1.70/1.81 YES 1.70/1.81 1.70/1.81 Problem 1: 1.70/1.81 1.70/1.81 (VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S) 1.70/1.81 (RULES 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 isNat(active(X:S)) -> isNat(X:S) 1.70/1.81 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.81 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.81 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.81 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.81 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.81 mark(0) -> active(0) 1.70/1.81 mark(tt) -> active(tt) 1.70/1.81 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 s(active(X:S)) -> s(X:S) 1.70/1.81 s(mark(X:S)) -> s(X:S) 1.70/1.81 ) 1.70/1.81 (STRATEGY INNERMOST) 1.70/1.81 1.70/1.81 Problem 1: 1.70/1.81 1.70/1.81 Dependency Pairs Processor: 1.70/1.81 -> Pairs: 1.70/1.81 U11#(active(X1:S),X2:S) -> U11#(X1:S,X2:S) 1.70/1.81 U11#(mark(X1:S),X2:S) -> U11#(X1:S,X2:S) 1.70/1.81 U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.81 U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.81 U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 ACTIVE(U11(tt,N:S)) -> MARK(N:S) 1.70/1.81 ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 1.70/1.81 ACTIVE(U21(tt,M:S,N:S)) -> PLUS(N:S,M:S) 1.70/1.81 ACTIVE(U21(tt,M:S,N:S)) -> S(plus(N:S,M:S)) 1.70/1.81 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.81 ACTIVE(isNat(plus(V1:S,V2:S))) -> AND(isNat(V1:S),isNat(V2:S)) 1.70/1.81 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNAT(V1:S) 1.70/1.81 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNAT(V2:S) 1.70/1.81 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 ACTIVE(isNat(s(V1:S))) -> ISNAT(V1:S) 1.70/1.81 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.81 ACTIVE(isNat(0)) -> MARK(tt) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> U21#(and(isNat(M:S),isNat(N:S)),M:S,N:S) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(M:S),isNat(N:S)) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(N:S) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 ACTIVE(plus(N:S,0)) -> U11#(isNat(N:S),N:S) 1.70/1.81 ACTIVE(plus(N:S,0)) -> ISNAT(N:S) 1.70/1.81 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.81 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) 1.70/1.81 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) 1.70/1.81 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) 1.70/1.81 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) 1.70/1.81 ISNAT(active(X:S)) -> ISNAT(X:S) 1.70/1.81 ISNAT(mark(X:S)) -> ISNAT(X:S) 1.70/1.81 MARK(U11(X1:S,X2:S)) -> U11#(mark(X1:S),X2:S) 1.70/1.81 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.81 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.81 MARK(U21(X1:S,X2:S,X3:S)) -> U21#(mark(X1:S),X2:S,X3:S) 1.70/1.81 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.81 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.81 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S) 1.70/1.81 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.81 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S)) 1.70/1.81 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.81 MARK(s(X:S)) -> MARK(X:S) 1.70/1.81 MARK(s(X:S)) -> S(mark(X:S)) 1.70/1.81 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) 1.70/1.81 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 1.70/1.81 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.81 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.81 S(active(X:S)) -> S(X:S) 1.70/1.81 S(mark(X:S)) -> S(X:S) 1.70/1.81 -> Rules: 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 isNat(active(X:S)) -> isNat(X:S) 1.70/1.81 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.81 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.81 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.81 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.81 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.81 mark(0) -> active(0) 1.70/1.81 mark(tt) -> active(tt) 1.70/1.81 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 s(active(X:S)) -> s(X:S) 1.70/1.81 s(mark(X:S)) -> s(X:S) 1.70/1.81 1.70/1.81 Problem 1: 1.70/1.81 1.70/1.81 SCC Processor: 1.70/1.81 -> Pairs: 1.70/1.81 U11#(active(X1:S),X2:S) -> U11#(X1:S,X2:S) 1.70/1.81 U11#(mark(X1:S),X2:S) -> U11#(X1:S,X2:S) 1.70/1.81 U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.81 U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.81 U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 ACTIVE(U11(tt,N:S)) -> MARK(N:S) 1.70/1.81 ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 1.70/1.81 ACTIVE(U21(tt,M:S,N:S)) -> PLUS(N:S,M:S) 1.70/1.81 ACTIVE(U21(tt,M:S,N:S)) -> S(plus(N:S,M:S)) 1.70/1.81 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.81 ACTIVE(isNat(plus(V1:S,V2:S))) -> AND(isNat(V1:S),isNat(V2:S)) 1.70/1.81 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNAT(V1:S) 1.70/1.81 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNAT(V2:S) 1.70/1.81 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 ACTIVE(isNat(s(V1:S))) -> ISNAT(V1:S) 1.70/1.81 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.81 