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diff --git a/doc/RefMan-cic.tex b/doc/RefMan-cic.tex new file mode 100755 index 000000000..51bbc506a --- /dev/null +++ b/doc/RefMan-cic.tex @@ -0,0 +1,1233 @@ +\chapter{The Calculus of Inductive Constructions} +\label{Cic} +\index{Cic@\textsc{CIC}} +\index{Calculus of (Co)Inductive Constructions} + +The underlying formal language of {\Coq} is the {\em Calculus of +(Co)Inductive Constructions}\index{Calculus of (Co)Inductive Constructions} +(\CIC\ in short). It is presented in this chapter. + +In \CIC\, all objects have a {\em type}. There are types for functions (or +programs), there are atomic types (especially datatypes)... but also +types for proofs and types for the types themselves. +Especially, any object handled in the formalism must belong to a +type. For instance, the statement {\it ``for all x, P''} is not +allowed in type theory; you must say instead: {\it ``for all x +belonging to T, P''}. The expression {\it ``x belonging to T''} is +written {\it ``x:T''}. One also says: {\it ``x has type T''}. +The terms of {\CIC} are detailed in section \ref{Terms}. + +In \CIC\, there is an internal reduction mechanism. In particular, it +allows to decide if two programs are {\em intentionally} equal (one +says {\em convertible}). Convertibility is presented in section +\ref{convertibility}. + +The remaining sections are concerned with the type-checking of terms. +The beginner can skip them. + +The reader seeking a background on the {\CIC} may read several +papers. Giménez~\cite{Gim98} +provides an introduction to inductive and coinductive +definitions in Coq, Werner~\cite{Wer94} and Paulin-Mohring~\cite{Moh97} are the +most recent theses on the +{\CIC}. Coquand-Huet~\cite{CoHu85a,CoHu85b,CoHu86} introduces the +Calculus of Constructions. Coquand-Paulin~\cite{CoPa89} introduces +inductive definitions. The {\CIC} is a formulation of type theory +including the possibility of inductive +constructions. Barendregt~\cite{Bar91} studies the modern form of type +theory. + +\section{The terms}\label{Terms} + +In most type theories, one usually makes a syntactic distinction +between types and terms. This is not the case for \CIC\ which defines +both types and terms in the same syntactical structure. This is +because the type-theory itself forces terms and types to be defined in +a mutual recursive way and also because similar constructions can be +applied to both terms and types and consequently can share the same +syntactic structure. + +For instance the type of functions will have several meanings. Assume +\nat\ is the type of natural numbers then $\nat\ra\nat$ is the type of +functions from \nat\ to \nat, $\nat \ra \Prop$ is the type of unary +predicates over the natural numbers. For instance $[x:\nat](x=x)$ will +represent a predicate $P$, informally written in mathematics +$P(x)\equiv x=x$. If $P$ has type $\nat \ra \Prop$, $(P~x)$ is a +proposition, furthermore $(x:\nat)(P~x)$ will represent the type of +functions which associate to each natural number $n$ an object of +type $(P~n)$ and consequently represent proofs of the formula +``$\forall x.P(x)$''. + +\subsection{Sorts}\label{Sorts} +\index{Sorts} +Types are seen as terms of the language and then should belong to +another type. The type of a type is always a constant of the language +called a sort. + +The two basic sorts in the language of \CIC\ are \Set\ and \Prop. + +The sort \Prop\ intends to be the type of logical propositions. If +$M$ is a logical proposition then it denotes a class, namely the class +of terms representing proofs of $M$. An object $m$ belonging to $M$ +witnesses the fact that $M$ is true. An object of type \Prop\ is +called a {\em proposition}. + +The sort \Set\ intends to be the type of specifications. This includes +programs and the usual sets such as booleans, naturals, lists +etc. + +These sorts themselves can be manipulated as ordinary terms. +Consequently sorts also should be given a type. Because assuming +simply that \Set\ has type \Set\ leads to an inconsistent theory, we +have infinitely many sorts in the language of \CIC\ . These are, in +addition to \Set\ and \Prop\, a hierarchy of universes \Type$(i)$ +for any integer $i$. We call \Sort\ the set of sorts +which is defined by: +\[\Sort \equiv \{\Prop,\Set,\Type(i)| i \in \NN\} \] +\index{Type@{\Type}} +\index{Prop@{\Prop}} +\index{Set@{\Set}} +The sorts enjoy the following properties: {\Prop:\Type(0)} and + {\Type$(i)$:\Type$(i+1)$}. + +The user will never mention explicitly the index $i$ when referring to +the universe \Type$(i)$. One only writes \Type. The +system itself generates for each instance of \Type\ a new +index for the universe and checks that the constraints between these +indexes can be solved. From the user point of view we consequently +have {\sf Type :Type}. + +We shall make precise in the typing rules the constraints between the +indexes. + +\medskip +\Rem +The extraction mechanism is not compatible with this universe +hierarchy. It is supposed to work only on terms which are explicitly +typed in the Calculus of Constructions without universes and with +Inductive Definitions at the Set level and only a small elimination. +In other cases, extraction may generate a dummy answer and sometimes +failed. To avoid failure when developing proofs, an error while +extracting the computational contents of a proof will not stop the +proof but only give a warning. + + +\subsection{Constants} +Besides the sorts, the language also contains constants denoting +objects in the environment. These constants may denote previously +defined objects but also objects related to inductive definitions +(either the type itself or one of its constructors or destructors). + +\medskip\noindent {\bf Remark. } In other presentations of \CIC, +the inductive objects are not seen as +external declarations but as first-class terms. Usually the +definitions are also completely ignored. This is a nice theoretical +point of view but not so practical. An inductive definition is +specified by a possibly huge set of declarations, clearly we want to +share this specification among the various inductive objects and not +to duplicate it. So the specification should exist somewhere and the +various objects should refer to it. We choose one more level of +indirection where the objects are just represented as constants and +the environment gives the information on the kind of object the +constant refers to. + +\medskip +Our inductive objects will be manipulated as constants declared in the +environment. This roughly corresponds to the way they are actually +implemented in the \Coq\ system. It is simple to map this presentation +in a theory where inductive objects are represented by terms. + +\subsection{Language} + +\paragraph{Types.