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


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 33 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(j(x, y), y) -> g(f(x, k(y)))
   f(x, h1(y, z)) -> h2(0, x, h1(y, z))
   g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
   i(f(x, h(y))) -> y
   i(h2(s(x), y, h1(x, z))) -> z
   k(h(x)) -> h1(0, x)
   k(h1(x, y)) -> h1(s(x), y)

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:h_1 > i_1 > s_1 > 0 > f_2 > g_1 > j_2 > h2_3 > k_1 > h1_2

and weight map:

   0=5
   g_1=4
   k_1=4
   s_1=4
   i_1=1
   h_1=1
   j_2=7
   f_2=5
   h2_3=0
   h1_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:

   f(j(x, y), y) -> g(f(x, k(y)))
   f(x, h1(y, z)) -> h2(0, x, h1(y, z))
   g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
   i(f(x, h(y))) -> y
   i(h2(s(x), y, h1(x, z))) -> z
   k(h(x)) -> h1(0, x)
   k(h1(x, y)) -> h1(s(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