3.17/3.70	YES
3.17/3.70	
3.17/3.70	Problem 1: 
3.17/3.70	
3.17/3.70	(VAR x y)
3.17/3.70	(THEORY 
3.17/3.70	(AC plus times))
3.17/3.70	(RULES
3.17/3.70	i(i(x)) -> x
3.17/3.70	i(p(x)) -> s(i(x))
3.17/3.70	i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	i(s(x)) -> p(i(x))
3.17/3.70	i(0) -> 0
3.17/3.70	p(s(x)) -> x
3.17/3.70	plus(i(x),x) -> 0
3.17/3.70	plus(p(x),y) -> p(plus(x,y))
3.17/3.70	plus(s(x),y) -> s(plus(x,y))
3.17/3.70	plus(0,y) -> y
3.17/3.70	plus(x,plus(i(x),y)) -> y
3.17/3.70	s(p(x)) -> x
3.17/3.70	times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	times(0,y) -> 0
3.17/3.70	)
3.17/3.70	
3.17/3.70	Problem 1: 
3.17/3.70	
3.17/3.70	Dependency Pairs Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.70	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 I(p(x)) -> I(x)
3.17/3.70	 I(p(x)) -> S(i(x))
3.17/3.70	 I(plus(x,y)) -> I(x)
3.17/3.70	 I(plus(x,y)) -> I(y)
3.17/3.70	 I(plus(x,y)) -> PLUS(i(y),i(x))
3.17/3.70	 I(s(x)) -> I(x)
3.17/3.70	 I(s(x)) -> P(i(x))
3.17/3.70	 PLUS(p(x),y) -> P(plus(x,y))
3.17/3.70	 PLUS(p(x),y) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(i(x),x),x2) -> PLUS(0,x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> P(plus(x,y))
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> S(plus(x,y))
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	 PLUS(s(x),y) -> S(plus(x,y))
3.17/3.70	 TIMES(p(x),y) -> I(y)
3.17/3.70	 TIMES(p(x),y) -> PLUS(times(x,y),i(y))
3.17/3.70	 TIMES(p(x),y) -> TIMES(x,y)
3.17/3.70	 TIMES(s(x),y) -> PLUS(times(x,y),y)
3.17/3.70	 TIMES(s(x),y) -> TIMES(x,y)
3.17/3.70	 TIMES(times(p(x),y),x2) -> I(y)
3.17/3.70	 TIMES(times(p(x),y),x2) -> PLUS(times(x,y),i(y))
3.17/3.70	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.70	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.70	 TIMES(times(s(x),y),x2) -> PLUS(times(x,y),y)
3.17/3.70	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.70	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.70	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.70	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.70	
3.17/3.70	Problem 1: 
3.17/3.70	
3.17/3.70	SCC Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.70	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 I(p(x)) -> I(x)
3.17/3.70	 I(p(x)) -> S(i(x))
3.17/3.70	 I(plus(x,y)) -> I(x)
3.17/3.70	 I(plus(x,y)) -> I(y)
3.17/3.70	 I(plus(x,y)) -> PLUS(i(y),i(x))
3.17/3.70	 I(s(x)) -> I(x)
3.17/3.70	 I(s(x)) -> P(i(x))
3.17/3.70	 PLUS(p(x),y) -> P(plus(x,y))
3.17/3.70	 PLUS(p(x),y) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(i(x),x),x2) -> PLUS(0,x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> P(plus(x,y))
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> S(plus(x,y))
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	 PLUS(s(x),y) -> S(plus(x,y))
3.17/3.70	 TIMES(p(x),y) -> I(y)
3.17/3.70	 TIMES(p(x),y) -> PLUS(times(x,y),i(y))
3.17/3.70	 TIMES(p(x),y) -> TIMES(x,y)
3.17/3.70	 TIMES(s(x),y) -> PLUS(times(x,y),y)
3.17/3.70	 TIMES(s(x),y) -> TIMES(x,y)
3.17/3.70	 TIMES(times(p(x),y),x2) -> I(y)
3.17/3.70	 TIMES(times(p(x),y),x2) -> PLUS(times(x,y),i(y))
3.17/3.70	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.70	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.70	 TIMES(times(s(x),y),x2) -> PLUS(times(x,y),y)
3.17/3.70	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.70	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.70	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.70	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.70	->Strongly Connected Components:
3.17/3.70	->->Cycle:
3.17/3.70	->->-> Pairs:
3.17/3.70	 PLUS(p(x),y) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(i(x),x),x2) -> PLUS(0,x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> FAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) -> plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) -> times(x3,x2)
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	->->-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->->Cycle:
3.17/3.70	->->-> Pairs:
3.17/3.70	 I(p(x)) -> I(x)
3.17/3.70	 I(plus(x,y)) -> I(x)
3.17/3.70	 I(plus(x,y)) -> I(y)
3.17/3.70	 I(s(x)) -> I(x)
