1.22/1.47	YES
1.22/1.47	
1.22/1.47	Problem 1: 
1.22/1.47	
1.22/1.47	(VAR x y)
1.22/1.47	(THEORY 
1.22/1.47	(AC plus times))
1.22/1.47	(RULES
1.22/1.47	div(0,s(y)) -> 0
1.22/1.47	div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	minus(x,0) -> x
1.22/1.47	plus(x,0) -> x
1.22/1.47	plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	times(x,0) -> 0
1.22/1.47	times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	)
1.22/1.47	
1.22/1.47	Problem 1: 
1.22/1.47	
1.22/1.47	Dependency Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
1.22/1.47	 DIV(s(x),s(y)) -> MINUS(x,y)
1.22/1.47	 MINUS(s(x),s(y)) -> MINUS(x,y)
1.22/1.47	 PLUS(plus(x,0),x2) -> PLUS(x,x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> PLUS(x,times(x,y))
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(plus(x,times(x,y)),x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
1.22/1.47	 TIMES(x,s(y)) -> PLUS(x,times(x,y))
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	
1.22/1.47	Problem 1: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
1.22/1.47	 DIV(s(x),s(y)) -> MINUS(x,y)
1.22/1.47	 MINUS(s(x),s(y)) -> MINUS(x,y)
1.22/1.47	 PLUS(plus(x,0),x2) -> PLUS(x,x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> PLUS(x,times(x,y))
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(plus(x,times(x,y)),x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
1.22/1.47	 TIMES(x,s(y)) -> PLUS(x,times(x,y))
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 PLUS(plus(x,0),x2) -> PLUS(x,x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) -> PLUS(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(plus(x,times(x,y)),x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) -> TIMES(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 MINUS(s(x),s(y)) -> MINUS(x,y)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 Empty
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 Empty
1.22/1.47	
1.22/1.47	
1.22/1.47	The problem is decomposed in 4 subproblems.
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 PLUS(plus(x,0),x2) -> PLUS(x,x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Interpretation type:
1.22/1.47	 Linear
1.22/1.47	->Coefficients:
1.22/1.47	 Natural Numbers
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 2
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 0
1.22/1.47	[plus](X1,X2) = X1 + X2 + 2
1.22/1.47	[times](X1,X2) = 0
1.22/1.47	[0] = 2
1.22/1.47	[s](X) = X + 1
1.22/1.47	[DIV](X1,X2) = 0
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 2.X1 + 2.X2
1.22/1.47	[TIMES](X1,X2) = 0
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) -> PLUS(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Interpretation type:
1.22/1.47	 Linear
1.22/1.47	->Coefficients:
1.22/1.47	 Natural Numbers
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 2
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 0
1.22/1.47	[plus](X1,X2) = X1 + X2 + 1
1.22/1.47	[times](X1,X2) = 0
1.22/1.47	[0] = 0
1.22/1.47	[s](X) = X + 2
1.22/1.47	[DIV](X1,X2) = 0
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 2.X1 + 2.X2
1.22/1.47	[TIMES](X1,X2) = 0
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) -> PLUS(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Interpretation type:
1.22/1.47	 Linear
1.22/1.47	->Coefficients:
1.22/1.47	 Natural Numbers
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 2
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 0
1.22/1.47	[plus](X1,X2) = X1 + X2 + 2
1.22/1.47	[times](X1,X2) = 0
1.22/1.47	[0] = 0
1.22/1.47	[s](X) = X + 1
1.22/1.47	[DIV](X1,X2) = 0
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 2.X1 + 2.X2
1.22/1.47	[TIMES](X1,X2) = 0
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) -> PLUS(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	 PLUS(x,s(y)) -> PLUS(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Interpretation type:
1.22/1.47	 Linear
1.22/1.47	->Coefficients:
1.22/1.47	 Natural Numbers
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 2
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 0
1.22/1.47	[plus](X1,X2) = X1 + X2 + 1
1.22/1.47	[times](X1,X2) = 0
1.22/1.47	[0] = 0
1.22/1.47	[s](X) = X + 2
1.22/1.47	[DIV](X1,X2) = 0
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 2.X1 + 2.X2
1.22/1.47	[TIMES](X1,X2) = 0
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) -> PLUS(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Interpretation type:
1.22/1.47	 Linear
1.22/1.47	->Coefficients:
1.22/1.47	 Natural Numbers
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 2
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 0
1.22/1.47	[plus](X1,X2) = X1 + X2 + 1
1.22/1.47	[times](X1,X2) = 0
1.22/1.47	[0] = 0
1.22/1.47	[s](X) = 2
1.22/1.47	[DIV](X1,X2) = 0
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 2.X1 + 2.X2
1.22/1.47	[TIMES](X1,X2) = 0
1.22/1.47	
1.22/1.47	Problem 1.1: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
1.22/1.47	 PLUS(x2,x3) = PLUS(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 Empty
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	 There is no strongly connected component
1.22/1.47	
1.22/1.47	The problem is finite.