} + +Roughly speaking types can be separated into atomic and composed +types. + +An atomic type of the {\em Calculus of Inductive Constructions} is +either a sort or is built from a type variable or an inductive +definition applied to some terms. + +A composed type will be a product $(x:T)U$ with $T$ and $U$ two types. + +\paragraph{Terms.} +A term is either a type or a term variable or a term constant of the +environment. + +As usual in $\lambda$-calculus, we combine objects using abstraction +and application. + +More precisely the language of the {\em Calculus of Inductive + Constructions} is built with the following rules: + +\begin{enumerate} +\item the sorts {\sf Set, Prop, Type} are terms. +\item constants of the environment are terms. +\item variables are terms. +\item if $x$ is a variable and $T$, $U$ are terms then $(x:T)U$ is a + term. If $x$ occurs in $U$, $(x:T)U$ reads as {\it ``for all x of + type T, U''}. As $U$ depends on $x$, one says that $(x:T)U$ is a + {\em dependent product}. If $x$ doesn't occurs in $U$ then $(x:T)U$ + reads as {\it ``if T then U''}. A non dependent product can be + written: $T \rightarrow U$. +\item if $x$ is a variable and $T$, $U$ are terms then $[x:T]U$ is a + term. This is a notation for the $\lambda$-abstraction of + $\lambda$-calculus\index{lambda-calculus@$\lambda$-calculus} + \cite{Bar81}. The term $[x:T]U$ is a function which maps elements of + $T$ to $U$. +\item if $T$ and $U$ are terms then $(T\ U)$ is a term. The term $(T\ + U)$ reads as {\it ``T applied to U''}. +\end{enumerate} + +\paragraph{Notations.} Application associates to the left such that +$(t~t_1\ldots t_n)$ represents $(\ldots (t~t_1)\ldots t_n)$. The +products and arrows associate to the right such that $(x:A)B\ra C\ra +D$ represents $(x:A)(B\ra (C\ra D))$. One uses sometimes $(x,y:A)B$ or +$[x,y:A]B$ to denote the abstraction or product of several variables +of the same type. The equivalent formulation is $(x:A)(y:A)B$ or +$[x:A][y:A]B$ +\paragraph{Free variables.} +The notion of free variables is defined as usual. In the expressions +$[x:T]U$ and $(x:T)U$ the occurrences of $x$ in $U$ are bound. They +are represented by de Bruijn indexes in the internal structure of +terms. + +\paragraph{Substitution.} \index{Substitution} +The notion of substituting a term $T$ to free occurrences of a +variable $x$ in a term $U$ is defined as usual. The resulting term +will be written $\subst{U}{x}{T}$. + + +\section{Typed terms}\label{Typed-terms} +As objects of type theory, terms are subjected to {\em type + discipline}. The well typing of a term depends on a set of +declarations of variables we call a {\em context}. A context $\Gamma$ +is written $[x_1:T_1;..; x_n:T_n]$ where the $x_i$'s are distinct +variables and the $T_i$'s are terms. If $\Gamma$ contains some $x:T$, +we write $(x:T)\in\Gamma$ and also $x \in\Gamma$. Contexts must be +themselves {\em well formed}. The notation $\Gamma::(y:T)$ denotes +the context $[x_1:T_1;..;x_n:T_n;y:T]$. The notation $[]$ denotes the +empty context. +\index{Context} + +We define the inclusion of two contexts $\Gamma$ and $\Delta$ (written +as $\Gamma \subset \Delta$) as the property, for all variable $x$ and +type $T$, if $(x:T) \in \Gamma$ then $(x:T)\in \Delta$. We write +$|\Delta|$ for the length of the context $\Delta$ which is $n$ if +$\Delta$ is $[x_1:T_1;..; x_n:T_n]$. + +A variable $x$ is said to be free in $\Gamma$ if $\Gamma$ contains a +declaration $y:T$ such that $x$ is free in $T$. + +\paragraph{Environment.}\index{Environment} +Because we are manipulating constants, we also need to consider an +environment $E$. We shall give afterwards the rules for introducing +new objects in the environment. For the typing relation of terms, it +is enough to introduce two notions. One which says if a name is +defined in the environment we shall write $c \in E$ and the other one +which gives the type of this constant in $E$. We shall write $(c : T) +\in E$. + +In the following, we assume $E$ is a valid environment. We define +simultaneously two judgments. The first one \WTEG{t}{T} means the +term $t$ is well-typed and has type $T$ in the environment $E$ and +context $\Gamma$. The second judgment \WFE{\Gamma} means that the +environment $E$ is well-formed and the context $\Gamma$ is a valid +context in this environment. It also means a third property which +makes sure that any constant in $E$ was defined in an environment +which is included in $\Gamma$ +\footnote{This requirement could be relaxed if we instead introduced + an explicit mechanism for instantiating constants. At the external + level, the Coq engine works accordingly to this view that all the + definitions in the environment were built in a sub-context of the + current context.}. + +A term $t$ is well typed in an environment $E$ iff there exists a +context $\Gamma$ and a term $T$ such that the judgment \WTEG{t}{T} can +be derived from the following rules. +\begin{itemize}\label{Typing-rules}\index{Typing rules} +\item [W-E] \inference{\WF{[]}{[]}} +\item [W-$s$] +\inference{\frac{\WTEG{T}{s}~~~~s\in \Sort~~~~x \not\in + \Gamma \cup E}{\WFE{\Gamma::(x:T)}}} +\item [Ax] \index{Typing rules!Ax} +\inference{\frac{\WFE{\Gamma}}{\WTEG{\Prop}{\Type(p)}}~~~~~ +\frac{\WFE{\Gamma}}{\WTEG{\Set}{\Type(q)}}}\\[3mm] +\inference{\frac{\WFE{\Gamma}~~~~i<j}{\WTEG{\Type(i)}{\Type(j)}}} +\item[Var]\index{Typing rules!Var} + \inference{\frac{ \WFE{\Gamma}~~~~~(x:T)\in\Gamma}{\WTEG{x}{T}}} +\item[Const] \index{Typing rules!Const} +\inference{\frac{\WFE{\Gamma}~~~~(c:T) \in E}{\WTEG{c}{T}}} +\item [Prod] \index{Typing rules!Prod} +\inference{\frac{\WTEG{T}{s_1}~~~~ + \WTE{\Gamma::(x:T)}{U}{s_2}~~~s_1\in\{\Prop, \Set\}~\mbox{or}~ + s_2\in\{\Prop, \Set\}} + { \WTEG{(x:T)U}{s_2}}} \\[3mm] +\inference{\frac{\WTEG{T}{\Type(i)}~~~~ + \WTE{\Gamma::(x:T)}{U}{\Type(j)}~~~i\leq + k~~~j \leq k}{ \WTEG{(x:T)U}{\Type(k)}}} +\item [Lam]\index{Typing rules!Lam} +\inference{\frac{\WTEG{(x:T)U}{s}~~~~ \WTE{\Gamma::(x:T)}{t}{U}} + {\WTEG{[x:T]t}{(x:T)U}}} +\item [App]\index{Typing rules!App} + \inference{\frac{\WTEG{t}{(x:U)T}~~~~\WTEG{u}{U}} + {\WTEG{(t\ u)}{\subst{T}{x}{u}}}} +\end{itemize} + +\section{Conversion rules} +\index{Conversion rules} +\label{conv-rules} +\paragraph{$\beta$-reduction.