YES Problem 1: (VAR v_NonEmpty:S I:S P:S V:S V1:S V2:S X:S X1:S X2:S Y:S Z:S) (RULES __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: __#(active(X1:S),X2:S) -> __#(X1:S,X2:S) __#(mark(X1:S),X2:S) -> __#(X1:S,X2:S) __#(X1:S,active(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) ACTIVE(__(__(X:S,Y:S),Z:S)) -> __#(X:S,__(Y:S,Z:S)) ACTIVE(__(__(X:S,Y:S),Z:S)) -> __#(Y:S,Z:S) ACTIVE(__(__(X:S,Y:S),Z:S)) -> MARK(__(X:S,__(Y:S,Z:S))) ACTIVE(__(nil,X:S)) -> MARK(X:S) ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> AND(isList(V1:S),isList(V2:S)) ACTIVE(isList(__(V1:S,V2:S))) -> ISLIST(V1:S) ACTIVE(isList(__(V1:S,V2:S))) -> ISLIST(V2:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(nil)) -> MARK(tt) ACTIVE(isList(V:S)) -> ISNELIST(V:S) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> AND(isList(V1:S),isNeList(V2:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> AND(isNeList(V1:S),isList(V2:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISLIST(V1:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISLIST(V2:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISNELIST(V1:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISNELIST(V2:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> ISQID(V:S) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> AND(isQid(I:S),isPal(P:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> ISPAL(P:S) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> ISQID(I:S) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> ISQID(V:S) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(nil)) -> MARK(tt) ACTIVE(isPal(V:S)) -> ISNEPAL(V:S) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) ACTIVE(isQid(a)) -> MARK(tt) ACTIVE(isQid(e)) -> MARK(tt) ACTIVE(isQid(i)) -> MARK(tt) ACTIVE(isQid(o)) -> MARK(tt) ACTIVE(isQid(u)) -> MARK(tt) 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) ISLIST(active(X:S)) -> ISLIST(X:S) ISLIST(mark(X:S)) -> ISLIST(X:S) ISNELIST(active(X:S)) -> ISNELIST(X:S) ISNELIST(mark(X:S)) -> ISNELIST(X:S) ISNEPAL(active(X:S)) -> ISNEPAL(X:S) ISNEPAL(mark(X:S)) -> ISNEPAL(X:S) ISPAL(active(X:S)) -> ISPAL(X:S) ISPAL(mark(X:S)) -> ISPAL(X:S) ISQID(active(X:S)) -> ISQID(X:S) ISQID(mark(X:S)) -> ISQID(X:S) MARK(__(X1:S,X2:S)) -> __#(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2: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(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) MARK(isQid(X:S)) -> ACTIVE(isQid(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1: SCC Processor: -> Pairs: __#(active(X1:S),X2:S) -> __#(X1:S,X2:S) __#(mark(X1:S),X2:S) -> __#(X1:S,X2:S) __#(X1:S,active(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) ACTIVE(__(__(X:S,Y:S),Z:S)) -> __#(X:S,__(Y:S,Z:S)) ACTIVE(__(__(X:S,Y:S),Z:S)) -> __#(Y:S,Z:S) ACTIVE(__(__(X:S,Y:S),Z:S)) -> MARK(__(X:S,__(Y:S,Z:S))) ACTIVE(__(nil,X:S)) -> MARK(X:S) ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> AND(isList(V1:S),isList(V2:S)) ACTIVE(isList(__(V1:S,V2:S))) -> ISLIST(V1:S) ACTIVE(isList(__(V1:S,V2:S))) -> ISLIST(V2:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(nil)) -> MARK(tt) ACTIVE(isList(V:S)) -> ISNELIST(V:S) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> AND(isList(V1:S),isNeList(V2:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> AND(isNeList(V1:S),isList(V2:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISLIST(V1:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISLIST(V2:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISNELIST(V1:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISNELIST(V2:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> ISQID(V:S) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> AND(isQid(I:S),isPal(P:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> ISPAL(P:S) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> ISQID(I:S) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> ISQID(V:S) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(nil)) -> MARK(tt) ACTIVE(isPal(V:S)) -> ISNEPAL(V:S) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) ACTIVE(isQid(a)) -> MARK(tt) ACTIVE(isQid(e)) -> MARK(tt) ACTIVE(isQid(i)) -> MARK(tt) ACTIVE(isQid(o)) -> MARK(tt) ACTIVE(isQid(u)) -> MARK(tt) 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) ISLIST(active(X:S)) -> ISLIST(X:S) ISLIST(mark(X:S)) -> ISLIST(X:S) ISNELIST(active(X:S)) -> ISNELIST(X:S) ISNELIST(mark(X:S)) -> ISNELIST(X:S) ISNEPAL(active(X:S)) -> ISNEPAL(X:S) ISNEPAL(mark(X:S)) -> ISNEPAL(X:S) ISPAL(active(X:S)) -> ISPAL(X:S) ISPAL(mark(X:S)) -> ISPAL(X:S) ISQID(active(X:S)) -> ISQID(X:S) ISQID(mark(X:S)) -> ISQID(X:S) MARK(__(X1:S,X2:S)) -> __#(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2: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(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) MARK(isQid(X:S)) -> ACTIVE(isQid(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISQID(active(X:S)) -> ISQID(X:S) ISQID(mark(X:S)) -> ISQID(X:S) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->->Cycle: ->->-> Pairs: ISPAL(active(X:S)) -> ISPAL(X:S) ISPAL(mark(X:S)) -> ISPAL(X:S) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->->Cycle: ->->-> Pairs: ISNEPAL(active(X:S)) -> ISNEPAL(X:S) ISNEPAL(mark(X:S)) -> ISNEPAL(X:S) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->->Cycle: ->->-> Pairs: ISNELIST(active(X:S)) -> ISNELIST(X:S) ISNELIST(mark(X:S)) -> ISNELIST(X:S) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->->Cycle: ->->-> Pairs: ISLIST(active(X:S)) -> ISLIST(X:S) ISLIST(mark(X:S)) -> ISLIST(X:S) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->->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: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->->Cycle: ->->-> Pairs: __#(active(X1:S),X2:S) -> __#(X1:S,X2:S) __#(mark(X1:S),X2:S) -> __#(X1:S,X2:S) __#(X1:S,active(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->->Cycle: ->->-> Pairs: ACTIVE(__(__(X:S,Y:S),Z:S)) -> MARK(__(X:S,__(Y:S,Z:S))) ACTIVE(__(nil,X:S)) -> MARK(X:S) ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) The problem is decomposed in 8 subproblems. Problem 1.1: Subterm Processor: -> Pairs: ISQID(active(X:S)) -> ISQID(X:S) ISQID(mark(X:S)) -> ISQID(X:S) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(ISQID) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: ISPAL(active(X:S)) -> ISPAL(X:S) ISPAL(mark(X:S)) -> ISPAL(X:S) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(ISPAL) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: ISNEPAL(active(X:S)) -> ISNEPAL(X:S) ISNEPAL(mark(X:S)) -> ISNEPAL(X:S) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(ISNEPAL) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: ISNELIST(active(X:S)) -> ISNELIST(X:S) ISNELIST(mark(X:S)) -> ISNELIST(X:S) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(ISNELIST) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: ISLIST(active(X:S)) -> ISLIST(X:S) ISLIST(mark(X:S)) -> ISLIST(X:S) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(ISLIST) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: 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: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(AND) = 1 Problem 1.6: 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: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->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: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.6: 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: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(AND) = 2 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: __#(active(X1:S),X2:S) -> __#(X1:S,X2:S) __#(mark(X1:S),X2:S) -> __#(X1:S,X2:S) __#(X1:S,active(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(__#) = 1 Problem 1.7: SCC Processor: -> Pairs: __#(X1:S,active(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: __#(X1:S,active(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.7: Subterm Processor: -> Pairs: __#(X1:S,active(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(__#) = 2 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(__(__(X:S,Y:S),Z:S)) -> MARK(__(X:S,__(Y:S,Z:S))) ACTIVE(__(nil,X:S)) -> MARK(X:S) ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = X1 + X2 + 2 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isNePal](X) = X + 2 [isPal](X) = 2.X + 2 [isQid](X) = 0 [mark](X) = X [a] = 0 [e] = 2 [fSNonEmpty] = 0 [i] = 0 [nil] = 2 [o] = 0 [tt] = 0 [u] = 2 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(__(nil,X:S)) -> MARK(X:S) ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(__(nil,X:S)) -> MARK(X:S) ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(__(nil,X:S)) -> MARK(X:S) ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = 2.X1 + X2 + 2 [isList](X) = 2.X + 1 [isNeList](X) = 2.X + 1 [isNePal](X) = 2.X + 2 [isPal](X) = 2.X + 2 [isQid](X) = 2.X + 1 [mark](X) = X [a] = 1 [e] = 0 [fSNonEmpty] = 0 [i] = 0 [nil] = 2 [o] = 1 [tt] = 0 [u] = 1 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(__(X:S,nil)) -> MARK(X:S) ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = X1 + X2 + 2 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isNePal](X) = 2.X + 2 [isPal](X) = 2.X + 2 [isQid](X) = X + 1 [mark](X) = X [a] = 1 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [nil] = 2 [o] = 2 [tt] = 1 [u] = 2 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = X1 + X2 + 2 [active](X) = X [and](X1,X2) = X1 + X2 + 2 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 1 [isNePal](X) = 2.X + 1 [isPal](X) = 2.X + 1 [isQid](X) = 1 [mark](X) = X [a] = 2 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [nil] = 2 [o] = 0 [tt] = 1 [u] = 1 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(isList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isList(V2:S))) ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = X1 + X2 + 2 [active](X) = X [and](X1,X2) = X1 + X2 + 1 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isNePal](X) = 2.X + 2 [isPal](X) = 2.X + 2 [isQid](X) = 2.X + 2 [mark](X) = X [a] = 1 [e] = 0 [fSNonEmpty] = 0 [i] = 1 [nil] = 2 [o] = 1 [tt] = 2 [u] = 2 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 