\chapter[Calculus of Inductive Constructions]{Calculus of Inductive Constructions \label{Cic} \index{Cic@\textsc{CIC}} \index{Calculus of Inductive Constructions}} The underlying formal language of {\Coq} is a {\em Calculus of Inductive Constructions} (\CIC) whose inference rules are presented in this chapter. The history of this formalism as well as pointers to related work are provided in a separate chapter; see {\em Credits}. \section[The terms]{The terms\label{Terms}} The expressions of the {\CIC} are {\em terms} and all terms 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, universal quantification is relative to a type and takes the form {\it ``for all x of type T, P''}. The expression {\it ``x of type T''} is written {\it ``x:T''}. Informally, {\it ``x:T''} can be thought as {\it ``x belongs to T''}. The types of types are {\em sorts}. Types and sorts are themselves terms so that terms, types and sorts are all components of a common syntactic language of terms which is described in Section~\ref{cic:terms} but, first, we describe sorts. \subsection[Sorts]{Sorts\label{Sorts} \index{Sorts}} All sorts have a type and there is an infinite well-founded typing hierarchy of sorts whose base sorts are {\Prop} and {\Set}. The sort {\Prop} intends to be the type of logical propositions. If $M$ is a logical proposition then it denotes the class of terms representing proofs of $M$. An object $m$ belonging to $M$ witnesses the fact that $M$ is provable. An object of type {\Prop} is called a proposition. The sort {\Set} intends to be the type of small sets. This includes data types such as booleans and naturals, but also products, subsets, and function types over these data types. {\Prop} and {\Set} themselves can be manipulated as ordinary terms. Consequently they also have a type. Because assuming simply that {\Set} has type {\Set} leads to an inconsistent theory~\cite{Coq86}, the language of {\CIC} has infinitely many sorts. There are, in addition to {\Set} and {\Prop} a hierarchy of universes {\Type$(i)$} for any integer $i$. Like {\Set}, all of the sorts {\Type$(i)$} contain small sets such as booleans, natural numbers, as well as products, subsets and function types over small sets. But, unlike {\Set}, they also contain large sets, namely the sorts {\Set} and {\Type$(j)$} for $j}~ u$ in \Coq{} concrete syntax) is a term. This is a notation for the $\lambda$-abstraction of $\lambda$-calculus\index{lambda-calculus@$\lambda$-calculus} \cite{Bar81}. The term $\lb x:T \mto u$ is a function which maps elements of $T$ to the expression $u$. \item if $t$ and $u$ are terms then $(t\ u)$ is a term ($t~u$ in \Coq{} concrete syntax). The term $(t\ u)$ reads as {\it ``t applied to u''}. \item if $x$ is a variable, and $t$, $T$ and $u$ are terms then $\kw{let}~x:=t:T~\kw{in}~u$ is a term which denotes the term $u$ where the variable $x$ is locally bound to $t$ of type $T$. This stands for the common ``let-in'' construction of functional programs such as ML or Scheme. %\item case ... %\item fixpoint ... %\item cofixpoint ... \end{enumerate} \paragraph{Free variables.} The notion of free variables is defined as usual. In the expressions $\lb x:T\mto U$ and $\forall x:T, U$ the occurrences of $x$ in $U$ are bound. \paragraph[Substitution.]{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 is written $\subst{u}{x}{t}$. \paragraph[The logical vs programming readings.]{The logical vs programming readings.} The constructions of the {\CIC} can be used to express both logical and programming notions, accordingly to the Curry-Howard correspondence between proofs and programs, and between propositions and types~\cite{Cur58,How80,Bru72}. For instance, let us assume that \nat\ is the type of natural numbers with zero element written $0$ and that ${\tt True}$ is the always true proposition. Then $\ra$ is used both to denote $\nat\ra\nat$ which is the type of functions from \nat\ to \nat, to denote ${\tt True}\ra{\tt True}$ which is an implicative proposition, to denote $\nat \ra \Prop$ which is the type of unary predicates over the natural numbers, etc. Let us assume that ${\tt mult}$ is a function of type $\nat\ra\nat\ra \nat$ and ${\tt eqnat}$ a predicate of type $\nat\ra\nat\ra \Prop$. The $\lambda$-abstraction can serve to build ``ordinary'' functions as in $\lambda x:\nat.({\tt mult}~x~x)$ (i.e. $\kw{fun}~x:\nat ~{\tt =>}~ {\tt mult} ~x~x$ in {\Coq} notation) but may build also predicates over the natural numbers. For instance $\lambda x:\nat.({\tt eqnat}~ x~0)$ (i.e. $\kw{fun}~x:\nat ~{\tt =>}~ {\tt eqnat}~ x~0$ in {\Coq} notation) will represent the predicate of one variable $x$ which asserts the equality of $x$ with $0$. This predicate has type $\nat \ra \Prop$ and it can be applied to any expression of type ${\nat}$, say $t$, to give an object $P~t$ of type \Prop, namely a proposition. Furthermore $\kw{forall}~x:\nat,\,P\;x$ will represent the type of functions which associate to each natural number $n$ an object of type $(P~n)$ and consequently represent the type of proofs of the formula ``$\forall x.\,P(x)$''. \section[Typing rules]{Typing rules\label{Typed-terms}} As objects of type theory, terms are subjected to {\em type discipline}. The well typing of a term depends on a global environment and a local context. \paragraph{Local context.\index{Local context}} A {\em local context} is an ordered list of {\em local declarations\index{declaration!local}} of names which we call {\em variables\index{variable}}. The declaration of some variable $x$ is either a {\em local assumption\index{assumption!local}}, written $x:T$ ($T$ is a type) or a {\em local definition\index{definition!local}}, written $x:=t:T$. We use brackets to write local contexts. A typical example is $[x:T;y:=u:U;z:V]$. Notice that the variables declared in a local context must be distinct. If $\Gamma$ declares some $x$, we write $x \in \Gamma$. By writing $(x:T) \in \Gamma$ we mean that either $x:T$ is an assumption in $\Gamma$ or that there exists some $t$ such that $x:=t:T$ is a definition in $\Gamma$. If $\Gamma$ defines some $x:=t:T$, we also write $(x:=t:T) \in \Gamma$. For the rest of the chapter, the $\Gamma::(y:T)$ denotes the local context $\Gamma$ enriched with the local assumption $y:T$. Similarly, $\Gamma::(y:=t:T)$ denotes the local context $\Gamma$ enriched with the local definition $(y:=t:T)$. The notation $[]$ denotes the empty local context. By $\Gamma_1; \Gamma_2$ we mean concatenation of the local context $\Gamma_1$ and the local context $\Gamma_2$. % Does not seem to be used further... % Si dans l'explication WF(E)[Gamma] concernant les constantes % definies ds un contexte %We define the inclusion of two local contexts $\Gamma$ and $\Delta$ (written %as $\Gamma \subset \Delta$) as the property, for all variable $x$, %type $T$ and term $t$, if $(x:T) \in \Gamma$ then $(x:T) \in \Delta$ %and if $(x:=t:T) \in \Gamma$ then $(x:=t:T) \in \Delta$. %We write % $|\Delta|$ for the length of the context $\Delta$, that is for the number % of declarations (assumptions or definitions) in $\Delta$. \paragraph[Global environment.]{Global environment.