The Gallina specification language — Coq 8.8.0 documentation

# The Gallina specification language

This chapter describes Gallina, the specification language of Coq. It allows developing mathematical theories and to prove specifications of programs. The theories are built from axioms, hypotheses, parameters, lemmas, theorems and definitions of constants, functions, predicates and sets. The syntax of logical objects involved in theories is described in Section Terms. The language of commands, called The Vernacular is described in Section The Vernacular.

In Coq, logical objects are typed to ensure their logical correctness. The rules implemented by the typing algorithm are described in Chapter Calculus of Inductive Constructions.

## About the grammars in the manual

Grammars are presented in Backus-Naur form (BNF). Terminal symbols are set in black `typewriter font`. In addition, there are special notations for regular expressions.

An expression enclosed in square brackets `[…]` means at most one occurrence of this expression (this corresponds to an optional component).

The notation “`entry sep … sep entry`” stands for a non empty sequence of expressions parsed by entry and separated by the literal “`sep`” [1].

Similarly, the notation “`entry … entry`” stands for a non empty sequence of expressions parsed by the “`entry`” entry, without any separator between.

At the end, the notation “`[entry sep … sep entry]`” stands for a possibly empty sequence of expressions parsed by the “`entry`” entry, separated by the literal “`sep`”.

## Lexical conventions

Blanks
  Space, newline and horizontal tabulation are considered as blanks. Blanks are ignored but they separate tokens.
Comments
  Comments in Coq are enclosed between `(*` and `*)`, and can be nested. They can contain any character. However, string literals must be correctly closed. Comments are treated as blanks.
Identifiers and access identifiers

  Identifiers, written ident, are sequences of letters, digits, `_` and `'`, that do not start with a digit or `'`. That is, they are recognized by the following lexical class:

      first_letter      ::=  a..z ∣ A..Z ∣ _ ∣ unicode-letter
      subsequent_letter ::=  a..z ∣ A..Z ∣ 0..9 ∣ _ ∣ ' ∣ unicode-letter ∣ unicode-id-part
      ident             ::=  first_letter [subsequent_letter … subsequent_letter]
      access_ident      ::=  . ident

  All characters are meaningful. In particular, identifiers are case- sensitive. The entry `unicode-letter` non-exhaustively includes Latin, Greek, Gothic, Cyrillic, Arabic, Hebrew, Georgian, Hangul, Hiragana and Katakana characters, CJK ideographs, mathematical letter-like symbols, hyphens, non-breaking space, … The entry `unicode-id-part` non- exhaustively includes symbols for prime letters and subscripts.

  Access identifiers, written `access_ident`, are identifiers prefixed by . (dot) without blank. They are used in the syntax of qualified identifiers.

Natural numbers and integers

  Numerals are sequences of digits. Integers are numerals optionally preceded by a minus sign.

      digit   ::=  0..9
      num     ::=  digit … digit
      integer ::=  [-] num

Strings
  Strings are delimited by `"` (double quote), and enclose a sequence of any characters different from `"` or the sequence `""` to denote the double quote character. In grammars, the entry for quoted strings is `string`.
Keywords

  The following identifiers are reserved keywords, and cannot be employed otherwise:

      _ as at cofix else end exists exists2 fix for
      forall fun if IF in let match mod Prop return
      Set then Type using where with

Special tokens

  The following sequences of characters are special tokens:

      ! % & && ( () ) * + ++ , - -> . .( ..
      / /\ : :: :< := :> ; < <- <-> <: <= <> =
      => =_D > >-> >= ? ?= @ [ \/ ] ^ { | |-
      || } ~

  Lexical ambiguities are resolved according to the “longest match” rule: when a sequence of non alphanumerical characters can be decomposed into several different ways, then the first token is the longest possible one (among all tokens defined at this moment), and so on.

## Terms

### Syntax of terms

The following grammars describe the basic syntax of the terms of the Calculus of Inductive Constructions (also called Cic). The formal presentation of Cic is given in Chapter Calculus of Inductive Constructions. Extensions of this syntax are given in Chapter Extensions of Gallina. How to customize the syntax is described in Chapter Syntax extensions and interpretation scopes.

    term         ::=  forall binders , term
                      | fun binders => term
                      | fix fix_bodies
                      | cofix cofix_bodies
                      | let ident [binders] [: term] := term in term
                      | let fix fix_body in term
                      | let cofix cofix_body in term
                      | let ( [name , … , name] ) [dep_ret_type] := term in term
                      | let ' pattern [in term] := term [return_type] in term
                      | if term [dep_ret_type] then term else term
                      | term : term
                      | term <: term
                      | term :>
                      | term -> term
                      | term arg … arg
                      | @ qualid [term … term]
                      | term % ident
                      | match match_item , … , match_item [return_type] with
                        [[|] equation | … | equation] end
                      | qualid
                      | sort
                      | num
                      | _
                      | ( term )
    arg          ::=  term
                      | ( ident := term )
    binders      ::=  binder … binder
    binder       ::=  name
                      | ( name … name : term )
                      | ( name [: term] := term )
    name         ::=  ident | _
    qualid       ::=  ident | qualid access_ident
    sort         ::=  Prop | Set | Type
    fix_bodies   ::=  fix_body
                      | fix_body with fix_body with … with fix_body for ident
    cofix_bodies ::=  cofix_body
                      | cofix_body with cofix_body with … with cofix_body for ident
    fix_body     ::=  ident binders [annotation] [: term] := term
    cofix_body   ::=  ident [binders] [: term] := term
    annotation   ::=  { struct ident }
    match_item   ::=  term [as name] [in qualid [pattern … pattern]]
    dep_ret_type ::=  [as name] return_type
    return_type  ::=  return term
    equation     ::=  mult_pattern | … | mult_pattern => term
    mult_pattern ::=  pattern , … , pattern
    pattern      ::=  qualid pattern … pattern
                      | @ qualid pattern … pattern
                      | pattern as ident
                      | pattern % ident
                      | qualid
                      | _
                      | num
                      | ( or_pattern , … , or_pattern )
    or_pattern   ::=  pattern | … | pattern

### Types

Coq terms are typed. Coq types are recognized by the same syntactic class as :token`term`. We denote by `type` the semantic subclass of types inside the syntactic class `term`.

