4.14/4.71	YES
4.14/4.71	
4.14/4.71	Problem 1: 
4.14/4.71	
4.14/4.71	(VAR x y z)
4.14/4.71	(THEORY 
4.14/4.71	(AC plus times))
4.14/4.71	(RULES
4.14/4.71	plus(x,0) -> x
4.14/4.71	plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	times(x,0) -> 0
4.14/4.71	times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	)
4.14/4.71	
4.14/4.71	Problem 1: 
4.14/4.71	
4.14/4.71	Dependency Pairs Processor:
4.14/4.71	-> FAxioms:
4.14/4.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.71	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.71	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.71	-> Pairs:
4.14/4.71	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(x,y)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z))
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y)
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z)
4.14/4.71	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.71	 TIMES(times(x,s(y)),x3) -> PLUS(times(x,y),x)
4.14/4.71	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.71	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.71	 TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z))
4.14/4.71	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.71	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.71	 TIMES(x,s(y)) -> PLUS(times(x,y),x)
4.14/4.71	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.71	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.71	
4.14/4.71	Problem 1: 
4.14/4.71	
4.14/4.71	SCC Processor:
4.14/4.71	-> FAxioms:
4.14/4.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.71	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.71	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.71	-> Pairs:
4.14/4.71	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(x,y)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z))
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y)
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z)
4.14/4.71	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.71	 TIMES(times(x,s(y)),x3) -> PLUS(times(x,y),x)
4.14/4.71	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.71	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.71	 TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z))
4.14/4.71	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.71	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.71	 TIMES(x,s(y)) -> PLUS(times(x,y),x)
4.14/4.71	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.71	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.71	->Strongly Connected Components:
4.14/4.71	->->Cycle:
4.14/4.71	->->-> Pairs:
4.14/4.71	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(x,y)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	-> FAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) -> plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) -> times(x4,x3)
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) -> PLUS(x4,x3)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	->->-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	->->Cycle:
4.14/4.71	->->-> Pairs:
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y)
4.14/4.71	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z)
4.14/4.71	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.71	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.71	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.71	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.71	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.71	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.71	-> FAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) -> plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) -> times(x4,x3)
4.14/4.71	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
4.14/4.71	 TIMES(x3,x4) -> TIMES(x4,x3)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	->->-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> SRules:
4.14/4.71	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.71	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.71	
4.14/4.71	
4.14/4.71	The problem is decomposed in 2 subproblems.
4.14/4.71	
4.14/4.71	Problem 1.1: 
4.14/4.71	
4.14/4.71	Reduction Pairs Processor:
4.14/4.71	-> FAxioms:
4.14/4.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.71	-> Pairs:
4.14/4.71	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(x,y)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	-> Usable Equations:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> Usable Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	->Interpretation type:
4.14/4.71	 Linear
4.14/4.71	->Coefficients:
4.14/4.71	 Natural Numbers
4.14/4.71	->Dimension:
4.14/4.71	 1
4.14/4.71	->Bound:
4.14/4.71	 2
4.14/4.71	->Interpretation:
4.14/4.71	 
4.14/4.71	[plus](X1,X2) = X1 + X2 + 2
4.14/4.71	[times](X1,X2) = 0
4.14/4.71	[0] = 2
4.14/4.71	[s](X) = X + 1
4.14/4.71	[PLUS](X1,X2) = 2.X1 + 2.X2
4.14/4.71	[TIMES](X1,X2) = 0
4.14/4.71	
4.14/4.71	Problem 1.1: 
4.14/4.71	
4.14/4.71	SCC Processor:
4.14/4.71	-> FAxioms:
4.14/4.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.71	-> Pairs:
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(x,y)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	->Strongly Connected Components:
4.14/4.71	->->Cycle:
