/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 55 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 2 ms]
(10) QTRS
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

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

   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   c -> k
   d -> k
   h(x) -> i(x, x)

The conditional TRS C consists of the following conditional rules:

   f(x) -> x <= x -> e
   g(x, x) -> C <= A -> B


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

(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(x, x) -> U2(A)
   U2(B) -> C
   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   c -> k
   d -> k
   h(x) -> i(x, x)

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(A) = 1
   POL(B) = 1
   POL(C) = 2
   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
   POL(U2(x_1)) = 2*x_1
   POL(a) = 1
   POL(b) = 1
   POL(c) = 1
   POL(d) = 1
   POL(e) = 0
   POL(f(x_1)) = 2 + 2*x_1
   POL(g(x_1, x_2)) = 2 + x_1 + x_2
   POL(h(x_1)) = 1 + 2*x_1
   POL(i(x_1, x_2)) = 1 + x_1 + x_2
   POL(k) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U1(e, x) -> x
   c -> e
   c -> k
   d -> k




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

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

   f(x) -> U1(x, x)
   g(x, x) -> U2(A)
   U2(B) -> C
   a -> c
   a -> d
   b -> c
   b -> d
   h(x) -> i(x, x)

Q is empty.

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

(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(A) = 1
   POL(B) = 1
   POL(C) = 2
   POL(U1(x_1, x_2)) = 1 + x_1 + x_2
   POL(U2(x_1)) = 2*x_1
   POL(a) = 1
   POL(b) = 1
   POL(c) = 0
   POL(d) = 0
   POL(f(x_1)) = 1 + 2*x_1
   POL(g(x_1, x_2)) = 2 + x_1 + x_2
   POL(h(x_1)) = 2 + 2*x_1
   POL(i(x_1, x_2)) = 2 + x_1 + x_2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a -> c
   a -> d
   b -> c
   b -> d




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

(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(x) -> U1(x, x)
   g(x, x) -> U2(A)
   U2(B) -> C
   h(x) -> i(x, x)

Q is empty.

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

(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(A) = 0
   POL(B) = 2
   POL(C) = 2
   POL(U1(x_1, x_2)) = 1 + x_1 + x_2
   POL(U2(x_1)) = 1 + x_1
   POL(f(x_1)) = 1 + 2*x_1
   POL(g(x_1, x_2)) = 2 + x_1 + x_2
   POL(h(x_1)) = 1 + 2*x_1
   POL(i(x_1, x_2)) = 1 + x_1 + x_2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   g(x, x) -> U2(A)
   U2(B) -> C




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

(8)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(x) -> U1(x, x)
   h(x) -> i(x, x)

Q is empty.

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

(9) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
f_1 > U1_2 > i_2
h_1 > i_2


Status:
f_1: multiset status
U1_2: multiset status
h_1: multiset status
i_2: 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)
   h(x) -> i(x, x)




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

(10)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

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

(12)
YES