/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 80 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

(0)
Obligation:
Conditional term rewrite system:
The TRS R consists of the following rules:

   g(s(x)) -> g(x)

The conditional TRS C consists of the following conditional rules:

   f(g(x)) -> x <= x -> s(0)


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

(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:

   f(g(x)) -> U1(x, x)
   U1(s(0), x) -> x
   g(s(x)) -> g(x)

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
f/1(YES)
g/1)YES(
U1/2(YES,YES)
s/1(YES)
0/0)

Quasi precedence:
f_1 > U1_2
s_1 > U1_2
0 > U1_2


Status:
f_1: multiset status
U1_2: multiset status
s_1: [1]
0: multiset status

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f(g(x)) -> U1(x, x)
   U1(s(0), x) -> x
   g(s(x)) -> g(x)




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

(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(6)
YES