4.14/4.71	->->-> Pairs:
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(x,y)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	-> FAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) -> plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) -> times(x4,x3)
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) -> PLUS(x4,x3)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	->->-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	
4.14/4.71	Problem 1.1: 
4.14/4.71	
4.14/4.71	Reduction Pairs Processor:
4.14/4.71	-> FAxioms:
4.14/4.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.71	-> Pairs:
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(x,y)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	-> Usable Equations:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> Usable Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	->Interpretation type:
4.14/4.71	 Linear
4.14/4.71	->Coefficients:
4.14/4.71	 Natural Numbers
4.14/4.71	->Dimension:
4.14/4.71	 1
4.14/4.71	->Bound:
4.14/4.71	 2
4.14/4.71	->Interpretation:
4.14/4.71	 
4.14/4.71	[plus](X1,X2) = X1 + X2 + 1
4.14/4.71	[times](X1,X2) = 0
4.14/4.71	[0] = 0
4.14/4.71	[s](X) = X + 2
4.14/4.71	[PLUS](X1,X2) = 2.X1 + 2.X2
4.14/4.71	[TIMES](X1,X2) = 0
4.14/4.71	
4.14/4.71	Problem 1.1: 
4.14/4.71	
4.14/4.71	SCC Processor:
4.14/4.71	-> FAxioms:
4.14/4.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.71	-> Pairs:
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	->Strongly Connected Components:
4.14/4.71	->->Cycle:
4.14/4.71	->->-> Pairs:
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	-> FAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) -> plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) -> times(x4,x3)
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) -> PLUS(x4,x3)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	->->-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	
4.14/4.71	Problem 1.1: 
4.14/4.71	
4.14/4.71	Reduction Pairs Processor:
4.14/4.71	-> FAxioms:
4.14/4.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.71	-> Pairs:
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.71	 times(x3,x4) = times(x4,x3)
4.14/4.71	-> Usable Equations:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.71	 plus(x3,x4) = plus(x4,x3)
4.14/4.71	-> Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.71	 times(x,0) -> 0
4.14/4.71	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.71	-> Usable Rules:
4.14/4.71	 plus(x,0) -> x
4.14/4.71	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.71	-> SRules:
4.14/4.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
4.14/4.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.71	->Interpretation type:
4.14/4.71	 Linear
4.14/4.71	->Coefficients:
4.14/4.71	 Natural Numbers
4.14/4.71	->Dimension:
4.14/4.71	 1
4.14/4.71	->Bound:
4.14/4.71	 2
4.14/4.71	->Interpretation:
4.14/4.71	 
4.14/4.71	[plus](X1,X2) = X1 + X2 + 2
4.14/4.71	[times](X1,X2) = 0
4.14/4.71	[0] = 0
4.14/4.71	[s](X) = X + 1
4.14/4.71	[PLUS](X1,X2) = 2.X1 + 2.X2
4.14/4.71	[TIMES](X1,X2) = 0
4.14/4.71	
4.14/4.71	Problem 1.1: 
4.14/4.71	
4.14/4.71	SCC Processor:
4.14/4.71	-> FAxioms:
4.14/4.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.71	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.71	-> Pairs:
4.14/4.71	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.71	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.71	-> EAxioms:
4.14/4.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.72	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
4.14/4.72	 PLUS(x3,x4) -> PLUS(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.1: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.72	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.72	 PLUS(x,s(y)) -> PLUS(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	-> SRules:
4.14/4.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Linear
4.14/4.72	->Coefficients:
4.14/4.72	 Natural Numbers
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 2
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 1
4.14/4.72	[times](X1,X2) = 0
4.14/4.72	[0] = 0
4.14/4.72	[s](X) = X + 2
4.14/4.72	[PLUS](X1,X2) = 2.X1 + 2.X2
4.14/4.72	[TIMES](X1,X2) = 0
4.14/4.72	
4.14/4.72	Problem 1.1: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.72	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
4.14/4.72	 PLUS(x3,x4) -> PLUS(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.1: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.72	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	-> SRules:
4.14/4.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Linear
4.14/4.72	->Coefficients:
4.14/4.72	 Natural Numbers
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 2
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 1
4.14/4.72	[times](X1,X2) = 0
4.14/4.72	[0] = 0
4.14/4.72	[s](X) = 2
4.14/4.72	[PLUS](X1,X2) = 2.X1 + 2.X2
4.14/4.72	[TIMES](X1,X2) = 0
4.14/4.72	
4.14/4.72	Problem 1.1: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
4.14/4.72	 PLUS(x3,x4) = PLUS(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 Empty
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	 There is no strongly connected component
4.14/4.72	
4.14/4.72	The problem is finite.
