/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
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
double(0) -> 0
double(s(x)) -> s(s(double(x)))
sqr(0) -> 0
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(x,s(y)) -> +#(x,y)
 DOUBLE(s(x)) -> DOUBLE(x)
 SQR(s(x)) -> +#(sqr(x),double(x))
 SQR(s(x)) -> +#(sqr(x),s(double(x)))
 SQR(s(x)) -> DOUBLE(x)
 SQR(s(x)) -> SQR(x)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))

Problem 1: 

SCC Processor:
-> Pairs:
 +#(x,s(y)) -> +#(x,y)
 DOUBLE(s(x)) -> DOUBLE(x)
 SQR(s(x)) -> +#(sqr(x),double(x))
 SQR(s(x)) -> +#(sqr(x),s(double(x)))
 SQR(s(x)) -> DOUBLE(x)
 SQR(s(x)) -> SQR(x)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 DOUBLE(s(x)) -> DOUBLE(x)
->->-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))
->->Cycle:
->->-> Pairs:
 +#(x,s(y)) -> +#(x,y)
->->-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))
->->Cycle:
->->-> Pairs:
 SQR(s(x)) -> SQR(x)
->->-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 DOUBLE(s(x)) -> DOUBLE(x)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))
->Projection:
 pi(DOUBLE) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 +#(x,s(y)) -> +#(x,y)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))
->Projection:
 pi(+#) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 SQR(s(x)) -> SQR(x)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))
->Projection:
 pi(SQR) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 sqr(0) -> 0
 sqr(s(x)) -> +(sqr(x),s(double(x)))
 sqr(s(x)) -> s(+(sqr(x),double(x)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.