ACTIVE(isNat(0)) -> MARK(tt) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> U21#(and(isNat(M:S),isNat(N:S)),M:S,N:S) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(M:S),isNat(N:S)) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(N:S) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 ACTIVE(plus(N:S,0)) -> U11#(isNat(N:S),N:S) 1.70/1.81 ACTIVE(plus(N:S,0)) -> ISNAT(N:S) 1.70/1.81 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.81 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) 1.70/1.81 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) 1.70/1.81 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) 1.70/1.81 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) 1.70/1.81 ISNAT(active(X:S)) -> ISNAT(X:S) 1.70/1.81 ISNAT(mark(X:S)) -> ISNAT(X:S) 1.70/1.81 MARK(U11(X1:S,X2:S)) -> U11#(mark(X1:S),X2:S) 1.70/1.81 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.81 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.81 MARK(U21(X1:S,X2:S,X3:S)) -> U21#(mark(X1:S),X2:S,X3:S) 1.70/1.81 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.81 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.81 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S) 1.70/1.81 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.81 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S)) 1.70/1.81 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.81 MARK(s(X:S)) -> MARK(X:S) 1.70/1.81 MARK(s(X:S)) -> S(mark(X:S)) 1.70/1.81 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) 1.70/1.81 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 1.70/1.81 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.81 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.81 S(active(X:S)) -> S(X:S) 1.70/1.81 S(mark(X:S)) -> S(X:S) 1.70/1.81 -> Rules: 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 isNat(active(X:S)) -> isNat(X:S) 1.70/1.81 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.81 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.81 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.81 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.81 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.81 mark(0) -> active(0) 1.70/1.81 mark(tt) -> active(tt) 1.70/1.81 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 s(active(X:S)) -> s(X:S) 1.70/1.81 s(mark(X:S)) -> s(X:S) 1.70/1.81 ->Strongly Connected Components: 1.70/1.81 ->->Cycle: 1.70/1.81 ->->-> Pairs: 1.70/1.81 S(active(X:S)) -> S(X:S) 1.70/1.81 S(mark(X:S)) -> S(X:S) 1.70/1.81 ->->-> Rules: 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 isNat(active(X:S)) -> isNat(X:S) 1.70/1.81 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.81 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.81 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.81 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.81 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.81 mark(0) -> active(0) 1.70/1.81 mark(tt) -> active(tt) 1.70/1.81 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 s(active(X:S)) -> s(X:S) 1.70/1.81 s(mark(X:S)) -> s(X:S) 1.70/1.81 ->->Cycle: 1.70/1.81 ->->-> Pairs: 1.70/1.81 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) 1.70/1.81 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 1.70/1.81 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.81 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.81 ->->-> Rules: 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 isNat(active(X:S)) -> isNat(X:S) 1.70/1.81 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.81 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.81 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.81 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.81 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.81 mark(0) -> active(0) 1.70/1.81 mark(tt) -> active(tt) 1.70/1.81 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 s(active(X:S)) -> s(X:S) 1.70/1.81 s(mark(X:S)) -> s(X:S) 1.70/1.81 ->->Cycle: 1.70/1.81 ->->-> Pairs: 1.70/1.81 ISNAT(active(X:S)) -> ISNAT(X:S) 1.70/1.81 ISNAT(mark(X:S)) -> ISNAT(X:S) 1.70/1.81 ->->-> Rules: 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 isNat(active(X:S)) -> isNat(X:S) 1.70/1.81 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.81 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.81 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.81 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.81 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.81 mark(0) -> active(0) 1.70/1.81 mark(tt) -> active(tt) 1.70/1.81 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 s(active(X:S)) -> s(X:S) 1.70/1.81 s(mark(X:S)) -> s(X:S) 1.70/1.81 ->->Cycle: 1.70/1.81 ->->-> Pairs: 1.70/1.81 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) 1.70/1.81 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) 1.70/1.81 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) 1.70/1.81 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) 1.70/1.81 ->->-> Rules: 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 isNat(active(X:S)) -> isNat(X:S) 1.70/1.81 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.81 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.81 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.81 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.81 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.81 mark(0) -> active(0) 1.70/1.81 mark(tt) -> active(tt) 1.70/1.81 