/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 79 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

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

   g(d, x, x) -> A
   h(x, x) -> g(x, x, f(k))
   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   c -> l
   k -> l
   k -> m
   d -> m

The conditional TRS C consists of the following conditional rules:

   f(x) -> x <= x -> e


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

(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(x) -> U1(x, x)
   U1(e, x) -> x
   g(d, x, x) -> A
   h(x, x) -> g(x, x, f(k))
   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   c -> l
   k -> l
   k -> m
   d -> m

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
a > c > e
a > c > [h_2, k, l] > f_1 > U1_2
a > c > [h_2, k, l] > g_3 > A
a > c > [h_2, k, l] > [d, m]
b > c > e
b > c > [h_2, k, l] > f_1 > U1_2
b > c > [h_2, k, l] > g_3 > A
b > c > [h_2, k, l] > [d, m]


Status:
f_1: multiset status
U1_2: [1,2]
e: multiset status
g_3: [2,1,3]
d: multiset status
A: multiset status
h_2: multiset status
k: multiset status
a: multiset status
c: multiset status
b: multiset status
l: multiset status
m: multiset status

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

   f(x) -> U1(x, x)
   U1(e, x) -> x
   g(d, x, x) -> A
   h(x, x) -> g(x, x, f(k))
   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   c -> l
   k -> m




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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   k -> l
   d -> m

Q is empty.

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

(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
k > l
d > m


Status:
k: multiset status
l: multiset status
d: multiset status
m: multiset status

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

   k -> l
   d -> m




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

(6)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

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

(8)
YES