Higher Order Rewriting Union Beta pair #487093823

details

property	value
status	complete
benchmark	prenex.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n146.star.cs.uiowa.edu
space	Mixed_HO_10

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.194118 seconds
cpu usage	0.194558
user time	0.159101
system time	0.035457
max virtual memory	113188.0
max residence set size	4680.0

stage attributes

key	value
starexec-result	YES

output

YES
We consider the system theBenchmark.

  Alphabet:

    and : [form * form] --> form 
    exists : [form -> form] --> form 
    forall : [form -> form] --> form 
    not : [form] --> form 
    or : [form * form] --> form 

  Rules:

    and(x, forall(/\y.f y)) => forall(/\z.and(x, f z)) 
    or(x, forall(/\y.f y)) => forall(/\z.or(x, f z)) 
    and(forall(/\x.f x), y) => forall(/\z.and(f z, y)) 
    or(forall(/\x.f x), y) => forall(/\z.or(f z, y)) 
    not(forall(/\x.f x)) => exists(/\y.not(f y)) 
    and(x, exists(/\y.f y)) => exists(/\z.and(x, f z)) 
    or(x, exists(/\y.f y)) => exists(/\z.or(x, f z)) 
    and(exists(/\x.f x), y) => exists(/\z.and(f z, y)) 
    or(exists(/\x.f x), y) => exists(/\z.or(f z, y)) 
    not(exists(/\x.f x)) => forall(/\y.not(f y)) 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    and : [form * form] --> form 
    exists : [form -> form] --> form 
    forall : [form -> form] --> form 
    not : [form] --> form 
    or : [form * form] --> form 
    ~AP1 : [form -> form * form] --> form 

  Rules:

    and(X, forall(/\x.~AP1(F, x))) => forall(/\y.and(X, ~AP1(F, y))) 
    or(X, forall(/\x.~AP1(F, x))) => forall(/\y.or(X, ~AP1(F, y))) 
    and(forall(/\x.~AP1(F, x)), X) => forall(/\y.and(~AP1(F, y), X)) 
    or(forall(/\x.~AP1(F, x)), X) => forall(/\y.or(~AP1(F, y), X)) 
    not(forall(/\x.~AP1(F, x))) => exists(/\y.not(~AP1(F, y))) 
    and(X, exists(/\x.~AP1(F, x))) => exists(/\y.and(X, ~AP1(F, y))) 
    or(X, exists(/\x.~AP1(F, x))) => exists(/\y.or(X, ~AP1(F, y))) 
    and(exists(/\x.~AP1(F, x)), X) => exists(/\y.and(~AP1(F, y), X)) 
    or(exists(/\x.~AP1(F, x)), X) => exists(/\y.or(~AP1(F, y), X)) 
    not(exists(/\x.~AP1(F, x))) => forall(/\y.not(~AP1(F, y))) 
    ~AP1(F, X) => F X 

Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:

  Alphabet:

    and : [form * form] --> form 
    exists : [form -> form] --> form 
    forall : [form -> form] --> form 
    not : [form] --> form 
    or : [form * form] --> form 

  Rules:

    and(X, forall(/\x.Y(x))) => forall(/\y.and(X, Y(y))) 
    or(X, forall(/\x.Y(x))) => forall(/\y.or(X, Y(y))) 
    and(forall(/\x.X(x)), Y) => forall(/\y.and(X(y), Y)) 
    or(forall(/\x.X(x)), Y) => forall(/\y.or(X(y), Y)) 
    not(forall(/\x.X(x))) => exists(/\y.not(X(y))) 
    and(X, exists(/\x.Y(x))) => exists(/\y.and(X, Y(y))) 
    or(X, exists(/\x.Y(x))) => exists(/\y.or(X, Y(y))) 
    and(exists(/\x.X(x)), Y) => exists(/\y.and(X(y), Y)) 
    or(exists(/\x.X(x)), Y) => exists(/\y.or(X(y), Y)) 
    not(exists(/\x.X(x))) => forall(/\y.not(X(y))) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  and(X, forall(/\x.Y(x))) >? forall(/\y.and(X, Y(y))) 
  or(X, forall(/\x.Y(x))) >? forall(/\y.or(X, Y(y))) 
  and(forall(/\x.X(x)), Y) >? forall(/\y.and(X(y), Y)) 
  or(forall(/\x.X(x)), Y) >? forall(/\y.or(X(y), Y)) 
  not(forall(/\x.X(x))) >? exists(/\y.not(X(y))) 
  and(X, exists(/\x.Y(x))) >? exists(/\y.and(X, Y(y))) 
  or(X, exists(/\x.Y(x))) >? exists(/\y.or(X, Y(y))) 
  and(exists(/\x.X(x)), Y) >? exists(/\y.and(X(y), Y)) 
  or(exists(/\x.X(x)), Y) >? exists(/\y.or(X(y), Y)) 
  not(exists(/\x.X(x))) >? forall(/\y.not(X(y))) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  and = \y0y1.y0 + 2y1 
  exists = \G0.1 + G0(0) 
  forall = \G0.1 + G0(0) 
  not = \y0.y0 
  or = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

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