0.00/0.01	YES
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	(VAR v_NonEmpty:S W:S X:S Y:S Z:S)
0.00/0.01	(RULES
0.00/0.01	fib(0) -> pair(s(0),0)
0.00/0.01	fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	plus(0,X:S) -> X:S
0.00/0.01	plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	)
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	Valid CTRS Processor:
0.00/0.01	-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	-> The system is a deterministic 3-CTRS.
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	Dependency Pairs Processor:
0.00/0.01	
0.00/0.01	Conditional Termination Problem 1:
0.00/0.01	-> Pairs:
0.00/0.01	 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
0.00/0.01	-> QPairs:
0.00/0.01	 Empty
0.00/0.01	-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	
0.00/0.01	Conditional Termination Problem 2:
0.00/0.01	-> Pairs:
0.00/0.01	 FIB(s(X:S)) -> FIB(X:S)
0.00/0.01	 FIB(s(X:S)) -> PLUS(Y:S,Z:S) | fib(X:S) ->* pair(Y:S,Z:S)
0.00/0.01	-> QPairs:
0.00/0.01	 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
0.00/0.01	-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	
0.00/0.01	
0.00/0.01	The problem is decomposed in 2 subproblems.
0.00/0.01	
0.00/0.01	Problem 1.1: 
0.00/0.01	
0.00/0.01	SCC Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
0.00/0.01	-> QPairs:
0.00/0.01	 Empty
0.00/0.01	-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	->Strongly Connected Components:
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
0.00/0.01	-> QPairs:
0.00/0.01	 Empty
0.00/0.01	->->-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	
0.00/0.01	Problem 1.1: 
0.00/0.01	
0.00/0.01	Conditional Subterm Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
0.00/0.01	-> QPairs:
0.00/0.01	 Empty
0.00/0.01	-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	->Projection:
0.00/0.01	 pi(PLUS) = 1
0.00/0.01	
0.00/0.01	Problem 1.1: 
0.00/0.01	
0.00/0.01	SCC Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 Empty
0.00/0.01	-> QPairs:
0.00/0.01	 Empty
0.00/0.01	-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	->Strongly Connected Components:
0.00/0.01	 There is no strongly connected component
0.00/0.01	
0.00/0.01	The problem is finite.
0.00/0.01	
0.00/0.01	Problem 1.2: 
0.00/0.01	
0.00/0.01	SCC Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 FIB(s(X:S)) -> FIB(X:S)
0.00/0.01	 FIB(s(X:S)) -> PLUS(Y:S,Z:S) | fib(X:S) ->* pair(Y:S,Z:S)
0.00/0.01	-> QPairs:
0.00/0.01	 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
0.00/0.01	-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	->Strongly Connected Components:
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 FIB(s(X:S)) -> FIB(X:S)
0.00/0.01	-> QPairs:
0.00/0.01	 Empty
0.00/0.01	->->-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	
0.00/0.01	Problem 1.2: 
0.00/0.01	
0.00/0.01	Conditional Subterm Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 FIB(s(X:S)) -> FIB(X:S)
0.00/0.01	-> QPairs:
0.00/0.01	 Empty
0.00/0.01	-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	->Projection:
0.00/0.01	 pi(FIB) = 1
0.00/0.01	
0.00/0.01	Problem 1.2: 
0.00/0.01	
0.00/0.01	SCC Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 Empty
0.00/0.01	-> QPairs:
0.00/0.01	 Empty
0.00/0.01	-> Rules:
0.00/0.01	 fib(0) -> pair(s(0),0)
0.00/0.01	 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
0.00/0.01	 plus(0,X:S) -> X:S
0.00/0.01	 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
0.00/0.01	->Strongly Connected Components:
0.00/0.01	 There is no strongly connected component
0.00/0.01	
0.00/0.01	The problem is finite.
0.00/0.01	EOF