is no such thing as the current goal. Goals are naturally ordered, though.

It is natural to do the same in Ltac2, but we must provide a way to get access to a given goal. This is the role of the enter primitive, which applies a tactic to each currently focused goal in turn:

val enter : (unit -> unit) -> unit


It is guaranteed that when evaluating enter f, f is called with exactly one goal under focus. Note that f may be called several times, or never, depending on the number of goals under focus before the call to enter.

Accessing the goal data is then implicit in the Ltac2 primitives, and may panic if the invariants are not respected. The two essential functions for observing goals are given below.:

val hyp : ident -> constr
val goal : unit -> constr


The two above functions panic if there is not exactly one goal under focus. In addition, hyp may also fail if there is no hypothesis with the corresponding name.

## Meta-programming¶

### Overview¶

One of the major implementation issues of Ltac1 is the fact that it is never clear whether an object refers to the object world or the meta-world. This is an incredible source of slowness, as the interpretation must be aware of bound variables and must use heuristics to decide whether a variable is a proper one or referring to something in the Ltac context.

Likewise, in Ltac1, constr parsing is implicit, so that foo 0 is not foo applied to the Ltac integer expression 0 (Ltac does have a notion of integers, though it is not first-class), but rather the Coq term Datatypes.O.

The implicit parsing is confusing to users and often gives unexpected results. Ltac2 makes these explicit using quoting and unquoting notation, although there are notations to do it in a short and elegant way so as not to be too cumbersome to the user.

### Quotations¶

#### Built-in quotations¶

::=ident : ( lident )|constr : ( term )|open_constr : ( term )|pattern : ( cpattern )|reference : ( qualid )|ltac1 : ( ltac1_expr_in_env )|ltac1val : ( ltac1_expr_in_env )::=ltac_expr|

The current implementation recognizes the following built-in quotations:

• ident, which parses identifiers (type Init.ident).

• constr, which parses Coq terms and produces an-evar free term at runtime (type Init.constr).

• open_constr, which parses Coq terms and produces a term potentially with holes at runtime (type Init.constr as well).

• pattern, which parses Coq patterns and produces a pattern used for term matching (type Init.pattern).

• reference Qualified names are globalized at internalization into the corresponding global reference, while &id is turned into Std.VarRef id. This produces at runtime a Std.reference.

• ltac1, for calling Ltac1 code, described in Simple API.

• ltac1val, for manipulating Ltac1 values, described in Low-level API.

The following syntactic sugar is provided for two common cases:

• @id is the same as ident:(id)

• 'term is the same as open_constr:(term)

#### Strict vs. non-strict mode¶

Depending on the context, quotation-producing terms (i.e. constr or open_constr) are not internalized in the same way. There are two possible modes, the strict and the non-strict mode.

• In strict mode, all simple identifiers appearing in a term quotation are required to be resolvable statically. That is, they must be the short name of a declaration which is defined globally, excluding section variables and hypotheses. If this doesn't hold, internalization will fail. To work around this error, one has to specifically use the & notation.

• In non-strict mode, any simple identifier appearing in a term quotation which is not bound in the global environment is turned into a dynamic reference to a hypothesis. That is to say, internalization will succeed, but the evaluation of the term at runtime will fail if there is no such variable in the dynamic context.

Strict mode is enforced by default, such as for all Ltac2 definitions. Non-strict mode is only set when evaluating Ltac2 snippets in interactive proof mode. The rationale is that it is cumbersome to explicitly add & interactively, while it is expected that global tactics enforce more invariants on their code.

### Term Antiquotations¶

#### Syntax¶

One can also insert Ltac2 code into Coq terms, similar to what is possible in Ltac1.

+=ltac2:( ltac2_expr )

Antiquoted terms are expected to have type unit, as they are only evaluated for their side-effects.

#### Semantics¶

A quoted Coq term is interpreted in two phases, internalization and evaluation.

• Internalization is part of the static semantics, that is, it is done at Ltac2 typing time.

• Evaluation is part of the dynamic semantics, that is, it is done when a term gets effectively computed by Ltac2.

Note that typing of Coq terms is a dynamic process occurring at Ltac2 evaluation time, and not at Ltac2 typing time.

