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\chapter[Calculus of Inductive Constructions]{Calculus of Inductive Constructions
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\index{Cic@\textsc{CIC}}
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\index{pCic@p\textsc{CIC}}
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\index{Calculus of (Co)Inductive Constructions}}
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The underlying formal language of {\Coq} is a {\em Calculus of
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Constructions} with {\em Inductive Definitions}. It is presented in
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For {\Coq} version V7, this Calculus was known as the
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{\em Calculus of (Co)Inductive Constructions}\index{Calculus of
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(Co)Inductive Constructions} (\iCIC\ in short).
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The underlying calculus of {\Coq} version V8.0 and up is a weaker
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calculus where the sort \Set{} satisfies predicative rules.
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We call this calculus the
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{\em Predicative Calculus of (Co)Inductive
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Constructions}\index{Predicative Calculus of
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(Co)Inductive Constructions} (\pCIC\ in short).
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In Section~\ref{impredicativity} we give the extra-rules for \iCIC. A
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compiling option of \Coq{} allows to type-check theories in this
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In \CIC\, all objects have a {\em type}. There are types for functions (or
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programs), there are atomic types (especially datatypes)... but also
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types for proofs and types for the types themselves.
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Especially, any object handled in the formalism must belong to a
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type. For instance, the statement {\it ``for all x, P''} is not
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allowed in type theory; you must say instead: {\it ``for all x
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belonging to T, P''}. The expression {\it ``x belonging to T''} is
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written {\it ``x:T''}. One also says: {\it ``x has type T''}.
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The terms of {\CIC} are detailed in Section~\ref{Terms}.
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In \CIC\, there is an internal reduction mechanism. In particular, it
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allows to decide if two programs are {\em intentionally} equal (one
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says {\em convertible}). Convertibility is presented in section
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The remaining sections are concerned with the type-checking of terms.
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The beginner can skip them.
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The reader seeking a background on the Calculus of Inductive
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Constructions may read several papers. Gim�nez and Cast�ran~\cite{GimCas05}
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an introduction to inductive and coinductive definitions in Coq. In
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their book~\cite{CoqArt}, Bertot and Cast�ran give a precise
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description of the \CIC{} based on numerous practical examples.
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Barras~\cite{Bar99}, Werner~\cite{Wer94} and
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Paulin-Mohring~\cite{Moh97} are the most recent theses dealing with
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Inductive Definitions. Coquand-Huet~\cite{CoHu85a,CoHu85b,CoHu86}
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introduces the Calculus of Constructions. Coquand-Paulin~\cite{CoPa89}
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extended this calculus to inductive definitions. The {\CIC} is a
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formulation of type theory including the possibility of inductive
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constructions, Barendregt~\cite{Bar91} studies the modern form of type
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\section[The terms]{The terms\label{Terms}}
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In most type theories, one usually makes a syntactic distinction
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between types and terms. This is not the case for \CIC\ which defines
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both types and terms in the same syntactical structure. This is
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because the type-theory itself forces terms and types to be defined in
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a mutual recursive way and also because similar constructions can be
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applied to both terms and types and consequently can share the same
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Consider for instance the $\ra$ constructor and assume \nat\ is the
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type of natural numbers. Then $\ra$ is used both to denote
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$\nat\ra\nat$ which is the type of functions from \nat\ to \nat, and
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to denote $\nat \ra \Prop$ which is the type of unary predicates over
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the natural numbers. Consider abstraction which builds functions. It
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serves to build ``ordinary'' functions as $\kw{fun}~x:\nat \Ra ({\tt mult} ~x~x)$ (assuming {\tt mult} is already defined) but may build also
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predicates over the natural numbers. For instance $\kw{fun}~x:\nat \Ra
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represent a predicate $P$, informally written in mathematics
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$P(x)\equiv x=x$. If $P$ has type $\nat \ra \Prop$, $(P~x)$ is a
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proposition, furthermore $\kw{forall}~x:\nat,(P~x)$ will represent the type of
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functions which associate to each natural number $n$ an object of
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type $(P~n)$ and consequently represent proofs of the formula
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\subsection[Sorts]{Sorts\label{Sorts}
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Types are seen as terms of the language and then should belong to
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another type. The type of a type is always a constant of the language
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The two basic sorts in the language of \CIC\ are \Set\ and \Prop.
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The sort \Prop\ intends to be the type of logical propositions. If
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$M$ is a logical proposition then it denotes a class, namely the class
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of terms representing proofs of $M$. An object $m$ belonging to $M$
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witnesses the fact that $M$ is true. An object of type \Prop\ is
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called a {\em proposition}.
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The sort \Set\ intends to be the type of specifications. This includes
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programs and the usual sets such as booleans, naturals, lists
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These sorts themselves can be manipulated as ordinary terms.
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Consequently sorts also should be given a type. Because assuming
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simply that \Set\ has type \Set\ leads to an inconsistent theory, we
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have infinitely many sorts in the language of \CIC. These are, in
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addition to \Set\ and \Prop\, a hierarchy of universes \Type$(i)$
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for any integer $i$. We call \Sort\ the set of sorts
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\[\Sort \equiv \{\Prop,\Set,\Type(i)| i \in \NN\} \]
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The sorts enjoy the following properties: {\Prop:\Type(0)}, {\Set:\Type(0)} and
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{\Type$(i)$:\Type$(i+1)$}.
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The user will never mention explicitly the index $i$ when referring to
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the universe \Type$(i)$. One only writes \Type. The
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system itself generates for each instance of \Type\ a new
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index for the universe and checks that the constraints between these
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indexes can be solved. From the user point of view we consequently
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have {\sf Type :Type}.
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We shall make precise in the typing rules the constraints between the
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\paragraph{Implementation issues}
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In practice, the {\Type} hierarchy is implemented using algebraic
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universes. An algebraic universe $u$ is either a variable (a qualified
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identifier with a number) or a successor of an algebraic universe (an
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expression $u+1$), or an upper bound of algebraic universes (an
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expression $max(u_1,...,u_n)$), or the base universe (the expression
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$0$) which corresponds, in the arity of sort-polymorphic inductive
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types, to the predicative sort {\Set}. A graph of constraints between
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the universe variables is maintained globally. To ensure the existence
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of a mapping of the universes to the positive integers, the graph of
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constraints must remain acyclic. Typing expressions that violate the
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acyclicity of the graph of constraints results in a \errindex{Universe
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inconsistency} error (see also Section~\ref{PrintingUniverses}).
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\subsection{Constants}
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Besides the sorts, the language also contains constants denoting
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objects in the environment. These constants may denote previously
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defined objects but also objects related to inductive definitions
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(either the type itself or one of its constructors or destructors).
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\medskip\noindent {\bf Remark. } In other presentations of \CIC,
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the inductive objects are not seen as
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external declarations but as first-class terms. Usually the
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definitions are also completely ignored. This is a nice theoretical
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point of view but not so practical. An inductive definition is
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specified by a possibly huge set of declarations, clearly we want to
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share this specification among the various inductive objects and not
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to duplicate it. So the specification should exist somewhere and the
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various objects should refer to it. We choose one more level of
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indirection where the objects are just represented as constants and
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the environment gives the information on the kind of object the
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Our inductive objects will be manipulated as constants declared in the
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environment. This roughly corresponds to the way they are actually
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implemented in the \Coq\ system. It is simple to map this presentation
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in a theory where inductive objects are represented by terms.
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Terms are built from variables, global names, constructors,
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abstraction, application, local declarations bindings (``let-in''
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expressions) and product.
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From a syntactic point of view, types cannot be distinguished from terms,
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except that they cannot start by an abstraction, and that if a term is
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a sort or a product, it should be a type.
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More precisely the language of the {\em Calculus of Inductive
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Constructions} is built from the following rules:
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\item the sorts {\sf Set, Prop, Type} are terms.
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\item names for global constants of the environment are terms.
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\item variables are terms.
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\item if $x$ is a variable and $T$, $U$ are terms then $\forall~x:T,U$
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($\kw{forall}~x:T,U$ in \Coq{} concrete syntax) is a term. If $x$
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occurs in $U$, $\forall~x:T,U$ reads as {\it ``for all x of type T,
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U''}. As $U$ depends on $x$, one says that $\forall~x:T,U$ is a
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{\em dependent product}. If $x$ doesn't occurs in $U$ then
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$\forall~x:T,U$ reads as {\it ``if T then U''}. A non dependent
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product can be written: $T \rightarrow U$.
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\item if $x$ is a variable and $T$, $U$ are terms then $\lb~x:T \mto U$
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($\kw{fun}~x:T\Ra U$ in \Coq{} concrete syntax) is a term. This is a
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notation for the $\lambda$-abstraction of
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$\lambda$-calculus\index{lambda-calculus@$\lambda$-calculus}
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\cite{Bar81}. The term $\lb~x:T \mto U$ is a function which maps
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elements of $T$ to $U$.
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\item if $T$ and $U$ are terms then $(T\ U)$ is a term
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($T~U$ in \Coq{} concrete syntax). The term $(T\
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U)$ reads as {\it ``T applied to U''}.
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\item if $x$ is a variable, and $T$, $U$ are terms then
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$\kw{let}~x:=T~\kw{in}~U$ is a
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term which denotes the term $U$ where the variable $x$ is locally
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bound to $T$. This stands for the common ``let-in'' construction of
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functional programs such as ML or Scheme.
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\paragraph{Notations.} Application associates to the left such that
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$(t~t_1\ldots t_n)$ represents $(\ldots (t~t_1)\ldots t_n)$. The
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products and arrows associate to the right such that $\forall~x:A,B\ra C\ra
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D$ represents $\forall~x:A,(B\ra (C\ra D))$. One uses sometimes
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$\lb~x~y:A\mto B$ to denote the abstraction or product of several variables
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of the same type. The equivalent formulation is $\forall~x:A, \forall y:A,B$ or
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$\lb~x:A \mto \lb y:A \mto B$
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\paragraph{Free variables.}
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The notion of free variables is defined as usual. In the expressions
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$\lb~x:T\mto U$ and $\forall x:T, U$ the occurrences of $x$ in $U$
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are bound. They are represented by de Bruijn indexes in the internal
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\paragraph[Substitution.]{Substitution.\index{Substitution}}
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The notion of substituting a term $t$ to free occurrences of a
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variable $x$ in a term $u$ is defined as usual. The resulting term
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is written $\subst{u}{x}{t}$.
