42.38/44.71	YES
42.38/44.71	
42.38/44.71	Problem 1: 
42.38/44.71	
42.38/44.71	(VAR x y z)
42.38/44.71	(THEORY 
42.38/44.71	(AC plus times))
42.38/44.71	(RULES
42.38/44.71	minus(x,y) -> plus(x,neg(y))
42.38/44.71	neg(zero(x)) -> zero(neg(x))
42.38/44.71	neg(0) -> 0
42.38/44.71	neg(j(x)) -> un(neg(x))
42.38/44.71	neg(un(x)) -> j(neg(x))
42.38/44.71	plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.71	plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.71	plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.71	plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.71	plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.71	plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.71	plus(x,0) -> x
42.38/44.71	times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.71	times(x,times(0,z)) -> times(0,z)
42.38/44.71	times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.71	times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.71	times(x,zero(y)) -> zero(times(x,y))
42.38/44.71	times(x,0) -> 0
42.38/44.71	times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.71	times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.71	zero(0) -> 0
42.38/44.71	)
42.38/44.71	
42.38/44.71	Problem 1: 
42.38/44.71	
42.38/44.71	Dependency Pairs Processor:
42.38/44.71	-> FAxioms:
42.38/44.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.71	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.71	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.71	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.71	-> Pairs:
42.38/44.71	 MINUS(x,y) -> NEG(y)
42.38/44.71	 MINUS(x,y) -> PLUS(x,neg(y))
42.38/44.71	 NEG(zero(x)) -> NEG(x)
42.38/44.71	 NEG(zero(x)) -> ZERO(neg(x))
42.38/44.71	 NEG(j(x)) -> NEG(x)
42.38/44.71	 NEG(un(x)) -> NEG(x)
42.38/44.71	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(zero(x),zero(y)),x3) -> ZERO(plus(x,y))
42.38/44.71	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.71	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.71	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.71	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(un(x),j(y)),x3) -> ZERO(plus(x,y))
42.38/44.71	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.71	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.71	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.71	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.71	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(zero(x),zero(y)) -> ZERO(plus(x,y))
42.38/44.71	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.71	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.71	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(un(x),j(y)) -> ZERO(plus(x,y))
42.38/44.71	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.71	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.71	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.71	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.71	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,times(zero(y),z)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> NEG(x)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> PLUS(zero(times(x,y)),neg(x))
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> PLUS(x,zero(times(x,y)))
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.71	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,zero(y)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.71	 TIMES(times(x,j(y)),x3) -> NEG(x)
42.38/44.71	 TIMES(times(x,j(y)),x3) -> PLUS(zero(times(x,y)),neg(x))
42.38/44.71	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.71	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,j(y)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,un(y)),x3) -> PLUS(x,zero(times(x,y)))
42.38/44.71	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.71	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,un(y)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.71	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,times(zero(y),z)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,times(j(y),z)) -> NEG(x)
42.38/44.71	 TIMES(x,times(j(y),z)) -> PLUS(zero(times(x,y)),neg(x))
42.38/44.71	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.71	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,times(j(y),z)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,times(un(y),z)) -> PLUS(x,zero(times(x,y)))
42.38/44.71	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.71	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,times(un(y),z)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,zero(y)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,j(y)) -> NEG(x)
42.38/44.71	 TIMES(x,j(y)) -> PLUS(zero(times(x,y)),neg(x))
42.38/44.71	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,j(y)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,un(y)) -> PLUS(x,zero(times(x,y)))
42.38/44.71	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,un(y)) -> ZERO(times(x,y))
42.38/44.71	-> EAxioms:
42.38/44.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.71	 plus(x3,x4) = plus(x4,x3)
42.38/44.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.71	 times(x3,x4) = times(x4,x3)
42.38/44.71	-> Rules:
42.38/44.71	 minus(x,y) -> plus(x,neg(y))
42.38/44.71	 neg(zero(x)) -> zero(neg(x))
42.38/44.71	 neg(0) -> 0
42.38/44.71	 neg(j(x)) -> un(neg(x))
42.38/44.71	 neg(un(x)) -> j(neg(x))
42.38/44.71	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.71	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.71	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.71	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.71	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.71	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.71	 plus(x,0) -> x
42.38/44.71	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.71	 times(x,times(0,z)) -> times(0,z)
42.38/44.71	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.71	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.71	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.71	 times(x,0) -> 0
42.38/44.71	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.71	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.71	 zero(0) -> 0
42.38/44.71	-> SRules:
42.38/44.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.71	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.71	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.71	
42.38/44.71	Problem 1: 
42.38/44.71	
42.38/44.71	SCC Processor:
42.38/44.71	-> FAxioms:
42.38/44.71	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.71	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.71	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.71	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.71	-> Pairs:
42.38/44.71	 MINUS(x,y) -> NEG(y)
42.38/44.71	 MINUS(x,y) -> PLUS(x,neg(y))
42.38/44.71	 NEG(zero(x)) -> NEG(x)
42.38/44.71	 NEG(zero(x)) -> ZERO(neg(x))
42.38/44.71	 NEG(j(x)) -> NEG(x)
42.38/44.71	 NEG(un(x)) -> NEG(x)
42.38/44.71	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(zero(x),zero(y)),x3) -> ZERO(plus(x,y))
42.38/44.71	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.71	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.71	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.71	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(un(x),j(y)),x3) -> ZERO(plus(x,y))
42.38/44.71	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.71	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.71	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.71	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.71	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(zero(x),zero(y)) -> ZERO(plus(x,y))
42.38/44.71	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.71	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.71	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(un(x),j(y)) -> ZERO(plus(x,y))
