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


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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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 153 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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

   f_0(x) -> a
   f_1(x) -> g_1(x, x)
   g_1(s(x), y) -> b(f_0(y), g_1(x, y))
   f_2(x) -> g_2(x, x)
   g_2(s(x), y) -> b(f_1(y), g_2(x, y))
   f_3(x) -> g_3(x, x)
   g_3(s(x), y) -> b(f_2(y), g_3(x, y))
   f_4(x) -> g_4(x, x)
   g_4(s(x), y) -> b(f_3(y), g_4(x, y))
   f_5(x) -> g_5(x, x)
   g_5(s(x), y) -> b(f_4(y), g_5(x, y))
   f_6(x) -> g_6(x, x)
   g_6(s(x), y) -> b(f_5(y), g_6(x, y))
   f_7(x) -> g_7(x, x)
   g_7(s(x), y) -> b(f_6(y), g_7(x, y))
   f_8(x) -> g_8(x, x)
   g_8(s(x), y) -> b(f_7(y), g_8(x, y))
   f_9(x) -> g_9(x, x)
   g_9(s(x), y) -> b(f_8(y), g_9(x, y))
   f_10(x) -> g_10(x, x)
   g_10(s(x), y) -> b(f_9(y), g_10(x, y))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
s_1 > [f_3_1, g_4_2] > [f_2_1, g_3_2] > [f_1_1, g_2_2] > g_1_2 > [f_0_1, a]
s_1 > [f_3_1, g_4_2] > [f_2_1, g_3_2] > [f_1_1, g_2_2] > g_1_2 > b_2
f_10_1 > [f_9_1, g_10_2] > g_9_2 > f_8_1 > [f_7_1, g_8_2] > [f_6_1, g_7_2] > [f_5_1, g_6_2] > [f_4_1, g_5_2] > [f_3_1, g_4_2] > [f_2_1, g_3_2] > [f_1_1, g_2_2] > g_1_2 > [f_0_1, a]
f_10_1 > [f_9_1, g_10_2] > g_9_2 > f_8_1 > [f_7_1, g_8_2] > [f_6_1, g_7_2] > [f_5_1, g_6_2] > [f_4_1, g_5_2] > [f_3_1, g_4_2] > [f_2_1, g_3_2] > [f_1_1, g_2_2] > g_1_2 > b_2


Status:
f_0_1: [1]
a: multiset status
f_1_1: multiset status
g_1_2: multiset status
s_1: multiset status
b_2: multiset status
f_2_1: multiset status
g_2_2: multiset status
f_3_1: multiset status
g_3_2: multiset status
f_4_1: multiset status
g_4_2: multiset status
f_5_1: multiset status
g_5_2: multiset status
f_6_1: multiset status
g_6_2: multiset status
f_7_1: multiset status
g_7_2: multiset status
f_8_1: [1]
g_8_2: multiset status
f_9_1: multiset status
g_9_2: multiset status
f_10_1: [1]
g_10_2: multiset status

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

   f_0(x) -> a
   f_1(x) -> g_1(x, x)
   g_1(s(x), y) -> b(f_0(y), g_1(x, y))
   f_2(x) -> g_2(x, x)
   g_2(s(x), y) -> b(f_1(y), g_2(x, y))
   f_3(x) -> g_3(x, x)
   g_3(s(x), y) -> b(f_2(y), g_3(x, y))
   f_4(x) -> g_4(x, x)
   g_4(s(x), y) -> b(f_3(y), g_4(x, y))
   f_5(x) -> g_5(x, x)
   g_5(s(x), y) -> b(f_4(y), g_5(x, y))
   f_6(x) -> g_6(x, x)
   g_6(s(x), y) -> b(f_5(y), g_6(x, y))
   f_7(x) -> g_7(x, x)
   g_7(s(x), y) -> b(f_6(y), g_7(x, y))
   f_8(x) -> g_8(x, x)
   g_8(s(x), y) -> b(f_7(y), g_8(x, y))
   f_9(x) -> g_9(x, x)
   g_9(s(x), y) -> b(f_8(y), g_9(x, y))
   f_10(x) -> g_10(x, x)
   g_10(s(x), y) -> b(f_9(y), g_10(x, y))




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES