TRS Standard pair #615296519

details

property	value
status	complete
benchmark	id_inc.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n051.star.cs.uiowa.edu
space	AProVE_08

run statistics

property	value
solver	AProVE23
configuration	standard
runtime (wallclock)	2.22597885132 seconds
cpu usage	6.194340729
max memory	3.23248128E8

stage attributes

key	value
output-size	9431
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# AProVE Commit ID: 363435ff5b19173ec8a83b45cfdec073678dabcc jan-christoph 20230815 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) UsableRulesReductionPairsProof [EQUIVALENT, 22 ms]
        (12) QDP
        (13) QDPOrderProof [EQUIVALENT, 56 ms]
        (14) QDP
        (15) DependencyGraphProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) PisEmptyProof [EQUIVALENT, 0 ms]
        (20) YES


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

   f(s(x)) -> f(id_inc(c(x, x)))
   f(c(s(x), y)) -> g(c(x, y))
   g(c(s(x), y)) -> g(c(y, x))
   g(c(x, s(y))) -> g(c(y, x))
   g(c(x, x)) -> f(x)
   id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
   id_inc(s(x)) -> s(id_inc(x))
   id_inc(0) -> 0
   id_inc(0) -> s(0)

Q is empty.

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(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(x)) -> F(id_inc(c(x, x)))
   F(s(x)) -> ID_INC(c(x, x))
   F(c(s(x), y)) -> G(c(x, y))
   G(c(s(x), y)) -> G(c(y, x))
   G(c(x, s(y))) -> G(c(y, x))
   G(c(x, x)) -> F(x)
   ID_INC(c(x, y)) -> ID_INC(x)
   ID_INC(c(x, y)) -> ID_INC(y)
   ID_INC(s(x)) -> ID_INC(x)

The TRS R consists of the following rules:

   f(s(x)) -> f(id_inc(c(x, x)))
   f(c(s(x), y)) -> g(c(x, y))
   g(c(s(x), y)) -> g(c(y, x))
   g(c(x, s(y))) -> g(c(y, x))
   g(c(x, x)) -> f(x)
   id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
   id_inc(s(x)) -> s(id_inc(x))
   id_inc(0) -> 0
   id_inc(0) -> s(0)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
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(4)
Complex Obligation (AND)

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