\index{Global environment}} %Because we are manipulating global declarations (global constants and global %assumptions), we also need to consider a global environment $E$. A {\em global environment} is an ordered list of {\em global declarations\index{declaration!global}}. Global declarations are either {\em global assumptions\index{assumption!global}} or {\em global definitions\index{definition!global}}, but also declarations of inductive objects. Inductive objects themselves declare both inductive or coinductive types and constructors (see Section~\ref{Cic-inductive-definitions}). A {\em global assumption} will be represented in the global environment as $(c:T)$ which assumes the name $c$ to be of some type $T$. A {\em global definition} will be represented in the global environment as $c:=t:T$ which defines the name $c$ to have value $t$ and type $T$. We shall call such names {\em constants}. For the rest of the chapter, the $E;c:T$ denotes the global environment $E$ enriched with the global assumption $c:T$. Similarly, $E;c:=t:T$ denotes the global environment $E$ enriched with the global definition $(c:=t:T)$. The rules for inductive definitions (see Section \ref{Cic-inductive-definitions}) have to be considered as assumption rules to which the following definitions apply: if the name $c$ is declared in $E$, we write $c \in E$ and if $c:T$ or $c:=t:T$ is declared in $E$, we write $(c : T) \in E$. \paragraph[Typing rules.]{Typing rules.\label{Typing-rules}\index{Typing rules}} In the following, we define simultaneously two judgments. The first one \WTEG{t}{T} means the term $t$ is well-typed and has type $T$ in the global environment $E$ and local context $\Gamma$. The second judgment \WFE{\Gamma} means that the global environment $E$ is well-formed and the local context $\Gamma$ is a valid local context in this global environment. % HH: This looks to me complicated. I think it would be better to talk % about ``discharge'' as a transformation of global environments, % rather than as keeping a local context next to global constants. % %% 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 local sub-context of the %% current local context.}. A term $t$ is well typed in a global environment $E$ iff there exists a local context $\Gamma$ and a term $T$ such that the judgment \WTEG{t}{T} can be derived from the following rules. \begin{description} \item[W-Empty] \inference{\WF{[]}{}} \item[W-Local-Assum] % Ce n'est pas vrai : x peut apparaitre plusieurs fois dans Gamma \inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~x \not\in \Gamma % \cup E }{\WFE{\Gamma::(x:T)}}} \item[W-Local-Def] \inference{\frac{\WTEG{t}{T}~~~~x \not\in \Gamma % \cup E }{\WFE{\Gamma::(x:=t:T)}}} \item[W-Global-Assum] \inference{\frac{\WTE{}{T}{s}~~~~s \in \Sort~~~~c \notin E} {\WF{E;c:T}{}}} \item[W-Global-Def] \inference{\frac{\WTE{}{t}{T}~~~c \notin E} {\WF{E;c:=t:T}{}}} \item[Ax-Prop] \index{Typing rules!Ax-Prop} \inference{\frac{\WFE{\Gamma}}{\WTEG{\Prop}{\Type(1)}}} \item[Ax-Set] \index{Typing rules!Ax-Set} \inference{\frac{\WFE{\Gamma}}{\WTEG{\Set}{\Type(1)}}} \item[Ax-Type] \index{Typing rules!Ax-Type} \inference{\frac{\WFE{\Gamma}}{\WTEG{\Type(i)}{\Type(i+1)}}} \item[Var]\index{Typing rules!Var} \inference{\frac{ \WFE{\Gamma}~~~~~(x:T) \in \Gamma~~\mbox{or}~~(x:=t:T) \in \Gamma~\mbox{for some $t$}}{\WTEG{x}{T}}} \item[Const] \index{Typing rules!Const} \inference{\frac{\WFE{\Gamma}~~~~(c:T) \in E~~\mbox{or}~~(c:=t:T) \in E~\mbox{for some $t$} }{\WTEG{c}{T}}} \item[Prod-Prop] \index{Typing rules!Prod-Prop} \inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~ \WTE{\Gamma::(x:T)}{U}{\Prop}} { \WTEG{\forall~x:T,U}{\Prop}}} \item[Prod-Set] \index{Typing rules!Prod-Set} \inference{\frac{\WTEG{T}{s}~~~~s \in\{\Prop, \Set\}~~~~~~ \WTE{\Gamma::(x:T)}{U}{\Set}} { \WTEG{\forall~x:T,U}{\Set}}} \item[Prod-Type] \index{Typing rules!Prod-Type} \inference{\frac{\WTEG{T}{\Type(i)}~~~~ \WTE{\Gamma::(x:T)}{U}{\Type(i)}} {\WTEG{\forall~x:T,U}{\Type(i)}}} \item[Lam]\index{Typing rules!Lam} \inference{\frac{\WTEG{\forall~x:T,U}{s}~~~~ \WTE{\Gamma::(x:T)}{t}{U}} {\WTEG{\lb x:T\mto t}{\forall x:T, U}}} \item[App]\index{Typing rules!App} \inference{\frac{\WTEG{t}{\forall~x:U,T}~~~~\WTEG{u}{U}} {\WTEG{(t\ u)}{\subst{T}{x}{u}}}} \item[Let]\index{Typing rules!Let} \inference{\frac{\WTEG{t}{T}~~~~ \WTE{\Gamma::(x:=t:T)}{u}{U}} {\WTEG{\letin{x}{t:T}{u}}{\subst{U}{x}{t}}}} \end{description} \Rem Prod$_1$ and Prod$_2$ typing-rules make sense if we consider the semantic difference between {\Prop} and {\Set}: \begin{itemize} \item All values of a type that has a sort {\Set} are extractable. \item No values of a type that has a sort {\Prop} are extractable. \end{itemize} \Rem We may have $\kw{let}~x:=t:T~\kw{in}~u$ well-typed without having $((\lb x:T\mto u)~t)$ well-typed (where $T$ is a type of $t$). This is because the value $t$ associated to $x$ may be used in a conversion rule (see Section~\ref{conv-rules}). \section[Conversion rules]{Conversion rules\index{Conversion rules} \label{conv-rules}} In \CIC, there is an internal reduction mechanism. In particular, it can decide if two programs are {\em intentionally} equal (one says {\em convertible}). Convertibility is described in this section. \paragraph[$\beta$-reduction.]{$\beta$-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 $\lb x:T\mto x$. In any global environment $E$ and local context $\Gamma$, we want to identify any object $a$ (of type $T$) with the application $((\lb x:T\mto x)~a)$. We define for this a {\em reduction} (or a {\em conversion}) rule we call $\beta$: \[ \WTEGRED{((\lb x:T\mto t)~u)}{\triangleright_{\beta}}{\subst{t}{x}{u}} \] We say that $\subst{t}{x}{u}$ is the {\em $\beta$-contraction} of $((\lb x:T\mto t)~u)$ and, conversely, that $((\lb x:T\mto t)~u)$ 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.]{$\iota$-reduction.\label{iota}\index{iota-reduction@$\iota$-reduction}} A specific conversion rule is associated to the inductive objects in the global environment. We shall give later on (see 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.]{$\delta$-reduction.\label{delta}\index{delta-reduction@$\delta$-reduction}} We may have variables defined in local contexts or constants defined in the global environment. It is legal to identify such a reference with its value, that is to expand (or unfold) it into its value. This reduction is called $\delta$-reduction and shows as follows. $$\WTEGRED{x}{\triangleright_{\delta}}{t}~~~~~\mbox{if $(x:=t:T) \in \Gamma$}~~~~~~~~~\WTEGRED{c}{\triangleright_{\delta}}{t}~~~~~\mbox{if $(c:=t:T) \in E$}$$ \paragraph[$\zeta$-reduction.]{$\zeta$-reduction.\label{zeta}\index{zeta-reduction@$\zeta$-reduction}} {\Coq} allows also to remove local definitions occurring in terms by replacing the defined variable by its value. The declaration being destroyed, this reduction differs from $\delta$-reduction. It is called $\zeta$-reduction and shows as follows. $$\WTEGRED{\kw{let}~x:=u~\kw{in}~t}{\triangleright_{\zeta}}{\subst{t}{x}{u}}$$ \paragraph{$\eta$-expansion.