### Qualified identifiers and simple identifiers

Qualified identifiers (`qualid`) denote global constants (definitions, lemmas, theorems, remarks or facts), global variables (parameters or axioms), inductive types or constructors of inductive types. Simple identifiers (or shortly `ident`) are a syntactic subset of qualified identifiers. Identifiers may also denote local variables, what qualified identifiers do not.

### Numerals

Numerals have no definite semantics in the calculus. They are mere notations that can be bound to objects through the notation mechanism (see Chapter Syntax extensions and interpretation scopes for details). Initially, numerals are bound to Peano’s representation of natural numbers (see Datatypes).

Note

negative integers are not at the same level as `num`, for this would make precedence unnatural.

### Sorts

There are three sorts `Set`, `Prop` and `Type`.

- `Prop` is the universe of logical propositions. The logical propositions themselves are typing the proofs. We denote propositions by form. This constitutes a semantic subclass of the syntactic class `term`.
- `Set` is is the universe of program types or specifications. The specifications themselves are typing the programs. We denote specifications by specif. This constitutes a semantic subclass of the syntactic class `term`.
- `Type` is the type of `Prop` and `Set`

More on sorts can be found in Section Sorts.

### Binders

Various constructions such as `fun`, `forall`, `fix` and `cofix` bind variables. A binding is represented by an identifier. If the binding variable is not used in the expression, the identifier can be replaced by the symbol `_`. When the type of a bound variable cannot be synthesized by the system, it can be specified with the notation `(ident:type)`. There is also a notation for a sequence of binding variables sharing the same type: `(``ident`\(_1\)…`ident`\(_n\):`type```)``. A binder can also be any pattern prefixed by a quote, e.g. `‘(x,y)`.

Some constructions allow the binding of a variable to value. This is called a “let-binder”. The entry `binder` of the grammar accepts either an assumption binder as defined above or a let-binder. The notation in the latter case is `(ident:=term)`. In a let-binder, only one variable can be introduced at the same time. It is also possible to give the type of the variable as follows: `(ident:term:=term)`.

Lists of `binder` are allowed. In the case of `fun` and `forall`, it is intended that at least one binder of the list is an assumption otherwise fun and forall gets identical. Moreover, parentheses can be omitted in the case of a single sequence of bindings sharing the same type (e.g.: `fun (x y z : A) => t` can be shortened in `fun x y z : A => t`).

### Abstractions

The expression `fun ident : type => term` defines the abstraction of the variable `ident`, of type `type`, over the term `term`. It denotes a function of the variable `ident` that evaluates to the expression `term` (e.g. `fun x : A => x` denotes the identity function on type `A`). The keyword `fun` can be followed by several binders as given in Section Binders. Functions over several variables are equivalent to an iteration of one-variable functions. For instance the expression “fun `ident`\(_{1}\) … `ident`\(_{n}\) : `type` => `term`” denotes the same function as “ fun `ident`\(_{1}\) : `type` => … fun `ident`\(_{n}\) : `type` => `term`”. If a let-binder occurs in the list of binders, it is expanded to a let-in definition (see Section Let-in definitions).

### Products

The expression `forall ident : type, term` denotes the product of the variable `ident` of type `type`, over the term `term`. As for abstractions, `forall` is followed by a binder list, and products over several variables are equivalent to an iteration of one-variable products. Note that `term` is intended to be a type.

If the variable `ident` occurs in `term`, the product is called dependent product. The intention behind a dependent product `forall x : A, B` is twofold. It denotes either the universal quantification of the variable `x` of type `A` in the proposition `B` or the functional dependent product from `A` to `B` (a construction usually written \(\Pi_{x:A}.B\) in set theory).

Non dependent product types have a special notation: `A -> B` stands for `forall _ : A, B`. The non dependent product is used both to denote the propositional implication and function types.

### Applications

The expression `term`\(_0\) `term`\(_1\) denotes the application of `term`\(_0\) to `term`\(_1\).

The expression `term`\(_0\) `term`\(_1\) … `term`\(_n\) denotes the application of the term `term`\(_0\) to the arguments `term`\(_1\) … then `term`\(_n\). It is equivalent to ( … ( `term`\(_0\) `term`\(_1\) ) … ) `term`\(_n\) : associativity is to the left.

The notation `(ident:=term)` for arguments is used for making explicit the value of implicit arguments (see Section Explicit applications).

### Type cast

The expression `term : type` is a type cast expression. It enforces the type of `term` to be `type`.

`term <: type` locally sets up the virtual machine for checking that `term` has type `type`.

### Inferable subterms

Expressions often contain redundant pieces of information. Subterms that can be automatically inferred by Coq can be replaced by the symbol `_` and Coq will guess the missing piece of information.

### Let-in definitions

`let` `ident` := `term`\(_1\) in `term`\(_2\) denotes the local binding of `term`\(_1\) to the variable `ident` in `term`\(_2\). There is a syntactic sugar for let-in definition of functions: `let` `ident` `binder`\(_1\) … `binder`\(_n\) := `term`\(_1\) in `term`\(_2\) stands for `let` `ident` := `fun` `binder`\(_1\) … `binder`\(_n\) => `term`\(_1\) in `term`\(_2\).

### Definition by case analysis

Objects of inductive types can be destructurated by a case-analysis construction called pattern-matching expression. A pattern-matching expression is used to analyze the structure of an inductive objects and to apply specific treatments accordingly.

