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4. Sets and Functions — Mathematics in Lean 0.1 documentation

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The vocabulary of sets, relations, and functions provides a uniform language for carrying out constructions in all the branches of mathematics. Since functions and relations can be defined in terms of sets, axiomatic set theory can be used as a foundation for mathematics. Lean’s foundation is based instead on the primitive notion of a type, and it includes ways of defining functions between types. Every expression in Lean has a type: there are natural numbers, real numbers, functions from reals to reals, groups, vector spaces, and so on. Some expressions are types, which is to say, their type is Type. Lean and Mathlib provide ways of defining new types, and ways of defining objects of those types. Conceptually, you can think of a type as just a set of objects. Requiring every object to have a type has some advantages. For example, it makes it possible to overload notation like +, and it sometimes makes input less verbose because Lean can infer a lot of information from an object’s type

4. Sets and Functions - Mathematics in Lean v4.19.0 documentation 4. Sets and Functions View page source 4. Sets and Functions  The vocabulary of sets, relations, and functions provides a uniform language for carrying out constructions in all the branches of mathematics. Since functions and relations can be defined in terms of sets, axiomatic set theory can be used as a foundation for mathematics. Lean’s foundation is based instead on the primitive notion of a type , and it includes ways of defining functions between types. Every expression in Lean has a type: there are natural n

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