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 s(active(X:S)) -> s(X:S) 1.70/1.81 s(mark(X:S)) -> s(X:S) 1.70/1.81 ->->Cycle: 1.70/1.81 ->->-> Pairs: 1.70/1.81 U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.81 ->->-> Rules: 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 isNat(active(X:S)) -> isNat(X:S) 1.70/1.81 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.81 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.81 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.81 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.81 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.81 mark(0) -> active(0) 1.70/1.81 mark(tt) -> active(tt) 1.70/1.81 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 s(active(X:S)) -> s(X:S) 1.70/1.81 s(mark(X:S)) -> s(X:S) 1.70/1.81 ->->Cycle: 1.70/1.81 ->->-> Pairs: 1.70/1.81 U11#(active(X1:S),X2:S) -> U11#(X1:S,X2:S) 1.70/1.81 U11#(mark(X1:S),X2:S) -> U11#(X1:S,X2:S) 1.70/1.81 U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.81 U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.81 ->->-> Rules: 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.81 isNat(active(X:S)) -> isNat(X:S) 1.70/1.81 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.81 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.81 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.81 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.81 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.81 mark(0) -> active(0) 1.70/1.81 mark(tt) -> active(tt) 1.70/1.81 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.81 s(active(X:S)) -> s(X:S) 1.70/1.81 s(mark(X:S)) -> s(X:S) 1.70/1.81 ->->Cycle: 1.70/1.81 ->->-> Pairs: 1.70/1.81 ACTIVE(U11(tt,N:S)) -> MARK(N:S) 1.70/1.81 ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 1.70/1.81 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.81 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.81 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.81 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.81 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.81 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.81 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.81 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.81 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.81 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.81 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.81 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.81 MARK(s(X:S)) -> MARK(X:S) 1.70/1.81 ->->-> Rules: 1.70/1.81 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.81 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.81 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.81 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.81 active(and(tt,X:S)) -> mark(X:S) 1.70/1.81 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.81 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.81 active(isNat(0)) -> mark(tt) 1.70/1.81 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.81 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.81 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 1.70/1.82 1.70/1.82 The problem is decomposed in 7 subproblems. 1.70/1.82 1.70/1.82 Problem 1.1: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 S(active(X:S)) -> S(X:S) 1.70/1.82 S(mark(X:S)) -> S(X:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(S) = 1 1.70/1.82 1.70/1.82 Problem 1.1: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 Empty 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 There is no strongly connected component 1.70/1.82 1.70/1.82 The problem is finite. 1.70/1.82 1.70/1.82 Problem 1.2: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) 1.70/1.82 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 1.70/1.82 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.82 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(PLUS) = 1 1.70/1.82 1.70/1.82 Problem 1.2: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.82 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 ->->Cycle: 1.70/1.82 ->->-> Pairs: 1.70/1.82 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.82 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.82 ->->-> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 1.70/1.82 Problem 1.2: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.82 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(PLUS) = 2 1.70/1.82 1.70/1.82 Problem 1.2: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 Empty 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 There is no strongly connected component 1.70/1.82 1.70/1.82 The problem is finite. 1.70/1.82 1.70/1.82 Problem 1.3: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 ISNAT(active(X:S)) -> ISNAT(X:S) 1.70/1.82 ISNAT(mark(X:S)) -> ISNAT(X:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(ISNAT) = 1 1.70/1.82 1.70/1.82 Problem 1.3: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 Empty 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 There is no strongly connected component 1.70/1.82 1.70/1.82 The problem is finite. 