This paragraph describes the basic form of pattern-matching. See Section Multiple and nested pattern-matching and Chapter Extended pattern-matching for the description of the general form. The basic form of pattern-matching is characterized by a single `match_item` expression, a `mult_pattern` restricted to a single `pattern` and `pattern` restricted to the form `qualid` `ident`.

The expression match `term`\(_0\) `return_type` with `pattern`\(_1\) => `term`\(_1\) \(|\) … \(|\) `pattern`\(_n\) => `term`\(_n\) end, denotes a `pattern-matching` over the term `term`\(_0\) (expected to be of an inductive type \(I\)). The terms `term`\(_1\)…`term`\(_n\) are the `branches` of the pattern-matching expression. Each of `pattern`\(_i\) has a form `qualid` `ident` where `qualid` must denote a constructor. There should be exactly one branch for every constructor of \(I\).

The `return_type` expresses the type returned by the whole match expression. There are several cases. In the non dependent case, all branches have the same type, and the `return_type` is the common type of branches. In this case, `return_type` can usually be omitted as it can be inferred from the type of the branches [2].

In the dependent case, there are three subcases. In the first subcase, the type in each branch may depend on the exact value being matched in the branch. In this case, the whole pattern-matching itself depends on the term being matched. This dependency of the term being matched in the return type is expressed with an “as `ident`” clause where `ident` is dependent in the return type. For instance, in the following example:

Inductive bool : Type := true : bool | false : bool.
  bool is defined bool_rect is defined bool_ind is defined bool_rec is defined
Inductive eq (A: Type) (x: A) : A -> Prop := eq_refl : eq A x x.
  eq is defined eq_rect is defined eq_ind is defined eq_rec is defined
Inductive or (A: Prop) (B: Prop) : Prop := | or_introl : A -> or A B | or_intror : B -> or A B.
  or is defined or_ind is defined
Definition bool_case (b: bool) : or (eq bool b true) (eq bool b false) := match b as x return or (eq bool x true) (eq bool x false) with | true => or_introl (eq bool true true) (eq bool true false) (eq_refl bool true) | false => or_intror (eq bool false true) (eq bool false false) (eq_refl bool false) end.
  bool_case is defined

the branches have respective types or `eq bool true true :g:`eq bool true false` and or `eq bool false true` `eq bool false false` while the whole pattern-matching expression has type or `eq bool b true` `eq bool b false`, the identifier `x` being used to represent the dependency. Remark that when the term being matched is a variable, the as clause can be omitted and the term being matched can serve itself as binding name in the return type. For instance, the following alternative definition is accepted and has the same meaning as the previous one.

Definition bool_case (b: bool) : or (eq bool b true) (eq bool b false) := match b return or (eq bool b true) (eq bool b false) with | true => or_introl (eq bool true true) (eq bool true false) (eq_refl bool true) | false => or_intror (eq bool false true) (eq bool false false) (eq_refl bool false) end.
  Toplevel input, characters 0-315: > Definition bool_case (b:bool) : or (eq bool b true) (eq bool b false) := match b return or (eq bool b true) (eq bool b false) with | true => or_introl (eq bool true true) (eq bool true false) (eq_refl bool true) | false => or_intror (eq bool false true) (eq bool false false) (eq_refl bool false) end. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Error: bool_case already exists.

The second subcase is only relevant for annotated inductive types such as the equality predicate (see Section Equality), the order predicate on natural numbers or the type of lists of a given length (see Section Matching objects of dependent types). In this configuration, the type of each branch can depend on the type dependencies specific to the branch and the whole pattern-matching expression has a type determined by the specific dependencies in the type of the term being matched. This dependency of the return type in the annotations of the inductive type is expressed using a “in I _ … _ `pattern`\(_1\) … `pattern`\(_n\)” clause, where

- \(I\) is the inductive type of the term being matched;
- the `_` are matching the parameters of the inductive type: the return type is not dependent on them.
- the `pattern`\(_i\) are matching the annotations of the inductive type: the return type is dependent on them
- in the basic case which we describe below, each `pattern`\(_i\) is a name `ident`\(_i\); see Patterns in in for the general case

For instance, in the following example:

Definition eq_sym (A: Type) (x y: A) (H: eq A x y) : eq A y x := match H in eq _ _ z return eq A z x with | eq_refl _ => eq_refl A x end.
  Toplevel input, characters 105-114: > Definition eq_sym (A:Type) (x y:A) (H:eq A x y) : eq A y x := match H in eq _ _ z return eq A z x with | eq_refl _ => eq_refl A x end. > ^^^^^^^^^ Error: The constructor eq_refl (in type eq) expects 2 arguments.

the type of the branch has type `eq A x x` because the third argument of g:eq is g:x in the type of the pattern `refl_equal`. On the contrary, the type of the whole pattern-matching expression has type `eq A y x` because the third argument of eq is y in the type of H. This dependency of the case analysis in the third argument of `eq` is expressed by the identifier g:z in the return type.

Finally, the third subcase is a combination of the first and second subcase. In particular, it only applies to pattern-matching on terms in a type with annotations. For this third subcase, both the clauses as and in are available.

There are specific notations for case analysis on types with one or two constructors: “if … then … else …” and “let (…, ” (see Sections Pattern-matching on boolean values: the if expression and Let-in definitions).

### Recursive functions

The expression “fix `ident`\(_1\) `binder`\(_1\) : `type`\(_1\) `:=` `term`\(_1\) with … with `ident`\(_n\) `binder`\(_n\) : `type`\(_n\) `:=` `term`\(_n\) for `ident`\(_i\)” denotes the \(i\)component of a block of functions defined by mutual well-founded recursion. It is the local counterpart of the `Fixpoint` command. When \(n=1\), the “for `ident`\(_i\)” clause is omitted.

