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Chapter 4  Calculus of Inductive Constructions

The underlying formal language of Coq is a Calculus of Constructions with Inductive Definitions. It is presented in this chapter. For Coq version V7, this Calculus was known as the Calculus of (Co)Inductive Constructions (Cic in short). The underlying calculus of Coq version V8.0 and up is a weaker calculus where the sort Set satisfies predicative rules. We call this calculus the Predicative Calculus of (Co)Inductive Constructions (pCic in short). In section 4.7 we give the extra-rules for Cic. A compiling option of Coq allows to type-check theories in this extended system.

In pCic  all objects have a type. There are types for functions (or programs), there are atomic types (especially datatypes)... but also types for proofs and types for the types themselves. Especially, any object handled in the formalism must belong to a type. For instance, the statement ``for all x, P'' is not allowed in type theory; you must say instead: ``for all x belonging to T, P''. The expression ``x belonging to T'' is written ``x:T''. One also says: ``x has type T''. The terms of pCic are detailed in section 4.1.

In pCic  there is an internal reduction mechanism. In particular, it allows to decide if two programs are intentionally equal (one says convertible). Convertibility is presented in section 4.3.

The remaining sections are concerned with the type-checking of terms. The beginner can skip them.

The reader seeking a background on the Calculus of Inductive Constructions may read several papers. Giménez [61] provides an introduction to inductive and coinductive definitions in Coq. In their book [13], Bertot and Castéran give a precise description of the pCic based on numerous practical examples. Barras [9], Werner [117] and Paulin-Mohring [104] are the most recent theses dealing with Inductive Definitions. Coquand-Huet [27, 28, 29] introduces the Calculus of Constructions. Coquand-Paulin [30] extended this calculus to inductive definitions. The pCic is a formulation of type theory including the possibility of inductive constructions, Barendregt [6] studies the modern form of type theory.

4.1  The terms

In most type theories, one usually makes a syntactic distinction between types and terms. This is not the case for pCic which defines both types and terms in the same syntactical structure. This is because the type-theory itself forces terms and types to be defined in a mutual recursive way and also because similar constructions can be applied to both terms and types and consequently can share the same syntactic structure.

Consider for instance the -> constructor and assume nat is the type of natural numbers. Then -> is used both to denote nat->nat which is the type of functions from nat to nat, and to denote nat -> Prop which is the type of unary predicates over the natural numbers. Consider abstraction which builds functions. It serves to build ``ordinary'' functions as fun x:nat => (mult  x x) (assuming mult is already defined) but may build also predicates over the natural numbers. For instance fun x:nat => (x=x) will represent a predicate P, informally written in mathematics P(x)same as x=x. If P has type nat -> Prop, (P x) is a proposition, furthermore forall x:nat,(P x) will represent the type of functions which associate to each natural number n an object of type (P n) and consequently represent proofs of the formula ``for all x.P(x)''.

4.1.1  Sorts

Types are seen as terms of the language and then should belong to another type. The type of a type is always a constant of the language called a sort.

The two basic sorts in the language of pCic are Set and Prop.

The sort Prop intends to be the type of logical propositions. If M is a logical proposition then it denotes a class, namely the class of terms representing proofs of M. An object m belonging to M witnesses the fact that M is true. An object of type Prop is called a proposition.

The sort Set intends to be the type of specifications. This includes programs and the usual sets such as booleans, naturals, lists etc.

These sorts themselves can be manipulated as ordinary terms. Consequently sorts also should be given a type. Because assuming simply that Set has type Set leads to an inconsistent theory, we have infinitely many sorts in the language of pCic. These are, in addition to Set and Prop  a hierarchy of universes Type(i) for any integer i. We call S the set of sorts which is defined by:
S same as {Prop,Set,Type(i)| i in N}
The sorts enjoy the following properties: Prop:Type(0), Set:Type(0) and Type(i):Type(i+1).

The user will never mention explicitly the index i when referring to the universe Type(i). One only writes Type. The system itself generates for each instance of Type a new index for the universe and checks that the constraints between these indexes can be solved. From the user point of view we consequently have Type :Type.

We shall make precise in the typing rules the constraints between the indexes.

4.1.2  Constants

Besides the sorts, the language also contains constants denoting objects in the environment. These constants may denote previously defined objects but also objects related to inductive definitions (either the type itself or one of its constructors or destructors).


Remark. In other presentations of pCic, the inductive objects are not seen as external declarations but as first-class terms. Usually the definitions are also completely ignored. This is a nice theoretical point of view but not so practical. An inductive definition is specified by a possibly huge set of declarations, clearly we want to share this specification among the various inductive objects and not to duplicate it. So the specification should exist somewhere and the various objects should refer to it. We choose one more level of indirection where the objects are just represented as constants and the environment gives the information on the kind of object the constant refers to.


Our inductive objects will be manipulated as constants declared in the environment. This roughly corresponds to the way they are actually implemented in the Coq system. It is simple to map this presentation in a theory where inductive objects are represented by terms.

4.1.3  Terms

Terms are built from variables, global names, constructors, abstraction, application, local declarations bindings (``let-in'' expressions) and product.

From a syntactic point of view, types cannot be distingued from terms, except that they cannot start by an abstraction, and that if a term is a sort or a product, it should be a type.