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\section[Typed terms]{Typed terms\label{Typed-terms}}
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As objects of type theory, terms are subjected to {\em type
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discipline}. The well typing of a term depends on an environment which
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consists in a global environment (see below) and a local context.
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\paragraph{Local context.}
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A {\em local context} (or shortly context) is an ordered list of
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declarations of variables. The declaration of some variable $x$ is
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either an assumption, written $x:T$ ($T$ is a type) or a definition,
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written $x:=t:T$. We use brackets to write contexts. A
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typical example is $[x:T;y:=u:U;z:V]$. Notice that the variables
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declared in a context must be distinct. If $\Gamma$ declares some $x$,
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we write $x \in \Gamma$. By writing $(x:T) \in \Gamma$ we mean that
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either $x:T$ is an assumption in $\Gamma$ or that there exists some $t$ such
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that $x:=t:T$ is a definition in $\Gamma$. If $\Gamma$ defines some
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$x:=t:T$, we also write $(x:=t:T) \in \Gamma$. Contexts must be
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themselves {\em well formed}. For the rest of the chapter, the
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notation $\Gamma::(y:T)$ (resp. $\Gamma::(y:=t:T)$) denotes the context
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$\Gamma$ enriched with the declaration $y:T$ (resp. $y:=t:T$). The
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notation $[]$ denotes the empty context. \index{Context}
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% Does not seem to be used further...
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% Si dans l'explication WF(E)[Gamma] concernant les constantes
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% definies ds un contexte
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We define the inclusion of two contexts $\Gamma$ and $\Delta$ (written
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as $\Gamma \subset \Delta$) as the property, for all variable $x$,
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type $T$ and term $t$, if $(x:T) \in \Gamma$ then $(x:T) \in \Delta$
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and if $(x:=t:T) \in \Gamma$ then $(x:=t:T) \in \Delta$.
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% $|\Delta|$ for the length of the context $\Delta$, that is for the number
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% of declarations (assumptions or definitions) in $\Delta$.
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A variable $x$ is said to be free in $\Gamma$ if $\Gamma$ contains a
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declaration $y:T$ such that $x$ is free in $T$.
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\paragraph[Environment.]{Environment.\index{Environment}}
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Because we are manipulating global declarations (constants and global
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assumptions), we also need to consider a global environment $E$.
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An environment is an ordered list of declarations of global
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names. Declarations are either assumptions or ``standard''
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definitions, that is abbreviations for well-formed terms
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but also definitions of inductive objects. In the latter
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case, an object in the environment will define one or more constants
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(that is types and constructors, see Section~\ref{Cic-inductive-definitions}).
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An assumption will be represented in the environment as
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\Assum{\Gamma}{c}{T} which means that $c$ is assumed of some type $T$
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well-defined in some context $\Gamma$. An (ordinary) definition will
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be represented in the environment as \Def{\Gamma}{c}{t}{T} which means
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that $c$ is a constant which is valid in some context $\Gamma$ whose
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value is $t$ and type is $T$.
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The rules for inductive definitions (see section
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\ref{Cic-inductive-definitions}) have to be considered as assumption
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rules to which the following definitions apply: if the name $c$ is
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declared in $E$, we write $c \in E$ and if $c:T$ or $c:=t:T$ is
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declared in $E$, we write $(c : T) \in E$.
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\paragraph[Typing rules.]{Typing rules.\label{Typing-rules}\index{Typing rules}}
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In the following, we assume $E$ is a valid environment wrt to
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inductive definitions. We define simultaneously two
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judgments. The first one \WTEG{t}{T} means the term $t$ is well-typed
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and has type $T$ in the environment $E$ and context $\Gamma$. The
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second judgment \WFE{\Gamma} means that the environment $E$ is
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well-formed and the context $\Gamma$ is a valid context in this
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environment. It also means a third property which makes sure that any
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constant in $E$ was defined in an environment which is included in
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\footnote{This requirement could be relaxed if we instead introduced
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an explicit mechanism for instantiating constants. At the external
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level, the Coq engine works accordingly to this view that all the
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definitions in the environment were built in a sub-context of the
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A term $t$ is well typed in an environment $E$ iff there exists a
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context $\Gamma$ and a term $T$ such that the judgment \WTEG{t}{T} can
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be derived from the following rules.
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\item[W-E] \inference{\WF{[]}{[]}}
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\item[W-S] % Ce n'est pas vrai : x peut apparaitre plusieurs fois dans Gamma
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\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~x \not\in
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{\WFE{\Gamma::(x:T)}}~~~~~
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\frac{\WTEG{t}{T}~~~~x \not\in
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}{\WFE{\Gamma::(x:=t:T)}}}
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\item[Def] \inference{\frac{\WTEG{t}{T}~~~c \notin E \cup \Gamma}
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{\WF{E;\Def{\Gamma}{c}{t}{T}}{\Gamma}}}
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\item[Assum] \inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~c \notin E \cup \Gamma}
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{\WF{E;\Assum{\Gamma}{c}{T}}{\Gamma}}}
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\item[Ax] \index{Typing rules!Ax}
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\inference{\frac{\WFE{\Gamma}}{\WTEG{\Prop}{\Type(p)}}~~~~~
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\frac{\WFE{\Gamma}}{\WTEG{\Set}{\Type(q)}}}
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\inference{\frac{\WFE{\Gamma}~~~~i<j}{\WTEG{\Type(i)}{\Type(j)}}}
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\item[Var]\index{Typing rules!Var}
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\inference{\frac{ \WFE{\Gamma}~~~~~(x:T) \in \Gamma~~\mbox{or}~~(x:=t:T) \in \Gamma~\mbox{for some $t$}}{\WTEG{x}{T}}}
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\item[Const] \index{Typing rules!Const}
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\inference{\frac{\WFE{\Gamma}~~~~(c:T) \in E~~\mbox{or}~~(c:=t:T) \in E~\mbox{for some $t$} }{\WTEG{c}{T}}}
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\item[Prod] \index{Typing rules!Prod}
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\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~
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\WTE{\Gamma::(x:T)}{U}{\Prop}}
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{ \WTEG{\forall~x:T,U}{\Prop}}}
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\inference{\frac{\WTEG{T}{s}~~~~s \in\{\Prop, \Set\}~~~~~~
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\WTE{\Gamma::(x:T)}{U}{\Set}}
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{ \WTEG{\forall~x:T,U}{\Set}}}
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\inference{\frac{\WTEG{T}{\Type(i)}~~~~i\leq k~~~
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\WTE{\Gamma::(x:T)}{U}{\Type(j)}~~~j \leq k}
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{\WTEG{\forall~x:T,U}{\Type(k)}}}
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\item[Lam]\index{Typing rules!Lam}
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\inference{\frac{\WTEG{\forall~x:T,U}{s}~~~~ \WTE{\Gamma::(x:T)}{t}{U}}
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{\WTEG{\lb~x:T\mto t}{\forall x:T, U}}}
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\item[App]\index{Typing rules!App}
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\inference{\frac{\WTEG{t}{\forall~x:U,T}~~~~\WTEG{u}{U}}
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{\WTEG{(t\ u)}{\subst{T}{x}{u}}}}
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\item[Let]\index{Typing rules!Let}
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\inference{\frac{\WTEG{t}{T}~~~~ \WTE{\Gamma::(x:=t:T)}{u}{U}}
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{\WTEG{\kw{let}~x:=t~\kw{in}~u}{\subst{U}{x}{t}}}}
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\Rem We may have $\kw{let}~x:=t~\kw{in}~u$
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well-typed without having $((\lb~x:T\mto u)~t)$ well-typed (where
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$T$ is a type of $t$). This is because the value $t$ associated to $x$
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may be used in a conversion rule (see Section~\ref{conv-rules}).
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\section[Conversion rules]{Conversion rules\index{Conversion rules}
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\paragraph[$\beta$-reduction.]{$\beta$-reduction.\label{beta}\index{beta-reduction@$\beta$-reduction}}
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We want to be able to identify some terms as we can identify the
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application of a function to a given argument with its result. For
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instance the identity function over a given type $T$ can be written
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$\lb~x:T\mto x$. In any environment $E$ and context $\Gamma$, we want to identify any object $a$ (of type $T$) with the
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application $((\lb~x:T\mto x)~a)$. We define for this a {\em reduction} (or a
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{\em conversion}) rule we call $\beta$:
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\[ \WTEGRED{((\lb~x:T\mto
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t)~u)}{\triangleright_{\beta}}{\subst{t}{x}{u}} \]
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We say that $\subst{t}{x}{u}$ is the {\em $\beta$-contraction} of
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$((\lb~x:T\mto t)~u)$ and, conversely, that $((\lb~x:T\mto t)~u)$
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is the {\em $\beta$-expansion} of $\subst{t}{x}{u}$.
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According to $\beta$-reduction, terms of the {\em Calculus of
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Inductive Constructions} enjoy some fundamental properties such as
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confluence, strong normalization, subject reduction. These results are
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theoretically of great importance but we will not detail them here and
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refer the interested reader to \cite{Coq85}.
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\paragraph[$\iota$-reduction.]{$\iota$-reduction.\label{iota}\index{iota-reduction@$\iota$-reduction}}
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A specific conversion rule is associated to the inductive objects in
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the environment. We shall give later on (see Section~\ref{iotared}) the
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precise rules but it just says that a destructor applied to an object
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built from a constructor behaves as expected. This reduction is
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called $\iota$-reduction and is more precisely studied in
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\paragraph[$\delta$-reduction.]{$\delta$-reduction.\label{delta}\index{delta-reduction@$\delta$-reduction}}
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We may have defined variables in contexts or constants in the global
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environment. It is legal to identify such a reference with its value,
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that is to expand (or unfold) it into its value. This
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reduction is called $\delta$-reduction and shows as follows.
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$$\WTEGRED{x}{\triangleright_{\delta}}{t}~~~~~\mbox{if $(x:=t:T) \in \Gamma$}~~~~~~~~~\WTEGRED{c}{\triangleright_{\delta}}{t}~~~~~\mbox{if $(c:=t:T) \in E$}$$
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\paragraph[$\zeta$-reduction.]{$\zeta$-reduction.\label{zeta}\index{zeta-reduction@$\zeta$-reduction}}
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Coq allows also to remove local definitions occurring in terms by
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replacing the defined variable by its value. The declaration being
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destroyed, this reduction differs from $\delta$-reduction. It is
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called $\zeta$-reduction and shows as follows.