% \label{eta}% \index{eta-expansion@$\eta$-expansion}% %\index{eta-reduction@$\eta$-reduction} }% Another important concept is $\eta$-expansion. It is legal to identify any term $t$ of functional type $\forall x:T, U$ with its so-called $\eta$-expansion $\lb x:T\mto (t\ x)$ for $x$ an arbitrary variable name fresh in $t$. \Rem We deliberately do not define $\eta$-reduction: \begin{latexonly}% $$\lb x:T\mto (t\ x)\not\triangleright_\eta\hskip.3em t$$ \end{latexonly}% \begin{htmlonly} $$\lb x:T\mto (t\ x)~\not\triangleright_\eta~t$$ \end{htmlonly} This is because, in general, the type of $t$ need not to be convertible to the type of $\lb x:T\mto (t\ x)$. E.g., if we take $f$ such that: \begin{latexonly}% $$f\hskip.5em:\hskip.5em\forall x:Type(2),Type(1)$$ \end{latexonly}% \begin{htmlonly} $$f~:~\forall x:Type(2),Type(1)$$ \end{htmlonly} then \begin{latexonly}% $$\lb x:Type(1),(f\, x)\hskip.5em:\hskip.5em\forall x:Type(1),Type(1)$$ \end{latexonly}% \begin{htmlonly} $$\lb x:Type(1),(f\, x)~:~\forall x:Type(1),Type(1)$$ \end{htmlonly} We could not allow \begin{latexonly}% $$\lb x:Type(1),(f\,x)\hskip.4em\not\triangleright_\eta\hskip.6em f$$ \end{latexonly}% \begin{htmlonly} $$\lb x:Type(1),(f\,x)~\not\triangleright_\eta~f$$ \end{htmlonly} because the type of the reduced term $\forall x:Type(2),Type(1)$ would not be convertible to the type of the original term $\forall x:Type(1),Type(1)$. \paragraph[Convertibility.]{Convertibility.\label{convertibility} \index{beta-reduction@$\beta$-reduction}\index{iota-reduction@$\iota$-reduction}\index{delta-reduction@$\delta$-reduction}\index{zeta-reduction@$\zeta$-reduction}} Let us write $\WTEGRED{t}{\triangleright}{u}$ for the contextual closure of the relation $t$ reduces to $u$ in the global environment $E$ and local context $\Gamma$ with one of the previous reduction $\beta$, $\iota$, $\delta$ or $\zeta$. We say that two terms $t_1$ and $t_2$ are {\em $\beta\iota\delta\zeta\eta$-convertible}, or simply {\em convertible}, or {\em equivalent}, in the global environment $E$ and local context $\Gamma$ iff there exist terms $u_1$ and $u_2$ such that $\WTEGRED{t_1}{\triangleright \ldots \triangleright}{u_1}$ and $\WTEGRED{t_2}{\triangleright \ldots \triangleright}{u_2}$ and either $u_1$ and $u_2$ are identical, or they are convertible up to $\eta$-expansion, i.e. $u_1$ is $\lb x:T\mto u'_1$ and $u_2\,x$ is recursively convertible to $u'_1$, or, symmetrically, $u_2$ is $\lb x:T\mto u'_2$ and $u_1\,x$ is recursively convertible to $u'_2$. We then write $\WTEGCONV{t_1}{t_2}$. The convertibility relation allows introducing a new typing rule which says that two convertible well-formed types have the same inhabitants. \section[Subtyping rules]{Subtyping rules\index{Subtyping rules} \label{subtyping-rules}} 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 is the {\em cumulativity} rule of {\CIC}). This property extends the equivalence relation of convertibility into a {\em subtyping} relation inductively defined by: \begin{enumerate} \item if $\WTEGCONV{t}{u}$ then $\WTEGLECONV{t}{u}$, \item if $i \leq j$ then $\WTEGLECONV{\Type(i)}{\Type(j)}$, \item for any $i$, $\WTEGLECONV{\Set}{\Type(i)}$, \item $\WTEGLECONV{\Prop}{\Set}$, hence, by transitivity, $\WTEGLECONV{\Prop}{\Type(i)}$, for any $i$ \item if $\WTEGCONV{T}{U}$ and $\WTELECONV{\Gamma::(x:T)}{T'}{U'}$ then $\WTEGLECONV{\forall~x:T, T'}{\forall~x:U, U'}$. \end{enumerate} The conversion rule up to subtyping is now exactly: \begin{description}\label{Conv} \item[Conv]\index{Typing rules!Conv} \inference{ \frac{\WTEG{U}{s}~~~~\WTEG{t}{T}~~~~\WTEGLECONV{T}{U}}{\WTEG{t}{U}}} \end{description} \paragraph[Normal form.]{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 rules. 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 $\lb x_1:T_1\mto \ldots \lb x_k:T_k \mto (t_0\ t_1\ldots t_n)$ 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 $\lb x:T\mto u_0$ then one step of $\beta$-head reduction of $t$ is: \[\lb x_1:T_1\mto \ldots \lb x_k:T_k\mto (\lb x:T\mto u_0\ t_1\ldots t_n) ~\triangleright ~ \lb (x_1:T_1)\ldots(x_k:T_k)\mto (\subst{u_0}{x}{t_1}\ t_2 \ldots t_n)\] 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 $t$: \[ t \triangleright \ldots \triangleright \lb x_1:T_1\mto \ldots\lb x_k:T_k\mto (v\ u_1 \ldots u_m)\] 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 $\delta$, $\iota$ and $\zeta$ reductions or any combination of those can also be defined. \section[Inductive definitions]{Inductive Definitions\label{Cic-inductive-definitions}} % Here we assume that the reader knows what is an inductive definition. Formally, we can represent any {\em inductive definition\index{definition!inductive}} as \Ind{}{p}{\Gamma_I}{\Gamma_C} where: \begin{itemize} \item $\Gamma_I$ determines the names and types of inductive types; \item $\Gamma_C$ determines the names and types of constructors of these inductive types; \item $p$ determines the number of parameters of these inductive types. \end{itemize} These inductive definitions, together with global assumptions and global definitions, then form the global environment. % Additionally, for any $p$ there always exists $\Gamma_P=[a_1:A_1;\dots;a_p:A_p]$ such that each $(t:T)\in\Gamma_I\cup\Gamma_C$ can be written as: $\forall\Gamma_P, T^\prime$ where $\Gamma_P$ is called the {\em context of parameters\index{context of parameters}}. \paragraph{Examples} \newcommand\ind[3]{$\mathsf{Ind}~[#1]\left(\hskip-.4em \begin{array}{r @{\mathrm{~:=~}} l} #2 & #3 \\ \end{array} \hskip-.4em \right)$} \def\colon{@{\hskip.5em:\hskip.5em}} The declaration for parameterized lists is: \begin{latexonly} \vskip.5em \ind{1}{[\List:\Set\ra\Set]}{\left[\begin{array}{r \colon l} \Nil & \forall A:\Set,\List~A \\ \cons & \forall A:\Set, A \ra \List~A \ra \List~A \end{array} \right]} \vskip.5em \end{latexonly} \begin{rawhtml}

  

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\end{rawhtml} \noindent which corresponds to the result of the \Coq\ declaration: \begin{coq_example*} Inductive list (A:Set) : Set := | nil : list A | cons : A -> list A -> list A. \end{coq_example*} \noindent The declaration for a mutual inductive definition of {\tree} and {\forest} is: \begin{latexonly} \vskip.5em \ind{~}{\left[\begin{array}{r \colon l}\tree&\Set\\\forest&\Set\end{array}\right]} {\left[\begin{array}{r \colon l} \node & \forest \ra \tree\\ \emptyf & \forest\\ \consf & \tree \ra \forest \ra \forest\\ \end{array}\right]} \vskip.5em \end{latexonly} \begin{rawhtml}

  

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\end{rawhtml} \noindent which corresponds to the result of the \Coq\ declaration: \begin{coq_example*} Inductive tree : Set := node : forest -> tree with forest : Set := | emptyf : forest | consf : tree -> forest -> forest. \end{coq_example*} \noindent The declaration for a mutual inductive definition of {\even} and {\odd} is: \begin{latexonly} \newcommand\GammaI{\left[\begin{array}{r \colon l} \even & \nat\ra\Prop \\ \odd & \nat\ra\Prop \end{array} \right]} \newcommand\GammaC{\left[\begin{array}{r \colon l} \evenO & \even~\nO \\ \evenS & \forall n : \nat, \odd~n \ra \even~(\nS~n)\\ \oddS & \forall n : \nat, \even~n \ra \odd~(\nS~n) \end{array} \right]} \vskip.5em \ind{1}{\GammaI}{\GammaC} \vskip.5em \end{latexonly} \begin{rawhtml}

  