3.17/3.70	-> FAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) -> plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) -> times(x3,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	->->-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 Empty
3.17/3.70	->->Cycle:
3.17/3.70	->->-> Pairs:
3.17/3.70	 TIMES(p(x),y) -> TIMES(x,y)
3.17/3.70	 TIMES(s(x),y) -> TIMES(x,y)
3.17/3.70	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.70	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.70	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.70	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.70	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.70	-> FAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) -> plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) -> times(x3,x2)
3.17/3.70	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
3.17/3.70	 TIMES(x2,x3) -> TIMES(x3,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	->->-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.70	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.70	
3.17/3.70	
3.17/3.70	The problem is decomposed in 3 subproblems.
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	Reduction Pairs Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(p(x),y) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(i(x),x),x2) -> PLUS(0,x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Usable Equations:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> Usable Rules:
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Interpretation type:
3.17/3.70	 Linear
3.17/3.70	->Coefficients:
3.17/3.70	 Natural Numbers
3.17/3.70	->Dimension:
3.17/3.70	 1
3.17/3.70	->Bound:
3.17/3.70	 2
3.17/3.70	->Interpretation:
3.17/3.70	 
3.17/3.70	[i](X) = 1
3.17/3.70	[p](X) = X + 2
3.17/3.70	[plus](X1,X2) = X1 + X2 + 2
3.17/3.70	[s](X) = X + 1
3.17/3.70	[times](X1,X2) = 0
3.17/3.70	[0] = 2
3.17/3.70	[I](X) = 0
3.17/3.70	[P](X) = 0
3.17/3.70	[PLUS](X1,X2) = X1 + X2
3.17/3.70	[S](X) = 0
3.17/3.70	[TIMES](X1,X2) = 0
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	SCC Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(i(x),x),x2) -> PLUS(0,x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Strongly Connected Components:
3.17/3.70	->->Cycle:
3.17/3.70	->->-> Pairs:
3.17/3.70	 PLUS(plus(i(x),x),x2) -> PLUS(0,x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> FAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) -> plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) -> times(x3,x2)
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	->->-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	Reduction Pairs Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(i(x),x),x2) -> PLUS(0,x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Usable Equations:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> Usable Rules:
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Interpretation type:
3.17/3.70	 Linear
3.17/3.70	->Coefficients:
3.17/3.70	 Natural Numbers
3.17/3.70	->Dimension:
3.17/3.70	 1
3.17/3.70	->Bound:
3.17/3.70	 2
3.17/3.70	->Interpretation:
3.17/3.70	 
3.17/3.70	[i](X) = 2
3.17/3.70	[p](X) = X + 2
3.17/3.70	[plus](X1,X2) = X1 + X2 + 1
3.17/3.70	[s](X) = X + 2
3.17/3.70	[times](X1,X2) = 0
3.17/3.70	[0] = 2
3.17/3.70	[I](X) = 0
3.17/3.70	[P](X) = 0
3.17/3.70	[PLUS](X1,X2) = 2.X1 + 2.X2
3.17/3.70	[S](X) = 0
3.17/3.70	[TIMES](X1,X2) = 0
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	SCC Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Strongly Connected Components:
3.17/3.70	->->Cycle:
3.17/3.70	->->-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> FAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) -> plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) -> times(x3,x2)
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	->->-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	Reduction Pairs Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Usable Equations:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> Usable Rules:
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Interpretation type:
3.17/3.70	 Linear
3.17/3.70	->Coefficients:
3.17/3.70	 Natural Numbers
3.17/3.70	->Dimension:
3.17/3.70	 1
3.17/3.70	->Bound:
3.17/3.70	 2
3.17/3.70	->Interpretation:
3.17/3.70	 
3.17/3.70	[i](X) = 1
3.17/3.70	[p](X) = X + 2
3.17/3.70	[plus](X1,X2) = X1 + X2 + 1
3.17/3.70	[s](X) = X + 2
3.17/3.70	[times](X1,X2) = 0
3.17/3.70	[0] = 2
3.17/3.70	[I](X) = 0
3.17/3.70	[P](X) = 0
3.17/3.70	[PLUS](X1,X2) = 2.X1 + 2.X2
3.17/3.70	[S](X) = 0
3.17/3.70	[TIMES](X1,X2) = 0
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	SCC Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Strongly Connected Components:
3.17/3.70	->->Cycle:
3.17/3.70	->->-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> FAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) -> plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) -> times(x3,x2)
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	->->-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	Reduction Pairs Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(x,y)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Usable Equations:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> Usable Rules:
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Interpretation type:
3.17/3.70	 Linear
3.17/3.70	->Coefficients:
3.17/3.70	 Natural Numbers
3.17/3.70	->Dimension:
3.17/3.70	 1
3.17/3.70	->Bound:
3.17/3.70	 2
3.17/3.70	->Interpretation:
3.17/3.70	 
3.17/3.70	[i](X) = 2
3.17/3.70	[p](X) = X + 2
3.17/3.70	[plus](X1,X2) = X1 + X2 + 2
3.17/3.70	[s](X) = X + 1
3.17/3.70	[times](X1,X2) = 0
3.17/3.70	[0] = 2
3.17/3.70	[I](X) = 0
3.17/3.70	[P](X) = 0
3.17/3.70	[PLUS](X1,X2) = 2.X1 + 2.X2
3.17/3.70	[S](X) = 0
3.17/3.70	[TIMES](X1,X2) = 0
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	SCC Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Strongly Connected Components:
3.17/3.70	->->Cycle:
3.17/3.70	->->-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> FAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) -> plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) -> times(x3,x2)
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	->->-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	Reduction Pairs Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(0,y),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Usable Equations:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> Usable Rules:
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Interpretation type:
3.17/3.70	 Linear
3.17/3.70	->Coefficients:
3.17/3.70	 Natural Numbers
3.17/3.70	->Dimension:
3.17/3.70	 1
3.17/3.70	->Bound:
3.17/3.70	 2
3.17/3.70	->Interpretation:
3.17/3.70	 
3.17/3.70	[i](X) = 2
3.17/3.70	[p](X) = X + 2
3.17/3.70	[plus](X1,X2) = X1 + X2 + 1
3.17/3.70	[s](X) = X + 2
3.17/3.70	[times](X1,X2) = 0
3.17/3.70	[0] = 2
3.17/3.70	[I](X) = 0
3.17/3.70	[P](X) = 0
3.17/3.70	[PLUS](X1,X2) = 2.X1 + 2.X2
3.17/3.70	[S](X) = 0
3.17/3.70	[TIMES](X1,X2) = 0
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	SCC Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Strongly Connected Components:
3.17/3.70	->->Cycle:
3.17/3.70	->->-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> FAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) -> plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) -> times(x3,x2)
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	->->-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	Reduction Pairs Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(x,plus(i(x),y)),x2) -> PLUS(y,x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Usable Equations:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> Usable Rules:
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Interpretation type:
3.17/3.70	 Linear
3.17/3.70	->Coefficients:
3.17/3.70	 Natural Numbers
3.17/3.70	->Dimension:
3.17/3.70	 1
3.17/3.70	->Bound:
3.17/3.70	 2
3.17/3.70	->Interpretation:
3.17/3.70	 
3.17/3.70	[i](X) = 2
3.17/3.70	[p](X) = X + 2
3.17/3.70	[plus](X1,X2) = X1 + X2
3.17/3.70	[s](X) = X + 2
3.17/3.70	[times](X1,X2) = 0
3.17/3.70	[0] = 0
3.17/3.70	[I](X) = 0
3.17/3.70	[P](X) = 0
3.17/3.70	[PLUS](X1,X2) = 2.X1 + 2.X2
3.17/3.70	[S](X) = 0
3.17/3.70	[TIMES](X1,X2) = 0
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	SCC Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	->Strongly Connected Components:
3.17/3.70	->->Cycle:
