62.65/65.54 YES 62.65/65.54 62.65/65.54 Problem 1: 62.65/65.54 62.65/65.54 (VAR v_NonEmpty:S M:S N:S X:S X1:S X2:S X3:S) 62.65/65.54 (RULES 62.65/65.54 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.54 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.54 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.54 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.54 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.54 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.54 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.54 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.54 a__plus(N:S,0) -> mark(N:S) 62.65/65.54 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.54 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.54 a__x(N:S,0) -> 0 62.65/65.54 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.54 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.54 mark(0) -> 0 62.65/65.54 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.54 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.54 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.54 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.54 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.54 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.54 mark(tt) -> tt 62.65/65.54 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.54 ) 62.65/65.54 (STRATEGY INNERMOST) 62.65/65.54 62.65/65.54 Problem 1: 62.65/65.54 62.65/65.54 Dependency Pairs Processor: 62.65/65.54 -> Pairs: 62.65/65.54 A__U11(tt,M:S,N:S) -> A__U12(tt,M:S,N:S) 62.65/65.54 A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 62.65/65.54 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.54 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.54 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.54 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.54 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.54 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.54 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.54 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.54 A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) 62.65/65.54 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.54 MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) 62.65/65.54 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.54 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.54 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.54 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.54 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.54 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.54 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.54 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U11(tt,M:S,N:S) -> A__U12(tt,M:S,N:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 A__U11(tt,M:S,N:S) -> A__U12(tt,M:S,N:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Reduction Pairs Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U11(tt,M:S,N:S) -> A__U12(tt,M:S,N:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 -> Usable rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Interpretation type: 62.65/65.55 Simple mixed 62.65/65.55 ->Coefficients: 62.65/65.55 Natural Numbers 62.65/65.55 ->Dimension: 62.65/65.55 1 62.65/65.55 ->Bound: 62.65/65.55 2 62.65/65.55 ->Interpretation: 62.65/65.55 62.65/65.55 [a__U11](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 62.65/65.55 [a__U12](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 62.65/65.55 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [a__U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 62.65/65.55 [a__plus](X1,X2) = X1 + X2 62.65/65.55 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 62.65/65.55 [mark](X) = X 62.65/65.55 [0] = 2 62.65/65.55 [U11](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 62.65/65.55 [U12](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 62.65/65.55 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 62.65/65.55 [fSNonEmpty] = 0 62.65/65.55 [plus](X1,X2) = X1 + X2 62.65/65.55 [s](X) = X + 2 62.65/65.55 [tt] = 0 62.65/65.55 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 62.65/65.55 [A__U11](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2 62.65/65.55 [A__U12](X1,X2,X3) = X1.X3 + 2.X1 + X2 + X3 + 1 62.65/65.55 [A__U21](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [A__U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X2 + 2.X3 + 1 62.65/65.55 [A__PLUS](X1,X2) = X1 + X2 62.65/65.55 [A__X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 62.65/65.55 [MARK](X) = X 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Reduction Pairs Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 -> Usable rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Interpretation type: 62.65/65.55 Simple mixed 62.65/65.55 ->Coefficients: 62.65/65.55 Natural Numbers 62.65/65.55 ->Dimension: 62.65/65.55 1 62.65/65.55 ->Bound: 62.65/65.55 2 62.65/65.55 ->Interpretation: 62.65/65.55 62.65/65.55 [a__U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 62.65/65.55 [a__U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [a__U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [a__U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [a__plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [mark](X) = X 62.65/65.55 [0] = 2 62.65/65.55 [U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 62.65/65.55 [U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [fSNonEmpty] = 0 62.65/65.55 [plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [s](X) = X + 2 62.65/65.55 [tt] = 2 62.65/65.55 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [A__U11](X1,X2,X3) = 0 62.65/65.55 [A__U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [A__U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [A__U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [A__PLUS](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [MARK](X) = X + 2 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Reduction Pairs Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 -> Usable rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Interpretation type: 62.65/65.55 Simple mixed 62.65/65.55 ->Coefficients: 62.65/65.55 Natural Numbers 62.65/65.55 ->Dimension: 62.65/65.55 1 62.65/65.55 ->Bound: 62.65/65.55 2 62.65/65.55 ->Interpretation: 62.65/65.55 62.65/65.55 [a__U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [a__U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [a__U21](X1,X2,X3) = 2.X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2 62.65/65.55 [a__U22](X1,X2,X3) = X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [a__plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [a__x](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [mark](X) = X 62.65/65.55 [0] = 2 62.65/65.55 [U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [U21](X1,X2,X3) = 2.X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2 62.65/65.55 [U22](X1,X2,X3) = X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [fSNonEmpty] = 0 62.65/65.55 [plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [s](X) = X + 2 62.65/65.55 [tt] = 2 62.65/65.55 [x](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [A__U11](X1,X2,X3) = 0 62.65/65.55 [A__U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2 62.65/65.55 [A__U21](X1,X2,X3) = X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2 62.65/65.55 [A__U22](X1,X2,X3) = X2.X3 + 2.X1 + 2.X2 + 2.X3 62.65/65.55 [A__PLUS](X1,X2) = X1 + 2 62.65/65.55 [A__X](X1,X2) = X1.X2 + 2.X2 + 2 62.65/65.55 [MARK](X) = X + 2 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Reduction Pairs Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U12(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 -> Usable rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Interpretation type: 62.65/65.55 Simple mixed 62.65/65.55 ->Coefficients: 62.65/65.55 Natural Numbers 62.65/65.55 ->Dimension: 62.65/65.55 1 62.65/65.55 ->Bound: 62.65/65.55 2 62.65/65.55 ->Interpretation: 62.65/65.55 62.65/65.55 [a__U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [a__U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 62.65/65.55 [a__U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 62.65/65.55 [a__U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 62.65/65.55 [a__plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [mark](X) = X 62.65/65.55 [0] = 1 62.65/65.55 [U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 62.65/65.55 [U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 62.65/65.55 [U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 62.65/65.55 [fSNonEmpty] = 0 62.65/65.55 [plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [s](X) = X + 2 62.65/65.55 [tt] = 2 62.65/65.55 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [A__U11](X1,X2,X3) = 0 62.65/65.55 [A__U12](X1,X2,X3) = X1.X2 + X1 + X3 + 2 62.65/65.55 [A__U21](X1,X2,X3) = X1.X2 + X1.X3 + 2.X2.X3 + X1 + 2 62.65/65.55 [A__U22](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [A__PLUS](X1,X2) = X1 + 2 62.65/65.55 [A__X](X1,X2) = 2.X1.X2 + 2.X2 + 2 62.65/65.55 [MARK](X) = X + 2 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Reduction Pairs Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 -> Usable rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Interpretation type: 62.65/65.55 Simple mixed 62.65/65.55 ->Coefficients: 62.65/65.55 Natural Numbers 62.65/65.55 ->Dimension: 62.65/65.55 