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\end{rawhtml} \noindent which corresponds to the result of the \Coq\ declaration: \begin{coq_example*} Inductive even : nat -> Prop := | even_O : even 0 | even_S : forall n, odd n -> even (S n) with odd : nat -> Prop := | odd_S : forall n, even n -> odd (S n). \end{coq_example*} \subsection{Types of inductive objects} We have to give the type of constants in a global environment $E$ which contains an inductive declaration. \begin{description} \item[Ind] \index{Typing rules!Ind} \inference{\frac{\WFE{\Gamma}~~~~~~~~\Ind{}{p}{\Gamma_I}{\Gamma_C} \in E~~~~~~~~(a:A)\in\Gamma_I}{\WTEG{a}{A}}} \item[Constr] \index{Typing rules!Constr} \inference{\frac{\WFE{\Gamma}~~~~~~~~\Ind{}{p}{\Gamma_I}{\Gamma_C} \in E~~~~~~~~(c:C)\in\Gamma_C}{\WTEG{c}{C}}} \end{description} \begin{latexonly}% \paragraph{Example.} Provided that our environment $E$ contains inductive definitions we showed before, these two inference rules above enable us to conclude that: \vskip.5em \newcommand\prefix{E[\Gamma]\vdash\hskip.25em} $\begin{array}{@{} l} \prefix\even : \nat\ra\Prop\\ \prefix\odd : \nat\ra\Prop\\ \prefix\evenO : \even~\nO\\ \prefix\evenS : \forall~n:\nat, \odd~n \ra \even~(\nS~n)\\ \prefix\oddS : \forall~n:\nat, \even~n \ra \odd~(\nS~n) \end{array}$ \end{latexonly}% %\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 \haslength\, $A$ is a parameter because %what is effectively inductively defined is $\ListA$ or $\haslengthA$ for %a given $A$ which is constant in the type of constructors. But when %we define $(\haslengthA~l~n)$, $l$ 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[Definition]{Definition\index{Arity}\label{Arity}} A type $T$ is an {\em arity of sort $s$} if it converts to the sort $s$ or to a product $\forall~x:T,U$ with $U$ an arity of sort $s$. \paragraph[Examples]{Examples} $A\ra \Set$ is an arity of sort $\Set$. $\forall~A:\Prop,A\ra \Prop$ is an arity of sort \Prop. \paragraph[Definition]{Definition} A type $T$ is an {\em arity} if there is a $s\in\Sort$ such that $T$ is an arity of sort $s$. \paragraph[Examples]{Examples} $A\ra \Set$ and $\forall~A:\Prop,A\ra \Prop$ are arities. \paragraph[Definition]{Definition\index{type of constructor}} We say that $T$ is a {\em type of constructor of $I$\index{type of constructor}} in one of the following two cases: \begin{itemize} \item $T$ is $(I~t_1\ldots ~t_n)$ \item $T$ is $\forall x:U,T^\prime$ where $T^\prime$ is also a type of constructor of $I$ \end{itemize} \paragraph[Examples]{Examples} $\nat$ and $\nat\ra\nat$ are types of constructors of $\nat$.\\ $\forall A:\Type,\List~A$ and $\forall A:\Type,A\ra\List~A\ra\List~A$ are constructors of $\List$. \paragraph[Definition]{Definition\index{Positivity}\label{Positivity}} 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 $T=\forall~x:U,V$ and $X$ occurs only strictly positively in $U$ and the type $V$ 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 $\forall~x:U,V$ 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 $\Ind{\Gamma}{m}{I:A}{c_1:\forall p_1:P_1,\ldots \forall p_m:P_m,C_1;\ldots;c_n:\forall p_1:P_1,\ldots \forall p_m:P_m,C_n}$ (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 (instantiated) types of constructor $C_i\{p_j/a_j\}_{j=1\ldots m}$ of $I$ satisfy the nested 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 of constructor $T$ of $I$ {\em satisfies the nested positivity condition} for a constant $X$ in the following cases: \begin{itemize} \item $T=(I~b_1\ldots b_m~u_1\ldots ~u_{p})$, $I$ is an inductive definition with $m$ parameters and $X$ does not occur in any $u_i$ \item $T=\forall~x:U,V$ and $X$ occurs only strictly positively in $U$ and the type $V$ satisfies the nested positivity condition for $X$ \end{itemize} \newcommand\vv{\textSFxi} % │ \newcommand\hh{\textSFx} % ─ \newcommand\vh{\textSFviii} % ├ \newcommand\hv{\textSFii} % └ \newlength\framecharacterwidth \settowidth\framecharacterwidth{\hh} \newcommand\ws{\hbox{}\hskip\the\framecharacterwidth} \newcommand\ruleref[1]{\hskip.25em\dots\hskip.2em{\em (bullet #1)}} \noindent For instance, if one considers the type \begin{verbatim} Inductive tree (A:Type) : Type := | leaf : list A | node : A -> (nat -> tree A) -> tree A \end{verbatim} \begin{latexonly} \noindent Then every instantiated constructor of $\ListA$ satisfies the nested positivity condition for $\List$\\ \noindent \ws\ws\vv\\ \ws\ws\vh\hh\ws concerning type $\ListA$ of constructor $\Nil$:\\ \ws\ws\vv\ws\ws\ws\ws Type $\ListA$ of constructor $\Nil$ satisfies the positivity condition for $\List$\\ \ws\ws\vv\ws\ws\ws\ws because $\List$ does not appear in any (real) arguments of the type of that constructor\\ \ws\ws\vv\ws\ws\ws\ws (primarily because $\List$ does not have any (real) arguments)\ruleref1\\ \ws\ws\vv\\ \ws\ws\hv\hh\ws concerning type $\forall~A\ra\ListA\ra\ListA$ of constructor $\cons$:\\ \ws\ws\ws\ws\ws\ws\ws Type $\forall~A:\Type,A\ra\ListA\ra\ListA$ of constructor $\cons$\\ \ws\ws\ws\ws\ws\ws\ws satisfies the positivity condition for $\List$ because:\\ \ws\ws\ws\ws\ws\ws\ws\vv\\ \ws\ws\ws\ws\ws\ws\ws\vh\hh\ws $\List$ occurs only strictly positively in $\Type$\ruleref3\\ \ws\ws\ws\ws\ws\ws\ws\vv\\ \ws\ws\ws\ws\ws\ws\ws\vh\hh\ws $\List$ occurs only strictly positively in $A$\ruleref3\\ \ws\ws\ws\ws\ws\ws\ws\vv\\ \ws\ws\ws\ws\ws\ws\ws\vh\hh\ws $\List$ occurs only strictly positively in $\ListA$\ruleref4\\ \ws\ws\ws\ws\ws\ws\ws\vv\\ \ws\ws\ws\ws\ws\ws\ws\hv\hh\ws $\List$ satisfies the positivity condition for $\ListA$\ruleref1 \end{latexonly} \begin{rawhtml} 
Then every instantiated constructor of list A satisfies the nested positivity condition for list
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  ├─ concerning type list A of constructor nil:
  │    Type list A of constructor nil satisfies the positivity condition for list
  │    because list does not appear in any (real) arguments of the type of that constructor
  │    (primarily because list does not have any (real) arguments) ... (bullet 1)
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  ╰─ concerning type ∀ A → list A → list A of constructor cons:
       Type ∀ A : Type, A → list A → list A of constructor cons
       satisfies the positivity condition for list because:
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       ├─ list occurs only strictly positively in Type ... (bullet 3)
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       ├─ list occurs only strictly positively in A ... (bullet 3)
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       ├─ list occurs only strictly positively in list A ... (bullet 4)
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       ╰─ list satisfies the positivity condition for list A ... (bullet 1)