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$$\WTEGRED{\kw{let}~x:=u~\kw{in}~t}{\triangleright_{\zeta}}{\subst{t}{x}{u}}$$
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\paragraph[Convertibility.]{Convertibility.\label{convertibility}
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\index{beta-reduction@$\beta$-reduction}\index{iota-reduction@$\iota$-reduction}\index{delta-reduction@$\delta$-reduction}\index{zeta-reduction@$\zeta$-reduction}}
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Let us write $\WTEGRED{t}{\triangleright}{u}$ 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 $\zeta$.
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We say that two terms $t_1$ and $t_2$ are {\em convertible} (or {\em
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equivalent)} in the environment $E$ and context $\Gamma$ iff there exists a term $u$ such that $\WTEGRED{t_1}{\triangleright \ldots \triangleright}{u}$
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and $\WTEGRED{t_2}{\triangleright \ldots \triangleright}{u}$.
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We then write $\WTEGCONV{t_1}{t_2}$.
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The convertibility relation allows to introduce a new typing rule
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which says that two convertible well-formed types have the same
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At the moment, we did not take into account one rule between universes
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which says that any term in a universe of index $i$ is also a term in
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the universe of index $i+1$. This property is included into the
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conversion rule by extending the equivalence relation of
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convertibility into an order inductively defined by:
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\item if $\WTEGCONV{t}{u}$ then $\WTEGLECONV{t}{u}$,
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\item if $i \leq j$ then $\WTEGLECONV{\Type(i)}{\Type(j)}$,
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\item for any $i$, $\WTEGLECONV{\Prop}{\Type(i)}$,
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\item for any $i$, $\WTEGLECONV{\Set}{\Type(i)}$,
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\item if $\WTEGCONV{T}{U}$ and $\WTELECONV{\Gamma::(x:T)}{T'}{U'}$ then $\WTEGLECONV{\forall~x:T,T'}{\forall~x:U,U'}$.
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The conversion rule is now exactly:
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\begin{description}\label{Conv}
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\item[Conv]\index{Typing rules!Conv}
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\frac{\WTEG{U}{s}~~~~\WTEG{t}{T}~~~~\WTEGLECONV{T}{U}}{\WTEG{t}{U}}}
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\paragraph{$\eta$-conversion.
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\index{eta-conversion@$\eta$-conversion}
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\index{eta-reduction@$\eta$-reduction}}
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An other important rule is the $\eta$-conversion. It is to identify
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terms over a dummy abstraction of a variable followed by an
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application of this variable. Let $T$ be a type, $t$ be a term in
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which the variable $x$ doesn't occurs free. We have
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\[ \WTEGRED{\lb~x:T\mto (t\ x)}{\triangleright}{t} \]
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Indeed, as $x$ doesn't occur free in $t$, for any $u$ one
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applies to $\lb~x:T\mto (t\ x)$, it $\beta$-reduces to $(t\ u)$. So
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$\lb~x:T\mto (t\ x)$ and $t$ can be identified.
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\Rem The $\eta$-reduction is not taken into account in the
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convertibility rule of \Coq.
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\paragraph[Normal form.]{Normal form.\index{Normal form}\label{Normal-form}\label{Head-normal-form}\index{Head normal form}}
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A term which cannot be any more reduced is said to be in {\em normal
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form}. There are several ways (or strategies) to apply the reduction
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rule. Among them, we have to mention the {\em head reduction} which
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will play an important role (see Chapter~\ref{Tactics}). Any term can
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be written as $\lb~x_1:T_1\mto \ldots \lb x_k:T_k \mto
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(t_0\ t_1\ldots t_n)$ where
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$t_0$ is not an application. We say then that $t_0$ is the {\em head
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of $t$}. If we assume that $t_0$ is $\lb~x:T\mto u_0$ then one step of
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$\beta$-head reduction of $t$ is:
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\[\lb~x_1:T_1\mto \ldots \lb x_k:T_k\mto (\lb~x:T\mto u_0\ t_1\ldots t_n)
465
~\triangleright ~ \lb~(x_1:T_1)\ldots(x_k:T_k)\mto
466
(\subst{u_0}{x}{t_1}\ t_2 \ldots t_n)\]
467
Iterating the process of head reduction until the head of the reduced
468
term is no more an abstraction leads to the {\em $\beta$-head normal
470
\[ t \triangleright \ldots \triangleright
471
\lb~x_1:T_1\mto \ldots\lb x_k:T_k\mto (v\ u_1
473
where $v$ is not an abstraction (nor an application). Note that the
474
head normal form must not be confused with the normal form since some
475
$u_i$ can be reducible.
477
Similar notions of head-normal forms involving $\delta$, $\iota$ and $\zeta$
478
reductions or any combination of those can also be defined.
480
\section{Derived rules for environments}
482
From the original rules of the type system, one can derive new rules
483
which change the context of definition of objects in the environment.
484
Because these rules correspond to elementary operations in the \Coq\
485
engine used in the discharge mechanism at the end of a section, we
486
state them explicitly.
488
\paragraph{Mechanism of substitution.}
490
One rule which can be proved valid, is to replace a term $c$ by its
491
value in the environment. As we defined the substitution of a term for
492
a variable in a term, one can define the substitution of a term for a
493
constant. One easily extends this substitution to contexts and
496
\paragraph{Substitution Property:}
497
\inference{\frac{\WF{E;\Def{\Gamma}{c}{t}{T}; F}{\Delta}}
498
{\WF{E; \subst{F}{c}{t}}{\subst{\Delta}{c}{t}}}}
501
\paragraph{Abstraction.}
503
One can modify the context of definition of a constant $c$ by
504
abstracting a constant with respect to the last variable $x$ of its
505
defining context. For doing that, we need to check that the constants
506
appearing in the body of the declaration do not depend on $x$, we need
507
also to modify the reference to the constant $c$ in the environment
508
and context by explicitly applying this constant to the variable $x$.
509
Because of the rules for building environments and terms we know the
510
variable $x$ is available at each stage where $c$ is mentioned.
512
\paragraph{Abstracting property:}
513
\inference{\frac{\WF{E; \Def{\Gamma::(x:U)}{c}{t}{T};
514
F}{\Delta}~~~~\WFE{\Gamma}}
515
{\WF{E;\Def{\Gamma}{c}{\lb~x:U\mto t}{\forall~x:U,T};
516
\subst{F}{c}{(c~x)}}{\subst{\Delta}{c}{(c~x)}}}}
518
\paragraph{Pruning the context.}
519
We said the judgment \WFE{\Gamma} means that the defining contexts of
520
constants in $E$ are included in $\Gamma$. If one abstracts or
521
substitutes the constants with the above rules then it may happen
522
that the context $\Gamma$ is now bigger than the one needed for
523
defining the constants in $E$. Because defining contexts are growing
524
in $E$, the minimum context needed for defining the constants in $E$
525
is the same as the one for the last constant. One can consequently
526
derive the following property.
528
\paragraph{Pruning property:}
529
\inference{\frac{\WF{E; \Def{\Delta}{c}{t}{T}}{\Gamma}}
530
{\WF{E;\Def{\Delta}{c}{t}{T}}{\Delta}}}
533
\section[Inductive Definitions]{Inductive Definitions\label{Cic-inductive-definitions}}
535
A (possibly mutual) inductive definition is specified by giving the
536
names and the type of the inductive sets or families to be
537
defined and the names and types of the constructors of the inductive
538
predicates. An inductive declaration in the environment can
539
consequently be represented with two contexts (one for inductive
540
definitions, one for constructors).
542
Stating the rules for inductive definitions in their general form
543
needs quite tedious definitions. We shall try to give a concrete
544
understanding of the rules by precising them on running examples. We
545
take as examples the type of natural numbers, the type of
546
parameterized lists over a type $A$, the relation which states that
547
a list has some given length and the mutual inductive definition of trees and
550
\subsection{Representing an inductive definition}
551
\subsubsection{Inductive definitions without parameters}
552
As for constants, inductive definitions can be defined in a non-empty
554
We write \NInd{\Gamma}{\Gamma_I}{\Gamma_C} an inductive
555
definition valid in a context $\Gamma$, a
556
context of definitions $\Gamma_I$ and a context of constructors
558
\paragraph{Examples.}
559
The inductive declaration for the type of natural numbers will be:
560
\[\NInd{}{\nat:\Set}{\nO:\nat,\nS:\nat\ra\nat}\]
561
In a context with a variable $A:\Set$, the lists of elements in $A$ are
563
\[\NInd{A:\Set}{\List:\Set}{\Nil:\List,\cons : A \ra \List \ra
566
$\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is
567
$[c_1:C_1;\ldots;c_n:C_n]$, the general typing rules are,
568
for $1\leq j\leq k$ and $1\leq i\leq n$:
571
\inference{\frac{\NInd{\Gamma}{\Gamma_I}{\Gamma_C} \in E}{(I_j:A_j) \in E}}
573
\inference{\frac{\NInd{\Gamma}{\Gamma_I}{\Gamma_C} \in E}{(c_i:C_i) \in E}}
575
\subsubsection{Inductive definitions with parameters}
577
We have to slightly complicate the representation above in order to handle
578
the delicate problem of parameters.
579
Let us explain that on the example of \List. With the above definition,
580
the type \List\ can only be used in an environment where we
581
have a variable $A:\Set$. Generally one want to consider lists of
582
elements in different types. For constants this is easily done by abstracting
583
the value over the parameter. In the case of inductive definitions we
584
have to handle the abstraction over several objects.
586
One possible way to do that would be to define the type \List\
587
inductively as being an inductive family of type $\Set\ra\Set$:
588
\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set,\List~A),
589
\cons : (\forall A:\Set, A \ra \List~A \ra \List~A)}\]
590
There are drawbacks to this point of view. The
591
information which says that for any $A$, $(\List~A)$ is an inductively defined
593
So we introduce two important definitions.
595
\paragraph{Inductive parameters, real arguments.}
596
An inductive definition $\NInd{\Gamma}{\Gamma_I}{\Gamma_C}$ admits
597
$r$ inductive parameters if each type of constructors $(c:C)$ in
598
$\Gamma_C$ is such that
600
p_1:P_1,\ldots,\forall p_r:P_r,\forall a_1:A_1, \ldots \forall a_n:A_n,
601
(I~p_1~\ldots p_r~t_1\ldots t_q)\]
602
with $I$ one of the inductive definitions in $\Gamma_I$.
