0.00/0.12	YES
0.00/0.12	
0.00/0.12	Problem 1: 
0.00/0.12	
0.00/0.12	(VAR v_NonEmpty:S x:S y:S)
0.00/0.12	(RULES
0.00/0.12	even(0) -> ttrue
0.00/0.12	even(s(0)) -> ffalse
0.00/0.12	even(s(s(x:S))) -> even(x:S)
0.00/0.12	half(0) -> 0
0.00/0.12	half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	plus(0,y:S) -> y:S
0.00/0.12	plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	times(0,y:S) -> 0
0.00/0.12	times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	) 
0.00/0.12	(STRATEGY INNERMOST)
0.00/0.12	
0.00/0.12	Problem 1: 
0.00/0.12	
0.00/0.12	Dependency Pairs Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 EVEN(s(s(x:S))) -> EVEN(x:S)
0.00/0.12	 HALF(s(s(x:S))) -> HALF(x:S)
0.00/0.12	 IF_TIMES(ffalse,s(x:S),y:S) -> PLUS(y:S,times(x:S,y:S))
0.00/0.12	 IF_TIMES(ffalse,s(x:S),y:S) -> TIMES(x:S,y:S)
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> HALF(s(x:S))
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> PLUS(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
0.00/0.12	 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
0.00/0.12	 TIMES(s(x:S),y:S) -> EVEN(s(x:S))
0.00/0.12	 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	
0.00/0.12	Problem 1: 
0.00/0.12	
0.00/0.12	SCC Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 EVEN(s(s(x:S))) -> EVEN(x:S)
0.00/0.12	 HALF(s(s(x:S))) -> HALF(x:S)
0.00/0.12	 IF_TIMES(ffalse,s(x:S),y:S) -> PLUS(y:S,times(x:S,y:S))
0.00/0.12	 IF_TIMES(ffalse,s(x:S),y:S) -> TIMES(x:S,y:S)
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> HALF(s(x:S))
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> PLUS(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
0.00/0.12	 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
0.00/0.12	 TIMES(s(x:S),y:S) -> EVEN(s(x:S))
0.00/0.12	 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->Strongly Connected Components:
0.00/0.12	->->Cycle:
0.00/0.12	->->-> Pairs:
0.00/0.12	 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
0.00/0.12	->->-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->->Cycle:
0.00/0.12	->->-> Pairs:
0.00/0.12	 HALF(s(s(x:S))) -> HALF(x:S)
0.00/0.12	->->-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->->Cycle:
0.00/0.12	->->-> Pairs:
0.00/0.12	 EVEN(s(s(x:S))) -> EVEN(x:S)
0.00/0.12	->->-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->->Cycle:
0.00/0.12	->->-> Pairs:
0.00/0.12	 IF_TIMES(ffalse,s(x:S),y:S) -> TIMES(x:S,y:S)
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
0.00/0.12	 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->->-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	
0.00/0.12	
0.00/0.12	The problem is decomposed in 4 subproblems.
0.00/0.12	
0.00/0.12	Problem 1.1: 
0.00/0.12	
0.00/0.12	Subterm Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->Projection:
0.00/0.12	 pi(PLUS) = 1
0.00/0.12	
0.00/0.12	Problem 1.1: 
0.00/0.12	
0.00/0.12	SCC Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 Empty
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->Strongly Connected Components:
0.00/0.12	 There is no strongly connected component
0.00/0.12	
0.00/0.12	The problem is finite.
0.00/0.12	
0.00/0.12	Problem 1.2: 
0.00/0.12	
0.00/0.12	Subterm Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 HALF(s(s(x:S))) -> HALF(x:S)
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->Projection:
0.00/0.12	 pi(HALF) = 1
0.00/0.12	
0.00/0.12	Problem 1.2: 
0.00/0.12	
0.00/0.12	SCC Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 Empty
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->Strongly Connected Components:
0.00/0.12	 There is no strongly connected component
0.00/0.12	
0.00/0.12	The problem is finite.
0.00/0.12	
0.00/0.12	Problem 1.3: 
0.00/0.12	
0.00/0.12	Subterm Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 EVEN(s(s(x:S))) -> EVEN(x:S)
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->Projection:
0.00/0.12	 pi(EVEN) = 1
0.00/0.12	
0.00/0.12	Problem 1.3: 
0.00/0.12	
0.00/0.12	SCC Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 Empty
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->Strongly Connected Components:
0.00/0.12	 There is no strongly connected component
0.00/0.12	
0.00/0.12	The problem is finite.