The expression “cofix `ident`\(_1\) `binder`\(_1\) : `type`\(_1\) with … with `ident`\(_n\) `binder`\(_n\) : `type`\(_n\) for `ident`\(_i\)” denotes the \(i\)component of a block of terms defined by a mutual guarded co-recursion. It is the local counterpart of the `CoFixpoint` command. See Section Definitions of recursive objects in co-inductive types for more details. When \(n=1\), the “ for `ident`\(_i\)” clause is omitted.

The association of a single fixpoint and a local definition have a special syntax: “let fix f … := … in …” stands for “let f := fix f … := … in …”. The same applies for co-fixpoints.

## The Vernacular

    sentence           ::=  assumption
                            | definition
                            | inductive
                            | fixpoint
                            | assertion proof
    assumption         ::=  assumption_keyword assums.
    assumption_keyword ::=  Axiom | Conjecture
                            | Parameter | Parameters
                            | Variable | Variables
                            | Hypothesis | Hypotheses
    assums             ::=  ident … ident : term
                            | ( ident … ident : term ) … ( ident … ident : term )
    definition         ::=  [Local] Definition ident [binders] [: term] := term .
                            | Let ident [binders] [: term] := term .
    inductive          ::=  Inductive ind_body with … with ind_body .
                            | CoInductive ind_body with … with ind_body .
    ind_body           ::=  ident [binders] : term :=
                            [[|] ident [binders] [:term] | … | ident [binders] [:term]]
    fixpoint           ::=  Fixpoint fix_body with … with fix_body .
                            | CoFixpoint cofix_body with … with cofix_body .
    assertion          ::=  assertion_keyword ident [binders] : term .
    assertion_keyword  ::=  Theorem | Lemma
                            | Remark | Fact
                            | Corollary | Proposition
                            | Definition | Example
    proof              ::=  Proof . … Qed .
                            | Proof . … Defined .
                            | Proof . … Admitted .

This grammar describes The Vernacular which is the language of commands of Gallina. A sentence of the vernacular language, like in many natural languages, begins with a capital letter and ends with a dot.

The different kinds of command are described hereafter. They all suppose that the terms occurring in the sentences are well-typed.

### Assumptions

Assumptions extend the environment with axioms, parameters, hypotheses or variables. An assumption binds an `ident` to a `type`. It is accepted by Coq if and only if this `type` is a correct type in the environment preexisting the declaration and if `ident` was not previously defined in the same module. This `type` is considered to be the type (or specification, or statement) assumed by `ident` and we say that `ident` has type `type`.

Command `Axiom ident : term.`

  This command links term to the name ident as its specification in the global context. The fact asserted by term is thus assumed as a postulate.

Error `ident already exists (Axiom)`

Variant `Parameter ident : term.`

  Is equivalent to `Axiom` `ident` : `term`

Variant `Parameter ident+ : term.`

  Adds parameters with specification `term`

Variant `Parameter(ident+ : term )+.`

  Adds blocks of parameters with different specifications.

Variant `Parameters (ident+ : term )+.`

  Synonym of `Parameter`.

Variant `Local Axiom ident : term.`

  Such axioms are never made accessible through their unqualified name by `Import` and its variants. You have to explicitly give their fully qualified name to refer to them.

Variant `Conjecture ident : term`

  Is equivalent to `Axiom` `ident` : `term`.

Command `Variable ident : term.`

This command links `term` to the name `ident` in the context of the current section (see Section Section mechanism for a description of the section mechanism). When the current section is closed, name `ident` will be unknown and every object using this variable will be explicitly parametrized (the variable is discharged). Using the `Variable` command out of any section is equivalent to using `Local Parameter`.

Error `ident already exists (Variable)`

Variant `Variable ident+ : term.`

  Links `term` to each `ident`.

Variant `Variable(ident+ : term)+.`

  Adds blocks of variables with different specifications.

Variant `Variables(ident+ : term)+.`

Variant `Hypothesis(ident+ : term)+.`

Variant `Hypotheses(ident+ : term)+.`

Synonyms of `Variable`.

It is advised to use the keywords `Axiom` and `Hypothesis` for logical postulates (i.e. when the assertion term is of sort `Prop`), and to use the keywords `Parameter` and `Variable` in other cases (corresponding to the declaration of an abstract mathematical entity).

### Definitions

Definitions extend the environment with associations of names to terms. A definition can be seen as a way to give a meaning to a name or as a way to abbreviate a term. In any case, the name can later be replaced at any time by its definition.

The operation of unfolding a name into its definition is called \(\delta\)-conversion (see Section δ-reduction). A definition is accepted by the system if and only if the defined term is well-typed in the current context of the definition and if the name is not already used. The name defined by the definition is called a constant and the term it refers to is its body. A definition has a type which is the type of its body.

A formal presentation of constants and environments is given in Section Typing rules.

Command `Definition ident := term.`

  This command binds `term` to the name `ident` in the environment, provided that `term` is well-typed.

Error `ident already exists (Definition)`

Variant `Definition ident : term := term.`

  It checks that the type of `term`\(_2\) is definitionally equal to `term`\(_1\), and registers `ident` as being of type `term`\(_1\), and bound to value `term`\(_2\).

Variant `Definition ident binder* : term := term.`

  This is equivalent to `Definition` `ident` : `forall` `binder`\(_1\) … `binder`\(_n\),`term`\(_1\):= fun`binder`\(_1\) … `binder`\(_n\)=>`term`\(_2\).

Variant `Local Definition ident := term.`

  Such definitions are never made accessible through their unqualified name by `Import` and its variants. You have to explicitly give their fully qualified name to refer to them.

Variant `Example ident := term.`

Variant `Example ident : term := term.`

Variant `Example ident binder* : term := term.`

These are synonyms of the Definition forms.