} +\label{beta}\index{beta-reduction@$\beta$-reduction} + +We want to be able to identify some terms as we can identify the +application of a function to a given argument with its result. For +instance the identity function over a given type $T$ can be written +$[x:T]x$. We want to identify any object $a$ (of type $T$) with the +application $([x:T]x~a)$. We define for this a {\em reduction} (or a +{\em conversion}) rule we call $\beta$: +\[ ([x:T]t~u) \triangleright_{\beta} \subst{t}{x}{u} \] + +We say that $\subst{t}{x}{u}$ is the {\em $\beta$-contraction} of +$([x:T]t~u)$ and, conversely, that $([x:T]t~u)$ is the {\em + $\beta$-expansion} of $\subst{t}{x}{u}$. + +According to $\beta$-reduction, terms of the {\em Calculus of + Inductive Constructions} enjoy some fundamental properties such as +confluence, strong normalization, subject reduction. These results are +theoretically of great importance but we will not detail them here and +refer the interested reader to \cite{Coq85}. + +\paragraph{$\iota$-reduction.} +\label{iota}\index{iota-reduction@$\iota$-reduction} +A specific conversion rule is associated to the inductive objects in +the environment. We shall give later on (section \ref{iotared}) the +precise rules but it just says that a destructor applied to an object +built from a constructor behaves as expected. This reduction is +called $\iota$-reduction and is more precisely studied in +\cite{Moh93,Wer94}. + + +\paragraph{$\delta$-reduction.} +\label{convertibility} +\label{delta}\index{delta-reduction@$\delta$-reduction} +In the environment we also have constants representing abbreviations +for terms. It is legal to identify a constant with its value. This +reduction will be precised in section \ref{deltared} where we define +well-formed environments. This reduction will be called +$\delta$-reduction. + +\paragraph{Convertibility.} +\index{beta-conversion@$\beta$-conversion}\index{iota-conversion@$\iota$-conversion}\index{delta-conversion@$\delta$-conversion} + + +Let us write $t \triangleright u$ for the relation $t$ reduces to $u$ +with one of the previous reduction $\beta,\iota$ or $\delta$. + +We say that two terms $t_1$ and $t_2$ are {\em convertible} (or {\em + equivalent)} iff there exists a term $u$ such that $t_1 +\triangleright \ldots \triangleright u$ and $t_2 \triangleright \ldots +\triangleright u$. We note $t_1 \convert t_2$. + +The convertibility relation allows to introduce a new typing rule +which says that two convertible well-formed types have the same +inhabitants. + +At the moment, we did not take into account one rule between universes +which says that any term in a universe of index $i$ is also a term in +the universe of index $i+1$. This property is included into the +conversion rule by extending the equivalence relation of +convertibility into an order inductively defined by: +\begin{enumerate} +\item if $M\convert N$ then $M\leconvert N$, +\item if $i \leq j$ then $\Type(i)\leconvert\Type(j)$, +\item if $T \convert U$ and $M\leconvert N$ then $(x:T)M\leconvert + (x:U)N$. +\end{enumerate} + +The conversion rule is now exactly: + +\begin{itemize}\label{Conv} +\item [Conv]\index{Typing rules!Conv} + \inference{ + \frac{\WTEG{U}{S}~~~~\WTEG{t}{T}~~~~T \leconvert U}{\WTEG{t}{U}}} +\end{itemize} + + +\paragraph{$\eta$-conversion.}\label{eta}\index{eta-conversion@$\eta$-conversion}\index{eta-reduction@$\eta$-reduction} +An other important rule is the $\eta$-conversion. It is to identify +terms over a dummy abstraction of a variable followed by an +application of this variable. Let $T$ be a type, $t$ be a term in +which the variable $x$ doesn't occurs free. We have +\[ [x:T](t\ x) \triangleright t \] +Indeed, as $x$ doesn't occurs free in $t$, for any $u$ one +applies to $[x:T](t\ x)$, it $\beta$-reduces to $(t\ u)$. So +$[x:T](t\ x)$ and $t$ can be identified. + +\Rem The $\eta$-reduction is not taken into account in the +convertibility rule of \Coq. + +\paragraph{Normal form.}\index{Normal form}\label{Normal-form}\label{Head-normal-form}\index{Head normal form} +A term which cannot be any more reduced is said to be in {\em normal + form}. There are several ways (or strategies) to apply the reduction +rule. Among them, we have to mention the {\em head reduction} which +will play an important role (see chapter \ref{Tactics}). Any term can +be written as $[x_1:T_1]\ldots[x_k:T_k](t_0\ t_1\ldots t_n)$ where +$t_0$ is not an application. We say then that $t_0$ is the {\em head + of $t$}. If we assume that $t_0$ is $[x:T]u_0$ then one step of +$\beta$-head reduction of $t$ is: +\[[x_1:T_1]\ldots[x_k:T_k]([x:T]u_0\ t_1\ldots t_n) +~\triangleright ~ [x_1:T_1]\ldots[x_k:T_k](\subst{u_0}{x}{t_1}\ t_2 \ldots t_n)\] +Iterating the process of head reduction until the head of the reduced +term is no more an abstraction leads to the {\em $\beta$-head normal + form} of $t$: +\[ t \triangleright \ldots \triangleright [x_1:T_1]\ldots[x_k:T_k](v\ u_1 +\ldots u_m)\] +where $v$ is not an abstraction (nor an application). Note that the +head normal form must not be confused with the normal form since some +$u_i$ can be reducible. + +Similar notions of head-normal forms involving $\delta$ and $\iota$ +reductions or any combination of those can also be defined. + +\section{Definitions in environments}\label{Cic-definitions} +We now give the rules for manipulating objects in the environment. +Because a constant can depend on previously introduced constants, the +environment will be an ordered list of declarations. When specifying +an inductive definition, several objects will be introduced at the +same time. So any object in the environment will define one or more +constants. + +In this presentation we introduce two different sorts of objects in +the environment. The first one is ordinary definitions which give a +name to a particular well-formed term, the second one is inductive +definitions which introduce new inductive objects. + +\subsection{Rules for definitions} + +\paragraph{Adding a new definition.} +The simplest objects in the environment are definitions which can be +seen as one possible mechanism for abbreviation. + +A definition will be represented in the environment as +\Def{\Gamma}{c}{t}{T} which means that $c$ is a constant which is +valid in the context $\Gamma$ whose value is $t$ and type is $T$. + +\paragraph{$\delta$-reduction.} \label{deltared} +If \Def{\Gamma}{c}{t}{T} is in the environment $E$ then in this +environment the $\delta$-reduc\-tion $c \triangleright_{\delta} t$ is +introduced. + +The rule for adding a new definition is simple: + +\begin{description} +\item[Def] \inference{\frac{\WTEG{t}{T}~~~c \notin E\cup \Gamma} + {\WF{E;\Def{\Gamma}{c}{t}{T}}{\Gamma}}} +\end{description} + +\subsection{Derived rules} + +From the original rules of the type system, one can derive new rules +which change the context of definition of objects in the environment. +Because these rules correspond to elementary operations in the \Coq\ +engine used in the discharge mechanism at the end of a section, we +state them explicitly. + +\paragraph{Mechanism of substitution.