/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
-(s(x),s(y)) -> -(x,y)
-(x,0) -> x
f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
p(s(x)) -> x
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
 f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
 p(s(x)) -> x
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
 F(s(x),y) -> -#(s(x),y)
 F(s(x),y) -> -#(y,s(x))
 F(s(x),y) -> F(p(-(s(x),y)),p(-(y,s(x))))
 F(s(x),y) -> P(-(s(x),y))
 F(s(x),y) -> P(-(y,s(x)))
 F(x,s(y)) -> -#(s(y),x)
 F(x,s(y)) -> -#(x,s(y))
 F(x,s(y)) -> F(p(-(x,s(y))),p(-(s(y),x)))
 F(x,s(y)) -> P(-(s(y),x))
 F(x,s(y)) -> P(-(x,s(y)))
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
 f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
 p(s(x)) -> x

Problem 1: 

SCC Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
 F(s(x),y) -> -#(s(x),y)
 F(s(x),y) -> -#(y,s(x))
 F(s(x),y) -> F(p(-(s(x),y)),p(-(y,s(x))))
 F(s(x),y) -> P(-(s(x),y))
 F(s(x),y) -> P(-(y,s(x)))
 F(x,s(y)) -> -#(s(y),x)
 F(x,s(y)) -> -#(x,s(y))
 F(x,s(y)) -> F(p(-(x,s(y))),p(-(s(y),x)))
 F(x,s(y)) -> P(-(s(y),x))
 F(x,s(y)) -> P(-(x,s(y)))
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
 f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
 p(s(x)) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
->->-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
 f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
 p(s(x)) -> x
->->Cycle:
->->-> Pairs:
 F(s(x),y) -> F(p(-(s(x),y)),p(-(y,s(x))))
 F(x,s(y)) -> F(p(-(x,s(y))),p(-(s(y),x)))
->->-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
 f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
 p(s(x)) -> x


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
 f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
 p(s(x)) -> x
->Projection:
 pi(-#) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.