Error `The term term has type type while it is expected to have type type`

See also `Opaque`, `Transparent`, `unfold`.

Command `Let ident := term.`

This command binds the value `term` to the name `ident` in the environment of the current section. The name `ident` disappears when the current section is eventually closed, and, all persistent objects (such as theorems) defined within the section and depending on `ident` are prefixed by the let-in definition `let` `ident` `:=` `term` `in`. Using the `Let` command out of any section is equivalent to using `Local Definition`.

Error `ident already exists (Let)`

Variant `Let ident : term := term.`

Variant `Let Fixpoint ident fix_body with fix_body*.`

Variant `Let CoFixpoint ident cofix_body with cofix_body*.`

See also Sections Section mechanism, commands `Opaque`, `Transparent`, and tactic `unfold`.

### Inductive definitions

We gradually explain simple inductive types, simple annotated inductive types, simple parametric inductive types, mutually inductive types. We explain also co-inductive types.

#### Simple inductive types

The definition of a simple inductive type has the following form:

Command `Inductive ident : sort := |? ident : type | ident : type*`

The name `ident` is the name of the inductively defined type and `sort` is the universes where it lives. The `ident` are the names of its constructors and `type` their respective types. The types of the constructors have to satisfy a positivity condition (see Section Positivity Condition) for `ident`. This condition ensures the soundness of the inductive definition. If this is the case, the `ident` are added to the environment with their respective types. Accordingly to the universe where the inductive type lives (e.g. its type `sort`), Coq provides a number of destructors for `ident`. Destructors are named `ident_ind`, `ident_rec` or `ident_rect` which respectively correspond to elimination principles on `Prop`, `Set` and `Type`. The type of the destructors expresses structural induction/recursion principles over objects of `ident`. We give below two examples of the use of the Inductive definitions.

The set of natural numbers is defined as:

Inductive nat : Set := | O : nat | S : nat -> nat.
  nat is defined nat_rect is defined nat_ind is defined nat_rec is defined

The type nat is defined as the least `Set` containing `O` and closed by the `S` constructor. The names `nat`, `O` and `S` are added to the environment.

Now let us have a look at the elimination principles. They are three of them: `nat_ind`, `nat_rec` and `nat_rect`. The type of `nat_ind` is:

Check nat_ind.
  nat_ind : forall P : nat -> Prop, P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n

This is the well known structural induction principle over natural numbers, i.e. the second-order form of Peano’s induction principle. It allows proving some universal property of natural numbers (`forall n:nat, P n`) by induction on `n`.

The types of `nat_rec` and `nat_rect` are similar, except that they pertain to `(P:nat->Set)` and `(P:nat->Type)` respectively. They correspond to primitive induction principles (allowing dependent types) respectively over sorts `Set` and `Type`. The constant `ident_ind` is always provided, whereas `ident_rec` and `ident_rect` can be impossible to derive (for example, when `ident` is a proposition).

Inductive nat : Set := O | S (_: nat).
  Toplevel input, characters 0-37: > Inductive nat : Set := O | S (_:nat). > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Error: nat already exists.

In the case where inductive types have no annotations (next section gives an example of such annotations), a constructor can be defined by only giving the type of its arguments.

#### Simple annotated inductive types

In an annotated inductive types, the universe where the inductive type is defined is no longer a simple sort, but what is called an arity, which is a type whose conclusion is a sort.

As an example of annotated inductive types, let us define the `even` predicate:

Inductive even : nat -> Prop := | even_0 : even O | even_SS : forall n: nat, even n -> even (S (S n)).
  even is defined even_ind is defined

The type `nat->Prop` means that even is a unary predicate (inductively defined) over natural numbers. The type of its two constructors are the defining clauses of the predicate even. The type of `even_ind` is:

Check even_ind.
  even_ind : forall P : nat -> Prop, P O -> (forall n : nat, even n -> P n -> P (S (S n))) -> forall n : nat, even n -> P n

From a mathematical point of view it asserts that the natural numbers satisfying the predicate even are exactly in the smallest set of naturals satisfying the clauses `even_0` or `even_SS`. This is why, when we want to prove any predicate `P` over elements of `even`, it is enough to prove it for `O` and to prove that if any natural number `n` satisfies `P` its double successor `(S (S n))` satisfies also `P`. This is indeed analogous to the structural induction principle we got for `nat`.

Error `Non strictly positive occurrence of ident in type`

Error `The conclusion of type is not valid; it must be built from ident`

#### Parametrized inductive types

In the previous example, each constructor introduces a different instance of the predicate even. In some cases, all the constructors introduces the same generic instance of the inductive definition, in which case, instead of an annotation, we use a context of parameters which are binders shared by all the constructors of the definition.

The general scheme is:

Variant `Inductive ident binder+ : term := |? ident : type | ident : type*`

Parameters differ from inductive type annotations in the fact that the conclusion of each type of constructor `term` invoke the inductive type with the same values of parameters as its specification.

A typical example is the definition of polymorphic lists:

Inductive list (A: Set) : Set := | nil : list A | cons : A -> list A -> list A.
  list is defined list_rect is defined list_ind is defined list_rec is defined

Note

In the type of `nil` and `cons`, we write `(list A)` and not just `list`. The constructors `nil` and `cons` will have respectively types:

Check nil.
  nil : forall A : Set, list A
Check cons.
  cons : forall A : Set, A -> list A -> list A

Types of destructors are also quantified with `(A:Set)`.

##### Variants

Inductive list (A: Set) : Set := nil | cons (_: A) (_: list A).
  Toplevel input, characters 0-60: > Inductive list (A:Set) : Set := nil | cons (_:A) (_:list A). > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Error: list already exists.

This is an alternative definition of lists where we specify the arguments of the constructors rather than their full type.