1.22/1.47	
1.22/1.47	Problem 1.2: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(plus(x,times(x,y)),x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->Interpretation type:
1.22/1.47	 Simple mixed
1.22/1.47	->Coefficients:
1.22/1.47	 Natural Numbers
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 1
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 0
1.22/1.47	[plus](X1,X2) = X1 + X2
1.22/1.47	[times](X1,X2) = X1.X2 + X1 + X2
1.22/1.47	[0] = 1
1.22/1.47	[s](X) = X + 1
1.22/1.47	[DIV](X1,X2) = 0
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 0
1.22/1.47	[TIMES](X1,X2) = X1.X2 + X1 + X2
1.22/1.47	
1.22/1.47	Problem 1.2: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) -> TIMES(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	
1.22/1.47	Problem 1.2: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->Interpretation type:
1.22/1.47	 Simple mixed
1.22/1.47	->Coefficients:
1.22/1.47	 Natural Numbers
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 1
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 0
1.22/1.47	[plus](X1,X2) = X1 + X2 + 1
1.22/1.47	[times](X1,X2) = X1.X2 + X1 + X2
1.22/1.47	[0] = 1
1.22/1.47	[s](X) = X + 1
1.22/1.47	[DIV](X1,X2) = 0
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 0
1.22/1.47	[TIMES](X1,X2) = X1.X2 + X1 + X2
1.22/1.47	
1.22/1.47	Problem 1.2: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) -> TIMES(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	
1.22/1.47	Problem 1.2: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	 TIMES(x,s(y)) -> TIMES(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->Interpretation type:
1.22/1.47	 Simple mixed
1.22/1.47	->Coefficients:
1.22/1.47	 Natural Numbers
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 1
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 0
1.22/1.47	[plus](X1,X2) = X1 + X2 + 1
1.22/1.47	[times](X1,X2) = X1.X2 + X1 + X2
1.22/1.47	[0] = 1
1.22/1.47	[s](X) = X + 1
1.22/1.47	[DIV](X1,X2) = 0
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 0
1.22/1.47	[TIMES](X1,X2) = X1.X2 + X1 + X2
1.22/1.47	
1.22/1.47	Problem 1.2: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	->->Cycle:
1.22/1.47	->->-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	-> FAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) -> plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) -> times(x3,x2)
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) -> TIMES(x3,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	->->-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	
1.22/1.47	Problem 1.2: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 TIMES(times(x,0),x2) -> TIMES(0,x2)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->Interpretation type:
1.22/1.47	 Simple mixed
1.22/1.47	->Coefficients:
1.22/1.47	 All rationals
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 3
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 0
1.22/1.47	[plus](X1,X2) = X1 + X2 + 2/3
1.22/1.47	[times](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3
1.22/1.47	[0] = 1/3
1.22/1.47	[s](X) = X + 2
1.22/1.47	[DIV](X1,X2) = 0
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 0
1.22/1.47	[TIMES](X1,X2) = 1/3.X1.X2 + 1/2.X1 + 1/2.X2
1.22/1.47	
1.22/1.47	Problem 1.2: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
1.22/1.47	 TIMES(x2,x3) = TIMES(x3,x2)
1.22/1.47	-> Pairs:
1.22/1.47	 Empty
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
1.22/1.47	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
1.22/1.47	->Strongly Connected Components:
1.22/1.47	 There is no strongly connected component
1.22/1.47	
1.22/1.47	The problem is finite.