More precisely the language of the Calculus of Inductive Constructions is built from the following rules:
  1. the sorts Set, Prop, Type are terms.
  2. names for global constants of the environment are terms.
  3. variables are terms.
  4. if x is a variable and T, U are terms then for all x:T,U (forall x:T,U in Coq concrete syntax) is a term. If x occurs in U, for all x:T,U reads as ``for all x of type T, U''. As U depends on x, one says that for all x:T,U is a dependent product. If x doesn't occurs in U then for all x:T,U reads as ``if T then U''. A non dependent product can be written: T -> U.
  5. if x is a variable and T, U are terms then lambda x:T , U (fun x:T=> U in Coq concrete syntax) is a term. This is a notation for the lambda-abstraction of lambda-calculus [8]. The term lambda x:T , U is a function which maps elements of T to U.
  6. if T and U are terms then (T U) is a term (T U in Coq concrete syntax). The term (T U) reads as ``T applied to U''.
  7. if x is a variable, and T, U are terms then let x:=T in U is a term which denotes the term U where the variable x is locally bound to T. This stands for the common ``let-in'' construction of functional programs such as ML or Scheme.
Notations.
Application associates to the left such that (t t1... tn) represents (... (t t1)... tn). The products and arrows associate to the right such that for all x:A,B-> C-> D represents for all x:A,(B-> (C-> D)). One uses sometimes for all x y:A,B or lambda x y:A, B to denote the abstraction or product of several variables of the same type. The equivalent formulation is for all x:A, for all y:A,B or lambda x:A , lambda y:A , B

Free variables.
The notion of free variables is defined as usual. In the expressions lambda x:T, U and for all x:T, U the occurrences of x in U are bound. They are represented by de Bruijn indexes in the internal structure of terms.

Substitution.
The notion of substituting a term t to free occurrences of a variable x in a term u is defined as usual. The resulting term is written u{x/t}.

4.2  Typed terms

As objects of type theory, terms are subjected to type discipline. The well typing of a term depends on an environment which consists in a global environment (see below) and a local context.

Local context.
A local context (or shortly context) is an ordered list of declarations of variables. The declaration of some variable x is either an assumption, written x:T (T is a type) or a definition, written x:=t:T. We use brackets to write contexts. A typical example is [x:T;y:=u:U;z:V]. Notice that the variables declared in a context must be distinct. If Gamma declares some x, we write x inGamma. By writing (x:T)inGamma we mean that either x:T is an assumption in Gamma or that there exists some t such that x:=t:T is a definition in Gamma. If Gamma defines some x:=t:T, we also write (x:=t:T)inGamma. Contexts must be themselves well formed. For the rest of the chapter, the notation Gamma::(y:T) (resp Gamma::(y:=t:T)) denotes the context Gamma enriched with the declaration y:T (resp y:=t:T). The notation [] denotes the empty context.

We define the inclusion of two contexts Gamma and Delta (written as Gamma included in Delta) as the property, for all variable x, type T and term t, if (x:T) in Gamma then (x:T)in Delta and if (x:=t:T) in Gamma then (x:=t:T)in Delta.

A variable x is said to be free in Gamma if Gamma contains a declaration y:T such that x is free in T.

Environment.
Because we are manipulating global declarations (constants and global assumptions), we also need to consider a global environment E.

An environment is an ordered list of declarations of global names. Declarations are either assumptions or ``standard'' definitions, that is abbreviations for well-formed terms but also definitions of inductive objects. In the latter case, an object in the environment will define one or more constants (that is types and constructors, see section 4.5).

An assumption will be represented in the environment as Assum(Gamma)(c:T) which means that c is assumed of some type T well-defined in some context Gamma. An (ordinary) definition will be represented in the environment as Def(Gamma)(c:=t:T) which means that c is a constant which is valid in some context Gamma whose value is t and type is T.

The rules for inductive definitions (see section 4.5) have to be considered as assumption rules to which the following definitions apply: if the name c is declared in E, we write c in E and if c:T or c:=t:T is declared in E, we write (c : T) in E.

Typing rules.
In the following, we assume E is a valid environment wrt to inductive definitions. We define simultaneously two judgments. The first one E[Gamma] |- t : T means the term t is well-typed and has type T in the environment E and context Gamma. The second judgment WF(E)[Gamma] means that the environment E is well-formed and the context Gamma is a valid context in this environment. It also means a third property which makes sure that any constant in E was defined in an environment which is included in Gamma 1.

A term t is well typed in an environment E iff there exists a context Gamma and a term T such that the judgment E[Gamma] |- t : T can be derived from the following rules.
W-E
WF([])[[]]
W-S
E[Gamma] |- T : s    sin S    x not in Gamma
WF(E)[Gamma::(x:T)]
     
E[Gamma] |- t : T    x not in Gamma
WF(E)[Gamma::(x:=t:T)]
Def
E[Gamma] |- t : T   c not in Eunion Gamma
WF(E;Def(Gamma)(c:=t:T))[Gamma]
Ax
WF(E)[Gamma]
E[Gamma] |- Prop : Type(p)
     
WF(E)[Gamma]
E[Gamma] |- Set : Type(q)
WF(E)[Gamma]    i<j
E[Gamma] |- Type(i) : Type(j)
Var
WF(E)[Gamma]     (x:T)inGamma  or  (x:=t:T)inGamma for some t
E[Gamma] |- x : T
Const
WF(E)[Gamma]    (c:T) in E
E[Gamma] |- c : T
Prod
E[Gamma] |- T : s    s in S    E[Gamma::(x:T)] |- U : Prop
E[Gamma] |- for all x:T,U : Prop
E[Gamma] |- T : s    sin{Prop, Set}       E[Gamma::(x:T)] |- U : Set
E[Gamma] |- for all x:T,U : Set
E[Gamma] |- T : Type(i)    i<= k    E[Gamma::(x:T)] |- U : Type(j)   j <= k
E[Gamma] |- for all x:T,U : Type(k)
Lam
E[Gamma] |- for all x:T,U : s     E[Gamma::(x:T)] |- t : U
E[Gamma] |- lambda x:T, t : for all x:T, U
App
E[Gamma] |- t : for all x:U,T    E[Gamma] |- u : U
E[Gamma] |- (t u) : T{x/u}
Let
E[Gamma] |- t : T     E[Gamma::(x:=t:T)] |- u : U
E[Gamma] |- let x:=t in u : U{x/t}

Remark: We may have let x:=t in u well-typed without having ((lambda x:T, u) t) well-typed (where T is a type of t). This is because the value t associated to x may be used in a conversion rule (see section 4.3).

