TRS Conditional pair #487562884

details

property	value
status	complete
benchmark	329.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n145.star.cs.uiowa.edu
space	COPS

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	2.24053096771 seconds
cpu usage	5.163485846
max memory	3.43769088E8

stage attributes

key	value
output-size	20825
starexec-result	YES

output

/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) Overlay + Local Confluence [EQUIVALENT, 0 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 14 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) UsableRulesProof [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) QReductionProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) QDP
        (24) UsableRulesProof [EQUIVALENT, 0 ms]
        (25) QDP
        (26) QReductionProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (29) YES
    (30) QDP
        (31) UsableRulesProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) QReductionProof [EQUIVALENT, 0 ms]
        (34) QDP
        (35) QDPOrderProof [EQUIVALENT, 41 ms]
        (36) QDP
        (37) PisEmptyProof [EQUIVALENT, 0 ms]
        (38) YES


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

(0)
Obligation:
Conditional term rewrite system:
The TRS R consists of the following rules:

   le(0, x) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, x) -> x
   app(cons(x, xs), ys) -> cons(x, app(xs, ys))
   split(x, nil) -> pair(nil, nil)
   qsort(nil) -> nil

The conditional TRS C consists of the following conditional rules:

   split(x, cons(y, ys)) -> pair(xs, cons(y, zs)) <= split(x, ys) -> pair(xs, zs), le(x, y) -> true
   split(x, cons(y, ys)) -> pair(cons(y, xs), zs) <= split(x, ys) -> pair(xs, zs), le(x, y) -> false
   qsort(cons(x, xs)) -> app(qsort(ys), cons(x, qsort(zs))) <= split(x, xs) -> pair(ys, zs)


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

(1) CTRSToQTRSProof (SOUND)
The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
----------------------------------------

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

   split(x, cons(y, ys)) -> U1(split(x, ys), x, y)
   U1(pair(xs, zs), x, y) -> U2(le(x, y), y, xs, zs)
   U2(false, y, xs, zs) -> pair(cons(y, xs), zs)
   U2(true, y, xs, zs) -> pair(xs, cons(y, zs))
   qsort(cons(x, xs)) -> U3(split(x, xs), x)
   U3(pair(ys, zs), x) -> app(qsort(ys), cons(x, qsort(zs)))
   le(0, x) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, x) -> x
   app(cons(x, xs), ys) -> cons(x, app(xs, ys))
   split(x, nil) -> pair(nil, nil)
   qsort(nil) -> nil

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