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ISABELLE/HOL

ISABELLE/HOL. CAS- 734 (Presentation -1) By : Vasudha Kapil. Outline. Intoduction Theory Format HOL Proof format Example. ISABELLE/HOL. Isabelle theorem prover is an interactive proof assistant. It is a Generic Proof Assistant.

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ISABELLE/HOL

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  1. 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 : https://www.cl.cam.ac.uk/research/hvg/Isabelle/ • 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 B1.......... Bn Begin (Declarations, Definitions & Proofs) end

  7. Brief Review of HOL • HOL has • 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 term ::= (term) | 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 xs) @ (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” where “equations” .... by pat_completeness auto Termination by lexicographic_order • User supplied termination proof.

  15. Proofs General format: lemma name : "..." apply (...) 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 • https://www.cl.cam.ac.uk/research/hvg/Isabelle/documentation.html • Theorem Proving with Isabelle/HOL : By Tobias Nipkow. http://isabelle.in.tum.de/coursematerial/PSV2009-1/ • Isabelle/HOL : A Proof Assistant for Higher Order Logic. By- Tobias Nipkow, Lawrence C. Paulson, Markus Wenzel

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