\[\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \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}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[4]{\kw{Ind}_{#4}[#1](#2:=#3)} \newcommand{\Indpstr}[5]{\kw{Ind}_{#4}[#1](#2:=#3)/{#5}} \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{\ModImp}[3]{{\kw{Mod}}({#1}:{#2}:={#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}\]

# Functions and assumptions

## Binders

open_binders`::=`name+ : type`````|`binder+``name`::=`_`````|`ident``binder`::=`name`````|`( name+ : type )`````|`( name : type? := term )`````|`implicit_binders`````|`generalizing_binder`````|`( name : type | term )`````|`' pattern0``

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+ : 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 : type := term)`.

Lists of `binder`s 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`).

## Functions (fun) and function types (forall)

term_forall_or_fun`::=`forall open_binders , type`````|`fun open_binders => term``

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 identi+ : type => term` denotes the same function as `fun identi : 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).

The expression `forall ident : type1, type2` denotes the product type (or product) of the variable `ident` of type `type1` over the type `type2`. If `ident` is used in `type2`, then we say the expression is a 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`. Non-dependent product is used to denote both propositional implication and function types.

These terms are also useful:

- `n : nat` is a dependent premise of `forall n:nat, n + 0 = n` because `n` appears both in the binder of the `forall` and in the quantified statement `n + 0 = n`.

- `A` and `B` are non-dependent premises (or, often, just "premises") of `A -> B -> C` because they don't appear in a `forall` binder. `C` is the conclusion of the type, which is a second meaning for the term conclusion. (As noted, `A -> B` is notation for the term `forall _ : A, B`; the wildcard `_` can't be referred to in the quantified statement.)

As for abstractions, `forall` is followed by a binder list, and products over several variables are equivalent to an iteration of one-variable products.

## Function application

term_application`::=`term1 arg+`````|`@ qualid_annotated term1+``arg`::=`( ident := term )`````|`( natural := term )`````|`term1``

`term1fun term1` denotes applying the function `term1fun` to `term1`.

`term1fun term1i+` denotes applying `term1fun` to the arguments `term1i`. It is equivalent to `( … ( term1fun term11 ) … ) term1n`: associativity is to the left.

The `@ qualid_annotated term1+` form requires specifying all arguments, including implicit ones. Otherwise, implicit arguments need not be given. See Implicit arguments.

The notations `(ident := term)` and `(natural := term)` for arguments are used for making explicit the value of implicit arguments. See Explicit applications.

## Assumptions

Assumptions extend the global environment with axioms, parameters, hypotheses or variables. An assumption binds an `ident` to a `type`. It is accepted by Coq only if `type` is a correct type in the global environment before 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 assumption_token Inline ( natural )?? ( assumpt )+​assumpt
  assumption_token`::=`Axiom​Axioms`````|`Conjecture​Conjectures`````|`Parameter​Parameters`````|`Hypothesis​Hypotheses`````|`Variable​Variables``assumpt`::=`ident_decl+ of_type``ident_decl`::=`ident univ_decl?``of_type`::=`:​:> type``

  These commands bind one or more `ident`(s) to specified `type`(s) as their specifications in the global environment. The fact asserted by `type` (or, equivalently, the existence of an object of this type) is accepted as a postulate. They accept the `program` attribute.

  `Axiom`, `Conjecture`, `Parameter` and their plural forms are equivalent. They can take the `local` attribute, which makes the defined `ident`s accessible by `Import` and its variants only through their fully qualified names.

  Similarly, `Hypothesis`, `Variable` and their plural forms are equivalent. Outside of a section, these are equivalent to `Local Parameter`. Inside a section, the `ident`s defined are only accessible within the section. When the current section is closed, the `ident`(s) become undefined and every object depending on them will be explicitly parameterized (i.e., the variables are discharged). See Section Section mechanism.

  `:>`

    If specified, `ident_decl` is automatically declared as a coercion to the class of its type. See Implicit Coercions.

  The `Inline` clause is only relevant inside functors. See `Module`.

Example: Simple assumptions

Parameter X Y : Set.
  X is declared Y is declared
Parameter (R : X -> Y -> Prop) (S : Y -> X -> Prop).
  R is declared S is declared
Axiom R_S_inv : forall x y, R x y <-> S y x.
  R_S_inv is declared

Error ident already exists.

Warning ident is declared as a local axiom

  Warning generated when using `Variable` or its equivalent instead of `Local Parameter` or its equivalent.

Note

We advise using the commands `Axiom`, `Conjecture` and `Hypothesis` (and their plural forms) for logical postulates (i.e. when the assertion `type` is of sort `Prop`), and to use the commands `Parameter` and `Variable` (and their plural forms) in other cases (corresponding to the declaration of an abstract object of the given type).

Other versions v: 8.16.1

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