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 x1:S)
0.00/0.01	(RULES
0.00/0.01	d(0(x1:S)) -> 0(x1:S)
0.00/0.01	d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	f(s(x1:S)) -> d(f(x1:S))
0.00/0.01	)
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	Innermost Equivalent Processor:
0.00/0.01	-> Rules:
0.00/0.01	 d(0(x1:S)) -> 0(x1:S)
0.00/0.01	 d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	 f(s(x1:S)) -> d(f(x1:S))
0.00/0.01	-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
0.00/0.01	 
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	-> Pairs:
0.00/0.01	 D(s(x1:S)) -> D(x1:S)
0.00/0.01	 F(s(x1:S)) -> D(f(x1:S))
0.00/0.01	 F(s(x1:S)) -> F(x1:S)
0.00/0.01	-> Rules:
0.00/0.01	 d(0(x1:S)) -> 0(x1:S)
0.00/0.01	 d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	 f(s(x1:S)) -> d(f(x1:S))
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	SCC Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 D(s(x1:S)) -> D(x1:S)
0.00/0.01	 F(s(x1:S)) -> D(f(x1:S))
0.00/0.01	 F(s(x1:S)) -> F(x1:S)
0.00/0.01	-> Rules:
0.00/0.01	 d(0(x1:S)) -> 0(x1:S)
0.00/0.01	 d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	 f(s(x1:S)) -> d(f(x1:S))
0.00/0.01	->Strongly Connected Components:
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 D(s(x1:S)) -> D(x1:S)
0.00/0.01	->->-> Rules:
0.00/0.01	 d(0(x1:S)) -> 0(x1:S)
0.00/0.01	 d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	 f(s(x1:S)) -> d(f(x1:S))
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 F(s(x1:S)) -> F(x1:S)
0.00/0.01	->->-> Rules:
0.00/0.01	 d(0(x1:S)) -> 0(x1:S)
0.00/0.01	 d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	 f(s(x1:S)) -> d(f(x1: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	Subterm Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 D(s(x1:S)) -> D(x1:S)
0.00/0.01	-> Rules:
0.00/0.01	 d(0(x1:S)) -> 0(x1:S)
0.00/0.01	 d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	 f(s(x1:S)) -> d(f(x1:S))
0.00/0.01	->Projection:
0.00/0.01	 pi(D) = 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	-> Rules:
0.00/0.01	 d(0(x1:S)) -> 0(x1:S)
0.00/0.01	 d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	 f(s(x1:S)) -> d(f(x1: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	Subterm Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 F(s(x1:S)) -> F(x1:S)
0.00/0.01	-> Rules:
0.00/0.01	 d(0(x1:S)) -> 0(x1:S)
0.00/0.01	 d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	 f(s(x1:S)) -> d(f(x1:S))
0.00/0.01	->Projection:
0.00/0.01	 pi(F) = 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	-> Rules:
0.00/0.01	 d(0(x1:S)) -> 0(x1:S)
0.00/0.01	 d(s(x1:S)) -> s(s(d(x1:S)))
0.00/0.01	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.01	 f(s(x1:S)) -> d(f(x1: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