603
We say that $q$ is the number of real arguments of the constructor
605
\paragraph{Context of parameters.}
606
If an inductive definition $\NInd{\Gamma}{\Gamma_I}{\Gamma_C}$ admits
607
$r$ inductive parameters, then there exists a context $\Gamma_P$ of
608
size $r$, such that $\Gamma_P=[p_1:P_1;\ldots;p_r:P_r]$ and
609
if $(t:A) \in \Gamma_I,\Gamma_C$ then $A$ can be written as
610
$\forall p_1:P_1,\ldots \forall p_r:P_r,A'$.
611
We call $\Gamma_P$ the context of parameters of the inductive
612
definition and use the notation $\forall \Gamma_P,A'$ for the term $A$.
614
If we have a term $t$ in an instance of an
615
inductive definition $I$ which starts with a constructor $c$, then the
616
$r$ first arguments of $c$ (the parameters) can be deduced from the
617
type $T$ of $t$: these are exactly the $r$ first arguments of $I$ in
618
the head normal form of $T$.
619
\paragraph{Examples.}
620
The \List{} definition has $1$ parameter:
621
\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set, \List~A),
622
\cons : (\forall A:\Set, A \ra \List~A \ra \List~A)}\]
623
This is also the case for this more complex definition where there is
624
a recursive argument on a different instance of \List:
625
\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set, \List~A),
626
\cons : (\forall A:\Set, A \ra \List~(A \ra A) \ra \List~A)}\]
627
But the following definition has $0$ parameters:
628
\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set, \List~A),
629
\cons : (\forall A:\Set, A \ra \List~A \ra \List~(A*A))}\]
632
%The interested reader may compare the above definition with the two
633
%following ones which have very different logical meaning:\\
634
%$\NInd{}{\List:\Set}{\Nil:\List,\cons : (A:\Set)A
635
% \ra \List \ra \List}$ \\
636
%$\NInd{}{\List:\Set\ra\Set}{\Nil:(A:\Set)(\List~A),\cons : (A:\Set)A
637
% \ra (\List~A\ra A) \ra (\List~A)}$.}
638
\paragraph{Concrete syntax.}
639
In the Coq system, the context of parameters is given explicitly
640
after the name of the inductive definitions and is shared between the
641
arities and the type of constructors.
642
% The vernacular declaration of polymorphic trees and forests will be:\\
643
% \begin{coq_example*}
644
% Inductive Tree (A:Set) : Set :=
645
% Node : A -> Forest A -> Tree A
646
% with Forest (A : Set) : Set :=
648
% | Cons : Tree A -> Forest A -> Forest A
650
% will correspond in our formalism to:
651
% \[\NInd{}{{\tt Tree}:\Set\ra\Set;{\tt Forest}:\Set\ra \Set}
652
% {{\tt Node} : \forall A:\Set, A \ra {\tt Forest}~A \ra {\tt Tree}~A,
653
% {\tt Empty} : \forall A:\Set, {\tt Forest}~A,
654
% {\tt Cons} : \forall A:\Set, {\tt Tree}~A \ra {\tt Forest}~A \ra
656
We keep track in the syntax of the number of
659
Formally the representation of an inductive declaration
661
\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} for an inductive
662
definition valid in a context $\Gamma$ with $p$ parameters, a
663
context of definitions $\Gamma_I$ and a context of constructors
666
The definition \Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} will be
667
well-formed exactly when \NInd{\Gamma}{\Gamma_I}{\Gamma_C} is and
668
when $p$ is (less or equal than) the number of parameters in
669
\NInd{\Gamma}{\Gamma_I}{\Gamma_C}.
672
The declaration for parameterized lists is:
673
\[\Ind{}{1}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set,\List~A),\cons :
674
(\forall A:\Set, A \ra \List~A \ra \List~A)}\]
676
The declaration for the length of lists is:
677
\[\Ind{}{1}{\Length:\forall A:\Set, (\List~A)\ra \nat\ra\Prop}
678
{\LNil:\forall A:\Set, \Length~A~(\Nil~A)~\nO,\\
679
\LCons :\forall A:\Set,\forall a:A, \forall l:(\List~A),\forall n:\nat, (\Length~A~l~n)\ra (\Length~A~(\cons~A~a~l)~(\nS~n))}\]
681
The declaration for a mutual inductive definition of forests and trees is:
682
\[\NInd{}{\tree:\Set,\forest:\Set}
683
{\\~~\node:\forest \ra \tree,
684
\emptyf:\forest,\consf:\tree \ra \forest \ra \forest\-}\]
686
These representations are the ones obtained as the result of the \Coq\
689
Inductive nat : Set :=
692
Inductive list (A:Set) : Set :=
694
| cons : A -> list A -> list A.
697
Inductive Length (A:Set) : list A -> nat -> Prop :=
698
| Lnil : Length A (nil A) O
700
forall (a:A) (l:list A) (n:nat),
701
Length A l n -> Length A (cons A a l) (S n).
702
Inductive tree : Set :=
703
node : forest -> tree
706
| consf : tree -> forest -> forest.
708
% The inductive declaration in \Coq\ is slightly different from the one
709
% we described theoretically. The difference is that in the type of
710
% constructors the inductive definition is explicitly applied to the
711
% parameters variables.
712
The \Coq\ type-checker verifies that all
713
parameters are applied in the correct manner in the conclusion of the
714
type of each constructors~:
716
In particular, the following definition will not be accepted because
717
there is an occurrence of \List\ which is not applied to the parameter
718
variable in the conclusion of the type of {\tt cons'}:
720
Set Printing Depth 50.
721
(********** The following is not correct and should produce **********)
722
(********* Error: The 1st argument of list' must be A in ... *********)
725
Inductive list' (A:Set) : Set :=
727
| cons' : A -> list' A -> list' (A*A).
729
Since \Coq{} version 8.1, there is no restriction about parameters in
730
the types of arguments of constructors. The following definition is
733
Inductive list' (A:Set) : Set :=
735
| cons' : A -> list' (A->A) -> list' A.
739
\subsection{Types of inductive objects}
740
We have to give the type of constants in an environment $E$ which
741
contains an inductive declaration.
744
\item[Ind-Const] Assuming
745
$\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is
746
$[c_1:C_1;\ldots;c_n:C_n]$,
748
\inference{\frac{\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} \in E
749
~~j=1\ldots k}{(I_j:A_j) \in E}}
751
\inference{\frac{\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} \in E
757
We have $(\List:\Set \ra \Set), (\cons:\forall~A:\Set,A\ra(\List~A)\ra
759
$(\Length:\forall~A:\Set, (\List~A)\ra\nat\ra\Prop)$, $\tree:\Set$ and $\forest:\Set$.
761
From now on, we write $\ListA$ instead of $(\List~A)$ and $\LengthA$
764
%\paragraph{Parameters.}
765
%%The parameters introduce a distortion between the inside specification
766
%%of the inductive declaration where parameters are supposed to be
767
%%instantiated (this representation is appropriate for checking the
768
%%correctness or deriving the destructor principle) and the outside
769
%%typing rules where the inductive objects are seen as objects
770
%%abstracted with respect to the parameters.
772
%In the definition of \List\ or \Length\, $A$ is a parameter because
773
%what is effectively inductively defined is $\ListA$ or $\LengthA$ for
774
%a given $A$ which is constant in the type of constructors. But when
775
%we define $(\LengthA~l~n)$, $l$ and $n$ are not parameters because the
776
%constructors manipulate different instances of this family.
778
\subsection{Well-formed inductive definitions}
779
We cannot accept any inductive declaration because some of them lead
780
to inconsistent systems. We restrict ourselves to definitions which
781
satisfy a syntactic criterion of positivity. Before giving the formal
782
rules, we need a few definitions:
784
\paragraph[Definitions]{Definitions\index{Positivity}\label{Positivity}}
786
A type $T$ is an {\em arity of sort $s$}\index{Arity} if it converts
787
to the sort $s$ or to a product $\forall~x:T,U$ with $U$ an arity
788
of sort $s$. (For instance $A\ra \Set$ or $\forall~A:\Prop,A\ra
789
\Prop$ are arities of sort respectively \Set\ and \Prop). A {\em type
790
of constructor of $I$}\index{Type of constructor} is either a term
791
$(I~t_1\ldots ~t_n)$ or $\fa x:T,C$ with $C$ recursively
792
a {\em type of constructor of $I$}.
796
The type of constructor $T$ will be said to {\em satisfy the positivity
797
condition} for a constant $X$ in the following cases:
800
\item $T=(X~t_1\ldots ~t_n)$ and $X$ does not occur free in
802
\item $T=\forall~x:U,V$ and $X$ occurs only strictly positively in $U$ and
803
the type $V$ satisfies the positivity condition for $X$
806
The constant $X$ {\em occurs strictly positively} in $T$ in the
810
\item $X$ does not occur in $T$
811
\item $T$ converts to $(X~t_1 \ldots ~t_n)$ and $X$ does not occur in
813
\item $T$ converts to $\forall~x:U,V$ and $X$ does not occur in
814
type $U$ but occurs strictly positively in type $V$
815
\item $T$ converts to $(I~a_1 \ldots ~a_m ~ t_1 \ldots ~t_p)$ where
816
$I$ is the name of an inductive declaration of the form
817
$\Ind{\Gamma}{m}{I:A}{c_1:\forall p_1:P_1,\ldots \forall
818
p_m:P_m,C_1;\ldots;c_n:\forall p_1:P_1,\ldots \forall
820
(in particular, it is not mutually defined and it has $m$
821
parameters) and $X$ does not occur in any of the $t_i$, and the
822
(instantiated) types of constructor $C_i\{p_j/a_j\}_{j=1\ldots m}$
824
the nested positivity condition for $X$
825
%\item more generally, when $T$ is not a type, $X$ occurs strictly
826
%positively in $T[x:U]u$ if $X$ does not occur in $U$ but occurs
827
%strictly positively in $u$
830
The type of constructor $T$ of $I$ {\em satisfies the nested
831
positivity condition} for a constant $X$ in the following
835
\item $T=(I~b_1\ldots b_m~u_1\ldots ~u_{p})$, $I$ is an inductive
836
definition with $m$ parameters and $X$ does not occur in
838
\item $T=\forall~x:U,V$ and $X$ occurs only strictly positively in $U$ and
839
the type $V$ satisfies the nested positivity condition for $X$
844
$X$ occurs strictly positively in $A\ra X$ or $X*A$ or $({\tt list}~
845
X)$ but not in $X \ra A$ or $(X \ra A)\ra A$ nor $({\tt neg}~A)$
846
assuming the notion of product and lists were already defined and {\tt
847
neg} is an inductive definition with declaration \Ind{}{A:\Set}{{\tt
848
neg}:\Set}{{\tt neg}:(A\ra{\tt False}) \ra {\tt neg}}. Assuming
849
$X$ has arity ${\tt nat \ra Prop}$ and {\tt ex} is the inductively
850
defined existential quantifier, the occurrence of $X$ in ${\tt (ex~
851
nat~ \lb~n:nat\mto (X~ n))}$ is also strictly positive.