4.3  Conversion rules

beta-reduction.
We want to be able to identify some terms as we can identify the application of a function to a given argument with its result. For instance the identity function over a given type T can be written lambda x:T, x. In any environment E and context Gamma, we want to identify any object a (of type T) with the application ((lambda x:T, x) a). We define for this a reduction (or a conversion) rule we call beta:
E[Gamma] |- ((lambda x:T, t) u) |>beta t{x/u}
We say that t{x/u} is the beta-contraction of ((lambda x:T, t) u) and, conversely, that ((lambda x:T, t) u) is the beta-expansion of t{x/u}.

According to beta-reduction, terms of the Calculus of Inductive Constructions enjoy some fundamental properties such as confluence, strong normalization, subject reduction. These results are theoretically of great importance but we will not detail them here and refer the interested reader to [21].

iota-reduction.
A specific conversion rule is associated to the inductive objects in the environment. We shall give later on (section 4.5.4) the precise rules but it just says that a destructor applied to an object built from a constructor behaves as expected. This reduction is called iota-reduction and is more precisely studied in [103, 117].

delta-reduction.
We may have defined variables in contexts or constants in the global environment. It is legal to identify such a reference with its value, that is to expand (or unfold) it into its value. This reduction is called delta-reduction and shows as follows.
E[Gamma] |- x |>delta t     if (x:=t:T)inGamma         E[Gamma] |- c |>delta t     if (c:=t:T)in E

zeta-reduction.
Coq allows also to remove local definitions occurring in terms by replacing the defined variable by its value. The declaration being destroyed, this reduction differs from delta-reduction. It is called zeta-reduction and shows as follows.
E[Gamma] |- let x:=u in t |>zeta t{x/u}

Convertibility.
Let us write E[Gamma] |- t |> u for the contextual closure of the relation t reduces to u in the environment E and context Gamma with one of the previous reduction beta, iota, delta or zeta.

We say that two terms t1 and t2 are convertible (or equivalent) in the environment E and context Gamma iff there exists a term u such that E[Gamma] |- t1 |> ... |> u and E[Gamma] |- t2 |> ... |> u. We then write E[Gamma] |- t1 =betadeltaiotazeta t2.

The convertibility relation allows to introduce a new typing rule which says that two convertible well-formed types have the same inhabitants.

At the moment, we did not take into account one rule between universes which says that any term in a universe of index i is also a term in the universe of index i+1. This property is included into the conversion rule by extending the equivalence relation of convertibility into an order inductively defined by:
  1. if E[Gamma] |- t =betadeltaiotazeta u then E[Gamma] |- t <=betadeltaiotazeta u,
  2. if i <= j then E[Gamma] |- Type(i) <=betadeltaiotazeta Type(j),
  3. for any i, E[Gamma] |- Prop <=betadeltaiotazeta Type(i),
  4. for any i, E[Gamma] |- Set <=betadeltaiotazeta Type(i),
  5. if E[Gamma] |- T =betadeltaiotazeta U and E[Gamma::(x:T)] |- T' <=betadeltaiotazeta U' then E[Gamma] |- for all x:T,T' <=betadeltaiotazeta for all x:U,U'.
The conversion rule is now exactly:
Conv
E[Gamma] |- U : s    E[Gamma] |- t : T    E[Gamma] |- T <=betadeltaiotazeta U
E[Gamma] |- t : U
eta-conversion.
An other important rule is the eta-conversion. It is to identify terms over a dummy abstraction of a variable followed by an application of this variable. Let T be a type, t be a term in which the variable x doesn't occurs free. We have
E[Gamma] |- lambda x:T, (t x) |> t
Indeed, as x doesn't occur free in t, for any u one applies to lambda x:T, (t x), it beta-reduces to (t u). So lambda x:T, (t x) and t can be identified.


Remark: The eta-reduction is not taken into account in the convertibility rule of Coq.

Normal form.
A term which cannot be any more reduced is said to be in normal form. There are several ways (or strategies) to apply the reduction rule. Among them, we have to mention the head reduction which will play an important role (see chapter 8). Any term can be written as lambda x1:T1, ... lambda xk:Tk , (t0 t1... tn) where t0 is not an application. We say then that t0 is the head of t. If we assume that t0 is lambda x:T, u0 then one step of beta-head reduction of t is:
lambda x1:T1, ... lambda xk:Tk, (lambda x:T, u0 t1... tn)  |>   lambda (x1:T1)...(xk:Tk), (u0{x/t1} t2 ... tn)
Iterating the process of head reduction until the head of the reduced term is no more an abstraction leads to the beta-head normal form of t:
t |> ... |> lambda x1:T1, ...lambda xk:Tk, (v u1 ... um)
where v is not an abstraction (nor an application). Note that the head normal form must not be confused with the normal form since some ui can be reducible.

Similar notions of head-normal forms involving delta, iota and zeta reductions or any combination of those can also be defined.

4.4  Derived rules for environments

From the original rules of the type system, one can derive new rules which change the context of definition of objects in the environment. Because these rules correspond to elementary operations in the Coq engine used in the discharge mechanism at the end of a section, we state them explicitly.

Mechanism of substitution.
One rule which can be proved valid, is to replace a term c by its value in the environment. As we defined the substitution of a term for a variable in a term, one can define the substitution of a term for a constant. One easily extends this substitution to contexts and environments.

