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


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR w ws xs y ys z zs)
(RULES
r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
r(xs,ys,zs,nil) -> xs
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 R(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 R(xs,cons(y,ys),nil,cons(w,ws)) -> R(xs,xs,cons(succ(zero),nil),ws)
 R(xs,nil,zs,cons(w,ws)) -> R(xs,xs,cons(succ(zero),zs),ws)
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs

Problem 1: 

SCC Processor:
-> Pairs:
 R(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 R(xs,cons(y,ys),nil,cons(w,ws)) -> R(xs,xs,cons(succ(zero),nil),ws)
 R(xs,nil,zs,cons(w,ws)) -> R(xs,xs,cons(succ(zero),zs),ws)
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 R(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 R(xs,cons(y,ys),nil,cons(w,ws)) -> R(xs,xs,cons(succ(zero),nil),ws)
 R(xs,nil,zs,cons(w,ws)) -> R(xs,xs,cons(succ(zero),zs),ws)
->->-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs

Problem 1: 

Instantiation Processor:
-> Pairs:
 R(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 R(xs,cons(y,ys),nil,cons(w,ws)) -> R(xs,xs,cons(succ(zero),nil),ws)
 R(xs,nil,zs,cons(w,ws)) -> R(xs,xs,cons(succ(zero),zs),ws)
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->Instantiated Pairs:
->->Original Pair:
 R(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
->-> Instantiated pairs:
 R(cons(y,ys),cons(y,ys),cons(succ(zero),nil),cons(w,ws)) -> R(ys,cons(y,ys),nil,cons(succ(zero),cons(w,ws)))
 R(cons(y,ys),cons(y,ys),cons(succ(zero),zs),cons(w,ws)) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 R(ys,cons(y,ys),cons(z,zs),cons(succ(zero),cons(x7,x8))) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(succ(zero),cons(x7,x8))))
->->Original Pair:
 R(xs,cons(y,ys),nil,cons(w,ws)) -> R(xs,xs,cons(succ(zero),nil),ws)
->-> Instantiated pairs:
 R(ys,cons(y,ys),nil,cons(succ(zero),cons(x7,x8))) -> R(ys,ys,cons(succ(zero),nil),cons(x7,x8))
->->Original Pair:
 R(xs,nil,zs,cons(w,ws)) -> R(xs,xs,cons(succ(zero),zs),ws)
->-> Instantiated pairs:
 R(nil,nil,cons(succ(zero),nil),cons(w,ws)) -> R(nil,nil,cons(succ(zero),cons(succ(zero),nil)),ws)
 R(nil,nil,cons(succ(zero),x16),cons(w,ws)) -> R(nil,nil,cons(succ(zero),cons(succ(zero),x16)),ws)

Problem 1: 

SCC Processor:
-> Pairs:
 R(cons(y,ys),cons(y,ys),cons(succ(zero),nil),cons(w,ws)) -> R(ys,cons(y,ys),nil,cons(succ(zero),cons(w,ws)))
 R(cons(y,ys),cons(y,ys),cons(succ(zero),zs),cons(w,ws)) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 R(nil,nil,cons(succ(zero),nil),cons(w,ws)) -> R(nil,nil,cons(succ(zero),cons(succ(zero),nil)),ws)
 R(nil,nil,cons(succ(zero),x16),cons(w,ws)) -> R(nil,nil,cons(succ(zero),cons(succ(zero),x16)),ws)
 R(ys,cons(y,ys),cons(z,zs),cons(succ(zero),cons(x7,x8))) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(succ(zero),cons(x7,x8))))
 R(ys,cons(y,ys),nil,cons(succ(zero),cons(x7,x8))) -> R(ys,ys,cons(succ(zero),nil),cons(x7,x8))
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 R(nil,nil,cons(succ(zero),x16),cons(w,ws)) -> R(nil,nil,cons(succ(zero),cons(succ(zero),x16)),ws)
->->-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->->Cycle:
->->-> Pairs:
 R(cons(y,ys),cons(y,ys),cons(succ(zero),nil),cons(w,ws)) -> R(ys,cons(y,ys),nil,cons(succ(zero),cons(w,ws)))
 R(cons(y,ys),cons(y,ys),cons(succ(zero),zs),cons(w,ws)) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 R(ys,cons(y,ys),cons(z,zs),cons(succ(zero),cons(x7,x8))) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(succ(zero),cons(x7,x8))))
 R(ys,cons(y,ys),nil,cons(succ(zero),cons(x7,x8))) -> R(ys,ys,cons(succ(zero),nil),cons(x7,x8))
->->-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 R(nil,nil,cons(succ(zero),x16),cons(w,ws)) -> R(nil,nil,cons(succ(zero),cons(succ(zero),x16)),ws)
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->Projection:
 pi(R) = 4

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 R(cons(y,ys),cons(y,ys),cons(succ(zero),nil),cons(w,ws)) -> R(ys,cons(y,ys),nil,cons(succ(zero),cons(w,ws)))
 R(cons(y,ys),cons(y,ys),cons(succ(zero),zs),cons(w,ws)) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 R(ys,cons(y,ys),cons(z,zs),cons(succ(zero),cons(x7,x8))) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(succ(zero),cons(x7,x8))))
 R(ys,cons(y,ys),nil,cons(succ(zero),cons(x7,x8))) -> R(ys,ys,cons(succ(zero),nil),cons(x7,x8))
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->Projection:
 pi(R) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 R(ys,cons(y,ys),cons(z,zs),cons(succ(zero),cons(x7,x8))) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(succ(zero),cons(x7,x8))))
 R(ys,cons(y,ys),nil,cons(succ(zero),cons(x7,x8))) -> R(ys,ys,cons(succ(zero),nil),cons(x7,x8))
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 R(ys,cons(y,ys),cons(z,zs),cons(succ(zero),cons(x7,x8))) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(succ(zero),cons(x7,x8))))
->->-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs

Problem 1.2: 

Subterm Processor:
-> Pairs:
 R(ys,cons(y,ys),cons(z,zs),cons(succ(zero),cons(x7,x8))) -> R(ys,cons(y,ys),zs,cons(succ(zero),cons(succ(zero),cons(x7,x8))))
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->Projection:
 pi(R) = 3

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 r(xs,cons(y,ys),cons(z,zs),cons(w,ws)) -> r(ys,cons(y,ys),zs,cons(succ(zero),cons(w,ws)))
 r(xs,cons(y,ys),nil,cons(w,ws)) -> r(xs,xs,cons(succ(zero),nil),ws)
 r(xs,nil,zs,cons(w,ws)) -> r(xs,xs,cons(succ(zero),zs),ws)
 r(xs,ys,zs,nil) -> xs
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.