::=let name : type? := term in term
|let name binder+ : type? := term in term
let ident := term1 in term2 represents the local binding of
ident to the value
let ident binder+ := term1 in term2 is an abbreviation
let ident := fun binder+ => term1 in term2.
Extensions of the
let ... in ... syntax are described in
Irrefutable patterns: the destructuring let variants.
::=term10 : type
|term10 <: type
|term10 <<: type
|term10 :> type
term10 : type is a type cast expression. It enforces
the type of
term10 to be
term10 <: type specifies that the virtual machine will be used
to type check that
term10 has type
term10 <<: type specifies that compilation to OCaml will be used
to type check that
term10 has type
term10 :> type enforces the type of
term10 to be
type without leaving a trace in the produced value.
This is a volatile cast.
Definitions extend the global environment by associating 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-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.
DefinitionExample ident_decl def_body¶
::=binder* : type? := reduce? term
|binder* : type
::=Eval red_expr in
These commands bind
termto the name
identin the global environment, provided that
termis well-typed. They can take the
localattribute, which makes the defined
Importand its variants only through their fully qualified names. If
reduceis present then
identis bound to the result of the specified computation on
These commands also support the
program(see Program Definition),
typeis required and Coq enters proof mode. This can be used to define a term incrementally, in particular by relying on the
refinetactic. In this case, the proof should be terminated with
Definedin order to define a constant for which the computational behavior is relevant. See Entering and exiting proof mode.
Definition ident : type := termchecks that the type of
termis definitionally equal to
type, and registers
identas being of type
type, and bound to value
Definition ident binder* : type := termis equivalent to
Definition ident : forall binder*, type := fun binder* => term.
Assertions and proofs¶
An assertion states a proposition (or a type) for which the proof (or an inhabitant of the type) is interactively built using tactics. Assertions cause Coq to enter proof mode (see Proof mode). Common tactics are described in the Basic proof writing chapter. The basic assertion command is:
thm_token ident_decl binder* : type with ident_decl binder* : type*¶
After the statement is asserted, Coq needs a proof. Once a proof of
typeunder the assumptions represented by
binders is given and validated, the proof is generalized into a proof of
forall binder*, typeand the theorem is bound to the name
identin the global environment.
These commands accept the
programattribute. See Program Lemma.
Forms using the
withclause are useful for theorems that are proved by simultaneous induction over a mutually inductive assumption, or that assert mutually dependent statements in some mutual coinductive type. It is equivalent to
CoFixpointbut using tactics to build the proof of the statements (or the body of the specification, depending on the point of view). The inductive or coinductive types on which the induction or coinduction has to be done is assumed to be unambiguous and is guessed by the system.
Like in a
CoFixpointdefinition, 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 global environment. To know if the use of induction hypotheses is correct at some time of the interactive development of a proof, use the command
This command accepts the
ident already exists.¶
The name you provided is already defined. You have then to choose another name.
Nested proofs are discouraged and not allowed by default. This error probably means that you forgot to close the last "Proof." with "Qed." or "Defined.". If you really intended to use nested proofs, you can do so by turning the "Nested Proofs Allowed" flag on.¶
You are asserting a new statement when you're already in proof mode. This feature, called nested proofs, is disabled by default. To activate it, turn the
Nested Proofs Allowedflag on.
Proofs start with the keyword
Proof. Then Coq enters the proof mode
until the proof is completed. In proof mode, the user primarily enters
tactics (see Basic proof writing). The user may also enter
commands to manage the proof mode (see Proof mode).
When the proof is complete, use the
Qed command so the kernel verifies
the proof and adds it to the global environment.
Several statements can be simultaneously asserted provided the
Nested Proofs Allowedflag was turned on.
Not only other assertions but any 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.
Proofs ended by
Qedare declared opaque. Their content cannot be unfolded (see Applying conversion rules), thus realizing some form of proof-irrelevance. Proofs that end with
Definedcan be unfolded.
Proofis recommended but can currently be omitted. On the opposite side,
Defined) is mandatory to validate a proof.
One can also use
Admittedin place of
Qedto turn the current asserted statement into an axiom and exit proof mode.