72.29/72.45	YES
72.29/72.45	
72.29/72.45	Problem 1: 
72.29/72.45	
72.29/72.45	(VAR v_NonEmpty:S x:S x':S y:S y':S z:S)
72.29/72.45	(RULES
72.29/72.45	fib(0) -> pair(0,s(0))
72.29/72.45	fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	)
72.29/72.45	
72.29/72.45	Problem 1: 
72.29/72.45	Valid CTRS Processor:
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	-> The system is a deterministic 3-CTRS.
72.29/72.45	
72.29/72.45	Problem 1: 
72.29/72.45	
72.29/72.45	Dependency Pairs Processor:
72.29/72.45	
72.29/72.45	Conditional Termination Problem 1:
72.29/72.45	-> Pairs:
72.29/72.45	 FIB(s(x:S)) -> PLUS(y:S,z:S) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 PLUS(x:S,y:S) -> PLUS(x':S,y':S) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	
72.29/72.45	Conditional Termination Problem 2:
72.29/72.45	-> Pairs:
72.29/72.45	 FIB(s(x:S)) -> FIB(x:S)
72.29/72.45	-> QPairs:
72.29/72.45	 FIB(s(x:S)) -> PLUS(y:S,z:S) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 PLUS(x:S,y:S) -> PLUS(x':S,y':S) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	
72.29/72.45	
72.29/72.45	The problem is decomposed in 2 subproblems.
72.29/72.45	
72.29/72.45	Problem 1.1: 
72.29/72.45	
72.29/72.45	SCC Processor:
72.29/72.45	-> Pairs:
72.29/72.45	 FIB(s(x:S)) -> PLUS(y:S,z:S) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 PLUS(x:S,y:S) -> PLUS(x':S,y':S) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	->Strongly Connected Components:
72.29/72.45	->->Cycle:
72.29/72.45	->->-> Pairs:
72.29/72.45	 PLUS(x:S,y:S) -> PLUS(x':S,y':S) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	->->-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	
72.29/72.45	Problem 1.1: 
72.29/72.45	
72.29/72.45	Simplifying Unifiable Condition Processor:
72.29/72.45	-> Pairs:
72.29/72.45	 PLUS(x:S,y:S) -> PLUS(x':S,y':S) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	->Transformed Pairs/Rules:
72.29/72.45	->->Original Pair/Rule:
72.29/72.45	 PLUS(x:S,y:S) -> PLUS(x':S,y':S) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	->-> Transformed Pair/Rule:
72.29/72.45	 PLUS(s(x':S),y':S) -> PLUS(x':S,y':S)
72.29/72.45	->->Original Pair/Rule:
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	->-> Transformed Pair/Rule:
72.29/72.45	 plus(s(x':S),y':S) -> s(plus(x':S,y':S))
72.29/72.45	->->Original Pair/Rule:
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	->-> Transformed Pair/Rule:
72.29/72.45	 plus(0,y':S) -> y':S
72.29/72.45	
72.29/72.45	Problem 1.1: 
72.29/72.45	
72.29/72.45	SCC Processor:
72.29/72.45	-> Pairs:
72.29/72.45	 PLUS(s(x':S),y':S) -> PLUS(x':S,y':S)
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(0,y':S) -> y':S
72.29/72.45	 plus(s(x':S),y':S) -> s(plus(x':S,y':S))
72.29/72.45	->Strongly Connected Components:
72.29/72.45	->->Cycle:
72.29/72.45	->->-> Pairs:
72.29/72.45	 PLUS(s(x':S),y':S) -> PLUS(x':S,y':S)
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	->->-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(0,y':S) -> y':S
72.29/72.45	 plus(s(x':S),y':S) -> s(plus(x':S,y':S))
72.29/72.45	
72.29/72.45	Problem 1.1: 
72.29/72.45	
72.29/72.45	Conditional Subterm Processor:
72.29/72.45	-> Pairs:
72.29/72.45	 PLUS(s(x':S),y':S) -> PLUS(x':S,y':S)
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(0,y':S) -> y':S
72.29/72.45	 plus(s(x':S),y':S) -> s(plus(x':S,y':S))
72.29/72.45	->Projection:
72.29/72.45	 pi(PLUS) = 1
72.29/72.45	
72.29/72.45	Problem 1.1: 
72.29/72.45	
72.29/72.45	SCC Processor:
72.29/72.45	-> Pairs:
72.29/72.45	 Empty
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(0,y':S) -> y':S
72.29/72.45	 plus(s(x':S),y':S) -> s(plus(x':S,y':S))
72.29/72.45	->Strongly Connected Components:
72.29/72.45	 There is no strongly connected component
72.29/72.45	
72.29/72.45	The problem is finite.
72.29/72.45	
72.29/72.45	Problem 1.2: 
72.29/72.45	
72.29/72.45	SCC Processor:
72.29/72.45	-> Pairs:
72.29/72.45	 FIB(s(x:S)) -> FIB(x:S)
72.29/72.45	-> QPairs:
72.29/72.45	 FIB(s(x:S)) -> PLUS(y:S,z:S) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 PLUS(x:S,y:S) -> PLUS(x':S,y':S) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	->Strongly Connected Components:
72.29/72.45	->->Cycle:
72.29/72.45	->->-> Pairs:
72.29/72.45	 FIB(s(x:S)) -> FIB(x:S)
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	->->-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	
72.29/72.45	Problem 1.2: 
72.29/72.45	
72.29/72.45	Conditional Subterm Processor:
72.29/72.45	-> Pairs:
72.29/72.45	 FIB(s(x:S)) -> FIB(x:S)
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	->Projection:
72.29/72.45	 pi(FIB) = 1
72.29/72.45	
72.29/72.45	Problem 1.2: 
72.29/72.45	
72.29/72.45	SCC Processor:
72.29/72.45	-> Pairs:
72.29/72.45	 Empty
72.29/72.45	-> QPairs:
72.29/72.45	 Empty
72.29/72.45	-> Rules:
72.29/72.45	 fib(0) -> pair(0,s(0))
72.29/72.45	 fib(s(x:S)) -> pair(z:S,plus(y:S,z:S)) | fib(x:S) ->* pair(y:S,z:S)
72.29/72.45	 plus(x:S,y:S) -> s(plus(x':S,y':S)) | x:S ->* s(x':S), y:S ->* y':S
72.29/72.45	 plus(x:S,y:S) -> y':S | x:S ->* 0, y:S ->* y':S
72.29/72.45	->Strongly Connected Components:
72.29/72.45	 There is no strongly connected component
72.29/72.45	
72.29/72.45	The problem is finite.
72.29/72.45	EOF