##### Static semantics¶

During internalization, Coq variables are resolved and antiquotations are type checked as Ltac2 terms, effectively producing a glob_constr in Coq implementation terminology. Note that although it went through the type checking of Ltac2, the resulting term has not been fully computed and is potentially ill-typed as a runtime Coq term.

Example

The following term is valid (with type unit -> constr), but will fail at runtime:

Ltac2 myconstr () := constr:(nat -> 0).

Term antiquotations are type checked in the enclosing Ltac2 typing context of the corresponding term expression.

Example

The following will type check, with type constr.

let x := '0 in constr:(1 + ltac2:(exact x))

Beware that the typing environment of antiquotations is not expanded by the Coq binders from the term.

Example

The following Ltac2 expression will not type check:

constr:(fun x : nat => ltac2:(exact x))
(* Error: Unbound variable 'x' *)


There is a simple reason for that, which is that the following expression would not make sense in general.

constr:(fun x : nat => ltac2:(clear @x; exact x))

Indeed, a hypothesis can suddenly disappear from the runtime context if some other tactic pulls the rug from under you.

Rather, the tactic writer has to resort to the dynamic goal environment, and must write instead explicitly that she is accessing a hypothesis, typically as follows.

constr:(fun x : nat => ltac2:(exact (hyp @x)))

This pattern is so common that we provide dedicated Ltac2 and Coq term notations for it.

• &x as an Ltac2 expression expands to hyp @x.

• &x as a Coq constr expression expands to ltac2:(Control.refine (fun () => hyp @x)).

In the special case where Ltac2 antiquotations appear inside a Coq term notation, the notation variables are systematically bound in the body of the tactic expression with type Ltac2.Init.preterm. Such a type represents untyped syntactic Coq expressions, which can by typed in the current context using the Ltac2.Constr.pretype function.

Example

The following notation is essentially the identity.

### Match over terms¶

Ltac2 features a construction similar to Ltac1 match over terms, although in a less hard-wired way.

Tactic ltac2_match_key ltac2_exprterm with ltac2_match_list end
::=lazy_match!|match!|multi_match!::=|? ::=::=cpattern|context [ cpattern ]

Evaluates ltac2_exprterm, which must yield a term, and matches it sequentially with the ltac2_match_patterns, which may contain metavariables. When a match is found, metavariable values are substituted into ltac2_expr, which is then applied.

Matching may continue depending on whether lazy_match!, match! or multi_match! is specified.

In the ltac2_match_patterns, metavariables have the form ?ident, whereas in the ltac2_exprs, the question mark is omitted.

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.

lazy_match!

Causes the match to commit to the first matching branch rather than trying a new match if ltac2_expr fails. Example.

match!

If ltac2_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.

multi_match!

If ltac2_expr fails, continue matching with the next branch. When a ltac2_expr succeeds for a branch, subsequent failures (after the multi_match!) causing consumption of all the successes of ltac2_expr trigger selection of a new matching branch. Example.

cpattern

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

Unlike Ltac1, Ltac2 ?id metavariables only match closed terms.

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. This hole in the matched context can be filled with the expression Pattern.instantiate ident cpattern.

For match! and multi_match!, if the evaluation of the ltac2_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.

ltac2_expr

The tactic to apply if the construct matches. Metavariable values from the pattern match are statically bound as Ltac2 variables in ltac2_expr before it is applied.

If ltac2_expr is a tactic with backtracking points, then subsequent failures after a lazy_match! or multi_match! (but not match!) can cause backtracking into ltac2_expr to select its next success.

Variables from the tac2pat1 are statically bound in the body of the branch. Variables from the term pattern have values of type constr. Variables from the ident in the context construct have values of type Pattern.context (defined in Pattern.v).

Note that unlike Ltac1, only lowercase identifiers are valid as Ltac2 bindings. Ltac2 will report an error if one of the bound variables starts with an uppercase character.

The semantics of this construction are otherwise the same as the corresponding one from Ltac1, except that it requires the goal to be focused.

Example: Ltac2 Comparison of lazy_match! and match!

(Equivalent to this Ltac1 example.)