Substitution Property:
WF(E;Def(Gamma)(c:=t:T); F)[Delta]
WF(E; F{c/t})[Delta{c/t}]

Abstraction.
One can modify the context of definition of a constant c by abstracting a constant with respect to the last variable x of its defining context. For doing that, we need to check that the constants appearing in the body of the declaration do not depend on x, we need also to modify the reference to the constant c in the environment and context by explicitly applying this constant to the variable x. Because of the rules for building environments and terms we know the variable x is available at each stage where c is mentioned.

Abstracting property:
WF(E; Def(Gamma::(x:U))(c:=t:T); F)[Delta]     WF(E)[Gamma]
WF(E;Def(Gamma)(c:=lambda x:U, t:for all x:U,T); F{c/(c x)})[Delta{c/(c x)}]

Pruning the context.
We said the judgment WF(E)[Gamma] means that the defining contexts of constants in E are included in Gamma. If one abstracts or substitutes the constants with the above rules then it may happen that the context Gamma is now bigger than the one needed for defining the constants in E. Because defining contexts are growing in E, the minimum context needed for defining the constants in E is the same as the one for the last constant. One can consequently derive the following property.

Pruning property:
WF(E; Def(Delta)(c:=t:T))[Gamma]
WF(E;Def(Delta)(c:=t:T))[Delta]

4.5  Inductive Definitions

A (possibly mutual) inductive definition is specified by giving the names and the type of the inductive sets or families to be defined and the names and types of the constructors of the inductive predicates. An inductive declaration in the environment can consequently be represented with two contexts (one for inductive definitions, one for constructors).

Stating the rules for inductive definitions in their general form needs quite tedious definitions. We shall try to give a concrete understanding of the rules by precising them on running examples. We take as examples the type of natural numbers, the type of parameterized lists over a type A, the relation which states that a list has some given length and the mutual inductive definition of trees and forests.

4.5.1  Representing an inductive definition

Inductive definitions without parameters

As for constants, inductive definitions can be defined in a non-empty context.
We write Ind(Gamma)(GammaI:=GammaC ) an inductive definition valid in a context Gamma, a context of definitions GammaI and a context of constructors GammaC.
Examples.
The inductive declaration for the type of natural numbers will be:
Ind()(nat:Set:=O:nat,S:nat->nat )
In a context with a variable A:Set, the lists of elements in A is represented by:
Ind(A:Set)(List:Set:=nil:List,cons : A -> List -> List )
Assuming GammaI is [I1:A1;...;Ik:Ak], and GammaC is [c1:C1;...;cn:Cn], the general typing rules are:



Ind(Gamma)(GammaI:=GammaC ) in E   j=1... k
(Ij:Aj) in E

Ind(Gamma)(GammaI:=GammaC ) in E     i=1.. n
(ci:Ci)in E

Inductive definitions with parameters

We have to slightly complicate the representation above in order to handle the delicate problem of parameters. Let us explain that on the example of List. As they were defined above, the type List can only be used in an environment where we have a variable A:Set. Generally one want to consider lists of elements in different types. For constants this is easily done by abstracting the value over the parameter. In the case of inductive definitions we have to handle the abstraction over several objects.

One possible way to do that would be to define the type List inductively as being an inductive family of type Set->Set:
Ind()(List:Set->Set:=nil:(A:Set)(List A),cons : (A:Set)A -> (List A) -> (List A) )
There are drawbacks to this point of view. The information which says that (List nat) is an inductively defined Set has been lost.

In the system, we keep track in the syntax of the context of parameters. The idea of these parameters is that they can be instantiated and still we have an inductive definition for which we know the specification.

Formally the representation of an inductive declaration will be Ind(Gamma)[GammaP](GammaI:=GammaC ) for an inductive definition valid in a context Gamma with parameters GammaP, a context of definitions GammaI and a context of constructors GammaC. The occurrences of the variables of GammaP in the contexts GammaI and GammaC are bound.

The definition Ind(Gamma)[GammaP](GammaI:=GammaC ) will be well-formed exactly when Ind(Gamma,GammaP)(GammaI:=GammaC ) is. If GammaP is [p1:P1;...;pr:Pr], an object in Ind(Gamma)[GammaP](GammaI:=GammaC ) applied to q1,...,qr will behave as the corresponding object of Ind(Gamma)(GammaI{(pi/qi)i=1..r}:=GammaC{(pi/qi)i=1..r} ).

Examples
The declaration for parameterized lists is:
Ind()[A:Set](List:Set:=nil:List,cons : A -> List -> List )

The declaration for the length of lists is:
Ind()[A:Set](Length:(List A)-> nat->Prop:=Lnil:(Length (nil A) O),
Lcons :for all a:A, for all l:(List A),for all n:nat, (Length l n)-> (Length (cons A a l) (S n)) )

The declaration for a mutual inductive definition of forests and trees is:
Ind()(tree:Set,forest:Set:=
  node:forest -> tree, emptyf:forest,consf:tree -> forest -> forest )

These representations are the ones obtained as the result of the Coq declaration:
Coq < Inductive nat : Set :=
Coq <   | O : nat
Coq <   | S : nat -> nat.

Coq < Inductive list (A:Set) : Set :=
Coq <   | nil : list A
Coq <   | cons : A -> list A -> list A.

Coq < Inductive Length (A:Set) : list A -> nat -> Prop :=
Coq <   | Lnil : Length A (nil A) O
Coq <   | Lcons :
Coq <       forall (a:A) (l:list A) (n:nat),
Coq <         Length A l n -> Length A (cons A a l) (S n).

