/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 x0 y z)
(STRATEGY CONTEXTSENSITIVE
(*top*_0 1)
(cons_0 1 2)
(cons_1)
(s_0 1)
(big_0)
(inf_1)
)
(RULES
*top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
cons_1(x,cons_0(y,z)) -> big_0
cons_1(x,cons_1(y,z)) -> big_0
s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 *TOP*_0(inf_1(x)) -> *TOP*_0(cons_0(x,inf_1(s_0(x))))
 *TOP*_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 *TOP*_0(inf_1(x)) -> x
 CONS_0(inf_1(x),x0) -> CONS_0(cons_0(x,inf_1(s_0(x))),x0)
 CONS_0(inf_1(x),x0) -> CONS_0(x,inf_1(s_0(x)))
 CONS_0(inf_1(x),x0) -> x
 CONS_0(x0,inf_1(x)) -> CONS_1(x0,cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding Rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)

Problem 1: 

SCC Processor:
-> Pairs:
 *TOP*_0(inf_1(x)) -> *TOP*_0(cons_0(x,inf_1(s_0(x))))
 *TOP*_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 *TOP*_0(inf_1(x)) -> x
 CONS_0(inf_1(x),x0) -> CONS_0(cons_0(x,inf_1(s_0(x))),x0)
 CONS_0(inf_1(x),x0) -> CONS_0(x,inf_1(s_0(x)))
 CONS_0(inf_1(x),x0) -> x
 CONS_0(x0,inf_1(x)) -> CONS_1(x0,cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CONS_0(inf_1(x),x0) -> CONS_0(cons_0(x,inf_1(s_0(x))),x0)
 CONS_0(inf_1(x),x0) -> CONS_0(x,inf_1(s_0(x)))
 CONS_0(inf_1(x),x0) -> x
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
->->-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->->-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
->->Cycle:
->->-> Pairs:
 *TOP*_0(inf_1(x)) -> *TOP*_0(cons_0(x,inf_1(s_0(x))))
->->-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->->-> Unhiding rules:
 Empty


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 CONS_0(inf_1(x),x0) -> CONS_0(cons_0(x,inf_1(s_0(x))),x0)
 CONS_0(inf_1(x),x0) -> CONS_0(x,inf_1(s_0(x)))
 CONS_0(inf_1(x),x0) -> x
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
-> Usable rules:
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons_0](X1,X2) = X1 + 1/2.X2
[cons_1](X1,X2) = 1/2.X2
[s_0](X) = X + 1/2
[big_0] = 0
[inf_1](X) = 2.X + 2
[CONS_0](X1,X2) = X1 + 1/2.X2
[S_0](X) = X + 1/2

Problem 1.1: 

SCC Processor:
-> Pairs:
 CONS_0(inf_1(x),x0) -> CONS_0(x,inf_1(s_0(x)))
 CONS_0(inf_1(x),x0) -> x
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CONS_0(inf_1(x),x0) -> CONS_0(x,inf_1(s_0(x)))
 CONS_0(inf_1(x),x0) -> x
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
->->-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->->-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 CONS_0(inf_1(x),x0) -> CONS_0(x,inf_1(s_0(x)))
 CONS_0(inf_1(x),x0) -> x
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
-> Usable rules:
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons_0](X1,X2) = X1 + X2
[cons_1](X1,X2) = X1 + 1/2.X2
[s_0](X) = 1/2.X
[big_0] = 0
[inf_1](X) = 2.X + 2
[CONS_0](X1,X2) = X1
[S_0](X) = 1/2.X

Problem 1.1: 

SCC Processor:
-> Pairs:
 CONS_0(inf_1(x),x0) -> x
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CONS_0(inf_1(x),x0) -> x
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
->->-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->->-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 CONS_0(inf_1(x),x0) -> x
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
-> Usable rules:
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons_0](X1,X2) = X1 + 1/2.X2
[cons_1](X1,X2) = X1 + 1/2.X2
[s_0](X) = X + 1
[big_0] = 0
[inf_1](X) = 2.X + 2
[CONS_0](X1,X2) = X1
[S_0](X) = 1/2.X + 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 S_0(inf_1(x)) -> CONS_0(x,inf_1(s_0(x)))
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x))) -> CONS_0(x,inf_1(s_0(x)))
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
->->-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->->-> Unhiding rules:
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 S_0(inf_1(x)) -> S_0(cons_0(x,inf_1(s_0(x))))
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
-> Usable rules:
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons_0](X1,X2) = X1 + 1/2.X2
[cons_1](X1,X2) = 1/2.X1 + 1/2
[s_0](X) = X + 1/2
[big_0] = 0
[inf_1](X) = 2.X + 2
[S_0](X) = 1/2.X + 1/2

Problem 1.1: 

SCC Processor:
-> Pairs:
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 S_0(inf_1(x)) -> x
->->-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->->-> Unhiding rules:
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 S_0(inf_1(x)) -> x
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
-> Usable rules:
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons_0](X1,X2) = X1 + 1/2.X2 + 1/2
[cons_1](X1,X2) = 1/2.X2
[s_0](X) = X
[big_0] = 0
[inf_1](X) = 2.X + 2
[S_0](X) = X

Problem 1.1: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 cons_0(x,inf_1(s_0(x4))) -> x4
 cons_0(x4,inf_1(s_0(x))) -> x4
 s_0(x) -> S_0(x)
-> Result:
 Set P is empty

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 *TOP*_0(inf_1(x)) -> *TOP*_0(cons_0(x,inf_1(s_0(x))))
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 Empty
-> Usable rules:
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons_0](X1,X2) = 0
[cons_1](X1,X2) = 2.X2
[s_0](X) = 1
[big_0] = 0
[inf_1](X) = 2
[*TOP*_0](X) = 2.X

Problem 1.2: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 *top*_0(inf_1(x)) -> *top*_0(cons_0(x,inf_1(s_0(x))))
 cons_0(inf_1(x),x0) -> cons_0(cons_0(x,inf_1(s_0(x))),x0)
 cons_0(x0,inf_1(x)) -> cons_1(x0,cons_0(x,inf_1(s_0(x))))
 cons_1(x,cons_0(y,z)) -> big_0
 cons_1(x,cons_1(y,z)) -> big_0
 s_0(inf_1(x)) -> s_0(cons_0(x,inf_1(s_0(x))))
-> Unhiding rules:
 Empty
-> Result:
 Set P is empty

The problem is finite.