42.38/44.71	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.71	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.71	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.71	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.71	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,times(zero(y),z)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> NEG(x)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> PLUS(zero(times(x,y)),neg(x))
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,times(j(y),z)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> PLUS(x,zero(times(x,y)))
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,times(un(y),z)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.71	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,zero(y)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.71	 TIMES(times(x,j(y)),x3) -> NEG(x)
42.38/44.71	 TIMES(times(x,j(y)),x3) -> PLUS(zero(times(x,y)),neg(x))
42.38/44.71	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.71	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,j(y)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(times(x,un(y)),x3) -> PLUS(x,zero(times(x,y)))
42.38/44.71	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.71	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.71	 TIMES(times(x,un(y)),x3) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.71	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,times(zero(y),z)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,times(j(y),z)) -> NEG(x)
42.38/44.71	 TIMES(x,times(j(y),z)) -> PLUS(zero(times(x,y)),neg(x))
42.38/44.71	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.71	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,times(j(y),z)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,times(un(y),z)) -> PLUS(x,zero(times(x,y)))
42.38/44.71	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.71	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,times(un(y),z)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,zero(y)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,j(y)) -> NEG(x)
42.38/44.71	 TIMES(x,j(y)) -> PLUS(zero(times(x,y)),neg(x))
42.38/44.71	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,j(y)) -> ZERO(times(x,y))
42.38/44.71	 TIMES(x,un(y)) -> PLUS(x,zero(times(x,y)))
42.38/44.71	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.71	 TIMES(x,un(y)) -> ZERO(times(x,y))
42.38/44.71	-> EAxioms:
42.38/44.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.71	 plus(x3,x4) = plus(x4,x3)
42.38/44.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.71	 times(x3,x4) = times(x4,x3)
42.38/44.71	-> Rules:
42.38/44.71	 minus(x,y) -> plus(x,neg(y))
42.38/44.71	 neg(zero(x)) -> zero(neg(x))
42.38/44.71	 neg(0) -> 0
42.38/44.71	 neg(j(x)) -> un(neg(x))
42.38/44.71	 neg(un(x)) -> j(neg(x))
42.38/44.71	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.71	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.71	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.71	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.71	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.71	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.71	 plus(x,0) -> x
42.38/44.71	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.71	 times(x,times(0,z)) -> times(0,z)
42.38/44.71	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.71	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.71	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.71	 times(x,0) -> 0
42.38/44.71	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.71	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.71	 zero(0) -> 0
42.38/44.71	-> SRules:
42.38/44.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.71	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.71	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.71	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.71	->Strongly Connected Components:
42.38/44.71	->->Cycle:
42.38/44.71	->->-> Pairs:
42.38/44.71	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.71	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.71	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.71	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.71	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.71	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.71	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.71	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.71	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.71	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.71	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.71	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.71	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.71	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.71	-> FAxioms:
42.38/44.71	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.71	 plus(x3,x4) -> plus(x4,x3)
42.38/44.71	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.71	 times(x3,x4) -> times(x4,x3)
42.38/44.71	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.71	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.71	-> EAxioms:
42.38/44.71	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.71	 plus(x3,x4) = plus(x4,x3)
42.38/44.71	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.71	 times(x3,x4) = times(x4,x3)
42.38/44.71	->->-> Rules:
42.38/44.71	 minus(x,y) -> plus(x,neg(y))
42.38/44.71	 neg(zero(x)) -> zero(neg(x))
42.38/44.71	 neg(0) -> 0
42.38/44.71	 neg(j(x)) -> un(neg(x))
42.38/44.71	 neg(un(x)) -> j(neg(x))
42.38/44.71	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.71	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.71	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.71	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.71	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.71	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.71	 plus(x,0) -> x
42.38/44.71	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.71	 times(x,times(0,z)) -> times(0,z)
42.38/44.71	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.71	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 NEG(zero(x)) -> NEG(x)
42.38/44.72	 NEG(j(x)) -> NEG(x)
42.38/44.72	 NEG(un(x)) -> NEG(x)
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 Empty
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.72	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.72	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y)
42.38/44.72	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.72	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.72	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.72	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.72	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.72	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.72	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.72	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.72	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.72	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.72	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.72	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.72	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.72	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.72	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.72	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.72	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.72	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.72	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.72	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.72	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.72	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.72	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.72	
42.38/44.72	
42.38/44.72	The problem is decomposed in 3 subproblems.
