author | wenzelm |
Sat, 10 May 2008 13:26:25 +0200 | |
changeset 26871 | 996add9defab |
parent 16417 | 9bc16273c2d4 |
permissions | -rw-r--r-- |
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header {* Syntactic classes *} |
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theory Product imports Main begin |
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text {* |
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\medskip\noindent There is still a feature of Isabelle's type system |
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left that we have not yet discussed. When declaring polymorphic |
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constants @{text "c \<Colon> \<sigma>"}, the type variables occurring in @{text \<sigma>} |
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may be constrained by type classes (or even general sorts) in an |
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arbitrary way. Note that by default, in Isabelle/HOL the |
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declaration @{text "\<odot> \<Colon> 'a \<Rightarrow> 'a \<Rightarrow> 'a"} is actually an abbreviation |
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for @{text "\<odot> \<Colon> 'a\<Colon>type \<Rightarrow> 'a \<Rightarrow> 'a"} Since class @{text type} is the |
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universal class of HOL, this is not really a constraint at all. |
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The @{text product} class below provides a less degenerate example of |
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syntactic type classes. |
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*} |
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axclass |
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product \<subseteq> type |
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consts |
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product :: "'a\<Colon>product \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<odot>" 70) |
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text {* |
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Here class @{text product} is defined as subclass of @{text type} |
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without any additional axioms. This effects in logical equivalence |
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of @{text product} and @{text type}, as is reflected by the trivial |
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introduction rule generated for this definition. |
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\medskip So what is the difference of declaring @{text "\<odot> \<Colon> |
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'a\<Colon>product \<Rightarrow> 'a \<Rightarrow> 'a"} vs.\ declaring @{text "\<odot> \<Colon> 'a\<Colon>type \<Rightarrow> 'a \<Rightarrow> |
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'a"} anyway? In this particular case where @{text "product \<equiv> |
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type"}, it should be obvious that both declarations are the same |
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from the logic's point of view. It even makes the most sense to |
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remove sort constraints from constant declarations, as far as the |
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purely logical meaning is concerned \cite{Wenzel:1997:TPHOL}. |
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On the other hand there are syntactic differences, of course. |
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Constants @{text \<odot>} on some type @{text \<tau>} are rejected by the |
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type-checker, unless the arity @{text "\<tau> \<Colon> product"} is part of the |
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type signature. In our example, this arity may be always added when |
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required by means of an \isakeyword{instance} with the default proof |
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(double-dot). |
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\medskip Thus, we may observe the following discipline of using |
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syntactic classes. Overloaded polymorphic constants have their type |
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arguments restricted to an associated (logically trivial) class |
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@{text c}. Only immediately before \emph{specifying} these |
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constants on a certain type @{text \<tau>} do we instantiate @{text "\<tau> \<Colon> |
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c"}. |
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This is done for class @{text product} and type @{typ bool} as |
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follows. |
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*} |
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instance bool :: product .. |
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defs (overloaded) |
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product_bool_def: "x \<odot> y \<equiv> x \<and> y" |
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text {* |
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The definition @{text prod_bool_def} becomes syntactically |
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well-formed only after the arity @{text "bool \<Colon> product"} is made |
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known to the type checker. |
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\medskip It is very important to see that above \isakeyword{defs} are |
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not directly connected with \isakeyword{instance} at all! We were |
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just following our convention to specify @{text \<odot>} on @{typ bool} |
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after having instantiated @{text "bool \<Colon> product"}. Isabelle does |
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not require these definitions, which is in contrast to programming |
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languages like Haskell \cite{haskell-report}. |
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\medskip While Isabelle type classes and those of Haskell are almost |
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the same as far as type-checking and type inference are concerned, |
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there are important semantic differences. Haskell classes require |
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their instances to \emph{provide operations} of certain \emph{names}. |
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Therefore, its \texttt{instance} has a \texttt{where} part that tells |
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the system what these ``member functions'' should be. |
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This style of \texttt{instance} would not make much sense in |
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Isabelle's meta-logic, because there is no internal notion of |
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``providing operations'' or even ``names of functions''. |
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*} |
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end |