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


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YES

Problem 1: 

(VAR x y z)
(RULES
f(.(.(x,y),z)) -> f(.(x,.(y,z)))
f(.(nil,y)) -> .(nil,f(y))
f(nil) -> nil
g(.(x,.(y,z))) -> g(.(.(x,y),z))
g(.(x,nil)) -> .(g(x),nil)
g(nil) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(.(.(x,y),z)) -> F(.(x,.(y,z)))
 F(.(nil,y)) -> F(y)
 G(.(x,.(y,z))) -> G(.(.(x,y),z))
 G(.(x,nil)) -> G(x)
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 F(.(.(x,y),z)) -> F(.(x,.(y,z)))
 F(.(nil,y)) -> F(y)
 G(.(x,.(y,z))) -> G(.(.(x,y),z))
 G(.(x,nil)) -> G(x)
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(.(x,.(y,z))) -> G(.(.(x,y),z))
 G(.(x,nil)) -> G(x)
->->-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
->->Cycle:
->->-> Pairs:
 F(.(.(x,y),z)) -> F(.(x,.(y,z)))
 F(.(nil,y)) -> F(y)
->->-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 G(.(x,.(y,z))) -> G(.(.(x,y),z))
 G(.(x,nil)) -> G(x)
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[.](X1,X2) = X1 + 2.X2 + 2
[nil] = 0
[G](X) = X

Problem 1.1: 

SCC Processor:
-> Pairs:
 G(.(x,nil)) -> G(x)
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(.(x,nil)) -> G(x)
->->-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil

Problem 1.1: 

Subterm Processor:
-> Pairs:
 G(.(x,nil)) -> G(x)
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
->Projection:
 pi(G) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 F(.(.(x,y),z)) -> F(.(x,.(y,z)))
 F(.(nil,y)) -> F(y)
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[.](X1,X2) = 2.X1 + X2 + 2
[nil] = 0
[F](X) = 2.X

Problem 1.2: 

SCC Processor:
-> Pairs:
 F(.(nil,y)) -> F(y)
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(.(nil,y)) -> F(y)
->->-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil

Problem 1.2: 

Subterm Processor:
-> Pairs:
 F(.(nil,y)) -> F(y)
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
->Projection:
 pi(F) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(.(.(x,y),z)) -> f(.(x,.(y,z)))
 f(.(nil,y)) -> .(nil,f(y))
 f(nil) -> nil
 g(.(x,.(y,z))) -> g(.(.(x,y),z))
 g(.(x,nil)) -> .(g(x),nil)
 g(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.