These lines define a msg tactic that's used in several examples as a more-succinct alternative to print (to_string "..."):

From Ltac2 Require Import Message.
Ltac2 msg x := print (of_string x).
Goal True.
1 subgoal ============================ True

In lazy_match!, if ltac2_expr fails, the lazy_match! fails; it doesn't look for further matches. In match!, if ltac2_expr fails in a matching branch, it will try to match on subsequent branches. Note that 'term below is equivalent to open_constr:(term).

Fail lazy_match! 'True with | True => msg "branch 1"; fail | _ => msg "branch 2" end.
branch 1 The command has indeed failed with message: Uncaught Ltac2 exception: Tactic_failure (None)
match! 'True with | True => msg "branch 1"; fail | _ => msg "branch 2" end.
branch 1 branch 2

Example: Ltac2 Comparison of match! and multi_match!

(Equivalent to this Ltac1 example.)

match! tactics are only evaluated once, whereas multi_match! tactics may be evaluated more than once if the following constructs trigger backtracking:

Fail match! 'True with | True => msg "branch 1" | _ => msg "branch 2" end ; msg "branch A"; fail.
branch 1 branch A The command has indeed failed with message: Uncaught Ltac2 exception: Tactic_failure (None)
Fail multi_match! 'True with | True => msg "branch 1" | _ => msg "branch 2" end ; msg "branch A"; fail.
branch 1 branch A branch 2 branch A The command has indeed failed with message: Uncaught Ltac2 exception: Match_failure

Example: Ltac2 Multiple matches for a "context" pattern.

(Equivalent to this Ltac1 example.)

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

Ltac2 f2 t := match! t with               | context [ (~ ?t) ] => print (of_constr t); fail               | _ => ()               end.
f2 constr:((~ True) <> (~ False)).
((~ True) = (~ False)) True False

### Match over goals¶

Tactic ltac2_match_key reverse? goal with goal_match_list end
::=|? ::=::=::=

Matches over goals, similar to Ltac1 match goal. 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 ltac2_match_key ltac2_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.

gmatch_pattern*,

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. Currently Ltac2 doesn't allow matching on or capturing the value of termbinder. It only supports matching on the name and the type, for example n : ?t.

If there are multiple gmatch_hyp_patterns in a branch, there may be multiple ways to match them to hypotheses. For match! goal and multi_match! goal, if the evaluation of the ltac2_expr fails, matching will continue with the next hypothesis combination. When those are exhausted, the next alternative from any context construct in the ltac2_match_patterns is tried and then, when the context alternatives are exhausted, the next branch is tried. Example.

reverse

Hypothesis matching for gmatch_hyp_patterns 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.

|- ltac2_match_pattern

A pattern to match with the current goal

Note that unlike Ltac1, only lowercase identifiers are valid as Ltac2 bindings. Ltac2 will report an error if you try to use a bound variable that starts with an uppercase character.

Variables from gmatch_hyp_pattern and ltac2_match_pattern are bound in the body of the branch. Their types are:

• constr for pattern variables appearing in a term

• Pattern.context for variables binding a context

• ident for variables binding a hypothesis name.

The same identifier caveat as in the case of matching over constr applies, and this feature has the same semantics as in Ltac1.

Example: Ltac2 Matching hypotheses

(Equivalent to this Ltac1 example.)

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 subgoal ============================ forall A B : Prop, A -> B -> A -> B
intros.
1 subgoal A, B : Prop H : A H0 : B H1 : A ============================ B
match! goal with | [ h : _ |- _ ] => let h := Control.hyp h in print (of_constr h); apply $h end. H1 H0 No more subgoals. Example: Matching hypotheses with reverse (Equivalent to this Ltac1 example.) Hypotheses are matched from the first hypothesis to the last until the apply succeeds. Goal forall A B : Prop, A -> B -> (A->B). 1 subgoal ============================ forall A B : Prop, A -> B -> A -> B intros. 1 subgoal A, B : Prop H : A H0 : B H1 : A ============================ B match! reverse goal with | [ h : _ |- _ ] => let h := Control.hyp h in print (of_constr h); apply$h end.
A B H H0 No more subgoals.

Example: Multiple ways to match a hypotheses

(Equivalent to this Ltac1 example.)

