20.15/20.21	YES
20.15/20.21	
20.15/20.21	Problem 1: 
20.15/20.21	
20.15/20.21	(VAR v_NonEmpty:S x:S y:S)
20.15/20.21	(RULES
20.15/20.21	le(0,y:S) -> ttrue
20.15/20.21	le(s(x:S),0) -> ffalse
20.15/20.21	le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	minus(x:S,0) -> x:S
20.15/20.21	quot(0,s(y:S)) -> 0
20.15/20.21	quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	) 
20.15/20.21	(STRATEGY INNERMOST)
20.15/20.21	
20.15/20.21	Problem 1: 
20.15/20.21	
20.15/20.21	Dependency Pairs Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
20.15/20.21	 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> MINUS(s(x:S),s(y:S))
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> QUOT(minus(s(x:S),s(y:S)),s(y:S))
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	
20.15/20.21	Problem 1: 
20.15/20.21	
20.15/20.21	SCC Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
20.15/20.21	 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> MINUS(s(x:S),s(y:S))
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> QUOT(minus(s(x:S),s(y:S)),s(y:S))
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->Strongly Connected Components:
20.15/20.21	->->Cycle:
20.15/20.21	->->-> Pairs:
20.15/20.21	 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
20.15/20.21	->->-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->->Cycle:
20.15/20.21	->->-> Pairs:
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> QUOT(minus(s(x:S),s(y:S)),s(y:S))
20.15/20.21	->->-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->->Cycle:
20.15/20.21	->->-> Pairs:
20.15/20.21	 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
20.15/20.21	->->-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	
20.15/20.21	
20.15/20.21	The problem is decomposed in 3 subproblems.
20.15/20.21	
20.15/20.21	Problem 1.1: 
20.15/20.21	
20.15/20.21	Subterm Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->Projection:
20.15/20.21	 pi(MINUS) = 1
20.15/20.21	
20.15/20.21	Problem 1.1: 
20.15/20.21	
20.15/20.21	SCC Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 Empty
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->Strongly Connected Components:
20.15/20.21	 There is no strongly connected component
20.15/20.21	
20.15/20.21	The problem is finite.
20.15/20.21	
20.15/20.21	Problem 1.2: 
20.15/20.21	
20.15/20.21	Narrowing Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> QUOT(minus(s(x:S),s(y:S)),s(y:S))
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->Narrowed Pairs:
20.15/20.21	->->Original Pair:
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> QUOT(minus(s(x:S),s(y:S)),s(y:S))
20.15/20.21	->-> Narrowed pairs:
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
20.15/20.21	
20.15/20.21	Problem 1.2: 
20.15/20.21	
20.15/20.21	SCC Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->Strongly Connected Components:
20.15/20.21	->->Cycle:
20.15/20.21	->->-> Pairs:
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
20.15/20.21	->->-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	
20.15/20.21	Problem 1.2: 
20.15/20.21	
20.15/20.21	Reduction Pairs Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	-> Usable rules:
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	->Interpretation type:
20.15/20.21	 Linear
20.15/20.21	->Coefficients:
20.15/20.21	 Natural Numbers
20.15/20.21	->Dimension:
20.15/20.21	 1
20.15/20.21	->Bound:
20.15/20.21	 2
20.15/20.21	->Interpretation:
20.15/20.21	 
20.15/20.21	[le](X1,X2) = 0
20.15/20.21	[minus](X1,X2) = 2.X1 + 1
20.15/20.21	[quot](X1,X2) = 0
20.15/20.21	[0] = 0
20.15/20.21	[fSNonEmpty] = 0
20.15/20.21	[false] = 0
20.15/20.21	[s](X) = 2.X + 2
20.15/20.21	[true] = 0
20.15/20.21	[LE](X1,X2) = 0
20.15/20.21	[MINUS](X1,X2) = 0
20.15/20.21	[QUOT](X1,X2) = 2.X1
20.15/20.21	
20.15/20.21	Problem 1.2: 
20.15/20.21	
20.15/20.21	SCC Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 Empty
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->Strongly Connected Components:
20.15/20.21	 There is no strongly connected component
20.15/20.21	
20.15/20.21	The problem is finite.
20.15/20.21	
20.15/20.21	Problem 1.3: 
20.15/20.21	
20.15/20.21	Subterm Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->Projection:
20.15/20.21	 pi(LE) = 1
20.15/20.21	
20.15/20.21	Problem 1.3: 
20.15/20.21	
20.15/20.21	SCC Processor:
20.15/20.21	-> Pairs:
20.15/20.21	 Empty
20.15/20.21	-> Rules:
20.15/20.21	 le(0,y:S) -> ttrue
20.15/20.21	 le(s(x:S),0) -> ffalse
20.15/20.21	 le(s(x:S),s(y:S)) -> le(x:S,y:S)
20.15/20.21	 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
20.15/20.21	 minus(x:S,0) -> x:S
20.15/20.21	 quot(0,s(y:S)) -> 0
20.15/20.21	 quot(s(x:S),s(y:S)) -> s(quot(minus(s(x:S),s(y:S)),s(y:S)))
20.15/20.21	->Strongly Connected Components:
20.15/20.21	 There is no strongly connected component
20.15/20.21	
20.15/20.21	The problem is finite.
20.15/20.21	EOF