1.22/1.47	
1.22/1.47	Problem 1.3: 
1.22/1.47	
1.22/1.47	Subterm Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 Empty
1.22/1.47	-> Pairs:
1.22/1.47	 MINUS(s(x),s(y)) -> MINUS(x,y)
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 Empty
1.22/1.47	->Projection:
1.22/1.47	 pi(MINUS) = [1]
1.22/1.47	
1.22/1.47	Problem 1.3: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 Empty
1.22/1.47	-> Pairs:
1.22/1.47	 Empty
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 Empty
1.22/1.47	->Strongly Connected Components:
1.22/1.47	 There is no strongly connected component
1.22/1.47	
1.22/1.47	The problem is finite.
1.22/1.47	
1.22/1.47	Problem 1.4: 
1.22/1.47	
1.22/1.47	Reduction Pairs Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 Empty
1.22/1.47	-> Pairs:
1.22/1.47	 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Usable Equations:
1.22/1.47	 Empty
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> Usable Rules:
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	-> SRules:
1.22/1.47	 Empty
1.22/1.47	->Interpretation type:
1.22/1.47	 Linear
1.22/1.47	->Coefficients:
1.22/1.47	 Natural Numbers
1.22/1.47	->Dimension:
1.22/1.47	 1
1.22/1.47	->Bound:
1.22/1.47	 2
1.22/1.47	->Interpretation:
1.22/1.47	 
1.22/1.47	[div](X1,X2) = 0
1.22/1.47	[minus](X1,X2) = 2.X1 + 1
1.22/1.47	[plus](X1,X2) = 0
1.22/1.47	[times](X1,X2) = 0
1.22/1.47	[0] = 0
1.22/1.47	[s](X) = 2.X + 2
1.22/1.47	[DIV](X1,X2) = 2.X1
1.22/1.47	[MINUS](X1,X2) = 0
1.22/1.47	[PLUS](X1,X2) = 0
1.22/1.47	[TIMES](X1,X2) = 0
1.22/1.47	
1.22/1.47	Problem 1.4: 
1.22/1.47	
1.22/1.47	SCC Processor:
1.22/1.47	-> FAxioms:
1.22/1.47	 Empty
1.22/1.47	-> Pairs:
1.22/1.47	 Empty
1.22/1.47	-> EAxioms:
1.22/1.47	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
1.22/1.47	 plus(x2,x3) = plus(x3,x2)
1.22/1.47	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
1.22/1.47	 times(x2,x3) = times(x3,x2)
1.22/1.47	-> Rules:
1.22/1.47	 div(0,s(y)) -> 0
1.22/1.47	 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
1.22/1.47	 minus(s(x),s(y)) -> minus(x,y)
1.22/1.47	 minus(x,0) -> x
1.22/1.47	 plus(x,0) -> x
1.22/1.47	 plus(x,s(y)) -> s(plus(x,y))
1.22/1.47	 times(x,0) -> 0
1.22/1.47	 times(x,s(y)) -> plus(x,times(x,y))
1.22/1.47	-> SRules:
1.22/1.47	 Empty
1.22/1.47	->Strongly Connected Components:
1.22/1.47	 There is no strongly connected component
1.22/1.47	
1.22/1.47	The problem is finite.
1.22/1.47	EOF