Coq < Inductive tree : Set :=
Coq <     node : forest -> tree
Coq < with forest : Set :=
Coq <   | emptyf : forest
Coq <   | consf : tree -> forest -> forest.
The inductive declaration in Coq is slightly different from the one we described theoretically. The difference is that in the type of constructors the inductive definition is explicitly applied to the parameters variables. The Coq type-checker verifies that all parameters are applied in the correct manner in each recursive call. In particular, the following definition will not be accepted because there is an occurrence of List which is not applied to the parameter variable:
Coq < Inductive list' (A:Set) : Set :=
Coq <   | nil' : list' A
Coq <   | cons' : A -> list' (A -> A) -> list' A.
Coq < Coq < Error: The 1st argument of "list'" must be "A" in
 A -> list' (A -> A) -> list' A

4.5.2  Types of inductive objects

We have to give the type of constants in an environment E which contains an inductive declaration.
Ind-Const
Assuming GammaP is [p1:P1;...;pr:Pr], GammaI is [I1:A1;...;Ik:Ak], and GammaC is [c1:C1;...;cn:Cn],
Ind(Gamma)[GammaP](GammaI:=GammaC ) in E   j=1... k
(Ij:for all p1:P1,...for all pr:Pr,Aj) in E

Ind(Gamma)[GammaP](GammaI:=GammaC ) in E     i=1.. n
(ci:for all p1:P1,... for all pr:Pr,Ci{Ij/(Ij p1... pr)}j=1... k)in E
Example.
We have (List:Set -> Set), (cons:for all A:Set,A->(List A)-> (List A)),
(Length:for all A:Set, (List A)->nat->Prop), tree:Set and forest:Set.

From now on, we write List_A instead of (List A) and Length_A for (Length A).

4.5.3  Well-formed inductive definitions

We cannot accept any inductive declaration because some of them lead to inconsistent systems. We restrict ourselves to definitions which satisfy a syntactic criterion of positivity. Before giving the formal rules, we need a few definitions:

Definitions
A type T is an arity of sort s if it converts to the sort s or to a product for all x:T,U with U an arity of sort s. (For instance A-> Set or for all A:Prop,A-> Prop are arities of sort respectively Set and Prop). A type of constructor of I is either a term (I t1...  tn) or for all x:T,C with C a type of constructor of I.




The type of constructor T will be said to satisfy the positivity condition for a constant X in the following cases: The constant X occurs strictly positively in T in the following cases: The type of constructor T of I satisfies the imbricated positivity condition for a constant X in the following cases:
Example
X occurs strictly positively in A-> X or X*A or (list X) but not in X -> A or (X -> A)-> A nor (neg A) assuming the notion of product and lists were already defined and neg is an inductive definition with declaration Ind()[A:Set](neg:Set:=neg:(A->False) -> neg ). Assuming X has arity nat -> Prop and ex is the inductively defined existential quantifier, the occurrence of X in (ex  nat  lambda n:nat, (X  n)) is also strictly positive.

Correctness rules.
We shall now describe the rules allowing the introduction of a new inductive definition.
W-Ind
Let E be an environment and Gamma,GammaP,GammaI,GammaC are contexts such that GammaI is [I1:A1;...;Ik:Ak] and GammaC is [c1:C1;...;cn:Cn].
(E[Gamma;GammaP] |- Aj : s'j)j=1... k    (E[Gamma;GammaP;GammaI] |- Ci : spi)i=1... n
WF(E;Ind(Gamma)[GammaP](GammaI:=GammaC ))[Gamma]
providing the following side conditions hold:
One can remark that there is a constraint between the sort of the arity of the inductive type and the sort of the type of its constructors which will always be satisfied for the impredicative sort (Prop) but may fail to define inductive definition on sort Set and generate constraints between universes for inductive definitions in types.

Examples
It is well known that existential quantifier can be encoded as an inductive definition. The following declaration introduces the second-order existential quantifier there exists X.P(X).
Coq < Inductive exProp (P:Prop->Prop) : Prop 
Coq <   := exP_intro : forall X:Prop, P X -> exProp P.
The same definition on Set is not allowed and fails :
Coq < Inductive exSet (P:Set->Prop) : Set 
Coq <   := exS_intro : forall X:Set, P X -> exSet P.
Coq < Coq < User error: Large non-propositional inductive types must be in Type
It is possible to declare the same inductive definition in the universe Type. The exType inductive definition has type (Typei ->Prop)-> Typej with the constraint i<j.
Coq < Inductive exType (P:Type->Prop) : Type
Coq <   := exT_intro : forall X:Type, P X -> exType P.

4.5.4  Destructors

The specification of inductive definitions with arities and constructors is quite natural. But we still have to say how to use an object in an inductive type.

This problem is rather delicate. There are actually several different ways to do that. Some of them are logically equivalent but not always equivalent from the computational point of view or from the user point of view.

From the computational point of view, we want to be able to define a function whose domain is an inductively defined type by using a combination of case analysis over the possible constructors of the object and recursion.

Because we need to keep a consistent theory and also we prefer to keep a strongly normalising reduction, we cannot accept any sort of recursion (even terminating). So the basic idea is to restrict ourselves to primitive recursive functions and functionals.

For instance, assuming a parameter A:Set exists in the context, we want to build a function length of type List_A-> nat which computes the length of the list, so such that (length nil) = O and (length (cons A a l)) = (S (length l)). We want these equalities to be recognized implicitly and taken into account in the conversion rule.

From the logical point of view, we have built a type family by giving a set of constructors. We want to capture the fact that we do not have any other way to build an object in this type. So when trying to prove a property (P m) for m in an inductive definition it is enough to enumerate all the cases where m starts with a different constructor.

In case the inductive definition is effectively a recursive one, we want to capture the extra property that we have built the smallest fixed point of this recursive equation. This says that we are only manipulating finite objects. This analysis provides induction principles.

For instance, in order to prove for all l:List_A,(Length_A l (length l)) it is enough to prove:

(Length_A nil (length nil)) and


for all a:A, for all l:List_A, (Length_A l (length l)) -> (Length_A (cons A a l) (length (cons A a l))).


which given the conversion equalities satisfied by length is the same as proving: (Length_A nil O) and for all a:A, for all l:List_A, (Length_A l (length l)) -> (Length_A (cons A a l) (S (length l))).

