Tutorial on Isabelle/HOL

4 1. The Basics1.2 TheoriesWorking with Isabelle means creating theories. Roughly speaking, a theoryis a named collection of types, functions, and theorems, much like a modulein a programming language or a specification in a specification language. Infact, theories in HOL can be either. The general format of a theory T istheory Timports B 1 . . . B nbegindeclarations, definitions, and proofsendwhere B 1 . . . B n are the names of existing theories that T is based on anddeclarations, definitions, and proofs represents the newly introduced concepts(types, functions etc.) and proofs about them. The B i are the direct parenttheories of T. Everything defined in the parent theories (and their parents,recursively) is automatically visible. To avoid name clashes, identifiers canbe qualified by theory names as in T.f and B.f. Each theory T must residein a theory file named T.thy.This tutorial is concerned with introducing you to the different linguisticconstructs that can fill the declarations, definitions, and proofs above. A completegrammar of the basic constructs is found in the Isabelle/Isar ReferenceManual [38].!!HOL contains a theory Main, the union of all the basic predefined theories likearithmetic, lists, sets, etc. Unless you know what you are doing, always includeMain as a direct or indirect parent of all your theories.HOL’s theory collection is available online athttp://isabelle.in.tum.de/library/HOL/and is recommended browsing. In subdirectory Library you find a growinglibrary of useful theories that are not part of Main but can be included amongthe parents of a theory and will then be loaded automatically.For the more adventurous, there is the Archive of Formal Proofs, ajournal-like collection of more advanced Isabelle theories:http://afp.sourceforge.net/We hope that you will contribute to it yourself one day.1.3 Types, Terms and FormulaeEmbedded in a theory are the types, terms and formulae of HOL. HOL isa typed logic whose type system resembles that of functional programminglanguages like ML or Haskell. Thus there are