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X + 1
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X + 1
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X + 2
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X + 2
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X + 1
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X + 1
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X + 2
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> FAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) -> plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) -> times(x4,x3)
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	->->-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	Reduction Pairs Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Usable Equations:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> Usable Rules:
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Interpretation type:
42.38/44.72	 Linear
42.38/44.72	->Coefficients:
42.38/44.72	 Natural Numbers
42.38/44.72	->Dimension:
42.38/44.72	 1
42.38/44.72	->Bound:
42.38/44.72	 2
42.38/44.72	->Interpretation:
42.38/44.72	 
42.38/44.72	[minus](X1,X2) = 0
42.38/44.72	[neg](X) = 0
42.38/44.72	[plus](X1,X2) = X1 + X2
42.38/44.72	[times](X1,X2) = 0
42.38/44.72	[zero](X) = X + 2
42.38/44.72	[0] = 0
42.38/44.72	[j](X) = X + 2
42.38/44.72	[un](X) = X + 2
42.38/44.72	[MINUS](X1,X2) = 0
42.38/44.72	[NEG](X) = 0
42.38/44.72	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.72	[TIMES](X1,X2) = 0
42.38/44.72	[ZERO](X) = 0
42.38/44.72	
42.38/44.72	Problem 1.1: 
42.38/44.72	
42.38/44.72	SCC Processor:
42.38/44.72	-> FAxioms:
42.38/44.72	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.72	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.72	-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.72	-> EAxioms:
42.38/44.72	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.72	 plus(x3,x4) = plus(x4,x3)
42.38/44.72	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.72	 times(x3,x4) = times(x4,x3)
42.38/44.72	-> Rules:
42.38/44.72	 minus(x,y) -> plus(x,neg(y))
42.38/44.72	 neg(zero(x)) -> zero(neg(x))
42.38/44.72	 neg(0) -> 0
42.38/44.72	 neg(j(x)) -> un(neg(x))
42.38/44.72	 neg(un(x)) -> j(neg(x))
42.38/44.72	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.72	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.72	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.72	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.72	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.72	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.72	 plus(x,0) -> x
42.38/44.72	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.72	 times(x,times(0,z)) -> times(0,z)
42.38/44.72	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.72	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.72	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.72	 times(x,0) -> 0
42.38/44.72	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.72	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.72	 zero(0) -> 0
42.38/44.72	-> SRules:
42.38/44.72	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.72	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.72	->Strongly Connected Components:
42.38/44.72	->->Cycle:
42.38/44.72	->->-> Pairs:
42.38/44.72	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.72	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.72	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.72	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.72	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.72	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.72	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X + 2
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 1
42.38/44.73	[un](X) = X + 1
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X + 1
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 2
42.38/44.73	[un](X) = X + 2
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),zero(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X + 1
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 2
42.38/44.73	[un](X) = X + 2
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X + 2
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 1
42.38/44.73	[un](X) = X + 1
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(zero(x),un(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X + 1
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 2
42.38/44.73	[un](X) = X + 2
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = X1 + X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X + 2
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 2
42.38/44.73	[un](X) = X + 2
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(j(x),j(y)) -> PLUS(y,j(0))
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X + 1
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 2
42.38/44.73	[un](X) = X + 2
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(un(x),j(y)) -> PLUS(x,y)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 2
42.38/44.73	[un](X) = X + 2
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = X1 + X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X + 2
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 2
42.38/44.73	[un](X) = X + 2
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	 PLUS(un(x),un(y)) -> PLUS(y,un(0))
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = X
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = X + 2
42.38/44.73	[un](X) = X + 2
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2 + 1
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = 0
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = 2
42.38/44.73	[un](X) = 1
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = X1 + X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Interpretation type:
42.38/44.73	 Linear
42.38/44.73	->Coefficients:
42.38/44.73	 Natural Numbers
42.38/44.73	->Dimension:
42.38/44.73	 1
42.38/44.73	->Bound:
42.38/44.73	 2
42.38/44.73	->Interpretation:
42.38/44.73	 
42.38/44.73	[minus](X1,X2) = 0
42.38/44.73	[neg](X) = 0
42.38/44.73	[plus](X1,X2) = X1 + X2
42.38/44.73	[times](X1,X2) = 0
42.38/44.73	[zero](X) = 2
42.38/44.73	[0] = 0
42.38/44.73	[j](X) = 2
42.38/44.73	[un](X) = 2
42.38/44.73	[MINUS](X1,X2) = 0
42.38/44.73	[NEG](X) = 0
42.38/44.73	[PLUS](X1,X2) = 2.X1 + 2.X2
42.38/44.73	[TIMES](X1,X2) = 0
42.38/44.73	[ZERO](X) = 0
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	SCC Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	->Strongly Connected Components:
42.38/44.73	->->Cycle:
42.38/44.73	->->-> Pairs:
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	-> FAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) -> plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) -> times(x4,x3)
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) -> PLUS(x4,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	->->-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> SRules:
42.38/44.73	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.73	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.73	
42.38/44.73	Problem 1.1: 
42.38/44.73	
42.38/44.73	Reduction Pairs Processor:
42.38/44.73	-> FAxioms:
42.38/44.73	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.73	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.73	-> Pairs:
42.38/44.73	 PLUS(plus(x,0),x3) -> PLUS(x,x3)
42.38/44.73	-> EAxioms:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.73	 times(x3,x4) = times(x4,x3)
42.38/44.73	-> Usable Equations:
42.38/44.73	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.73	 plus(x3,x4) = plus(x4,x3)
42.38/44.73	-> Rules:
42.38/44.73	 minus(x,y) -> plus(x,neg(y))
42.38/44.73	 neg(zero(x)) -> zero(neg(x))
42.38/44.73	 neg(0) -> 0
42.38/44.73	 neg(j(x)) -> un(neg(x))
42.38/44.73	 neg(un(x)) -> j(neg(x))
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.73	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.73	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.73	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.73	 plus(x,0) -> x
42.38/44.73	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.73	 times(x,times(0,z)) -> times(0,z)
42.38/44.73	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.73	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.73	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.73	 times(x,0) -> 0
42.38/44.73	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.73	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.73	 zero(0) -> 0
42.38/44.73	-> Usable Rules:
42.38/44.73	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.73	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.73	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.74	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Linear
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 2
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = 0
42.38/44.74	[plus](X1,X2) = X1 + X2 + 1
42.38/44.74	[times](X1,X2) = 0
42.38/44.74	[zero](X) = 2
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = 1
42.38/44.74	[un](X) = 2
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = X1 + X2
42.38/44.74	[TIMES](X1,X2) = 0
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.1: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
42.38/44.74	 PLUS(x3,x4) = PLUS(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 Empty
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
42.38/44.74	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
42.38/44.74	->Strongly Connected Components:
42.38/44.74	 There is no strongly connected component
42.38/44.74	
42.38/44.74	The problem is finite.
42.38/44.74	
42.38/44.74	Problem 1.2: 
42.38/44.74	
42.38/44.74	Subterm Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 Empty
42.38/44.74	-> Pairs:
42.38/44.74	 NEG(zero(x)) -> NEG(x)
42.38/44.74	 NEG(j(x)) -> NEG(x)
42.38/44.74	 NEG(un(x)) -> NEG(x)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 Empty
42.38/44.74	->Projection:
42.38/44.74	 pi(NEG) = [1]
42.38/44.74	
42.38/44.74	Problem 1.2: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 Empty
42.38/44.74	-> Pairs:
42.38/44.74	 Empty
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 Empty
42.38/44.74	->Strongly Connected Components:
42.38/44.74	 There is no strongly connected component
42.38/44.74	
42.38/44.74	The problem is finite.
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	Reduction Pairs Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Usable Equations:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> Usable Rules:
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Simple mixed
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 1
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = X
42.38/44.74	[plus](X1,X2) = X1 + X2
42.38/44.74	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[zero](X) = X + 1
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = X + 1
42.38/44.74	[un](X) = X + 1
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = 0
42.38/44.74	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Strongly Connected Components:
42.38/44.74	->->Cycle:
42.38/44.74	->->-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> FAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) -> plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) -> times(x4,x3)
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	->->-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	Reduction Pairs Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Usable Equations:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> Usable Rules:
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Simple mixed
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 1
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = X
42.38/44.74	[plus](X1,X2) = X1 + X2
42.38/44.74	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[zero](X) = X
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = X + 1
42.38/44.74	[un](X) = X + 1
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = 0
42.38/44.74	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Strongly Connected Components:
42.38/44.74	->->Cycle:
42.38/44.74	->->-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> FAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) -> plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) -> times(x4,x3)
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	->->-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	Reduction Pairs Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Usable Equations:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> Usable Rules:
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Simple mixed
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 1
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = X
42.38/44.74	[plus](X1,X2) = X1 + X2
42.38/44.74	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[zero](X) = X
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = X + 1
42.38/44.74	[un](X) = X + 1
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = 0
42.38/44.74	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Strongly Connected Components:
42.38/44.74	->->Cycle:
42.38/44.74	->->-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> FAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) -> plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) -> times(x4,x3)
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	->->-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	Reduction Pairs Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Usable Equations:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> Usable Rules:
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Simple mixed
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 1
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = X
42.38/44.74	[plus](X1,X2) = X1 + X2
42.38/44.74	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[zero](X) = X + 1
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = X + 1
42.38/44.74	[un](X) = X + 1
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = 0
42.38/44.74	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Strongly Connected Components:
42.38/44.74	->->Cycle:
42.38/44.74	->->-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> FAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) -> plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) -> times(x4,x3)
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	->->-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	Reduction Pairs Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Usable Equations:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> Usable Rules:
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Simple mixed
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 1
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = X + 1
42.38/44.74	[plus](X1,X2) = X1 + X2
42.38/44.74	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[zero](X) = X
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = X + 1
42.38/44.74	[un](X) = X + 1
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = 0
42.38/44.74	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Strongly Connected Components:
42.38/44.74	->->Cycle:
42.38/44.74	->->-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> FAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) -> plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) -> times(x4,x3)
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	->->-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	Reduction Pairs Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Usable Equations:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> Usable Rules:
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Simple mixed
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 1
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = X + 1
42.38/44.74	[plus](X1,X2) = X1 + X2
42.38/44.74	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[zero](X) = X
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = X + 1
42.38/44.74	[un](X) = X + 1
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = 0
42.38/44.74	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Strongly Connected Components:
42.38/44.74	->->Cycle:
42.38/44.74	->->-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> FAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) -> plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) -> times(x4,x3)
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	->->-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	Reduction Pairs Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Usable Equations:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> Usable Rules:
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Simple mixed
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 1
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = X
42.38/44.74	[plus](X1,X2) = X1 + X2
42.38/44.74	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[zero](X) = X
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = X + 1
42.38/44.74	[un](X) = X + 1
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = 0
42.38/44.74	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Strongly Connected Components:
42.38/44.74	->->Cycle:
42.38/44.74	->->-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> FAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) -> plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) -> times(x4,x3)
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	->->-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	Reduction Pairs Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Usable Equations:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> Usable Rules:
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Simple mixed
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 1
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = X
42.38/44.74	[plus](X1,X2) = X1 + X2
42.38/44.74	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[zero](X) = X + 1
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = X + 1
42.38/44.74	[un](X) = X + 1
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = 0
42.38/44.74	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Strongly Connected Components:
42.38/44.74	->->Cycle:
42.38/44.74	->->-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> FAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) -> plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) -> times(x4,x3)
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	->->-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	Reduction Pairs Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.74	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.74	-> EAxioms:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Usable Equations:
42.38/44.74	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.74	 plus(x3,x4) = plus(x4,x3)
42.38/44.74	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.74	 times(x3,x4) = times(x4,x3)
42.38/44.74	-> Rules:
42.38/44.74	 minus(x,y) -> plus(x,neg(y))
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> Usable Rules:
42.38/44.74	 neg(zero(x)) -> zero(neg(x))
42.38/44.74	 neg(0) -> 0
42.38/44.74	 neg(j(x)) -> un(neg(x))
42.38/44.74	 neg(un(x)) -> j(neg(x))
42.38/44.74	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.74	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.74	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.74	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.74	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.74	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.74	 plus(x,0) -> x
42.38/44.74	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.74	 times(x,times(0,z)) -> times(0,z)
42.38/44.74	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.74	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.74	 times(x,0) -> 0
42.38/44.74	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.74	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.74	 zero(0) -> 0
42.38/44.74	-> SRules:
42.38/44.74	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.74	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.74	->Interpretation type:
42.38/44.74	 Simple mixed
42.38/44.74	->Coefficients:
42.38/44.74	 Natural Numbers
42.38/44.74	->Dimension:
42.38/44.74	 1
42.38/44.74	->Bound:
42.38/44.74	 1
42.38/44.74	->Interpretation:
42.38/44.74	 
42.38/44.74	[minus](X1,X2) = 0
42.38/44.74	[neg](X) = X
42.38/44.74	[plus](X1,X2) = X1 + X2
42.38/44.74	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[zero](X) = X
42.38/44.74	[0] = 0
42.38/44.74	[j](X) = X + 1
42.38/44.74	[un](X) = X + 1
42.38/44.74	[MINUS](X1,X2) = 0
42.38/44.74	[NEG](X) = 0
42.38/44.74	[PLUS](X1,X2) = 0
42.38/44.74	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.74	[ZERO](X) = 0
42.38/44.74	
42.38/44.74	Problem 1.3: 
42.38/44.74	
42.38/44.74	SCC Processor:
42.38/44.74	-> FAxioms:
42.38/44.74	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.74	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.74	-> Pairs:
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.74	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.74	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.74	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.74	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.74	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.74	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.74	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X + 1
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X + 1
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X + 1
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X + 1
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(j(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X + 1
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X + 1
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,times(un(y),z)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X + 1
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,zero(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X + 1
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,j(y)) -> TIMES(x,y)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> Usable Rules:
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Interpretation type:
42.38/44.75	 Simple mixed
42.38/44.75	->Coefficients:
42.38/44.75	 Natural Numbers
42.38/44.75	->Dimension:
42.38/44.75	 1
42.38/44.75	->Bound:
42.38/44.75	 1
42.38/44.75	->Interpretation:
42.38/44.75	 
42.38/44.75	[minus](X1,X2) = 0
42.38/44.75	[neg](X) = X
42.38/44.75	[plus](X1,X2) = X1 + X2
42.38/44.75	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[zero](X) = X + 1
42.38/44.75	[0] = 0
42.38/44.75	[j](X) = X + 1
42.38/44.75	[un](X) = X + 1
42.38/44.75	[MINUS](X1,X2) = 0
42.38/44.75	[NEG](X) = 0
42.38/44.75	[PLUS](X1,X2) = 0
42.38/44.75	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.75	[ZERO](X) = 0
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	SCC Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	->Strongly Connected Components:
42.38/44.75	->->Cycle:
42.38/44.75	->->-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> FAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) -> plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) -> times(x4,x3)
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	->->-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.75	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.75	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.75	 times(x,0) -> 0
42.38/44.75	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.75	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.75	 zero(0) -> 0
42.38/44.75	-> SRules:
42.38/44.75	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.75	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.75	
42.38/44.75	Problem 1.3: 
42.38/44.75	
42.38/44.75	Reduction Pairs Processor:
42.38/44.75	-> FAxioms:
42.38/44.75	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.75	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.75	-> Pairs:
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.75	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.75	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.75	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.75	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.75	 TIMES(x,un(y)) -> TIMES(x,y)
42.38/44.75	-> EAxioms:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Usable Equations:
42.38/44.75	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.75	 plus(x3,x4) = plus(x4,x3)
42.38/44.75	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.75	 times(x3,x4) = times(x4,x3)
42.38/44.75	-> Rules:
42.38/44.75	 minus(x,y) -> plus(x,neg(y))
42.38/44.75	 neg(zero(x)) -> zero(neg(x))
42.38/44.75	 neg(0) -> 0
42.38/44.75	 neg(j(x)) -> un(neg(x))
42.38/44.75	 neg(un(x)) -> j(neg(x))
42.38/44.75	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.75	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.75	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.75	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.75	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.75	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.75	 plus(x,0) -> x