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y)
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Simple mixed
4.14/4.72	->Coefficients:
4.14/4.72	 Natural Numbers
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 1
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 1
4.14/4.72	[times](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	[0] = 1
4.14/4.72	[s](X) = X + 1
4.14/4.72	[PLUS](X1,X2) = 0
4.14/4.72	[TIMES](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) -> TIMES(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Simple mixed
4.14/4.72	->Coefficients:
4.14/4.72	 Natural Numbers
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 1
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 1
4.14/4.72	[times](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	[0] = 1
4.14/4.72	[s](X) = X + 1
4.14/4.72	[PLUS](X1,X2) = 0
4.14/4.72	[TIMES](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) -> TIMES(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Simple mixed
4.14/4.72	->Coefficients:
4.14/4.72	 Natural Numbers
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 1
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 1
4.14/4.72	[times](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	[0] = 1
4.14/4.72	[s](X) = X + 1
4.14/4.72	[PLUS](X1,X2) = 0
4.14/4.72	[TIMES](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) -> TIMES(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,y)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Simple mixed
4.14/4.72	->Coefficients:
4.14/4.72	 Natural Numbers
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 1
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 1
4.14/4.72	[times](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	[0] = 1
4.14/4.72	[s](X) = X + 1
4.14/4.72	[PLUS](X1,X2) = 0
4.14/4.72	[TIMES](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) -> TIMES(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(x,plus(y,z)) -> TIMES(x,z)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Simple mixed
4.14/4.72	->Coefficients:
4.14/4.72	 Natural Numbers
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 1
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 1
4.14/4.72	[times](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	[0] = 1
4.14/4.72	[s](X) = X + 1
4.14/4.72	[PLUS](X1,X2) = 0
4.14/4.72	[TIMES](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) -> TIMES(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	 TIMES(x,s(y)) -> TIMES(x,y)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Simple mixed
4.14/4.72	->Coefficients:
4.14/4.72	 Natural Numbers
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 1
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 1
4.14/4.72	[times](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	[0] = 1
4.14/4.72	[s](X) = X + 1
4.14/4.72	[PLUS](X1,X2) = 0
4.14/4.72	[TIMES](X1,X2) = X1.X2 + X1 + X2
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) -> TIMES(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3)
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Simple mixed
4.14/4.72	->Coefficients:
4.14/4.72	 All rationals
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 3
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 3/2
4.14/4.72	[times](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2
4.14/4.72	[0] = 3
4.14/4.72	[s](X) = X + 1
4.14/4.72	[PLUS](X1,X2) = 0
4.14/4.72	[TIMES](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) -> TIMES(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Simple mixed
4.14/4.72	->Coefficients:
4.14/4.72	 All rationals
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 3
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 2/3
4.14/4.72	[times](X1,X2) = 3.X1.X2 + 2.X1 + 2.X2 + 2/3
4.14/4.72	[0] = 0
4.14/4.72	[s](X) = X + 1
4.14/4.72	[PLUS](X1,X2) = 0
4.14/4.72	[TIMES](X1,X2) = 1/2.X1.X2 + 1/3.X1 + 1/3.X2
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	->->Cycle:
4.14/4.72	->->-> Pairs:
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	-> FAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) -> plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) -> times(x4,x3)
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) -> TIMES(x4,x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	->->-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	Reduction Pairs Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3)
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Usable Equations:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> Usable Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Interpretation type:
4.14/4.72	 Simple mixed
4.14/4.72	->Coefficients:
4.14/4.72	 All rationals
4.14/4.72	->Dimension:
4.14/4.72	 1
4.14/4.72	->Bound:
4.14/4.72	 3
4.14/4.72	->Interpretation:
4.14/4.72	 
4.14/4.72	[plus](X1,X2) = X1 + X2 + 2
4.14/4.72	[times](X1,X2) = 3.X1.X2 + X1 + X2
4.14/4.72	[0] = 3/2
4.14/4.72	[s](X) = X + 3
4.14/4.72	[PLUS](X1,X2) = 0
4.14/4.72	[TIMES](X1,X2) = X1.X2 + 1/3.X1 + 1/3.X2
4.14/4.72	
4.14/4.72	Problem 1.2: 
4.14/4.72	
4.14/4.72	SCC Processor:
4.14/4.72	-> FAxioms:
4.14/4.72	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
4.14/4.72	 TIMES(x3,x4) = TIMES(x4,x3)
4.14/4.72	-> Pairs:
4.14/4.72	 Empty
4.14/4.72	-> EAxioms:
4.14/4.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
4.14/4.72	 plus(x3,x4) = plus(x4,x3)
4.14/4.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
4.14/4.72	 times(x3,x4) = times(x4,x3)
4.14/4.72	-> Rules:
4.14/4.72	 plus(x,0) -> x
4.14/4.72	 plus(x,s(y)) -> s(plus(x,y))
4.14/4.72	 times(x,plus(y,z)) -> plus(times(x,y),times(x,z))
4.14/4.72	 times(x,0) -> 0
4.14/4.72	 times(x,s(y)) -> plus(times(x,y),x)
4.14/4.72	-> SRules:
4.14/4.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
4.14/4.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
4.14/4.72	->Strongly Connected Components:
4.14/4.72	 There is no strongly connected component
4.14/4.72	
4.14/4.72	The problem is finite.
4.14/4.72	EOF