TRS Standard pair #487073083

details

property	value
status	complete
benchmark	thiemann17.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n175.star.cs.uiowa.edu
space	AProVE_07

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	2.71394 seconds
cpu usage	12.1733
user time	11.7086
system time	0.464682
max virtual memory	1.8609852E7
max residence set size	376624.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QReductionProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms]
        (13) QDP
        (14) PisEmptyProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) UsableRulesProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) QReductionProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QDPOrderProof [EQUIVALENT, 92 ms]
        (22) QDP
        (23) DependencyGraphProof [EQUIVALENT, 0 ms]
        (24) TRUE


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

   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
   sum(cons(0, x), y) -> sum(x, y)
   sum(nil, y) -> y
   empty(nil) -> true
   empty(cons(n, x)) -> false
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   head(cons(n, x)) -> n
   weight(x) -> if(empty(x), empty(tail(x)), x)
   if(true, b, x) -> weight_undefined_error
   if(false, b, x) -> if2(b, x)
   if2(true, x) -> head(x)
   if2(false, x) -> weight(sum(x, cons(0, tail(tail(x)))))

Q is empty.

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(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:

   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
   sum(cons(0, x), y) -> sum(x, y)
   sum(nil, y) -> y
   empty(nil) -> true
   empty(cons(n, x)) -> false
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   head(cons(n, x)) -> n
   weight(x) -> if(empty(x), empty(tail(x)), x)
   if(true, b, x) -> weight_undefined_error
   if(false, b, x) -> if2(b, x)
   if2(true, x) -> head(x)
   if2(false, x) -> weight(sum(x, cons(0, tail(tail(x)))))

The set Q consists of the following terms:

   sum(cons(s(x0), x1), cons(x2, x3))
   sum(cons(0, x0), x1)
   sum(nil, x0)
   empty(nil)
   empty(cons(x0, x1))
   tail(nil)
   tail(cons(x0, x1))
   head(cons(x0, x1))
   weight(x0)
   if(true, x0, x1)
   if(false, x0, x1)
   if2(true, x0)
   if2(false, x0)


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