42.38/44.75	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.75	 times(x,times(0,z)) -> times(0,z)
42.38/44.75	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> Usable Rules:
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Interpretation type:
42.38/44.76	 Simple mixed
42.38/44.76	->Coefficients:
42.38/44.76	 Natural Numbers
42.38/44.76	->Dimension:
42.38/44.76	 1
42.38/44.76	->Bound:
42.38/44.76	 1
42.38/44.76	->Interpretation:
42.38/44.76	 
42.38/44.76	[minus](X1,X2) = 0
42.38/44.76	[neg](X) = X + 1
42.38/44.76	[plus](X1,X2) = X1 + X2
42.38/44.76	[times](X1,X2) = X1.X2 + X1 + X2
42.38/44.76	[zero](X) = X
42.38/44.76	[0] = 0
42.38/44.76	[j](X) = X + 1
42.38/44.76	[un](X) = X + 1
42.38/44.76	[MINUS](X1,X2) = 0
42.38/44.76	[NEG](X) = 0
42.38/44.76	[PLUS](X1,X2) = 0
42.38/44.76	[TIMES](X1,X2) = X1.X2 + X1 + X2
42.38/44.76	[ZERO](X) = 0
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	SCC Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.76	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.76	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Strongly Connected Components:
42.38/44.76	->->Cycle:
42.38/44.76	->->-> Pairs:
42.38/44.76	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.76	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.76	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> FAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) -> plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) -> times(x4,x3)
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	->->-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	Reduction Pairs Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
42.38/44.76	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.76	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Usable Equations:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> Usable Rules:
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Interpretation type:
42.38/44.76	 Linear
42.38/44.76	->Coefficients:
42.38/44.76	 Natural Numbers
42.38/44.76	->Dimension:
42.38/44.76	 1
42.38/44.76	->Bound:
42.38/44.76	 2
42.38/44.76	->Interpretation:
42.38/44.76	 
42.38/44.76	[minus](X1,X2) = 0
42.38/44.76	[neg](X) = 2
42.38/44.76	[plus](X1,X2) = X1 + X2
42.38/44.76	[times](X1,X2) = X1 + X2 + 2
42.38/44.76	[zero](X) = 0
42.38/44.76	[0] = 0
42.38/44.76	[j](X) = 2
42.38/44.76	[un](X) = 2
42.38/44.76	[MINUS](X1,X2) = 0
42.38/44.76	[NEG](X) = 0
42.38/44.76	[PLUS](X1,X2) = 0
42.38/44.76	[TIMES](X1,X2) = 2.X1 + 2.X2
42.38/44.76	[ZERO](X) = 0
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	SCC Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.76	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Strongly Connected Components:
42.38/44.76	->->Cycle:
42.38/44.76	->->-> Pairs:
42.38/44.76	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.76	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> FAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) -> plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) -> times(x4,x3)
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	->->-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	Reduction Pairs Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
42.38/44.76	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Usable Equations:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> Usable Rules:
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Interpretation type:
42.38/44.76	 Linear
42.38/44.76	->Coefficients:
42.38/44.76	 Natural Numbers
42.38/44.76	->Dimension:
42.38/44.76	 1
42.38/44.76	->Bound:
42.38/44.76	 2
42.38/44.76	->Interpretation:
42.38/44.76	 
42.38/44.76	[minus](X1,X2) = 0
42.38/44.76	[neg](X) = X + 1
42.38/44.76	[plus](X1,X2) = X1 + X2 + 2
42.38/44.76	[times](X1,X2) = X1 + X2 + 1
42.38/44.76	[zero](X) = 0
42.38/44.76	[0] = 0
42.38/44.76	[j](X) = 2
42.38/44.76	[un](X) = 2
42.38/44.76	[MINUS](X1,X2) = 0
42.38/44.76	[NEG](X) = 0
42.38/44.76	[PLUS](X1,X2) = 0
42.38/44.76	[TIMES](X1,X2) = 2.X1 + 2.X2
42.38/44.76	[ZERO](X) = 0
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	SCC Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Strongly Connected Components:
42.38/44.76	->->Cycle:
42.38/44.76	->->-> Pairs:
42.38/44.76	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> FAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) -> plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) -> times(x4,x3)
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	->->-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	Reduction Pairs Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Usable Equations:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> Usable Rules:
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Interpretation type:
42.38/44.76	 Linear
42.38/44.76	->Coefficients:
42.38/44.76	 Natural Numbers
42.38/44.76	->Dimension:
42.38/44.76	 1
42.38/44.76	->Bound:
42.38/44.76	 2
42.38/44.76	->Interpretation:
42.38/44.76	 
42.38/44.76	[minus](X1,X2) = 0
42.38/44.76	[neg](X) = X + 2
42.38/44.76	[plus](X1,X2) = X1 + X2 + 2
42.38/44.76	[times](X1,X2) = X1 + X2 + 2
42.38/44.76	[zero](X) = 0
42.38/44.76	[0] = 0
42.38/44.76	[j](X) = 2
42.38/44.76	[un](X) = 0
42.38/44.76	[MINUS](X1,X2) = 0
42.38/44.76	[NEG](X) = 0
42.38/44.76	[PLUS](X1,X2) = 0
42.38/44.76	[TIMES](X1,X2) = 2.X1 + 2.X2
42.38/44.76	[ZERO](X) = 0
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	SCC Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Strongly Connected Components:
42.38/44.76	->->Cycle:
42.38/44.76	->->-> Pairs:
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> FAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) -> plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) -> times(x4,x3)
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	->->-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	Reduction Pairs Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Usable Equations:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> Usable Rules:
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Interpretation type:
42.38/44.76	 Linear
42.38/44.76	->Coefficients:
42.38/44.76	 Natural Numbers
42.38/44.76	->Dimension:
42.38/44.76	 1