One conceptually simple way to do that, following the basic scheme proposed by Martin-Löf in his Intuitionistic Type Theory, is to introduce for each inductive definition an elimination operator. At the logical level it is a proof of the usual induction principle and at the computational level it implements a generic operator for doing primitive recursion over the structure.

But this operator is rather tedious to implement and use. We choose in this version of Coq to factorize the operator for primitive recursion into two more primitive operations as was first suggested by Th. Coquand in [25]. One is the definition by pattern-matching. The second one is a definition by guarded fixpoints.

The match...with ...end construction.

The basic idea of this destructor operation is that we have an object m in an inductive type I and we want to prove a property (P m) which in general depends on m. For this, it is enough to prove the property for m = (ci u1... upi) for each constructor of I.

The Coq term for this proof will be written :
match m with  (c1 x11 ... x1p1) => f1  | ... |  (cn xn1...xnpn) => fn  end
In this expression, if m is a term built from a constructor (ci u1... upi) then the expression will behave as it is specified with i-th branch and will reduce to fi where the xi1...xipi are replaced by the u1... up according to the iota-reduction.

Actually, for type-checking a match...with...end expression we also need to know the predicate P to be proved by case analysis. Coq can sometimes infer this predicate but sometimes not. The concrete syntax for describing this predicate uses the as...return construction. The predicate will be explicited using the syntax :
match m as  x  return  (P  x)  with  (c1 x11 ... x1p1) => f1  | ... |  (cn xn1...xnpn) => fn end
For the purpose of presenting the inference rules, we use a more compact notation :
case(m,(lambda x , P), lambda x11 ... x1p1 , f1  | ... |  lambda xn1...xnpn , fn)

This is the basic idea which is generalized to the case where I is an inductively defined n-ary relation (in which case the property P to be proved will be a n+1-ary relation).

Non-dependent elimination.
When defining a function by case analysis, we build an object of type I -> C and the minimality principle on an inductively defined logical predicate of type A -> Prop is often used to prove a property for all x:A,(I x)-> (C x). This is a particular case of the dependent principle that we stated before with a predicate which does not depend explicitly on the object in the inductive definition.

For instance, a function testing whether a list is empty can be defined as:
lambda l:List_A ,case(l,bool,nil  => true  |  (cons a m)  => false)

Allowed elimination sorts.
An important question for building the typing rule for match is what can be the type of P with respect to the type of the inductive definitions.

We define now a relation [I:A|B] between an inductive definition I of type A, an arity B which says that an object in the inductive definition I can be eliminated for proving a property P of type B.

The case of inductive definitions in sorts Set or Type is simple. There is no restriction on the sort of the predicate to be eliminated.

Notations.
The [I:A|B] is defined as the smallest relation satisfying the following rules: We write [I|B] for [I:A|B] where A is the type of I.
Prod
[(I x):A'|B']
[I:(x:A)A'|(x:A)B']
Set& Type
s1 in {Set,Type(j)}, s2 in S
[I:s1|I-> s2]
The case of Inductive Definitions of sort Prop is a bit more complicated, because of our interpretation of this sort. The only harmless allowed elimination, is the one when predicate P is also of sort Prop.
Prop
[I:Prop|I->Prop]
Prop is the type of logical propositions, the proofs of properties P in Prop could not be used for computation and are consequentely ignored by the extraction mechanism. Assume A and B are two propositions, and the logical disjunction A\/ B is defined inductively by :
Coq < Inductive or (A B:Prop) : Prop :=
Coq <   lintro : A -> or A B | rintro : B -> or A B.
The following definition which computes a boolean value by case over the proof of or A B is not accepted :
Coq < Definition choice (A B: Prop) (x:or A B) := 
Coq <    match x with lintro a => true | rintro b => false end.
Coq < Coq < Error: Incorrect elimination of "x" in the inductive type
 or
The elimination predicate "fun _ : or A B => bool" has type
 "or A B -> Set"
It should be one of :
 "Prop"
Elimination of an inductive object of sort : "Prop"
is not allowed on a predicate in sort : "Set"
because non-informative objects may not construct informative ones.
From the computational point of view, the structure of the proof of (or A B) in this term is needed for computing the boolean value.

In general, if I has type Prop then P cannot have type I-> Set, because it will mean to build an informative proof of type (P m) doing a case analysis over a non-computational object that will disappear in the extracted program. But the other way is safe with respect to our interpretation we can have I a computational object and P a non-computational one, it just corresponds to proving a logical property of a computational object.

In the same spirit, elimination on P of type I-> Type cannot be allowed because it trivially implies the elimination on P of type I-> Set by cumulativity. It also implies that there is two proofs of the same property which are provably different, contradicting the proof-irrelevance property which is sometimes a useful axiom :
Coq < Axiom proof_irrelevance : forall (P : Prop) (x y : P), x=y.
proof_irrelevance is assumed
The elimination of an inductive definition of type Prop on a predicate P of type I-> Type leads to a paradox when applied to impredicative inductive definition like the second-order existential quantifier exProp defined above, because it give access to the two projections on this type.

Empty and singleton elimination
There are special inductive definitions in Prop for which more eliminations are allowed.
Prop-extended
I  is an empty or singleton definition   sin S
[I:Prop|I-> s]
A singleton definition has only one constructor and all the arguments of this constructor have type Prop. In that case, there is a canonical way to interpret the informative extraction on an object in that type, such that the elimination on any sort s is legal. Typical examples are the conjunction of non-informative propositions and the equality. If there is an hypothesis h:a=b in the context, it can be used for rewriting not only in logical propositions but also in any type.
Coq < Print eq_rec.
eq_rec = 
fun (A : Type) (x : A) (P : A -> Set) => eq_rect x P
     : forall (A : Type) (x : A) (P : A -> Set),
       P x -> forall y : A, x = y -> P y
Argument A is implicit
Argument scopes are [type_scope _ _ _ _ _]

Coq < Extraction eq_rec.
(** val eq_rec : 'a1 -> 'a2 -> 'a1 -> 'a2 **)
let eq_rec x f y =
  f
An empty definition has no constructors, in that case also, elimination on any sort is allowed.

