YES Problem 1: (VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S) (RULES U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(active(X1:S),X2:S) -> U11#(X1:S,X2:S) U11#(mark(X1:S),X2:S) -> U11#(X1:S,X2:S) U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) ACTIVE(U11(tt,N:S)) -> MARK(N:S) ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(U21(tt,M:S,N:S)) -> PLUS(N:S,M:S) ACTIVE(U21(tt,M:S,N:S)) -> S(plus(N:S,M:S)) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> AND(isNat(V1:S),isNat(V2:S)) ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNAT(V1:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNAT(V2:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> ISNAT(V1:S) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(plus(N:S,s(M:S))) -> U21#(and(isNat(M:S),isNat(N:S)),M:S,N:S) ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(M:S),isNat(N:S)) ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S) ACTIVE(plus(N:S,s(M:S))) -> ISNAT(N:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> U11#(isNat(N:S),N:S) ACTIVE(plus(N:S,0)) -> ISNAT(N:S) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) MARK(U11(X1:S,X2:S)) -> U11#(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> U21#(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1: SCC Processor: -> Pairs: U11#(active(X1:S),X2:S) -> U11#(X1:S,X2:S) U11#(mark(X1:S),X2:S) -> U11#(X1:S,X2:S) U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) ACTIVE(U11(tt,N:S)) -> MARK(N:S) ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(U21(tt,M:S,N:S)) -> PLUS(N:S,M:S) ACTIVE(U21(tt,M:S,N:S)) -> S(plus(N:S,M:S)) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> AND(isNat(V1:S),isNat(V2:S)) ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNAT(V1:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNAT(V2:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> ISNAT(V1:S) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(plus(N:S,s(M:S))) -> U21#(and(isNat(M:S),isNat(N:S)),M:S,N:S) ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(M:S),isNat(N:S)) ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S) ACTIVE(plus(N:S,s(M:S))) -> ISNAT(N:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> U11#(isNat(N:S),N:S) ACTIVE(plus(N:S,0)) -> ISNAT(N:S) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) MARK(U11(X1:S,X2:S)) -> U11#(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> U21#(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U11#(active(X1:S),X2:S) -> U11#(X1:S,X2:S) U11#(mark(X1:S),X2:S) -> U11#(X1:S,X2:S) U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,N:S)) -> MARK(N:S) ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) The problem is decomposed in 7 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(PLUS) = 1 Problem 1.2: SCC Processor: -> Pairs: PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.2: Subterm Processor: -> Pairs: PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(PLUS) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(ISNAT) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(AND) = 1 Problem 1.4: SCC Processor: -> Pairs: AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.4: Subterm Processor: -> Pairs: AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(AND) = 2 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: U21#(active(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(mark(X1:S),X2:S,X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U21#) = 1 Problem 1.5: SCC Processor: -> Pairs: U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Subterm Processor: -> Pairs: U21#(X1:S,active(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,mark(X2:S),X3:S) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U21#) = 2 Problem 1.5: SCC Processor: -> Pairs: U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Subterm Processor: -> Pairs: U21#(X1:S,X2:S,active(X3:S)) -> U21#(X1:S,X2:S,X3:S) U21#(X1:S,X2:S,mark(X3:S)) -> U21#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U21#) = 3 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: U11#(active(X1:S),X2:S) -> U11#(X1:S,X2:S) U11#(mark(X1:S),X2:S) -> U11#(X1:S,X2:S) U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U11#) = 1 Problem 1.6: SCC Processor: -> Pairs: U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.6: Subterm Processor: -> Pairs: U11#(X1:S,active(X2:S)) -> U11#(X1:S,X2:S) U11#(X1:S,mark(X2:S)) -> U11#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U11#) = 2 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,N:S)) -> MARK(N:S) ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 2.X1 + X2 + 2 [U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(U21(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 2.X1 + 2.X2 + 1 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,s(M:S))) -> MARK(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 2.X1 + 2.X2 + 2 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2.X [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) ACTIVE(plus(N:S,0)) -> MARK(U11(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 2.X1 + X2 + 1 [U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = X1 + 2.X2 [isNat](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U11(X1:S,X2:S)) -> ACTIVE(U11(mark(X1:S),X2:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 0 [U21](X1,X2,X3) = 2 [active](X) = 2 [and](X1,X2) = 2 [isNat](X) = 2 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = X [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 1 [0] = 1 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U21(X1:S,X2:S,X3:S)) -> ACTIVE(U21(mark(X1:S),X2:S,X3:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = X2 + 2 [U21](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [mark](X) = X [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = X + 2 [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 2.X1 + 2.X2 + 2 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = X1 + 2.X2 [isNat](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 2.X2 + 2 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 2.X [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 1 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(and(isNat(V1:S),isNat(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = 2.X2 + 2 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = X1 + X2 + 1 [isNat](X) = 2.X + 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = 2.X [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(isNat(s(V1:S))) -> MARK(isNat(V1:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2) = X2 + 1 [U21](X1,X2,X3) = X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = X1 + X2 [isNat](X) = 2.X + 2 [mark](X) = X [plus](X1,X2) = 2.X1 + X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2) = 0 [U21#](X1,X2,X3) = 0 [ACTIVE](X) = X + 2 [AND](X1,X2) = 0 [ISNAT](X) = 0 [MARK](X) = X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.7: SCC Processor: -> Pairs: MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Subterm Processor: -> Pairs: MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(MARK) = 1 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S) -> U11(X1:S,X2:S) U11(mark(X1:S),X2:S) -> U11(X1:S,X2:S) U11(X1:S,active(X2:S)) -> U11(X1:S,X2:S) U11(X1:S,mark(X2:S)) -> U11(X1:S,X2:S) U21(active(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(mark(X1:S),X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,active(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,mark(X2:S),X3:S) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,active(X3:S)) -> U21(X1:S,X2:S,X3:S) U21(X1:S,X2:S,mark(X3:S)) -> U21(X1:S,X2:S,X3:S) active(U11(tt,N:S)) -> mark(N:S) active(U21(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(and(tt,X:S)) -> mark(X:S) active(isNat(plus(V1:S,V2:S))) -> mark(and(isNat(V1:S),isNat(V2:S))) active(isNat(s(V1:S))) -> mark(isNat(V1:S)) active(isNat(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)) active(plus(N:S,0)) -> mark(U11(isNat(N:S),N:S)) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) mark(U11(X1:S,X2:S)) -> active(U11(mark(X1:S),X2:S)) mark(U21(X1:S,X2:S,X3:S)) -> active(U21(mark(X1:S),X2:S,X3:S)) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.