TRS Standard pair #487066738

details

property	value
status	complete
benchmark	10.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n072.star.cs.uiowa.edu
space	Various_04

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	2.07813 seconds
cpu usage	4.9061
user time	4.67114
system time	0.234955
max virtual memory	1.961014E7
max residence set size	399792.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) UsableRulesProof [EQUIVALENT, 0 ms]
        (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) QDPOrderProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) PisEmptyProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) UsableRulesProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES
    (25) QDP
        (26) UsableRulesProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (29) YES
    (30) QDP
        (31) UsableRulesProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) MNOCProof [EQUIVALENT, 0 ms]
        (34) QDP
        (35) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
        (36) QDP
        (37) PisEmptyProof [EQUIVALENT, 0 ms]
        (38) YES
    (39) QDP
        (40) UsableRulesProof [EQUIVALENT, 0 ms]
        (41) QDP
        (42) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
        (43) QDP
        (44) MRRProof [EQUIVALENT, 7 ms]
        (45) QDP
        (46) MRRProof [EQUIVALENT, 0 ms]
        (47) QDP
        (48) DependencyGraphProof [EQUIVALENT, 0 ms]
        (49) TRUE


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

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

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

Q is empty.

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

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

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   +^1(s(x), y) -> +^1(x, y)
   ++^1(:(x, xs), ys) -> ++^1(xs, ys)
   SUM(:(x, :(y, xs))) -> SUM(:(+(x, y), xs))
   SUM(:(x, :(y, xs))) -> +^1(x, y)

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