78.71/81.60 YES 78.71/81.60 78.71/81.60 Problem 1: 78.71/81.60 78.71/81.60 (VAR v_NonEmpty:S M:S N:S X:S X1:S X2:S X3:S) 78.71/81.60 (RULES 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 ) 78.71/81.60 (STRATEGY INNERMOST) 78.71/81.60 78.71/81.60 Problem 1: 78.71/81.60 78.71/81.60 Dependency Pairs Processor: 78.71/81.60 -> Pairs: 78.71/81.60 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(active(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(active(X1:S),X2:S,X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(mark(X1:S),X2:S,X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,active(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,mark(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 ACTIVE(U11(tt,M:S,N:S)) -> U12#(tt,M:S,N:S) 78.71/81.60 ACTIVE(U11(tt,M:S,N:S)) -> MARK(U12(tt,M:S,N:S)) 78.71/81.60 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.60 ACTIVE(U12(tt,M:S,N:S)) -> PLUS(N:S,M:S) 78.71/81.60 ACTIVE(U12(tt,M:S,N:S)) -> S(plus(N:S,M:S)) 78.71/81.60 ACTIVE(U21(tt,M:S,N:S)) -> U22#(tt,M:S,N:S) 78.71/81.60 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.60 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.60 ACTIVE(U22(tt,M:S,N:S)) -> PLUS(x(N:S,M:S),N:S) 78.71/81.60 ACTIVE(U22(tt,M:S,N:S)) -> X(N:S,M:S) 78.71/81.60 ACTIVE(plus(N:S,s(M:S))) -> U11#(tt,M:S,N:S) 78.71/81.60 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.60 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.60 ACTIVE(x(N:S,s(M:S))) -> U21#(tt,M:S,N:S) 78.71/81.60 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.60 ACTIVE(x(N:S,0)) -> MARK(0) 78.71/81.60 MARK(U11(X1:S,X2:S,X3:S)) -> U11#(mark(X1:S),X2:S,X3:S) 78.71/81.60 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(U12(X1:S,X2:S,X3:S)) -> U12#(mark(X1:S),X2:S,X3:S) 78.71/81.60 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(U21(X1:S,X2:S,X3:S)) -> U21#(mark(X1:S),X2:S,X3:S) 78.71/81.60 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(U22(X1:S,X2:S,X3:S)) -> U22#(mark(X1:S),X2:S,X3:S) 78.71/81.60 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S)) 78.71/81.60 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 78.71/81.60 MARK(s(X:S)) -> MARK(X:S) 78.71/81.60 MARK(s(X:S)) -> S(mark(X:S)) 78.71/81.60 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.60 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.60 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.60 MARK(x(X1:S,X2:S)) -> X(mark(X1:S),mark(X2:S)) 78.71/81.60 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) 78.71/81.60 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 78.71/81.60 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.60 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.60 S(active(X:S)) -> S(X:S) 78.71/81.60 S(mark(X:S)) -> S(X:S) 78.71/81.60 X(active(X1:S),X2:S) -> X(X1:S,X2:S) 78.71/81.60 X(mark(X1:S),X2:S) -> X(X1:S,X2:S) 78.71/81.60 X(X1:S,active(X2:S)) -> X(X1:S,X2:S) 78.71/81.60 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) 78.71/81.60 -> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 78.71/81.60 Problem 1: 78.71/81.60 78.71/81.60 SCC Processor: 78.71/81.60 -> Pairs: 78.71/81.60 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(active(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(active(X1:S),X2:S,X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(mark(X1:S),X2:S,X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,active(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,mark(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 ACTIVE(U11(tt,M:S,N:S)) -> U12#(tt,M:S,N:S) 78.71/81.60 ACTIVE(U11(tt,M:S,N:S)) -> MARK(U12(tt,M:S,N:S)) 78.71/81.60 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.60 ACTIVE(U12(tt,M:S,N:S)) -> PLUS(N:S,M:S) 78.71/81.60 ACTIVE(U12(tt,M:S,N:S)) -> S(plus(N:S,M:S)) 78.71/81.60 ACTIVE(U21(tt,M:S,N:S)) -> U22#(tt,M:S,N:S) 78.71/81.60 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.60 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.60 ACTIVE(U22(tt,M:S,N:S)) -> PLUS(x(N:S,M:S),N:S) 78.71/81.60 ACTIVE(U22(tt,M:S,N:S)) -> X(N:S,M:S) 78.71/81.60 ACTIVE(plus(N:S,s(M:S))) -> U11#(tt,M:S,N:S) 78.71/81.60 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.60 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.60 ACTIVE(x(N:S,s(M:S))) -> U21#(tt,M:S,N:S) 78.71/81.60 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.60 ACTIVE(x(N:S,0)) -> MARK(0) 78.71/81.60 MARK(U11(X1:S,X2:S,X3:S)) -> U11#(mark(X1:S),X2:S,X3:S) 78.71/81.60 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(U12(X1:S,X2:S,X3:S)) -> U12#(mark(X1:S),X2:S,X3:S) 78.71/81.60 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(U21(X1:S,X2:S,X3:S)) -> U21#(mark(X1:S),X2:S,X3:S) 78.71/81.60 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(U22(X1:S,X2:S,X3:S)) -> U22#(mark(X1:S),X2:S,X3:S) 78.71/81.60 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S)) 78.71/81.60 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 78.71/81.60 MARK(s(X:S)) -> MARK(X:S) 78.71/81.60 MARK(s(X:S)) -> S(mark(X:S)) 78.71/81.60 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.60 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.60 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.60 MARK(x(X1:S,X2:S)) -> X(mark(X1:S),mark(X2:S)) 78.71/81.60 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) 78.71/81.60 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 78.71/81.60 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.60 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.60 S(active(X:S)) -> S(X:S) 78.71/81.60 S(mark(X:S)) -> S(X:S) 78.71/81.60 X(active(X1:S),X2:S) -> X(X1:S,X2:S) 78.71/81.60 X(mark(X1:S),X2:S) -> X(X1:S,X2:S) 78.71/81.60 X(X1:S,active(X2:S)) -> X(X1:S,X2:S) 78.71/81.60 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) 78.71/81.60 -> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 ->Strongly Connected Components: 78.71/81.60 ->->Cycle: 78.71/81.60 ->->-> Pairs: 78.71/81.60 X(active(X1:S),X2:S) -> X(X1:S,X2:S) 78.71/81.60 X(mark(X1:S),X2:S) -> X(X1:S,X2:S) 78.71/81.60 X(X1:S,active(X2:S)) -> X(X1:S,X2:S) 78.71/81.60 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) 