} + +One rule which can be proved valid, is to replace a term $c$ by its +value in the environment. As we defined the substitution of a term for +a variable in a term, one can define the substitution of a term for a +constant. One easily extends this substitution to contexts and +environments. + +\paragraph{Substitution Property:} +\inference{\frac{\WF{E;\Def{\Gamma}{c}{t}{T}; F}{\Delta}} + {\WF{E; \subst{F}{c}{t}}{\subst{\Delta}{c}{t}}}} + + +\paragraph{Abstraction.} + +One can modify the context of definition of a constant $c$ by +abstracting a constant with respect to the last variable $x$ of its +defining context. For doing that, we need to check that the constants +appearing in the body of the declaration do not depend on $x$, we need +also to modify the reference to the constant $c$ in the environment +and context by explicitly applying this constant to the variable $x$. +Because of the rules for building environments and terms we know the +variable $x$ is available at each stage where $c$ is mentioned. + +\paragraph{Abstracting property:} + \inference{\frac{\WF{E; \Def{\Gamma::(x:U)}{c}{t}{T}; + F}{\Delta}~~~~\WFE{\Gamma}} + {\WF{E;\Def{\Gamma}{c}{[x:U]t}{(x:U)T}; + \subst{F}{c}{(c~x)}}{\subst{\Delta}{c}{(c~x)}}}} + +\paragraph{Pruning the context.} +We said the judgment \WFE{\Gamma} means that the defining contexts of +constants in $E$ are included in $\Gamma$. If one abstracts or +substitutes the constants with the above rules then it may happen +that the context $\Gamma$ is now bigger than the one needed for +defining the constants in $E$. Because defining contexts are growing +in $E$, the minimum context needed for defining the constants in $E$ +is the same as the one for the last constant. One can consequently +derive the following property. + +\paragraph{Pruning property:} +\inference{\frac{\WF{E; \Def{\Delta}{c}{t}{T}}{\Gamma}} + {\WF{E;\Def{\Delta}{c}{t}{T}}{\Delta}}} + + +\section{Inductive Definitions}\label{Cic-inductive-definitions} + +A (possibly mutual) inductive definition is specified by giving the +names and the type of the inductive sets or families to be +defined and the names and types of the constructors of the inductive +predicates. An inductive declaration in the environment can +consequently be represented with two contexts (one for inductive +definitions, one for constructors). + +Stating the rules for inductive definitions in their general form +needs quite tedious definitions. We shall try to give a concrete +understanding of the rules by precising them on running examples. We +take as examples the type of natural numbers, the type of +parameterized lists over a type $A$, the relation which state that +a list has some given length and the mutual inductive definition of trees and +forests. + +\subsection{Representing an inductive definition} +\subsubsection{Inductive definitions without parameters} +As for constants, inductive definitions can be defined in a non-empty +context. \\ +We write \NInd{\Gamma}{\Gamma_I}{\Gamma_C} an inductive +definition valid in a context $\Gamma$, a +context of definitions $\Gamma_I$ and a context of constructors +$\Gamma_C$. +\paragraph{Examples.} +The inductive declaration for the type of natural numbers will be: +\[\NInd{}{\nat:\Set}{\nO:\nat,\nS:\nat\ra\nat}\] +In a context with a variable $A:\Set$, the lists of elements in $A$ is +represented by: +\[\NInd{A:\Set}{\List:\Set}{\Nil:\List,\cons : A \ra \List \ra + \List}\] + Assuming + $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is + $[c_1:C_1;\ldots;c_n:C_n]$, the general typing rules are: + +\bigskip +\inference{\frac{\NInd{\Gamma}{\Gamma_I}{\Gamma_C} \in E + ~~j=1\ldots k}{(I_j:A_j) \in E}} + +\inference{\frac{\NInd{\Gamma}{\Gamma_I}{\Gamma_C} \in E + ~~~~i=1.. n} + {(c_i:\subst{C_i}{I_j}{I_j}_{j=1\ldots k})\in E}} + +\subsubsection{Inductive definitions with parameters} + +We have to slightly complicate the representation above in order to handle +the delicate problem of parameters. +Let us explain that on the example of \List. As they were defined +above, the type \List\ can only be used in an environment where we +have a variable $A:\Set$. Generally one want to consider lists of +elements in different types. For constants this is easily done by abstracting +the value over the parameter. In the case of inductive definitions we +have to handle the abstraction over several objects. + +One possible way to do that would be to define the type \List\ +inductively as being an inductive family of type $\Set\ra\Set$: +\[\NInd{}{\List:\Set\ra\Set}{\Nil:(A:\Set)(\List~A),\cons : (A:\Set)A + \ra (\List~A) \ra (\List~A)}\] +There are drawbacks to this point of view. The +information which says that $(\List~\nat)$ is an inductively defined +\Set\ has been lost. +%\footnote{ +%The interested reader may look at the compare the above definition with the two +%following ones which have very different logical meaning:\\ +%$\NInd{}{\List:\Set}{\Nil:\List,\cons : (A:\Set)A +% \ra \List \ra \List}$ \\ +%$\NInd{}{\List:\Set\ra\Set}{\Nil:(A:\Set)(\List~A),\cons : (A:\Set)A +% \ra (\List~A\ra A) \ra (\List~A)}$.} + +In the system, we keep track in the syntax of the context of +parameters. The idea of these parameters is that they can be +instantiated and still we have an inductive definition for which we +know the specification. + +Formally the representation of an inductive declaration +will be +\Ind{\Gamma}{\Gamma_P}{\Gamma_I}{\Gamma_C} for an inductive +definition valid in a context $\Gamma$ with parameters $\Gamma_P$, a +context of definitions $\Gamma_I$ and a context of constructors +$\Gamma_C$. +The occurrences of the variables of $\Gamma_P$ in the contexts +$\Gamma_I$ and $\Gamma_C$ are bound. + +The definition \Ind{\Gamma}{\Gamma_P}{\;\Gamma_I}{\Gamma_C\;} will be +well-formed exactly when \NInd{\Gamma,\Gamma_P}{\;\Gamma_I}{\Gamma_C\;} is. +If $\Gamma_P$ is $[p_1:P_1;\ldots;p_r:P_r]$, an object in +\Ind{\Gamma}{\Gamma_P}{\Gamma_I}{\Gamma_C} applied to $q_1,\ldots,q_r$ +will behave as the corresponding object of +\NInd{\Gamma}{\substs{\Gamma_I}{p_i}{q_i}{i=1..r}}{\substs{\Gamma_C}{p_i}{q_i}{i=1..r}}. + +\paragraph{Examples} +The declaration for parameterized lists is: +\[\Ind{}{A:\Set}{\List:\Set}{\Nil:\List,\cons : A \ra \List \ra + \List}\] + +The declaration for the length of lists is: +\[\Ind{}{A:\Set}{\Length:(\List~A)\ra \nat\ra\Prop} + {\LNil:(\Length~(\Nil~A)~\nO),\\ + \LCons :(a:A)(l:(\List~A))(n:\nat)(\Length~l~n)\ra (\Length~(\cons~A~a~l)~(\nS~n))}\] + +The declaration for a mutual inductive definition of forests and trees is: +\[\NInd{[]}{\tree:\Set,\forest:\Set} + {\node:\forest \ra \tree, + \emptyf:\forest,\consf:\tree \ra \forest \ra \forest\-}\] + +These representations