0.00/0.06	YES
0.00/0.06	
0.00/0.06	Problem 1: 
0.00/0.06	
0.00/0.06	(VAR v_NonEmpty:S X:S)
0.00/0.06	(RULES
0.00/0.06	active(f(X:S)) -> f(active(X:S))
0.00/0.06	active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.06	active(h(X:S)) -> h(active(X:S))
0.00/0.06	f(mark(X:S)) -> mark(f(X:S))
0.00/0.06	f(ok(X:S)) -> ok(f(X:S))
0.00/0.06	g(ok(X:S)) -> ok(g(X:S))
0.00/0.06	h(mark(X:S)) -> mark(h(X:S))
0.00/0.06	h(ok(X:S)) -> ok(h(X:S))
0.00/0.06	proper(f(X:S)) -> f(proper(X:S))
0.00/0.06	proper(g(X:S)) -> g(proper(X:S))
0.00/0.06	proper(h(X:S)) -> h(proper(X:S))
0.00/0.06	top(mark(X:S)) -> top(proper(X:S))
0.00/0.06	top(ok(X:S)) -> top(active(X:S))
0.00/0.06	) 
0.00/0.06	(STRATEGY INNERMOST)
0.00/0.06	
0.00/0.06	Problem 1: 
0.00/0.06	
0.00/0.06	Dependency Pairs Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 ACTIVE(f(X:S)) -> ACTIVE(X:S)
0.00/0.06	 ACTIVE(f(X:S)) -> F(active(X:S))
0.00/0.06	 ACTIVE(f(X:S)) -> G(h(f(X:S)))
0.00/0.06	 ACTIVE(f(X:S)) -> H(f(X:S))
0.00/0.06	 ACTIVE(h(X:S)) -> ACTIVE(X:S)
0.00/0.06	 ACTIVE(h(X:S)) -> H(active(X:S))
0.00/0.06	 F(mark(X:S)) -> F(X:S)
0.00/0.06	 F(ok(X:S)) -> F(X:S)
0.00/0.06	 G(ok(X:S)) -> G(X:S)
0.00/0.06	 H(mark(X:S)) -> H(X:S)
0.00/0.06	 H(ok(X:S)) -> H(X:S)
0.00/0.06	 PROPER(f(X:S)) -> F(proper(X:S))
0.00/0.06	 PROPER(f(X:S)) -> PROPER(X:S)
0.00/0.06	 PROPER(g(X:S)) -> G(proper(X:S))
0.00/0.06	 PROPER(g(X:S)) -> PROPER(X:S)
0.00/0.06	 PROPER(h(X:S)) -> H(proper(X:S))
0.00/0.06	 PROPER(h(X:S)) -> PROPER(X:S)
0.00/0.06	 TOP(mark(X:S)) -> PROPER(X:S)
0.00/0.06	 TOP(mark(X:S)) -> TOP(proper(X:S))
0.00/0.06	 TOP(ok(X:S)) -> ACTIVE(X:S)
0.00/0.06	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.06	-> Rules:
0.00/0.06	 active(f(X:S)) -> f(active(X:S))
0.00/0.06	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.06	 active(h(X:S)) -> h(active(X:S))
0.00/0.06	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.06	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.06	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.06	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.06	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.06	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.06	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.06	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.06	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.06	 top(ok(X:S)) -> top(active(X:S))
0.00/0.06	
0.00/0.06	Problem 1: 
0.00/0.06	
0.00/0.06	SCC Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 ACTIVE(f(X:S)) -> ACTIVE(X:S)
0.00/0.06	 ACTIVE(f(X:S)) -> F(active(X:S))
0.00/0.06	 ACTIVE(f(X:S)) -> G(h(f(X:S)))
0.00/0.06	 ACTIVE(f(X:S)) -> H(f(X:S))
0.00/0.06	 ACTIVE(h(X:S)) -> ACTIVE(X:S)
0.00/0.06	 ACTIVE(h(X:S)) -> H(active(X:S))
0.00/0.06	 F(mark(X:S)) -> F(X:S)
0.00/0.06	 F(ok(X:S)) -> F(X:S)
0.00/0.06	 G(ok(X:S)) -> G(X:S)
0.00/0.06	 H(mark(X:S)) -> H(X:S)
0.00/0.06	 H(ok(X:S)) -> H(X:S)
0.00/0.06	 PROPER(f(X:S)) -> F(proper(X:S))
0.00/0.06	 PROPER(f(X:S)) -> PROPER(X:S)
0.00/0.06	 PROPER(g(X:S)) -> G(proper(X:S))
0.00/0.06	 PROPER(g(X:S)) -> PROPER(X:S)
0.00/0.06	 PROPER(h(X:S)) -> H(proper(X:S))
0.00/0.06	 PROPER(h(X:S)) -> PROPER(X:S)
0.00/0.06	 TOP(mark(X:S)) -> PROPER(X:S)
0.00/0.06	 TOP(mark(X:S)) -> TOP(proper(X:S))
0.00/0.06	 TOP(ok(X:S)) -> ACTIVE(X:S)
0.00/0.06	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.06	-> Rules:
0.00/0.06	 active(f(X:S)) -> f(active(X:S))
0.00/0.06	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.06	 active(h(X:S)) -> h(active(X:S))
0.00/0.06	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.06	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.06	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.06	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.06	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.06	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.06	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.06	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.06	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.06	 top(ok(X:S)) -> top(active(X:S))
