/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)
(RULES
f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))
)

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(a,f(x,a)) -> F(f(a,x),f(a,a))
 F(a,f(x,a)) -> F(a,f(f(a,x),f(a,a)))
 F(a,f(x,a)) -> F(a,x)
-> Rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))

Problem 1: 

SCC Processor:
-> Pairs:
 F(a,f(x,a)) -> F(f(a,x),f(a,a))
 F(a,f(x,a)) -> F(a,f(f(a,x),f(a,a)))
 F(a,f(x,a)) -> F(a,x)
-> Rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(a,f(x,a)) -> F(f(a,x),f(a,a))
 F(a,f(x,a)) -> F(a,f(f(a,x),f(a,a)))
 F(a,f(x,a)) -> F(a,x)
->->-> Rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))

Problem 1: 

Reduction Pairs Processor:
-> Pairs:
 F(a,f(x,a)) -> F(f(a,x),f(a,a))
 F(a,f(x,a)) -> F(a,f(f(a,x),f(a,a)))
 F(a,f(x,a)) -> F(a,x)
-> Rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))
-> Usable rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X1,X2) = 0
[a] = 1
[F](X1,X2) = X1

Problem 1: 

SCC Processor:
-> Pairs:
 F(a,f(x,a)) -> F(a,f(f(a,x),f(a,a)))
 F(a,f(x,a)) -> F(a,x)
-> Rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(a,f(x,a)) -> F(a,f(f(a,x),f(a,a)))
 F(a,f(x,a)) -> F(a,x)
->->-> Rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))

Problem 1: 

Reduction Pairs Processor:
-> Pairs:
 F(a,f(x,a)) -> F(a,f(f(a,x),f(a,a)))
 F(a,f(x,a)) -> F(a,x)
-> Rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))
-> Usable rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[f](X1,X2) = [1 1;0 0].X1 + [0 1;0 0].X2
[a] = [0;1]
[F](X1,X2) = [1 0;1 1].X2

Problem 1: 

SCC Processor:
-> Pairs:
 F(a,f(x,a)) -> F(a,f(f(a,x),f(a,a)))
-> Rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(a,f(x,a)) -> F(a,f(f(a,x),f(a,a)))
->->-> Rules:
 f(a,f(x,a)) -> f(a,f(f(a,x),f(a,a)))

Problem 1: 

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

Problem 1: 

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

The problem is finite.