TRS Standard pair #615303743

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	n119.star.cs.uiowa.edu
space	AProVE_07

run statistics

property	value
solver	AProVE23
configuration	standard
runtime (wallclock)	2.41624808311 seconds
cpu usage	6.848387265
max memory	3.80674048E8

stage attributes

key	value
output-size	13898
starexec-result	"YES"

output

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


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


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 363435ff5b19173ec8a83b45cfdec073678dabcc jan-christoph 20230815 unpublished


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, 4 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, 0 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, 81 ms]
        (22) QDP
        (23) DependencyGraphProof [EQUIVALENT, 0 ms]
        (24) TRUE


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

(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.

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

(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)

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