Variant sum (A B: Set) : Set := left : A -> sum A B | right : B -> sum A B.
  sum is defined

The `Variant` keyword is identical to the `Inductive` keyword, except that it disallows recursive definition of types (in particular lists cannot be defined with the Variant keyword). No induction scheme is generated for this variant, unless `Nonrecursive Elimination Schemes` is set.

Error `The num th argument of ident must be ident in type`

##### New from Coq V8.1

The condition on parameters for inductive definitions has been relaxed since Coq V8.1. It is now possible in the type of a constructor, to invoke recursively the inductive definition on an argument which is not the parameter itself.

One can define :

Inductive list2 (A: Set) : Set := | nil2 : list2 A | cons2 : A -> list2 (A*A) -> list2 A.
  list2 is defined list2_rect is defined list2_ind is defined list2_rec is defined

that can also be written by specifying only the type of the arguments:

Inductive list2 (A: Set) : Set := nil2 | cons2 (_: A) (_: list2 (A*A)).
  list2 is defined list2_rect is defined list2_ind is defined list2_rec is defined

But the following definition will give an error:

Fail Inductive listw (A: Set) : Set := | nilw : listw (A*A) | consw : A -> listw (A*A) -> listw (A*A).
  The command has indeed failed with message: Last occurrence of “listw” must have “A” as 1st argument in “listw (A * A)%type”.

Because the conclusion of the type of constructors should be `listw A` in both cases.

A parametrized inductive definition can be defined using annotations instead of parameters but it will sometimes give a different (bigger) sort for the inductive definition and will produce a less convenient rule for case elimination.

See also Section Inductive Definitions and the `induction` tactic.

#### Mutually defined inductive types

The definition of a block of mutually inductive types has the form:

Variant `Inductive ident : term := |? ident : type | ident : type* with ident : term := |? ident : type | ident : type**.`

It has the same semantics as the above `Inductive` definition for each `ident` All `ident` are simultaneously added to the environment. Then well-typing of constructors can be checked. Each one of the `ident` can be used on its own.

It is also possible to parametrize these inductive definitions. However, parameters correspond to a local context in which the whole set of inductive declarations is done. For this reason, the parameters must be strictly the same for each inductive types The extended syntax is:

Variant `Inductive ident binder+ : term := |? ident : type | ident : type* with ident binder+ : term := |? ident : type | ident : type**.`

The typical example of a mutual inductive data type is the one for trees and forests. We assume given two types `A` and `B` as variables. It can be declared the following way.

Variables A B : Set.
  Toplevel input, characters 0-20: > Variables A B : Set. > ^^^^^^^^^^^^^^^^^^^^ Warning: A is declared as a local axiom [local-declaration,scope] A is declared Toplevel input, characters 0-20: > Variables A B : Set. > ^^^^^^^^^^^^^^^^^^^^ Warning: B is declared as a local axiom [local-declaration,scope] B is declared
Inductive tree : Set := node : A -> forest -> tree with forest : Set := | leaf : B -> forest | cons : tree -> forest -> forest.
  tree, forest are defined tree_rect is defined tree_ind is defined tree_rec is defined forest_rect is defined forest_ind is defined forest_rec is defined

This declaration generates automatically six induction principles. They are respectively called `tree_rec`, `tree_ind`, `tree_rect`, `forest_rec`, `forest_ind`, `forest_rect`. These ones are not the most general ones but are just the induction principles corresponding to each inductive part seen as a single inductive definition.

To illustrate this point on our example, we give the types of `tree_rec` and `forest_rec`.

Check tree_rec.
  tree_rec : forall P : tree -> Set, (forall (a : A) (f : forest), P (node a f)) -> forall t : tree, P t
Check forest_rec.
  forest_rec : forall P : forest -> Set, (forall b : B, P (leaf b)) -> (forall (t : tree) (f0 : forest), P f0 -> P (cons t f0)) -> forall f1 : forest, P f1

Assume we want to parametrize our mutual inductive definitions with the two type variables `A` and `B`, the declaration should be done the following way:

Inductive tree (A B: Set) : Set := node : A -> forest A B -> tree A B with forest (A B: Set) : Set := | leaf : B -> forest A B | cons : tree A B -> forest A B -> forest A B.
  Toplevel input, characters 0-177: > Inductive tree (A B:Set) : Set := node : A -> forest A B -> tree A B with forest (A B:Set) : Set := | leaf : B -> forest A B | cons : tree A B -> forest A B -> forest A B. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Error: forest already exists.

Assume we define an inductive definition inside a section. When the section is closed, the variables declared in the section and occurring free in the declaration are added as parameters to the inductive definition.

See also Section Section mechanism.

#### Co-inductive types

The objects of an inductive type are well-founded with respect to the constructors of the type. In other words, such objects contain only a finite number of constructors. Co-inductive types arise from relaxing this condition, and admitting types whose objects contain an infinity of constructors. Infinite objects are introduced by a non-ending (but effective) process of construction, defined in terms of the constructors of the type.

An example of a co-inductive type is the type of infinite sequences of natural numbers, usually called streams. It can be introduced in Coq using the `CoInductive` command:

CoInductive Stream : Set := Seq : nat -> Stream -> Stream.
  Stream is defined

The syntax of this command is the same as the command `Inductive`. Notice that no principle of induction is derived from the definition of a co-inductive type, since such principles only make sense for inductive ones. For co-inductive ones, the only elimination principle is case analysis. For example, the usual destructors on streams `hd:Stream->nat` and `tl:Str->Str` can be defined as follows:

Definition hd (x: Stream) := let (a,s) := x in a.
  hd is defined
Definition tl (x: Stream) := let (a,s) := x in s.
  tl is defined

Definition of co-inductive predicates and blocks of mutually co-inductive definitions are also allowed. An example of a co-inductive predicate is the extensional equality on streams:

CoInductive EqSt : Stream -> Stream -> Prop := eqst : forall s1 s2: Stream, hd s1 = hd s2 -> EqSt (tl s1) (tl s2) -> EqSt s1 s2.
  EqSt is defined

In order to prove the extensionally equality of two streams `s1` and `s2` we have to construct an infinite proof of equality, that is, an infinite object of type `(EqSt s1 s2)`. We will see how to introduce infinite objects in Section Definitions of recursive objects in co-inductive types.