Type of branches.
Let c be a term of type C, we assume C is a type of constructor for an inductive definition I. Let P be a term that represents the property to be proved. We assume r is the number of parameters.

We define a new type {c:C}P which represents the type of the branch corresponding to the c:C constructor.
{c:(Ii p1... pr t1 ... tp)}P same as (P t1...  tp c)
{c:for all x:T,C}P same as for all x:T,{(c x):C}P
We write {c}P for {c:C}P with C the type of c.

Examples.
For List_A the type of P will be List_A-> s for s in S.
{(cons A)}P same as for all a:A, for all l:List_A,(P (cons A a l)).

For Length_A, the type of P will be for all l:List_A,for all n:nat, (Length_A l n)-> Prop and the expression {(Lcons A)}P is defined as:
for all a:A, for all l:List_A, for all n:nat, for all h:(Length_A l n), (P (cons A a l) (S n) (Lcons A a l n l)).
If P does not depend on its third argument, we find the more natural expression:
for all a:A, for all l:List_A, for all n:nat, (Length_A l n)->(P (cons A a l) (S n)).

Typing rule.
Our very general destructor for inductive definition enjoys the following typing rule
match
E[Gamma] |- c : (I q1... qr t1... ts)   E[Gamma] |- P : B  [(I q1... qr)|B]    (E[Gamma] |- fi : {(cpi q1... qr)}P)i=1... l
E[Gamma] |- case(c,P,f1... fl) : (P t1... ts c)

provided I is an inductive type in a declaration Ind(Delta)[GammaP](GammaI:=GammaC ) with |GammaP| = r, GammaC = [c1:C1;...;cn:Cn] and cp1... cpl are the only constructors of I.
Example.
For List and Length the typing rules for the match expression are (writing just t:M instead of E[Gamma] |- t : M, the environment and context being the same in all the judgments).
l:List_A  P:List_A-> s   f1:(P (nil A))   f2:for all a:A, for all l:List_A, (P (cons A a l))
case(l,P,f1 f2):(P l)

H:(Length_A L N)
P:for all l:List_A, for all n:nat, (Length_A l n)-> Prop
f1:(P (nil A) O Lnil)
f2:for all a:A, for all l:List_A, for all n:nat, for all h:(Length_A l n), (P (cons A a n) (S n) (Lcons A a l n h))
case(H,P,f1 f2):(P L N H)

Definition of iota-reduction.
We still have to define the iota-reduction in the general case.

A iota-redex is a term of the following form:
case((cpi q1... qr a1... am),P,f1... fl)
with cpi the i-th constructor of the inductive type I with r parameters.

The iota-contraction of this term is (fi a1... am) leading to the general reduction rule:
case((cpi q1... qr a1... am),P,f1... fn) |>iota (fi a1... am)

4.5.5  Fixpoint definitions

The second operator for elimination is fixpoint definition. This fixpoint may involve several mutually recursive definitions. The basic concrete syntax for a recursive set of mutually recursive declarations is (with Gammai contexts) :
fix f1 (Gamma1) :A1:=t1 with ... with  fn (Gamman) :An:=tn
The terms are obtained by projections from this set of declarations and are written
fix f1 (Gamma1) :A1:=t1 with ... with  fn (Gamman) :An:=tn for fi
In the inference rules, we represent such a term by
Fix fi{f1:A1':=t1' ... fn:An':=tn'}
with ti' (resp. Ai') representing the term ti abstracted (resp. generalised) with respect to the bindings in the context Gammai, namely ti'=lambda Gammai , ti and Ai'=for all Gammai, Ai.

Typing rule

The typing rule is the expected one for a fixpoint.
Fix
(E[Gamma] |- Ai : si)i=1... n     (E[Gamma,f1:A1,...,fn:An] |- ti : Ai)i=1... n
E[Gamma] |- Fix fi{f1:A1:=t1 ... fn:An:=tn} : Ai
Any fixpoint definition cannot be accepted because non-normalizing terms will lead to proofs of absurdity.

The basic scheme of recursion that should be allowed is the one needed for defining primitive recursive functionals. In that case the fixpoint enjoys a special syntactic restriction, namely one of the arguments belongs to an inductive type, the function starts with a case analysis and recursive calls are done on variables coming from patterns and representing subterms.

For instance in the case of natural numbers, a proof of the induction principle of type
for all P:nat->Prop, (P O)->((n:nat)(P n)->(P (S n)))-> for all n:nat, (P n)
can be represented by the term:
lambda P:nat->Prop,lambda f:(P O), lambda g:(for all n:nat, (P n)->(P (S n))) ,
Fix h{h:for all n:nat, (P n):=lambda n:nat, case(n,P,f lambda p:nat, (g p (h p)))}

Before accepting a fixpoint definition as being correctly typed, we check that the definition is ``guarded''. A precise analysis of this notion can be found in [59].

The first stage is to precise on which argument the fixpoint will be decreasing. The type of this argument should be an inductive definition.

For doing this the syntax of fixpoints is extended and becomes
Fix fi{f1/k1:A1:=t1 ... fn/kn:An:=tn}
where ki are positive integers. Each Ai should be a type (reducible to a term) starting with at least ki products for all y1:B1,... for all yki:Bki, A'i and Bki being an instance of an inductive definition.

