TRS Standard pair #487073558

details

property	value
status	complete
benchmark	Ex7_9.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n190.star.cs.uiowa.edu
space	Applicative_05

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	2.1092 seconds
cpu usage	5.3305
user time	5.12127
system time	0.209223
max virtual memory	1.8810344E7
max residence set size	374488.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 12 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 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) UsableRulesProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) ATransformationProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) QReductionProof [EQUIVALENT, 0 ms]
        (31) QDP
        (32) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (33) YES
    (34) QDP
        (35) UsableRulesProof [EQUIVALENT, 0 ms]
        (36) QDP
        (37) ATransformationProof [EQUIVALENT, 0 ms]
        (38) QDP
        (39) QReductionProof [EQUIVALENT, 0 ms]
        (40) QDP
        (41) QDPOrderProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) PisEmptyProof [EQUIVALENT, 0 ms]
        (44) YES
    (45) QDP
        (46) UsableRulesProof [EQUIVALENT, 0 ms]
        (47) QDP
        (48) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (49) YES


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

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

   app(app(app(if, true), xs), ys) -> xs
   app(app(app(if, false), xs), ys) -> ys
   app(app(sub, x), 0) -> x
   app(app(sub, app(s, x)), app(s, y)) -> app(app(sub, x), y)
   app(app(gtr, 0), y) -> false
   app(app(gtr, app(s, x)), 0) -> true
   app(app(gtr, app(s, x)), app(s, y)) -> app(app(gtr, x), y)
   app(app(d, x), 0) -> true
   app(app(d, app(s, x)), app(s, y)) -> app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
   app(len, nil) -> 0
   app(len, app(app(cons, x), xs)) -> app(s, app(len, xs))
   app(app(filter, p), nil) -> nil
   app(app(filter, p), app(app(cons, x), xs)) -> app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), 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:

   app(app(app(if, true), xs), ys) -> xs
   app(app(app(if, false), xs), ys) -> ys
   app(app(sub, x), 0) -> x
   app(app(sub, app(s, x)), app(s, y)) -> app(app(sub, x), y)
   app(app(gtr, 0), y) -> false
   app(app(gtr, app(s, x)), 0) -> true
   app(app(gtr, app(s, x)), app(s, y)) -> app(app(gtr, x), y)

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