1. CAS- 734 (Presentation -1) By : Vasudha Kapil
ISABELLE/HOL
CAS- 734 (Presentation -1)
By : Vasudha Kapil

2. Outline
Intoduction
Theory Format
HOL
Proof format
Example

3. ISABELLE/HOL
Isabelle theorem prover is an interactive proof assistant.
It is a Generic Proof Assistant.
It was developed at University of Cambridge (Laury Paulson) , TU Munchen (Tobias Nipkow) and Universt Paris Sud (Makarius Wenzel)
Isabelle/HOL is specialization of Isabelle for HOL (Higher Order Logic).

4. INSTALLATION Download system from :
It is currently available for three platforms :
WINDOWS
MAC OS X
LINUX
Platform specific application bundle includes sources, documentation and add on components required.

5. INTERFACE
Isabelle jEdit is the interface for current version of Isabelle/HOL.
Interactive Development Environment
Parses and interprets file while it is typed.
List of mathematical symbols provided.

6. THEORIES General format of theory T in Isabelle/HOL is : Theory T
Imports B Bn
Begin
(Declarations, Definitions & Proofs)
end

7. Brief Review of HOL HOL has HOL = functional programming + logic
datatypes
recursive functions
logical operators (∧, −→, ∀, ∃, )
HOL = functional programming + logic

8. Types Basic Syntax – τ ::= (τ )
| bool | nat | base types
| ’a | ’b | type variables
| τ ⇒ τ total functions
| sets, lists type constructors
| user-defined types
All terms and formulae should be well typed in Isabelle.

9. Type Inference and Type Annotation
Type Inference : Isabelle automatically computes the type of each variable in a term.
Type Annotations : In the presence of overloaded functions type inference is not always possible. Type constraints are needed in such cases.
Syntax : f (x::nat)

10. Terms Syntax Terms must be well-typed
| a constant or variable (identifier)
| term term function application
| λx. term function “abstraction”
| lots of syntactic sugar
Terms must be well-typed
Notation: t :: τ means t is a well-typed term of type τ .

11. Formulae
They are terms of type bool (True & False) and usual logical connectives.
Syntax : form ::=
(form) | term = term | ￢form| form ∧ form | form ∨ form | form −→ form| ∀x. form |
∃x. form

12. Variables Isabelle has three kinds of variables : Bound Variables
Free Variables
Schematic variables or unknown. Example : ?x
It has ‘?’ as its first character.

13. Functions Function definition schemas in Isabelle/HOL
Non Recursive with definition
definition name :: “domain” where “fun_def”
Example : definition sq :: “nat => nat” where “sq n= n*n”
Primitive Recursive with primrec
primrec name :: “domain” where “fun_def1| fun_def2| |fun_defn”
Example : primrec rev :: "'a list =>'a list“ where
"rev [] = []" |
"rev (x # xs) = (rev (x # [])"

14. Functions (continued)
Well founded recursion with fun
Syntax : fun f :: “τ”
where
“equations”
Fun has automatic termination proof.
Well founded recursion with function.
Syntax : function f :: “τ”
....
by pat_completeness auto
Termination by lexicographic_order
User supplied termination proof.

15. Proofs General format:
lemma name : "..."
apply (...) .
done
If the lemma is suitable as a simplification rule:
lemma name [simp]: "..."

16. Automated Methods Methods are commands to work on proof state.
Syntax : apply (method <parameters>)
assumption : It solves a sub goal if consequent is contained in set of assumptions.
auto : Instructs Isabelle to try and prove all subgoals automatically essentially by simplifying them.
simp : Same as auto but act on subgoal 1 only.
[simp] : It can be used to make a theorem simplification rule. Example : prove rev(rev x) = x
lemma rev_rev [simp] : “rev(rev x) = x”

17. Methods (continued) blast : Covers logic, sets, relations
Doesn’t support equality.
arith : Covers linear arithmetic.
Supports int, reals as well
Doesn’t support complex multiplication (*)
Induction : apply (induction m) : Tells Isabelle to start a proof by induction on m.

18. EXAMPLE
theory addition imports Main begin fun add :: "nat⇒ nat ⇒ nat" where "add 0 n = n" | "add (Suc m) n = Suc(add m n)" lemma add_ex [simp]: "add m 0 = m" apply(induction m) apply(auto) done end

19. Bibliography
Theorem Proving with Isabelle/HOL : By Tobias Nipkow.
Isabelle/HOL : A Proof Assistant for Higher Order Logic. By- Tobias Nipkow, Lawrence C. Paulson, Markus Wenzel