1.70/1.82 1.70/1.82 Problem 1.4: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) 1.70/1.82 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) 1.70/1.82 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) 1.70/1.82 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(AND) = 1 1.70/1.82 1.70/1.82 Problem 1.4: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) 1.70/1.82 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 ->->Cycle: 1.70/1.82 ->->-> Pairs: 1.70/1.82 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) 1.70/1.82 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) 1.70/1.82 ->->-> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 1.70/1.82 Problem 1.4: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) 1.70/1.82 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(AND) = 2 1.70/1.82 1.70/1.82 Problem 1.4: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 Empty 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 There is no strongly connected component 1.70/1.82 1.70/1.82 The problem is finite. 1.70/1.82 1.70/1.82 Problem 1.5: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(U21#) = 1 1.70/1.82 1.70/1.82 Problem 1.5: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 ->->Cycle: 1.70/1.82 ->->-> Pairs: 1.70/1.82 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 ->->-> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 1.70/1.82 Problem 1.5: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(U21#) = 2 1.70/1.82 1.70/1.82 Problem 1.5: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 ->->Cycle: 1.70/1.82 ->->-> Pairs: 1.70/1.82 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 ->->-> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 1.70/1.82 Problem 1.5: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(U21#) = 3 1.70/1.82 1.70/1.82 Problem 1.5: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 Empty 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 There is no strongly connected component 1.70/1.82 1.70/1.82 The problem is finite. 1.70/1.82 1.70/1.82 Problem 1.6: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 U11#(active(X1:S),X2:S) -> U11#(X1:S,X2:S) 1.70/1.82 U11#(mark(X1:S),X2:S) -> U11#(X1:S,X2:S) 1.70/1.82 U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.82 U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(U11#) = 1 1.70/1.82 1.70/1.82 Problem 1.6: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.82 U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 ->->Cycle: 1.70/1.82 ->->-> Pairs: 1.70/1.82 U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.82 U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.82 ->->-> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 1.70/1.82 Problem 1.6: 1.70/1.82 1.70/1.82 Subterm Processor: 1.70/1.82 -> Pairs: 1.70/1.82 U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.82 U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Projection: 1.70/1.82 pi(U11#) = 2 1.70/1.82 1.70/1.82 Problem 1.6: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 Empty 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 There is no strongly connected component 1.70/1.82 1.70/1.82 The problem is finite. 1.70/1.82 1.70/1.82 Problem 1.7: 1.70/1.82 1.70/1.82 Reduction Pairs Processor: 1.70/1.82 -> Pairs: 1.70/1.82 ACTIVE(U11(tt,N:S)) -> MARK(N:S) 1.70/1.82 ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 1.70/1.82 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.82 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.82 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.82 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.82 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.82 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.82 MARK(s(X:S)) -> MARK(X:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 -> Usable rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Interpretation type: 1.70/1.82 Linear 1.70/1.82 ->Coefficients: 1.70/1.82 Natural Numbers 1.70/1.82 ->Dimension: 1.70/1.82 1 1.70/1.82 ->Bound: 1.70/1.82 2 1.70/1.82 ->Interpretation: 1.70/1.82 1.70/1.82 [U11](X1,X2) = 2.X1 + X2 + 2 1.70/1.82 [U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.70/1.82 [active](X) = X 1.70/1.82 [and](X1,X2) = 2.X1 + 2.X2 1.70/1.82 [isNat](X) = 0 1.70/1.82 [mark](X) = X 1.70/1.82 [plus](X1,X2) = 2.X1 + 2.X2 1.70/1.82 [s](X) = X + 2 1.70/1.82 [0] = 2 1.70/1.82 [fSNonEmpty] = 0 1.70/1.82 [tt] = 0 1.70/1.82 [U11#](X1,X2) = 0 1.70/1.82 [U21#](X1,X2,X3) = 0 1.70/1.82 [ACTIVE](X) = 2.X + 2 1.70/1.82 [AND](X1,X2) = 0 1.70/1.82 [ISNAT](X) = 0 1.70/1.82 [MARK](X) = 2.X + 2 1.70/1.82 [PLUS](X1,X2) = 0 1.70/1.82 [S](X) = 0 1.70/1.82 1.70/1.82 Problem 1.7: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 1.70/1.82 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.82 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.82 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.82 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.82 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.82 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.82 MARK(s(X:S)) -> MARK(X:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 ->->Cycle: 1.70/1.82 ->->-> Pairs: 1.70/1.82 ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 1.70/1.82 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.82 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.82 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.82 