\end{rawhtml} \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 a global environment and $\Gamma_P,\Gamma_I,\Gamma_C$ are contexts such that $\Gamma_I$ is $[I_1:\forall \Gamma_P,A_1;\ldots;I_k:\forall \Gamma_P,A_k]$ and $\Gamma_C$ is $[c_1:\forall \Gamma_P,C_1;\ldots;c_n:\forall \Gamma_P,C_n]$. \inference{ \frac{ (\WTE{\Gamma_P}{A_j}{s'_j})_{j=1\ldots k} ~~~~~~~~ (\WTE{\Gamma_I;\Gamma_P}{C_i}{s_{q_i}})_{i=1\ldots n} } {\WF{E;\Ind{}{p}{\Gamma_I}{\Gamma_C}}{\Gamma}}} provided that the following side conditions hold: \begin{itemize} \item $k>0$ and all of $I_j$ and $c_i$ are distinct names for $j=1\ldots k$ and $i=1\ldots n$, \item $p$ is the number of parameters of \NInd{}{\Gamma_I}{\Gamma_C} and $\Gamma_P$ is the context of parameters, \item for $j=1\ldots k$ we have that $A_j$ is an arity of sort $s_j$ and $I_j \notin E$, \item for $i=1\ldots n$ we have that $C_i$ is a type of constructor of $I_{q_i}$ 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 the impredicative sort {\Prop} but may fail to define inductive definition on sort \Set{} and generate constraints between universes for inductive definitions in the {\Type} hierarchy. \paragraph{Examples.} It is well known that existential quantifier can be encoded as an inductive definition. The following declaration introduces the second-order existential quantifier $\exists X.P(X)$. \begin{coq_example*} Inductive exProp (P:Prop->Prop) : Prop := exP_intro : forall X:Prop, P X -> exProp P. \end{coq_example*} The same definition on \Set{} is not allowed and fails: % (********** The following is not correct and should produce **********) % (*** Error: Large non-propositional inductive types must be in Type***) \begin{coq_example} Fail Inductive exSet (P:Set->Prop) : Set := exS_intro : forall X:Set, P X -> exSet P. \end{coq_example} It is possible to declare the same inductive definition in the universe \Type. The \texttt{exType} inductive definition has type $(\Type_i \ra\Prop)\ra \Type_j$ with the constraint that the parameter \texttt{X} of \texttt{exT\_intro} has type $\Type_k$ with $kProp) : Type := exT_intro : forall X:Type, P X -> exType P. \end{coq_example*} %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. \paragraph[Template polymorphism.]{Template polymorphism.\index{Template polymorphism}} \label{Template-polymorphism} Inductive types declared in {\Type} are polymorphic over their arguments in {\Type}. If $A$ is an arity of some sort and $s$ is a sort, we write $A_{/s}$ for the arity obtained from $A$ by replacing its sort with $s$. Especially, if $A$ is well-typed in some global environment and local context, then $A_{/s}$ is typable by typability of all products in the Calculus of Inductive Constructions. The following typing rule is added to the theory. \begin{description} \item[Ind-Family] Let $\Ind{}{p}{\Gamma_I}{\Gamma_C}$ be an inductive definition. Let $\Gamma_P = [p_1:P_1;\ldots;p_{p}:P_{p}]$ be its context of parameters, $\Gamma_I = [I_1:\forall \Gamma_P,A_1;\ldots;I_k:\forall \Gamma_P,A_k]$ its context of definitions and $\Gamma_C = [c_1:\forall \Gamma_P,C_1;\ldots;c_n:\forall \Gamma_P,C_n]$ its context of constructors, with $c_i$ a constructor of $I_{q_i}$. Let $m \leq p$ be the length of the longest prefix of parameters such that the $m$ first arguments of all occurrences of all $I_j$ in all $C_k$ (even the occurrences in the hypotheses of $C_k$) are exactly applied to $p_1~\ldots~p_m$ ($m$ is the number of {\em recursively uniform parameters} and the $p-m$ remaining parameters are the {\em recursively non-uniform parameters}). Let $q_1$, \ldots, $q_r$, with $0\leq r\leq m$, be a (possibly) partial instantiation of the recursively uniform parameters of $\Gamma_P$. We have: \inference{\frac {\left\{\begin{array}{l} \Ind{}{p}{\Gamma_I}{\Gamma_C} \in E\\ (E[] \vdash q_l : P'_l)_{l=1\ldots r}\\ (\WTELECONV{}{P'_l}{\subst{P_l}{p_u}{q_u}_{u=1\ldots l-1}})_{l=1\ldots r}\\ 1 \leq j \leq k \end{array} \right.} {E[] \vdash I_j\,q_1\,\ldots\,q_r:\forall [p_{r+1}:P_{r+1};\ldots;p_{p}:P_{p}], (A_j)_{/s_j}} } provided that the following side conditions hold: \begin{itemize} \item $\Gamma_{P'}$ is the context obtained from $\Gamma_P$ by replacing each $P_l$ that is an arity with $P'_l$ for $1\leq l \leq r$ (notice that $P_l$ arity implies $P'_l$ arity since $\WTELECONV{}{P'_l}{ \subst{P_l}{p_u}{q_u}_{u=1\ldots l-1}}$); \item there are sorts $s_i$, for $1 \leq i \leq k$ such that, for $\Gamma_{I'} = [I_1:\forall \Gamma_{P'},(A_1)_{/s_1};\ldots;I_k:\forall \Gamma_{P'},(A_k)_{/s_k}]$ we have $(\WTE{\Gamma_{I'};\Gamma_{P'}}{C_i}{s_{q_i}})_{i=1\ldots n}$; \item the sorts $s_i$ are such that all eliminations, to {\Prop}, {\Set} and $\Type(j)$, are allowed (see Section~\ref{allowedeleminationofsorts}). \end{itemize} \end{description} % Notice that if $I_j\,q_1\,\ldots\,q_r$ is typable using the rules {\bf Ind-Const} and {\bf App}, then it is typable using the rule {\bf Ind-Family}. Conversely, the extended theory is not stronger than the theory without {\bf Ind-Family}. We get an equiconsistency result by mapping each $\Ind{}{p}{\Gamma_I}{\Gamma_C}$ occurring into a given derivation into as many different inductive types and constructors as the number of different (partial) replacements of sorts, needed for this derivation, in the parameters that are arities (this is possible because $\Ind{}{p}{\Gamma_I}{\Gamma_C}$ well-formed implies that $\Ind{}{p}{\Gamma_{I'}}{\Gamma_{C'}}$ is well-formed and has the same allowed eliminations, where $\Gamma_{I'}$ is defined as above and $\Gamma_{C'} = [c_1:\forall \Gamma_{P'},C_1;\ldots;c_n:\forall \Gamma_{P'},C_n]$). That is, the changes in the types of each partial instance $q_1\,\ldots\,q_r$ can be characterized by the ordered sets of arity sorts among the types of parameters, and to each signature is associated a new inductive definition with fresh names. Conversion is preserved as any (partial) instance $I_j\,q_1\,\ldots\,q_r$ or $C_i\,q_1\,\ldots\,q_r$ is mapped to the names chosen in the specific instance of $\Ind{}{p}{\Gamma_I}{\Gamma_C}$. \newcommand{\Single}{\mbox{\textsf{Set}}} In practice, the rule {\bf Ind-Family} is used by {\Coq} only when all the inductive types of the inductive definition are declared with an arity whose sort is in the $\Type$ hierarchy. Then, the polymorphism is over the parameters whose type is an arity of sort in the {\Type} hierarchy. The sort $s_j$ are chosen canonically so that each $s_j$ is minimal with respect to the hierarchy ${\Prop}\subset{\Set_p}\subset\Type$ where $\Set_p$ is predicative {\Set}. %and ${\Prop_u}$ is the sort of small singleton %inductive types (i.e. of inductive types with one single constructor %and that contains either proofs or inhabitants of singleton types %only). More precisely, an empty or small singleton inductive definition (i.e. an inductive definition of which all inductive types are singleton -- see paragraph~\ref{singleton}) is set in {\Prop}, a small non-singleton inductive type is set in {\Set} (even in case {\Set} is impredicative -- see Section~\ref{impredicativity}), and otherwise in the {\Type} hierarchy. Note that the side-condition about allowed elimination sorts in the rule~{\bf Ind-Family} is just to avoid to recompute the allowed elimination sorts at each instance of a pattern-matching (see section~\ref{elimdep}). As an example, let us consider the following definition: \begin{coq_example*} Inductive option (A:Type) : Type := | None : option A | Some : A -> option A. \end{coq_example*} % As the definition is set in the {\Type} hierarchy, it is used polymorphically over its parameters whose types are arities of a sort in the {\Type} hierarchy. Here, the parameter $A$ has this property, hence, if \texttt{option} is applied to a type in {\Set}, the result is in {\Set}. Note that if \texttt{option} is applied to a type in {\Prop}, then, the result is not set in \texttt{Prop} but in \texttt{Set} still. This is because \texttt{option} is not a singleton type (see section~\ref{singleton}) and it would lose the elimination to {\Set} and {\Type} if set in {\Prop}. \begin{coq_example} Check (fun A:Set => option A). Check (fun A:Prop => option A). \end{coq_example} % Here is another