/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
log(s(0)) -> 0
log(s(s(X))) -> s(log(s(quot(X,s(s(0))))))
min(s(X),s(Y)) -> min(X,Y)
min(X,0) -> X
quot(0,s(Y)) -> 0
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 log(s(0)) -> 0
 log(s(s(X))) -> s(log(s(quot(X,s(s(0))))))
 min(s(X),s(Y)) -> min(X,Y)
 min(X,0) -> X
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
-> 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))) -> LOG(s(quot(X,s(s(0)))))
 LOG(s(s(X))) -> QUOT(X,s(s(0)))
 MIN(s(X),s(Y)) -> MIN(X,Y)
 QUOT(s(X),s(Y)) -> MIN(X,Y)
 QUOT(s(X),s(Y)) -> QUOT(min(X,Y),s(Y))
-> Rules:
 log(s(0)) -> 0
 log(s(s(X))) -> s(log(s(quot(X,s(s(0))))))
 min(s(X),s(Y)) -> min(X,Y)
 min(X,0) -> X
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.