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

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

PVS

Isabelle

Coq

ACL2