example. % \begin{coq_example*} Inductive prod (A B:Type) : Type := pair : A -> B -> prod A B. \end{coq_example*} % As \texttt{prod} is a singleton type, it will be in {\Prop} if applied twice to propositions, in {\Set} if applied twice to at least one type in {\Set} and none in {\Type}, and in {\Type} otherwise. In all cases, the three kind of eliminations schemes are allowed. \begin{coq_example} Check (fun A:Set => prod A). Check (fun A:Prop => prod A A). Check (fun (A:Prop) (B:Set) => prod A B). Check (fun (A:Type) (B:Prop) => prod A B). \end{coq_example} \Rem Template polymorphism used to be called ``sort-polymorphism of inductive types'' before universe polymorphism (see Chapter~\ref{Universes-full}) was introduced. \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 normalizing 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 local context, we want to build a function \length\ of type $\ListA\ra \nat$ which computes the length of the list, so such that $(\length~(\Nil~A)) = \nO$ 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 about an object $m$ 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 $\forall l:\ListA,(\haslengthA~l~(\length~l))$ it is enough to prove: % \begin{itemize} \item $(\haslengthA~(\Nil~A)~(\length~(\Nil~A)))$ \item $\forall a:A, \forall l:\ListA, (\haslengthA~l~(\length~l)) \ra\\ \ra (\haslengthA~(\cons~A~a~l)~(\length~(\cons~A~a~l)))$ \end{itemize} % which given the conversion equalities satisfied by \length\ is the same as proving: % \begin{itemize} \item $(\haslengthA~(\Nil~A)~\nO)$ \item $\forall a:A, \forall l:\ListA, (\haslengthA~l~(\length~l)) \ra\\ \ra (\haslengthA~(\cons~A~a~l)~(\nS~(\length~l)))$ \end{itemize} % 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 pattern-matching. The second one is a definition by guarded fixpoints. \subsubsection[The {\tt match\ldots with \ldots end} construction.]{The {\tt match\ldots with \ldots end} construction.\label{Caseexpr} \index{match@{\tt match\ldots with\ldots end}}} The basic idea of this operator is that we have an object $m$ in an inductive type $I$ and we want to prove a property which possibly 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 $I$. The \Coq{} term for this proof will be written: \[\kw{match}~m~\kw{with}~ (c_1~x_{11}~...~x_{1p_1}) \Ra f_1 ~|~\ldots~|~ (c_n~x_{n1}~...~x_{np_n}) \Ra f_n~ \kw{end}\] In this expression, if $m$ eventually happens to evaluate to $(c_i~u_1\ldots u_{p_i})$ then the expression will behave as specified in its $i$-th branch and it will reduce to $f_i$ where the $x_{i1}$\ldots $x_{ip_i}$ are replaced by the $u_1\ldots u_{p_i}$ according to the $\iota$-reduction. Actually, for type-checking a \kw{match\ldots with\ldots end} expression we also need to know the predicate $P$ to be proved by case analysis. In the general case where $I$ is an inductively defined $n$-ary relation, $P$ is a predicate over $n+1$ arguments: the $n$ first ones correspond to the arguments of $I$ (parameters excluded), and the last one corresponds to object $m$. \Coq{} can sometimes infer this predicate but sometimes not. The concrete syntax for describing this predicate uses the \kw{as\ldots in\ldots return} construction. For instance, let us assume that $I$ is an unary predicate with one parameter and one argument. The predicate is made explicit using the syntax: \[\kw{match}~m~\kw{as}~ x~ \kw{in}~ I~\verb!_!~a~ \kw{return}~ P ~\kw{with}~ (c_1~x_{11}~...~x_{1p_1}) \Ra f_1 ~|~\ldots~|~ (c_n~x_{n1}~...~x_{np_n}) \Ra f_n \kw{end}\] The \kw{as} part can be omitted if either the result type does not depend on $m$ (non-dependent elimination) or $m$ is a variable (in this case, $m$ can occur in $P$ where it is considered a bound variable). The \kw{in} part can be omitted if the result type does not depend on the arguments of $I$. Note that the arguments of $I$ corresponding to parameters \emph{must} be \verb!_!, because the result type is not generalized to all possible values of the parameters. The other arguments of $I$ (sometimes called indices in the literature) % NOTE: e.g. http://www.qatar.cmu.edu/~sacchini/papers/types08.pdf have to be variables ($a$ above) and these variables can occur in $P$. The expression after \kw{in} must be seen as an \emph{inductive type pattern}. Notice that expansion of implicit arguments and notations apply to this pattern. % For the purpose of presenting the inference rules, we use a more compact notation: \[ \Case{(\lb a x \mto P)}{m}{ \lb x_{11}~...~x_{1p_1} \mto f_1 ~|~\ldots~|~ \lb x_{n1}...x_{np_n} \mto f_n}\] %% CP 06/06 Obsolete avec la nouvelle syntaxe et incompatible avec la %% presentation theorique qui suit % \paragraph{Non-dependent elimination.} % % When defining a function of codomain $C$ by case analysis over an % object in an inductive type $I$, we build an object of type $I % \ra C$. The minimality principle on an inductively defined logical % predicate $I$ of type $A \ra \Prop$ is often used to prove a property % $\forall x:A,(I~x)\ra (C~x)$. These are particular cases 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: % \[\kw{fun} l:\ListA \Ra \kw{match}~l~\kw{with}~ \Nil \Ra \true~ % |~(\cons~a~m) \Ra \false \kw{end}\] % represented by % \[\lb l:\ListA \mto\Case{\bool}{l}{\true~ |~ \lb a~m,~\false}\] %\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.]{Allowed elimination sorts.\index{Elimination sorts}} \label{allowedeleminationofsorts} An important question for building the typing rule for \kw{match} is what can be the type of $\lb a x \mto P$ with respect to the type of $m$. If $m:I$ and $I:A$ and $\lb a x \mto P : B$ then by \compat{I:A}{B} we mean that one can use $\lb a x \mto P$ with $m$ in the above match-construct. \paragraph{Notations.} The \compat{I:A}{B} is defined as the smallest relation satisfying the following rules: We write \compat{I}{B} for \compat{I:A}{B} where $A$ is the type of $I$. The case of inductive definitions in sorts \Set\ or \Type{} is simple. There is no restriction on the sort of the predicate to be eliminated. % \begin{description} \item[Prod] \inference{\frac{\compat{(I~x):A'}{B'}} {\compat{I:\forall x:A, A'}{\forall x:A, B'}}} \item[{\Set} \& \Type] \inference{\frac{ s_1 \in \{\Set,\Type(j)\}~~~~~~~~s_2 \in \Sort}{\compat{I:s_1}{I\ra s_2}}} \end{description} % The case of Inductive definitions of sort \Prop{} is a bit more complicated, because of our interpretation of this sort. The only harmless allowed elimination, is the one when predicate $P$ is also of sort \Prop. \begin{description} \item[\Prop] \inference{\compat{I:\Prop}{I\ra\Prop}} \end{description} \Prop{} is the type of logical propositions, the proofs of properties $P$ in \Prop{} could not be used for computation and are consequently ignored by the extraction mechanism. Assume $A$ and $B$ are two propositions, and the logical disjunction $A\vee B$ is defined inductively by: \begin{coq_example*} Inductive or (A B:Prop) : Prop := or_introl : A -> or A B | or_intror : B -> or A B. \end{coq_example*} The following definition which computes a boolean value by case over the proof of \texttt{or A B} is not accepted: % (***************************************************************) % (*** This example should fail with ``Incorrect elimination'' ***) \begin{coq_example} Fail Definition choice (A B: Prop) (x:or A B) := match x with or_introl _ _ a => true | or_intror _ _ b => false end. \end{coq_example} From the computational point of view, the structure of the proof of \texttt{(or A B)} in this term is needed for computing the boolean value. In general, if $I$ has type \Prop\ then $P$ cannot have type $I\ra \Set$, 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 % $\forall~x:T,C$ 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}. In the same spirit, elimination on $P$ of type $I\ra \Type$ cannot be allowed because it trivially implies the elimination on $P$ of type $I\ra \Set$ by cumulativity. It also implies that there are two proofs of the same property which are provably different, contradicting the proof-irrelevance property which is sometimes a useful axiom: \begin{coq_example} Axiom proof_irrelevance : forall (P : Prop) (x y : P), x=y. \end{coq_example} \begin{coq_eval} Reset proof_irrelevance. \end{coq_eval} The elimination of an inductive definition of type \Prop\ on a predicate $P$ of type $I\ra \Type$ leads to a paradox when applied to impredicative inductive definition like the second-order existential quantifier \texttt{exProp} defined above, because it give access to the two projections on this type. %\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[Empty and singleton elimination]{Empty and singleton elimination\label{singleton} \index{Elimination!Singleton elimination} \index{Elimination!Empty elimination}} There are special inductive definitions in \Prop\ for which more eliminations are allowed. \begin{description} \item[\Prop-extended] \inference{ \frac{I \mbox{~is an empty or singleton definition}~~~s \in \Sort}{\compat{I:\Prop}{I\ra s}} } \end{description} % % 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 arguments of this constructor have type \Prop. In that case, there is a canonical way to interpret the informative extraction on an object in that type, such that the elimination on any sort $s$ is legal. Typical examples are the conjunction of non-informative propositions and the equality. If there is an hypothesis $h:a=b$ in the local context, it can be used for rewriting not only in logical propositions but also in any type. % 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} An empty definition has no constructors, in that case also, elimination on any sort is allowed. \paragraph{Type of branches.} Let $c$ be a term of type $C$, we assume $C$ is a type of constructor for an inductive type $I$. Let $P$ be a term that represents the property to be proved. We assume $r$ is the number of parameters and $p$ is the number of arguments. We define a new type \CI{c:C}{P} which represents the type of the branch corresponding to the $c:C$ constructor. \[ \begin{array}{ll} \CI{c:(I~p_1\ldots p_r\ t_1 \ldots t_p)}{P} &\equiv (P~t_1\ldots ~t_p~c) \\[2mm] \CI{c:\forall~x:T,C}{P} &\equiv \forall~x:T,\CI{(c~x):C}{P} \end{array} \] We write \CI{c}{P} for \CI{c:C}{P} with $C$ the type of $c$. \paragraph{Example.} The following term in concrete syntax: \begin{verbatim} match t as l return P' with | nil _ => t1 | cons _ hd tl => t2 end \end{verbatim} can be represented in abstract syntax as $$\Case{P}{t}{f_1\,|\,f_2}$$ where \begin{eqnarray*} P & = & \lambda~l~.~P^\prime\\ f_1 & = & t_1\\ f_2 & = & \lambda~(hd:\nat)~.~\lambda~(tl:\List~\nat)~.~t_2 \end{eqnarray*} According to the definition: \begin{latexonly}\vskip.5em\noindent\end{latexonly}% \begin{htmlonly} \end{htmlonly} $ \CI{(\Nil~\nat)}{P} \equiv \CI{(\Nil~\nat) : (\List~\nat)}{P} \equiv (P~(\Nil~\nat))$ \begin{latexonly}\vskip.5em\noindent\end{latexonly}% \begin{htmlonly} \end{htmlonly} $ \CI{(\cons~\nat)}{P} \equiv\CI{(\cons~\nat) : (\nat\ra\List~\nat\ra\List~\nat)}{P} \equiv\\ \equiv\forall n:\nat, \CI{(\cons~\nat~n) : \List~\nat\ra\List~\nat)}{P} \equiv\\ \equiv\forall n:\nat, \forall l:\List~\nat, \CI{(\cons~\nat~n~l) : \List~\nat)}{P} \equiv\\ \equiv\forall n:\nat, \forall l:\List~\nat,(P~(\cons~\nat~n~l))$. \begin{latexonly}\vskip.5em\noindent\end{latexonly}% \begin{htmlonly} \end{htmlonly} Given some $P$, then \CI{(\Nil~\nat)}{P} represents the expected type of $f_1$, and \CI{(\cons~\nat)}{P} represents the expected type of $f_2$. \paragraph{Typing rule.} Our very general destructor for inductive definition enjoys the following typing rule % , where we write % \[ % \Case{P}{c}{[x_{11}:T_{11}]\ldots[x_{1p_1}:T_{1p_1}]g_1\ldots % [x_{n1}:T_{n1}]\ldots[x_{np_n}:T_{np_n}]g_n} % \] % for % \[ % \Case{P}{c}{(c_1~x_{11}~...~x_{1p_1}) \Ra g_1 ~|~\ldots~|~ % (c_n~x_{n1}...x_{np_n}) \Ra g_n } % \] \begin{description} \item[match] \label{elimdep} \index{Typing rules!match} \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 definition \Ind{}{r}{\Gamma_I}{\Gamma_C} with $\Gamma_C = [c_1:C_1;\ldots;c_n:C_n]$ and $c_{p_1}\ldots c_{p_l}$ are the only constructors of $I$. \end{description} \paragraph{Example.} Below is a typing rule for the term shown in the previous example: \inference{ \frac{% \WTEG{t}{(\List~\nat)}~~~~% \WTEG{P}{B}~~~~% \compat{(\List~\nat)}{B}~~~~% \WTEG{f_1}{\CI{(\Nil~\nat)}{P}}~~~~% \WTEG{f_2}{\CI{(\cons~\nat)}{P}}% } {\WTEG{\Case{P}{t}{f_1|f_2}}{(P~t)}}} \paragraph[Definition of $\iota$-reduction.]{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]{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 concrete syntax for a recursive set of mutually recursive declarations is (with $\Gamma_i$ contexts): \[\kw{fix}~f_1 (\Gamma_1) :A_1:=t_1~\kw{with} \ldots \kw{with}~ f_n (\Gamma_n) :A_n:=t_n\] The terms are obtained by projections from this set of declarations and are written \[\kw{fix}~f_1 (\Gamma_1) :A_1:=t_1~\kw{with} \ldots \kw{with}~ f_n (\Gamma_n) :A_n:=t_n~\kw{for}~f_i\] In the inference rules, we represent such a term by \[\Fix{f_i}{f_1:A_1':=t_1' \ldots f_n:A_n':=t_n'}\] with $t_i'$ (resp. $A_i'$) representing the term $t_i$ abstracted (resp. generalized) with respect to the bindings in the context $\Gamma_i$, namely $t_i'=\lb \Gamma_i \mto t_i$ and $A_i'=\forall \Gamma_i, A_i$. \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 allow proofs of absurdity. % The basic scheme of recursion that should be allowed is the one needed for defining primitive recursive functionals. In that case the fixpoint enjoys a special syntactic restriction, namely one of the arguments belongs to an inductive type, the function starts with a case analysis and recursive calls are done on variables coming from patterns and representing subterms. % For instance in the case of natural numbers, a proof of the induction principle of type \[\forall P:\nat\ra\Prop, (P~\nO)\ra(\forall n:\nat, (P~n)\ra(P~(\nS~n)))\ra \forall n:\nat, (P~n)\] can be represented by the term: \[\begin{array}{l} \lb P:\nat\ra\Prop\mto\lb f:(P~\nO)\mto \lb g:(\forall n:\nat, (P~n)\ra(P~(\nS~n))) \mto\\ \Fix{h}{h:\forall n:\nat, (P~n):=\lb n:\nat\mto \Case{P}{n}{f~|~\lb p:\nat\mto (g~p~(h~p))}} \end{array} \] % 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 $k_i$ represents the index of pararameter of $f_i$, on which $f_i$ is decreasing. Each $A_i$ should be a type (reducible to a term) starting with at least $k_i$ products $\forall y_1:B_1,\ldots \forall y_{k_i}:B_{k_i}, A'_i$ and $B_{k_i}$ an is unductive type. 