are the ones obtained as the result of the \Coq\ +declaration: +\begin{coq_example*} +Inductive Set nat := O : nat | S : nat -> nat. +Inductive list [A : Set] : Set := + nil : (list A) | cons : A -> (list A) -> (list A). +\end{coq_example*} +\begin{coq_example*} +Inductive Length [A:Set] : (list A) -> nat -> Prop := + Lnil : (Length A (nil A) O) + | Lcons : (a:A)(l:(list A))(n:nat) + (Length A l n)->(Length A (cons A a l) (S n)). +Mutual Inductive tree : Set := node : forest -> tree +with forest : Set := emptyf : forest | consf : tree -> forest -> forest. +\end{coq_example*} +The inductive declaration in \Coq\ is slightly different from the one +we described theoretically. The difference is that in the type of +constructors the inductive definition is explicitly applied to the +parameters variables. The \Coq\ type-checker verifies that all +parameters are applied in the correct manner in each recursive call. +In particular, the following definition will not be accepted because +there is an occurrence of \List\ which is not applied to the parameter +variable: +\begin{coq_example} +Inductive list [A : Set] : Set := + nil : (list A) | cons : A -> (list A->A) -> (list A). +\end{coq_example} + +\subsection{Types of inductive objects} +We have to give the type of constants in an environment $E$ which +contains an inductive declaration. + +\begin{description} +\item[Ind-Const] Assuming $\Gamma_P$ is $[p_1:P_1;\ldots;p_r:P_r]$, + $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is + $[c_1:C_1;\ldots;c_n:C_n]$, + +\inference{\frac{\Ind{\Gamma}{\Gamma_P}{\Gamma_I}{\Gamma_C} \in E + ~~j=1\ldots k}{(I_j:(p_1:P_1)\ldots(p_r:P_r)A_j) \in E}} + +\inference{\frac{\Ind{\Gamma}{\Gamma_P}{\Gamma_I}{\Gamma_C} \in E + ~~~~i=1.. n} + {(c_i:(p_1:P_1)\ldots(p_r:P_r)\subst{C_i}{I_j}{(I_j~p_1\ldots + p_r)}_{j=1\ldots k})\in E}} +\end{description} + +\paragraph{Example.} +We have $(\List:\Set \ra \Set), (\cons:(A:\Set)A\ra(\List~A)\ra +(\List~A))$, \\ +$(\Length:(A:\Set)(\List~A)\ra\nat\ra\Prop)$, $\tree:\Set$ and $\forest:\Set$. + +From now on, we write $\ListA$ instead of $(\List~A)$ and $\LengthA$ +for $(\Length~A)$. + +%\paragraph{Parameters.} +%%The parameters introduce a distortion between the inside specification +%%of the inductive declaration where parameters are supposed to be +%%instantiated (this representation is appropriate for checking the +%%correctness or deriving the destructor principle) and the outside +%%typing rules where the inductive objects are seen as objects +%%abstracted with respect to the parameters. + +%In the definition of \List\ or \Length\, $A$ is a parameter because +%what is effectively inductively defined is $\ListA$ or $\LengthA$ for +%a given $A$ which is constant in the type of constructors. But when +%we define $(\LengthA~l~n)$, $l$ and $n$ are not parameters because the +%constructors manipulate different instances of this family. + +\subsection{Well-formed inductive definitions} +We cannot accept any inductive declaration because some of them lead +to inconsistent systems. We restrict ourselves to definitions which +satisfy a syntactic criterion of positivity. Before giving the formal +rules, we need a few definitions: + +\paragraph{Definitions}\index{Positivity}\label{Positivity} + +A type $T$ is an {\em arity of sort $s$}\index{Arity} +if it converts to the sort $s$ or to a +product $(x:T)U$ with $U$ an arity of sort $s$. (For instance $A\ra +\Set$ or $(A:\Prop)A\ra \Prop$ are arities of sort respectively \Set\ +and \Prop). +A {\em type of constructor of $I$}\index{Type of constructor} + is either a term $(I~t_1\ldots ~t_n)$ or +$(x:T)C$ with $C$ a {\em type of constructor of $I$}. + +\smallskip + +The type of constructor $T$ will be said to {\em satisfy the positivity +condition} for a constant $X$ in the following cases: + +\begin{itemize} +\item $T=(X~t_1\ldots ~t_n)$ and $X$ does not occur free in +any $t_i$ +\item $T=(x:T)U$ and $X$ occurs only strictly positively in $T$ and +the type $U$ satisfies the positivity condition for $X$ +\end{itemize} + +The constant $X$ {\em occurs strictly positively} in $T$ in the +following cases: + +\begin{itemize} +\item $X$ does not occur in $T$ +\item $T$ converts to $(X~t_1 \ldots ~t_n)$ and $X$ does not occur in any of $t_i$ +\item $T$ converts to $(x:U)V$ and $X$ does not occur in type $U$ but occurs strictly +positively in type $V$ +\item $T$ converts to $(I~a_1 \ldots ~a_m ~ t_1 \ldots ~t_p)$ where $I$ is the name of an +inductive declaration of the form $\Ind{\Gamma}{[p_1:P_1;\ldots;p_m:P_m]}{[I:A]}{[c_1:C_1;\ldots;c_n:C_n]}$ +(in particular, it is not mutually defined and it has $m$ parameters) +and $X$ does not occur in any of the $t_i$, and +the types of constructor $C_i\{p_j/a_j\}_{j=1\ldots m}$ of $I$ +satisfy the imbricated positivity condition for $X$ +%\item more generally, when $T$ is not a type, $X$ occurs strictly +%positively in $T[x:U]u$ if $X$ does not occur in $U$ but occurs +%strictly positively in $u$ +\end{itemize} + +The type constructor $T$ of $I$ {\em satisfies the imbricated +positivity condition} for a constant $X$ in the following +cases: + +\begin{itemize} +\item $T=(I~t_1\ldots ~t_n)$ and $X$ does not occur in +any $t_i$ +\item $T=(x:T)U$ and $X$ occurs only strictly positively in $T$ and +the type $U$ satisfies the imbricated positivity condition for $X$ +\end{itemize} + +\paragraph{Example} +$X$ occurs strictly positively in $A\ra X$ or $X*A$ or $({\tt list} +X)$ but not in $X \ra A$ or $(X \ra A)\ra A$ assuming the notion of +product and lists were already defined. Assuming $X$ has arity ${\tt +nat \ra Prop}$ and {\tt ex} is inductively defined existential +quantifier, the occurrence of $X$ in ${\tt (ex~ nat~ [n:nat](X~ n))}$ is +also strictly positive.\\ + +\paragraph{Correctness rules.