42.38/44.76	->Bound:
42.38/44.76	 2
42.38/44.76	->Interpretation:
42.38/44.76	 
42.38/44.76	[minus](X1,X2) = 0
42.38/44.76	[neg](X) = X + 1
42.38/44.76	[plus](X1,X2) = X1 + X2 + 2
42.38/44.76	[times](X1,X2) = X1 + X2 + 2
42.38/44.76	[zero](X) = 0
42.38/44.76	[0] = 0
42.38/44.76	[j](X) = 1
42.38/44.76	[un](X) = 0
42.38/44.76	[MINUS](X1,X2) = 0
42.38/44.76	[NEG](X) = 0
42.38/44.76	[PLUS](X1,X2) = 0
42.38/44.76	[TIMES](X1,X2) = X1 + X2
42.38/44.76	[ZERO](X) = 0
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	SCC Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Strongly Connected Components:
42.38/44.76	->->Cycle:
42.38/44.76	->->-> Pairs:
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> FAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) -> plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) -> times(x4,x3)
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	->->-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	Reduction Pairs Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(times(x,0),x3) -> TIMES(0,x3)
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Usable Equations:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> Usable Rules:
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Interpretation type:
42.38/44.76	 Linear
42.38/44.76	->Coefficients:
42.38/44.76	 Natural Numbers
42.38/44.76	->Dimension:
42.38/44.76	 1
42.38/44.76	->Bound:
42.38/44.76	 2
42.38/44.76	->Interpretation:
42.38/44.76	 
42.38/44.76	[minus](X1,X2) = 0
42.38/44.76	[neg](X) = X
42.38/44.76	[plus](X1,X2) = X1 + X2 + 2
42.38/44.76	[times](X1,X2) = X1 + X2 + 2
42.38/44.76	[zero](X) = 0
42.38/44.76	[0] = 0
42.38/44.76	[j](X) = 0
42.38/44.76	[un](X) = 0
42.38/44.76	[MINUS](X1,X2) = 0
42.38/44.76	[NEG](X) = 0
42.38/44.76	[PLUS](X1,X2) = 0
42.38/44.76	[TIMES](X1,X2) = X1 + X2
42.38/44.76	[ZERO](X) = 0
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	SCC Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Strongly Connected Components:
42.38/44.76	->->Cycle:
42.38/44.76	->->-> Pairs:
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> FAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) -> plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) -> times(x4,x3)
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	->->-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	Reduction Pairs Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Usable Equations:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> Usable Rules:
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Interpretation type:
42.38/44.76	 Linear
42.38/44.76	->Coefficients:
42.38/44.76	 Natural Numbers
42.38/44.76	->Dimension:
42.38/44.76	 1
42.38/44.76	->Bound:
42.38/44.76	 2
42.38/44.76	->Interpretation:
42.38/44.76	 
42.38/44.76	[minus](X1,X2) = 0
42.38/44.76	[neg](X) = X
42.38/44.76	[plus](X1,X2) = X1 + X2 + 2
42.38/44.76	[times](X1,X2) = X1 + X2 + 2
42.38/44.76	[zero](X) = 2
42.38/44.76	[0] = 2
42.38/44.76	[j](X) = 2
42.38/44.76	[un](X) = 2
42.38/44.76	[MINUS](X1,X2) = 0
42.38/44.76	[NEG](X) = 0
42.38/44.76	[PLUS](X1,X2) = 0
42.38/44.76	[TIMES](X1,X2) = 2.X1 + 2.X2
42.38/44.76	[ZERO](X) = 0
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	SCC Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Strongly Connected Components:
42.38/44.76	->->Cycle:
42.38/44.76	->->-> Pairs:
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> FAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) -> plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) -> times(x4,x3)
42.38/44.76	 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) -> TIMES(x4,x3)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	->->-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	Reduction Pairs Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Usable Equations:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> Usable Rules:
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Interpretation type:
42.38/44.76	 Linear
42.38/44.76	->Coefficients:
42.38/44.76	 Natural Numbers
42.38/44.76	->Dimension:
42.38/44.76	 1
42.38/44.76	->Bound:
42.38/44.76	 2
42.38/44.76	->Interpretation:
42.38/44.76	 
42.38/44.76	[minus](X1,X2) = 0
42.38/44.76	[neg](X) = X + 2
42.38/44.76	[plus](X1,X2) = X1 + X2
42.38/44.76	[times](X1,X2) = X1 + X2 + 2
42.38/44.76	[zero](X) = 2
42.38/44.76	[0] = 2
42.38/44.76	[j](X) = 2
42.38/44.76	[un](X) = 2
42.38/44.76	[MINUS](X1,X2) = 0
42.38/44.76	[NEG](X) = 0
42.38/44.76	[PLUS](X1,X2) = 0
42.38/44.76	[TIMES](X1,X2) = 2.X1 + 2.X2
42.38/44.76	[ZERO](X) = 0
42.38/44.76	
42.38/44.76	Problem 1.3: 
42.38/44.76	
42.38/44.76	SCC Processor:
42.38/44.76	-> FAxioms:
42.38/44.76	 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
42.38/44.76	 TIMES(x3,x4) = TIMES(x4,x3)
42.38/44.76	-> Pairs:
42.38/44.76	 Empty
42.38/44.76	-> EAxioms:
42.38/44.76	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
42.38/44.76	 plus(x3,x4) = plus(x4,x3)
42.38/44.76	 times(times(x3,x4),x5) = times(x3,times(x4,x5))
42.38/44.76	 times(x3,x4) = times(x4,x3)
42.38/44.76	-> Rules:
42.38/44.76	 minus(x,y) -> plus(x,neg(y))
42.38/44.76	 neg(zero(x)) -> zero(neg(x))
42.38/44.76	 neg(0) -> 0
42.38/44.76	 neg(j(x)) -> un(neg(x))
42.38/44.76	 neg(un(x)) -> j(neg(x))
42.38/44.76	 plus(zero(x),zero(y)) -> zero(plus(x,y))
42.38/44.76	 plus(zero(x),j(y)) -> j(plus(x,y))
42.38/44.76	 plus(zero(x),un(y)) -> un(plus(x,y))
42.38/44.76	 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
42.38/44.76	 plus(un(x),j(y)) -> zero(plus(x,y))
42.38/44.76	 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
42.38/44.76	 plus(x,0) -> x
42.38/44.76	 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
42.38/44.76	 times(x,times(0,z)) -> times(0,z)
42.38/44.76	 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),z)
42.38/44.76	 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
42.38/44.76	 times(x,zero(y)) -> zero(times(x,y))
42.38/44.76	 times(x,0) -> 0
42.38/44.76	 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
42.38/44.76	 times(x,un(y)) -> plus(x,zero(times(x,y)))
42.38/44.76	 zero(0) -> 0
42.38/44.76	-> SRules:
42.38/44.76	 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
42.38/44.76	->Strongly Connected Components:
42.38/44.76	 There is no strongly connected component
42.38/44.76	
42.38/44.76	The problem is finite.
42.38/44.76	EOF