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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(x:S) -> g(d,x:S)
 f(s(0),y:S) -> y:S
 f(s(x:S),y:S) -> f(half(s(x:S)),double(y:S))
 g(d,s(x:S)) -> s(s(g(d,x:S)))
 g(h,s(0)) -> 0
 g(h,s(s(x:S))) -> s(g(h,x:S))
 g(x:S,0) -> 0
 half(x:S) -> g(h,x:S)
 id(x:S) -> f(x:S,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:S),y:S) -> F(half(s(x:S)),double(y:S))
-> Rules:
 double(x:S) -> g(d,x:S)
 f(s(0),y:S) -> y:S
 f(s(x:S),y:S) -> f(half(s(x:S)),double(y:S))
 g(d,s(x:S)) -> s(s(g(d,x:S)))
 g(h,s(0)) -> 0
 g(h,s(s(x:S))) -> s(g(h,x:S))
 g(x:S,0) -> 0
 half(x:S) -> g(h,x:S)
 id(x:S) -> f(x:S,s(0))
-> Usable rules:
 double(x:S) -> g(d,x:S)
 g(d,s(x:S)) -> s(s(g(d,x:S)))
 g(h,s(0)) -> 0
 g(h,s(s(x:S))) -> s(g(h,x:S))
 g(x:S,0) -> 0
 half(x:S) -> g(h,x:S)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[double](X) = 2.X + 1
[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:S) -> g(d,x:S)
 f(s(0),y:S) -> y:S
 f(s(x:S),y:S) -> f(half(s(x:S)),double(y:S))
 g(d,s(x:S)) -> s(s(g(d,x:S)))
 g(h,s(0)) -> 0
 g(h,s(s(x:S))) -> s(g(h,x:S))
 g(x:S,0) -> 0
 half(x:S) -> g(h,x:S)
 id(x:S) -> f(x:S,s(0))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.