/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
id_inc(x) -> s(x)
id_inc(x) -> x
if(false,x,y) -> y
if(true,x,y) -> rand(p(x),id_inc(y))
nonZero(0) -> false
nonZero(s(x)) -> true
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))
rand(x,y) -> if(nonZero(x),x,y)
random(x) -> rand(x,0)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 IF(true,x,y) -> ID_INC(y)
 IF(true,x,y) -> P(x)
 IF(true,x,y) -> RAND(p(x),id_inc(y))
 P(s(s(x))) -> P(s(x))
 RAND(x,y) -> IF(nonZero(x),x,y)
 RAND(x,y) -> NONZERO(x)
 RANDOM(x) -> RAND(x,0)
-> Rules:
 id_inc(x) -> s(x)
 id_inc(x) -> x
 if(false,x,y) -> y
 if(true,x,y) -> rand(p(x),id_inc(y))
 nonZero(0) -> false
 nonZero(s(x)) -> true
 p(s(0)) -> 0
 p(s(s(x))) -> s(p(s(x)))
 rand(x,y) -> if(nonZero(x),x,y)
 random(x) -> rand(x,0)

Problem 1: 

SCC Processor:
-> Pairs:
 IF(true,x,y) -> ID_INC(y)
 IF(true,x,y) -> P(x)
 IF(true,x,y) -> RAND(p(x),id_inc(y))
 P(s(s(x))) -> P(s(x))
 RAND(x,y) -> IF(nonZero(x),x,y)
 RAND(x,y) -> NONZERO(x)
 RANDOM(x) -> RAND(x,0)
-> Rules:
 id_inc(x) -> s(x)
 id_inc(x) -> x
 if(false,x,y) -> y
 if(true,x,y) -> rand(p(x),id_inc(y))
 nonZero(0) -> false
 nonZero(s(x)) -> true
 p(s(0)) -> 0
 p(s(s(x))) -> s(p(s(x)))
 rand(x,y) -> if(nonZero(x),x,y)
 random(x) -> rand(x,0)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 P(s(s(x))) -> P(s(x))
->->-> Rules:
 id_inc(x) -> s(x)
 id_inc(x) -> x
 if(false,x,y) -> y
 if(true,x,y) -> rand(p(x),id_inc(y))
 nonZero(0) -> false
 nonZero(s(x)) -> true
 p(s(0)) -> 0
 p(s(s(x))) -> s(p(s(x)))
 rand(x,y) -> if(nonZero(x),x,y)
 random(x) -> rand(x,0)
->->Cycle:
->->-> Pairs:
 IF(true,x,y) -> RAND(p(x),id_inc(y))
 RAND(x,y) -> IF(nonZero(x),x,y)
->->-> Rules:
 id_inc(x) -> s(x)
 id_inc(x) -> x
 if(false,x,y) -> y
 if(true,x,y) -> rand(p(x),id_inc(y))
 nonZero(0) -> false
 nonZero(s(x)) -> true
 p(s(0)) -> 0
 p(s(s(x))) -> s(p(s(x)))
 rand(x,y) -> if(nonZero(x),x,y)
 random(x) -> rand(x,0)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 P(s(s(x))) -> P(s(x))
-> Rules:
 id_inc(x) -> s(x)
 id_inc(x) -> x
 if(false,x,y) -> y
 if(true,x,y) -> rand(p(x),id_inc(y))
 nonZero(0) -> false
 nonZero(s(x)) -> true
 p(s(0)) -> 0
 p(s(s(x))) -> s(p(s(x)))
 rand(x,y) -> if(nonZero(x),x,y)
 random(x) -> rand(x,0)
->Projection:
 pi(P) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 id_inc(x) -> s(x)
 id_inc(x) -> x
 if(false,x,y) -> y
 if(true,x,y) -> rand(p(x),id_inc(y))
 nonZero(0) -> false
 nonZero(s(x)) -> true
 p(s(0)) -> 0
 p(s(s(x))) -> s(p(s(x)))
 rand(x,y) -> if(nonZero(x),x,y)
 random(x) -> rand(x,0)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 IF(true,x,y) -> RAND(p(x),id_inc(y))
 RAND(x,y) -> IF(nonZero(x),x,y)
-> Rules:
 id_inc(x) -> s(x)
 id_inc(x) -> x
 if(false,x,y) -> y
 if(true,x,y) -> rand(p(x),id_inc(y))
 nonZero(0) -> false
 nonZero(s(x)) -> true
 p(s(0)) -> 0
 p(s(s(x))) -> s(p(s(x)))
 rand(x,y) -> if(nonZero(x),x,y)
 random(x) -> rand(x,0)
-> Usable rules:
 id_inc(x) -> s(x)
 id_inc(x) -> x
 nonZero(0) -> false
 nonZero(s(x)) -> true
 p(s(0)) -> 0
 p(s(s(x))) -> s(p(s(x)))
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[id_inc](X) = 2.X.X + 2.X + 2
[nonZero](X) = 1/2.X + 1/2
[p](X) = 1/2.X
[0] = 1
[false] = 0
[s](X) = 2.X.X + 1
[true] = 1
[IF](X1,X2,X3) = 2.X1 + X2 + 1/2
[RAND](X1,X2) = 2.X1 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 RAND(x,y) -> IF(nonZero(x),x,y)
-> Rules:
 id_inc(x) -> s(x)
 id_inc(x) -> x
 if(false,x,y) -> y
 if(true,x,y) -> rand(p(x),id_inc(y))
 nonZero(0) -> false
 nonZero(s(x)) -> true
 p(s(0)) -> 0
 p(s(s(x))) -> s(p(s(x)))
 rand(x,y) -> if(nonZero(x),x,y)
 random(x) -> rand(x,0)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.