Isabelle/HOL and Proof General Reference

This site is intended to help getting started with using Isabelle/HOL and the Isabelle jEdit editor. This page in particular is the quick cheat sheet and can be used as a reference.

Please use the FAQ - Ask Questions page to post and view questions and the Exam Questions page to collect possible questions for the exams. The Goals of the Exercises page summarizes what you should have learned by doing them.

For best practices in Isabelle/HOL, see the corresponding page.

Please also check out and add stuff concerning the Editor, Isabelle/HOL Syntax and useful External References. Last but not least, you can post and check out Example Code.

Help us to improve the Wiki by checking and complementing existing content, as well as creating wanted pages!

You can download and install Isabelle from http://isabelle.in.tum.de/. To run Isabelle you can run `isabelle jedit -m brackets`.

Isabelle is already installed on the SCI machines tux1, tux2, tux3 and tux9. On those machines you can start it by simply typing `isabelle` into a terminal.

Every Isabelle/HOL theory looks like this:

theory MyTheory imports Main
begin

  ...

end

It is important that the name of the theory is equal to the name of the file.

Isabelle uses many symbols which are not available on a normal keyboard. You can either use the “Symbols” tab in jedit or type the following shortcuts and use autocomplete with “TAB” to get the correct symbol.

If you know LaTeX, try using commands you know from there.

Heuristic: try ML! Major differences:

• Function definition with primrec, fun and function
• List constructor is #, not ::.
• Types and expressions have to be enclosed in double quotes

• Constants
  • definition $ident where “$ident == $expression” – assigns the value of $expression to identifier $ident
• Theorems
  • lemma ($name :)? “$formula” – states $formula as lemma, assigns it a name (if given) and creates a proof goal
  • theorem ($name :)? “$formula” – states $formula as theorem, assigns it a name (if given) and creates a proof goal
• Datatypes
  • datatype
  • type_synonym
• Functions
  • primrec – used for primitive recursive functions (nothing to prove, but heavy constraints)
  • fun – used for more general functions, but have to prove termination (e.g. via well-founded order)
  • function – used for arbitrary function definitions; have to prove all kinds of stuff
• (Co)Inductive Stuff

Apply method $method with parameters $params by entering apply ($method $params). If you have no parameters, you can just write apply $method. For example, a (very simple) proof can look like this:

  lemma "[| A; A --> B |] ==> B"
  apply (frule mp)
  apply assumption+
  done

Note that safety of automated proof methods can be sabotaged by adding unsafe rules to rule sets used.

There are many more methods. You can print them by issuing the command print_methods, key combination C-c C-a h m or via [Isabelle → Show Me → Methods].

The sequencing of methods has the additional effect that backtracking is used to make the whole sequence work. As many methods could be applied in different ways, e.g. by matching the premise with a different assumption, failure of one step of the sequence just leads to trying another possibility for one of the steps before.

Attributes (also: directives) can be used to obtain new, more specific theorems from already proven, more general ones. In other words, they allow you to adapt theorems to your current needs. There are two major uses:

1. $thm [$attr1, $attr2, …] – can be used in any place where you would put a theorem or rule. Instead of $thm itself, the result of applying the given attributes from left to right is used.
2. lemmas $name = $thm [$attr1, $attr2, …] – assigns a new name to the modified theorem, enabling later (re)use.
3. (lemma|theorem) $name [$attr1, $attr2, …] : $formula – applies the given attributes from left to right after the proof is finished, assigning the result to the given name.

Other attributes perform some action with a theorem. They probably only make sense in lemma/theorem definitions or together with lemmas (see above):

There are many more attributes. You can print them by issuing the command print_attributes, key combination C-c C-a h a or via [Isabelle → Show Me → Attributes].

• conjI, conjE, conjunct1, conjunct2
• disjI1, disjI2, disjE
• impI, impE, mp
• iffI, iffE
• allI, spec, bspec
• exI, exE, bexI
• notI, notE
• FalseE
• classical
• contrapos_(pp|np|pn|nn)
• Basically any theorem!

• thm $name – command that shows theorem with name $name
• value $expr – command that evaluates the specified expression and prints the result
• prefer $k – command that moves subgoal number $k to the top of the list
• defer – command that moves the current subgoal to the bottom of the list
• quickcheck – command that tries to find a counterexample to the current subgoal. Useful to check wether an unsafe rule did harm. Note: it might not find a counterexample even if the goal can not be proven!
• refute and nitpick – similar to quickcheck but try to find counterexample models, not only variable assignments. Can handle more constructs.
• sledgehammer – command that invokes fully automated theorem provers both locally and on remote clusters. Tries to find a (minimal) set of theorems needed to solve the current goal.
• lfp $function – a function that yields the least fixpoint of the given function
• undefined – a distinguished value for any type
• f(x := y) – the function update: the result of this expression is the function f updated such that it now returns y for parameter x; the other values do not change.
• {x. P x} – the set of values fulfilling predicate P. For instance, {x::nat. x dvd 125} is the set of (natural) divisors of $125$.
• {E x | x. P x} – the set of values created by expression E, for all values fulfilling predicate P, {x + y | x y. x < 10 /\ y < 10} is the set of sums of all pairs of natural numbers with a single digit.