} +We shall now describe the rules allowing the introduction of a new +inductive definition. + +\begin{description} +\item[W-Ind] Let $E$ be an environment and + $\Gamma,\Gamma_P,\Gamma_I,\Gamma_C$ are contexts such that + $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$ and $\Gamma_C$ is + $[c_1:C_1;\ldots;c_n:C_n]$. \\[3mm] +\inference{ + \frac{ + (\WTE{\Gamma;\Gamma_P}{A_j}{s'_j})_{j=1\ldots k} + ~~ (\WTE{\Gamma;\Gamma_P;\Gamma_I}{C_i}{s_{p_i}})_{i=1\ldots n} +} + {\WF{E;\Ind{\Gamma}{\Gamma_P}{\Gamma_I}{\Gamma_C}}{\Gamma}}}\\ +providing the following side conditions hold: +\begin{itemize} +\item $k>0$, $I_j$, $c_i$ are different names for $j=1\ldots k$ and $i=1\ldots n$, +\item for $j=1\ldots k$ we have $A_j$ is an arity of sort $s_j$ and $I_j + \notin \Gamma \cup E$, +\item for $i=1\ldots n$ we have $C_i$ is a type of constructor of + $I_{p_i}$ which satisfies the positivity condition for $I_1 \ldots I_k$ + and $c_i \notin \Gamma \cup E$. +\end{itemize} +\end{description} +One can remark that there is a constraint between the sort of the +arity of the inductive type and the sort of the type of its +constructors which will always be satisfied for impredicative sorts +(\Prop\ or \Set) but may generate constraints between universes. + + +%We shall assume for the following definitions that, if necessary, we +%annotated the type of constructors such that we know if the argument +%is recursive or not. We shall write the type $(x:_R T)C$ if it is +%a recursive argument and $(x:_P T)C$ if the argument is not recursive. + +\subsection{Destructors} +The specification of inductive definitions with arities and +constructors is quite natural. But we still have to say how to use an +object in an inductive type. + +This problem is rather delicate. There are actually several different +ways to do that. Some of them are logically equivalent but not always +equivalent from the computational point of view or from the user point +of view. + +From the computational point of view, we want to be able to define a +function whose domain is an inductively defined type by using a +combination of case analysis over the possible constructors of the +object and recursion. + +Because we need to keep a consistent theory and also we prefer to keep +a strongly normalising reduction, we cannot accept any sort of +recursion (even terminating). So the basic idea is to restrict +ourselves to primitive recursive functions and functionals. + +For instance, assuming a parameter $A:\Set$ exists in the context, we +want to build a function \length\ of type $\ListA\ra \nat$ which +computes the length of the list, so such that $(\length~\Nil) = \nO$ +and $(\length~(\cons~A~a~l)) = (\nS~(\length~l))$. We want these +equalities to be recognized implicitly and taken into account in the +conversion rule. + +From the logical point of view, we have built a type family by giving +a set of constructors. We want to capture the fact that we do not +have any other way to build an object in this type. So when trying to +prove a property $(P~m)$ for $m$ in an inductive definition it is +enough to enumerate all the cases where $m$ starts with a different +constructor. + +In case the inductive definition is effectively a recursive one, we +want to capture the extra property that we have built the smallest +fixed point of this recursive equation. This says that we are only +manipulating finite objects. This analysis provides induction +principles. + +For instance, in order to prove $(l:\ListA)(\LengthA~l~(\length~l))$ +it is enough to prove: + +\noindent $(\LengthA~\Nil~(\length~\Nil))$ and + +\smallskip +$(a:A)(l:\ListA)(\LengthA~l~(\length~l)) \ra +(\LengthA~(\cons~A~a~l)~(\length~(\cons~A~a~l)))$. +\smallskip + +\noindent which given the conversion equalities satisfied by \length\ is the +same as proving: +$(\LengthA~\Nil~\nO)$ and $(a:A)(l:\ListA)(\LengthA~l~(\length~l)) \ra +(\LengthA~(\cons~A~a~l)~(\nS~(\length~l)))$. + +One conceptually simple way to do that, following the basic scheme +proposed by Martin-L\"of in his Intuitionistic Type Theory, is to +introduce for each inductive definition an elimination operator. At +the logical level it is a proof of the usual induction principle and +at the computational level it implements a generic operator for doing +primitive recursion over the structure. + +But this operator is rather tedious to implement and use. We choose in +this version of Coq to factorize the operator for primitive recursion +into two more primitive operations as was first suggested by Th. Coquand +in~\cite{Coq92}. One is the definition by case +analysis. The second one is a definition by guarded fixpoints. + +\subsubsection{The {\tt Cases\ldots of \ldots end} construction.} +\label{Caseexpr} +\index{Cases@{\tt Cases\ldots of\ldots end}} + +The basic idea of this destructor operation is that we have an object +$m$ in an inductive type $I$ and we want to prove a property $(P~m)$ +which in general depends on $m$. For this, it is enough to prove the +property for $m = (c_i~u_1\ldots u_{p_i})$ for each constructor of $I$. + +This proof will be denoted by a generic term: +\[\Case{P}{m}{(c_1~x_{11}~...~x_{1p_1}) \mbox{\tt =>} f_1 \ldots +(c_n~x_{n1}~|~...~|~x_{np_n}) \mbox{\tt =>} f_n }\] +In this expression, if $m$ is a term built from a constructor +$(c_i~u_1\ldots u_{p_i})$ then the expression will behave as it is specified +with $i$-th branch and will reduce to $f_i$ where the $x_{i1}$\ldots $x_{ip_i}$ are +replaced by the $u_1\ldots u_p$ according to +the $\iota$-reduction.\\ + +This is the basic idea which is generalized to the case where $I$ is +an inductively defined $n$-ary relation (in which case the property +$P$ to be proved will be a $n+1$-ary relation). + +\paragraph{Non-dependent elimination.} +When defining a function by case analysis, we build an object of type $I +\ra C$ and the minimality principle on an inductively defined logical +predicate of type $A \ra \Prop$ is often used to prove a property +$(x:A)(I~x)\ra (C~x)$. This is a particular case of the dependent +principle that we stated before with a predicate which does not depend +explicitly on the object in the inductive definition. + +For instance, a function testing whether a list is empty +can be +defined as: + +\[[l:\ListA]\Case{[H:\ListA]\bool}{l}{\Nil~ \mbox{\tt =>}~\true~ |~ (\cons~a~m)~ \mbox{\tt =>}~\false}\] +\noindent {\bf Remark. } In the system \Coq\ the expression above, can be +written without mentioning +the dummy abstraction: +\Case{\bool}{l}{\Nil~ \mbox{\tt =>}~\true~ |~ (\cons~a~m)~ + \mbox{\tt =>}~ \false} + +\paragraph{Allowed elimination sorts.} +\index{Elimination sorts} + +An important question for building the typing rule for {\tt Case} is +what can be the type of $P$ with respect to the type of the inductive +definitions. + +Remembering that the elimination builds an object in $(P~m)$ from an +object in $m$ in type $I$ it is clear that we cannot allow any combination. + +For instance we cannot in general have $I$ has type \Prop\ +and $P$ has type $I\ra \Set$, because it will mean to build an informative +proof of type $(P~m)$ doing a case analysis over a non-computational +object that will disappear in the extracted program. +But the other way is safe +with respect to our interpretation we can have $I$ a computational +object and $P$ a non-computational one, it just corresponds to proving a +logical property of a computational object. + +Also if $I$ is in one of the sorts \{\Prop, \Set\}, one cannot in +general allow an elimination over a bigger sort such as \Type. +But this operation is safe whenever $I$ is a {\em small + inductive} type, which means that all