0.00/0.06	->Strongly Connected Components:
0.00/0.06	->->Cycle:
0.00/0.06	->->-> Pairs:
0.00/0.06	 H(mark(X:S)) -> H(X:S)
0.00/0.06	 H(ok(X:S)) -> H(X:S)
0.00/0.06	->->-> Rules:
0.00/0.06	 active(f(X:S)) -> f(active(X:S))
0.00/0.06	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.06	 active(h(X:S)) -> h(active(X:S))
0.00/0.06	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.06	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.06	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.06	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.06	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.06	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 G(ok(X:S)) -> G(X:S)
0.00/0.07	->->-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 F(mark(X:S)) -> F(X:S)
0.00/0.07	 F(ok(X:S)) -> F(X:S)
0.00/0.07	->->-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 PROPER(f(X:S)) -> PROPER(X:S)
0.00/0.07	 PROPER(g(X:S)) -> PROPER(X:S)
0.00/0.07	 PROPER(h(X:S)) -> PROPER(X:S)
0.00/0.07	->->-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 ACTIVE(f(X:S)) -> ACTIVE(X:S)
0.00/0.07	 ACTIVE(h(X:S)) -> ACTIVE(X:S)
0.00/0.07	->->-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 TOP(mark(X:S)) -> TOP(proper(X:S))
0.00/0.07	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.07	->->-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	
0.00/0.07	
0.00/0.07	The problem is decomposed in 6 subproblems.
0.00/0.07	
0.00/0.07	Problem 1.1: 
0.00/0.07	
0.00/0.07	Subterm Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 H(mark(X:S)) -> H(X:S)
0.00/0.07	 H(ok(X:S)) -> H(X:S)
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Projection:
0.00/0.07	 pi(H) = 1
0.00/0.07	
0.00/0.07	Problem 1.1: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 Empty
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Strongly Connected Components:
0.00/0.07	 There is no strongly connected component
0.00/0.07	
0.00/0.07	The problem is finite.
0.00/0.07	
0.00/0.07	Problem 1.2: 
0.00/0.07	
0.00/0.07	Subterm Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 G(ok(X:S)) -> G(X:S)
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Projection:
0.00/0.07	 pi(G) = 1
0.00/0.07	
0.00/0.07	Problem 1.2: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 Empty
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Strongly Connected Components:
0.00/0.07	 There is no strongly connected component
0.00/0.07	
0.00/0.07	The problem is finite.
0.00/0.07	
0.00/0.07	Problem 1.3: 
0.00/0.07	
0.00/0.07	Subterm Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 F(mark(X:S)) -> F(X:S)
0.00/0.07	 F(ok(X:S)) -> F(X:S)
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Projection:
0.00/0.07	 pi(F) = 1
0.00/0.07	
0.00/0.07	Problem 1.3: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 Empty
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Strongly Connected Components:
0.00/0.07	 There is no strongly connected component
0.00/0.07	
0.00/0.07	The problem is finite.
0.00/0.07	
0.00/0.07	Problem 1.4: 
0.00/0.07	
0.00/0.07	Subterm Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 PROPER(f(X:S)) -> PROPER(X:S)
0.00/0.07	 PROPER(g(X:S)) -> PROPER(X:S)
0.00/0.07	 PROPER(h(X:S)) -> PROPER(X:S)
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Projection:
0.00/0.07	 pi(PROPER) = 1
0.00/0.07	
0.00/0.07	Problem 1.4: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 Empty
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Strongly Connected Components:
0.00/0.07	 There is no strongly connected component
0.00/0.07	
0.00/0.07	The problem is finite.
