Definitions — Coq 8.12.0 documentation

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# Definitions

## Let-in definitions

term_let`::=`let name : type? := term in term```|`let name binder+ : type? := term in term```|`let ( name*, ) as name? return term100? := term in term```|`let ' pattern := term return term100? in term```|`let ' pattern in pattern := term return term100 in term

`let ident := term in term’` denotes the local binding of `term` to the variable `ident` in `term`’. There is a syntactic sugar for let-in definition of functions: `let ident binder+ := term in term’` stands for `let ident := fun binder+ => term in term’`.

## Type cast

term_cast`::=`term10 <: type```|`term10 <<: type```|`term10 : type```|`term10 :>

The expression `term10 : type` is a type cast expression. It enforces the type of `term10` to be `type`.

`term10 <: type` locally sets up the virtual machine for checking that `term10` has type `type`.

`term10 <<: type` uses native compilation for checking that `term10` has type `type`.

## Top-level 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​Example ident_decl def_body`
  def_body`::=`binder* : type? := reduce? term```|`binder* : type reduce`::=`Eval red_expr in

  These commands bind `term` to the name `ident` in the environment, provided that `term` is well-typed. They can take the `local` attribute, which makes the defined `ident` accessible by `Import` and its variants only through their fully qualified names. If `reduce` is present then `ident` is bound to the result of the specified computation on `term`.

  These commands also support the `universes(polymorphic)`, `universes(monomorphic)`, `program` and `canonical` attributes.

  If `term` is omitted, `type` is required and Coq enters proof editing mode. This can be used to define a term incrementally, in particular by relying on the `refine` tactic. In this case, the proof should be terminated with `Defined` in order to define a constant for which the computational behavior is relevant. See Entering and leaving proof editing mode.

  The form `Definition ident : type := term` checks that the type of `term` is definitionally equal to `type`, and registers `ident` as being of type `type`, and bound to value `term`.

  The form `Definition ident binder* : type := term` is equivalent to `Definition ident : forall binder*, type := fun binder* => term`.

  See also

  `Opaque`, `Transparent`, `unfold`.

  Error `ident already exists.`
  Error `The term term has type type while it is expected to have type type'.`

## 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 `thm_token ident_decl binder* : type with ident_decl binder* : type*`
  thm_token`::=`Theorem```|`Lemma```|`Fact```|`Remark```|`Corollary```|`Proposition```|`Property

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

  Forms using the `with` clause are 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`.

  Error `The term term has type type which should be Set, Prop or Type.`
  Error `ident already exists.`

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

  Error `Nested proofs are not allowed unless you turn the Nested Proofs Allowed flag on.`

    You are asserting a new statement while already being in proof editing mode. This feature, called nested proofs, is disabled by default. To activate it, turn the `Nested Proofs Allowed` flag on.

Proofs start with the keyword `Proof`. Then Coq enters the proof editing mode until the proof is completed. In proof editing mode, the user primarily enters tactics, which are described in chapter Tactics. The user may also enter commands to manage the proof editing mode. They are described in Chapter Proof handling.

When the proof is complete, use the `Qed` command so the kernel verifies the proof and adds it to the environment.

Note

1. Several statements can be simultaneously asserted provided the `Nested Proofs Allowed` flag was turned on.

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. Nonetheless, this practice is discouraged and may stop working in future versions.

3. 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`.

4. `Proof` is recommended but can currently be omitted. On the opposite side, `Qed` (or `Defined`) is mandatory to validate a proof.

5. One can also use `Admitted` in place of `Qed` to turn the current asserted statement into an axiom and exit the proof editing mode.

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