1/24 An Introduction to PVS Charngki PSWLAB An Introduction to PVS Judy Crow, Sam Owre, John Rushby, Natarajan Shankar, Mandayam Srivas Computer Science Laboratory, SRI International

2/24 An Introduction to PVSCharngki PSWLAB Table of Contents  Introduction  A brief tour of PVS  PVS language  More examples  References

3/24 An Introduction to PVSCharngki PSWLAB Introduction  PVS stands for “Prototype Verification System”  PVS  consists of a specification language integrated with support tools and theorem prover  is both interactive and highly mechanized: the user chooses each proving step and PVS performs it, displays the result, and then waits for the next command  The goal of PVS  PVS is designed to help in the detection of errors as well as in the confirmation of correctness

4/24 An Introduction to PVSCharngki PSWLAB Table of Contents  Introduction  A brief tour of PVS  PVS language  More examples  References

5/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  PVS has three steps to prove target specifications 1. Creating a specification 2. Typechecking 3. Proving

6/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  Creating a specification 1. Use M-x new-pvs-file command to create a new PVS file, and type a name of the file 2. or you can simply load a existing PVS file using M-x find-pvs- file command

7/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  Create a sum.pvs file  specification for summation of the first n natural numbers sum : THEORY BEGIN n : VAR nat sum (n) : RECURSIVE nat = (IF n = 0 THEN 0 ELSE n + sum(n-1) ENDIF) MEASURE (LAMBDA n: n) closed_form: THEOREM sum(n) = (n * (n+1) ) / 2 END sum used to show that the definition terminates

8/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  Typechecking 1. M-x typecheck command to typecheck 2. M-x show-tccs command to see TCCs 3. M-x typecheck-prove to prove TCCs  TCC  Type Correctness Condition  TCCs must be proved in order to show that the theory is type correct  The proofs of the TCCs may be postponed indefinately

9/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  Typechecking  TCCs  sum takes an argument of type nat, but the type of the argument in the recursive call to sum is integer, since nat is not closed under subtraction  Since sum is recursive form, we need to ensure this function terminates % Subtype TCC generated (line 7) for n-1 % unchecked sum_TCC1 : OBLIGATION (FORALL (n: nat): NOT n=0 IMPLIES n-1 >= 0) % Termination TCC generated (line 7) for sum % unchecked sum_TCC2 : OBLIGATION (FORALL (n: nat): NOT n=0 IMPLIES n-1 < n)

10/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  Proving  Place the cursor on the line containing the theorem, and type M-x prove  A new buffer will pop up, the formula will be displayed, and the cursor will appear at the Rule? prompt, indicating that users can interact with the prover  The proving process is completed if there are no more unproven subgoals

11/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  Proving 1. Prove formula by induction on n Generate 2 subgoals 1.base case 2.inductive step

12/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  Proving simplifies the formula send the proof to the PVS decision procedure

13/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  Proving  To eliminate the FORALL quantifier  skolem! command  Provide new constants for the bound variables  flatten command  break up the succedent into a new antecedent and consequent antecedent consequent

14/24 An Introduction to PVSCharngki PSWLAB A brief tour of PVS  Proving

15/24 An Introduction to PVSCharngki PSWLAB Table of Contents  Introduction  A brief tour of PVS  PVS language  More examples  References

16/24 An Introduction to PVSCharngki PSWLAB PVS language  A simple example : the rational numbers  predicate subtype rats : THEORY BEGIN rat : TYPE zero : rat / : [rat, rat  rat] * : [rat, rat  rat] x, y : VAR rat left_canclelation : AXIOM x * (y/x) = y zero_times : AXIOM zero * x = zero END rats We need to consider divide by zero

17/24 An Introduction to PVSCharngki PSWLAB PVS language  A simple example : the rational numbers  predicate subtypes rats : THEORY BEGIN rat : TYPE zero : rat nonzero : TYPE = { x | x /= zero } / : [rat, nonzero  rat] * : [rat, rat  rat] x, y : VAR rat left_canclelation : AXIOM x /= zero IMPLIES x * (y/x) = y zero_times : AXIOM zero * x = zero END rats predicate subtype

18/24 An Introduction to PVSCharngki PSWLAB PVS language  Example : Stacks  Generic type stacks [t : TYPE] : THEORY BEGIN stack : TYPE empty : stack s : VAR stack x : VAR t push : [t, stack  stack] pop : [stack  stack] top : [stack  t] pop_push : AXIOM pop(push(x, s)) = s top_push : AXIOM top(push(x, s)) = x END stacks Generic type

19/24 An Introduction to PVSCharngki PSWLAB PVS language  Example : factorial  Recursive  The MEASURE function is used to show that the definition terminates, by generating an obligation that the MEASURE decreases with each call factorial : THEORY BEGIN fac(x: nat) : RECURSIVE nat = IF x = 0 THEN 1 ELSE x * fac(x-1) ENDIF MEASURE (LAMBDA (x: nat): x) END factorial

20/24 An Introduction to PVSCharngki PSWLAB Table of Contents  Introduction  A brief tour of PVS  PVS language  More examples  References

21/24 An Introduction to PVSCharngki PSWLAB More examples  Quantifier Proof  Original goal : FORALL x : P(x) AND Q(x)  (FORALL x : P(x)) AND (FORALL x : Q(x))  After split command  Subgoal 1 : FORALL x : P(x) AND Q(x)  (FORALL x : P(x))  Subgoal 2 : FORALL x : P(x) AND Q(x)  (FORALL x : Q(x)) predicate : THEORY BEGIN T : TYPE x, y, z : VAR T P, Q : [T  bool] pred_calc : THEOREM (FORALL x : P(x) AND Q(x)) IMPLIES (FORALL x : P(x)) AND (FORALL x : Q(x)) END predicate

22/24 An Introduction to PVSCharngki PSWLAB More examples  Decision Procedures  i + 8 can be expressed as 3*m + 5*n  i = 3*m’ + 5*n’  case n=0  i = 3*(m-3) + 5*2  subgoal 2.1  case n>0  i = 3*(m+2) + 5(n-1)  subgoal 2.2 stamps : THEORY BEGIN i, three, five : VAR nat stamps : THEOREM ( FORALL i : (EXISTS three, five : i+8 = 3 * three + 5 * five )) END stamps

23/24 An Introduction to PVSCharngki PSWLAB Table of Contents  Introduction  A brief tour of PVS  PVS language  More examples  References

24/24 An Introduction to PVSCharngki PSWLAB References  A Tutorial Introduction to PVS by Judy Crow, Sam Owre, John Rushby, Natarajan Shankar and Mandayam Srivas, WIFT ‘95