/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
f(0,1,x) -> f(h(x),h(x),x)
h(0) -> 0
h(g(x,y)) -> y
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(0,1,x) -> F(h(x),h(x),x)
 F(0,1,x) -> H(x)
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y

Problem 1: 

SCC Processor:
-> Pairs:
 F(0,1,x) -> F(h(x),h(x),x)
 F(0,1,x) -> H(x)
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(0,1,x) -> F(h(x),h(x),x)
->->-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y

Problem 1: 

Narrowing Processor:
-> Pairs:
 F(0,1,x) -> F(h(x),h(x),x)
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->Narrowed Pairs:
->->Original Pair:
 F(0,1,x) -> F(h(x),h(x),x)
->-> Narrowed pairs:
 F(0,1,0) -> F(h(0),0,0)
 F(0,1,0) -> F(0,h(0),0)
 F(0,1,g(x,y)) -> F(h(g(x,y)),y,g(x,y))
 F(0,1,g(x,y)) -> F(y,h(g(x,y)),g(x,y))

Problem 1: 

SCC Processor:
-> Pairs:
 F(0,1,0) -> F(h(0),0,0)
 F(0,1,0) -> F(0,h(0),0)
 F(0,1,g(x,y)) -> F(h(g(x,y)),y,g(x,y))
 F(0,1,g(x,y)) -> F(y,h(g(x,y)),g(x,y))
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(0,1,g(x,y)) -> F(h(g(x,y)),y,g(x,y))
 F(0,1,g(x,y)) -> F(y,h(g(x,y)),g(x,y))
->->-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->->Cycle:
->->-> Pairs:
 F(0,1,0) -> F(0,h(0),0)
->->-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Instantiation Processor:
-> Pairs:
 F(0,1,g(x,y)) -> F(h(g(x,y)),y,g(x,y))
 F(0,1,g(x,y)) -> F(y,h(g(x,y)),g(x,y))
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->Instantiated Pairs:
->->Original Pair:
 F(0,1,g(x,y)) -> F(h(g(x,y)),y,g(x,y))
->-> Instantiated pairs:
 F(0,1,g(x,0)) -> F(h(g(x,0)),0,g(x,0))
 F(0,1,g(x,1)) -> F(h(g(x,1)),1,g(x,1))
->->Original Pair:
 F(0,1,g(x,y)) -> F(y,h(g(x,y)),g(x,y))
->-> Instantiated pairs:
 F(0,1,g(x,0)) -> F(0,h(g(x,0)),g(x,0))
 F(0,1,g(x,1)) -> F(1,h(g(x,1)),g(x,1))

Problem 1.1: 

SCC Processor:
-> Pairs:
 F(0,1,g(x,0)) -> F(h(g(x,0)),0,g(x,0))
 F(0,1,g(x,0)) -> F(0,h(g(x,0)),g(x,0))
 F(0,1,g(x,1)) -> F(h(g(x,1)),1,g(x,1))
 F(0,1,g(x,1)) -> F(1,h(g(x,1)),g(x,1))
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(0,1,g(x,1)) -> F(h(g(x,1)),1,g(x,1))
->->-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->->Cycle:
->->-> Pairs:
 F(0,1,g(x,0)) -> F(0,h(g(x,0)),g(x,0))
->->-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y


The problem is decomposed in 2 subproblems.

Problem 1.1.1: 

Reduction Pairs Processor:
-> Pairs:
 F(0,1,g(x,1)) -> F(h(g(x,1)),1,g(x,1))
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
-> Usable rules:
 h(0) -> 0
 h(g(x,y)) -> y
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[h](X) = X
[0] = 2
[1] = 1
[g](X1,X2) = X2
[F](X1,X2,X3) = 2.X1

Problem 1.1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.1.2: 

Reduction Pairs Processor:
-> Pairs:
 F(0,1,g(x,0)) -> F(0,h(g(x,0)),g(x,0))
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
-> Usable rules:
 h(0) -> 0
 h(g(x,y)) -> y
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[h](X) = X
[0] = 1
[1] = 2
[g](X1,X2) = X2
[F](X1,X2,X3) = 2.X2

Problem 1.1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 F(0,1,0) -> F(0,h(0),0)
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
-> Usable rules:
 h(0) -> 0
 h(g(x,y)) -> y
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[h](X) = 2.X + 1
[0] = 0
[1] = 2
[g](X1,X2) = 2.X2
[F](X1,X2,X3) = 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0,1,x) -> f(h(x),h(x),x)
 h(0) -> 0
 h(g(x,y)) -> y
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.