0.00/0.36	YES
0.00/0.36	
0.00/0.36	Problem 1: 
0.00/0.36	
0.00/0.36	(VAR x y z)
0.00/0.36	(THEORY 
0.00/0.36	(AC plus))
0.00/0.36	(RULES
0.00/0.36	minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	minus(x,0) -> x
0.00/0.36	plus(0,y) -> y
0.00/0.36	plus(s(x),y) -> s(plus(x,y))
0.00/0.36	quot(0,s(y)) -> 0
0.00/0.36	quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	)
0.00/0.36	
0.00/0.36	Problem 1: 
0.00/0.36	
0.00/0.36	Dependency Pairs Processor:
0.00/0.36	-> FAxioms:
0.00/0.36	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.36	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.36	-> Pairs:
0.00/0.36	 MINUS(minus(x,y),z) -> MINUS(x,plus(y,z))
0.00/0.36	 MINUS(minus(x,y),z) -> PLUS(y,z)
0.00/0.36	 MINUS(s(x),s(y)) -> MINUS(x,y)
0.00/0.36	 PLUS(plus(0,y),x3) -> PLUS(y,x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
0.00/0.36	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.36	 QUOT(s(x),s(y)) -> MINUS(x,y)
0.00/0.36	 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> SRules:
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.36	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.36	
0.00/0.36	Problem 1: 
0.00/0.36	
0.00/0.36	SCC Processor:
0.00/0.36	-> FAxioms:
0.00/0.36	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.36	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.36	-> Pairs:
0.00/0.36	 MINUS(minus(x,y),z) -> MINUS(x,plus(y,z))
0.00/0.36	 MINUS(minus(x,y),z) -> PLUS(y,z)
0.00/0.36	 MINUS(s(x),s(y)) -> MINUS(x,y)
0.00/0.36	 PLUS(plus(0,y),x3) -> PLUS(y,x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
0.00/0.36	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.36	 QUOT(s(x),s(y)) -> MINUS(x,y)
0.00/0.36	 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> SRules:
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.36	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.36	->Strongly Connected Components:
0.00/0.36	->->Cycle:
0.00/0.36	->->-> Pairs:
0.00/0.36	 PLUS(plus(0,y),x3) -> PLUS(y,x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
0.00/0.36	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.36	-> FAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) -> plus(x4,x3)
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
0.00/0.36	 PLUS(x3,x4) -> PLUS(x4,x3)
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	->->-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> SRules:
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.36	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.36	->->Cycle:
0.00/0.36	->->-> Pairs:
0.00/0.36	 MINUS(minus(x,y),z) -> MINUS(x,plus(y,z))
0.00/0.36	 MINUS(s(x),s(y)) -> MINUS(x,y)
0.00/0.36	-> FAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) -> plus(x4,x3)
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	->->-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> SRules:
0.00/0.36	 Empty
0.00/0.36	->->Cycle:
0.00/0.36	->->-> Pairs:
0.00/0.36	 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))
0.00/0.36	-> FAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) -> plus(x4,x3)
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	->->-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> SRules:
0.00/0.36	 Empty
0.00/0.36	
0.00/0.36	
0.00/0.36	The problem is decomposed in 3 subproblems.