1 62.65/65.55 ->Bound: 62.65/65.55 2 62.65/65.55 ->Interpretation: 62.65/65.55 62.65/65.55 [a__U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 62.65/65.55 [a__U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [a__U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 62.65/65.55 [a__U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2 62.65/65.55 [a__plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [mark](X) = X 62.65/65.55 [0] = 2 62.65/65.55 [U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 62.65/65.55 [U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 62.65/65.55 [U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 62.65/65.55 [U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2 62.65/65.55 [fSNonEmpty] = 0 62.65/65.55 [plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [s](X) = X + 2 62.65/65.55 [tt] = 2 62.65/65.55 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [A__U11](X1,X2,X3) = 0 62.65/65.55 [A__U12](X1,X2,X3) = 0 62.65/65.55 [A__U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 62.65/65.55 [A__U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X2 + 2 62.65/65.55 [A__PLUS](X1,X2) = X1 + 2.X2 62.65/65.55 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 62.65/65.55 [MARK](X) = X + 2 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Reduction Pairs Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 -> Usable rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Interpretation type: 62.65/65.55 Simple mixed 62.65/65.55 ->Coefficients: 62.65/65.55 Natural Numbers 62.65/65.55 ->Dimension: 62.65/65.55 1 62.65/65.55 ->Bound: 62.65/65.55 2 62.65/65.55 ->Interpretation: 62.65/65.55 62.65/65.55 [a__U11](X1,X2,X3) = X1.X2 + X1 + X3 + 2 62.65/65.55 [a__U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 62.65/65.55 [a__U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [a__U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + X1 + 2.X3 + 2 62.65/65.55 [a__plus](X1,X2) = X1 + 2.X2 62.65/65.55 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 62.65/65.55 [mark](X) = X 62.65/65.55 [0] = 0 62.65/65.55 [U11](X1,X2,X3) = X1.X2 + X1 + X3 + 2 62.65/65.55 [U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 62.65/65.55 [U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + X1 + 2.X3 + 2 62.65/65.55 [fSNonEmpty] = 0 62.65/65.55 [plus](X1,X2) = X1 + 2.X2 62.65/65.55 [s](X) = X + 2 62.65/65.55 [tt] = 2 62.65/65.55 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 62.65/65.55 [A__U11](X1,X2,X3) = 0 62.65/65.55 [A__U12](X1,X2,X3) = 0 62.65/65.55 [A__U21](X1,X2,X3) = 0 62.65/65.55 [A__U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + 1 62.65/65.55 [A__PLUS](X1,X2) = 2.X1 + 2 62.65/65.55 [A__X](X1,X2) = 0 62.65/65.55 [MARK](X) = 2.X + 2 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Reduction Pairs Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(M:S) 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 -> Usable rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Interpretation type: 62.65/65.55 Simple mixed 62.65/65.55 ->Coefficients: 62.65/65.55 Natural Numbers 62.65/65.55 ->Dimension: 62.65/65.55 1 62.65/65.55 ->Bound: 62.65/65.55 2 62.65/65.55 ->Interpretation: 62.65/65.55 62.65/65.55 [a__U11](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 62.65/65.55 [a__U12](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 62.65/65.55 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [a__U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [a__plus](X1,X2) = X1 + X2 62.65/65.55 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 62.65/65.55 [mark](X) = X 62.65/65.55 [0] = 2 62.65/65.55 [U11](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 62.65/65.55 [U12](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 62.65/65.55 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [fSNonEmpty] = 0 62.65/65.55 [plus](X1,X2) = X1 + X2 62.65/65.55 [s](X) = X + 2 62.65/65.55 [tt] = 0 62.65/65.55 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 62.65/65.55 [A__U11](X1,X2,X3) = 0 62.65/65.55 [A__U12](X1,X2,X3) = 0 62.65/65.55 [A__U21](X1,X2,X3) = 0 62.65/65.55 [A__U22](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X2 + 2.X3 + 2 62.65/65.55 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 1 62.65/65.55 [A__X](X1,X2) = 0 62.65/65.55 [MARK](X) = 2.X + 1 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Reduction Pairs Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__U22(tt,M:S,N:S) -> MARK(N:S) 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 -> Usable rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Interpretation type: 62.65/65.55 Simple mixed 62.65/65.55 ->Coefficients: 62.65/65.55 Natural Numbers 62.65/65.55 ->Dimension: 