/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 x1)
(RULES
d(0(x1)) -> 0(x1)
d(s(x1)) -> s(s(d(x1)))
f(0(x1)) -> s(0(x1))
f(s(x1)) -> d(f(x1))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 d(0(x1)) -> 0(x1)
 d(s(x1)) -> s(s(d(x1)))
 f(0(x1)) -> s(0(x1))
 f(s(x1)) -> d(f(x1))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 D(s(x1)) -> D(x1)
 F(s(x1)) -> D(f(x1))
 F(s(x1)) -> F(x1)
-> Rules:
 d(0(x1)) -> 0(x1)
 d(s(x1)) -> s(s(d(x1)))
 f(0(x1)) -> s(0(x1))
 f(s(x1)) -> d(f(x1))

Problem 1: 

SCC Processor:
-> Pairs:
 D(s(x1)) -> D(x1)
 F(s(x1)) -> D(f(x1))
 F(s(x1)) -> F(x1)
-> Rules:
 d(0(x1)) -> 0(x1)
 d(s(x1)) -> s(s(d(x1)))
 f(0(x1)) -> s(0(x1))
 f(s(x1)) -> d(f(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 D(s(x1)) -> D(x1)
->->-> Rules:
 d(0(x1)) -> 0(x1)
 d(s(x1)) -> s(s(d(x1)))
 f(0(x1)) -> s(0(x1))
 f(s(x1)) -> d(f(x1))
->->Cycle:
->->-> Pairs:
 F(s(x1)) -> F(x1)
->->-> Rules:
 d(0(x1)) -> 0(x1)
 d(s(x1)) -> s(s(d(x1)))
 f(0(x1)) -> s(0(x1))
 f(s(x1)) -> d(f(x1))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 D(s(x1)) -> D(x1)
-> Rules:
 d(0(x1)) -> 0(x1)
 d(s(x1)) -> s(s(d(x1)))
 f(0(x1)) -> s(0(x1))
 f(s(x1)) -> d(f(x1))
->Projection:
 pi(D) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 d(0(x1)) -> 0(x1)
 d(s(x1)) -> s(s(d(x1)))
 f(0(x1)) -> s(0(x1))
 f(s(x1)) -> d(f(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 F(s(x1)) -> F(x1)
-> Rules:
 d(0(x1)) -> 0(x1)
 d(s(x1)) -> s(s(d(x1)))
 f(0(x1)) -> s(0(x1))
 f(s(x1)) -> d(f(x1))
->Projection:
 pi(F) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 d(0(x1)) -> 0(x1)
 d(s(x1)) -> s(s(d(x1)))
 f(0(x1)) -> s(0(x1))
 f(s(x1)) -> d(f(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.