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 subgoal ============================ forall A B : Prop, A -> B -> A -> B
intros A B H.
1 subgoal A, B : Prop H : A ============================ B -> A -> B
match! goal with | [ h1 : _, h2 : _ |- _ ] =>    print (concat (of_string "match ")          (concat (of_constr (Control.hyp h1))          (concat (of_string " ")          (of_constr (Control.hyp h2)))));    fail | [ |- _ ] => () end.
match B H match A H match H B match A B match H A match B A

### Match on values¶

Tactic match ltac2_expr5 with ltac2_branches? end

Matches a value, akin to the OCaml match construct. By itself, it doesn't cause backtracking as do the *match*! and *match*! goal constructs.

::=|? +|::=|qualid|[ ]||tac2pat0::=_|()|qualid|::=||tac2pat1
Tactic if ltac2_expr5test then ltac2_expr5then else ltac2_expr5else

Equivalent to a match on a boolean value. If the ltac2_expr5test evaluates to true, ltac2_expr5then is evaluated. Otherwise ltac2_expr5else is evaluated.

Note

For now, deep pattern matching is not implemented.

## Notations¶

Command Ltac2 Notation ltac2_scope+ : natural? := ltac2_expr

Ltac2 Notation provides a way to extend the syntax of Ltac2 tactics. The left-hand side (before the :=) defines the syntax to recognize and gives formal parameter names for the syntactic values. integer is the level of the notation. When the notation is used, the values are substituted into the right-hand side. The right-hand side is typechecked when the notation is used, not when it is defined. In the following example, x is the formal parameter name and constr is its syntactic class. print and of_constr are functions provided by Coq through Message.v.

Example: Printing a term

Goal True.
1 subgoal ============================ True
From Ltac2 Require Import Message.
Ltac2 Notation "ex1" x(constr) := print (of_constr x).
ex1 (1 + 2).
(1 + 2)

You can also print terms with a regular Ltac2 definition, but then the term must be in the quotation constr:( … ):

Ltac2 ex2 x := print (of_constr x).
ex2 constr:(1+2).
(1 + 2)

There are also metasyntactic classes described here that combine other items. For example, list1(constr, ",") recognizes a comma-separated list of one or more terms.

Example: Parsing a list of terms

Ltac2 rec print_list x := match x with | a :: t => print (of_constr a); print_list t | [] => () end.
Ltac2 Notation "ex2" x(list1(constr, ",")) := print_list x.
ex2 1, 2, 3.
1 2 3

An Ltac2 notation adds a parsing rule to the Ltac2 grammar, which is expanded to the provided body where every token from the notation is let-bound to the corresponding generated expression.

Example

Assume we perform:

Ltac2 Notation "foo" c(thunk(constr)) ids(list0(ident)) := Bar.f c ids.