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{}{r}{\Gamma_I}{\Gamma_C}, if the type of a constructor $c$ has the form $\forall p_1:P_1,\ldots \forall p_r:P_r, \forall x_1:T_1, \ldots \forall x_r:T_r, (I_j~p_1\ldots p_r~t_1\ldots t_s)$, then 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{}{r}{\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 $(t~u)$ and $\lb x:u \mto t$ 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 $C_q \equiv \forall p_1:P_1,\ldots,\forall p_r:P_r, \forall y_1:B_1, \ldots \forall y_k:B_k, (I~a_1\ldots a_k)$ and can consequently be written $\lb y_1:B'_1\mto \ldots \lb y_k:B'_k\mto g_i$. ($B'_i$ 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 m:nat) {struct n} : nat := match n with | O => m | S p => S (plus p m) end. Print plus. Fixpoint lgth (A:Set) (l:list A) {struct l} : nat := match l with | nil _ => O | cons _ a l' => S (lgth A l') end. Print lgth. Fixpoint sizet (t:tree) : nat := let (f) := t in S (sizef f) with sizef (f:forest) : nat := match f with | emptyf => O | consf t f => plus (sizet t) (sizef f) end. Print sizet. \end{coq_example} \subsubsection[Reduction rule]{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: \[ (\Fix{f_i}{F}~a_1\ldots a_{k_i}) \triangleright_{\iota} \substs{t_i}{f_k}{\Fix{f_k}{F}}{k=1\ldots n} ~a_1\ldots a_{k_i}\] 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. % The following reductions are now possible: \def\plus{\mathsf{plus}} \def\tri{\triangleright_\iota} \begin{eqnarray*} \plus~(\nS~(\nS~\nO))~(\nS~\nO) & \tri & \nS~(\plus~(\nS~\nO)~(\nS~\nO))\\ & \tri & \nS~(\nS~(\plus~\nO~(\nS~\nO)))\\ & \tri & \nS~(\nS~(\nS~\nO))\\ \end{eqnarray*} % La disparition de Program devrait rendre la construction Match obsolete % \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} \subsubsection{Mutual induction} The principles of mutual induction can be automatically generated using the {\tt Scheme} command described in Section~\ref{Scheme}. \section{Admissible rules for global environments} From the original rules of the type system, one can show the admissibility of rules which change the local context of definition of objects in the global environment. We show here the admissible rules that are used used in the discharge mechanism at the end of a section. % This is obsolete: Abstraction over defined constants actually uses a % let-in since there are let-ins in Coq %% \paragraph{Mechanism of substitution.} %% One rule which can be proved valid, is to replace a term $c$ by its %% value in the global 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 local contexts and global %% environments. %% \paragraph{Substitution Property:} %% \inference{\frac{\WF{E;c:=t:T; E'}{\Gamma}} %% {\WF{E; \subst{E'}{c}{t}}{\subst{\Gamma}{c}{t}}}} \paragraph{Abstraction.} One can modify a global declaration by generalizing it over a previously assumed constant $c$. For doing that, we need to modify the reference to the global declaration in the subsequent global environment and local context by explicitly applying this constant to the constant $c'$. Below, if $\Gamma$ is a context of the form $[y_1:A_1;\ldots;y_n:A_n]$, we write $\forall x:U,\subst{\Gamma}{c}{x}$ to mean $[y_1:\forall~x:U,\subst{A_1}{c}{x};\ldots;y_n:\forall~x:U,\subst{A_n}{c}{x}]$ and $\subst{E}{|\Gamma|}{|\Gamma|c}$. to mean the parallel substitution $\subst{\subst{E}{y_1}{(y_1~c)}\ldots}{y_n}{(y_n~c)}$. \paragraph{First abstracting property:} \inference{\frac{\WF{E;c:U;E';c':=t:T;E''}{\Gamma}} {\WF{E;c:U;E';c':=\lb x:U\mto \subst{t}{c}{x}:\forall~x:U,\subst{T}{c}{x}; \subst{E''}{c'}{(c'~c)}}{\subst{\Gamma}{c}{(c~c')}}}} \inference{\frac{\WF{E;c:U;E';c':T;E''}{\Gamma}} {\WF{E;c:U;E';c':\forall~x:U,\subst{T}{c}{x}; \subst{E''}{c'}{(c'~c)}}{\subst{\Gamma}{c}{(c~c')}}}} \inference{\frac{\WF{E;c:U;E';\Ind{}{p}{\Gamma_I}{\Gamma_C};E''}{\Gamma}} {\WFTWOLINES{E;c:U;E';\Ind{}{p+1}{\forall x:U,\subst{\Gamma_I}{c}{x}}{\forall x:U,\subst{\Gamma_C}{c}{x}};\subst{E''}{|\Gamma_I,\Gamma_C|}{|\Gamma_I,\Gamma_C|~c}}{\subst{\Gamma}{|\Gamma_I,\Gamma_C|}{|\Gamma_I,\Gamma_C|~c}}}} % One can similarly modify a global declaration by generalizing it over a previously defined constant~$c'$. Below, if $\Gamma$ is a context of the form $[y_1:A_1;\ldots;y_n:A_n]$, we write $ \subst{\Gamma}{c}{u}$ to mean $[y_1:\subst{A_1}{c}{u};\ldots;y_n:\subst{A_n}{c}{u}]$. \paragraph{Second abstracting property:} \inference{\frac{\WF{E;c:=u:U;E';c':=t:T;E''}{\Gamma}} {\WF{E;c:=u:U;E';c':=(\letin{x}{u:U}{\subst{t}{c}{x}}):\subst{T}{c}{u};E''}{\Gamma}}} \inference{\frac{\WF{E;c:=u:U;E';c':T;E''}{\Gamma}} {\WF{E;c:=u:U;E';c':\subst{T}{c}{u};E''}{\Gamma}}} \inference{\frac{\WF{E;c:=u:U;E';\Ind{}{p}{\Gamma_I}{\Gamma_C};E''}{\Gamma}} {\WF{E;c:=u:U;E';\Ind{}{p}{\subst{\Gamma_I}{c}{u}}{\subst{\Gamma_C}{c}{u}};E''}{\Gamma}}} \paragraph{Pruning the local context.} If one abstracts or substitutes constants with the above rules then it may happen that some declared or defined constant does not occur any more in the subsequent global environment and in the local context. One can consequently derive the following property. \paragraph{First pruning property:} \inference{\frac{\WF{E;c:U;E'}{\Gamma} \qquad c \mbox{ does not occur in $E'$ and $\Gamma$}} {\WF{E;E'}{\Gamma}}} \paragraph{Second pruning property:} \inference{\frac{\WF{E;c:=u:U;E'}{\Gamma} \qquad c \mbox{ does not occur in $E'$ and $\Gamma$}} {\WF{E;E'}{\Gamma}}} \section{Co-inductive types} The implementation contains also co-inductive definitions, which are types inhabited by infinite objects. More information on co-inductive definitions can be found in~\cite{Gimenez95b,Gim98,GimCas05}. %They are described in Chapter~\ref{Co-inductives}. \section[The Calculus of Inductive Construction with impredicative \Set]{The Calculus of Inductive Construction with impredicative \Set\label{impredicativity}} \Coq{} can be used as a type-checker for the Calculus of Inductive Constructions with an impredicative sort \Set{} by using the compiler option \texttt{-impredicative-set}. % For example, using the ordinary \texttt{coqtop} command, the following is rejected. % (** This example should fail ******************************* % Error: The term forall X:Set, X -> X has type Type % while it is expected to have type Set ***) \begin{coq_example} Fail Definition id: Set := forall X:Set,X->X. \end{coq_example} while it will type-check, if one uses instead the \texttt{coqtop -impredicative-set} command. The major change in the theory concerns the rule for product formation in the sort \Set, which is extended to a domain in any sort: \begin{description} \item [Prod] \index{Typing rules!Prod (impredicative Set)} \inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~~~ \WTE{\Gamma::(x:T)}{U}{\Set}} { \WTEG{\forall~x:T,U}{\Set}}} \end{description} This extension has consequences on the inductive definitions which are allowed. In the impredicative system, one can build so-called {\em large inductive definitions} like the example of second-order existential quantifier (\texttt{exSet}). There should be restrictions on the eliminations which can be performed on such definitions. The eliminations rules in the impredicative system for sort \Set{} become: \begin{description} \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}}} \end{description} %%% Local Variables: %%% mode: latex %%% TeX-master: "Reference-Manual" %%% End: