/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 z)
(RULES
plus(x,0) -> x
plus(x,s(y)) -> s(plus(x,y))
times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
times(x,0) -> 0
times(x,s(y)) -> plus(times(x,y),x)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 PLUS(x,s(y)) -> PLUS(x,y)
 TIMES(x,plus(y,s(z))) -> PLUS(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 TIMES(x,plus(y,s(z))) -> PLUS(y,times(s(z),0))
 TIMES(x,plus(y,s(z))) -> TIMES(s(z),0)
 TIMES(x,plus(y,s(z))) -> TIMES(x,plus(y,times(s(z),0)))
 TIMES(x,plus(y,s(z))) -> TIMES(x,s(z))
 TIMES(x,s(y)) -> PLUS(times(x,y),x)
 TIMES(x,s(y)) -> TIMES(x,y)
-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)

Problem 1: 

SCC Processor:
-> Pairs:
 PLUS(x,s(y)) -> PLUS(x,y)
 TIMES(x,plus(y,s(z))) -> PLUS(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 TIMES(x,plus(y,s(z))) -> PLUS(y,times(s(z),0))
 TIMES(x,plus(y,s(z))) -> TIMES(s(z),0)
 TIMES(x,plus(y,s(z))) -> TIMES(x,plus(y,times(s(z),0)))
 TIMES(x,plus(y,s(z))) -> TIMES(x,s(z))
 TIMES(x,s(y)) -> PLUS(times(x,y),x)
 TIMES(x,s(y)) -> TIMES(x,y)
-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(x,s(y)) -> PLUS(x,y)
->->-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)
->->Cycle:
->->-> Pairs:
 TIMES(x,plus(y,s(z))) -> TIMES(x,plus(y,times(s(z),0)))
 TIMES(x,plus(y,s(z))) -> TIMES(x,s(z))
 TIMES(x,s(y)) -> TIMES(x,y)
->->-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(x,s(y)) -> PLUS(x,y)
-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)
->Projection:
 pi(PLUS) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 TIMES(x,plus(y,s(z))) -> TIMES(x,plus(y,times(s(z),0)))
 TIMES(x,plus(y,s(z))) -> TIMES(x,s(z))
 TIMES(x,s(y)) -> TIMES(x,y)
-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)
-> Usable rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 2.X1.X2
[0] = 0
[s](X) = X + 2
[TIMES](X1,X2) = 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 TIMES(x,plus(y,s(z))) -> TIMES(x,s(z))
 TIMES(x,s(y)) -> TIMES(x,y)
-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(x,plus(y,s(z))) -> TIMES(x,s(z))
 TIMES(x,s(y)) -> TIMES(x,y)
->->-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)

Problem 1.2: 

Subterm Processor:
-> Pairs:
 TIMES(x,plus(y,s(z))) -> TIMES(x,s(z))
 TIMES(x,s(y)) -> TIMES(x,y)
-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)
->Projection:
 pi(TIMES) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0))),times(x,s(z)))
 times(x,0) -> 0
 times(x,s(y)) -> plus(times(x,y),x)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.