1 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 1 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(isList(V:S)) -> MARK(isNeList(V:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = 2.X1 + X2 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 1 [isNePal](X) = 2.X + 2 [isPal](X) = 2.X + 2 [isQid](X) = X + 1 [mark](X) = X [a] = 2 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [nil] = 1 [o] = 2 [tt] = 2 [u] = 1 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isList(X:S)) -> ACTIVE(isList(X:S)) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isList(V1:S),isNeList(V2:S))) ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = 2.X1 + X2 + 1 [isList](X) = 2.X + 1 [isNeList](X) = 2.X [isNePal](X) = 2.X + 2 [isPal](X) = 2.X + 2 [isQid](X) = 2.X [mark](X) = X [a] = 2 [e] = 2 [fSNonEmpty] = 0 [i] = 0 [nil] = 2 [o] = 2 [tt] = 0 [u] = 2 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X Problem 1.8: SCC Processor: -> Pairs: ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(isNeList(__(V1:S,V2:S))) -> MARK(and(isNeList(V1:S),isList(V2:S))) ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 1 [active](X) = X [and](X1,X2) = X1 + X2 [isList](X) = 2.X + 1 [isNeList](X) = 2.X + 1 [isNePal](X) = X + 2 [isPal](X) = X + 2 [isQid](X) = X + 1 [mark](X) = X [a] = 1 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [nil] = 1 [o] = 2 [tt] = 1 [u] = 2 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(isNeList(V:S)) -> MARK(isQid(V:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = 2.X1 + X2 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isNePal](X) = 2.X + 1 [isPal](X) = 2.X + 1 [isQid](X) = 2.X + 1 [mark](X) = X [a] = 2 [e] = 0 [fSNonEmpty] = 0 [i] = 2 [nil] = 2 [o] = 2 [tt] = 0 [u] = 1 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 1 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 1 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNeList(X:S)) -> ACTIVE(isNeList(X:S)) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> MARK(and(isQid(I:S),isPal(P:S))) ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = X1 + X2 + 1 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isNePal](X) = 2.X [isPal](X) = 2.X [isQid](X) = 2.X [mark](X) = X [a] = 1 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [nil] = 2 [o] = 2 [tt] = 2 [u] = 1 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(isNePal(V:S)) -> MARK(isQid(V:S)) ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isList](X) = 0 [isNeList](X) = 0 [isNePal](X) = 2.X + 2 [isPal](X) = 2.X + 2 [isQid](X) = 0 [mark](X) = X [a] = 0 [e] = 1 [fSNonEmpty] = 0 [i] = 0 [nil] = 2 [o] = 2 [tt] = 0 [u] = 2 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isNePal(X:S)) -> ACTIVE(isNePal(X:S)) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Reduction Pairs Processor: -> Pairs: ACTIVE(isPal(V:S)) -> MARK(isNePal(V:S)) MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) -> Usable rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = X1 + X2 + 2 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isNePal](X) = 2.X + 1 [isPal](X) = 2.X + 2 [isQid](X) = 2.X + 1 [mark](X) = X [a] = 2 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [nil] = 2 [o] = 2 [tt] = 0 [u] = 2 [__#](X1,X2) = 0 [ACTIVE](X) = 2.X [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [MARK](X) = 2.X + 1 Problem 1.8: SCC Processor: -> Pairs: MARK(__(X1:S,X2:S)) -> ACTIVE(__(mark(X1:S),mark(X2:S))) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(isPal(X:S)) -> ACTIVE(isPal(X:S)) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) ->->-> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) Problem 1.8: Subterm Processor: -> Pairs: MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Projection: pi(MARK) = 1 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: __(active(X1:S),X2:S) -> __(X1:S,X2:S) __(mark(X1:S),X2:S) -> __(X1:S,X2:S) __(X1:S,active(X2:S)) -> __(X1:S,X2:S) __(X1:S,mark(X2:S)) -> __(X1:S,X2:S) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(and(tt,X:S)) -> mark(X:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) 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) isList(active(X:S)) -> isList(X:S) isList(mark(X:S)) -> isList(X:S) isNeList(active(X:S)) -> isNeList(X:S) isNeList(mark(X:S)) -> isNeList(X:S) isNePal(active(X:S)) -> isNePal(X:S) isNePal(mark(X:S)) -> isNePal(X:S) isPal(active(X:S)) -> isPal(X:S) isPal(mark(X:S)) -> isPal(X:S) isQid(active(X:S)) -> isQid(X:S) isQid(mark(X:S)) -> isQid(X:S) mark(__(X1:S,X2:S)) -> active(__(mark(X1:S),mark(X2:S))) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(isList(X:S)) -> active(isList(X:S)) mark(isNeList(X:S)) -> active(isNeList(X:S)) mark(isNePal(X:S)) -> active(isNePal(X:S)) mark(isPal(X:S)) -> active(isPal(X:S)) mark(isQid(X:S)) -> active(isQid(X:S)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(nil) -> active(nil) mark(o) -> active(o) mark(tt) -> active(tt) mark(u) -> active(u) ->Strongly Connected Components: There is no strongly connected component The problem is finite.