/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
double(x) -> g(d,x)
f(s(0),y) -> y
f(s(x),y) -> f(half(s(x)),double(y))
g(d,s(x)) -> s(s(g(d,x)))
g(h,s(0)) -> 0
g(h,s(s(x))) -> s(g(h,x))
g(x,0) -> 0
half(x) -> g(h,x)
id(x) -> f(x,s(0))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 DOUBLE(x) -> G(d,x)
 F(s(x),y) -> DOUBLE(y)
 F(s(x),y) -> F(half(s(x)),double(y))
 F(s(x),y) -> HALF(s(x))
 G(d,s(x)) -> G(d,x)
 G(h,s(s(x))) -> G(h,x)
 HALF(x) -> G(h,x)
 ID(x) -> F(x,s(0))
-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))

Problem 1: 

SCC Processor:
-> Pairs:
 DOUBLE(x) -> G(d,x)
 F(s(x),y) -> DOUBLE(y)
 F(s(x),y) -> F(half(s(x)),double(y))
 F(s(x),y) -> HALF(s(x))
 G(d,s(x)) -> G(d,x)
 G(h,s(s(x))) -> G(h,x)
 HALF(x) -> G(h,x)
 ID(x) -> F(x,s(0))
-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(h,s(s(x))) -> G(h,x)
->->-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))
->->Cycle:
->->-> Pairs:
 G(d,s(x)) -> G(d,x)
->->-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))
->->Cycle:
->->-> Pairs:
 F(s(x),y) -> F(half(s(x)),double(y))
->->-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 G(h,s(s(x))) -> G(h,x)
-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))
->Projection:
 pi(G) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 G(d,s(x)) -> G(d,x)
-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))
->Projection:
 pi(G) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 F(s(x),y) -> F(half(s(x)),double(y))
-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))
-> Usable rules:
 double(x) -> g(d,x)
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[double](X) = 2.X
[g](X1,X2) = X1.X2
[half](X) = 1/2.X
[0] = 0
[d] = 2
[h] = 1/2
[s](X) = X + 1/2
[F](X1,X2) = X1.X2 + 2.X1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(x) -> g(d,x)
 f(s(0),y) -> y
 f(s(x),y) -> f(half(s(x)),double(y))
 g(d,s(x)) -> s(s(g(d,x)))
 g(h,s(0)) -> 0
 g(h,s(s(x))) -> s(g(h,x))
 g(x,0) -> 0
 half(x) -> g(h,x)
 id(x) -> f(x,s(0))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.