TRS Standard pair #487072983

details

property	value
status	complete
benchmark	thiemann35.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n182.star.cs.uiowa.edu
space	AProVE_07

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	2.25706 seconds
cpu usage	5.73572
user time	5.50788
system time	0.227842
max virtual memory	1.8477524E7
max residence set size	397624.0

stage attributes

key	value
starexec-result	YES

output

YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 64 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) ATransformationProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QReductionProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) UsableRulesProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) ATransformationProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QReductionProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES
    (25) QDP
        (26) QDPApplicativeOrderProof [EQUIVALENT, 41 ms]
        (27) QDP
        (28) UsableRulesProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (31) YES


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

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

   a(a(append, nil), ys) -> ys
   a(a(append, a(a(cons, x), xs)), ys) -> a(a(cons, x), a(a(append, xs), ys))
   a(a(filter, f), nil) -> nil
   a(a(filter, f), a(a(cons, x), xs)) -> a(a(a(if, a(f, x)), x), a(a(filter, f), xs))
   a(a(le, 0), y) -> true
   a(a(le, a(s, x)), 0) -> false
   a(a(le, a(s, x)), a(s, y)) -> a(a(le, x), y)
   a(a(a(if, true), x), xs) -> a(a(cons, x), xs)
   a(a(a(if, false), x), xs) -> xs
   a(a(not, f), b) -> a(not2, a(f, b))
   a(not2, true) -> false
   a(not2, false) -> true
   a(qs, nil) -> nil
   a(qs, a(a(cons, x), xs)) -> a(a(append, a(qs, a(a(filter, a(le, x)), xs))), a(a(cons, x), a(qs, a(a(filter, a(not, a(le, x))), xs))))

Q is empty.

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

(1) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(append, nil), ys) -> ys
   a(a(append, a(a(cons, x), xs)), ys) -> a(a(cons, x), a(a(append, xs), ys))
   a(a(filter, f), nil) -> nil
   a(a(filter, f), a(a(cons, x), xs)) -> a(a(a(if, a(f, x)), x), a(a(filter, f), xs))
   a(a(le, 0), y) -> true
   a(a(le, a(s, x)), 0) -> false
   a(a(le, a(s, x)), a(s, y)) -> a(a(le, x), y)
   a(a(a(if, true), x), xs) -> a(a(cons, x), xs)
   a(a(a(if, false), x), xs) -> xs
   a(a(not, f), b) -> a(not2, a(f, b))
   a(not2, true) -> false
   a(not2, false) -> true
   a(qs, nil) -> nil
   a(qs, a(a(cons, x), xs)) -> a(a(append, a(qs, a(a(filter, a(le, x)), xs))), a(a(cons, x), a(qs, a(a(filter, a(not, a(le, x))), xs))))

The set Q consists of the following terms:

   a(a(append, nil), x0)
   a(a(append, a(a(cons, x0), x1)), x2)
   a(a(filter, x0), nil)
   a(a(filter, x0), a(a(cons, x1), x2))
   a(a(le, 0), x0)
   a(a(le, a(s, x0)), 0)
   a(a(le, a(s, x0)), a(s, x1))

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