0.00/0.12	
0.00/0.12	Problem 1.4: 
0.00/0.12	
0.00/0.12	Reduction Pairs Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 IF_TIMES(ffalse,s(x:S),y:S) -> TIMES(x:S,y:S)
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
0.00/0.12	 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	-> Usable rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	->Interpretation type:
0.00/0.12	 Linear
0.00/0.12	->Coefficients:
0.00/0.12	 Natural Numbers
0.00/0.12	->Dimension:
0.00/0.12	 1
0.00/0.12	->Bound:
0.00/0.12	 2
0.00/0.12	->Interpretation:
0.00/0.12	 
0.00/0.12	[even](X) = 1
0.00/0.12	[half](X) = X
0.00/0.12	[if_times](X1,X2,X3) = 0
0.00/0.12	[plus](X1,X2) = 0
0.00/0.12	[times](X1,X2) = 0
0.00/0.12	[0] = 0
0.00/0.12	[fSNonEmpty] = 0
0.00/0.12	[false] = 1
0.00/0.12	[s](X) = 2.X + 2
0.00/0.12	[true] = 1
0.00/0.12	[EVEN](X) = 0
0.00/0.12	[HALF](X) = 0
0.00/0.12	[IF_TIMES](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
0.00/0.12	[PLUS](X1,X2) = 0
0.00/0.12	[TIMES](X1,X2) = 2.X1 + 2.X2 + 2
0.00/0.12	
0.00/0.12	Problem 1.4: 
0.00/0.12	
0.00/0.12	SCC Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
0.00/0.12	 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->Strongly Connected Components:
0.00/0.12	->->Cycle:
0.00/0.12	->->-> Pairs:
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
0.00/0.12	 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->->-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	
0.00/0.12	Problem 1.4: 
0.00/0.12	
0.00/0.12	Reduction Pairs Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
0.00/0.12	 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	-> Usable rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	->Interpretation type:
0.00/0.12	 Linear
0.00/0.12	->Coefficients:
0.00/0.12	 All rationals
0.00/0.12	->Dimension:
0.00/0.12	 1
0.00/0.12	->Bound:
0.00/0.12	 2
0.00/0.12	->Interpretation:
0.00/0.12	 
0.00/0.12	[even](X) = 1/2.X + 1/2
0.00/0.12	[half](X) = 1/2.X + 1/2
0.00/0.12	[if_times](X1,X2,X3) = 0
0.00/0.12	[plus](X1,X2) = 0
0.00/0.12	[times](X1,X2) = 0
0.00/0.12	[0] = 1
0.00/0.12	[fSNonEmpty] = 0
0.00/0.12	[false] = 1/2
0.00/0.12	[s](X) = 2.X + 2
0.00/0.12	[true] = 1
0.00/0.12	[EVEN](X) = 0
0.00/0.12	[HALF](X) = 0
0.00/0.12	[IF_TIMES](X1,X2,X3) = X1 + X2 + 1/2.X3 + 1/2
0.00/0.12	[PLUS](X1,X2) = 0
0.00/0.12	[TIMES](X1,X2) = 2.X1 + 1/2.X2
0.00/0.12	
0.00/0.12	Problem 1.4: 
0.00/0.12	
0.00/0.12	SCC Processor:
0.00/0.12	-> Pairs:
0.00/0.12	 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
0.00/0.12	-> Rules:
0.00/0.12	 even(0) -> ttrue
0.00/0.12	 even(s(0)) -> ffalse
0.00/0.12	 even(s(s(x:S))) -> even(x:S)
0.00/0.12	 half(0) -> 0
0.00/0.12	 half(s(s(x:S))) -> s(half(x:S))
0.00/0.12	 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
0.00/0.12	 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
0.00/0.12	 plus(0,y:S) -> y:S
0.00/0.12	 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
0.00/0.12	 times(0,y:S) -> 0
0.00/0.12	 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
0.00/0.12	->Strongly Connected Components:
0.00/0.12	 There is no strongly connected component
0.00/0.12	
0.00/0.12	The problem is finite.
0.00/0.12	EOF