/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)
(RULES
+(X,0) -> X
+(X,s(Y)) -> s(+(X,Y))
double(X) -> +(X,X)
f(0,s(0),X) -> f(X,double(X),X)
g(X,Y) -> X
g(X,Y) -> Y
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(X,s(Y)) -> +#(X,Y)
 DOUBLE(X) -> +#(X,X)
 F(0,s(0),X) -> DOUBLE(X)
 F(0,s(0),X) -> F(X,double(X),X)
-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y

Problem 1: 

SCC Processor:
-> Pairs:
 +#(X,s(Y)) -> +#(X,Y)
 DOUBLE(X) -> +#(X,X)
 F(0,s(0),X) -> DOUBLE(X)
 F(0,s(0),X) -> F(X,double(X),X)
-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(X,s(Y)) -> +#(X,Y)
->->-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y
->->Cycle:
->->-> Pairs:
 F(0,s(0),X) -> F(X,double(X),X)
->->-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 +#(X,s(Y)) -> +#(X,Y)
-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y
->Projection:
 pi(+#) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Instantiation Processor:
-> Pairs:
 F(0,s(0),X) -> F(X,double(X),X)
-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y
->Instantiated Pairs:
->->Original Pair:
 F(0,s(0),X) -> F(X,double(X),X)
->-> Instantiated pairs:
 F(0,s(0),0) -> F(0,double(0),0)

Problem 1.2: 

SCC Processor:
-> Pairs:
 F(0,s(0),0) -> F(0,double(0),0)
-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(0,s(0),0) -> F(0,double(0),0)
->->-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 F(0,s(0),0) -> F(0,double(0),0)
-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y
-> Usable rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[double](X) = 2.X
[0] = 1
[s](X) = X + 2
[F](X1,X2,X3) = 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(X,0) -> X
 +(X,s(Y)) -> s(+(X,Y))
 double(X) -> +(X,X)
 f(0,s(0),X) -> f(X,double(X),X)
 g(X,Y) -> X
 g(X,Y) -> Y
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.