/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
f(c(s(x),y)) -> g(c(x,y))
f(s(x)) -> f(id_inc(c(x,x)))
g(c(s(x),y)) -> g(c(y,x))
g(c(x,s(y))) -> g(c(y,x))
g(c(x,x)) -> f(x)
id_inc(0) -> 0
id_inc(0) -> s(0)
id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
id_inc(s(x)) -> s(id_inc(x))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(c(s(x),y)) -> G(c(x,y))
 F(s(x)) -> F(id_inc(c(x,x)))
 F(s(x)) -> ID_INC(c(x,x))
 G(c(s(x),y)) -> G(c(y,x))
 G(c(x,s(y))) -> G(c(y,x))
 G(c(x,x)) -> F(x)
 ID_INC(c(x,y)) -> ID_INC(x)
 ID_INC(c(x,y)) -> ID_INC(y)
 ID_INC(s(x)) -> ID_INC(x)
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))

Problem 1: 

SCC Processor:
-> Pairs:
 F(c(s(x),y)) -> G(c(x,y))
 F(s(x)) -> F(id_inc(c(x,x)))
 F(s(x)) -> ID_INC(c(x,x))
 G(c(s(x),y)) -> G(c(y,x))
 G(c(x,s(y))) -> G(c(y,x))
 G(c(x,x)) -> F(x)
 ID_INC(c(x,y)) -> ID_INC(x)
 ID_INC(c(x,y)) -> ID_INC(y)
 ID_INC(s(x)) -> ID_INC(x)
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ID_INC(c(x,y)) -> ID_INC(x)
 ID_INC(c(x,y)) -> ID_INC(y)
 ID_INC(s(x)) -> ID_INC(x)
->->-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->->Cycle:
->->-> Pairs:
 F(c(s(x),y)) -> G(c(x,y))
 F(s(x)) -> F(id_inc(c(x,x)))
 G(c(s(x),y)) -> G(c(y,x))
 G(c(x,s(y))) -> G(c(y,x))
 G(c(x,x)) -> F(x)
->->-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ID_INC(c(x,y)) -> ID_INC(x)
 ID_INC(c(x,y)) -> ID_INC(y)
 ID_INC(s(x)) -> ID_INC(x)
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->Projection:
 pi(ID_INC) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 F(c(s(x),y)) -> G(c(x,y))
 F(s(x)) -> F(id_inc(c(x,x)))
 G(c(s(x),y)) -> G(c(y,x))
 G(c(x,s(y))) -> G(c(y,x))
 G(c(x,x)) -> F(x)
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
-> Usable rules:
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[id_inc](X) = X + 2
[0] = 2
[c](X1,X2) = 1/2.X1 + 1/2.X2
[s](X) = X + 2
[F](X) = X
[G](X) = X

Problem 1.2: 

SCC Processor:
-> Pairs:
 F(s(x)) -> F(id_inc(c(x,x)))
 G(c(s(x),y)) -> G(c(y,x))
 G(c(x,s(y))) -> G(c(y,x))
 G(c(x,x)) -> F(x)
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(x)) -> F(id_inc(c(x,x)))
->->-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->->Cycle:
->->-> Pairs:
 G(c(s(x),y)) -> G(c(y,x))
 G(c(x,s(y))) -> G(c(y,x))
->->-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))


The problem is decomposed in 2 subproblems.

Problem 1.2.1: 

Reduction Pair Processor:
-> Pairs:
 F(s(x)) -> F(id_inc(c(x,x)))
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
-> Usable rules:
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[id_inc](X) = X
[0] = 2
[c](X1,X2) = 1
[s](X) = 2
[F](X) = X

Problem 1.2.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2.2: 

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

Problem 1.2.2: 

SCC Processor:
-> Pairs:
 G(c(x,s(y))) -> G(c(y,x))
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(c(x,s(y))) -> G(c(y,x))
->->-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))

Problem 1.2.2: 

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

Problem 1.2.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(c(s(x),y)) -> g(c(x,y))
 f(s(x)) -> f(id_inc(c(x,x)))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(0) -> 0
 id_inc(0) -> s(0)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.