TRS Standard pair #487068143

details

property	value
status	complete
benchmark	4.31.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n074.star.cs.uiowa.edu
space	SK90

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	1.75171 seconds
cpu usage	3.82315
user time	3.66156
system time	0.161584
max virtual memory	1.8408912E7
max residence set size	241696.0

stage attributes

key	value
starexec-result	YES

output

YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QDP
(7) UsableRulesProof [EQUIVALENT, 0 ms]
(8) QDP
(9) QReductionProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(12) YES


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

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

   purge(nil) -> nil
   purge(.(x, y)) -> .(x, purge(remove(x, y)))
   remove(x, nil) -> nil
   remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))

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:

   purge(nil) -> nil
   purge(.(x, y)) -> .(x, purge(remove(x, y)))
   remove(x, nil) -> nil
   remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))

The set Q consists of the following terms:

   purge(nil)
   purge(.(x0, x1))
   remove(x0, nil)
   remove(x0, .(x1, x2))


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

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PURGE(.(x, y)) -> PURGE(remove(x, y))
   PURGE(.(x, y)) -> REMOVE(x, y)
   REMOVE(x, .(y, z)) -> REMOVE(x, z)

The TRS R consists of the following rules:

   purge(nil) -> nil
   purge(.(x, y)) -> .(x, purge(remove(x, y)))
   remove(x, nil) -> nil
   remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))

The set Q consists of the following terms:

   purge(nil)
   purge(.(x0, x1))
   remove(x0, nil)
   remove(x0, .(x1, x2))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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