### Definition of recursive functions

#### Definition of functions by recursion over inductive objects

This section describes the primitive form of definition by recursion over inductive objects. See the `Function` command for more advanced constructions.

Command `Fixpoint ident params {struct ident} : type := term.`

This command allows defining functions by pattern-matching over inductive objects using a fixed point construction. The meaning of this declaration is to define `ident` a recursive function with arguments specified by the binders in `params` such that `ident` applied to arguments corresponding to these binders has type `type`\(_0\), and is equivalent to the expression `term`\(_0\). The type of the `ident` is consequently `forall` `params`, `type`\(_0\) and the value is equivalent to `fun` `params` `=>` `term`\(_0\).

To be accepted, a `Fixpoint` definition has to satisfy some syntactical constraints on a special argument called the decreasing argument. They are needed to ensure that the Fixpoint definition always terminates. The point of the {struct `ident`} annotation is to let the user tell the system which argument decreases along the recursive calls. For instance, one can define the addition function as :

Fixpoint add (n m: nat) {struct n} : nat := match n with | O => m | S p => S (add p m) end.
  add is defined add is recursively defined (decreasing on 1st argument)

The `{struct` `ident```}`` annotation may be left implicit, in this case the system try successively arguments from left to right until it finds one that satisfies the decreasing condition.

Note

Some fixpoints may have several arguments that fit as decreasing arguments, and this choice influences the reduction of the fixpoint. Hence an explicit annotation must be used if the leftmost decreasing argument is not the desired one. Writing explicit annotations can also speed up type-checking of large mutual fixpoints.

The match operator matches a value (here `n`) with the various constructors of its (inductive) type. The remaining arguments give the respective values to be returned, as functions of the parameters of the corresponding constructor. Thus here when `n` equals `O` we return `m`, and when `n` equals `(S p)` we return `(S (add p m))`.

The match operator is formally described in detail in Section The match … with … end construction. The system recognizes that in the inductive call `(add p m)` the first argument actually decreases because it is a pattern variable coming from `match n with`.

Example

The following definition is not correct and generates an error message:

Fail Fixpoint wrongplus (n m: nat) {struct n} : nat := match m with | O => n | S p => S (wrongplus n p) end.
  The command has indeed failed with message: Recursive definition of wrongplus is ill-formed. In environment wrongplus : nat -> nat -> nat n : nat m : nat p : nat Recursive call to wrongplus has principal argument equal to “n” instead of a subterm of “n”. Recursive definition is: “fun n m : nat => match m with | 0 => n | S p => S (wrongplus n p) end“.

because the declared decreasing argument n actually does not decrease in the recursive call. The function computing the addition over the second argument should rather be written:

Fixpoint plus (n m: nat) {struct m} : nat := match m with | O => n | S p => S (plus n p) end.
  plus is defined plus is recursively defined (decreasing on 2nd argument)

Example

The ordinary match operation on natural numbers can be mimicked in the following way.

Fixpoint nat_match (C: Set) (f0: C) (fS: nat -> C -> C) (n: nat) {struct n} : C := match n with | O => f0 | S p => fS p (nat_match C f0 fS p) end.
  nat_match is defined nat_match is recursively defined (decreasing on 4th argument)

Example

The recursive call may not only be on direct subterms of the recursive variable n but also on a deeper subterm and we can directly write the function mod2 which gives the remainder modulo 2 of a natural number.

Fixpoint mod2 (n: nat) : nat := match n with | O => O | S p => match p with | O => S O | S q => mod2 q end end.
  mod2 is defined mod2 is recursively defined (decreasing on 1st argument)

In order to keep the strong normalization property, the fixed point reduction will only be performed when the argument in position of the decreasing argument (which type should be in an inductive definition) starts with a constructor.

The `Fixpoint` construction enjoys also the with extension to define functions over mutually defined inductive types or more generally any mutually recursive definitions.

Variant `Fixpoint ident params {struct ident} : type := term with ident params+ : type := term*.`

allows to define simultaneously fixpoints.

The size of trees and forests can be defined the following way:

Fixpoint tree_size (t: tree) : nat := match t with | node a f => S (forest_size f) end with forest_size (f: forest) : nat := match f with | leaf b => 1 | cons t f’ => (tree_size t + forest_size f’) end.
  tree_size is defined forest_size is defined tree_size, forest_size are recursively defined (decreasing respectively on 1st, 1st arguments)

A generic command Scheme is useful to build automatically various mutual induction principles. It is described in Section Generation of induction principles with Scheme.

#### Definitions of recursive objects in co-inductive types

Command `CoFixpoint ident : type := term.`

introduces a method for constructing an infinite object of a coinductive type. For example, the stream containing all natural numbers can be introduced applying the following method to the number `O` (see Section Co-inductive types for the definition of `Stream`, `hd` and `tl`):

CoFixpoint from (n: nat) : Stream := Seq n (from (S n)).
  from is defined from is corecursively defined

Oppositely to recursive ones, there is no decreasing argument in a co-recursive definition. To be admissible, a method of construction must provide at least one extra constructor of the infinite object for each iteration. A syntactical guard condition is imposed on co-recursive definitions in order to ensure this: each recursive call in the definition must be protected by at least one constructor, and only by constructors. That is the case in the former definition, where the single recursive call of `from` is guarded by an application of `Seq`. On the contrary, the following recursive function does not satisfy the guard condition:

Fail CoFixpoint filter (p: nat -> bool) (s: Stream) : Stream := if p (hd s) then Seq (hd s) (filter p (tl s)) else filter p (tl s).
  The command has indeed failed with message: Recursive definition of filter is ill-formed. In environment filter : (nat -> bool) -> Stream -> Stream p : nat -> bool s : Stream Unguarded recursive call in “filter p (tl s)”. Recursive definition is: “fun (p : nat -> bool) (s : Stream) => if p (hd s) then Seq (hd s) (filter p (tl s)) else filter p (tl s)”.

