... ... @@ -4,15 +4,17 @@ This chapter describes the input syntax, and informally gives its semantics, illustrated by examples. A \why\ text contains a list of \emph{theories}. A \why\ text contains a list of \emph{theories}. A theory is a list of \emph{declarations}. Declarations introduce new types, functions and predicates, state axioms, lemmas and goals. types, functions and predicates, state axioms, lemmas and goals. These declarations can be directly written in the theory or taken from existing theories. The base logic of \why\ is a first-order polymorphic logic. \subsection{Example 1: lists} The Figure~\ref{fig:tutorial1} contains an example of \why\ input text, containing three theories. The first theory, \texttt{List}, text, containing three theories. The first theory, \texttt{List}, declares a new algebraic type for polymorphic lists, \texttt{list 'a}. As in ML, \texttt{'a} stands for a type variable. The type \texttt{list 'a} has two constructors, \texttt{Nil} and ... ... @@ -23,43 +25,37 @@ We deliberately make this theory that short, for reasons which will be discussed later. \begin{figure} \centering \centering \begin{verbatim} theory List type list 'a = Nil | Cons 'a (list 'a) end theory Length use import List use import int.Int logic length (l : list 'a) : int = logic length (l : list 'a) : int = match l with | Nil -> 0 | Cons _ r -> 1 + length r end lemma Length_nonnegative : forall l:list 'a. length(l) >= 0 end theory Sorted use import List use import int.Int inductive sorted (list int) = | Sorted_Nil : | Sorted_Nil : sorted Nil | Sorted_One : | Sorted_One : forall x:int. sorted (Cons x Nil) | Sorted_Two : forall x y : int, l : list int. | Sorted_Two : forall x y : int, l : list int. x <= y -> sorted (Cons y l) -> sorted (Cons x (Cons y l)) end \end{verbatim} \caption{Example of Why3 text.} ... ... @@ -78,19 +74,22 @@ context the theory \texttt{int.Int} from the standard library. The prefix \texttt{int} indicates the file in the standard library containing theory \texttt{Int}. Theories referred to without prefix either appear earlier in the current file, \eg\ \texttt{List}, or are predefined. predefined. The next declaration defines a recursive function, \emph{length}, which computes the length of a list. The \texttt{logic} keyword is used to introduce or define both function and predicate symbols. used to introduce or define both function and predicate symbols. \why\ checks every recursive, or mutually recursive, definition for termination. Basically, we require a lexicographic and structural descent for every recursive call for some reordering of arguments. Note that matching must be exhaustive and that every \texttt{match} descent for every recursive call for some reordering of arguments. Notice that matching must be exhaustive and that every \texttt{match} expression must be terminated by the \texttt{end} keyword. Despite using higher-order curried'' syntax, \why\ does not permit partial application: function and predicate arities must be respected. The last declaration in theory \texttt{Length} is a lemma stating that the length of a list is non-negative. the length of a list is non-negative. The third theory, \texttt{Sorted}, demonstrates the definition of an inductive predicate. Every such definition is a list of clauses: ... ... @@ -109,7 +108,192 @@ Note that the type signature of \texttt{sorted} predicate does not include the name of a parameter (see \texttt{l} in the definition of \texttt{length}): it is unused and therefore optional. \section*{Another Example} \subsection{Example 1 (continued): lists and abstract orderings} In the previous section we have seen how a theory can reuse the declarations of another theory, coming either from the same input text or from the library. Another way to referring to a theory is by <>. A \texttt{clone} declaration constructs a local copy of the cloned theory, possibly instantiating some of its abstract (i.e.~declared but not defined) symbols. \begin{figure} \centering \begin{verbatim} theory Order type t logic (<=) t t axiom Le_refl : forall x : t. x <= x axiom Le_asym : forall x y : t. x <= y -> y <= x -> x = y axiom Le_trans: forall x y z : t. x <= y -> y <= z -> x <= z end theory SortedGen use import List clone import Order as O inductive sorted (l : list t) = | Sorted_Nil : sorted Nil | Sorted_One : forall x:t. sorted (Cons x Nil) | Sorted_Two : forall x y : t, l : list t. x <= y -> sorted (Cons y l) -> sorted (Cons x (Cons y l)) end theory SortedIntList use import int.Int clone SortedGen with type O.t = int, logic O.(<=) = (<=) end \end{verbatim} \caption{Example of Why3 text (continued).} \label{fig:tutorial2} \end{figure} Consider the continued example in Figure~\ref{fig:tutorial2}. We write an abstract theory of partial orders, declaring an abstract type \texttt{t} and an abstract binary predicate \texttt{<=}. Notice that an infix operation must be enclosed in parentheses when used outside a term. We also specify three axioms of a partial order. There is little value in \texttt{use}'ing such a theory: this would constrain us to stay with the type \texttt{t}. However, we can construct an instance of theory \texttt{Order} for any suitable type and predicate. Moreover, we can build some further abstract theories using order, and then instantiate those theories. Consider theory \texttt{SortedGen}. In the beginning, we \texttt{use} the earlier theory \texttt{List}. Then we make a simple \texttt{clone} theory \texttt{Order}. This is pretty much equivalent to copy-pasting every declaration from \texttt{Order} to \texttt{SortedGen}; the only difference is that \why\ traces the history of cloning and transformations and drivers often make use of it (see Section~\ref{sec:drivers}). Notice an important difference between \texttt{use} and \texttt{clone}. If we \texttt{use} a theory, say \texttt{List}, twice (directly or indirectly: e.g.