TRS Standard pair #487073868

details

property	value
status	complete
benchmark	nrvsq.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n185.star.cs.uiowa.edu
space	MNZ_10

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	28.9678 seconds
cpu usage	81.7491
user time	78.677
system time	3.07215
max virtual memory	1.8819056E7
max residence set size	5646296.0

stage attributes

key	value
starexec-result	YES

output

YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 27 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) TransformationProof [EQUIVALENT, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) TransformationProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) DependencyGraphProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) TransformationProof [EQUIVALENT, 2 ms]
        (15) QDP
        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) QDPOrderProof [EQUIVALENT, 182 ms]
        (22) QDP
        (23) QDPOrderProof [EQUIVALENT, 106 ms]
        (24) QDP
        (25) DependencyGraphProof [EQUIVALENT, 0 ms]
        (26) AND
            (27) QDP
                (28) QDPOrderProof [EQUIVALENT, 7437 ms]
                (29) QDP
                (30) DependencyGraphProof [EQUIVALENT, 0 ms]
                (31) QDP
                (32) UsableRulesProof [EQUIVALENT, 0 ms]
                (33) QDP
                (34) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (35) YES
            (36) QDP
                (37) UsableRulesProof [EQUIVALENT, 0 ms]
                (38) QDP
                (39) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (40) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   s(x) -> h(0, x)
   s(x) -> h(x, 0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   h(f(x), g(x)) -> f(s(x))
   g(x) -> h(h(h(h(x, x), x), x), x)
   f(s(s(x))) -> h(f(x), g(h(x, x)))
   s(0) -> r(0)
   s(s(s(0))) -> r(s(0))
   r(s(0)) -> s(0)
   g(x) -> r(x)
   s(0) -> p(0)
   s(s(0)) -> p(s(0))
   p(s(0)) -> 0
   s(s(s(s(s(0))))) -> p(s(s(0)))
   p(s(s(0))) -> s(s(s(0)))
   h(p(x), g(x)) -> p(s(x))
   s(0) -> k(0)
   s(s(p(p(a)))) -> s(k(p(a)))
   s(k(p(a))) -> p(p(a))
   g(x) -> k(x)
   a -> 0
   s(h(r(k(p(x))), r(x))) -> h(r(r(p(x))), k(x))

Q is empty.

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(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(g(x)) -> G(g(f(x)))
   F(g(x)) -> G(f(x))
   F(g(x)) -> F(x)

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