0.00/0.33	YES
0.00/0.33	
0.00/0.33	Problem 1: 
0.00/0.33	
0.00/0.33	(VAR m n x y)
0.00/0.33	(THEORY 
0.00/0.33	(C eq))
0.00/0.33	(RULES
0.00/0.33	eq(0,0) -> true
0.00/0.33	eq(0,s(x)) -> false
0.00/0.33	eq(s(x),0) -> false
0.00/0.33	eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	purge(nil) -> nil
0.00/0.33	rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	rm(n,nil) -> nil
0.00/0.33	)
0.00/0.33	
0.00/0.33	Problem 1: 
0.00/0.33	
0.00/0.33	Dependency Pairs Processor:
0.00/0.33	-> FAxioms:
0.00/0.33	 EQ(x4,x5) = EQ(x5,x4)
0.00/0.33	-> Pairs:
0.00/0.33	 EQ(s(x),s(y)) -> EQ(x,y)
0.00/0.33	 IF_RM(false,n,add(m,x)) -> RM(n,x)
0.00/0.33	 IF_RM(true,n,add(m,x)) -> RM(n,x)
0.00/0.33	 PURGE(add(n,x)) -> PURGE(rm(n,x))
0.00/0.33	 PURGE(add(n,x)) -> RM(n,x)
0.00/0.33	 RM(n,add(m,x)) -> EQ(n,m)
0.00/0.33	 RM(n,add(m,x)) -> IF_RM(eq(n,m),n,add(m,x))
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	
0.00/0.33	Problem 1: 
0.00/0.33	
0.00/0.33	SCC Processor:
0.00/0.33	-> FAxioms:
0.00/0.33	 EQ(x4,x5) = EQ(x5,x4)
0.00/0.33	-> Pairs:
0.00/0.33	 EQ(s(x),s(y)) -> EQ(x,y)
0.00/0.33	 IF_RM(false,n,add(m,x)) -> RM(n,x)
0.00/0.33	 IF_RM(true,n,add(m,x)) -> RM(n,x)
0.00/0.33	 PURGE(add(n,x)) -> PURGE(rm(n,x))
0.00/0.33	 PURGE(add(n,x)) -> RM(n,x)
0.00/0.33	 RM(n,add(m,x)) -> EQ(n,m)
0.00/0.33	 RM(n,add(m,x)) -> IF_RM(eq(n,m),n,add(m,x))
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	->Strongly Connected Components:
0.00/0.33	->->Cycle:
0.00/0.33	->->-> Pairs:
0.00/0.33	 EQ(s(x),s(y)) -> EQ(x,y)
0.00/0.33	-> FAxioms:
0.00/0.33	 eq(x4,x5) -> eq(x5,x4)
0.00/0.33	 EQ(x4,x5) -> EQ(x5,x4)
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	->->-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	->->Cycle:
0.00/0.33	->->-> Pairs:
0.00/0.33	 IF_RM(false,n,add(m,x)) -> RM(n,x)
0.00/0.33	 IF_RM(true,n,add(m,x)) -> RM(n,x)
0.00/0.33	 RM(n,add(m,x)) -> IF_RM(eq(n,m),n,add(m,x))
0.00/0.33	-> FAxioms:
0.00/0.33	 eq(x4,x5) -> eq(x5,x4)
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	->->-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	->->Cycle:
0.00/0.33	->->-> Pairs:
0.00/0.33	 PURGE(add(n,x)) -> PURGE(rm(n,x))
0.00/0.33	-> FAxioms:
0.00/0.33	 eq(x4,x5) -> eq(x5,x4)
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	->->-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	
0.00/0.33	
0.00/0.33	The problem is decomposed in 3 subproblems.
0.00/0.33	
0.00/0.33	Problem 1.1: 
0.00/0.33	
0.00/0.33	Subterm Processor:
0.00/0.33	-> FAxioms:
0.00/0.33	 EQ(x4,x5) = EQ(x5,x4)
0.00/0.33	-> Pairs:
0.00/0.33	 EQ(s(x),s(y)) -> EQ(x,y)
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	->Projection:
0.00/0.33	 pi(EQ) = [1,2]
0.00/0.33	
0.00/0.33	Problem 1.1: 
0.00/0.33	
0.00/0.33	SCC Processor:
0.00/0.33	-> FAxioms:
0.00/0.33	 EQ(x4,x5) = EQ(x5,x4)
0.00/0.33	-> Pairs:
0.00/0.33	 Empty
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	->Strongly Connected Components:
0.00/0.33	 There is no strongly connected component
0.00/0.33	
0.00/0.33	The problem is finite.