853
\paragraph{Correctness rules.}
854
We shall now describe the rules allowing the introduction of a new
855
inductive definition.
858
\item[W-Ind] Let $E$ be an environment and
859
$\Gamma,\Gamma_P,\Gamma_I,\Gamma_C$ are contexts such that
860
$\Gamma_I$ is $[I_1:\forall \Gamma_P,A_1;\ldots;I_k:\forall
861
\Gamma_P,A_k]$ and $\Gamma_C$ is
862
$[c_1:\forall \Gamma_P,C_1;\ldots;c_n:\forall \Gamma_P,C_n]$.
865
(\WTE{\Gamma;\Gamma_P}{A_j}{s'_j})_{j=1\ldots k}
866
~~ (\WTE{\Gamma;\Gamma_I;\Gamma_P}{C_i}{s_{q_i}})_{i=1\ldots n}
868
{\WF{E;\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}}{\Gamma}}}
869
provided that the following side conditions hold:
871
\item $k>0$ and all of $I_j$ and $c_i$ are distinct names for $j=1\ldots k$ and $i=1\ldots n$,
872
\item $p$ is the number of parameters of \NInd{\Gamma}{\Gamma_I}{\Gamma_C}
873
and $\Gamma_P$ is the context of parameters,
874
\item for $j=1\ldots k$ we have that $A_j$ is an arity of sort $s_j$ and $I_j
875
\notin \Gamma \cup E$,
876
\item for $i=1\ldots n$ we have that $C_i$ is a type of constructor of
877
$I_{q_i}$ which satisfies the positivity condition for $I_1 \ldots I_k$
878
and $c_i \notin \Gamma \cup E$.
881
One can remark that there is a constraint between the sort of the
882
arity of the inductive type and the sort of the type of its
883
constructors which will always be satisfied for the impredicative sort
884
(\Prop) but may fail to define inductive definition
885
on sort \Set{} and generate constraints between universes for
886
inductive definitions in the {\Type} hierarchy.
888
\paragraph{Examples.}
889
It is well known that existential quantifier can be encoded as an
890
inductive definition.
891
The following declaration introduces the second-order existential
892
quantifier $\exists X.P(X)$.
894
Inductive exProp (P:Prop->Prop) : Prop
895
:= exP_intro : forall X:Prop, P X -> exProp P.
897
The same definition on \Set{} is not allowed and fails~:
899
(********** The following is not correct and should produce **********)
900
(*** Error: Large non-propositional inductive types must be in Type***)
903
Inductive exSet (P:Set->Prop) : Set
904
:= exS_intro : forall X:Set, P X -> exSet P.
906
It is possible to declare the same inductive definition in the
908
The \texttt{exType} inductive definition has type $(\Type_i \ra\Prop)\ra
909
\Type_j$ with the constraint that the parameter \texttt{X} of \texttt{exT\_intro} has type $\Type_k$ with $k<j$ and $k\leq i$.
911
Inductive exType (P:Type->Prop) : Type
912
:= exT_intro : forall X:Type, P X -> exType P.
914
%We shall assume for the following definitions that, if necessary, we
915
%annotated the type of constructors such that we know if the argument
916
%is recursive or not. We shall write the type $(x:_R T)C$ if it is
917
%a recursive argument and $(x:_P T)C$ if the argument is not recursive.
919
\paragraph[Sort-polymorphism of inductive families.]{Sort-polymorphism of inductive families.\index{Sort-polymorphism of inductive families}}
921
From {\Coq} version 8.1, inductive families declared in {\Type} are
922
polymorphic over their arguments in {\Type}.
924
If $A$ is an arity and $s$ a sort, we write $A_{/s}$ for the arity
925
obtained from $A$ by replacing its sort with $s$. Especially, if $A$
926
is well-typed in some environment and context, then $A_{/s}$ is typable
927
by typability of all products in the Calculus of Inductive Constructions.
928
The following typing rule is added to the theory.
931
\item[Ind-Family] Let $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$ be an
932
inductive definition. Let $\Gamma_P = [p_1:P_1;\ldots;p_{p}:P_{p}]$
933
be its context of parameters, $\Gamma_I = [I_1:\forall
934
\Gamma_P,A_1;\ldots;I_k:\forall \Gamma_P,A_k]$ its context of
935
definitions and $\Gamma_C = [c_1:\forall
936
\Gamma_P,C_1;\ldots;c_n:\forall \Gamma_P,C_n]$ its context of
937
constructors, with $c_i$ a constructor of $I_{q_i}$.
939
Let $m \leq p$ be the length of the longest prefix of parameters
940
such that the $m$ first arguments of all occurrences of all $I_j$ in
941
all $C_k$ (even the occurrences in the hypotheses of $C_k$) are
942
exactly applied to $p_1~\ldots~p_m$ ($m$ is the number of {\em
943
recursively uniform parameters} and the $p-m$ remaining parameters
944
are the {\em recursively non-uniform parameters}). Let $q_1$,
945
\ldots, $q_r$, with $0\leq r\leq m$, be a (possibly) partial
946
instantiation of the recursively uniform parameters of
950
{\left\{\begin{array}{l}
951
\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} \in E\\
952
(E[\Gamma] \vdash q_l : P'_l)_{l=1\ldots r}\\
953
(\WTEGLECONV{P'_l}{\subst{P_l}{p_u}{q_u}_{u=1\ldots l-1}})_{l=1\ldots r}\\
957
{E[\Gamma] \vdash (I_j\,q_1\,\ldots\,q_r:\forall [p_{r+1}:P_{r+1};\ldots;p_{p}:P_{p}], (A_j)_{/s_j})}
960
provided that the following side conditions hold:
963
\item $\Gamma_{P'}$ is the context obtained from $\Gamma_P$ by
964
replacing each $P_l$ that is an arity with $P'_l$ for $1\leq l \leq r$ (notice that
965
$P_l$ arity implies $P'_l$ arity since $\WTEGLECONV{P'_l}{ \subst{P_l}{p_u}{q_u}_{u=1\ldots l-1}}$);
966
\item there are sorts $s_i$, for $1 \leq i \leq k$ such that, for
967
$\Gamma_{I'} = [I_1:\forall
968
\Gamma_{P'},(A_1)_{/s_1};\ldots;I_k:\forall \Gamma_{P'},(A_k)_{/s_k}]$
969
we have $(\WTE{\Gamma;\Gamma_{I'};\Gamma_{P'}}{C_i}{s_{q_i}})_{i=1\ldots n}$;
970
\item the sorts are such that all eliminations, to {\Prop}, {\Set} and
971
$\Type(j)$, are allowed (see section~\ref{elimdep}).
975
Notice that if $I_j\,q_1\,\ldots\,q_r$ is typable using the rules {\bf
976
Ind-Const} and {\bf App}, then it is typable using the rule {\bf
977
Ind-Family}. Conversely, the extended theory is not stronger than the
978
theory without {\bf Ind-Family}. We get an equiconsistency result by
979
mapping each $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$ occurring into a
980
given derivation into as many different inductive types and constructors
981
as the number of different (partial) replacements of sorts, needed for
982
this derivation, in the parameters that are arities (this is possible
983
because $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$ well-formed implies
984
that $\Ind{\Gamma}{p}{\Gamma_{I'}}{\Gamma_{C'}}$ is well-formed and
985
has the same allowed eliminations, where
986
$\Gamma_{I'}$ is defined as above and $\Gamma_{C'} = [c_1:\forall
987
\Gamma_{P'},C_1;\ldots;c_n:\forall \Gamma_{P'},C_n]$). That is,
988
the changes in the types of each partial instance
989
$q_1\,\ldots\,q_r$ can be characterized by the ordered sets of arity
990
sorts among the types of parameters, and to each signature is
991
associated a new inductive definition with fresh names. Conversion is
992
preserved as any (partial) instance $I_j\,q_1\,\ldots\,q_r$ or
993
$C_i\,q_1\,\ldots\,q_r$ is mapped to the names chosen in the specific
994
instance of $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$.
996
\newcommand{\Single}{\mbox{\textsf{Set}}}
998
In practice, the rule {\bf Ind-Family} is used by {\Coq} only when all the
999
inductive types of the inductive definition are declared with an arity whose
1000
sort is in the $\Type$
1001
hierarchy. Then, the polymorphism is over the parameters whose
1002
type is an arity of sort in the {\Type} hierarchy.
1004
chosen canonically so that each $s_j$ is minimal with respect to the
1005
hierarchy ${\Prop}\subset{\Set_p}\subset\Type$ where $\Set_p$ is
1007
%and ${\Prop_u}$ is the sort of small singleton
1008
%inductive types (i.e. of inductive types with one single constructor
1009
%and that contains either proofs or inhabitants of singleton types
1011
More precisely, an empty or small singleton inductive definition
1012
(i.e. an inductive definition of which all inductive types are
1013
singleton -- see paragraph~\ref{singleton}) is set in
1014
{\Prop}, a small non-singleton inductive family is set in {\Set} (even
1015
in case {\Set} is impredicative -- see Section~\ref{impredicativity}),
1016
and otherwise in the {\Type} hierarchy.
1017
% TODO: clarify the case of a partial application ??
1019
Note that the side-condition about allowed elimination sorts in the
1020
rule~{\bf Ind-Family} is just to avoid to recompute the allowed
1021
elimination sorts at each instance of a pattern-matching (see
1022
section~\ref{elimdep}).
1024
As an example, let us consider the following definition:
1025
\begin{coq_example*}
1026
Inductive option (A:Type) : Type :=
1028
| Some : A -> option A.
1031
As the definition is set in the {\Type} hierarchy, it is used
1032
polymorphically over its parameters whose types are arities of a sort
1033
in the {\Type} hierarchy. Here, the parameter $A$ has this property,
1034
hence, if \texttt{option} is applied to a type in {\Set}, the result is
1035
in {\Set}. Note that if \texttt{option} is applied to a type in {\Prop},
1036
then, the result is not set in \texttt{Prop} but in \texttt{Set}
1037
still. This is because \texttt{option} is not a singleton type (see
1038
section~\ref{singleton}) and it would loose the elimination to {\Set} and
1039
{\Type} if set in {\Prop}.
