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sav08:syntax_and_shorthands_of_hol [2008/05/28 01:05] vkuncak |
sav08:syntax_and_shorthands_of_hol [2008/05/29 17:00] 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 $i \Rightarrow i \Rightarrow o$ | ||
| Line 35: | Line 35: | ||
| \] | \] | ||
| Note that we have defined negation, conjunction, and universal quantification, so all propositional operations and existential quantification are expressible. | Note that we have defined negation, conjunction, and universal quantification, so all propositional operations and existential quantification are expressible. | ||
| + | |||
| + | Note also that we can express | ||
| + | \[ | ||
| + | \mbox{ let } x = E \mbox{ in } F | ||
| + | \] | ||
| + | by $(\lambda x.F)E$. | ||
| + | |||
| + | The selection operator $\iota$ is not essential, but is a useful notation. By writing | ||
| + | \[ | ||
| + | \mbox{ let } y = \iota(\lambda x. P(x)) \mbox{ in } F | ||
| + | \] | ||
| + | we can express the phrase "let y in F denote the unique x such that P(x) holds". | ||