0.00/0.07	
0.00/0.07	Problem 1.5: 
0.00/0.07	
0.00/0.07	Subterm Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 ACTIVE(f(X:S)) -> ACTIVE(X:S)
0.00/0.07	 ACTIVE(h(X:S)) -> ACTIVE(X:S)
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Projection:
0.00/0.07	 pi(ACTIVE) = 1
0.00/0.07	
0.00/0.07	Problem 1.5: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 Empty
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Strongly Connected Components:
0.00/0.07	 There is no strongly connected component
0.00/0.07	
0.00/0.07	The problem is finite.
0.00/0.07	
0.00/0.07	Problem 1.6: 
0.00/0.07	
0.00/0.07	Reduction Pairs Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 TOP(mark(X:S)) -> TOP(proper(X:S))
0.00/0.07	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	-> Usable rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	->Interpretation type:
0.00/0.07	 Linear
0.00/0.07	->Coefficients:
0.00/0.07	 Natural Numbers
0.00/0.07	->Dimension:
0.00/0.07	 1
0.00/0.07	->Bound:
0.00/0.07	 2
0.00/0.07	->Interpretation:
0.00/0.07	 
0.00/0.07	[active](X) = 2.X + 2
0.00/0.07	[f](X) = X
0.00/0.07	[g](X) = X
0.00/0.07	[h](X) = X
0.00/0.07	[proper](X) = 1
0.00/0.07	[top](X) = 0
0.00/0.07	[fSNonEmpty] = 0
0.00/0.07	[mark](X) = 2
0.00/0.07	[ok](X) = 2.X + 2
0.00/0.07	[ACTIVE](X) = 0
0.00/0.07	[F](X) = 0
0.00/0.07	[G](X) = 0
0.00/0.07	[H](X) = 0
0.00/0.07	[PROPER](X) = 0
0.00/0.07	[TOP](X) = 2.X
0.00/0.07	
0.00/0.07	Problem 1.6: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Strongly Connected Components:
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.07	->->-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	
0.00/0.07	Problem 1.6: 
0.00/0.07	
0.00/0.07	Reduction Pairs Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	-> Usable rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	->Interpretation type:
0.00/0.07	 Linear
0.00/0.07	->Coefficients:
0.00/0.07	 Natural Numbers
0.00/0.07	->Dimension:
0.00/0.07	 1
0.00/0.07	->Bound:
0.00/0.07	 2
0.00/0.07	->Interpretation:
0.00/0.07	 
0.00/0.07	[active](X) = X
0.00/0.07	[f](X) = 2.X + 1
0.00/0.07	[g](X) = 2.X
0.00/0.07	[h](X) = 2.X + 1
0.00/0.07	[proper](X) = 0
0.00/0.07	[top](X) = 0
0.00/0.07	[fSNonEmpty] = 0
0.00/0.07	[mark](X) = 1
0.00/0.07	[ok](X) = X + 2
0.00/0.07	[ACTIVE](X) = 0
0.00/0.07	[F](X) = 0
0.00/0.07	[G](X) = 0
0.00/0.07	[H](X) = 0
0.00/0.07	[PROPER](X) = 0
0.00/0.07	[TOP](X) = X
0.00/0.07	
0.00/0.07	Problem 1.6: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 Empty
0.00/0.07	-> Rules:
0.00/0.07	 active(f(X:S)) -> f(active(X:S))
0.00/0.07	 active(f(X:S)) -> mark(g(h(f(X:S))))
0.00/0.07	 active(h(X:S)) -> h(active(X:S))
0.00/0.07	 f(mark(X:S)) -> mark(f(X:S))
0.00/0.07	 f(ok(X:S)) -> ok(f(X:S))
0.00/0.07	 g(ok(X:S)) -> ok(g(X:S))
0.00/0.07	 h(mark(X:S)) -> mark(h(X:S))
0.00/0.07	 h(ok(X:S)) -> ok(h(X:S))
0.00/0.07	 proper(f(X:S)) -> f(proper(X:S))
0.00/0.07	 proper(g(X:S)) -> g(proper(X:S))
0.00/0.07	 proper(h(X:S)) -> h(proper(X:S))
0.00/0.07	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.07	 top(ok(X:S)) -> top(active(X:S))
0.00/0.07	->Strongly Connected Components:
0.00/0.07	 There is no strongly connected component
0.00/0.07	
0.00/0.07	The problem is finite.
0.00/0.07	EOF