the types of constructors of +$I$ are small with the following definition:\\ +$(I~t_1\ldots t_s)$ is a {\em small type of constructor} and $(x:T)C$ is a +small type of constructor if $C$ is and if $T$ has type \Prop\ or +\Set. +\index{Small inductive type} + +We call this particular elimination which gives the possibility to +compute a type by induction on the structure of a term, a {\em strong + elimination}\index{Strong elimination}. + + +We define now a relation \compat{I:A}{B} between an inductive +definition $I$ of type $A$, an arity $B$ which says that an object in +the inductive definition $I$ can be eliminated for proving a property +$P$ of type $B$. + +The \compat{I:A}{B} is defined as the smallest relation satisfying the +following rules: + +\begin{description} +\item[Prod] \inference{\frac{\compat{(I~x):A'}{B'}} + {\compat{I:(x:A)A'}{(x:A)B'}}} +\item[\Prop] \inference{\compat{I:\Prop}{I\ra\Prop}~~~~~ + \frac{I \mbox{~is a singleton definition}}{\compat{I:\Prop}{I\ra \Set}}} + +\item[\Set] \inference{\frac{s \in + \{\Prop, \Set\}}{\compat{I:\Set}{I\ra s}} +~~~~\frac{I \mbox{~is a small inductive definition}~~~~s \in + \{\Type(i)\}} + {\compat{I:\Set}{I\ra s}}} +\item[\Type] \inference{\frac{ + s \in \{\Prop,\Set,\Type(j)\}}{\compat{I:\Type(i)}{I\ra s}}} +\end{description} + +\paragraph{Notations.} +We write \compat{I}{B} for \compat{I:A}{B} where $A$ is the type of +$I$. + +%\paragraph{Warning: strong elimination} +%\index{Elimination!Strong elimination} +%In previous versions of Coq, for a small inductive definition, only the +%non-informative strong elimination on \Type\ was allowed, because +%strong elimination on \Typeset\ was not compatible with the current +%extraction procedure. In this version, strong elimination on \Typeset\ +%is accepted but a dummy element is extracted from it and may generate +%problems if extracted terms are explicitly used such as in the +%{\tt Program} tactic or when extracting ML programs. + +\paragraph{Singleton elimination} +\index{Elimination!Singleton elimination} + +A {\em singleton definition} has always an informative content, +even if it is a proposition. + +A {\em singleton +definition} has only one constructor and all the argument of this +constructor are non informative. In that case, there is a canonical +way to interpret the informative extraction on an object in that type, +such that the elimination on sort $s$ is legal. Typical examples are +the conjunction of non-informative propositions and the equality. In +that case, the term \verb!eq_rec! which was defined as an axiom, is +now a term of the calculus. +\begin{coq_example} +Print eq_rec. +Extraction eq_rec. +\end{coq_example} + +\paragraph{Type of branches.} +Let $c$ be a term of type $C$, we assume $C$ is a type of constructor +for an inductive definition $I$. Let $P$ be a term that represents the +property to be proved. +We assume $r$ is the number of parameters. + +We define a new type \CI{c:C}{P} which represents the type of the +branch corresponding to the $c:C$ constructor. +\[ +\begin{array}{ll} +\CI{c:(I_i~p_1\ldots p_r\ t_1 \ldots t_p)}{P} &\equiv (P~t_1\ldots ~t_p~c) \\[2mm] +\CI{c:(x: T)C}{P} &\equiv (x:T)\CI{(c~x):C}{P} +\end{array} +\] +We write \CI{c}{P} for \CI{c:C}{P} with $C$ the type of $c$. + +\paragraph{Examples.} +For $\ListA$ the type of $P$ will be $\ListA\ra s$ for $s \in +\{\Prop,\Set,\Type(i)\}$. \\ +$ \CI{(\cons~A)}{P} \equiv +(a:A)(l:\ListA)(P~(\cons~A~a~l))$. + +For $\LengthA$, the type of $P$ will be +$(l:\ListA)(n:\nat)(\LengthA~l~n)\ra \Prop$ and the expression +\CI{(\LCons~A)}{P} is defined as:\\ +$(a:A)(l:\ListA)(n:\nat)(h:(\LengthA~l~n))(P~(\cons~A~a~l)~(\nS~n)~(\LCons~A~a~l~n~l))$.\\ +If $P$ does not depend on its third argument, we find the more natural +expression:\\ +$(a:A)(l:\ListA)(n:\nat)(\LengthA~l~n)\ra(P~(\cons~A~a~l)~(\nS~n))$. + +\paragraph{Typing rule.} + +Our very general destructor for inductive definition enjoys the +following typing rule (we write {\Case{P}{c}{[x_{11}:T_{11}]\ldots[x_{1p_1}:T_{1p_1}]f_1\ldots [x_{n1}:T_{n1}]\ldots[x_{np_n}:T_{np_n}f_n} for \Case{P}{m}{(c_1~x_{11}~...~x_{1p_1}) \mbox{\tt =>} f_1 \ldots +(c_n~x_{n1}~|~...~|~x_{np_n}) \mbox{\tt =>} f_n }}): + +\begin{description} +\item[Case] \label{elimdep} \index{Typing rules!Case} +\inference{ +\frac{\WTEG{c}{(I~q_1\ldots q_r~t_1\ldots t_s)}~~ + \WTEG{P}{B}~~\compat{(I~q_1\ldots q_r)}{B} + ~~ +(\WTEG{f_i}{\CI{(c_{p_i}~q_1\ldots q_r)}{P}})_{i=1\ldots l}} +{\WTEG{\Case{P}{c}{f_1\ldots f_l}}{(P\ t_1\ldots t_s\ c)}}}%\\[3mm] + +provided $I$ is an inductive type in a declaration +\Ind{\Delta}{\Gamma_P}{\Gamma_I}{\Gamma_C} with $|\Gamma_P| = r$, +$\Gamma_C = [c_1:C_1;\ldots;c_n:C_n]$ and $c_{p_1}\ldots c_{p_l}$ are the +only constructors of $I$. +\end{description} + +\paragraph{Example.} +For \List\ and \Length\ the typing rules for the {\tt Case} expression +are (writing just $t:M$ instead of \WTEG{t}{M}, the environment and +context being the same in all the judgments). + +\[\frac{l:\ListA~~P:\ListA\ra s~~~f_1:(P~(\Nil~A))~~ + f_2:(a:A)(l:\ListA)(P~(\cons~A~a~l))} + {\Case{P}{l}{f_1~f_2}:(P~l)}\] + +\[\frac{ + \begin{array}[b]{c} +H:(\LengthA~L~N) \\ P:(l:\ListA)(n:\nat)(\LengthA~l~n)\ra + \Prop\\ + f_1:(P~(\Nil~A)~\nO~\LNil) \\ + f_2:(a:A)(l:\ListA)(n:\nat)(h:(\LengthA~l~n))(P~(\cons~A~a~n)~(\nS~n)~(\LCons~A~a~l~n~h)) + \end{array}} + {\Case{P}{H}{f_1~f_2}:(P~L~N~H)}\] + +\paragraph{Definition of $\iota$-reduction.}\label{iotared} +\index{iota-reduction@$\iota$-reduction} +We still have to define the $\iota$-reduction in the general case. + +A $\iota$-redex is a term of the following form: +\[\Case{P}{(c_{p_i}~q_1\ldots q_r~a_1\ldots a_m)}{f_1\ldots + f_l}\] +with $c_{p_i}$ the $i$-th constructor of the inductive type $I$ with $r$ +parameters. + +The $\iota$-contraction of this term is $(f_i~a_1\ldots a_m)$ leading +to the general reduction rule: +\[ \Case{P}{(c_{p_i}~q_1\ldots q_r~a_1\ldots a_m)}{f_1\ldots + f_n} \triangleright_{\iota} (f_i~a_1\ldots a_m) \] + +\subsection{Fixpoint definitions} +\label{Fix-term} \index{Fix@{\tt Fix}} +The second operator for elimination is fixpoint definition. +This fixpoint may involve several mutually recursive definitions. +The basic syntax for a recursive set of declarations is +\[\Fix{}{f_1:A_1:=t_1 \ldots f_n:A_n:=t_n}\] +The terms are obtained by projections from this set of declarations +and are written $\Fix{f_i}{f_1:A_1:=t_1 \ldots f_n:A_n:=t_n}$ + +\subsubsection{Typing rule} +The typing rule is the expected one for a fixpoint. + +\begin{description} +\item[Fix] \index{Typing rules!Fix} +\inference{\frac{(\WTEG{A_i}{s_i})_{i=1\ldots n}~~~~ + (\WTE{\Gamma,f_1:A_1,\ldots,f_n:A_n}{t_i}{A_i})_{i=1\ldots n}} + {\WTEG{\Fix{f_i}{f_1:A_1:=t_1 \ldots f_n:A_n:=t_n}}{A_i}}} +\end{description} + +Any fixpoint definition cannot be accepted because non-normalizing terms +will lead to proofs of absurdity. + +The basic scheme of recursion that should be allowed is the one needed for +defining primitive +recursive functionals. In that case the fixpoint enjoys special +syntactic restriction, namely one of the arguments belongs to an +inductive type, the function starts with a case analysis and recursive +calls are done on variables coming from patterns and representing subterms. + +For instance in the case of natural numbers, a proof of the induction +principle of type +\[(P:\nat\ra\Prop)(P~\nO)\ra((n:\nat)(P~n)\ra(P~(\nS~n)))\ra(n:\nat)(P~n)\] +can be represented by the term: +\[\begin{array}{l} +[P:\nat\ra\Prop][f:(P~\nO)][g:(n:\nat)(P~n)\ra(P~(\nS~n))]\\ +\Fix{h}{h:(n:\nat)(P~n):=[n:\nat]\Case{P}{n}{f~[p:\nat](g~p~(h~p))}} +\end{array} +\] + +Before accepting a fixpoint definition as being correctly typed, we +check that the definition is ``guarded''. A precise analysis of this +notion can be found in~\cite{Gim94}. + +The first stage is to precise on which argument the fixpoint will be +decreasing. The type of this argument should be an inductive +definition. + +For doing this the syntax of fixpoints is extended and becomes + \[\Fix{f_i}{f_1/k_1:A_1:=t_1 \ldots f_n/k_n:A_n:=t_n}\] +where $k_i$ are positive integers. +Each $A_i$ should be a type (reducible to a term) starting with at least +$k_i$ products $(y_1:B_1)\ldots (y_{k_i}:B_{k_i}) A'_i$ and $B_{k_i}$ +being an instance of an inductive definition. + +Now in the definition $t_i$, if $f_j$ occurs then it should be applied +to at least $k_j$ arguments and the $k_j$-th argument should be +syntactically recognized as structurally smaller than $y_{k_i}$ + + +The definition of being structurally smaller is a bit technical. +One needs first to define the notion of +{\em recursive arguments of a constructor}\index{Recursive arguments}. +For an inductive definition \Ind{\Gamma}{\Gamma_P}{\Gamma_I}{\Gamma_C}, +the type of a constructor $c$ have the form +$(p_1:P_1)\ldots(p_r:P_r)(x_1:T_1)\ldots (x_r:T_r)(I_j~p_1\ldots +p_r~t_1\ldots t_s)$ the recursive arguments will correspond to $T_i$ in +which one of the $I_l$ occurs. + + +The main rules for being structurally smaller are the following:\\ +Given a variable $y$ of type an inductive +definition in a declaration +\Ind{\Gamma}{\Gamma_P}{\Gamma_I}{\Gamma_C} +where $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is + $[c_1:C_1;\ldots;c_n:C_n]$. +The terms structurally smaller than $y$ are: +\begin{itemize} +\item $(t~u), [x:u]t$ when $t$ is structurally smaller than $y$ . +\item \Case{P}{c}{f_1\ldots f_n} when each $f_i$ is structurally + smaller than $y$. \\ + If $c$ is $y$ or is structurally smaller than $y$, its type is an inductive + definition $I_p$ part of the inductive + declaration corresponding to $y$. + Each $f_i$ corresponds to a type of constructor $C_q \equiv + (y_1:B_1)\ldots (y_k:B_k)(I~a_1\ldots a_k)$ and can consequently be + written $[y_1:B'_1]\ldots [y_k:B'_k]g_i$. + ($B'_i$ is obtained from $B_i$ by substituting parameters variables) + the variables $y_j$ occurring + in $g_i$ corresponding to recursive arguments $B_i$ (the ones in + which one of the $I_l$ occurs) are structurally smaller than $y$. +\end{itemize} +The following definitions are correct, we enter them using the +{\tt Fixpoint} command as described in section~\ref{Fixpoint} and show +the internal representation. +\begin{coq_example} +Fixpoint plus [n:nat] : nat -> nat := + [m:nat]Case n of m [p:nat](S (plus p m)) end. +Print plus. +Fixpoint lgth [A:Set;l:(list A)] : nat := + Case l of O [a:A][l':(list A)](S (lgth A l')) end. +Print lgth. +Fixpoint sizet [t:tree] : nat + := Case t of [f:forest](S (sizef f)) end +with sizef [f:forest] : nat + := Case f of O [t:tree][f:forest](plus (sizet t) (sizef f)) end. +Print sizet. +\end{coq_example} + + +\subsubsection{Reduction rule} +\index{iota-reduction@$\iota$-reduction} +Let $F$ be the set of declarations: $f_1/k_1:A_1:=t_1 \ldots +f_n/k_n:A_n:=t_n$. +The reduction for fixpoints is: +\[ (\Fix{f_i}{F}~a_1\ldots +a_{k_i}) \triangleright_{\iota} \substs{t_i}{f_k}{\Fix{f_k}{F}}{k=1\ldots n}\] +when $a_{k_i}$ starts with a constructor. +This last restriction is needed in order to keep strong normalization +and corresponds to the reduction for primitive recursive operators. + +We can illustrate this behavior on examples. +\begin{coq_example} +Goal (n,m:nat)(plus (S n) m)=(S (plus n m)). +Reflexivity. +Abort. +Goal (f:forest)(sizet (node f))=(S (sizef f)). +Reflexivity. +Abort. +\end{coq_example} +But assuming the definition of a son function from \tree\ to \forest: +\begin{coq_example} + Definition sont : tree -> forest := [t]Case t of [f]f end. +\end{coq_example} +The following is not a conversion but can be proved after a case analysis. +\begin{coq_example} +Goal (t:tree)(sizet t)=(S (sizef (sont t))). +(* this one fails *) +Reflexivity. +Destruct t. +Reflexivity. +\end{coq_example} +\begin{coq_eval} +Abort. +\end{coq_eval} + +\subsubsection{The {\tt Match \ldots with \ldots end} expression} +\label{Matchexpr} +%\paragraph{A unary {\tt Match\ldots with \ldots end}.} +\index{Match...with...end@{\tt Match \ldots with \ldots end}} +The {\tt Match} operator which was a primitive notion in older +presentations of the Calculus of Inductive Constructions is now just a +macro definition which generates the good combination of {\tt Case} +and {\tt Fix} operators in order to generate an operator for primitive +recursive definitions. It always considers an inductive definition as +a single inductive definition. + +The following examples illustrates this feature. +\begin{coq_example} +Definition nat_pr : (C:Set)C->(nat->C->C)->nat->C + :=[C,x,g,n]Match n with x g end. +Print nat_pr. +\end{coq_example} +\begin{coq_example} +Definition forest_pr + : (C:Set)C->(tree->forest->C->C)->forest->C + := [C,x,g,n]Match n with x g end. +\end{coq_example} + +% Cet exemple faisait error (HH le 12/12/96), j'ai change pour une +% version plus simple +%\begin{coq_example} +%Definition forest_pr +% : (P:forest->Set)(P emptyf)->((t:tree)(f:forest)(P f)->(P (consf t f))) +% ->(f:forest)(P f) +% := [C,x,g,n]Match n with x g end. +%\end{coq_example} + +The principles of mutual induction can be automatically generated +using the {\tt Scheme} command described in section~\ref{Scheme}. + +\section{Coinductive types} +The implementation contains also coinductive definitions, which are +types inhabited by infinite objects. +%They are described inchapter~\ref{Coinductives}. + +% $Id$ + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "Reference-Manual" +%%% End: + + |