Back to Basics: Algebraic Data Types I also do C++ training! arthur.j.odwyer@gmail.com Arthur O’Dwyer 2020-09-16Outline ● Why the name “algebraic data types”? [3–18] ○ Memory layout diagrams. Why More about pair and tuple [57–69] Questions? 2What do I mean by algebraic types? pair C++98. The original algebraic data type. tuple C++11. optional C++17. variant C++17, with with minor tweaks to its constructors in C++20. 3Why do we say “algebraic”? It’s about the type’s number of possible values, a.k.a. the size of its domain. How many possible states might the object take

User-Defined Datatypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2.1 Algebraic Datatype Declarations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2.2 Type Synonym provides higher-order functions, non-strict semantics, static poly- morphic typing, user-defined algebraic datatypes, pattern-matching, list comprehensions, a module system, a monadic I/O system, and a rich sequences of guards respecively; p and pi are patterns; v, x, and xi are variables; K and K ′ are algebraic datatype (data) constructors (including tuple constructors); and N is a newtype constructor. Rule

used by functional programming languages like Haskell, to share notation and properties across algebraic structures.) Lean uses the Unicode character N as alternative notation for the type nat. You can booleans, characters and strings, lists, products, disjoint sums, and so on. We can also define algebraic structures like groups, rings, fields, vector spaces, and categories. In fact, dependent type theory objects. It can define the number systems, ranging from the natural numbers to the complex numbers; algebraic structures, from semi- groups to categories and modules over an arbitrary ring; limits, derivatives

readable, fairly complete documentation. It is open source. Lean’s core library includes only basic algebraic definitions. More setup and proof automation are found in mathlib. It is more than a mathematical acyclic data in a program. You may know them under some other, largely synonymous names, including algebraic data types, inductive data types, freely generated data types, recursive data types, and data types of 1. The main purpose of these type classes is to form the foundation for a rich hierar- chy of algebraic type classes (groups, monoids, ring, fields, etc.) and to allow the overloading of common mathematical

discuss methods of elaboration and type inference, which can be used to support flexible forms of algebraic reasoning. Finally, of course, we will discuss features that are specific to Lean, including the equivalent int. Identities like these are designed to work in arbitrary instances of the relevant algebraic structures, using the type class mechanism that is described in Chapter 10. In particular, all these previous page) exact t end Note that the rewrite tactic can carry out generic calculations in any algebraic structure. The following examples involve an arbitrary ring and an arbitrary group, respectively

for example define the interface Algebraic: type Algebraic interface { Add(b Algebraic) Algebraic Min(b Algebraic) Algebraic Mult(b Algebraic) Algebraic … Elements() } and define the the methods Add(), Min(), Mult(), … for our matrix types. Each type which implements the Algebraic interface above will allow for method chaining. Each method implementation should use a type-switch specified which relies only on the methods in the interface: func (a *denseMatrix) Add(b Algebraic) Algebraic { switch b.(type) { case sparseMatrix: return addDenseToSparse(a, b.(sparseMatrix))

mathematical objects: number sys- tems, ranging from the natural numbers to the complex numbers; algebraic structures, from semigroups to categories and modules over an arbitrary ring; limits, derivatives #eval mauve.red #eval mauve.green Lean provides extensive support for reasoning generically about algebraic structures, in particular, allowing the inheritance and sharing of notation and facts. Chief among the argument to mul. With has_mul and has_one in place, some of the most basic objects of the algebraic hierarchy are defined as follows: universe u variables {α : Type u} (continues on next page)

• Euclid’s GCD algorithm
 • Abstraction in mathematics
 • Deriving generic algorithms
 • Algebraic structures
 • Programming concepts
 • Permutation algorithms
 • Cryptology (RSA) ~ 1977 AD2023 • Transformations and Their Orbits • Associative Operations • Linear Orderings • Ordered Algebraic Structures • Iterators • Coordinate Structures • Coordinates with Mutable Successors • Copying

.......................................................................... 223 Miscellaneous Algebraic Functions ..................................................................................... 1): 1 Math.log1p(17.5): 2.917770732084279 Miscellaneous Algebraic Functions Following is a list of miscellaneous algebraic functions with their description. Function Description

determinism From Yossi Kreinin - Consistency: how to defeat the purpose of IEEE floating point 1. Algebraic compiler optimizations 2. "Complex" instructions 3. x86-specific pain. SG14: P3375R0 tries to address