78.71/81.60 ->->-> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 ->->Cycle: 78.71/81.60 ->->-> Pairs: 78.71/81.60 S(active(X:S)) -> S(X:S) 78.71/81.60 S(mark(X:S)) -> S(X:S) 78.71/81.60 ->->-> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 ->->Cycle: 78.71/81.60 ->->-> Pairs: 78.71/81.60 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) 78.71/81.60 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 78.71/81.60 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.60 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.60 ->->-> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 ->->Cycle: 78.71/81.60 ->->-> Pairs: 78.71/81.60 U22#(active(X1:S),X2:S,X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(mark(X1:S),X2:S,X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,active(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,mark(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.60 ->->-> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 ->->Cycle: 78.71/81.60 ->->-> Pairs: 78.71/81.60 U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.60 ->->-> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 ->->Cycle: 78.71/81.60 ->->-> Pairs: 78.71/81.60 U12#(active(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.60 ->->-> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 ->->Cycle: 78.71/81.60 ->->-> Pairs: 78.71/81.60 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.60 ->->-> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.60 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.60 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.60 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.60 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.60 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.60 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.60 active(plus(N:S,0)) -> mark(N:S) 78.71/81.60 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.60 active(x(N:S,0)) -> mark(0) 78.71/81.60 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.60 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.60 mark(0) -> active(0) 78.71/81.60 mark(tt) -> active(tt) 78.71/81.60 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.60 s(active(X:S)) -> s(X:S) 78.71/81.60 s(mark(X:S)) -> s(X:S) 78.71/81.60 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.60 ->->Cycle: 78.71/81.60 ->->-> Pairs: 78.71/81.60 ACTIVE(U11(tt,M:S,N:S)) -> MARK(U12(tt,M:S,N:S)) 78.71/81.60 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.60 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.60 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.60 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.60 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.60 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.60 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.60 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.60 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.60 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 78.71/81.60 MARK(s(X:S)) -> MARK(X:S) 78.71/81.60 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.60 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.60 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.60 ->->-> Rules: 78.71/81.60 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.60 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.60 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 78.71/81.61 78.71/81.61 The problem is decomposed in 8 subproblems. 78.71/81.61 78.71/81.61 Problem 1.1: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 X(active(X1:S),X2:S) -> X(X1:S,X2:S) 78.71/81.61 X(mark(X1:S),X2:S) -> X(X1:S,X2:S) 78.71/81.61 X(X1:S,active(X2:S)) -> X(X1:S,X2:S) 78.71/81.61 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(X) = 1 78.71/81.61 78.71/81.61 Problem 1.1: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 X(X1:S,active(X2:S)) -> X(X1:S,X2:S) 78.71/81.61 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 ->->Cycle: 78.71/81.61 ->->-> Pairs: 78.71/81.61 X(X1:S,active(X2:S)) -> X(X1:S,X2:S) 78.71/81.61 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) 78.71/81.61 ->->-> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 78.71/81.61 Problem 1.1: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 X(X1:S,active(X2:S)) -> X(X1:S,X2:S) 78.71/81.61 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(X) = 2 78.71/81.61 78.71/81.61 Problem 1.1: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 Empty 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 There is no strongly connected component 78.71/81.61 78.71/81.61 The problem is finite. 78.71/81.61 78.71/81.61 Problem 1.2: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 S(active(X:S)) -> S(X:S) 78.71/81.61 S(mark(X:S)) -> S(X:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(S) = 1 78.71/81.61 78.71/81.61 Problem 1.2: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 Empty 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 There is no strongly connected component 78.71/81.61 78.71/81.61 The problem is finite. 78.71/81.61 78.71/81.61 Problem 1.3: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) 78.71/81.61 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 78.71/81.61 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.61 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(PLUS) = 1 78.71/81.61 78.71/81.61 Problem 1.3: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.61 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 ->->Cycle: 78.71/81.61 ->->-> Pairs: 78.71/81.61 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.61 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.61 ->->-> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 78.71/81.61 Problem 1.3: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.61 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(PLUS) = 2 78.71/81.61 78.71/81.61 Problem 1.3: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 Empty 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 There is no strongly connected component 78.71/81.61 78.71/81.61 The problem is finite. 