3.17/3.70	->->-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> FAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) -> plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) -> times(x3,x2)
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.70	 times(x2,x3) = times(x3,x2)
3.17/3.70	->->-> Rules:
3.17/3.70	 i(i(x)) -> x
3.17/3.70	 i(p(x)) -> s(i(x))
3.17/3.70	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.70	 i(s(x)) -> p(i(x))
3.17/3.70	 i(0) -> 0
3.17/3.70	 p(s(x)) -> x
3.17/3.70	 plus(i(x),x) -> 0
3.17/3.70	 plus(p(x),y) -> p(plus(x,y))
3.17/3.70	 plus(s(x),y) -> s(plus(x,y))
3.17/3.70	 plus(0,y) -> y
3.17/3.70	 plus(x,plus(i(x),y)) -> y
3.17/3.70	 s(p(x)) -> x
3.17/3.70	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.70	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.70	 times(0,y) -> 0
3.17/3.70	-> SRules:
3.17/3.70	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.70	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.70	
3.17/3.70	Problem 1.1: 
3.17/3.70	
3.17/3.70	Reduction Pairs Processor:
3.17/3.70	-> FAxioms:
3.17/3.70	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.70	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.70	-> Pairs:
3.17/3.70	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.70	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.70	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.70	-> EAxioms:
3.17/3.70	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.70	 plus(x2,x3) = plus(x3,x2)
3.17/3.70	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	-> SRules:
3.17/3.71	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
3.17/3.71	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Linear
3.17/3.71	->Coefficients:
3.17/3.71	 Natural Numbers
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 2
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = 2
3.17/3.71	[p](X) = X
3.17/3.71	[plus](X1,X2) = X1 + X2 + 2
3.17/3.71	[s](X) = X
3.17/3.71	[times](X1,X2) = 0
3.17/3.71	[0] = 0
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 2.X1 + 2.X2
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = 0
3.17/3.71	
3.17/3.71	Problem 1.1: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.71	
3.17/3.71	Problem 1.1: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	 PLUS(s(x),y) -> PLUS(x,y)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	-> SRules:
3.17/3.71	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Linear
3.17/3.71	->Coefficients:
3.17/3.71	 Natural Numbers
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 2
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = 2
3.17/3.71	[p](X) = X + 2
3.17/3.71	[plus](X1,X2) = X1 + X2 + 2
3.17/3.71	[s](X) = X + 2
3.17/3.71	[times](X1,X2) = 0
3.17/3.71	[0] = 1
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 2.X1 + 2.X2
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = 0
3.17/3.71	
3.17/3.71	Problem 1.1: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.71	
3.17/3.71	Problem 1.1: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	-> SRules:
3.17/3.71	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Linear
3.17/3.71	->Coefficients:
3.17/3.71	 Natural Numbers
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 2
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = 2
3.17/3.71	[p](X) = X + 1
3.17/3.71	[plus](X1,X2) = X1 + X2 + 2
3.17/3.71	[s](X) = X + 2
3.17/3.71	[times](X1,X2) = 0
3.17/3.71	[0] = 2
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 2.X1 + 2.X2
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = 0
3.17/3.71	
3.17/3.71	Problem 1.1: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 Empty
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 Empty
3.17/3.71	
3.17/3.71	Problem 1.1: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 PLUS(plus(p(x),y),x2) -> PLUS(p(plus(x,y)),x2)
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	-> SRules:
3.17/3.71	 Empty
3.17/3.71	->Interpretation type:
3.17/3.71	 Simple mixed
3.17/3.71	->Coefficients:
3.17/3.71	 All rationals
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 3
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = 1/2
3.17/3.71	[p](X) = 2/3.X + 2/3
3.17/3.71	[plus](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2
