Tutorial on Isabelle/HOL

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1.4 Variables 71.4 VariablesIsabelle distinguishes free and bound variables, as is customary. Bound variablesare automatically renamed to avoid clashes with free variables. In addition,Isabelle has a third kind of variable, called a schematic variable orunknown, which must have a ? as its first character. Logically, an unknownis a free variable. But it may be instantiated by another term during the proofprocess. For example, the mathematical theorem x = x is represented in Isabelleas ?x = ?x, which means that Isabelle can instantiate it arbitrarily. Thisis in contrast to ordinary variables, which remain fixed. The programminglanguage Prolog calls unknowns logical variables.Most of the time you can and should ignore unknowns and work withordinary variables. Just don’t be surprised that after you have finished theproof of a theorem, Isabelle will turn your free variables into unknowns. Itindicates that Isabelle will automatically instantiate those unknowns suitablywhen the theorem is used in some other proof. Note that for readability weoften drop the ?s when displaying a theorem.!!For historical reasons, Isabelle accepts ? as an ASCII representation of the∃ symbol. However, the ? character must then be followed by a space, as in? x. f(x) = 0. Otherwise, ?x is interpreted as a schematic variable. The preferredASCII representation of the ∃ symbol is EX.1.5 Interaction and InterfacesThe recommended interface for Isabelle/Isar is the (X)Emacs-based ProofGeneral [1, 2]. Interaction with Isabelle at the shell level, although possible,should be avoided. Most of the tutorial is independent of the interface and isphrased in a neutral language. For example, the phrase “to abandon a proof”corresponds to the obvious action of clicking on the Undo symbol in ProofGeneral. Proof General specific information is often displayed in paragraphsidentified by a miniature Proof General icon. Here are two examples:Proof General supports a special font with mathematical symbols known as “xsymbols”.All symbols have ascii-equivalents: for example, you can enter either& or \ to obtain ∧. For a list of the most frequent symbols see table A.1 inthe appendix.Note that by default x-symbols are not enabled. You have to switch them onvia the menu item Proof-General > Options > X-Symbols (and save the option viathe top-level Options menu).Proof General offers the Isabelle menu for displaying information and settingflags. A particularly useful flag is Isabelle > Settings > Show Types which causesIsabelle to output the type information that is usually suppressed. This is indispensiblein case of errors of all kinds because often the types reveal the source ofthe problem. Once you have diagnosed the problem you may no longer want to seethe types because they clutter all output. Simply reset the flag.
8 1. The Basics1.6 Getting StartedAssuming you have installed Isabelle and Proof General, you start it by typingIsabelle in a shell window. This launches a Proof General window. Bydefault, you are in HOL 1 .You can choose a different logic via the Isabelle > Logics menu. For example, youmay want to work in the real numbers, an extension of HOL (see Sect. 8.4.4).1 This is controlled by the ISABELLE_LOGIC setting, see The Isabelle System Manualfor more details.
Page 1 and 2: Tobias NipkowMarkus WenzelLawrence
Page 3 and 4: viPrefacegives the best performance
Page 5 and 6: viiiContents3.1.3 The simp Method .
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6. Sets, Functions and RelationsThi
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6.1 Sets 1096.1.1 Finite Set Notati
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6.2 Functions 111The internal form
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6.3 Relations 1131. ∀ x y. f x =
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Induction over the reflexive transi
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inv_image r f ≡ {(x,y). (f x, f y
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6.6 Case Study: Verified Model Chec
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6.6.2 Computation Tree Logic — CT
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Part IIIAdvanced Material
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152 8. More about TypesThe intuitiv
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160 8. More about Types8.3 Axiomati
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162 8. More about Typesprefix_def:"
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164 8. More about TypesLinear Order
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166 8. More about Types8.4 NumbersU
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168 8. More about Types2 + n = Suc
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178 9. Advanced Simplification and
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10. Case Study: Verifying a Securit
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10.2 Agents and Messages 1911. A
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analz (synth H) = analz H ∪ synth
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10.6 Proving Elementary Properties
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10.7 Proving Secrecy Theorems 197No
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10.7 Proving Secrecy Theorems 199ca
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201You know my methods. Apply them!
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204 A. AppendixConstant Type Syntax
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206 BIBLIOGRAPHY[13] Florian Haftma
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Index!, 203?, 203∃! , 203?!, 203&
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210 Index- defining inductively, 12
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Tutorial on Isabelle/HOL

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