78.71/81.61 78.71/81.61 Problem 1.4: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U22#(active(X1:S),X2:S,X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(mark(X1:S),X2:S,X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,active(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,mark(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(U22#) = 1 78.71/81.61 78.71/81.61 Problem 1.4: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U22#(X1:S,active(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,mark(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 ->->Cycle: 78.71/81.61 ->->-> Pairs: 78.71/81.61 U22#(X1:S,active(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,mark(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 ->->-> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 78.71/81.61 Problem 1.4: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U22#(X1:S,active(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,mark(X2:S),X3:S) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(U22#) = 2 78.71/81.61 78.71/81.61 Problem 1.4: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 ->->Cycle: 78.71/81.61 ->->-> Pairs: 78.71/81.61 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 ->->-> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 78.71/81.61 Problem 1.4: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U22#(X1:S,X2:S,active(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 U22#(X1:S,X2:S,mark(X3:S)) -> U22#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(U22#) = 3 78.71/81.61 78.71/81.61 Problem 1.4: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 Empty 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 There is no strongly connected component 78.71/81.61 78.71/81.61 The problem is finite. 78.71/81.61 78.71/81.61 Problem 1.5: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(U21#) = 1 78.71/81.61 78.71/81.61 Problem 1.5: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 ->->Cycle: 78.71/81.61 ->->-> Pairs: 78.71/81.61 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 ->->-> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 78.71/81.61 Problem 1.5: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(U21#) = 2 78.71/81.61 78.71/81.61 Problem 1.5: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 ->->Cycle: 78.71/81.61 ->->-> Pairs: 78.71/81.61 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 ->->-> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 78.71/81.61 Problem 1.5: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(U21#) = 3 78.71/81.61 78.71/81.61 Problem 1.5: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 Empty 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 There is no strongly connected component 78.71/81.61 78.71/81.61 The problem is finite. 78.71/81.61 78.71/81.61 Problem 1.6: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U12#(active(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(U12#) = 1 78.71/81.61 78.71/81.61 Problem 1.6: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Strongly Connected Components: 78.71/81.61 ->->Cycle: 78.71/81.61 ->->-> Pairs: 78.71/81.61 U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 ->->-> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 78.71/81.61 Problem 1.6: 78.71/81.61 78.71/81.61 Subterm Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.61 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.61 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.61 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.61 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.61 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.61 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.61 active(plus(N:S,0)) -> mark(N:S) 78.71/81.61 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.61 active(x(N:S,0)) -> mark(0) 78.71/81.61 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.61 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.61 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.61 mark(0) -> active(0) 78.71/81.61 mark(tt) -> active(tt) 78.71/81.61 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.61 s(active(X:S)) -> s(X:S) 78.71/81.61 s(mark(X:S)) -> s(X:S) 78.71/81.61 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.61 ->Projection: 78.71/81.61 pi(U12#) = 2 78.71/81.61 78.71/81.61 Problem 1.6: 78.71/81.61 78.71/81.61 SCC Processor: 78.71/81.61 -> Pairs: 78.71/81.61 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.61 -> Rules: 78.71/81.61 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.61 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.61 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Strongly Connected Components: 78.71/81.62 ->->Cycle: 78.71/81.62 ->->-> Pairs: 78.71/81.62 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.62 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.62 ->->-> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 78.71/81.62 Problem 1.6: 78.71/81.62 78.71/81.62 Subterm Processor: 78.71/81.62 -> Pairs: 78.71/81.62 U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.62 U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Projection: 78.71/81.62 pi(U12#) = 3 78.71/81.62 78.71/81.62 Problem 1.6: 78.71/81.62 78.71/81.62 SCC Processor: 78.71/81.62 -> Pairs: 78.71/81.62 Empty 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Strongly Connected Components: 78.71/81.62 There is no strongly connected component 78.71/81.62 78.71/81.62 The problem is finite. 