The elimination of co-recursive definition is done lazily, i.e. the definition is expanded only when it occurs at the head of an application which is the argument of a case analysis expression. In any other context, it is considered as a canonical expression which is completely evaluated. We can test this using the command `Eval`, which computes the normal forms of a term:

Eval compute in (from 0).
  = (cofix from (n : nat) : Stream := Seq n (from (S n))) 0 : Stream
Eval compute in (hd (from 0)).
  = 0 : nat
Eval compute in (tl (from 0)).
  = (cofix from (n : nat) : Stream := Seq n (from (S n))) 1 : Stream

Variant `CoFixpoint ident params : type := term`

  As for most constructions, arguments of co-fixpoints expressions can be introduced before the `:=` sign.

Variant `CoFixpoint ident : type := term with ident : type := term+`

  As in the `Fixpoint` command, it is possible to introduce a block of mutually dependent methods.

### Assertions and proofs

An assertion states a proposition (or a type) of which the proof (or an inhabitant of the type) is interactively built using tactics. The interactive proof mode is described in Chapter Proof handling and the tactics in Chapter Tactics. The basic assertion command is:

Command `Theorem ident : type.`

After the statement is asserted, Coq needs a proof. Once a proof of `type` under the assumptions represented by `binders` is given and validated, the proof is generalized into a proof of forall , `type` and the theorem is bound to the name `ident` in the environment.

Error `The term term has type type which should be Set, Prop or Type`

Error `ident already exists (Theorem)`

  The name you provided is already defined. You have then to choose another name.

Variant `Lemma ident : type.`

Variant `Remark ident : type.`

Variant `Fact ident : type.`

Variant `Corollary ident : type.`

Variant `Proposition ident : type.`

  These commands are synonyms of `Theorem` `ident` : `type`.

Variant `Theorem ident : type with ident : type*.`

  This command is useful for theorems that are proved by simultaneous induction over a mutually inductive assumption, or that assert mutually dependent statements in some mutual co-inductive type. It is equivalent to `Fixpoint` or `CoFixpoint` but using tactics to build the proof of the statements (or the body of the specification, depending on the point of view). The inductive or co-inductive types on which the induction or coinduction has to be done is assumed to be non ambiguous and is guessed by the system.

  Like in a `Fixpoint` or `CoFixpoint` definition, the induction hypotheses have to be used on structurally smaller arguments (for a `Fixpoint`) or be guarded by a constructor (for a `CoFixpoint`). The verification that recursive proof arguments are correct is done only at the time of registering the lemma in the environment. To know if the use of induction hypotheses is correct at some time of the interactive development of a proof, use the command `Guarded`.

  The command can be used also with `Lemma`, `Remark`, etc. instead of `Theorem`.

Variant `Definition ident : type.`

  This allows defining a term of type `type` using the proof editing mode. It behaves as Theorem but is intended to be used in conjunction with `Defined` in order to define a constant of which the computational behavior is relevant.

  The command can be used also with `Example` instead of `Definition`.

  See also `Opaque`, `Transparent`, `unfold`.

Variant `Let ident : type.`

  Like Definition `ident` : `type`. except that the definition is turned into a let-in definition generalized over the declarations depending on it after closing the current section.

Variant `Fixpoint ident binders with .`

  This generalizes the syntax of Fixpoint so that one or more bodies can be defined interactively using the proof editing mode (when a body is omitted, its type is mandatory in the syntax). When the block of proofs is completed, it is intended to be ended by Defined.

Variant `CoFixpoint ident with.`

  This generalizes the syntax of CoFixpoint so that one or more bodies can be defined interactively using the proof editing mode.

Command `Proof`

  A proof starts by the keyword Proof. Then Coq enters the proof editing mode until the proof is completed. The proof editing mode essentially contains tactics that are described in chapter Tactics. Besides tactics, there are commands to manage the proof editing mode. They are described in Chapter Proof handling.

Command `Qed`

  When the proof is completed it should be validated and put in the environment using the keyword Qed.

Error `ident already exists (Qed)`

Note

1. Several statements can be simultaneously asserted.
2. Not only other assertions but any vernacular command can be given while in the process of proving a given assertion. In this case, the command is understood as if it would have been given before the statements still to be proved.
3. Proof is recommended but can currently be omitted. On the opposite side, Qed (or Defined, see below) is mandatory to validate a proof.
4. Proofs ended by Qed are declared opaque. Their content cannot be unfolded (see Performing computations), thus realizing some form of proof-irrelevance. To be able to unfold a proof, the proof should be ended by Defined (see below).

Variant `Defined`

  Same as `Qed` but the proof is then declared transparent, which means that its content can be explicitly used for type-checking and that it can be unfolded in conversion tactics (see Performing computations, `Opaque`, `Transparent`).

Variant `Admitted.`

  Turns the current asserted statement into an axiom and exits the proof mode.

[1] | This is similar to the expression “entry \(\{\) sep entry \(\}\)” in standard BNF, or “entry \((\) sep entry \()\)*” in the syntax of regular expressions.

[2] | Except if the inductive type is empty in which case there is no equation that can be used to infer the return type.