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.82 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.82 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.82 MARK(s(X:S)) -> MARK(X:S) 1.70/1.82 ->->-> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 1.70/1.82 Problem 1.7: 1.70/1.82 1.70/1.82 Reduction Pairs Processor: 1.70/1.82 -> Pairs: 1.70/1.82 ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) 1.70/1.82 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.82 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.82 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.82 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.82 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.82 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.82 MARK(s(X:S)) -> MARK(X:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 -> Usable rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Interpretation type: 1.70/1.82 Linear 1.70/1.82 ->Coefficients: 1.70/1.82 Natural Numbers 1.70/1.82 ->Dimension: 1.70/1.82 1 1.70/1.82 ->Bound: 1.70/1.82 2 1.70/1.82 ->Interpretation: 1.70/1.82 1.70/1.82 [U11](X1,X2) = 2.X1 + 2.X2 + 1 1.70/1.82 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.70/1.82 [active](X) = X 1.70/1.82 [and](X1,X2) = 2.X1 + 2.X2 1.70/1.82 [isNat](X) = 0 1.70/1.82 [mark](X) = X 1.70/1.82 [plus](X1,X2) = 2.X1 + 2.X2 1.70/1.82 [s](X) = X + 1 1.70/1.82 [0] = 2 1.70/1.82 [fSNonEmpty] = 0 1.70/1.82 [tt] = 0 1.70/1.82 [U11#](X1,X2) = 0 1.70/1.82 [U21#](X1,X2,X3) = 0 1.70/1.82 [ACTIVE](X) = 2.X + 2 1.70/1.82 [AND](X1,X2) = 0 1.70/1.82 [ISNAT](X) = 0 1.70/1.82 [MARK](X) = 2.X + 2 1.70/1.82 [PLUS](X1,X2) = 0 1.70/1.82 [S](X) = 0 1.70/1.82 1.70/1.82 Problem 1.7: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.82 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.82 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.82 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.82 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.82 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.82 MARK(s(X:S)) -> MARK(X:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 ->->Cycle: 1.70/1.82 ->->-> Pairs: 1.70/1.82 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.82 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.82 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.82 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.82 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.82 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.82 MARK(s(X:S)) -> MARK(X:S) 1.70/1.82 ->->-> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 1.70/1.82 Problem 1.7: 1.70/1.82 1.70/1.82 Reduction Pairs Processor: 1.70/1.82 -> Pairs: 1.70/1.82 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.82 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.82 ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.82 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.82 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.82 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.82 MARK(s(X:S)) -> MARK(X:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 -> Usable rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Interpretation type: 1.70/1.82 Linear 1.70/1.82 ->Coefficients: 1.70/1.82 Natural Numbers 1.70/1.82 ->Dimension: 1.70/1.82 1 1.70/1.82 ->Bound: 1.70/1.82 2 1.70/1.82 ->Interpretation: 1.70/1.82 1.70/1.82 [U11](X1,X2) = 2.X1 + 2.X2 + 2 1.70/1.82 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.70/1.82 [active](X) = X 1.70/1.82 [and](X1,X2) = 2.X1 + 2.X2 1.70/1.82 [isNat](X) = 0 1.70/1.82 [mark](X) = X 1.70/1.82 [plus](X1,X2) = 2.X1 + 2.X2 + 1 1.70/1.82 [s](X) = X + 1 1.70/1.82 [0] = 2 1.70/1.82 [fSNonEmpty] = 0 1.70/1.82 [tt] = 0 1.70/1.82 [U11#](X1,X2) = 0 1.70/1.82 [U21#](X1,X2,X3) = 0 1.70/1.82 [ACTIVE](X) = 2.X 1.70/1.82 [AND](X1,X2) = 0 1.70/1.82 [ISNAT](X) = 0 1.70/1.82 [MARK](X) = 2.X 1.70/1.82 [PLUS](X1,X2) = 0 1.70/1.82 [S](X) = 0 1.70/1.82 1.70/1.82 Problem 1.7: 1.70/1.82 1.70/1.82 SCC Processor: 1.70/1.82 -> Pairs: 1.70/1.82 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.82 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.82 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.82 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.82 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.82 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.82 MARK(s(X:S)) -> MARK(X:S) 1.70/1.82 -> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.82 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.82 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.82 active(and(tt,X:S)) -> mark(X:S) 1.70/1.82 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.82 active(isNat(0)) -> mark(tt) 1.70/1.82 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.82 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.82 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.82 isNat(active(X:S)) -> isNat(X:S) 1.70/1.82 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.82 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.82 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.82 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.82 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.82 mark(0) -> active(0) 1.70/1.82 mark(tt) -> active(tt) 1.70/1.82 