78.71/81.62 78.71/81.62 Problem 1.7: 78.71/81.62 78.71/81.62 Subterm Processor: 78.71/81.62 -> Pairs: 78.71/81.62 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Projection: 78.71/81.62 pi(U11#) = 1 78.71/81.62 78.71/81.62 Problem 1.7: 78.71/81.62 78.71/81.62 SCC Processor: 78.71/81.62 -> Pairs: 78.71/81.62 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Strongly Connected Components: 78.71/81.62 ->->Cycle: 78.71/81.62 ->->-> Pairs: 78.71/81.62 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 ->->-> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 78.71/81.62 Problem 1.7: 78.71/81.62 78.71/81.62 Subterm Processor: 78.71/81.62 -> Pairs: 78.71/81.62 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Projection: 78.71/81.62 pi(U11#) = 2 78.71/81.62 78.71/81.62 Problem 1.7: 78.71/81.62 78.71/81.62 SCC Processor: 78.71/81.62 -> Pairs: 78.71/81.62 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Strongly Connected Components: 78.71/81.62 ->->Cycle: 78.71/81.62 ->->-> Pairs: 78.71/81.62 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 ->->-> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 78.71/81.62 Problem 1.7: 78.71/81.62 78.71/81.62 Subterm Processor: 78.71/81.62 -> Pairs: 78.71/81.62 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Projection: 78.71/81.62 pi(U11#) = 3 78.71/81.62 78.71/81.62 Problem 1.7: 78.71/81.62 78.71/81.62 SCC Processor: 78.71/81.62 -> Pairs: 78.71/81.62 Empty 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Strongly Connected Components: 78.71/81.62 There is no strongly connected component 78.71/81.62 78.71/81.62 The problem is finite. 78.71/81.62 78.71/81.62 Problem 1.8: 78.71/81.62 78.71/81.62 Reduction Pairs Processor: 78.71/81.62 -> Pairs: 78.71/81.62 ACTIVE(U11(tt,M:S,N:S)) -> MARK(U12(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 -> Usable rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Interpretation type: 78.71/81.62 Linear 78.71/81.62 ->Coefficients: 78.71/81.62 Natural Numbers 78.71/81.62 ->Dimension: 78.71/81.62 1 78.71/81.62 ->Bound: 78.71/81.62 2 78.71/81.62 ->Interpretation: 78.71/81.62 78.71/81.62 [U11](X1,X2,X3) = 2 78.71/81.62 [U12](X1,X2,X3) = 2 78.71/81.62 [U21](X1,X2,X3) = 2 78.71/81.62 [U22](X1,X2,X3) = 2 78.71/81.62 [active](X) = 2 78.71/81.62 [mark](X) = 2 78.71/81.62 [plus](X1,X2) = 2 78.71/81.62 [s](X) = 1 78.71/81.62 [x](X1,X2) = 2 78.71/81.62 [0] = 2 78.71/81.62 [fSNonEmpty] = 0 78.71/81.62 [tt] = 2 78.71/81.62 [U11#](X1,X2,X3) = 0 78.71/81.62 [U12#](X1,X2,X3) = 0 78.71/81.62 [U21#](X1,X2,X3) = 0 78.71/81.62 [U22#](X1,X2,X3) = 0 78.71/81.62 [ACTIVE](X) = X 78.71/81.62 [MARK](X) = 2 78.71/81.62 [PLUS](X1,X2) = 0 78.71/81.62 [S](X) = 0 78.71/81.62 [X](X1,X2) = 0 78.71/81.62 78.71/81.62 Problem 1.8: 78.71/81.62 78.71/81.62 SCC Processor: 78.71/81.62 -> Pairs: 78.71/81.62 ACTIVE(U11(tt,M:S,N:S)) -> MARK(U12(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Strongly Connected Components: 78.71/81.62 ->->Cycle: 78.71/81.62 ->->-> Pairs: 78.71/81.62 ACTIVE(U11(tt,M:S,N:S)) -> MARK(U12(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 ->->-> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 78.71/81.62 Problem 1.8: 78.71/81.62 78.71/81.62 Reduction Pairs Processor: 78.71/81.62 -> Pairs: 78.71/81.62 ACTIVE(U11(tt,M:S,N:S)) -> MARK(U12(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 -> Usable rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Interpretation type: 78.71/81.62 Simple mixed 78.71/81.62 ->Coefficients: 78.71/81.62 Natural Numbers 78.71/81.62 ->Dimension: 78.71/81.62 1 78.71/81.62 ->Bound: 78.71/81.62 2 78.71/81.62 ->Interpretation: 78.71/81.62 78.71/81.62 [U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 78.71/81.62 [U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 78.71/81.62 [U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 78.71/81.62 [U22](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + X1 + 2.X3 + 2 78.71/81.62 [active](X) = X 78.71/81.62 [mark](X) = X 78.71/81.62 [plus](X1,X2) = X1 + 2.X2 + 2 78.71/81.62 [s](X) = X + 2 78.71/81.62 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 78.71/81.62 [0] = 2 78.71/81.62 [fSNonEmpty] = 0 78.71/81.62 [tt] = 2 78.71/81.62 [U11#](X1,X2,X3) = 0 78.71/81.62 [U12#](X1,X2,X3) = 0 78.71/81.62 [U21#](X1,X2,X3) = 0 78.71/81.62 [U22#](X1,X2,X3) = 0 78.71/81.62 [ACTIVE](X) = 2.X + 2 78.71/81.62 [MARK](X) = 2.X + 2 78.71/81.62 [PLUS](X1,X2) = 0 78.71/81.62 [S](X) = 0 78.71/81.62 [X](X1,X2) = 0 78.71/81.62 78.71/81.62 Problem 1.8: 78.71/81.62 78.71/81.62 SCC Processor: 78.71/81.62 -> Pairs: 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Strongly Connected Components: 78.71/81.62 ->->Cycle: 78.71/81.62 ->->-> Pairs: 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 ->->-> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 78.71/81.62 Problem 1.8: 78.71/81.62 78.71/81.62 Reduction Pairs Processor: 78.71/81.62 -> Pairs: 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 -> Usable rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Interpretation type: 78.71/81.62 Linear 78.71/81.62 ->Coefficients: 78.71/81.62 Natural Numbers 78.71/81.62 ->Dimension: 78.71/81.62 1 78.71/81.62 ->Bound: 78.71/81.62 2 78.71/81.62 ->Interpretation: 78.71/81.62 78.71/81.62 [U11](X1,X2,X3) = 1 78.71/81.62 [U12](X1,X2,X3) = 2 78.71/81.62 [U21](X1,X2,X3) = 2 78.71/81.62 [U22](X1,X2,X3) = 2 78.71/81.62 [active](X) = 0 78.71/81.62 [mark](X) = 0 78.71/81.62 [plus](X1,X2) = 2 78.71/81.62 [s](X) = 2 78.71/81.62 [x](X1,X2) = 2 78.71/81.62 [0] = 0 78.71/81.62 [fSNonEmpty] = 0 78.71/81.62 [tt] = 2 78.71/81.62 [U11#](X1,X2,X3) = 0 78.71/81.62 [U12#](X1,X2,X3) = 0 78.71/81.62 [U21#](X1,X2,X3) = 0 78.71/81.62 [U22#](X1,X2,X3) = 0 78.71/81.62 [ACTIVE](X) = X 78.71/81.62 [MARK](X) = 2 78.71/81.62 [PLUS](X1,X2) = 0 78.71/81.62 [S](X) = 0 78.71/81.62 [X](X1,X2) = 0 78.71/81.62 78.71/81.62 Problem 1.8: 78.71/81.62 78.71/81.62 SCC Processor: 78.71/81.62 -> Pairs: 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Strongly Connected Components: 78.71/81.62 ->->Cycle: 78.71/81.62 ->->-> Pairs: 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 ->->-> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 78.71/81.62 Problem 1.8: 78.71/81.62 78.71/81.62 Reduction Pairs Processor: 78.71/81.62 -> Pairs: 78.71/81.62 ACTIVE(U12(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 -> Usable rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Interpretation type: 78.71/81.62 Simple mixed 78.71/81.62 ->Coefficients: 78.71/81.62 Natural Numbers 78.71/81.62 ->Dimension: 78.71/81.62 1 78.71/81.62 ->Bound: 78.71/81.62 2 78.71/81.62 ->Interpretation: 78.71/81.62 78.71/81.62 [U11](X1,X2,X3) = X1 + 2.X2 + X3 + 2 78.71/81.62 [U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2 78.71/81.62 [U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + X1 + 2.X3 + 2 78.71/81.62 [U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 78.71/81.62 [active](X) = X 78.71/81.62 [mark](X) = X 78.71/81.62 [plus](X1,X2) = X1 + 2.X2 + 1 78.71/81.62 [s](X) = X + 2 78.71/81.62 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 78.71/81.62 [0] = 0 78.71/81.62 [fSNonEmpty] = 0 78.71/81.62 [tt] = 2 78.71/81.62 [U11#](X1,X2,X3) = 0 78.71/81.62 [U12#](X1,X2,X3) = 0 78.71/81.62 [U21#](X1,X2,X3) = 0 78.71/81.62 [U22#](X1,X2,X3) = 0 78.71/81.62 [ACTIVE](X) = 2.X.X + 2.X + 1 78.71/81.62 [MARK](X) = 2.X.X + 2.X + 1 78.71/81.62 [PLUS](X1,X2) = 0 78.71/81.62 [S](X) = 0 78.71/81.62 [X](X1,X2) = 0 78.71/81.62 78.71/81.62 Problem 1.8: 78.71/81.62 78.71/81.62 SCC Processor: 78.71/81.62 -> Pairs: 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 -> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 ->Strongly Connected Components: 78.71/81.62 ->->Cycle: 78.71/81.62 ->->-> Pairs: 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 MARK(s(X:S)) -> MARK(X:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.62 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.62 ->->-> Rules: 78.71/81.62 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.62 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.62 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.62 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.62 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.62 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.62 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.62 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.62 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.62 active(plus(N:S,0)) -> mark(N:S) 78.71/81.62 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.62 active(x(N:S,0)) -> mark(0) 78.71/81.62 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.62 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.62 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.62 mark(0) -> active(0) 78.71/81.62 mark(tt) -> active(tt) 78.71/81.62 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.62 s(active(X:S)) -> s(X:S) 78.71/81.62 s(mark(X:S)) -> s(X:S) 78.71/81.62 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.62 78.71/81.62 Problem 1.8: 78.71/81.62 78.71/81.62 Reduction Pairs Processor: 78.71/81.62 -> Pairs: 78.71/81.62 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.62 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.62 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.62 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.62 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.62 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.62 MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.63 MARK(s(X:S)) -> MARK(X:S) 78.71/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.63 -> Rules: 78.71/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.63 active(plus(N:S,0)) -> mark(N:S) 78.71/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.63 active(x(N:S,0)) -> mark(0) 78.71/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(0) -> active(0) 78.71/81.63 mark(tt) -> active(tt) 78.71/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 s(active(X:S)) -> s(X:S) 78.71/81.63 s(mark(X:S)) -> s(X:S) 78.71/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 -> Usable rules: 78.71/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.63 active(plus(N:S,0)) -> mark(N:S) 78.71/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.63 active(x(N:S,0)) -> mark(0) 78.71/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(0) -> active(0) 78.71/81.63 mark(tt) -> active(tt) 78.71/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 s(active(X:S)) -> s(X:S) 78.71/81.63 s(mark(X:S)) -> s(X:S) 78.71/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 ->Interpretation type: 78.71/81.63 Linear 78.71/81.63 ->Coefficients: 78.71/81.63 Natural Numbers 78.71/81.63 ->Dimension: 78.71/81.63 1 78.71/81.63 ->Bound: 78.71/81.63 2 78.71/81.63 ->Interpretation: 78.71/81.63 78.71/81.63 [U11](X1,X2,X3) = 0 78.71/81.63 [U12](X1,X2,X3) = 0 78.71/81.63 [U21](X1,X2,X3) = 1 78.71/81.63 [U22](X1,X2,X3) = 1 78.71/81.63 [active](X) = 2 78.71/81.63 [mark](X) = 2 78.71/81.63 [plus](X1,X2) = 1 78.71/81.63 [s](X) = 2 78.71/81.63 [x](X1,X2) = 1 78.71/81.63 [0] = 2 78.71/81.63 [fSNonEmpty] = 0 78.71/81.63 [tt] = 2 78.71/81.63 [U11#](X1,X2,X3) = 0 78.71/81.63 [U12#](X1,X2,X3) = 0 78.71/81.63 [U21#](X1,X2,X3) = 0 78.71/81.63 [U22#](X1,X2,X3) = 0 78.71/81.63 [ACTIVE](X) = 2.X 78.71/81.63 [MARK](X) = 2 78.71/81.63 [PLUS](X1,X2) = 0 78.71/81.63 [S](X) = 0 78.71/81.63 [X](X1,X2) = 0 78.71/81.63 78.71/81.63 Problem 1.8: 78.71/81.63 78.71/81.63 SCC Processor: 78.71/81.63 -> Pairs: 78.71/81.63 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.63 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.63 MARK(s(X:S)) -> MARK(X:S) 78.71/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.63 -> Rules: 78.71/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.63 active(plus(N:S,0)) -> mark(N:S) 78.71/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.63 active(x(N:S,0)) -> mark(0) 78.71/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(0) -> active(0) 78.71/81.63 mark(tt) -> active(tt) 78.71/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 s(active(X:S)) -> s(X:S) 78.71/81.63 s(mark(X:S)) -> s(X:S) 78.71/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 ->Strongly Connected Components: 78.71/81.63 ->->Cycle: 78.71/81.63 ->->-> Pairs: 78.71/81.63 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.63 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.63 MARK(s(X:S)) -> MARK(X:S) 78.71/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.63 ->->-> Rules: 78.71/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.63 active(plus(N:S,0)) -> mark(N:S) 78.71/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.63 active(x(N:S,0)) -> mark(0) 78.71/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(0) -> active(0) 78.71/81.63 mark(tt) -> active(tt) 78.71/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 s(active(X:S)) -> s(X:S) 78.71/81.63 s(mark(X:S)) -> s(X:S) 78.71/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 