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 a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: A__U21(tt,V2:S) -> A__U22(a__isList(V2:S)) A__U21(tt,V2:S) -> A__ISLIST(V2:S) A__U41(tt,V2:S) -> A__U42(a__isNeList(V2:S)) A__U41(tt,V2:S) -> A__ISNELIST(V2:S) A__U51(tt,V2:S) -> A__U52(a__isList(V2:S)) A__U51(tt,V2:S) -> A__ISLIST(V2:S) A__U71(tt,P:S) -> A__U72(a__isPal(P:S)) A__U71(tt,P:S) -> A__ISPAL(P:S) A____(__(X:S,Y:S),Z:S) -> A____(mark(X:S),a____(mark(Y:S),mark(Z:S))) A____(__(X:S,Y:S),Z:S) -> A____(mark(Y:S),mark(Z:S)) A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) A__ISLIST(__(V1:S,V2:S)) -> A__U21(a__isList(V1:S),V2:S) A__ISLIST(__(V1:S,V2:S)) -> A__ISLIST(V1:S) A__ISLIST(V:S) -> A__U11(a__isNeList(V:S)) A__ISLIST(V:S) -> A__ISNELIST(V:S) A__ISNELIST(__(V1:S,V2:S)) -> A__U41(a__isList(V1:S),V2:S) A__ISNELIST(__(V1:S,V2:S)) -> A__U51(a__isNeList(V1:S),V2:S) A__ISNELIST(__(V1:S,V2:S)) -> A__ISLIST(V1:S) A__ISNELIST(__(V1:S,V2:S)) -> A__ISNELIST(V1:S) A__ISNELIST(V:S) -> A__U31(a__isQid(V:S)) A__ISNELIST(V:S) -> A__ISQID(V:S) A__ISNEPAL(__(I:S,__(P:S,I:S))) -> A__U71(a__isQid(I:S),P:S) A__ISNEPAL(__(I:S,__(P:S,I:S))) -> A__ISQID(I:S) A__ISNEPAL(V:S) -> A__U61(a__isQid(V:S)) A__ISNEPAL(V:S) -> A__ISQID(V:S) A__ISPAL(V:S) -> A__U81(a__isNePal(V:S)) A__ISPAL(V:S) -> A__ISNEPAL(V:S) MARK(U11(X:S)) -> A__U11(mark(X:S)) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> A__U21(mark(X1:S),X2:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> A__U22(mark(X:S)) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> A__U31(mark(X:S)) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> A__U42(mark(X:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> A__U52(mark(X:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> A__U61(mark(X:S)) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> A__U71(mark(X1:S),X2:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> A__U72(mark(X:S)) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> A__U81(mark(X:S)) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(isList(X:S)) -> A__ISLIST(X:S) MARK(isNeList(X:S)) -> A__ISNELIST(X:S) MARK(isNePal(X:S)) -> A__ISNEPAL(X:S) MARK(isPal(X:S)) -> A__ISPAL(X:S) MARK(isQid(X:S)) -> A__ISQID(X:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u Problem 1: SCC Processor: -> Pairs: A__U21(tt,V2:S) -> A__U22(a__isList(V2:S)) A__U21(tt,V2:S) -> A__ISLIST(V2:S) A__U41(tt,V2:S) -> A__U42(a__isNeList(V2:S)) A__U41(tt,V2:S) -> A__ISNELIST(V2:S) A__U51(tt,V2:S) -> A__U52(a__isList(V2:S)) A__U51(tt,V2:S) -> A__ISLIST(V2:S) A__U71(tt,P:S) -> A__U72(a__isPal(P:S)) A__U71(tt,P:S) -> A__ISPAL(P:S) A____(__(X:S,Y:S),Z:S) -> A____(mark(X:S),a____(mark(Y:S),mark(Z:S))) A____(__(X:S,Y:S),Z:S) -> A____(mark(Y:S),mark(Z:S)) A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) A__ISLIST(__(V1:S,V2:S)) -> A__U21(a__isList(V1:S),V2