1042
Check (fun A:Set => option A).
1043
Check (fun A:Prop => option A).
1046
Here is another example.
1048
\begin{coq_example*}
1049
Inductive prod (A B:Type) : Type := pair : A -> B -> prod A B.
1052
As \texttt{prod} is a singleton type, it will be in {\Prop} if applied
1053
twice to propositions, in {\Set} if applied twice to at least one type
1054
in {\Set} and none in {\Type}, and in {\Type} otherwise. In all cases,
1055
the three kind of eliminations schemes are allowed.
1058
Check (fun A:Set => prod A).
1059
Check (fun A:Prop => prod A A).
1060
Check (fun (A:Prop) (B:Set) => prod A B).
1061
Check (fun (A:Type) (B:Prop) => prod A B).
1064
\subsection{Destructors}
1065
The specification of inductive definitions with arities and
1066
constructors is quite natural. But we still have to say how to use an
1067
object in an inductive type.
1069
This problem is rather delicate. There are actually several different
1070
ways to do that. Some of them are logically equivalent but not always
1071
equivalent from the computational point of view or from the user point
1074
From the computational point of view, we want to be able to define a
1075
function whose domain is an inductively defined type by using a
1076
combination of case analysis over the possible constructors of the
1077
object and recursion.
1079
Because we need to keep a consistent theory and also we prefer to keep
1080
a strongly normalizing reduction, we cannot accept any sort of
1081
recursion (even terminating). So the basic idea is to restrict
1082
ourselves to primitive recursive functions and functionals.
1084
For instance, assuming a parameter $A:\Set$ exists in the context, we
1085
want to build a function \length\ of type $\ListA\ra \nat$ which
1086
computes the length of the list, so such that $(\length~(\Nil~A)) = \nO$
1087
and $(\length~(\cons~A~a~l)) = (\nS~(\length~l))$. We want these
1088
equalities to be recognized implicitly and taken into account in the
1091
From the logical point of view, we have built a type family by giving
1092
a set of constructors. We want to capture the fact that we do not
1093
have any other way to build an object in this type. So when trying to
1094
prove a property $(P~m)$ for $m$ in an inductive definition it is
1095
enough to enumerate all the cases where $m$ starts with a different
1098
In case the inductive definition is effectively a recursive one, we
1099
want to capture the extra property that we have built the smallest
1100
fixed point of this recursive equation. This says that we are only
1101
manipulating finite objects. This analysis provides induction
1104
For instance, in order to prove $\forall l:\ListA,(\LengthA~l~(\length~l))$
1105
it is enough to prove:
1107
\noindent $(\LengthA~(\Nil~A)~(\length~(\Nil~A)))$ and
1110
$\forall a:A, \forall l:\ListA, (\LengthA~l~(\length~l)) \ra
1111
(\LengthA~(\cons~A~a~l)~(\length~(\cons~A~a~l)))$.
1114
\noindent which given the conversion equalities satisfied by \length\ is the
1116
$(\LengthA~(\Nil~A)~\nO)$ and $\forall a:A, \forall l:\ListA,
1117
(\LengthA~l~(\length~l)) \ra
1118
(\LengthA~(\cons~A~a~l)~(\nS~(\length~l)))$.
1120
One conceptually simple way to do that, following the basic scheme
1121
proposed by Martin-L\"of in his Intuitionistic Type Theory, is to
1122
introduce for each inductive definition an elimination operator. At
1123
the logical level it is a proof of the usual induction principle and
1124
at the computational level it implements a generic operator for doing
1125
primitive recursion over the structure.
1127
But this operator is rather tedious to implement and use. We choose in
1128
this version of Coq to factorize the operator for primitive recursion
1129
into two more primitive operations as was first suggested by Th. Coquand
1130
in~\cite{Coq92}. One is the definition by pattern-matching. The second one is a definition by guarded fixpoints.
1132
\subsubsection[The {\tt match\ldots with \ldots end} construction.]{The {\tt match\ldots with \ldots end} construction.\label{Caseexpr}
1133
\index{match@{\tt match\ldots with\ldots end}}}
1135
The basic idea of this destructor operation is that we have an object
1136
$m$ in an inductive type $I$ and we want to prove a property $(P~m)$
1137
which in general depends on $m$. For this, it is enough to prove the
1138
property for $m = (c_i~u_1\ldots u_{p_i})$ for each constructor of $I$.
1140
The \Coq{} term for this proof will be written~:
1141
\[\kw{match}~m~\kw{with}~ (c_1~x_{11}~...~x_{1p_1}) \Ra f_1 ~|~\ldots~|~
1142
(c_n~x_{n1}...x_{np_n}) \Ra f_n~ \kw{end}\]
1143
In this expression, if
1144
$m$ is a term built from a constructor $(c_i~u_1\ldots u_{p_i})$ then
1145
the expression will behave as it is specified with $i$-th branch and
1146
will reduce to $f_i$ where the $x_{i1}$\ldots $x_{ip_i}$ are replaced
1147
by the $u_1\ldots u_p$ according to the $\iota$-reduction.
1149
Actually, for type-checking a \kw{match\ldots with\ldots end}
1150
expression we also need to know the predicate $P$ to be proved by case
1151
analysis. In the general case where $I$ is an inductively defined
1152
$n$-ary relation, $P$ is a $n+1$-ary relation: the $n$ first arguments
1153
correspond to the arguments of $I$ (parameters excluded), and the last
1154
one corresponds to object $m$. \Coq{} can sometimes infer this
1155
predicate but sometimes not. The concrete syntax for describing this
1156
predicate uses the \kw{as\ldots in\ldots return} construction. For
1157
instance, let us assume that $I$ is an unary predicate with one
1158
parameter. The predicate is made explicit using the syntax~:
1159
\[\kw{match}~m~\kw{as}~ x~ \kw{in}~ I~\verb!_!~a~ \kw{return}~ (P~ x)
1160
~\kw{with}~ (c_1~x_{11}~...~x_{1p_1}) \Ra f_1 ~|~\ldots~|~
1161
(c_n~x_{n1}...x_{np_n}) \Ra f_n \kw{end}\]
1162
The \kw{as} part can be omitted if either the result type does not
1163
depend on $m$ (non-dependent elimination) or $m$ is a variable (in
1164
this case, the result type can depend on $m$). The \kw{in} part can be
1165
omitted if the result type does not depend on the arguments of
1166
$I$. Note that the arguments of $I$ corresponding to parameters
1167
\emph{must} be \verb!_!, because the result type is not generalized to
1168
all possible values of the parameters. The expression after \kw{in}
1169
must be seen as an \emph{inductive type pattern}. As a final remark,
1170
expansion of implicit arguments and notations apply to this pattern.
1172
For the purpose of presenting the inference rules, we use a more
1174
\[ \Case{(\lb a x \mto P)}{m}{ \lb x_{11}~...~x_{1p_1} \mto f_1 ~|~\ldots~|~
1175
\lb x_{n1}...x_{np_n} \mto f_n}\]
1177
%% CP 06/06 Obsolete avec la nouvelle syntaxe et incompatible avec la
1178
%% presentation theorique qui suit
1179
% \paragraph{Non-dependent elimination.}
1181
% When defining a function of codomain $C$ by case analysis over an
1182
% object in an inductive type $I$, we build an object of type $I
1183
% \ra C$. The minimality principle on an inductively defined logical
1184
% predicate $I$ of type $A \ra \Prop$ is often used to prove a property
1185
% $\forall x:A,(I~x)\ra (C~x)$. These are particular cases of the dependent
1186
% principle that we stated before with a predicate which does not depend
1187
% explicitly on the object in the inductive definition.
1189
% For instance, a function testing whether a list is empty
1192
% \[\kw{fun} l:\ListA \Ra \kw{match}~l~\kw{with}~ \Nil \Ra \true~
1193
% |~(\cons~a~m) \Ra \false \kw{end}\]
1195
% \[\lb~l:\ListA \mto\Case{\bool}{l}{\true~ |~ \lb a~m,~\false}\]
1196
%\noindent {\bf Remark. }
1198
% In the system \Coq\ the expression above, can be
1199
% written without mentioning
1200
% the dummy abstraction:
1201
% \Case{\bool}{l}{\Nil~ \mbox{\tt =>}~\true~ |~ (\cons~a~m)~
1202
% \mbox{\tt =>}~ \false}
1204
\paragraph[Allowed elimination sorts.]{Allowed elimination sorts.\index{Elimination sorts}}
1206
An important question for building the typing rule for \kw{match} is
1207
what can be the type of $P$ with respect to the type of the inductive
1210
We define now a relation \compat{I:A}{B} between an inductive
1211
definition $I$ of type $A$ and an arity $B$. This relation states that
1212
an object in the inductive definition $I$ can be eliminated for
1213
proving a property $P$ of type $B$.
1215
The case of inductive definitions in sorts \Set\ or \Type{} is simple.
1216
There is no restriction on the sort of the predicate to be
1219
\paragraph{Notations.}
1220
The \compat{I:A}{B} is defined as the smallest relation satisfying the
1222
We write \compat{I}{B} for \compat{I:A}{B} where $A$ is the type of
1226
\item[Prod] \inference{\frac{\compat{(I~x):A'}{B'}}
1227
{\compat{I:(x:A)A'}{(x:A)B'}}}
1228
\item[\Set \& \Type] \inference{\frac{
1229
s_1 \in \{\Set,\Type(j)\},
1230
s_2 \in \Sort}{\compat{I:s_1}{I\ra s_2}}}
1233
The case of Inductive definitions of sort \Prop{} is a bit more
1234
complicated, because of our interpretation of this sort. The only
1235
harmless allowed elimination, is the one when predicate $P$ is also of
1238
\item[\Prop] \inference{\compat{I:\Prop}{I\ra\Prop}}
1240
\Prop{} is the type of logical propositions, the proofs of properties
1241
$P$ in \Prop{} could not be used for computation and are consequently
1242
ignored by the extraction mechanism.
1243
Assume $A$ and $B$ are two propositions, and the logical disjunction
1244
$A\vee B$ is defined inductively by~:
1245
\begin{coq_example*}
1246
Inductive or (A B:Prop) : Prop :=
1247
lintro : A -> or A B | rintro : B -> or A B.
