TRS Standard pair #487066763

details

property	value
status	complete
benchmark	zerolist.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n177.star.cs.uiowa.edu
space	AProVE_09_Inductive

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	2.37312 seconds
cpu usage	5.67714
user time	5.4276
system time	0.249538
max virtual memory	1.8544088E7
max residence set size	370796.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) Overlay + Local Confluence [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 15 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QReductionProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QReductionProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) QDPOrderProof [EQUIVALENT, 36 ms]
        (20) QDP
        (21) Induction-Processor [SOUND, 68 ms]
        (22) AND
            (23) QDP
                (24) DependencyGraphProof [EQUIVALENT, 0 ms]
                (25) TRUE
            (26) QTRS
                (27) QTRSRRRProof [EQUIVALENT, 0 ms]
                (28) QTRS
                (29) RisEmptyProof [EQUIVALENT, 0 ms]
                (30) YES


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

   sub(0, 0) -> 0
   sub(s(x), 0) -> s(x)
   sub(0, s(x)) -> 0
   sub(s(x), s(y)) -> sub(x, y)
   zero(nil) -> zero2(0, nil)
   zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs))
   zero2(0, nil) -> nil
   zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs))
   zero2(s(y), nil) -> zero(nil)
   zero2(s(y), cons(x, xs)) -> zero(cons(x, xs))

Q is empty.

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(1) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   sub(0, 0) -> 0
   sub(s(x), 0) -> s(x)
   sub(0, s(x)) -> 0
   sub(s(x), s(y)) -> sub(x, y)
   zero(nil) -> zero2(0, nil)
   zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs))
   zero2(0, nil) -> nil
   zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs))
   zero2(s(y), nil) -> zero(nil)
   zero2(s(y), cons(x, xs)) -> zero(cons(x, xs))

The set Q consists of the following terms:

   sub(0, 0)
   sub(s(x0), 0)
   sub(0, s(x0))
   sub(s(x0), s(x1))
   zero(nil)
   zero(cons(x0, x1))
   zero2(0, nil)
   zero2(0, cons(x0, x1))
   zero2(s(x0), nil)
   zero2(s(x0), cons(x1, x2))


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(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

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