$\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\Assum}[3]{\kw{Assum}(#1)(#2:#3)} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\Def}[4]{\kw{Def}(#1)(#2:=#3:#4)} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\In}{\kw{in}} \newcommand{\Ind}[4]{\kw{Ind}[#2](#3:=#4)} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[5]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)} \newcommand{\Indpstr}[6]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)/{#6}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\Type}{\textsf{Type}} \newcommand{\WEV}[3]{\mbox{#1[] \vdash #2 \lra #3}} \newcommand{\WEVT}[3]{\mbox{#1[] \vdash #2 \lra}\\ \mbox{ #3}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \end{split}$

# Definitions¶

## Let-in definitions¶

::=
let name : type? := term in term
|
let name : 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 := term in term stands for let ident := fun => term in term.

## Type cast¶

::=
|
|
|
term10 :>

The expression is a type cast expression. It enforces the type of term10 to be 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 DefinitionExample ident_decl def_body
::=
: type? := term
|
::=
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 (see Program Definition) 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 : type := term is equivalent to Definition ident : forall , type := fun  => term.

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 : type with ident_decl : type*
::=
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 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.

These commands accept the program attribute. See Program Lemma.

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.

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.