~by making \texttt{use} of both \texttt{Length} and \texttt{Sorted}), there is no duplication: there is still only one type of lists and a unique pair of constructors. On the contrary, when we \texttt{clone} a theory, we create a local copy of every cloned declaration, and the newly created symbols, despite having the same names, are different from their originals. Returning to the example, we finish theory \texttt{SortedGen} with a familiar definition of predicate \texttt{sorted}; this time we use the abstract order on the values of type \texttt{t}. Now, we can instantiate theory \texttt{SortedGen} to any ordered type, without having to retype the definition of \texttt{sorted}. For example, theory \texttt{SortedIntList} makes \texttt{clone} of \texttt{SortedGen} (i.e.~copies its declarations) substituting type \texttt{int} for type \texttt{O.t} of \texttt{SortedGen} and the default order on integers for predicate \texttt{O.(<=)}. \why\ will control that the result of cloning is well-typed. Several remarks ought to be made here. First of all, why should we clone theory \texttt{Order} in \texttt{SortedGen} if we make no instantiation? Couldn't we write \texttt{use import Order as O} instead? The answer is no, we could not. When cloning a theory, we only can instantiate the symbols declared locally in this theory, not the symbols imported with \texttt{use}. Therefore, we create a local copy of \texttt{Order} in \texttt{SortedGen} to be able to instantiate \texttt{t} and \texttt{(<=)} later. Secondly, when we instantiate an abstract symbol, its declaration is not copied from the theory being cloned. Thus, we will not create a second declaration of type \texttt{int} in \texttt{SortedIntList}. The mechanism of cloning bears some resemblance to modules and functors of ML-like languages. Unlike those languages, \why\ makes no distinction between modules and module signatures, modules and functors. Any \why\ theory can be \texttt{use}'d directly or instantiated in any of its abstract symbols. The command-line tool \texttt{why3} (described in Section~\ref{sec:batch}), allows us to see the effect of cloning. If the input file containing our example is called \texttt{lists.why}, we can launch the following command: \begin{verbatim} > why3 lists.why -T SortedIntList \end{verbatim} to see the resulting theory \texttt{SortedIntList}: \begin{verbatim} theory SortedIntList (* use BuiltIn *) (* use Int *) (* use List *) axiom Le_refl : forall x:int. x <= x axiom Le_asym : forall x:int, y:int. x <= y -> y <= x -> x = y axiom Le_trans : forall x:int, y:int, z:int. x <= y -> y <= z -> x <= z (* clone Order with type t = int, logic (<=) = (<=), prop Le_trans1 = Le_trans, prop Le_asym1 = Le_asym, prop Le_refl1 = Le_refl *) inductive sorted (list int) = | Sorted_Nil : sorted (Nil:list int) | Sorted_One : forall x:int. sorted (Cons x (Nil:list int)) | Sorted_Two : forall x:int, y:int, l:list int. x <= y -> sorted (Cons y l) -> sorted (Cons x (Cons y l)) (* clone SortedGen with type t1 = int, logic sorted1 = sorted, logic (<=) = (<=), prop Sorted_Two1 = Sorted_Two, prop Sorted_One1 = Sorted_One, prop Sorted_Nil1 = Sorted_Nil, prop Le_trans2 = Le_trans, prop Le_asym2 = Le_asym, prop Le_refl2 = Le_refl *) end \end{verbatim} In conclusion, let us briefly explain the concept of namespaces in \why. Both \texttt{use} and \texttt{clone} instructions can be used in three forms (the examples below are given for \texttt{use}, the semantics for \texttt{clone} is the same): \begin{itemize} \item \texttt{use List as L} --- every symbol $s$ of theory \texttt{List} is accessible under the name \texttt{L.$s$}. The \texttt{as L} part is optional, if it is omitted, the name of the symbol is \texttt{List.$s$}. \item \texttt{use import List as L} --- every symbol $s$ from \texttt{List} is accessible under the name \texttt{L.$s$}. It is also accessible simply as \texttt{$s$}, but only up to the end of the current namespace, e.g.~the current theory. If the current theory, that is the one making \texttt{use}, is later used under the name \texttt{T}, the name of the symbol would be \texttt{T.L.$s$}. (This is why we could refer directly to the symbols of \texttt{Order} in theory \texttt{SortedGen}, but had to qualify them with \texttt{O.} in \texttt{SortedIntList}.) As in the previous case, \texttt{as L} part is optional. \item \texttt{use export List} --- every symbol $s$ from \texttt{List} is accessible simply as \texttt{$s$}. If the current theory is later used under the name \texttt{T}, the name of the symbol would be \texttt{T.$s$}. \end{itemize} \why\ allows to open new namespaces explicitly in the text. In particular, the instruction \texttt{clone import Order as O}'' can be equivalently written as: \begin{verbatim} namespace import O clone export Order end \end{verbatim} However, since \why\ favours short theories over long and complex ones, this feature is rarely used. \subsection{Example 2: Einstein's problem} \index{Einstein's logic problem} We now consider another, slightly more complex example: to use \why\ ... ... @@ -210,9 +394,9 @@ first define a predicate \texttt{leftof} over two houses. \begin{verbatim} logic leftof (h1 h2 : house) = match h1, h2 with | H1, H2 | H2, H3 | H3, H4 | H1, H2 | H2, H3 | H3, H4 | H4, H5 -> true | _ -> false end ... ...