/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 v_NonEmpty:S x:S y:S)
(RULES
fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
p(s(0)) -> 0
p(s(s(x:S))) -> s(p(s(x:S)))
plus(x:S,0) -> x:S
plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
times(0,y:S) -> 0
times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
times(x:S,0) -> 0
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 FAC(s(x:S)) -> FAC(p(s(x:S)))
 FAC(s(x:S)) -> P(s(x:S))
 FAC(s(x:S)) -> TIMES(fac(p(s(x:S))),s(x:S))
 P(s(s(x:S))) -> P(s(x:S))
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
 TIMES(s(x:S),y:S) -> PLUS(times(x:S,y:S),y:S)
 TIMES(s(x:S),y:S) -> TIMES(x:S,y:S)
-> Rules:
 fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
 times(x:S,0) -> 0

Problem 1: 

SCC Processor:
-> Pairs:
 FAC(s(x:S)) -> FAC(p(s(x:S)))
 FAC(s(x:S)) -> P(s(x:S))
 FAC(s(x:S)) -> TIMES(fac(p(s(x:S))),s(x:S))
 P(s(s(x:S))) -> P(s(x:S))
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
 TIMES(s(x:S),y:S) -> PLUS(times(x:S,y:S),y:S)
 TIMES(s(x:S),y:S) -> TIMES(x:S,y:S)
-> Rules:
 fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
 times(x:S,0) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
->->-> Rules:
 fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
 times(x:S,0) -> 0
->->Cycle:
->->-> Pairs:
 TIMES(s(x:S),y:S) -> TIMES(x:S,y:S)
->->-> Rules:
 fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
 times(x:S,0) -> 0
->->Cycle:
->->-> Pairs:
 P(s(s(x:S))) -> P(s(x:S))
->->-> Rules:
 fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
 times(x:S,0) -> 0
->->Cycle:
->->-> Pairs:
 FAC(s(x:S)) -> FAC(p(s(x:S)))
->->-> Rules:
 fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
 times(x:S,0) -> 0


The problem is decomposed in 4 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 TIMES(s(x:S),y:S) -> TIMES(x:S,y:S)
-> Rules:
 fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
 times(x:S,0) -> 0
->Projection:
 pi(TIMES) = 1

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 P(s(s(x:S))) -> P(s(x:S))
-> Rules:
 fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
 times(x:S,0) -> 0
->Projection:
 pi(P) = 1

Problem 1.3: 

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

The problem is finite.

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 FAC(s(x:S)) -> FAC(p(s(x:S)))
-> Rules:
 fac(s(x:S)) -> times(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> plus(times(x:S,y:S),y:S)
 times(x:S,0) -> 0
-> Usable rules:
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[p](X) = 1/2.X
[0] = 1
[s](X) = 2.X.X + 2.X + 1
[FAC](X) = 2.X

Problem 1.4: 

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

The problem is finite.