/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR v_NonEmpty:S X:S Y:S)
(RULES
log(s(0)) -> 0
log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
min(X:S,0) -> X:S
quot(0,s(Y:S)) -> 0
quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 LOG(s(s(X:S))) -> LOG(s(quot(X:S,s(s(0)))))
 LOG(s(s(X:S))) -> QUOT(X:S,s(s(0)))
 MIN(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
 QUOT(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
 QUOT(s(X:S),s(Y:S)) -> QUOT(min(X:S,Y:S),s(Y:S))
-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 LOG(s(s(X:S))) -> LOG(s(quot(X:S,s(s(0)))))
 LOG(s(s(X:S))) -> QUOT(X:S,s(s(0)))
 MIN(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
 QUOT(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
 QUOT(s(X:S),s(Y:S)) -> QUOT(min(X:S,Y:S),s(Y:S))
-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MIN(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
->->-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->->Cycle:
->->-> Pairs:
 QUOT(s(X:S),s(Y:S)) -> QUOT(min(X:S,Y:S),s(Y:S))
->->-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->->Cycle:
->->-> Pairs:
 LOG(s(s(X:S))) -> LOG(s(quot(X:S,s(s(0)))))
->->-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MIN(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Projection:
 pi(MIN) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 QUOT(s(X:S),s(Y:S)) -> QUOT(min(X:S,Y:S),s(Y:S))
-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
-> Usable rules:
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[log](X) = 0
[min](X1,X2) = 2.X1 + 1
[quot](X1,X2) = 0
[0] = 0
[fSNonEmpty] = 0
[s](X) = 2.X + 2
[LOG](X) = 0
[MIN](X1,X2) = 0
[QUOT](X1,X2) = 2.X1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 LOG(s(s(X:S))) -> LOG(s(quot(X:S,s(s(0)))))
-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
-> Usable rules:
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[log](X) = 0
[min](X1,X2) = X1
[quot](X1,X2) = 2.X1 + 1
[0] = 0
[fSNonEmpty] = 0
[s](X) = 2.X + 2
[LOG](X) = 2.X
[MIN](X1,X2) = 0
[QUOT](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 log(s(0)) -> 0
 log(s(s(X:S))) -> s(log(s(quot(X:S,s(s(0))))))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.