1249
The following definition which computes a boolean value by case over
1250
the proof of \texttt{or A B} is not accepted~:
1252
(***************************************************************)
1253
(*** This example should fail with ``Incorrect elimination'' ***)
1256
Definition choice (A B: Prop) (x:or A B) :=
1257
match x with lintro a => true | rintro b => false end.
1259
From the computational point of view, the structure of the proof of
1260
\texttt{(or A B)} in this term is needed for computing the boolean
1263
In general, if $I$ has type \Prop\ then $P$ cannot have type $I\ra
1264
\Set$, because it will mean to build an informative proof of type
1265
$(P~m)$ doing a case analysis over a non-computational object that
1266
will disappear in the extracted program. But the other way is safe
1267
with respect to our interpretation we can have $I$ a computational
1268
object and $P$ a non-computational one, it just corresponds to proving
1269
a logical property of a computational object.
1271
% Also if $I$ is in one of the sorts \{\Prop, \Set\}, one cannot in
1272
% general allow an elimination over a bigger sort such as \Type. But
1273
% this operation is safe whenever $I$ is a {\em small inductive} type,
1274
% which means that all the types of constructors of
1275
% $I$ are small with the following definition:\\
1276
% $(I~t_1\ldots t_s)$ is a {\em small type of constructor} and
1277
% $\forall~x:T,C$ is a small type of constructor if $C$ is and if $T$
1278
% has type \Prop\ or \Set. \index{Small inductive type}
1280
% We call this particular elimination which gives the possibility to
1281
% compute a type by induction on the structure of a term, a {\em strong
1282
% elimination}\index{Strong elimination}.
1284
In the same spirit, elimination on $P$ of type $I\ra
1285
\Type$ cannot be allowed because it trivially implies the elimination
1286
on $P$ of type $I\ra \Set$ by cumulativity. It also implies that there
1287
is two proofs of the same property which are provably different,
1288
contradicting the proof-irrelevance property which is sometimes a
1291
Axiom proof_irrelevance : forall (P : Prop) (x y : P), x=y.
1294
Reset proof_irrelevance.
1296
The elimination of an inductive definition of type \Prop\ on a
1297
predicate $P$ of type $I\ra \Type$ leads to a paradox when applied to
1298
impredicative inductive definition like the second-order existential
1299
quantifier \texttt{exProp} defined above, because it give access to
1300
the two projections on this type.
1302
%\paragraph{Warning: strong elimination}
1303
%\index{Elimination!Strong elimination}
1304
%In previous versions of Coq, for a small inductive definition, only the
1305
%non-informative strong elimination on \Type\ was allowed, because
1306
%strong elimination on \Typeset\ was not compatible with the current
1307
%extraction procedure. In this version, strong elimination on \Typeset\
1308
%is accepted but a dummy element is extracted from it and may generate
1309
%problems if extracted terms are explicitly used such as in the
1310
%{\tt Program} tactic or when extracting ML programs.
1312
\paragraph[Empty and singleton elimination]{Empty and singleton elimination\label{singleton}
1313
\index{Elimination!Singleton elimination}
1314
\index{Elimination!Empty elimination}}
1316
There are special inductive definitions in \Prop\ for which more
1317
eliminations are allowed.
1319
\item[\Prop-extended]
1321
\frac{I \mbox{~is an empty or singleton
1322
definition}~~~s \in \Sort}{\compat{I:\Prop}{I\ra s}}
1326
% A {\em singleton definition} has always an informative content,
1327
% even if it is a proposition.
1330
definition} has only one constructor and all the arguments of this
1331
constructor have type \Prop. In that case, there is a canonical
1332
way to interpret the informative extraction on an object in that type,
1333
such that the elimination on any sort $s$ is legal. Typical examples are
1334
the conjunction of non-informative propositions and the equality.
1335
If there is an hypothesis $h:a=b$ in the context, it can be used for
1336
rewriting not only in logical propositions but also in any type.
1337
% In that case, the term \verb!eq_rec! which was defined as an axiom, is
1338
% now a term of the calculus.
1343
An empty definition has no constructors, in that case also,
1344
elimination on any sort is allowed.
1346
\paragraph{Type of branches.}
1347
Let $c$ be a term of type $C$, we assume $C$ is a type of constructor
1348
for an inductive definition $I$. Let $P$ be a term that represents the
1349
property to be proved.
1350
We assume $r$ is the number of parameters.
1352
We define a new type \CI{c:C}{P} which represents the type of the
1353
branch corresponding to the $c:C$ constructor.
1356
\CI{c:(I_i~p_1\ldots p_r\ t_1 \ldots t_p)}{P} &\equiv (P~t_1\ldots ~t_p~c) \\[2mm]
1357
\CI{c:\forall~x:T,C}{P} &\equiv \forall~x:T,\CI{(c~x):C}{P}
1360
We write \CI{c}{P} for \CI{c:C}{P} with $C$ the type of $c$.
1362
\paragraph{Examples.}
1363
For $\ListA$ the type of $P$ will be $\ListA\ra s$ for $s \in \Sort$. \\
1364
$ \CI{(\cons~A)}{P} \equiv
1365
\forall a:A, \forall l:\ListA,(P~(\cons~A~a~l))$.
1367
For $\LengthA$, the type of $P$ will be
1368
$\forall l:\ListA,\forall n:\nat, (\LengthA~l~n)\ra \Prop$ and the expression
1369
\CI{(\LCons~A)}{P} is defined as:\\
1370
$\forall a:A, \forall l:\ListA, \forall n:\nat, \forall
1371
h:(\LengthA~l~n), (P~(\cons~A~a~l)~(\nS~n)~(\LCons~A~a~l~n~l))$.\\
1372
If $P$ does not depend on its third argument, we find the more natural
1374
$\forall a:A, \forall l:\ListA, \forall n:\nat,
1375
(\LengthA~l~n)\ra(P~(\cons~A~a~l)~(\nS~n))$.
1377
\paragraph{Typing rule.}
1379
Our very general destructor for inductive definition enjoys the
1380
following typing rule
1383
% \Case{P}{c}{[x_{11}:T_{11}]\ldots[x_{1p_1}:T_{1p_1}]g_1\ldots
1384
% [x_{n1}:T_{n1}]\ldots[x_{np_n}:T_{np_n}]g_n}
1388
% \Case{P}{c}{(c_1~x_{11}~...~x_{1p_1}) \Ra g_1 ~|~\ldots~|~
1389
% (c_n~x_{n1}...x_{np_n}) \Ra g_n }
1393
\item[match] \label{elimdep} \index{Typing rules!match}
1395
\frac{\WTEG{c}{(I~q_1\ldots q_r~t_1\ldots t_s)}~~
1396
\WTEG{P}{B}~~\compat{(I~q_1\ldots q_r)}{B}
1398
(\WTEG{f_i}{\CI{(c_{p_i}~q_1\ldots q_r)}{P}})_{i=1\ldots l}}
1399
{\WTEG{\Case{P}{c}{f_1|\ldots |f_l}}{(P\ t_1\ldots t_s\ c)}}}%\\[3mm]
1401
provided $I$ is an inductive type in a declaration
1402
\Ind{\Delta}{r}{\Gamma_I}{\Gamma_C} with
1403
$\Gamma_C = [c_1:C_1;\ldots;c_n:C_n]$ and $c_{p_1}\ldots c_{p_l}$ are the
1404
only constructors of $I$.
1407
\paragraph{Example.}
1408
For \List\ and \Length\ the typing rules for the {\tt match} expression
1409
are (writing just $t:M$ instead of \WTEG{t}{M}, the environment and
1410
context being the same in all the judgments).
1412
\[\frac{l:\ListA~~P:\ListA\ra s~~~f_1:(P~(\Nil~A))~~
1413
f_2:\forall a:A, \forall l:\ListA, (P~(\cons~A~a~l))}
1414
{\Case{P}{l}{f_1~|~f_2}:(P~l)}\]
1417
\begin{array}[b]{@{}c@{}}
1418
H:(\LengthA~L~N) \\ P:\forall l:\ListA, \forall n:\nat, (\LengthA~l~n)\ra
1420
f_1:(P~(\Nil~A)~\nO~\LNil) \\
1421
f_2:\forall a:A, \forall l:\ListA, \forall n:\nat, \forall
1422
h:(\LengthA~l~n), (P~(\cons~A~a~n)~(\nS~n)~(\LCons~A~a~l~n~h))
1424
{\Case{P}{H}{f_1~|~f_2}:(P~L~N~H)}\]
1426
\paragraph[Definition of $\iota$-reduction.]{Definition of $\iota$-reduction.\label{iotared}
1427
\index{iota-reduction@$\iota$-reduction}}
1428
We still have to define the $\iota$-reduction in the general case.
1430
A $\iota$-redex is a term of the following form:
1431
\[\Case{P}{(c_{p_i}~q_1\ldots q_r~a_1\ldots a_m)}{f_1|\ldots |
1433
with $c_{p_i}$ the $i$-th constructor of the inductive type $I$ with $r$
1436
The $\iota$-contraction of this term is $(f_i~a_1\ldots a_m)$ leading
1437
to the general reduction rule:
1438
\[ \Case{P}{(c_{p_i}~q_1\ldots q_r~a_1\ldots a_m)}{f_1|\ldots |
1439
f_n} \triangleright_{\iota} (f_i~a_1\ldots a_m) \]
1441
\subsection[Fixpoint definitions]{Fixpoint definitions\label{Fix-term} \index{Fix@{\tt Fix}}}
1442
The second operator for elimination is fixpoint definition.
1443
This fixpoint may involve several mutually recursive definitions.
1444
The basic concrete syntax for a recursive set of mutually recursive
1445
declarations is (with $\Gamma_i$ contexts)~:
1446
\[\kw{fix}~f_1 (\Gamma_1) :A_1:=t_1~\kw{with} \ldots \kw{with}~ f_n
1447
(\Gamma_n) :A_n:=t_n\]
1448
The terms are obtained by projections from this set of declarations
1450
\[\kw{fix}~f_1 (\Gamma_1) :A_1:=t_1~\kw{with} \ldots \kw{with}~ f_n
1451
(\Gamma_n) :A_n:=t_n~\kw{for}~f_i\]
1452
In the inference rules, we represent such a
1454
\[\Fix{f_i}{f_1:A_1':=t_1' \ldots f_n:A_n':=t_n'}\]
1455
with $t_i'$ (resp. $A_i'$) representing the term $t_i$ abstracted
1456
(resp. generalized) with
1457
respect to the bindings in the context $\Gamma_i$, namely
1458
$t_i'=\lb \Gamma_i \mto t_i$ and $A_i'=\forall \Gamma_i, A_i$.
