Type theory

Induction-recursion

In intuitionistic type theory (ITT), a discipline within mathematical logic, induction-recursion is a feature for simultaneously declaring a type and function on that type. It allows the creation of larger types, such as universes, than inductive types. The types created still remain predicative inside ITT. An inductive definition is given by rules for generating elements of a type. One can then define functions from that type by induction on the way the elements of the type are generated. Induction-recursion generalizes this situation since one can simultaneously define the type and the function, because the rules for generating elements of the type are allowed to refer to the function. Induction-recursion can be used to define large types including various universe constructions. It increases the proof-theoretic strength of type theory substantially. Nevertheless, inductive-recursive recursive definitions are still considered predicative. (Wikipedia).

Video thumbnail

Induction, bijections, products -- Proofs

This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.

From playlist Proofs

Video thumbnail

Recursive Factorial Function

Introduction to recursion.

From playlist Computer Science

Video thumbnail

Sequences: Introduction to Solving Recurrence Relations

This video introduces solving recurrence relations by the methods of inspection, telescoping, and characteristic root technique. mathispower4u.com

From playlist Sequences (Discrete Math)

Video thumbnail

Java Recursion

Get the Code: http://goo.gl/S8GBL Welcome to my Java Recursion tutorial. In this video, I'm going to cover java recursion in 5 different ways. I figured if I show it using many different diagrams that it will make complete sense. A recursive method is just a method that calls itself. As

From playlist Java Algorithms

Video thumbnail

Applying the recursive formula to a sequence to determine the first five terms

👉 Learn all about recursive sequences. Recursive form is a way of expressing sequences apart from the explicit form. In the recursive form of defining sequences, each term of a sequence is expressed in terms of the preceding term unlike in the explicit form where each term is expressed in

From playlist Sequences

Video thumbnail

How to use the recursive formula to evaluate the first five terms

👉 Learn all about recursive sequences. Recursive form is a way of expressing sequences apart from the explicit form. In the recursive form of defining sequences, each term of a sequence is expressed in terms of the preceding term unlike in the explicit form where each term is expressed in

From playlist Sequences

Video thumbnail

The foundation -- Number Theory 1

Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcollege.edu/ma

From playlist Number Theory v2

Video thumbnail

Peano Arithmetic -- Number Theory 1

⭐Support the channel⭐ Patreon: https://www.patreon.com/michaelpennmath Merch: https://teespring.com/stores/michael-penn-math My amazon shop: https://www.amazon.com/shop/michaelpenn ⭐my other channels⭐ Main Channel: https://www.youtube.com/michaelpennmath non-math podcast: http

From playlist Number Theory

Video thumbnail

C9 Lectures: Dr. Erik Meijer - Functional Programming Fundamentals Chapter 13 of 13

In Chapter 13, Equational Reasoning (and also revealing why Erik says 'uhm' and 'you know' so often), the grand finale, Dr. Meijer digs into referential transparency and being able to replace equals by equals in all contexts. In some sense, the purity inherent in functional languages like

From playlist Haskell - Functional Programming Fundamentals (Dr. Erik Meijer )

Video thumbnail

Discrete Math - 5.3.2 Structural Induction

Several proofs using structural induction. These examples revolve around trees. Textbook: Rosen, Discrete Mathematics and Its Applications, 7e Playlist: https://www.youtube.com/playlist?list=PLl-gb0E4MII28GykmtuBXNUNoej-vY5Rz

From playlist Discrete Math I (Entire Course)

Video thumbnail

Discrete Math II - 5.3.2 Structural Induction

We finish our study in Chapter 5 by examining proof by structural induction. You will find that this method ties in well to the recursive definitions we just studied, and in fact is the best method to prove a characteristic of a recursively defined function. Please be sure you've watched 5

From playlist Discrete Math II/Combinatorics (entire course)

Video thumbnail

Mathematical Induction Proof with Recursively Defined Function

Mathematical Induction Proof with Recursively Defined Function If you enjoyed this video please consider liking, sharing, and subscribing. Udemy Courses Via My Website: https://mathsorcerer.com My FaceBook Page: https://www.facebook.com/themathsorcerer There are several ways that you c

From playlist Principle of Mathematical Induction

Video thumbnail

Foundations - Seminar 14 - Gödel's incompleteness theorem Part 6

Billy Price and Will Troiani present a series of seminars on foundations of mathematics. In this seminar Will Troiani continues with the proof of Gödel's incompleteness theorem. You can join this seminar from anywhere, on any device, at https://www.metauni.org. This video was filmed in D

From playlist Foundations seminar

Video thumbnail

C9 Lectures: Dr. Erik Meijer - Functional Programming Fundamentals Chapter 6 of 13

We've kicked off C9 Lectures with a journey into the world of Functional Programming with functional language purist and high priest of the lambda calculus, Dr. Erik Meijer (you can thank Erik for many of the functional constructs that have shown up in languages like C# and VB.NET. When yo

From playlist Haskell - Functional Programming Fundamentals (Dr. Erik Meijer )

Video thumbnail

1.10.7 Recursive Functions: Video

MIT 6.042J Mathematics for Computer Science, Spring 2015 View the complete course: http://ocw.mit.edu/6-042JS15 Instructor: Albert R. Meyer License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 6.042J Mathematics for Computer Science, Spring 2015

Video thumbnail

Introduction to the Coq Proof Assistant - Andrew Appel

Introduction to the Coq Proof Assistant - Andrew Appel Princeton University December 7, 2010 A "proof assistant" is a software package comprising a validity checker for proofs in a particular logic, accompanied by semi-decision procedures called "tactics" that assist the mathematician in

From playlist Mathematics

Video thumbnail

How to determine the first five terms for a recursive sequence

👉 Learn all about recursive sequences. Recursive form is a way of expressing sequences apart from the explicit form. In the recursive form of defining sequences, each term of a sequence is expressed in terms of the preceding term unlike in the explicit form where each term is expressed in

From playlist Sequences

Related pages

Impredicativity | Intuitionistic type theory | Mathematical logic | Induction-induction | ALF (proof assistant) | Agda (programming language)