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


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YES

Problem 1: 

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

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F_1(x) -> G_1(x,x)
 F_10(x) -> G_10(x,x)
 F_2(x) -> G_2(x,x)
 F_3(x) -> G_3(x,x)
 F_4(x) -> G_4(x,x)
 F_5(x) -> G_5(x,x)
 F_6(x) -> G_6(x,x)
 F_7(x) -> G_7(x,x)
 F_8(x) -> G_8(x,x)
 F_9(x) -> G_9(x,x)
 G_1(s(x),y) -> F_0(y)
 G_1(s(x),y) -> G_1(x,y)
 G_10(s(x),y) -> F_9(y)
 G_10(s(x),y) -> G_10(x,y)
 G_2(s(x),y) -> F_1(y)
 G_2(s(x),y) -> G_2(x,y)
 G_3(s(x),y) -> F_2(y)
 G_3(s(x),y) -> G_3(x,y)
 G_4(s(x),y) -> F_3(y)
 G_4(s(x),y) -> G_4(x,y)
 G_5(s(x),y) -> F_4(y)
 G_5(s(x),y) -> G_5(x,y)
 G_6(s(x),y) -> F_5(y)
 G_6(s(x),y) -> G_6(x,y)
 G_7(s(x),y) -> F_6(y)
 G_7(s(x),y) -> G_7(x,y)
 G_8(s(x),y) -> F_7(y)
 G_8(s(x),y) -> G_8(x,y)
 G_9(s(x),y) -> F_8(y)
 G_9(s(x),y) -> G_9(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))

Problem 1: 

SCC Processor:
-> Pairs:
 F_1(x) -> G_1(x,x)
 F_10(x) -> G_10(x,x)
 F_2(x) -> G_2(x,x)
 F_3(x) -> G_3(x,x)
 F_4(x) -> G_4(x,x)
 F_5(x) -> G_5(x,x)
 F_6(x) -> G_6(x,x)
 F_7(x) -> G_7(x,x)
 F_8(x) -> G_8(x,x)
 F_9(x) -> G_9(x,x)
 G_1(s(x),y) -> F_0(y)
 G_1(s(x),y) -> G_1(x,y)
 G_10(s(x),y) -> F_9(y)
 G_10(s(x),y) -> G_10(x,y)
 G_2(s(x),y) -> F_1(y)
 G_2(s(x),y) -> G_2(x,y)
 G_3(s(x),y) -> F_2(y)
 G_3(s(x),y) -> G_3(x,y)
 G_4(s(x),y) -> F_3(y)
 G_4(s(x),y) -> G_4(x,y)
 G_5(s(x),y) -> F_4(y)
 G_5(s(x),y) -> G_5(x,y)
 G_6(s(x),y) -> F_5(y)
 G_6(s(x),y) -> G_6(x,y)
 G_7(s(x),y) -> F_6(y)
 G_7(s(x),y) -> G_7(x,y)
 G_8(s(x),y) -> F_7(y)
 G_8(s(x),y) -> G_8(x,y)
 G_9(s(x),y) -> F_8(y)
 G_9(s(x),y) -> G_9(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G_1(s(x),y) -> G_1(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->->Cycle:
->->-> Pairs:
 G_2(s(x),y) -> G_2(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->->Cycle:
->->-> Pairs:
 G_3(s(x),y) -> G_3(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->->Cycle:
->->-> Pairs:
 G_4(s(x),y) -> G_4(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->->Cycle:
->->-> Pairs:
 G_5(s(x),y) -> G_5(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->->Cycle:
->->-> Pairs:
 G_6(s(x),y) -> G_6(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->->Cycle:
->->-> Pairs:
 G_7(s(x),y) -> G_7(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->->Cycle:
->->-> Pairs:
 G_8(s(x),y) -> G_8(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->->Cycle:
->->-> Pairs:
 G_9(s(x),y) -> G_9(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->->Cycle:
->->-> Pairs:
 G_10(s(x),y) -> G_10(x,y)
->->-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))


The problem is decomposed in 10 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 G_1(s(x),y) -> G_1(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_1) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 G_2(s(x),y) -> G_2(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_2) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 G_3(s(x),y) -> G_3(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_3) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 G_4(s(x),y) -> G_4(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_4) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 G_5(s(x),y) -> G_5(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_5) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Subterm Processor:
-> Pairs:
 G_6(s(x),y) -> G_6(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_6) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Subterm Processor:
-> Pairs:
 G_7(s(x),y) -> G_7(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_7) = 1

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.8: 

Subterm Processor:
-> Pairs:
 G_8(s(x),y) -> G_8(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_8) = 1

Problem 1.8: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.9: 

Subterm Processor:
-> Pairs:
 G_9(s(x),y) -> G_9(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_9) = 1

Problem 1.9: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.10: 

Subterm Processor:
-> Pairs:
 G_10(s(x),y) -> G_10(x,y)
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Projection:
 pi(G_10) = 1

Problem 1.10: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f_0(x) -> a
 f_1(x) -> g_1(x,x)
 f_10(x) -> g_10(x,x)
 f_2(x) -> g_2(x,x)
 f_3(x) -> g_3(x,x)
 f_4(x) -> g_4(x,x)
 f_5(x) -> g_5(x,x)
 f_6(x) -> g_6(x,x)
 f_7(x) -> g_7(x,x)
 f_8(x) -> g_8(x,x)
 f_9(x) -> g_9(x,x)
 g_1(s(x),y) -> b(f_0(y),g_1(x,y))
 g_10(s(x),y) -> b(f_9(y),g_10(x,y))
 g_2(s(x),y) -> b(f_1(y),g_2(x,y))
 g_3(s(x),y) -> b(f_2(y),g_3(x,y))
 g_4(s(x),y) -> b(f_3(y),g_4(x,y))
 g_5(s(x),y) -> b(f_4(y),g_5(x,y))
 g_6(s(x),y) -> b(f_5(y),g_6(x,y))
 g_7(s(x),y) -> b(f_6(y),g_7(x,y))
 g_8(s(x),y) -> b(f_7(y),g_8(x,y))
 g_9(s(x),y) -> b(f_8(y),g_9(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.