Intuition for HOL

Untyped Lambda Calculus

Simply Typed Lambda Calculus

Further reading:

• Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984)
• H. P. Barendregt: Lambda Calculi with Types

Syntax and Shorthands of HOL

Standard-Model Semantics of HOL

Axioms of HOL

Results Proved in HOL

Further reading:

• Peter B. Andrews: An Introduction to Mathematical Logic and Type theory: To Truth through Proof, Springer 2002 (Chapter 5: Type Theory)
• The Seven Virtues of Simple Type Theory

Approaches to Reliable Complex Proofs

Immutable Abstract Data Types

Consequence of Type Safety and Abstraction

Theorems as Abstract Data Types

Concrete Example in Scala

Further reading:

• A Metalanguage for interactive proof in LCF
• HOL Light
• Upcoming book “Introduction to Logic and Automated Theorem Proving” by John Harrison

HOL, Isabelle

PVS - decision procedures

Coq, NuPRL - more complex type theory, constructive mathematics

ACL2 - emphasis on executable functions, quantifier-free statements, automated induction, pioneering industrial-scale case studies