CppCon 2020 Template Metaprogramming: Type Traits Part 1 Jody Hagins jhagins@maystreet.com coachhagins@gmail.com ## CppCon 2020 Template Metaprogramming: Type Traits Introduction ## I ntended Audience Not necessarily beginner to C++, but beginner to traditional template metaprogramming techniques • Type traits part of standard library for ~10 years ## I ntended Audience • Beginner/Intermediate • Gentle Not necessarily beginner to C++, but beginner to traditional template metaprogramming techniques • Type traits part of standard library for ~10 years • Fundamentals have been in use for ~20 years ## I

3.6 An Example: Abstract Syntax 36 4 Theorem Proving in Lean 38 4.1 Assertions in Dependent Type Theory 38 4.2 Propositions as Types 39 4.3 Induction and Calculation 42 4.4 Axioms Lean is an implementation of a logical foundation known as dependent type theory. Specifically, it implements a version of dependent type theory known as the Calculus of Inductive Constructions. The CIC expression has a type. The type of expression indicates what sort of object the expression denotes. For example, an expression may denote a mathematical object like a natural number, a data type, an assertion

Agda page [https://github.com/agda/agda]. This is the manual for the Agda programming language, its type checking, compilation and editing system and related resources/tools. The latest PDF version of this system can be used can be found in chapter Tools. ## Getting Started • What is Agda? Dependent types • Installation Installation from source ○ Prebuilt Packages and System-Specific Instructions in Agda Hello, Agda! o Hello, World! • A Taste of Agda o Preliminaries ○ Programming With Dependent Types: Vectors Agda as a Proof Assistant: Proving Associativity of Addition Building an Executable

'a', "Hello world!", [2,3,4,5,6] In a language with dependent types, however, the distinction is less clear. Dependent types allow types to "depend" on values — in other words, types manipulated like any other value. The standard example is the type of lists of a given length $ ^{1} $ , Vect n a, where a is the element type and n is the length of the list and can be an arbitrary term contain values, and where those values describe properties, for example the length of a list, the type of a function can begin to describe its own properties. Take for example the concatenation of two

System Syntactic Sugar Syntax Declarations Telescopes Termination Checking Two-Level Type Theory Universe Levels With-Abstraction Without K Tools Automatic Proof Search (Auto) Command-line Agda page [https://github.com/agda/agda]. This is the manual for the Agda programming language, its type checking, compilation and editing system and related resources/tools. The latest PDF version of this compilation system can be used can be found in chapter Tools. ## Getting Started • What is Agda? Dependent types • Installation Installation from source ○ Prebuilt Packages and System-Specific Instructions

What is Agda? Prerequisites ○ Installation ○ ‘Hello world’ in Agda ○ Quick Guide to Editing, Type Checking and Compiling Agda Code ○ A List of Tutorials • Language Reference ☐ Abstract definitions Agda page [https://github.com/agda/agda]. This is the manual for the Agda programming language, its type checking, compilation and editing system and related resources/tools. The latest PDF version of this system can be used can be found in chapter Tools. ## Getting Started • What is Agda? Dependent types • Prerequisites Installing Emacs under Windows • Installation Installation from Hackage

it the most destructive type of work. It’s not really work at all, like the others. The others are what you planned on doing, allegedly because you needed to do it.” ## Theory of Constraints “Eliyahu “Eliyahu M. Goldratt, who created the Theory of Constraints, showed us how any improvements made anywhere besides the bottleneck are an illusion. Astonishing, but true! Any improvement made after the bottleneck even have what I need. It’s why I’ve stopped even putting in change requests.” “I have to manually type in hundreds of server names in one of the text boxes. Most of the time, there’s not enough room in

'a', "Hello world!", [2,3,4,5,6] In a language with dependent types, however, the distinction is less clear. Dependent types allow types to “depend” on values — in other words, types are manipulated like any other value. The standard example is the type of lists of a given length $ ^{1} $ , Vect n a, where a is the element type and n is the length of the list and can be an arbitrary term contain values, and where those values describe properties, for example the length of a list, the type of a function can begin to describe its own properties. Take for example the concatenation of two

Functions 8 3.4 Dependent Types 10 3.5 I/O 12 3.6 "do" notation 13 3.7 Laziness 13 3.8 Useful Data Types 14 3.9 More Expressions 17 3.10 Dependent Records 18 4 Type Classes 21 4 ..... 39 8.1 Dependent pattern matching ..... 39 8.2 The with rule — matching intermediate values ..... 39 9 Theorem Proving ..... 41 9.1 Equality ..... 41 9.2 The Empty Type ..... 41 9.3 Simple conversions ..... 58 13.3 Literate programming ..... 58 13.4 Foreign function calls ..... 58 13.5 Type Providers ..... 60 13.6 C Target ..... 60 13.7 JavaScript Target ..... 61 13.8 Cumulativity

learn it by heart and use it. The author of this paper being an example of this. Machine learning is a type of stochastic algorithms that try to find a solution based upon statistical data. A stochastic algorithm because the cards in each hand are hidden. Furthermore the shuffling and dealing of the deck is a type of random element, if done correctly $ ^{6} $ . ### 3.4 The Sprague-Grundy Theorem The Sprague-Grundy relevant, mostly those in use. $ ^{9} $ ### 4.1 Reinforcement learning Reinforcement learning is a type of machine learning where a program tries to find a solution for a problem, and is taught how to do