/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(s(X)) -> f(X)
g(cons(0,Y)) -> g(Y)
g(cons(s(X),Y)) -> s(X)
h(cons(X,Y)) -> h(g(cons(X,Y)))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,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(s(X)) -> F(X)
 G(cons(0,Y)) -> G(Y)
 H(cons(X,Y)) -> G(cons(X,Y))
 H(cons(X,Y)) -> H(g(cons(X,Y)))
-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(X)) -> F(X)
 G(cons(0,Y)) -> G(Y)
 H(cons(X,Y)) -> G(cons(X,Y))
 H(cons(X,Y)) -> H(g(cons(X,Y)))
-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(cons(0,Y)) -> G(Y)
->->-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))
->->Cycle:
->->-> Pairs:
 H(cons(X,Y)) -> H(g(cons(X,Y)))
->->-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))
->->Cycle:
->->-> Pairs:
 F(s(X)) -> F(X)
->->-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 G(cons(0,Y)) -> G(Y)
-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))
->Projection:
 pi(G) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 F(s(X)) -> F(X)
-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))
->Projection:
 pi(F) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(s(X)) -> f(X)
 g(cons(0,Y)) -> g(Y)
 g(cons(s(X),Y)) -> s(X)
 h(cons(X,Y)) -> h(g(cons(X,Y)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.