0.00/0.33	
0.00/0.33	Problem 1.2: 
0.00/0.33	
0.00/0.33	Subterm Processor:
0.00/0.33	-> FAxioms:
0.00/0.33	 Empty
0.00/0.33	-> Pairs:
0.00/0.33	 IF_RM(false,n,add(m,x)) -> RM(n,x)
0.00/0.33	 IF_RM(true,n,add(m,x)) -> RM(n,x)
0.00/0.33	 RM(n,add(m,x)) -> IF_RM(eq(n,m),n,add(m,x))
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	->Projection:
0.00/0.33	 pi(IF_RM) = [3]
0.00/0.33	 pi(RM) = [2]
0.00/0.33	
0.00/0.33	Problem 1.2: 
0.00/0.33	
0.00/0.33	SCC Processor:
0.00/0.33	-> FAxioms:
0.00/0.33	 Empty
0.00/0.33	-> Pairs:
0.00/0.33	 RM(n,add(m,x)) -> IF_RM(eq(n,m),n,add(m,x))
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	->Strongly Connected Components:
0.00/0.33	 There is no strongly connected component
0.00/0.33	
0.00/0.33	The problem is finite.
0.00/0.33	
0.00/0.33	Problem 1.3: 
0.00/0.33	
0.00/0.33	Reduction Pairs Processor:
0.00/0.33	-> FAxioms:
0.00/0.33	 Empty
0.00/0.33	-> Pairs:
0.00/0.33	 PURGE(add(n,x)) -> PURGE(rm(n,x))
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	-> Usable Equations:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> Usable Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	->Interpretation type:
0.00/0.33	 Linear
0.00/0.33	->Coefficients:
0.00/0.33	 Natural Numbers
0.00/0.33	->Dimension:
0.00/0.33	 1
0.00/0.33	->Bound:
0.00/0.33	 2
0.00/0.33	->Interpretation:
0.00/0.33	 
0.00/0.33	[eq](X1,X2) = 2
0.00/0.33	[if_rm](X1,X2,X3) = 2.X3 + 1
0.00/0.33	[purge](X) = 0
0.00/0.33	[rm](X1,X2) = 2.X2 + 1
0.00/0.33	[0] = 2
0.00/0.33	[add](X1,X2) = 2.X1 + 2.X2 + 2
0.00/0.33	[false] = 0
0.00/0.33	[nil] = 2
0.00/0.33	[s](X) = 0
0.00/0.33	[true] = 1
0.00/0.33	[EQ](X1,X2) = 0
0.00/0.33	[IF_RM](X1,X2,X3) = 0
0.00/0.33	[PURGE](X) = 2.X
0.00/0.33	[RM](X1,X2) = 0
0.00/0.33	
0.00/0.33	Problem 1.3: 
0.00/0.33	
0.00/0.33	SCC Processor:
0.00/0.33	-> FAxioms:
0.00/0.33	 Empty
0.00/0.33	-> Pairs:
0.00/0.33	 Empty
0.00/0.33	-> EAxioms:
0.00/0.33	 eq(x4,x5) = eq(x5,x4)
0.00/0.33	-> Rules:
0.00/0.33	 eq(0,0) -> true
0.00/0.33	 eq(0,s(x)) -> false
0.00/0.33	 eq(s(x),0) -> false
0.00/0.33	 eq(s(x),s(y)) -> eq(x,y)
0.00/0.33	 if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
0.00/0.33	 if_rm(true,n,add(m,x)) -> rm(n,x)
0.00/0.33	 purge(add(n,x)) -> add(n,purge(rm(n,x)))
0.00/0.33	 purge(nil) -> nil
0.00/0.33	 rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
0.00/0.33	 rm(n,nil) -> nil
0.00/0.33	-> SRules:
0.00/0.33	 Empty
0.00/0.33	->Strongly Connected Components:
0.00/0.33	 There is no strongly connected component
0.00/0.33	
0.00/0.33	The problem is finite.
0.00/0.33	EOF