Then the following expression

let y := @X in foo (nat -> nat) x $y will expand at parsing time to let y := @X in let c := fun () => constr:(nat -> nat) with ids := [@x; y] in Bar.f c ids Beware that the order of evaluation of multiple let-bindings is not specified, so that you may have to resort to thunking to ensure that side-effects are performed at the right time. ### Abbreviations¶ Command Ltac2 Notation string​lident := ltac2_expr Introduces a special kind of notation, called an abbreviation, that does not add any parsing rules. It is similar in spirit to Coq abbreviations (see Notation (abbreviation), insofar as its main purpose is to give an absolute name to a piece of pure syntax, which can be transparently referred to by this name as if it were a proper definition. The abbreviation can then be manipulated just like a normal Ltac2 definition, except that it is expanded at internalization time into the given expression. Furthermore, in order to make this kind of construction useful in practice in an effectful language such as Ltac2, any syntactic argument to an abbreviation is thunked on-the-fly during its expansion. For instance, suppose that we define the following. Ltac2 Notation foo := fun x => x (). Then we have the following expansion at internalization time. foo 0 ↦ (fun x => x ()) (fun _ => 0) Note that abbreviations are not type checked at all, and may result in typing errors after expansion. ### Defining tactics¶ Built-in tactics (those defined in OCaml code in the Coq executable) and Ltac1 tactics, which are defined in .v files, must be defined through notations. (Ltac2 tactics can be defined with Ltac2. Notations for many but not all built-in tactics are defined in Notations.v, which is automatically loaded with Ltac2. The Ltac2 syntax for these tactics is often identical or very similar to the tactic syntax described in other chapters of this documentation. These notations rely on tactic functions declared in Std.v. Functions corresponding to some built-in tactics may not yet be defined in the Coq executable or declared in Std.v. Adding them may require code changes to Coq or defining workarounds through Ltac1 (described below). Two examples of syntax differences: • There is no notation defined that's equivalent to intros until ident​natural. There is, however, already an intros_until tactic function defined Std.v, so it may be possible for a user to add the necessary notation. • The built-in simpl tactic in Ltac1 supports the use of scope keys in delta flags, e.g. simpl ["+"%nat] which is not accepted by Ltac2. This is because Ltac2 uses a different definition for delta_flag; compare it to ltac2_delta_flag. This also affects compute. Ltac1 tactics are not automatically available in Ltac2. (Note that some of the tactics described in the documentation are defined with Ltac1.) You can make them accessible in Ltac2 with commands similar to the following: From Coq Require Import Lia. [Loading ML file ring_plugin.cmxs ... done] [Loading ML file zify_plugin.cmxs ... done] [Loading ML file micromega_plugin.cmxs ... done] Local Ltac2 lia_ltac1 () := ltac1:(lia). Ltac2 Notation "lia" := lia_ltac1 (). A similar approach can be used to access missing built-in tactics. See Simple API for an example that passes two parameters to a missing build-in tactic. ### Syntactic classes¶ The simplest syntactic classes in Ltac2 notations represent individual nonterminals from the Coq grammar. Only a few selected nonterminals are available as syntactic classes. In addition, there are metasyntactic operations for describing more complex syntax, such as making an item optional or representing a list of items. When parsing, each syntactic class expression returns a value that's bound to a name in the notation definition. Syntactic classes are described with a form of S-expression: ::=string|integer|name|name ( ) Metasyntactic operations that can be applied to other syntactic classes are: opt(ltac2_scope) Parses an optional ltac2_scope. The associated value is either None or enclosed in Some list1(ltac2_scope , string?) Parses a list of one or more ltac2_scopes. If string is specified, items must be separated by string. list0(ltac2_scope , string?) Parses a list of zero or more ltac2_scopes. If string is specified, items must be separated by string. For zero items, the associated value is an empty list. seq(ltac2_scope+,) Parses the ltac2_scopes in order. The associated value is a tuple, omitting ltac2_scopes that are strings. self and next are not permitted within seq. The following classes represent nonterminals with some special handling. The table further down lists the classes that that are handled plainly. constr ( scope_key+, )? Parses a term. If specified, the scope_keys are used to interpret the term (as described in Local interpretation rules for notations). The last scope_key is the top of the scope stack that's applied to the term. open_constr Parses an open term. ident Parses ident or $ident. The first form returns ident:(ident), while the latter form returns the variable ident.

string

Accepts the specified string that is not a keyword, returning a value of ().

keyword(string)

Accepts the specified string that is a keyword, returning a value of ().

terminal(string)

Accepts the specified string whether it's a keyword or not, returning a value of ().

tactic (integer)?

Parses an ltac2_expr. If integer is specified, the construct parses a ltac2_exprinteger, for example tactic(5) parses ltac2_expr5. tactic(6) parses ltac2_expr. integer must be in the range 0 .. 6.

You can also use tactic to accept an integer or a string, but there's no syntactic class that accepts only an integer or a string.

self

parses an Ltac2 expression at the current level and returns it as is.

next

parses an Ltac2 expression at the next level and returns it as is.

thunk(ltac2_scope)

Used for semantic effect only, parses the same as ltac2_scope. If e is the parsed expression for ltac2_scope, thunk returns fun () => e.

