0.00/0.01	YES
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	(VAR M N)
0.00/0.01	(STRATEGY CONTEXTSENSITIVE
0.00/0.01	(U11 1)
0.00/0.01	(U12 1)
0.00/0.01	(U21 1)
0.00/0.01	(U22 1)
0.00/0.01	(plus 1 2)
0.00/0.01	(x 1 2)
0.00/0.01	(0)
0.00/0.01	(s 1)
0.00/0.01	(tt)
0.00/0.01	)
0.00/0.01	(RULES
0.00/0.01	U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	plus(N,0) -> N
0.00/0.01	plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	x(N,0) -> 0
0.00/0.01	x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	)
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	Innermost Equivalent Processor:
0.00/0.01	-> Rules:
0.00/0.01	 U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	 U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	 U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	 U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	 plus(N,0) -> N
0.00/0.01	 plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	 x(N,0) -> 0
0.00/0.01	 x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	-> The context-sensitive term rewriting system is an orthogonal system. Therefore, innermost cs-termination implies cs-termination.
0.00/0.01	 
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	Dependency Pairs Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 U11#(tt,M,N) -> U12#(tt,M,N)
0.00/0.01	 U12#(tt,M,N) -> PLUS(N,M)
0.00/0.01	 U12#(tt,M,N) -> M
0.00/0.01	 U12#(tt,M,N) -> N
0.00/0.01	 U21#(tt,M,N) -> U22#(tt,M,N)
0.00/0.01	 U22#(tt,M,N) -> PLUS(x(N,M),N)
0.00/0.01	 U22#(tt,M,N) -> X(N,M)
0.00/0.01	 U22#(tt,M,N) -> M
0.00/0.01	 U22#(tt,M,N) -> N
0.00/0.01	 PLUS(N,s(M)) -> U11#(tt,M,N)
0.00/0.01	 X(N,s(M)) -> U21#(tt,M,N)
0.00/0.01	-> Rules:
0.00/0.01	 U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	 U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	 U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	 U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	 plus(N,0) -> N
0.00/0.01	 plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	 x(N,0) -> 0
0.00/0.01	 x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	-> Unhiding Rules:
0.00/0.01	 Empty
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	SCC Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 U11#(tt,M,N) -> U12#(tt,M,N)
0.00/0.01	 U12#(tt,M,N) -> PLUS(N,M)
0.00/0.01	 U12#(tt,M,N) -> M
0.00/0.01	 U12#(tt,M,N) -> N
0.00/0.01	 U21#(tt,M,N) -> U22#(tt,M,N)
0.00/0.01	 U22#(tt,M,N) -> PLUS(x(N,M),N)
0.00/0.01	 U22#(tt,M,N) -> X(N,M)
0.00/0.01	 U22#(tt,M,N) -> M
0.00/0.01	 U22#(tt,M,N) -> N
0.00/0.01	 PLUS(N,s(M)) -> U11#(tt,M,N)
0.00/0.01	 X(N,s(M)) -> U21#(tt,M,N)
0.00/0.01	-> Rules:
0.00/0.01	 U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	 U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	 U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	 U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	 plus(N,0) -> N
0.00/0.01	 plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	 x(N,0) -> 0
0.00/0.01	 x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Strongly Connected Components:
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 U11#(tt,M,N) -> U12#(tt,M,N)
0.00/0.01	 U12#(tt,M,N) -> PLUS(N,M)
0.00/0.01	 PLUS(N,s(M)) -> U11#(tt,M,N)
0.00/0.01	->->-> Rules:
0.00/0.01	 U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	 U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	 U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	 U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	 plus(N,0) -> N
0.00/0.01	 plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	 x(N,0) -> 0
0.00/0.01	 x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	->->-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 U21#(tt,M,N) -> U22#(tt,M,N)
0.00/0.01	 U22#(tt,M,N) -> X(N,M)
0.00/0.01	 X(N,s(M)) -> U21#(tt,M,N)
0.00/0.01	->->-> Rules:
0.00/0.01	 U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	 U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	 U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	 U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	 plus(N,0) -> N
0.00/0.01	 plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	 x(N,0) -> 0
0.00/0.01	 x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	->->-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	
0.00/0.01	
0.00/0.01	The problem is decomposed in 2 subproblems.
0.00/0.01	
0.00/0.01	Problem 1.1: 
0.00/0.01	
0.00/0.01	SubNColl Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 U11#(tt,M,N) -> U12#(tt,M,N)
0.00/0.01	 U12#(tt,M,N) -> PLUS(N,M)
0.00/0.01	 PLUS(N,s(M)) -> U11#(tt,M,N)
0.00/0.01	-> Rules:
0.00/0.01	 U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	 U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	 U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	 U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	 plus(N,0) -> N
0.00/0.01	 plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	 x(N,0) -> 0
0.00/0.01	 x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Projection:
0.00/0.01	 pi(U11#) = 2
0.00/0.01	 pi(U12#) = 2
0.00/0.01	 pi(PLUS) = 2
0.00/0.01	
0.00/0.01	Problem 1.1: 
0.00/0.01	
0.00/0.01	SCC Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 U11#(tt,M,N) -> U12#(tt,M,N)
0.00/0.01	 U12#(tt,M,N) -> PLUS(N,M)
0.00/0.01	-> Rules:
0.00/0.01	 U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	 U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	 U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	 U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	 plus(N,0) -> N
0.00/0.01	 plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	 x(N,0) -> 0
0.00/0.01	 x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Strongly Connected Components:
0.00/0.01	 There is no strongly connected component
0.00/0.01	
0.00/0.01	The problem is finite.
0.00/0.01	
0.00/0.01	Problem 1.2: 
0.00/0.01	
0.00/0.01	SubNColl Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 U21#(tt,M,N) -> U22#(tt,M,N)
0.00/0.01	 U22#(tt,M,N) -> X(N,M)
0.00/0.01	 X(N,s(M)) -> U21#(tt,M,N)
0.00/0.01	-> Rules:
0.00/0.01	 U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	 U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	 U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	 U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	 plus(N,0) -> N
0.00/0.01	 plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	 x(N,0) -> 0
0.00/0.01	 x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Projection:
0.00/0.01	 pi(U21#) = 2
0.00/0.01	 pi(U22#) = 2
0.00/0.01	 pi(X) = 2
0.00/0.01	
0.00/0.01	Problem 1.2: 
0.00/0.01	
0.00/0.01	SCC Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 U21#(tt,M,N) -> U22#(tt,M,N)
0.00/0.01	 U22#(tt,M,N) -> X(N,M)
0.00/0.01	-> Rules:
0.00/0.01	 U11(tt,M,N) -> U12(tt,M,N)
0.00/0.01	 U12(tt,M,N) -> s(plus(N,M))
0.00/0.01	 U21(tt,M,N) -> U22(tt,M,N)
0.00/0.01	 U22(tt,M,N) -> plus(x(N,M),N)
0.00/0.01	 plus(N,0) -> N
0.00/0.01	 plus(N,s(M)) -> U11(tt,M,N)
0.00/0.01	 x(N,0) -> 0
0.00/0.01	 x(N,s(M)) -> U21(tt,M,N)
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Strongly Connected Components:
0.00/0.01	 There is no strongly connected component
0.00/0.01	
0.00/0.01	The problem is finite.
0.00/0.01	EOF