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.82 s(active(X:S)) -> s(X:S) 1.70/1.82 s(mark(X:S)) -> s(X:S) 1.70/1.82 ->Strongly Connected Components: 1.70/1.82 ->->Cycle: 1.70/1.82 ->->-> Pairs: 1.70/1.82 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.82 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.82 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.82 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.82 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.82 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.82 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.82 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.82 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.82 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.82 MARK(s(X:S)) -> MARK(X:S) 1.70/1.82 ->->-> Rules: 1.70/1.82 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.82 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 1.70/1.83 Problem 1.7: 1.70/1.83 1.70/1.83 Reduction Pairs Processor: 1.70/1.83 -> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 -> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 -> Usable rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 ->Interpretation type: 1.70/1.83 Linear 1.70/1.83 ->Coefficients: 1.70/1.83 Natural Numbers 1.70/1.83 ->Dimension: 1.70/1.83 1 1.70/1.83 ->Bound: 1.70/1.83 2 1.70/1.83 ->Interpretation: 1.70/1.83 1.70/1.83 [U11](X1,X2) = 2.X1 + X2 + 1 1.70/1.83 [U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.70/1.83 [active](X) = X 1.70/1.83 [and](X1,X2) = X1 + 2.X2 1.70/1.83 [isNat](X) = 0 1.70/1.83 [mark](X) = X 1.70/1.83 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.70/1.83 [s](X) = X 1.70/1.83 [0] = 2 1.70/1.83 [fSNonEmpty] = 0 1.70/1.83 [tt] = 0 1.70/1.83 [U11#](X1,X2) = 0 1.70/1.83 [U21#](X1,X2,X3) = 0 1.70/1.83 [ACTIVE](X) = 2.X + 2 1.70/1.83 [AND](X1,X2) = 0 1.70/1.83 [ISNAT](X) = 0 1.70/1.83 [MARK](X) = 2.X + 2 1.70/1.83 [PLUS](X1,X2) = 0 1.70/1.83 [S](X) = 0 1.70/1.83 1.70/1.83 Problem 1.7: 1.70/1.83 1.70/1.83 SCC Processor: 1.70/1.83 -> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 -> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 ->Strongly Connected Components: 1.70/1.83 ->->Cycle: 1.70/1.83 ->->-> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 ->->-> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 1.70/1.83 Problem 1.7: 1.70/1.83 1.70/1.83 Reduction Pairs Processor: 1.70/1.83 -> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 -> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 -> Usable rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 ->Interpretation type: 1.70/1.83 Linear 1.70/1.83 ->Coefficients: 1.70/1.83 Natural Numbers 1.70/1.83 ->Dimension: 1.70/1.83 1 1.70/1.83 ->Bound: 1.70/1.83 2 1.70/1.83 ->Interpretation: 1.70/1.83 1.70/1.83 [U11](X1,X2) = 0 1.70/1.83 [U21](X1,X2,X3) = 2 1.70/1.83 [active](X) = 2 1.70/1.83 [and](X1,X2) = 2 1.70/1.83 [isNat](X) = 2 1.70/1.83 [mark](X) = 2 1.70/1.83 [plus](X1,X2) = 1 1.70/1.83 [s](X) = 2 1.70/1.83 [0] = 2 1.70/1.83 [fSNonEmpty] = 0 1.70/1.83 [tt] = 2 1.70/1.83 [U11#](X1,X2) = 0 1.70/1.83 [U21#](X1,X2,X3) = 0 1.70/1.83 [ACTIVE](X) = X 1.70/1.83 [AND](X1,X2) = 0 1.70/1.83 [ISNAT](X) = 0 1.70/1.83 [MARK](X) = 2 1.70/1.83 [PLUS](X1,X2) = 0 1.70/1.83 [S](X) = 0 1.70/1.83 1.70/1.83 Problem 1.7: 1.70/1.83 1.70/1.83 SCC Processor: 1.70/1.83 -> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 -> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 ->Strongly Connected Components: 1.70/1.83 ->->Cycle: 1.70/1.83 ->->-> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 ->->-> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 1.70/1.83 Problem 1.7: 1.70/1.83 1.70/1.83 Reduction Pairs Processor: 1.70/1.83 -> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 -> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 -> Usable rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 ->Interpretation type: 1.70/1.83 Linear 1.70/1.83 ->Coefficients: 1.70/1.83 Natural Numbers 1.70/1.83 ->Dimension: 1.70/1.83 1 1.70/1.83 ->Bound: 1.70/1.83 2 1.70/1.83 ->Interpretation: 1.70/1.83 1.70/1.83 [U11](X1,X2) = X1 + X2 + 2 1.70/1.83 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.70/1.83 [active](X) = X 1.70/1.83 [and](X1,X2) = 2.X1 + 2.X2 1.70/1.83 [isNat](X) = 0 1.70/1.83 [mark](X) = X 1.70/1.83 [plus](X1,X2) = 2.X1 + 2.X2 1.70/1.83 [s](X) = X + 1 1.70/1.83 [0] = 1 1.70/1.83 [fSNonEmpty] = 0 1.70/1.83 [tt] = 0 1.70/1.83 [U11#](X1,X2) = 0 1.70/1.83 [U21#](X1,X2,X3) = 0 1.70/1.83 [ACTIVE](X) = 2.X + 2 1.70/1.83 [AND](X1,X2) = 0 1.70/1.83 [ISNAT](X) = 0 1.70/1.83 [MARK](X) = 2.X + 2 1.70/1.83 [PLUS](X1,X2) = 0 1.70/1.83 [S](X) = 0 1.70/1.83 1.70/1.83 Problem 1.7: 1.70/1.83 1.70/1.83 SCC Processor: 1.70/1.83 -> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 -> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 ->Strongly Connected Components: 1.70/1.83 ->->Cycle: 1.70/1.83 ->->-> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 ->->-> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 1.70/1.83 Problem 1.7: 1.70/1.83 1.70/1.83 Reduction Pairs Processor: 1.70/1.83 -> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 -> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 -> Usable rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 ->Interpretation type: 1.70/1.83 Linear 1.70/1.83 ->Coefficients: 1.70/1.83 Natural Numbers 1.70/1.83 ->Dimension: 1.70/1.83 1 1.70/1.83 ->Bound: 1.70/1.83 2 1.70/1.83 ->Interpretation: 1.70/1.83 1.70/1.83 [U11](X1,X2) = X2 + 2 1.70/1.83 [U21](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 1.70/1.83 [active](X) = X 1.70/1.83 [and](X1,X2) = 2.X1 + 2.X2 1.70/1.83 [isNat](X) = 0 1.70/1.83 [mark](X) = X 1.70/1.83 [plus](X1,X2) = X1 + 2.X2 + 2 1.70/1.83 [s](X) = X 1.70/1.83 [0] = 2 1.70/1.83 [fSNonEmpty] = 0 1.70/1.83 [tt] = 0 1.70/1.83 [U11#](X1,X2) = 0 1.70/1.83 [U21#](X1,X2,X3) = 0 1.70/1.83 [ACTIVE](X) = X + 2 1.70/1.83 [AND](X1,X2) = 0 1.70/1.83 [ISNAT](X) = 0 1.70/1.83 [MARK](X) = 2.X + 2 1.70/1.83 [PLUS](X1,X2) = 0 1.70/1.83 [S](X) = 0 1.70/1.83 1.70/1.83 Problem 1.7: 1.70/1.83 1.70/1.83 SCC Processor: 1.70/1.83 -> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 -> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 ->Strongly Connected Components: 1.70/1.83 ->->Cycle: 1.70/1.83 ->->-> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 ->->-> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.70/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.70/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.70/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.70/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.70/1.83 mark(0) -> active(0) 1.70/1.83 mark(tt) -> active(tt) 1.70/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.70/1.83 s(active(X:S)) -> s(X:S) 1.70/1.83 s(mark(X:S)) -> s(X:S) 1.70/1.83 1.70/1.83 Problem 1.7: 1.70/1.83 1.70/1.83 Reduction Pairs Processor: 1.70/1.83 -> Pairs: 1.70/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.70/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.70/1.83 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.70/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.70/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.70/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.70/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.70/1.83 MARK(s(X:S)) -> MARK(X:S) 1.70/1.83 -> Rules: 1.70/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.70/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.70/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.70/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.70/1.83 active(and(tt,X:S)) -> mark(X:S) 1.70/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.70/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.70/1.83 active(isNat(0)) -> mark(tt) 1.70/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.70/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.70/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.70/1.83 isNat(active(X:S)) -> isNat(X:S) 1.70/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.70/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 -> Usable rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Interpretation type: 1.76/1.83 Linear 1.76/1.83 ->Coefficients: 1.76/1.83 Natural Numbers 1.76/1.83 ->Dimension: 1.76/1.83 1 1.76/1.83 ->Bound: 1.76/1.83 2 1.76/1.83 ->Interpretation: 1.76/1.83 1.76/1.83 [U11](X1,X2) = 2.X1 + 2.X2 + 2 1.76/1.83 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.76/1.83 [active](X) = X 1.76/1.83 [and](X1,X2) = X1 + 2.X2 1.76/1.83 [isNat](X) = 0 1.76/1.83 [mark](X) = X 1.76/1.83 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.76/1.83 [s](X) = X 1.76/1.83 [0] = 2 1.76/1.83 [fSNonEmpty] = 0 1.76/1.83 [tt] = 0 1.76/1.83 [U11#](X1,X2) = 0 1.76/1.83 [U21#](X1,X2,X3) = 0 1.76/1.83 [ACTIVE](X) = 2.X + 2 1.76/1.83 [AND](X1,X2) = 0 1.76/1.83 [ISNAT](X) = 0 1.76/1.83 [MARK](X) = 2.X + 2 1.76/1.83 [PLUS](X1,X2) = 0 1.76/1.83 [S](X) = 0 1.76/1.83 1.76/1.83 Problem 1.7: 1.76/1.83 1.76/1.83 SCC Processor: 1.76/1.83 -> Pairs: 1.76/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.76/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 -> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Strongly Connected Components: 1.76/1.83 ->->Cycle: 1.76/1.83 ->->-> Pairs: 1.76/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.76/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 ->->-> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 1.76/1.83 Problem 1.7: 1.76/1.83 1.76/1.83 Reduction Pairs Processor: 1.76/1.83 -> Pairs: 1.76/1.83 ACTIVE(and(tt,X:S)) -> MARK(X:S) 1.76/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 -> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 -> Usable rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Interpretation type: 1.76/1.83 Linear 1.76/1.83 ->Coefficients: 1.76/1.83 Natural Numbers 1.76/1.83 ->Dimension: 1.76/1.83 1 1.76/1.83 ->Bound: 1.76/1.83 2 1.76/1.83 ->Interpretation: 1.76/1.83 1.76/1.83 [U11](X1,X2) = 2.X2 + 2 1.76/1.83 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 1.76/1.83 [active](X) = X 1.76/1.83 [and](X1,X2) = 2.X1 + 2.X2 + 2 1.76/1.83 [isNat](X) = 2.X 1.76/1.83 [mark](X) = X 1.76/1.83 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.76/1.83 [s](X) = X 1.76/1.83 [0] = 2 1.76/1.83 [fSNonEmpty] = 0 1.76/1.83 [tt] = 1 1.76/1.83 [U11#](X1,X2) = 0 1.76/1.83 [U21#](X1,X2,X3) = 0 1.76/1.83 [ACTIVE](X) = 2.X + 2 1.76/1.83 [AND](X1,X2) = 0 1.76/1.83 [ISNAT](X) = 0 1.76/1.83 [MARK](X) = 2.X + 2 1.76/1.83 [PLUS](X1,X2) = 0 1.76/1.83 [S](X) = 0 