3.17/3.71	[s](X) = 3/2.X + 2
3.17/3.71	[times](X1,X2) = 0
3.17/3.71	[0] = 1/3
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 2/3.X1.X2 + 2/3.X1 + 2/3.X2
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = 0
3.17/3.71	
3.17/3.71	Problem 1.1: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 Empty
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) -> PLUS(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 Empty
3.17/3.71	
3.17/3.71	Problem 1.1: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	-> SRules:
3.17/3.71	 Empty
3.17/3.71	->Interpretation type:
3.17/3.71	 Simple mixed
3.17/3.71	->Coefficients:
3.17/3.71	 All rationals
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 3
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = 2
3.17/3.71	[p](X) = 3/2.X + 3/2
3.17/3.71	[plus](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2
3.17/3.71	[s](X) = 2/3.X
3.17/3.71	[times](X1,X2) = 0
3.17/3.71	[0] = 0
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = 0
3.17/3.71	
3.17/3.71	Problem 1.1: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
3.17/3.71	 PLUS(x2,x3) = PLUS(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 Empty
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 Empty
3.17/3.71	->Strongly Connected Components:
3.17/3.71	 There is no strongly connected component
3.17/3.71	
3.17/3.71	The problem is finite.
3.17/3.71	
3.17/3.71	Problem 1.2: 
3.17/3.71	
3.17/3.71	Subterm Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 Empty
3.17/3.71	-> Pairs:
3.17/3.71	 I(p(x)) -> I(x)
3.17/3.71	 I(plus(x,y)) -> I(x)
3.17/3.71	 I(plus(x,y)) -> I(y)
3.17/3.71	 I(s(x)) -> I(x)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 Empty
3.17/3.71	->Projection:
3.17/3.71	 pi(I) = [1]
3.17/3.71	
3.17/3.71	Problem 1.2: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 Empty
3.17/3.71	-> Pairs:
3.17/3.71	 Empty
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 Empty
3.17/3.71	->Strongly Connected Components:
3.17/3.71	 There is no strongly connected component
3.17/3.71	
3.17/3.71	The problem is finite.
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(p(x),y) -> TIMES(x,y)
3.17/3.71	 TIMES(s(x),y) -> TIMES(x,y)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Simple mixed
3.17/3.71	->Coefficients:
3.17/3.71	 Natural Numbers
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 1
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = X
3.17/3.71	[p](X) = X + 1
3.17/3.71	[plus](X1,X2) = X1 + X2 + 1
3.17/3.71	[s](X) = X + 1
3.17/3.71	[times](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	[0] = 1
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 0
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(s(x),y) -> TIMES(x,y)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 TIMES(s(x),y) -> TIMES(x,y)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) -> TIMES(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(s(x),y) -> TIMES(x,y)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Simple mixed
3.17/3.71	->Coefficients:
3.17/3.71	 Natural Numbers
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 1
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = X
3.17/3.71	[p](X) = X + 1
3.17/3.71	[plus](X1,X2) = X1 + X2 + 1
3.17/3.71	[s](X) = X + 1
3.17/3.71	[times](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	[0] = 1
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 0
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) -> TIMES(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(plus(times(x,y),i(y)),x2)
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Simple mixed
3.17/3.71	->Coefficients:
3.17/3.71	 Natural Numbers
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 1
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = X
3.17/3.71	[p](X) = X + 1
3.17/3.71	[plus](X1,X2) = X1 + X2
3.17/3.71	[s](X) = X + 1
3.17/3.71	[times](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	[0] = 0
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 0
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) -> TIMES(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(p(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Simple mixed
3.17/3.71	->Coefficients:
3.17/3.71	 Natural Numbers
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 1
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = X
3.17/3.71	[p](X) = X + 1
3.17/3.71	[plus](X1,X2) = X1 + X2 + 1
3.17/3.71	[s](X) = X + 1
3.17/3.71	[times](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	[0] = 1
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 0
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) -> TIMES(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(plus(times(x,y),y),x2)
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Simple mixed
3.17/3.71	->Coefficients:
3.17/3.71	 Natural Numbers
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 1
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = X
3.17/3.71	[p](X) = X + 1
3.17/3.71	[plus](X1,X2) = X1 + X2
3.17/3.71	[s](X) = X + 1
3.17/3.71	[times](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	[0] = 0
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 0
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) -> TIMES(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(s(x),y),x2) -> TIMES(x,y)
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Simple mixed
3.17/3.71	->Coefficients:
3.17/3.71	 Natural Numbers
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 1
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = X
3.17/3.71	[p](X) = X + 1
3.17/3.71	[plus](X1,X2) = X1 + X2 + 1
3.17/3.71	[s](X) = X + 1
3.17/3.71	[times](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	[0] = 1
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 0
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = X1.X2 + X1 + X2
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Strongly Connected Components:
3.17/3.71	->->Cycle:
3.17/3.71	->->-> Pairs:
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> FAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) -> plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) -> times(x3,x2)
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) -> TIMES(x3,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	->->-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	Reduction Pairs Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 TIMES(times(0,y),x2) -> TIMES(0,x2)
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Usable Equations:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> Usable Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Interpretation type:
3.17/3.71	 Simple mixed
3.17/3.71	->Coefficients:
3.17/3.71	 All rationals
3.17/3.71	->Dimension:
3.17/3.71	 1
3.17/3.71	->Bound:
3.17/3.71	 3
3.17/3.71	->Interpretation:
3.17/3.71	 
3.17/3.71	[i](X) = 3/2.X + 1/2
3.17/3.71	[p](X) = X + 3
3.17/3.71	[plus](X1,X2) = X1 + X2 + 2
3.17/3.71	[s](X) = X + 3
3.17/3.71	[times](X1,X2) = 1/2.X1.X2 + 3/2.X1 + 3/2.X2 + 3/2
3.17/3.71	[0] = 1/3
3.17/3.71	[I](X) = 0
3.17/3.71	[P](X) = 0
3.17/3.71	[PLUS](X1,X2) = 0
3.17/3.71	[S](X) = 0
3.17/3.71	[TIMES](X1,X2) = 1/3.X1.X2 + X1 + X2
3.17/3.71	
3.17/3.71	Problem 1.3: 
3.17/3.71	
3.17/3.71	SCC Processor:
3.17/3.71	-> FAxioms:
3.17/3.71	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
3.17/3.71	 TIMES(x2,x3) = TIMES(x3,x2)
3.17/3.71	-> Pairs:
3.17/3.71	 Empty
3.17/3.71	-> EAxioms:
3.17/3.71	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
3.17/3.71	 plus(x2,x3) = plus(x3,x2)
3.17/3.71	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
3.17/3.71	 times(x2,x3) = times(x3,x2)
3.17/3.71	-> Rules:
3.17/3.71	 i(i(x)) -> x
3.17/3.71	 i(p(x)) -> s(i(x))
3.17/3.71	 i(plus(x,y)) -> plus(i(y),i(x))
3.17/3.71	 i(s(x)) -> p(i(x))
3.17/3.71	 i(0) -> 0
3.17/3.71	 p(s(x)) -> x
3.17/3.71	 plus(i(x),x) -> 0
3.17/3.71	 plus(p(x),y) -> p(plus(x,y))
3.17/3.71	 plus(s(x),y) -> s(plus(x,y))
3.17/3.71	 plus(0,y) -> y
3.17/3.71	 plus(x,plus(i(x),y)) -> y
3.17/3.71	 s(p(x)) -> x
3.17/3.71	 times(p(x),y) -> plus(times(x,y),i(y))
3.17/3.71	 times(s(x),y) -> plus(times(x,y),y)
3.17/3.71	 times(0,y) -> 0
3.17/3.71	-> SRules:
3.17/3.71	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
3.17/3.71	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
3.17/3.71	->Strongly Connected Components:
3.17/3.71	 There is no strongly connected component
3.17/3.71	
3.17/3.71	The problem is finite.
3.17/3.71	EOF