pattern

parses a cpattern

A few syntactic classes contain antiquotation features. For the sake of uniformity, all antiquotations are introduced by the syntax $lident. A few other specific syntactic classes exist to handle Ltac1-like syntax, but their use is discouraged and they are thus not documented. For now there is no way to declare new syntactic classes from the Ltac2 side, but this is planned. Other nonterminals that have syntactic classes are listed here. Syntactic class name Nonterminal Similar non-Ltac2 syntax intropatterns ltac2_intropatterns intropattern_list intropattern ltac2_simple_intropattern simple_intropattern ident ident_or_anti ident destruction_arg ltac2_destruction_arg destruction_arg with_bindings q_with_bindings with bindings? bindings ltac2_bindings bindings strategy ltac2_strategy_flag strategy_flag reference refglobal reference clause ltac2_clause occurrences occurrences q_occurrences at occs_nums? induction_clause ltac2_induction_clause induction_clause conversion ltac2_conversion rewriting ltac2_oriented_rewriter oriented_rewriter dispatch ltac2_for_each_goal for_each_goal hintdb hintdb hintbases move_location move_location where pose pose bindings_with_parameters assert assertion ( ident := term ) constr_matching ltac2_match_list goal_matching goal_match_list Here is the syntax for the q_* nonterminals: ::=::=*|**|ltac2_simple_intropattern::=ltac2_naming_intropattern|_|ltac2_or_and_intropattern|ltac2_equality_intropattern::=|()||::=->|<-|::=|?$ lident|?|ident_or_anti::=lident|::=natural|lident|ltac2_constr_with_bindings::=term with ltac2_bindings?::=with ltac2_bindings?::=|term+::=( qhyp := term )::=|natural|lident::=|::=beta|iota|match|fix|cofix|zeta|delta ::=-? [ ]::=|qualid|::=|::=|* |- |::=::=::=-? natural+::=::=::=ident_or_anti|( type of ident_or_anti )|( value of ident_or_anti )::=::=::=term|term with term::=-><-? ltac2_rewriter::=::=ltac2_goal_tactics|::=*|::=*|::=at top|at bottom|after ident_or_anti|before ident_or_anti::=|::=::=||::=::=

## Evaluation¶

Ltac2 features a toplevel loop that can be used to evaluate expressions.

Command Ltac2 Eval ltac2_expr

This command evaluates the term in the current proof if there is one, or in the global environment otherwise, and displays the resulting value to the user together with its type. This command is pure in the sense that it does not modify the state of the proof, and in particular all side-effects are discarded.

## Debug¶

Flag Ltac2 Backtrace

When this flag is set, toplevel failures will be printed with a backtrace.

## Compatibility layer with Ltac1¶

### Ltac1 from Ltac2¶

#### Simple API¶

One can call Ltac1 code from Ltac2 by using the ltac1:(ltac1_expr_in_env) quotation. See Built-in quotations. It parses a Ltac1 expression, and semantics of this quotation is the evaluation of the corresponding code for its side effects. In particular, it cannot return values, and the quotation has type unit.

Ltac1 cannot implicitly access variables from the Ltac2 scope, but this can be done with an explicit annotation on the ltac1:(ident* |- ltac_expr) quotation. See Built-in quotations. For example:

Local Ltac2 replace_with (lhs: constr) (rhs: constr) :=   ltac1:(lhs rhs |- replace lhs with rhs) (Ltac1.of_constr lhs) (Ltac1.of_constr rhs).
Ltac2 Notation "replace" lhs(constr) "with" rhs(constr) := replace_with lhs rhs.

The return type of this expression is a function of the same arity as the number of identifiers, with arguments of type Ltac2.Ltac1.t (see below). This syntax will bind the variables in the quoted Ltac1 code as if they had been bound from Ltac1 itself. Similarly, the arguments applied to the quotation will be passed at runtime to the Ltac1 code.

#### Low-level API¶

There exists a lower-level FFI into Ltac1 that is not recommended for daily use, which is available in the Ltac2.Ltac1 module. This API allows to directly manipulate dynamically-typed Ltac1 values, either through the function calls, or using the ltac1val quotation. The latter parses the same as ltac1, but has type Ltac2.Ltac1.t instead of unit, and dynamically behaves as an Ltac1 thunk, i.e. ltac1val:(foo) corresponds to the tactic closure that Ltac1 would generate from idtac; foo.

Due to intricate dynamic semantics, understanding when Ltac1 value quotations focus is very hard. This is why some functions return a continuation-passing style value, as it can dispatch dynamically between focused and unfocused behaviour.

The same mechanism for explicit binding of variables as described in the previous section applies.

### Ltac2 from Ltac1¶

Same as above by switching Ltac1 by Ltac2 and using the ltac2 quotation instead.

+=ltac2 : ( ltac2_expr )|ltac2 : ( |- ltac2_expr )

The typing rules are dual, that is, the optional identifiers are bound with type Ltac2.Ltac1.t in the Ltac2 expression, which is expected to have type unit. The value returned by this quotation is an Ltac1 function with the same arity as the number of bound variables.

