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| -rw-r--r-- | CHANGES | 6 | ||||
| -rw-r--r-- | doc/refman/RefMan-cic.tex | 24 | ||||
| -rw-r--r-- | doc/refman/Universes.tex | 46 |
3 files changed, 75 insertions, 1 deletions
@@ -94,6 +94,12 @@ Build Infrastructure access to the same .cmi files. In short, use "make -j && make -j byte" instead of "make -j world byte". +Universes + +- Cumulative inductive types. see prefixes "Cumulative", "NonCumulative" + for inductive definitions and the option "Set Inductive Cumulativity" + in the reference manual. + Changes from V8.6beta1 to V8.6 ============================== diff --git a/doc/refman/RefMan-cic.tex b/doc/refman/RefMan-cic.tex index fdd272581..96fb1eb75 100644 --- a/doc/refman/RefMan-cic.tex +++ b/doc/refman/RefMan-cic.tex @@ -461,6 +461,13 @@ 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 $\WTEGCONV{t_1}{t_2}$. +Apart from this we consider two instances of polymorphic and cumulative (see Chapter~\ref{Universes-full}) inductive types (see below) +convertible $\WTEGCONV{t\ w_1 \dots w_m}{t\ w_1' \dots w_m'}$ if we have subtypings (see below) in both directions, i.e., +$\WTEGLECONV{t\ w_1 \dots w_m}{t\ w_1' \dots w_m'}$ and $\WTEGLECONV{t\ w_1' \dots w_m'}{t\ w_1 \dots w_m}$. +Furthermore, we consider $\WTEGCONV{c\ v_1 \dots v_m}{c'\ v_1' \dots v_m'}$ convertible if $\WTEGCONV{v_i}{v_i'}$ +and we have that $c$ and $c'$ are the same constructors of different instances the same inductive types (differing only in universe levels) +such that $\WTEG{c\ v_1 \dots v_m}{t\ w_1 \dots w_m}$ and $\WTEG{c'\ v_1' \dots v_m'}{t'\ w_1' \dots w_m'}$ and we have $\WTEGCONV{t\ w_1 \dots w_m}{t\ w_1' \dots w_m'}$. + The convertibility relation allows introducing a new typing rule which says that two convertible well-formed types have the same inhabitants. @@ -480,6 +487,17 @@ convertibility into a {\em subtyping} relation inductively defined by: \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 $\WTEGLECONV{\forall~x:T, T'}{\forall~x:U, U'}$. +\item if $\Ind{}{p}{\Gamma_I}{\Gamma_C}$ is a universe polymorphic and cumulative (see Chapter~\ref{Universes-full}) + inductive type (see below) and $(t : \forall\Gamma_P,\forall\Gamma_{\mathit{Arr}(t)}, \Sort)\in\Gamma_I$ + and $(t' : \forall\Gamma_P',\forall\Gamma_{\mathit{Arr}(t)}', \Sort')\in\Gamma_I$ + are two different instances of \emph{the same} inductive type (differing only in universe levels) with constructors + \[[c_1: \forall\Gamma_P,\forall T_{1,1} \dots T_{1,n_1},t\ v_{1,1} \dots v_{1,m}; \dots; c_k: \forall\Gamma_P,\forall T_{k, 1} \dots T_{k,n_k},t\ v_{n,1}\dots v_{n,m}]\] + and + \[[c_1: \forall\Gamma_P',\forall T_{1,1}' \dots T_{1,n_1}',t'\ v_{1,1}' \dots v_{1,m}'; \dots; c_k: \forall\Gamma_P',\forall T_{k, 1}' \dots T_{k,n_k}',t\ v_{n,1}'\dots v_{n,m}']\] + respectively then $\WTEGLECONV{t\ w_1 \dots w_m}{t\ w_1' \dots w_m'}$ (notice that $t$ and $t'$ are both fully applied, i.e., they have a sort as a type) + if $\WTEGCONV{w_i}{w_i'}$ for $1 \le i \le m$ and we have + \[ \WTEGLECONV{T_{i,j}}{T_{i,j}'} \text{ and } \WTEGLECONV{A_i}{A_i'}\] + where $\Gamma_{\mathit{Arr}(t)} = [a_1 : A_1; a_1 : A_l]$ and $\Gamma_{\mathit{Arr}(t)} = [a_1 : A_1'; a_1 : A_l']$. \end{enumerate} The conversion rule up to subtyping is now exactly: @@ -530,8 +548,12 @@ Formally, we can represent any {\em inductive definition\index{definition!induct 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: +such that each $T$ in $(t:T)\in\Gamma_I\cup\Gamma_C$ can be written as: $\forall\Gamma_P, T^\prime$ where $\Gamma_P$ is called the {\em context of parameters\index{context of parameters}}. +Furthermore, we must have that each $T$ in $(t:T)\in\Gamma_I$ can be written as: +$\forall\Gamma_P,\forall\Gamma_{\mathit{Arr}(t)}, \Sort$ where $\Gamma_{\mathit{Arr}(t)}$ is called the +{\em Arity} of the inductive type\index{arity of inductive type} $t$ and +$\Sort$ is called the sort of the inductive type $t$. \paragraph{Examples} diff --git a/doc/refman/Universes.tex b/doc/refman/Universes.tex index 36518e6fa..2bb1301c7 100644 --- a/doc/refman/Universes.tex +++ b/doc/refman/Universes.tex @@ -131,6 +131,52 @@ producing global universe constraints, one can use the polymorphically, not at a single instance. \end{itemize} +\asection{{\tt Cumulative, NonCumulative}} +\comindex{Cumulative} +\comindex{NonCumulative} +\optindex{Inductive Cumulativity} + +Inductive types, coinductive types, variants and records can be +declared cumulative using the \texttt{Cumulative}. Alternatively, +there is an option \texttt{Set Inductive Cumulativity} which when set, +makes all subsequent inductive definitions cumulative. Consider the examples below. +\begin{coq_example*} +Polymorphic Cumulative Inductive list {A : Type} := +| nil : list +| cons : A -> list -> list. +\end{coq_example*} +\begin{coq_example} +Print list. +\end{coq_example} +When printing \texttt{list}, the part of the output of the form +\texttt{$\mathtt{\sim}$@\{i\} <= $\mathtt{\sim}$@\{j\} iff } +indicates the universe constraints in order to have the subtyping +$\WTEGLECONV{\mathtt{list@\{i\}} A}{\mathtt{list@\{j\}} B}$ +(for fully applied instances of \texttt{list}) whenever $\WTEGCONV{A}{B}$. +In the case of \texttt{list} there is no constraint! +This also means that any two instances of \texttt{list} are convertible: +$\WTEGCONV{\mathtt{list@\{i\}} A}{\mathtt{list@\{j\}} B}$ whenever $\WTEGCONV{A}{B}$ and +furthermore their corresponding (when fully applied to convertible arguments) constructors. +See Chapter~\ref{Cic} for more details on convertibility and subtyping. +Also notice the subtyping constraints for the \emph{non-cumulative} version of list: +\begin{coq_example*} +Polymorphic NonCumulative Inductive list' {A : Type} := +| nil' : list' +| cons' : A -> list' -> list'. +\end{coq_example*} +\begin{coq_example} +Print list'. +\end{coq_example} +The following is an example of a record with non-trivial subtyping relation: +\begin{coq_example*} +Polymorphic Cumulative Record packType := {pk : Type}. +\end{coq_example*} +\begin{coq_example} +Print packType. +\end{coq_example} +Notice that as expected, \texttt{packType@\{i\}} and \texttt{packType@\{j\}} are convertible if and only if \texttt{i $=$ j}. + + \asection{Global and local universes} Each universe is declared in a global or local environment before it can |