78.71/81.63 Problem 1.8: 78.71/81.63 78.71/81.63 Reduction Pairs Processor: 78.71/81.63 -> Pairs: 78.71/81.63 ACTIVE(U21(tt,M:S,N:S)) -> MARK(U22(tt,M:S,N:S)) 78.71/81.63 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.71/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.71/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.71/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.71/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.63 MARK(s(X:S)) -> MARK(X:S) 78.71/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.71/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.71/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.71/81.63 -> Rules: 78.71/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.63 active(plus(N:S,0)) -> mark(N:S) 78.71/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.63 active(x(N:S,0)) -> mark(0) 78.71/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.71/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.71/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.71/81.63 mark(0) -> active(0) 78.71/81.63 mark(tt) -> active(tt) 78.71/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.71/81.63 s(active(X:S)) -> s(X:S) 78.71/81.63 s(mark(X:S)) -> s(X:S) 78.71/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.71/81.63 -> Usable rules: 78.71/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.71/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.71/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.71/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.71/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.71/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.71/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.71/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.71/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.71/81.63 active(plus(N:S,0)) -> mark(N:S) 78.71/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.71/81.63 active(x(N:S,0)) -> mark(0) 78.71/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Interpretation type: 78.77/81.63 Simple mixed 78.77/81.63 ->Coefficients: 78.77/81.63 Natural Numbers 78.77/81.63 ->Dimension: 78.77/81.63 1 78.77/81.63 ->Bound: 78.77/81.63 2 78.77/81.63 ->Interpretation: 78.77/81.63 78.77/81.63 [U11](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + X1 + X2 + X3 + 2 78.77/81.63 [U12](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2 78.77/81.63 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 78.77/81.63 [U22](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 78.77/81.63 [active](X) = X 78.77/81.63 [mark](X) = X 78.77/81.63 [plus](X1,X2) = X1 + X2 78.77/81.63 [s](X) = X + 2 78.77/81.63 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 78.77/81.63 [0] = 2 78.77/81.63 [fSNonEmpty] = 0 78.77/81.63 [tt] = 0 78.77/81.63 [U11#](X1,X2,X3) = 0 78.77/81.63 [U12#](X1,X2,X3) = 0 78.77/81.63 [U21#](X1,X2,X3) = 0 78.77/81.63 [U22#](X1,X2,X3) = 0 78.77/81.63 [ACTIVE](X) = 2.X.X + 2.X + 2 78.77/81.63 [MARK](X) = 2.X.X + 2.X + 2 78.77/81.63 [PLUS](X1,X2) = 0 78.77/81.63 [S](X) = 0 78.77/81.63 [X](X1,X2) = 0 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 SCC Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Strongly Connected Components: 78.77/81.63 ->->Cycle: 78.77/81.63 ->->-> Pairs: 78.77/81.63 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 ->->-> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 Reduction Pairs Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 -> Usable rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Interpretation type: 78.77/81.63 Linear 78.77/81.63 ->Coefficients: 78.77/81.63 Natural Numbers 78.77/81.63 ->Dimension: 78.77/81.63 1 78.77/81.63 ->Bound: 78.77/81.63 2 78.77/81.63 ->Interpretation: 78.77/81.63 78.77/81.63 [U11](X1,X2,X3) = 1 78.77/81.63 [U12](X1,X2,X3) = 2 78.77/81.63 [U21](X1,X2,X3) = 0 78.77/81.63 [U22](X1,X2,X3) = 1 78.77/81.63 [active](X) = 2 78.77/81.63 [mark](X) = 2 78.77/81.63 [plus](X1,X2) = 1 78.77/81.63 [s](X) = 0 78.77/81.63 [x](X1,X2) = 1 78.77/81.63 [0] = 2 78.77/81.63 [fSNonEmpty] = 0 78.77/81.63 [tt] = 2 78.77/81.63 [U11#](X1,X2,X3) = 0 78.77/81.63 [U12#](X1,X2,X3) = 0 78.77/81.63 [U21#](X1,X2,X3) = 0 78.77/81.63 [U22#](X1,X2,X3) = 0 78.77/81.63 [ACTIVE](X) = 2.X 78.77/81.63 [MARK](X) = 2 78.77/81.63 [PLUS](X1,X2) = 0 78.77/81.63 [S](X) = 0 78.77/81.63 [X](X1,X2) = 0 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 SCC Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Strongly Connected Components: 78.77/81.63 ->->Cycle: 78.77/81.63 ->->-> Pairs: 78.77/81.63 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 ->->-> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 Reduction Pairs Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(U22(tt,M:S,N:S)) -> MARK(plus(x(N:S,M:S),N:S)) 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 -> Usable rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Interpretation type: 78.77/81.63 Simple mixed 78.77/81.63 ->Coefficients: 78.77/81.63 Natural Numbers 78.77/81.63 ->Dimension: 78.77/81.63 1 78.77/81.63 ->Bound: 78.77/81.63 2 78.77/81.63 ->Interpretation: 78.77/81.63 78.77/81.63 [U11](X1,X2,X3) = X1 + 2.X2 + X3 + 2 78.77/81.63 [U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 78.77/81.63 [U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 78.77/81.63 [U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 78.77/81.63 [active](X) = X 78.77/81.63 [mark](X) = X 78.77/81.63 [plus](X1,X2) = X1 + 2.X2 + 2 78.77/81.63 [s](X) = X + 2 78.77/81.63 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 78.77/81.63 [0] = 1 78.77/81.63 [fSNonEmpty] = 0 78.77/81.63 [tt] = 2 78.77/81.63 [U11#](X1,X2,X3) = 0 78.77/81.63 [U12#](X1,X2,X3) = 0 78.77/81.63 [U21#](X1,X2,X3) = 0 78.77/81.63 [U22#](X1,X2,X3) = 0 78.77/81.63 [ACTIVE](X) = 2.X.X + 2.X + 2 78.77/81.63 [MARK](X) = 2.X.X + 2.X + 2 78.77/81.63 [PLUS](X1,X2) = 0 78.77/81.63 [S](X) = 0 78.77/81.63 [X](X1,X2) = 0 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 SCC Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Strongly Connected Components: 78.77/81.63 ->->Cycle: 78.77/81.63 ->->-> Pairs: 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 ->->-> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 Reduction Pairs Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> ACTIVE(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 -> Usable rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Interpretation type: 78.77/81.63 Linear 78.77/81.63 ->Coefficients: 78.77/81.63 Natural Numbers 78.77/81.63 ->Dimension: 78.77/81.63 1 78.77/81.63 ->Bound: 78.77/81.63 2 78.77/81.63 ->Interpretation: 78.77/81.63 78.77/81.63 [U11](X1,X2,X3) = 0 78.77/81.63 [U12](X1,X2,X3) = 2 78.77/81.63 [U21](X1,X2,X3) = 2 78.77/81.63 [U22](X1,X2,X3) = 0 78.77/81.63 [active](X) = 2 78.77/81.63 [mark](X) = 2 78.77/81.63 [plus](X1,X2) = 1 78.77/81.63 [s](X) = 1 78.77/81.63 [x](X1,X2) = 1 78.77/81.63 [0] = 0 78.77/81.63 [fSNonEmpty] = 0 78.77/81.63 [tt] = 0 78.77/81.63 [U11#](X1,X2,X3) = 0 78.77/81.63 [U12#](X1,X2,X3) = 0 78.77/81.63 [U21#](X1,X2,X3) = 0 78.77/81.63 [U22#](X1,X2,X3) = 0 78.77/81.63 [ACTIVE](X) = X + 1 78.77/81.63 [MARK](X) = 2 78.77/81.63 [PLUS](X1,X2) = 0 78.77/81.63 [S](X) = 0 78.77/81.63 [X](X1,X2) = 0 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 SCC Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Strongly Connected Components: 78.77/81.63 ->->Cycle: 78.77/81.63 ->->-> Pairs: 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 ->->-> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 Reduction Pairs Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(plus(N:S,s(M:S))) -> MARK(U11(tt,M:S,N:S)) 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 -> Usable rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Interpretation type: 78.77/81.63 Simple mixed 78.77/81.63 ->Coefficients: 78.77/81.63 Natural Numbers 78.77/81.63 ->Dimension: 78.77/81.63 1 78.77/81.63 ->Bound: 78.77/81.63 2 78.77/81.63 ->Interpretation: 78.77/81.63 78.77/81.63 [U11](X1,X2,X3) = X1.X2 + X1 + X3 + 2 78.77/81.63 [U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 78.77/81.63 [U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 78.77/81.63 [U22](X1,X2,X3) = X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 78.77/81.63 [active](X) = X 78.77/81.63 [mark](X) = X 78.77/81.63 [plus](X1,X2) = X1 + 2.X2 + 2 78.77/81.63 [s](X) = X + 2 78.77/81.63 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 78.77/81.63 [0] = 2 78.77/81.63 [fSNonEmpty] = 0 78.77/81.63 [tt] = 2 78.77/81.63 [U11#](X1,X2,X3) = 0 78.77/81.63 [U12#](X1,X2,X3) = 0 78.77/81.63 [U21#](X1,X2,X3) = 0 78.77/81.63 [U22#](X1,X2,X3) = 0 78.77/81.63 [ACTIVE](X) = 2.X.X + 2 78.77/81.63 [MARK](X) = 2.X.X + 2 78.77/81.63 [PLUS](X1,X2) = 0 78.77/81.63 [S](X) = 0 78.77/81.63 [X](X1,X2) = 0 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 SCC Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Strongly Connected Components: 78.77/81.63 ->->Cycle: 78.77/81.63 ->->-> Pairs: 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 ->->-> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 Reduction Pairs Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(plus(N:S,0)) -> MARK(N:S) 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 -> Usable rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.63 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.63 active(x(N:S,0)) -> mark(0) 78.77/81.63 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.63 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.63 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.63 mark(0) -> active(0) 78.77/81.63 mark(tt) -> active(tt) 78.77/81.63 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.63 s(active(X:S)) -> s(X:S) 78.77/81.63 s(mark(X:S)) -> s(X:S) 78.77/81.63 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.63 ->Interpretation type: 78.77/81.63 Simple mixed 78.77/81.63 ->Coefficients: 78.77/81.63 Natural Numbers 78.77/81.63 ->Dimension: 78.77/81.63 1 78.77/81.63 ->Bound: 78.77/81.63 2 78.77/81.63 ->Interpretation: 78.77/81.63 78.77/81.63 [U11](X1,X2,X3) = X1.X2 + X1 + X3 + 2 78.77/81.63 [U12](X1,X2,X3) = X1.X2 + X1 + X3 + 2 78.77/81.63 [U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 78.77/81.63 [U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 78.77/81.63 [active](X) = X 78.77/81.63 [mark](X) = X 78.77/81.63 [plus](X1,X2) = X1 + 2.X2 78.77/81.63 [s](X) = X + 2 78.77/81.63 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 78.77/81.63 [0] = 2 78.77/81.63 [fSNonEmpty] = 0 78.77/81.63 [tt] = 2 78.77/81.63 [U11#](X1,X2,X3) = 0 78.77/81.63 [U12#](X1,X2,X3) = 0 78.77/81.63 [U21#](X1,X2,X3) = 0 78.77/81.63 [U22#](X1,X2,X3) = 0 78.77/81.63 [ACTIVE](X) = 2.X 78.77/81.63 [MARK](X) = 2.X 78.77/81.63 [PLUS](X1,X2) = 0 78.77/81.63 [S](X) = 0 78.77/81.63 [X](X1,X2) = 0 78.77/81.63 78.77/81.63 Problem 1.8: 78.77/81.63 78.77/81.63 SCC Processor: 78.77/81.63 -> Pairs: 78.77/81.63 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.63 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 MARK(s(X:S)) -> MARK(X:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.63 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.63 -> Rules: 78.77/81.63 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.63 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.63 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.63 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.63 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.63 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.63 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.63 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.63 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.63 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 ->Strongly Connected Components: 78.77/81.64 ->->Cycle: 78.77/81.64 ->->-> Pairs: 78.77/81.64 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.64 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 MARK(s(X:S)) -> MARK(X:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 ->->-> Rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 78.77/81.64 Problem 1.8: 78.77/81.64 78.77/81.64 Reduction Pairs Processor: 78.77/81.64 -> Pairs: 78.77/81.64 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.64 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 