TRS Conditional pair #516978543

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	n061.star.cs.uiowa.edu
space	COPS

run statistics

property	value
solver	AProVE21
configuration	standard
runtime (wallclock)	2.52087903023 seconds
cpu usage	6.73270092
max memory	4.88787968E8

stage attributes

key	value
output-size	20820
starexec-result	YES

output

/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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, 0 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, 9 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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