TRS Standard pair #487073208

details

property	value
status	complete
benchmark	thiemann39.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n178.star.cs.uiowa.edu
space	AProVE_07

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	2.97763 seconds
cpu usage	6.01391
user time	5.77584
system time	0.238075
max virtual memory	1.8412052E7
max residence set size	395316.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 20 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) QDPApplicativeOrderProof [EQUIVALENT, 22 ms]
        (18) QDP
        (19) UsableRulesProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) YES


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

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

   a(a(divides, 0), a(s, y)) -> true
   a(a(divides, a(s, x)), a(s, y)) -> a(a(a(div2, x), a(s, y)), y)
   a(a(a(div2, x), y), 0) -> a(a(divides, x), y)
   a(a(a(div2, 0), y), a(s, z)) -> false
   a(a(a(div2, a(s, x)), y), a(s, z)) -> a(a(a(div2, x), y), z)
   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(a(if, true), x), xs) -> a(a(cons, x), xs)
   a(a(a(if, false), x), xs) -> xs
   a(a(not, f), x) -> a(not2, a(f, x))
   a(not2, true) -> false
   a(not2, false) -> true
   a(sieve, nil) -> nil
   a(sieve, a(a(cons, x), xs)) -> a(a(cons, x), a(sieve, a(a(filter, a(not, a(divides, 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(divides, 0), a(s, y)) -> true
   a(a(divides, a(s, x)), a(s, y)) -> a(a(a(div2, x), a(s, y)), y)
   a(a(a(div2, x), y), 0) -> a(a(divides, x), y)
   a(a(a(div2, 0), y), a(s, z)) -> false
   a(a(a(div2, a(s, x)), y), a(s, z)) -> a(a(a(div2, x), y), z)
   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(a(if, true), x), xs) -> a(a(cons, x), xs)
   a(a(a(if, false), x), xs) -> xs
   a(a(not, f), x) -> a(not2, a(f, x))
   a(not2, true) -> false
   a(not2, false) -> true
   a(sieve, nil) -> nil
   a(sieve, a(a(cons, x), xs)) -> a(a(cons, x), a(sieve, a(a(filter, a(not, a(divides, x))), xs)))

The set Q consists of the following terms:

   a(a(divides, 0), a(s, x0))
   a(a(divides, a(s, x0)), a(s, x1))
   a(a(a(div2, x0), x1), 0)
   a(a(a(div2, 0), x0), a(s, x1))
   a(a(a(div2, a(s, x0)), x1), a(s, x2))
   a(a(filter, x0), nil)
   a(a(filter, x0), a(a(cons, x1), x2))
   a(a(a(if, true), x0), x1)
   a(a(a(if, false), x0), x1)
   a(a(not, x0), x1)
   a(not2, true)
   a(not2, false)
   a(sieve, nil)
   a(sieve, a(a(cons, x0), x1))

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