Now in the definition ti, if fj occurs then it should be applied to at least kj arguments and the kj-th argument should be syntactically recognized as structurally smaller than yki

The definition of being structurally smaller is a bit technical. One needs first to define the notion of recursive arguments of a constructor. For an inductive definition Ind(Gamma)[GammaP](GammaI:=GammaC ), the type of a constructor c have the form for all p1:P1,... for all pr:Pr, for all x1:T1, ... for all xr:Tr, (Ij p1... pr t1... ts) the recursive arguments will correspond to Ti in which one of the Il occurs.

The main rules for being structurally smaller are the following:
Given a variable y of type an inductive definition in a declaration Ind(Gamma)[GammaP](GammaI:=GammaC ) where GammaI is [I1:A1;...;Ik:Ak], and GammaC is [c1:C1;...;cn:Cn]. The terms structurally smaller than y are: The following definitions are correct, we enter them using the Fixpoint command as described in section 1.3.4 and show the internal representation.
Coq < Fixpoint plus (n m:nat) {struct n} : nat :=
Coq <   match n with
Coq <   | O => m
Coq <   | S p => S (plus p m)
Coq <   end.
plus is recursively defined

Coq < Print plus.
plus = 
(fix plus (n m : nat) {struct n} : nat :=
   match n with
   | O => m
   | S p => S (plus p m)
   end)
     : nat -> nat -> nat

Coq < Fixpoint lgth (A:Set) (l:list A) {struct l} : nat :=
Coq <   match l with
Coq <   | nil => O
Coq <   | cons a l' => S (lgth A l')
Coq <   end.
lgth is recursively defined

Coq < Print lgth.
lgth = 
(fix lgth (A : Set) (l : list A) {struct l} : nat :=
   match l with
   | nil => O
   | cons _ l' => S (lgth A l')
   end)
     : forall A : Set, list A -> nat
Argument scopes are [type_scope _]

Coq < Fixpoint sizet (t:tree) : nat := let (f) := t in S (sizef f)
Coq <  with sizef (f:forest) : nat :=
Coq <   match f with
Coq <   | emptyf => O
Coq <   | consf t f => plus (sizet t) (sizef f)
Coq <   end.
sizet, sizef are recursively defined

Coq < Print sizet.
sizet = 
fix sizet (t : tree) : nat :=
  let (f) := t in S (sizef f)
with sizef (f : forest) : nat :=
  match f with
  | emptyf => O
  | consf t f0 => plus (sizet t) (sizef f0)
  end
for sizet
     : tree -> nat

Reduction rule

Let F be the set of declarations: f1/k1:A1:=t1 ... fn/kn:An:=tn. The reduction for fixpoints is:
(Fix fi{F} a1... aki) |>iota ti{(fk/Fix fk{F})k=1... n}
when aki starts with a constructor. This last restriction is needed in order to keep strong normalization and corresponds to the reduction for primitive recursive operators.

We can illustrate this behavior on examples.
Coq < Goal forall n m:nat, plus (S n) m = S (plus n m).
1 subgoal
  
  ============================
   forall n m : nat, plus (S n) m = S (plus n m)

Coq < reflexivity.
Proof completed.

Coq < Abort.
Current goal aborted

Coq < Goal forall f:forest, sizet (node f) = S (sizef f).
1 subgoal
  
  ============================
   forall f : forest, sizet (node f) = S (sizef f)

Coq < reflexivity.
Proof completed.

Coq < Abort.
Current goal aborted
But assuming the definition of a son function from tree to forest:
Coq < Definition sont (t:tree) : forest 
Coq <    := let (f) := t in f.
sont is defined
The following is not a conversion but can be proved after a case analysis.
Coq < Goal forall t:tree, sizet t = S (sizef (sont t)).
Coq < Coq < 1 subgoal
  
  ============================
   forall t : tree, sizet t = S (sizef (sont t))

Coq < reflexivity. (** this one fails **)
Toplevel input, characters 0-11
> reflexivity.
> ^^^^^^^^^^^
Error: Impossible to unify "S (sizef (sont t))" with "sizet t"

Coq < destruct t.
1 subgoal
  
  f : forest
  ============================
   sizet (node f) = S (sizef (sont (node f)))

Coq < reflexivity.
Proof completed.

Mutual induction

The principles of mutual induction can be automatically generated using the Scheme command described in section 8.13.

4.6  Coinductive types

The implementation contains also coinductive definitions, which are types inhabited by infinite objects. More information on coinductive definitions can be found in [60, 61].

4.7  Cic: the Calculus of Inductive Construction with impredicative Set

Coq can be used as a type-checker for Cic, the original Calculus of Inductive Constructions with an impredicative sort Set by using the compiler option -impredicative-set.

For example, using the ordinary coqtop command, the following is rejected.
Coq < Definition id: Set := forall X:Set,X->X.
Coq < Coq < Coq < Coq < Toplevel input, characters 192-202
> Definition id: Set := forall X:Set,X->X.
>                              ^^^^^^^^^^
Error: The term "forall X : Set, X -> X" has type "Type"
 while it is expected to have type "Set"
while it will type-check, if one use instead the coqtop -impredicative-set command.

The major change in the theory concerns the rule for product formation in the sort Set, which is extended to a domain in any sort :
Prod
E[Gamma] |- T : s    sin S       E[Gamma::(x:T)] |- U : Set
E[Gamma] |- for all x:T,U : Set
This extension has consequences on the inductive definitions which are allowed. In the impredicative system, one can build so-called large inductive definitions like the example of second-order existential quantifier (exSet).

There should be restrictions on the eliminations which can be performed on such definitions. The eliminations rules in the impredicative system for sort Set become :
Set
s in {Prop, Set}
[I:Set|I-> s]
    
I  is a small inductive definition    s in {Type(i)}
[I:Set|I-> s]
1
This requirement could be relaxed if we instead introduced an explicit mechanism for instantiating constants. At the external level, the Coq engine works accordingly to this view that all the definitions in the environment were built in a sub-context of the current context.
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