1460
\subsubsection{Typing rule}
1461
The typing rule is the expected one for a fixpoint.
1464
\item[Fix] \index{Typing rules!Fix}
1465
\inference{\frac{(\WTEG{A_i}{s_i})_{i=1\ldots n}~~~~
1466
(\WTE{\Gamma,f_1:A_1,\ldots,f_n:A_n}{t_i}{A_i})_{i=1\ldots n}}
1467
{\WTEG{\Fix{f_i}{f_1:A_1:=t_1 \ldots f_n:A_n:=t_n}}{A_i}}}
1470
Any fixpoint definition cannot be accepted because non-normalizing terms
1471
will lead to proofs of absurdity.
1473
The basic scheme of recursion that should be allowed is the one needed for
1475
recursive functionals. In that case the fixpoint enjoys a special
1476
syntactic restriction, namely one of the arguments belongs to an
1477
inductive type, the function starts with a case analysis and recursive
1478
calls are done on variables coming from patterns and representing subterms.
1480
For instance in the case of natural numbers, a proof of the induction
1482
\[\forall P:\nat\ra\Prop, (P~\nO)\ra(\forall n:\nat, (P~n)\ra(P~(\nS~n)))\ra
1483
\forall n:\nat, (P~n)\]
1484
can be represented by the term:
1486
\lb P:\nat\ra\Prop\mto\lb f:(P~\nO)\mto \lb g:(\forall n:\nat,
1487
(P~n)\ra(P~(\nS~n))) \mto\\
1488
\Fix{h}{h:\forall n:\nat, (P~n):=\lb n:\nat\mto \Case{P}{n}{f~|~\lb
1489
p:\nat\mto (g~p~(h~p))}}
1493
Before accepting a fixpoint definition as being correctly typed, we
1494
check that the definition is ``guarded''. A precise analysis of this
1495
notion can be found in~\cite{Gim94}.
1497
The first stage is to precise on which argument the fixpoint will be
1498
decreasing. The type of this argument should be an inductive
1501
For doing this the syntax of fixpoints is extended and becomes
1502
\[\Fix{f_i}{f_1/k_1:A_1:=t_1 \ldots f_n/k_n:A_n:=t_n}\]
1503
where $k_i$ are positive integers.
1504
Each $A_i$ should be a type (reducible to a term) starting with at least
1505
$k_i$ products $\forall y_1:B_1,\ldots \forall y_{k_i}:B_{k_i}, A'_i$
1507
being an instance of an inductive definition.
1509
Now in the definition $t_i$, if $f_j$ occurs then it should be applied
1510
to at least $k_j$ arguments and the $k_j$-th argument should be
1511
syntactically recognized as structurally smaller than $y_{k_i}$
1514
The definition of being structurally smaller is a bit technical.
1515
One needs first to define the notion of
1516
{\em recursive arguments of a constructor}\index{Recursive arguments}.
1517
For an inductive definition \Ind{\Gamma}{r}{\Gamma_I}{\Gamma_C},
1518
the type of a constructor $c$ has the form
1519
$\forall p_1:P_1,\ldots \forall p_r:P_r,
1520
\forall x_1:T_1, \ldots \forall x_r:T_r, (I_j~p_1\ldots
1521
p_r~t_1\ldots t_s)$ the recursive arguments will correspond to $T_i$ in
1522
which one of the $I_l$ occurs.
1525
The main rules for being structurally smaller are the following:\\
1526
Given a variable $y$ of type an inductive
1527
definition in a declaration
1528
\Ind{\Gamma}{r}{\Gamma_I}{\Gamma_C}
1529
where $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is
1530
$[c_1:C_1;\ldots;c_n:C_n]$.
1531
The terms structurally smaller than $y$ are:
1533
\item $(t~u), \lb x:u \mto t$ when $t$ is structurally smaller than $y$ .
1534
\item \Case{P}{c}{f_1\ldots f_n} when each $f_i$ is structurally
1535
smaller than $y$. \\
1536
If $c$ is $y$ or is structurally smaller than $y$, its type is an inductive
1537
definition $I_p$ part of the inductive
1538
declaration corresponding to $y$.
1539
Each $f_i$ corresponds to a type of constructor $C_q \equiv
1540
\forall p_1:P_1,\ldots,\forall p_r:P_r, \forall y_1:B_1, \ldots \forall y_k:B_k, (I~a_1\ldots a_k)$
1541
and can consequently be
1542
written $\lb y_1:B'_1\mto \ldots \lb y_k:B'_k\mto g_i$.
1543
($B'_i$ is obtained from $B_i$ by substituting parameters variables)
1544
the variables $y_j$ occurring
1545
in $g_i$ corresponding to recursive arguments $B_i$ (the ones in
1546
which one of the $I_l$ occurs) are structurally smaller than $y$.
1548
The following definitions are correct, we enter them using the
1549
{\tt Fixpoint} command as described in Section~\ref{Fixpoint} and show
1550
the internal representation.
1552
Fixpoint plus (n m:nat) {struct n} : nat :=
1555
| S p => S (plus p m)
1558
Fixpoint lgth (A:Set) (l:list A) {struct l} : nat :=
1561
| cons a l' => S (lgth A l')
1564
Fixpoint sizet (t:tree) : nat := let (f) := t in S (sizef f)
1565
with sizef (f:forest) : nat :=
1568
| consf t f => plus (sizet t) (sizef f)
1574
\subsubsection[Reduction rule]{Reduction rule\index{iota-reduction@$\iota$-reduction}}
1575
Let $F$ be the set of declarations: $f_1/k_1:A_1:=t_1 \ldots
1577
The reduction for fixpoints is:
1578
\[ (\Fix{f_i}{F}~a_1\ldots
1579
a_{k_i}) \triangleright_{\iota} \substs{t_i}{f_k}{\Fix{f_k}{F}}{k=1\ldots n}
1580
~a_1\ldots a_{k_i}\]
1581
when $a_{k_i}$ starts with a constructor.
1582
This last restriction is needed in order to keep strong normalization
1583
and corresponds to the reduction for primitive recursive operators.
1585
We can illustrate this behavior on examples.
1587
Goal forall n m:nat, plus (S n) m = S (plus n m).
1590
Goal forall f:forest, sizet (node f) = S (sizef f).
1594
But assuming the definition of a son function from \tree\ to \forest:
1596
Definition sont (t:tree) : forest
1597
:= let (f) := t in f.
1599
The following is not a conversion but can be proved after a case analysis.
1601
(******************************************************************)
1602
(** Error: Impossible to unify .... **)
1605
Goal forall t:tree, sizet t = S (sizef (sont t)).
1606
reflexivity. (** this one fails **)
1614
% La disparition de Program devrait rendre la construction Match obsolete
1615
% \subsubsection{The {\tt Match \ldots with \ldots end} expression}
1617
% %\paragraph{A unary {\tt Match\ldots with \ldots end}.}
1618
% \index{Match...with...end@{\tt Match \ldots with \ldots end}}
1619
% The {\tt Match} operator which was a primitive notion in older
1620
% presentations of the Calculus of Inductive Constructions is now just a
1621
% macro definition which generates the good combination of {\tt Case}
1622
% and {\tt Fix} operators in order to generate an operator for primitive
1623
% recursive definitions. It always considers an inductive definition as
1624
% a single inductive definition.
1626
% The following examples illustrates this feature.
1627
% \begin{coq_example}
1628
% Definition nat_pr : (C:Set)C->(nat->C->C)->nat->C
1629
% :=[C,x,g,n]Match n with x g end.
1632
% \begin{coq_example}
1633
% Definition forest_pr
1634
% : (C:Set)C->(tree->forest->C->C)->forest->C
1635
% := [C,x,g,n]Match n with x g end.
1638
% Cet exemple faisait error (HH le 12/12/96), j'ai change pour une
1639
% version plus simple
1640
%\begin{coq_example}
1641
%Definition forest_pr
1642
% : (P:forest->Set)(P emptyf)->((t:tree)(f:forest)(P f)->(P (consf t f)))
1644
% := [C,x,g,n]Match n with x g end.
1647
\subsubsection{Mutual induction}
1649
The principles of mutual induction can be automatically generated
1650
using the {\tt Scheme} command described in Section~\ref{Scheme}.
1652
\section{Coinductive types}
1653
The implementation contains also coinductive definitions, which are
1654
types inhabited by infinite objects.
1655
More information on coinductive definitions can be found
1656
in~\cite{Gimenez95b,Gim98,GimCas05}.
1657
%They are described in Chapter~\ref{Coinductives}.
1659
\section[\iCIC : the Calculus of Inductive Construction with
1660
impredicative \Set]{\iCIC : the Calculus of Inductive Construction with
1661
impredicative \Set\label{impredicativity}}
1663
\Coq{} can be used as a type-checker for \iCIC{}, the original
1664
Calculus of Inductive Constructions with an impredicative sort \Set{}
1665
by using the compiler option \texttt{-impredicative-set}.
1667
For example, using the ordinary \texttt{coqtop} command, the following
1670
(** This example should fail *******************************
1671
Error: The term forall X:Set, X -> X has type Type
1672
while it is expected to have type Set
1676
Definition id: Set := forall X:Set,X->X.
1678
while it will type-check, if one use instead the \texttt{coqtop
1679
-impredicative-set} command.
1681
The major change in the theory concerns the rule for product formation
1682
in the sort \Set, which is extended to a domain in any sort~:
1684
\item [Prod] \index{Typing rules!Prod (impredicative Set)}
1685
\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~~~
1686
\WTE{\Gamma::(x:T)}{U}{\Set}}
1687
{ \WTEG{\forall~x:T,U}{\Set}}}
1689
This extension has consequences on the inductive definitions which are
1691
In the impredicative system, one can build so-called {\em large inductive
1692
definitions} like the example of second-order existential
1693
quantifier (\texttt{exSet}).
1695
There should be restrictions on the eliminations which can be
1696
performed on such definitions. The eliminations rules in the
1697
impredicative system for sort \Set{} become~:
1699
\item[\Set] \inference{\frac{s \in
1700
\{\Prop, \Set\}}{\compat{I:\Set}{I\ra s}}
1701
~~~~\frac{I \mbox{~is a small inductive definition}~~~~s \in
1703
{\compat{I:\Set}{I\ra s}}}
1708
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1710
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1712
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