0.00/0.36	
0.00/0.36	Problem 1.1: 
0.00/0.36	
0.00/0.36	Reduction Pairs Processor:
0.00/0.36	-> FAxioms:
0.00/0.36	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.36	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.36	-> Pairs:
0.00/0.36	 PLUS(plus(0,y),x3) -> PLUS(y,x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
0.00/0.36	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	-> Usable Equations:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> Usable Rules:
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	-> SRules:
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.36	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.36	->Interpretation type:
0.00/0.36	 Linear
0.00/0.36	->Coefficients:
0.00/0.36	 Natural Numbers
0.00/0.36	->Dimension:
0.00/0.36	 1
0.00/0.36	->Bound:
0.00/0.36	 2
0.00/0.36	->Interpretation:
0.00/0.36	 
0.00/0.36	[minus](X1,X2) = 0
0.00/0.36	[plus](X1,X2) = X1 + X2
0.00/0.36	[quot](X1,X2) = 0
0.00/0.36	[0] = 2
0.00/0.36	[s](X) = X
0.00/0.36	[MINUS](X1,X2) = 0
0.00/0.36	[PLUS](X1,X2) = X1 + X2
0.00/0.36	[QUOT](X1,X2) = 0
0.00/0.36	
0.00/0.36	Problem 1.1: 
0.00/0.36	
0.00/0.36	SCC Processor:
0.00/0.36	-> FAxioms:
0.00/0.36	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.36	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.36	-> Pairs:
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
0.00/0.36	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> SRules:
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.36	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.36	->Strongly Connected Components:
0.00/0.36	->->Cycle:
0.00/0.36	->->-> Pairs:
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
0.00/0.36	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.36	-> FAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) -> plus(x4,x3)
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
0.00/0.36	 PLUS(x3,x4) -> PLUS(x4,x3)
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	->->-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> SRules:
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.36	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.36	
0.00/0.36	Problem 1.1: 
0.00/0.36	
0.00/0.36	Reduction Pairs Processor:
0.00/0.36	-> FAxioms:
0.00/0.36	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.36	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.36	-> Pairs:
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
0.00/0.36	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	-> Usable Equations:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> Usable Rules:
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	-> SRules:
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.36	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.36	->Interpretation type:
0.00/0.36	 Linear
0.00/0.36	->Coefficients:
0.00/0.36	 Natural Numbers
0.00/0.36	->Dimension:
0.00/0.36	 1
0.00/0.36	->Bound:
0.00/0.36	 2
0.00/0.36	->Interpretation:
0.00/0.36	 
0.00/0.36	[minus](X1,X2) = 0
0.00/0.36	[plus](X1,X2) = X1 + X2 + 2
0.00/0.36	[quot](X1,X2) = 0
0.00/0.36	[0] = 0
0.00/0.36	[s](X) = X + 2
0.00/0.36	[MINUS](X1,X2) = 0
0.00/0.36	[PLUS](X1,X2) = 2.X1 + 2.X2
0.00/0.36	[QUOT](X1,X2) = 0
0.00/0.36	
0.00/0.36	Problem 1.1: 
0.00/0.36	
0.00/0.36	SCC Processor:
0.00/0.36	-> FAxioms:
0.00/0.36	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.36	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.36	-> Pairs:
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.36	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.36	-> Rules:
0.00/0.36	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.36	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.36	 minus(x,0) -> x
0.00/0.36	 plus(0,y) -> y
0.00/0.36	 plus(s(x),y) -> s(plus(x,y))
0.00/0.36	 quot(0,s(y)) -> 0
0.00/0.36	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.36	-> SRules:
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.36	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.36	->Strongly Connected Components:
0.00/0.36	->->Cycle:
0.00/0.36	->->-> Pairs:
0.00/0.36	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.36	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.36	-> FAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) -> plus(x4,x3)
0.00/0.36	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
0.00/0.36	 PLUS(x3,x4) -> PLUS(x4,x3)
0.00/0.36	-> EAxioms:
0.00/0.36	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.36	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	->->-> Rules:
0.00/0.37	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.37	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.37	 minus(x,0) -> x
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.37	 quot(0,s(y)) -> 0
0.00/0.37	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.37	-> SRules:
0.00/0.37	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.37	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.37	
0.00/0.37	Problem 1.1: 
0.00/0.37	
0.00/0.37	Reduction Pairs Processor:
0.00/0.37	-> FAxioms:
0.00/0.37	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.37	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.37	-> Pairs:
0.00/0.37	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.37	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.37	-> EAxioms:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	-> Usable Equations:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	-> Rules:
0.00/0.37	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.37	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.37	 minus(x,0) -> x
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.37	 quot(0,s(y)) -> 0
0.00/0.37	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.37	-> Usable Rules:
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.37	-> SRules:
0.00/0.37	 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
0.00/0.37	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.37	->Interpretation type:
0.00/0.37	 Linear
0.00/0.37	->Coefficients:
0.00/0.37	 Natural Numbers
0.00/0.37	->Dimension:
0.00/0.37	 1
0.00/0.37	->Bound:
0.00/0.37	 2
0.00/0.37	->Interpretation:
0.00/0.37	 
0.00/0.37	[minus](X1,X2) = 0
0.00/0.37	[plus](X1,X2) = X1 + X2 + 1
0.00/0.37	[quot](X1,X2) = 0
0.00/0.37	[0] = 0
0.00/0.37	[s](X) = X + 2
0.00/0.37	[MINUS](X1,X2) = 0
0.00/0.37	[PLUS](X1,X2) = 2.X1 + 2.X2
0.00/0.37	[QUOT](X1,X2) = 0
0.00/0.37	
0.00/0.37	Problem 1.1: 
0.00/0.37	
0.00/0.37	SCC Processor:
0.00/0.37	-> FAxioms:
0.00/0.37	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.37	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.37	-> Pairs:
0.00/0.37	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.37	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.37	-> EAxioms:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	-> Rules:
0.00/0.37	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.37	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.37	 minus(x,0) -> x
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.37	 quot(0,s(y)) -> 0
0.00/0.37	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.37	-> SRules:
0.00/0.37	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.37	->Strongly Connected Components:
0.00/0.37	->->Cycle:
0.00/0.37	->->-> Pairs:
0.00/0.37	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.37	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.37	-> FAxioms:
0.00/0.37	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) -> plus(x4,x3)
0.00/0.37	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
0.00/0.37	 PLUS(x3,x4) -> PLUS(x4,x3)
0.00/0.37	-> EAxioms:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	->->-> Rules:
0.00/0.37	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.37	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.37	 minus(x,0) -> x
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.37	 quot(0,s(y)) -> 0
0.00/0.37	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.37	-> SRules:
0.00/0.37	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.37	
0.00/0.37	Problem 1.1: 
0.00/0.37	
0.00/0.37	Reduction Pairs Processor:
0.00/0.37	-> FAxioms:
0.00/0.37	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.37	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.37	-> Pairs:
0.00/0.37	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.37	 PLUS(s(x),y) -> PLUS(x,y)
0.00/0.37	-> EAxioms:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	-> Usable Equations:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	-> Rules:
0.00/0.37	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.37	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.37	 minus(x,0) -> x
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.37	 quot(0,s(y)) -> 0
0.00/0.37	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.37	-> Usable Rules:
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.37	-> SRules:
0.00/0.37	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.37	->Interpretation type:
0.00/0.37	 Linear
0.00/0.37	->Coefficients:
0.00/0.37	 Natural Numbers
0.00/0.37	->Dimension:
0.00/0.37	 1
0.00/0.37	->Bound:
0.00/0.37	 2
0.00/0.37	->Interpretation:
0.00/0.37	 
0.00/0.37	[minus](X1,X2) = 0
0.00/0.37	[plus](X1,X2) = X1 + X2 + 2
0.00/0.37	[quot](X1,X2) = 0
0.00/0.37	[0] = 0
0.00/0.37	[s](X) = X + 2
0.00/0.37	[MINUS](X1,X2) = 0
0.00/0.37	[PLUS](X1,X2) = 2.X1 + 2.X2
0.00/0.37	[QUOT](X1,X2) = 0
0.00/0.37	
0.00/0.37	Problem 1.1: 
0.00/0.37	
0.00/0.37	SCC Processor:
0.00/0.37	-> FAxioms:
0.00/0.37	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.37	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.37	-> Pairs:
0.00/0.37	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.37	-> EAxioms:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	-> Rules:
0.00/0.37	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.37	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.37	 minus(x,0) -> x
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.37	 quot(0,s(y)) -> 0
0.00/0.37	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.37	-> SRules:
0.00/0.37	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.37	->Strongly Connected Components:
0.00/0.37	->->Cycle:
0.00/0.37	->->-> Pairs:
0.00/0.37	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.37	-> FAxioms:
0.00/0.37	 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) -> plus(x4,x3)
0.00/0.37	 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
0.00/0.37	 PLUS(x3,x4) -> PLUS(x4,x3)
0.00/0.37	-> EAxioms:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	->->-> Rules:
0.00/0.37	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.37	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.37	 minus(x,0) -> x
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.37	 quot(0,s(y)) -> 0
0.00/0.37	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.37	-> SRules:
0.00/0.37	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.37	
0.00/0.37	Problem 1.1: 
0.00/0.37	
0.00/0.37	Reduction Pairs Processor:
0.00/0.37	-> FAxioms:
0.00/0.37	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.37	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.37	-> Pairs:
0.00/0.37	 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
0.00/0.37	-> EAxioms:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	-> Usable Equations:
0.00/0.37	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.37	 plus(x3,x4) = plus(x4,x3)
0.00/0.37	-> Rules:
0.00/0.37	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.37	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.37	 minus(x,0) -> x
0.00/0.37	 plus(0,y) -> y
0.00/0.37	 plus(s(x),y) -> s(plus(x,y))
0.00/0.38	 quot(0,s(y)) -> 0
0.00/0.38	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.38	-> Usable Rules:
0.00/0.38	 plus(0,y) -> y
0.00/0.38	 plus(s(x),y) -> s(plus(x,y))
0.00/0.38	-> SRules:
0.00/0.38	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.38	->Interpretation type:
0.00/0.38	 Linear
0.00/0.38	->Coefficients:
0.00/0.38	 Natural Numbers
0.00/0.38	->Dimension:
0.00/0.38	 1
0.00/0.38	->Bound:
0.00/0.38	 2
0.00/0.38	->Interpretation:
0.00/0.38	 
0.00/0.38	[minus](X1,X2) = 0
0.00/0.38	[plus](X1,X2) = X1 + X2 + 2
0.00/0.38	[quot](X1,X2) = 0
0.00/0.38	[0] = 0
0.00/0.38	[s](X) = 2
0.00/0.38	[MINUS](X1,X2) = 0
0.00/0.38	[PLUS](X1,X2) = 2.X1 + 2.X2
0.00/0.38	[QUOT](X1,X2) = 0
0.00/0.38	
0.00/0.38	Problem 1.1: 
0.00/0.38	
0.00/0.38	SCC Processor:
0.00/0.38	-> FAxioms:
0.00/0.38	 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
0.00/0.38	 PLUS(x3,x4) = PLUS(x4,x3)
0.00/0.38	-> Pairs:
0.00/0.38	 Empty
0.00/0.38	-> EAxioms:
0.00/0.38	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.38	 plus(x3,x4) = plus(x4,x3)
0.00/0.38	-> Rules:
0.00/0.38	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.38	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.38	 minus(x,0) -> x
0.00/0.38	 plus(0,y) -> y
0.00/0.38	 plus(s(x),y) -> s(plus(x,y))
0.00/0.38	 quot(0,s(y)) -> 0
0.00/0.38	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.38	-> SRules:
0.00/0.38	 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
0.00/0.38	->Strongly Connected Components:
0.00/0.38	 There is no strongly connected component
0.00/0.38	
0.00/0.38	The problem is finite.
