/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 70 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

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

   f(x, y) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)
   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)
   h(x, x) -> x

Q is empty.

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

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

   POL(f(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(g1(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(g2(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(h(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:

   h(x, x) -> x




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

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(x, y) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)
   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)

Q is empty.

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

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

   POL(f(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(g1(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(g2(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(h(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:

   f(x, y) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)




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

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

   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)

Q is empty.

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

(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:g2_3 > g1_3 > h_2

and weight map:

   g1_3=0
   h_2=1
   g2_3=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:

   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)




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

(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