Note that when no variables are bound, the inner tactic expression is evaluated eagerly, if one wants to use it as an argument to a Ltac1 function, one has to resort to the good old idtac; ltac2:(foo) trick. For instance, the code below will fail immediately and won't print anything.

From Ltac2 Require Import Ltac2.
Set Default Proof Mode "Classic".
Ltac mytac tac := idtac "I am being evaluated"; tac.
mytac is defined
Goal True.
1 subgoal ============================ True
Proof.
Fail mytac ltac2:(fail).
The command has indeed failed with message: Uncaught Ltac2 exception: Tactic_failure (None)
Fail mytac ltac:(idtac; ltac2:(fail)).
I am being evaluated The command has indeed failed with message: Uncaught Ltac2 exception: Tactic_failure (None)

In any case, the value returned by the fully applied quotation is an unspecified dummy Ltac1 closure and should not be further used.

### Switching between Ltac languages¶

We recommend using the Default Proof Mode option to switch between tactic languages with a proof-based granularity. This allows to incrementally port the proof scripts.

## Transition from Ltac1¶

Owing to the use of a lot of notations, the transition should not be too difficult. In particular, it should be possible to do it incrementally. That said, we do not guarantee it will be a blissful walk either. Hopefully, owing to the fact Ltac2 is typed, the interactive dialogue with Coq will help you.

We list the major changes and the transition strategies hereafter.

### Syntax changes¶

Due to conflicts, a few syntactic rules have changed.

• The dispatch tactical tac; [foo|bar] is now written tac > [foo|bar].

• Levels of a few operators have been revised. Some tacticals now parse as if they were normal functions. Parentheses are now required around complex arguments, such as abstractions. The tacticals affected are: try, repeat, do, once, progress, time, abstract.

• idtac is no more. Either use () if you expect nothing to happen, (fun () => ()) if you want a thunk (see next section), or use printing primitives from the Message module if you want to display something.

### Tactic delay¶

Tactics are not magically delayed anymore, neither as functions nor as arguments. It is your responsibility to thunk them beforehand and apply them at the call site.

A typical example of a delayed function:

Ltac foo := blah.

becomes

Ltac2 foo () := blah.

All subsequent calls to foo must be applied to perform the same effect as before.

Likewise, for arguments:

Ltac bar tac := tac; tac; tac.

becomes

Ltac2 bar tac := tac (); tac (); tac ().

We recommend the use of syntactic notations to ease the transition. For instance, the first example can alternatively be written as:

Ltac2 foo0 () := blah. Ltac2 Notation foo := foo0 ().

This allows to keep the subsequent calls to the tactic as-is, as the expression foo will be implicitly expanded everywhere into foo0 (). Such a trick also works for arguments, as arguments of syntactic notations are implicitly thunked. The second example could thus be written as follows.

Ltac2 bar0 tac := tac (); tac (); tac (). Ltac2 Notation bar := bar0.

### Variable binding¶

Ltac1 relies on complex dynamic trickery to be able to tell apart bound variables from terms, hypotheses, etc. There is no such thing in Ltac2, as variables are recognized statically and other constructions do not live in the same syntactic world. Due to the abuse of quotations, it can sometimes be complicated to know what a mere identifier represents in a tactic expression. We recommend tracking the context and letting the compiler print typing errors to understand what is going on.

We list below the typical changes one has to perform depending on the static errors produced by the typechecker.

#### In Ltac expressions¶

Error Unbound value​constructor X
• if X is meant to be a term from the current static environment, replace the problematic use by 'X.

• if X is meant to be a hypothesis from the local context, replace the problematic use by &X.

#### In quotations¶

Error The reference X was not found in the current environment
• if X is meant to be a tactic expression bound by a Ltac2 let or function, replace the problematic use by \$X.

• if X is meant to be a hypothesis from the local context, replace the problematic use by &X.

### Exception catching¶

Ltac2 features a proper exception-catching mechanism. For this reason, the Ltac1 mechanism relying on fail taking integers, and tacticals decreasing it, has been removed. Now exceptions are preserved by all tacticals, and it is your duty to catch them and re-raise them as needed.