/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 W X Y Z)
(RULES
fib(0) -> pair(s(0),0)
fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
plus(0,X) -> X
plus(s(X),Y) -> plus(X,s(Y))
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 PLUS(s(X),Y) -> PLUS(X,s(Y))
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))

Conditional Termination Problem 2:
-> Pairs:
 FIB(s(X)) -> FIB(X)
 FIB(s(X)) -> PLUS(Y,Z) | fib(X) -> pair(Y,Z)
-> QPairs:
 PLUS(s(X),Y) -> PLUS(X,s(Y))
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 PLUS(s(X),Y) -> PLUS(X,s(Y))
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(X),Y) -> PLUS(X,s(Y))
->->-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))

Problem 1.1: 

Conditional Subterm Processor:
-> Pairs:
 PLUS(s(X),Y) -> PLUS(X,s(Y))
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 FIB(s(X)) -> FIB(X)
 FIB(s(X)) -> PLUS(Y,Z) | fib(X) -> pair(Y,Z)
-> QPairs:
 PLUS(s(X),Y) -> PLUS(X,s(Y))
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 FIB(s(X)) -> FIB(X)
->->-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))

Problem 1.2: 

Conditional Subterm Processor:
-> Pairs:
 FIB(s(X)) -> FIB(X)
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))
->Projection:
 pi(FIB) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X)) -> pair(W,Y) | fib(X) -> pair(Y,Z), plus(Y,Z) -> W
 plus(0,X) -> X
 plus(s(X),Y) -> plus(X,s(Y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.