MARK(s(X:S)) -> MARK(X:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 -> Rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 -> Usable rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 ->Interpretation type: 78.77/81.64 Linear 78.77/81.64 ->Coefficients: 78.77/81.64 Natural Numbers 78.77/81.64 ->Dimension: 78.77/81.64 1 78.77/81.64 ->Bound: 78.77/81.64 2 78.77/81.64 ->Interpretation: 78.77/81.64 78.77/81.64 [U11](X1,X2,X3) = 2 78.77/81.64 [U12](X1,X2,X3) = 1 78.77/81.64 [U21](X1,X2,X3) = 2 78.77/81.64 [U22](X1,X2,X3) = 2 78.77/81.64 [active](X) = 2 78.77/81.64 [mark](X) = 2 78.77/81.64 [plus](X1,X2) = 0 78.77/81.64 [s](X) = 2 78.77/81.64 [x](X1,X2) = 2 78.77/81.64 [0] = 2 78.77/81.64 [fSNonEmpty] = 0 78.77/81.64 [tt] = 1 78.77/81.64 [U11#](X1,X2,X3) = 0 78.77/81.64 [U12#](X1,X2,X3) = 0 78.77/81.64 [U21#](X1,X2,X3) = 0 78.77/81.64 [U22#](X1,X2,X3) = 0 78.77/81.64 [ACTIVE](X) = X 78.77/81.64 [MARK](X) = 2 78.77/81.64 [PLUS](X1,X2) = 0 78.77/81.64 [S](X) = 0 78.77/81.64 [X](X1,X2) = 0 78.77/81.64 78.77/81.64 Problem 1.8: 78.77/81.64 78.77/81.64 SCC Processor: 78.77/81.64 -> Pairs: 78.77/81.64 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.64 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 MARK(s(X:S)) -> MARK(X:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 -> Rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 ->Strongly Connected Components: 78.77/81.64 ->->Cycle: 78.77/81.64 ->->-> Pairs: 78.77/81.64 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.64 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 MARK(s(X:S)) -> MARK(X:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 ->->-> Rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 78.77/81.64 Problem 1.8: 78.77/81.64 78.77/81.64 Reduction Pairs Processor: 78.77/81.64 -> Pairs: 78.77/81.64 ACTIVE(x(N:S,s(M:S))) -> MARK(U21(tt,M:S,N:S)) 78.77/81.64 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 MARK(s(X:S)) -> MARK(X:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 -> Rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 -> Usable rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 ->Interpretation type: 78.77/81.64 Simple mixed 78.77/81.64 ->Coefficients: 78.77/81.64 Natural Numbers 78.77/81.64 ->Dimension: 78.77/81.64 1 78.77/81.64 ->Bound: 78.77/81.64 2 78.77/81.64 ->Interpretation: 78.77/81.64 78.77/81.64 [U11](X1,X2,X3) = 2.X1 + X2 + X3 78.77/81.64 [U12](X1,X2,X3) = 2.X1 + X2 + X3 78.77/81.64 [U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 1 78.77/81.64 [U22](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 1 78.77/81.64 [active](X) = X 78.77/81.64 [mark](X) = X 78.77/81.64 [plus](X1,X2) = X1 + X2 + 2 78.77/81.64 [s](X) = X + 2 78.77/81.64 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 78.77/81.64 [0] = 2 78.77/81.64 [fSNonEmpty] = 0 78.77/81.64 [tt] = 2 78.77/81.64 [U11#](X1,X2,X3) = 0 78.77/81.64 [U12#](X1,X2,X3) = 0 78.77/81.64 [U21#](X1,X2,X3) = 0 78.77/81.64 [U22#](X1,X2,X3) = 0 78.77/81.64 [ACTIVE](X) = X + 2 78.77/81.64 [MARK](X) = X + 2 78.77/81.64 [PLUS](X1,X2) = 0 78.77/81.64 [S](X) = 0 78.77/81.64 [X](X1,X2) = 0 78.77/81.64 78.77/81.64 Problem 1.8: 78.77/81.64 78.77/81.64 SCC Processor: 78.77/81.64 -> Pairs: 78.77/81.64 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 MARK(s(X:S)) -> MARK(X:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 -> Rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 ->Strongly Connected Components: 78.77/81.64 ->->Cycle: 78.77/81.64 ->->-> Pairs: 78.77/81.64 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 MARK(s(X:S)) -> MARK(X:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 ->->-> Rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 78.77/81.64 Problem 1.8: 78.77/81.64 78.77/81.64 Subterm Processor: 78.77/81.64 -> Pairs: 78.77/81.64 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 MARK(s(X:S)) -> MARK(X:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 78.77/81.64 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 78.77/81.64 -> Rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 ->Projection: 78.77/81.64 pi(MARK) = 1 78.77/81.64 78.77/81.64 Problem 1.8: 78.77/81.64 78.77/81.64 SCC Processor: 78.77/81.64 -> Pairs: 78.77/81.64 Empty 78.77/81.64 -> Rules: 78.77/81.64 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) 78.77/81.64 U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) 78.77/81.64 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 78.77/81.64 U22(active(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(mark(X1:S),X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,active(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,mark(X2:S),X3:S) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,active(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 U22(X1:S,X2:S,mark(X3:S)) -> U22(X1:S,X2:S,X3:S) 78.77/81.64 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 78.77/81.64 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 78.77/81.64 active(U21(tt,M:S,N:S)) -> mark(U22(tt,M:S,N:S)) 78.77/81.64 active(U22(tt,M:S,N:S)) -> mark(plus(x(N:S,M:S),N:S)) 78.77/81.64 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 78.77/81.64 active(plus(N:S,0)) -> mark(N:S) 78.77/81.64 active(x(N:S,s(M:S))) -> mark(U21(tt,M:S,N:S)) 78.77/81.64 active(x(N:S,0)) -> mark(0) 78.77/81.64 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(U22(X1:S,X2:S,X3:S)) -> active(U22(mark(X1:S),X2:S,X3:S)) 78.77/81.64 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(s(X:S)) -> active(s(mark(X:S))) 78.77/81.64 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) 78.77/81.64 mark(0) -> active(0) 78.77/81.64 mark(tt) -> active(tt) 78.77/81.64 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 78.77/81.64 s(active(X:S)) -> s(X:S) 78.77/81.64 s(mark(X:S)) -> s(X:S) 78.77/81.64 x(active(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(mark(X1:S),X2:S) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,active(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) 78.77/81.64 ->Strongly Connected Components: 78.77/81.64 There is no strongly connected component 78.77/81.64 78.77/81.64 The problem is finite. 78.77/81.64 EOF