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sav08:syntax_and_shorthands_of_hol [2008/05/28 02:58] vkuncak |
sav08:syntax_and_shorthands_of_hol [2008/05/29 17:02] vkuncak |
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| All types are built from i and o using function type constructor $\Rightarrow$. For example, | All types are built from i and o using function type constructor $\Rightarrow$. For example, | ||
| - | * binary functions are terms of type $i \Rightarow i \Rightarrow i$ (arrow associates to the right, we use currying to represent functions with multiple arguments) | + | * binary functions are terms of type $i \Rightarrow i \Rightarrow i$ (arrow associates to the right, we use currying to represent functions with multiple arguments) |
| * binary predicates are terms of type $i \Rightarrow i \Rightarrow o$ | * binary predicates are terms of type $i \Rightarrow i \Rightarrow o$ | ||
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| To obtain a logic based on such lambda calculus, we add two family of constants: | To obtain a logic based on such lambda calculus, we add two family of constants: | ||
| * equality $=_{t \Rightarrow t \Rightarrow o}$ for each type $t$, denoting equality on elements of type $t$ | * equality $=_{t \Rightarrow t \Rightarrow o}$ for each type $t$, denoting equality on elements of type $t$ | ||
| - | * selection operator $\iota_{(i \Rightarrow o) \Rightarrow o}$, denoting 'choice function' that takes a predicate and returns one element satisfying the predicate (if such element exists) | + | * selection operator $\iota_{(i \Rightarrow o) \Rightarrow i}$, denoting 'choice function' that takes a predicate and returns one element satisfying the predicate (if exactly one such element exists) |
| Everything else (most of mathematics) can be defined in this logic, and to a large extent has been done in systems such as Isabelle and HOL. | Everything else (most of mathematics) can be defined in this logic, and to a large extent has been done in systems such as Isabelle and HOL. | ||