62.65/65.55 1 62.65/65.55 ->Bound: 62.65/65.55 2 62.65/65.55 ->Interpretation: 62.65/65.55 62.65/65.55 [a__U11](X1,X2,X3) = X1 + X2 + X3 + 2 62.65/65.55 [a__U12](X1,X2,X3) = X1 + X2 + X3 + 2 62.65/65.55 [a__U21](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1 + 2.X3 + 2 62.65/65.55 [a__U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1 + 2.X3 + 1 62.65/65.55 [a__plus](X1,X2) = X1 + X2 + 2 62.65/65.55 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 62.65/65.55 [mark](X) = X 62.65/65.55 [0] = 2 62.65/65.55 [U11](X1,X2,X3) = X1 + X2 + X3 + 2 62.65/65.55 [U12](X1,X2,X3) = X1 + X2 + X3 + 2 62.65/65.55 [U21](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1 + 2.X3 + 2 62.65/65.55 [U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1 + 2.X3 + 1 62.65/65.55 [fSNonEmpty] = 0 62.65/65.55 [plus](X1,X2) = X1 + X2 + 2 62.65/65.55 [s](X) = X + 2 62.65/65.55 [tt] = 2 62.65/65.55 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 62.65/65.55 [A__U11](X1,X2,X3) = 0 62.65/65.55 [A__U12](X1,X2,X3) = 0 62.65/65.55 [A__U21](X1,X2,X3) = 0 62.65/65.55 [A__U22](X1,X2,X3) = 2.X1.X2.X3 + X1 + 2.X3 + 2 62.65/65.55 [A__PLUS](X1,X2) = 2.X1 + 2 62.65/65.55 [A__X](X1,X2) = 0 62.65/65.55 [MARK](X) = 2.X + 2 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Reduction Pairs Processor: 62.65/65.55 -> Pairs: 62.65/65.55 A__PLUS(N:S,0) -> MARK(N:S) 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 -> Usable rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Interpretation type: 62.65/65.55 Simple mixed 62.65/65.55 ->Coefficients: 62.65/65.55 Natural Numbers 62.65/65.55 ->Dimension: 62.65/65.55 1 62.65/65.55 ->Bound: 62.65/65.55 2 62.65/65.55 ->Interpretation: 62.65/65.55 62.65/65.55 [a__U11](X1,X2,X3) = 2.X1.X2 + 2.X1 + X3 + 2 62.65/65.55 [a__U12](X1,X2,X3) = X1.X3 + 2.X1 + 2.X2 + 2 62.65/65.55 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [a__U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 62.65/65.55 [a__plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 62.65/65.55 [mark](X) = X 62.65/65.55 [0] = 2 62.65/65.55 [U11](X1,X2,X3) = 2.X1.X2 + 2.X1 + X3 + 2 62.65/65.55 [U12](X1,X2,X3) = X1.X3 + 2.X1 + 2.X2 + 2 62.65/65.55 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + X1 + 2.X2 + 2.X3 + 2 62.65/65.55 [U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 62.65/65.55 [fSNonEmpty] = 0 62.65/65.55 [plus](X1,X2) = X1 + 2.X2 + 2 62.65/65.55 [s](X) = X + 2 62.65/65.55 [tt] = 1 62.65/65.55 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 62.65/65.55 [A__U11](X1,X2,X3) = 0 62.65/65.55 [A__U12](X1,X2,X3) = 0 62.65/65.55 [A__U21](X1,X2,X3) = 0 62.65/65.55 [A__U22](X1,X2,X3) = 0 62.65/65.55 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 1 62.65/65.55 [A__X](X1,X2) = 0 62.65/65.55 [MARK](X) = 2.X + 1 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 ->->Cycle: 62.65/65.55 ->->-> Pairs: 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 ->->-> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 Subterm Processor: 62.65/65.55 -> Pairs: 62.65/65.55 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 MARK(s(X:S)) -> MARK(X:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 62.65/65.55 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Projection: 62.65/65.55 pi(MARK) = 1 62.65/65.55 62.65/65.55 Problem 1: 62.65/65.55 62.65/65.55 SCC Processor: 62.65/65.55 -> Pairs: 62.65/65.55 Empty 62.65/65.55 -> Rules: 62.65/65.55 a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) 62.65/65.55 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 62.65/65.55 a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 62.65/65.55 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 62.65/65.55 a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) 62.65/65.55 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 62.65/65.55 a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 62.65/65.55 a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) 62.65/65.55 a__plus(N:S,0) -> mark(N:S) 62.65/65.55 a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) 62.65/65.55 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 62.65/65.55 a__x(N:S,0) -> 0 62.65/65.55 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) 62.65/65.55 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 62.65/65.55 mark(0) -> 0 62.65/65.55 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) 62.65/65.55 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 62.65/65.55 mark(s(X:S)) -> s(mark(X:S)) 62.65/65.55 mark(tt) -> tt 62.65/65.55 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 62.65/65.55 ->Strongly Connected Components: 62.65/65.55 There is no strongly connected component 62.65/65.55 62.65/65.55 The problem is finite. 62.65/65.55 EOF