0.00/0.38	
0.00/0.38	Problem 1.2: 
0.00/0.38	
0.00/0.38	Subterm Processor:
0.00/0.38	-> FAxioms:
0.00/0.38	 Empty
0.00/0.38	-> Pairs:
0.00/0.38	 MINUS(minus(x,y),z) -> MINUS(x,plus(y,z))
0.00/0.38	 MINUS(s(x),s(y)) -> MINUS(x,y)
0.00/0.38	-> EAxioms:
0.00/0.38	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.38	 plus(x3,x4) = plus(x4,x3)
0.00/0.38	-> Rules:
0.00/0.38	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.38	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.38	 minus(x,0) -> x
0.00/0.38	 plus(0,y) -> y
0.00/0.38	 plus(s(x),y) -> s(plus(x,y))
0.00/0.38	 quot(0,s(y)) -> 0
0.00/0.38	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.38	-> SRules:
0.00/0.38	 Empty
0.00/0.38	->Projection:
0.00/0.38	 pi(MINUS) = [1]
0.00/0.38	
0.00/0.38	Problem 1.2: 
0.00/0.38	
0.00/0.38	SCC Processor:
0.00/0.38	-> FAxioms:
0.00/0.38	 Empty
0.00/0.38	-> Pairs:
0.00/0.38	 Empty
0.00/0.38	-> EAxioms:
0.00/0.38	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.38	 plus(x3,x4) = plus(x4,x3)
0.00/0.38	-> Rules:
0.00/0.38	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.38	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.38	 minus(x,0) -> x
0.00/0.38	 plus(0,y) -> y
0.00/0.38	 plus(s(x),y) -> s(plus(x,y))
0.00/0.38	 quot(0,s(y)) -> 0
0.00/0.38	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.38	-> SRules:
0.00/0.38	 Empty
0.00/0.38	->Strongly Connected Components:
0.00/0.38	 There is no strongly connected component
0.00/0.38	
0.00/0.38	The problem is finite.
0.00/0.38	
0.00/0.38	Problem 1.3: 
0.00/0.38	
0.00/0.38	Reduction Pairs Processor:
0.00/0.38	-> FAxioms:
0.00/0.38	 Empty
0.00/0.38	-> Pairs:
0.00/0.38	 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))
0.00/0.38	-> EAxioms:
0.00/0.38	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.38	 plus(x3,x4) = plus(x4,x3)
0.00/0.38	-> Usable Equations:
0.00/0.38	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.38	 plus(x3,x4) = plus(x4,x3)
0.00/0.38	-> Rules:
0.00/0.38	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.38	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.38	 minus(x,0) -> x
0.00/0.38	 plus(0,y) -> y
0.00/0.38	 plus(s(x),y) -> s(plus(x,y))
0.00/0.38	 quot(0,s(y)) -> 0
0.00/0.38	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.38	-> Usable Rules:
0.00/0.38	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.38	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.38	 minus(x,0) -> x
0.00/0.38	 plus(0,y) -> y
0.00/0.38	 plus(s(x),y) -> s(plus(x,y))
0.00/0.38	-> SRules:
0.00/0.38	 Empty
0.00/0.38	->Interpretation type:
0.00/0.38	 Linear
0.00/0.38	->Coefficients:
0.00/0.38	 Natural Numbers
0.00/0.38	->Dimension:
0.00/0.38	 1
0.00/0.38	->Bound:
0.00/0.38	 2
0.00/0.38	->Interpretation:
0.00/0.38	 
0.00/0.38	[minus](X1,X2) = X1 + 1
0.00/0.38	[plus](X1,X2) = X1 + X2 + 2
0.00/0.38	[quot](X1,X2) = 0
0.00/0.38	[0] = 0
0.00/0.38	[s](X) = X + 2
0.00/0.38	[MINUS](X1,X2) = 0
0.00/0.38	[PLUS](X1,X2) = 0
0.00/0.38	[QUOT](X1,X2) = 2.X1
0.00/0.38	
0.00/0.38	Problem 1.3: 
0.00/0.38	
0.00/0.38	SCC Processor:
0.00/0.38	-> FAxioms:
0.00/0.38	 Empty
0.00/0.38	-> Pairs:
0.00/0.38	 Empty
0.00/0.38	-> EAxioms:
0.00/0.38	 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
0.00/0.38	 plus(x3,x4) = plus(x4,x3)
0.00/0.38	-> Rules:
0.00/0.38	 minus(minus(x,y),z) -> minus(x,plus(y,z))
0.00/0.38	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.38	 minus(x,0) -> x
0.00/0.38	 plus(0,y) -> y
0.00/0.38	 plus(s(x),y) -> s(plus(x,y))
0.00/0.38	 quot(0,s(y)) -> 0
0.00/0.38	 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
0.00/0.38	-> SRules:
0.00/0.38	 Empty
0.00/0.38	->Strongly Connected Components:
0.00/0.38	 There is no strongly connected component
0.00/0.38	
0.00/0.38	The problem is finite.
0.00/0.38	EOF