/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 48 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 is empty.
The conditional TRS C consists of the following conditional rules:

   f(x, y) -> x <= g(x) -> z, g(y) -> z
   g(x) -> c <= d -> c


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

(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, y) -> U1(g(x), x, y)
   U1(z, x, y) -> U2(g(y), x)
   U2(z, x) -> x
   g(x) -> U3(d)
   U3(c) -> c

Q is empty.

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

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

   POL(U1(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
   POL(U2(x_1, x_2)) = x_1 + x_2
   POL(U3(x_1)) = 2*x_1
   POL(c) = 2
   POL(d) = 0
   POL(f(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(g(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f(x, y) -> U1(g(x), x, y)
   U3(c) -> c




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

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

   U1(z, x, y) -> U2(g(y), x)
   U2(z, x) -> x
   g(x) -> U3(d)

Q is empty.

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

(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:d > g_1 > U3_1 > U1_3 > U2_2

and weight map:

   d=2
   g_1=2
   U3_1=1
   U1_3=1
   U2_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U1(z, x, y) -> U2(g(y), x)
   U2(z, x) -> x
   g(x) -> U3(d)




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

(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