1.76/1.83 1.76/1.83 Problem 1.7: 1.76/1.83 1.76/1.83 SCC Processor: 1.76/1.83 -> Pairs: 1.76/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 -> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Strongly Connected Components: 1.76/1.83 ->->Cycle: 1.76/1.83 ->->-> Pairs: 1.76/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 ->->-> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 1.76/1.83 Problem 1.7: 1.76/1.83 1.76/1.83 Reduction Pairs Processor: 1.76/1.83 -> Pairs: 1.76/1.83 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 -> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 -> Usable rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Interpretation type: 1.76/1.83 Linear 1.76/1.83 ->Coefficients: 1.76/1.83 Natural Numbers 1.76/1.83 ->Dimension: 1.76/1.83 1 1.76/1.83 ->Bound: 1.76/1.83 2 1.76/1.83 ->Interpretation: 1.76/1.83 1.76/1.83 [U11](X1,X2) = 2.X2 + 2 1.76/1.83 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 1.76/1.83 [active](X) = X 1.76/1.83 [and](X1,X2) = X1 + X2 + 1 1.76/1.83 [isNat](X) = 2.X + 2 1.76/1.83 [mark](X) = X 1.76/1.83 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.76/1.83 [s](X) = X 1.76/1.83 [0] = 2 1.76/1.83 [fSNonEmpty] = 0 1.76/1.83 [tt] = 2 1.76/1.83 [U11#](X1,X2) = 0 1.76/1.83 [U21#](X1,X2,X3) = 0 1.76/1.83 [ACTIVE](X) = 2.X 1.76/1.83 [AND](X1,X2) = 0 1.76/1.83 [ISNAT](X) = 0 1.76/1.83 [MARK](X) = 2.X 1.76/1.83 [PLUS](X1,X2) = 0 1.76/1.83 [S](X) = 0 1.76/1.83 1.76/1.83 Problem 1.7: 1.76/1.83 1.76/1.83 SCC Processor: 1.76/1.83 -> Pairs: 1.76/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 -> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Strongly Connected Components: 1.76/1.83 ->->Cycle: 1.76/1.83 ->->-> Pairs: 1.76/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 ->->-> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 1.76/1.83 Problem 1.7: 1.76/1.83 1.76/1.83 Reduction Pairs Processor: 1.76/1.83 -> Pairs: 1.76/1.83 ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 -> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 -> Usable rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Interpretation type: 1.76/1.83 Linear 1.76/1.83 ->Coefficients: 1.76/1.83 Natural Numbers 1.76/1.83 ->Dimension: 1.76/1.83 1 1.76/1.83 ->Bound: 1.76/1.83 2 1.76/1.83 ->Interpretation: 1.76/1.83 1.76/1.83 [U11](X1,X2) = X2 + 1 1.76/1.83 [U21](X1,X2,X3) = X2 + 2.X3 + 2 1.76/1.83 [active](X) = X 1.76/1.83 [and](X1,X2) = X1 + X2 1.76/1.83 [isNat](X) = 2.X + 2 1.76/1.83 [mark](X) = X 1.76/1.83 [plus](X1,X2) = 2.X1 + X2 + 1 1.76/1.83 [s](X) = X + 1 1.76/1.83 [0] = 2 1.76/1.83 [fSNonEmpty] = 0 1.76/1.83 [tt] = 0 1.76/1.83 [U11#](X1,X2) = 0 1.76/1.83 [U21#](X1,X2,X3) = 0 1.76/1.83 [ACTIVE](X) = X + 2 1.76/1.83 [AND](X1,X2) = 0 1.76/1.83 [ISNAT](X) = 0 1.76/1.83 [MARK](X) = X + 2 1.76/1.83 [PLUS](X1,X2) = 0 1.76/1.83 [S](X) = 0 1.76/1.83 1.76/1.83 Problem 1.7: 1.76/1.83 1.76/1.83 SCC Processor: 1.76/1.83 -> Pairs: 1.76/1.83 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 -> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Strongly Connected Components: 1.76/1.83 ->->Cycle: 1.76/1.83 ->->-> Pairs: 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 ->->-> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 1.76/1.83 Problem 1.7: 1.76/1.83 1.76/1.83 Subterm Processor: 1.76/1.83 -> Pairs: 1.76/1.83 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.76/1.83 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.76/1.83 MARK(s(X:S)) -> MARK(X:S) 1.76/1.83 -> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Projection: 1.76/1.83 pi(MARK) = 1 1.76/1.83 1.76/1.83 Problem 1.7: 1.76/1.83 1.76/1.83 SCC Processor: 1.76/1.83 -> Pairs: 1.76/1.83 Empty 1.76/1.83 -> Rules: 1.76/1.83 U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) 1.76/1.83 U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) 1.76/1.83 active(U11(tt,N:S)) -> mark(N:S) 1.76/1.83 active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 1.76/1.83 active(and(tt,X:S)) -> mark(X:S) 1.76/1.83 active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) 1.76/1.83 active(isNat(s(V1:S))) -> mark(isNat(V1:S)) 1.76/1.83 active(isNat(0)) -> mark(tt) 1.76/1.83 active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) 1.76/1.83 active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) 1.76/1.83 and(active(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(mark(X1:S),X2:S) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,active(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) 1.76/1.83 isNat(active(X:S)) -> isNat(X:S) 1.76/1.83 isNat(mark(X:S)) -> isNat(X:S) 1.76/1.83 mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) 1.76/1.83 mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) 1.76/1.83 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) 1.76/1.83 mark(isNat(X:S)) -> active(isNat(X:S)) 1.76/1.83 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) 1.76/1.83 mark(s(X:S)) -> active(s(mark(X:S))) 1.76/1.83 mark(0) -> active(0) 1.76/1.83 mark(tt) -> active(tt) 1.76/1.83 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) 1.76/1.83 s(active(X:S)) -> s(X:S) 1.76/1.83 s(mark(X:S)) -> s(X:S) 1.76/1.83 ->Strongly Connected Components: 1.76/1.83 There is no strongly connected component 1.76/1.83 1.76/1.83 The problem is finite. 1.76/1.83 EOF