:S) A__ISLIST(__(V1:S,V2:S)) -> A__ISLIST(V1:S) A__ISLIST(V:S) -> A__U11(a__isNeList(V:S)) A__ISLIST(V:S) -> A__ISNELIST(V:S) A__ISNELIST(__(V1:S,V2:S)) -> A__U41(a__isList(V1:S),V2:S) A__ISNELIST(__(V1:S,V2:S)) -> A__U51(a__isNeList(V1:S),V2:S) A__ISNELIST(__(V1:S,V2:S)) -> A__ISLIST(V1:S) A__ISNELIST(__(V1:S,V2:S)) -> A__ISNELIST(V1:S) A__ISNELIST(V:S) -> A__U31(a__isQid(V:S)) A__ISNELIST(V:S) -> A__ISQID(V:S) A__ISNEPAL(__(I:S,__(P:S,I:S))) -> A__U71(a__isQid(I:S),P:S) A__ISNEPAL(__(I:S,__(P:S,I:S))) -> A__ISQID(I:S) A__ISNEPAL(V:S) -> A__U61(a__isQid(V:S)) A__ISNEPAL(V:S) -> A__ISQID(V:S) A__ISPAL(V:S) -> A__U81(a__isNePal(V:S)) A__ISPAL(V:S) -> A__ISNEPAL(V:S) MARK(U11(X:S)) -> A__U11(mark(X:S)) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> A__U21(mark(X1:S),X2:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> A__U22(mark(X:S)) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> A__U31(mark(X:S)) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> A__U42(mark(X:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> A__U52(mark(X:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> A__U61(mark(X:S)) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> A__U71(mark(X1:S),X2:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> A__U72(mark(X:S)) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> A__U81(mark(X:S)) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) MARK(isList(X:S)) -> A__ISLIST(X:S) MARK(isNeList(X:S)) -> A__ISNELIST(X:S) MARK(isNePal(X:S)) -> A__ISNEPAL(X:S) MARK(isPal(X:S)) -> A__ISPAL(X:S) MARK(isQid(X:S)) -> A__ISQID(X:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U71(tt,P:S) -> A__ISPAL(P:S) A__ISNEPAL(__(I:S,__(P:S,I:S))) -> A__U71(a__isQid(I:S),P:S) A__ISPAL(V:S) -> A__ISNEPAL(V:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->->Cycle: ->->-> Pairs: A__U21(tt,V2:S) -> A__ISLIST(V2:S) A__U41(tt,V2:S) -> A__ISNELIST(V2:S) A__U51(tt,V2:S) -> A__ISLIST(V2:S) A__ISLIST(__(V1:S,V2:S)) -> A__U21(a__isList(V1:S),V2:S) A__ISLIST(__(V1:S,V2:S)) -> A__ISLIST(V1:S) A__ISLIST(V:S) -> A__ISNELIST(V:S) A__ISNELIST(__(V1:S,V2:S)) -> A__U41(a__isList(V1:S),V2:S) A__ISNELIST(__(V1:S,V2:S)) -> A__U51(a__isNeList(V1:S),V2:S) A__ISNELIST(__(V1:S,V2:S)) -> A__ISLIST(V1:S) A__ISNELIST(__(V1:S,V2:S)) -> A__ISNELIST(V1:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->->Cycle: ->->-> Pairs: A____(__(X:S,Y:S),Z:S) -> A____(mark(X:S),a____(mark(Y:S),mark(Z:S))) A____(__(X:S,Y:S),Z:S) -> A____(mark(Y:S),mark(Z:S)) A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u The problem is decomposed in 3 subproblems. Problem 1.1: Subterm Processor: -> Pairs: A__U71(tt,P:S) -> A__ISPAL(P:S) A__ISNEPAL(__(I:S,__(P:S,I:S))) -> A__U71(a__isQid(I:S),P:S) A__ISPAL(V:S) -> A__ISNEPAL(V:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Projection: pi(A__U71) = 2 pi(A__ISNEPAL) = 1 pi(A__ISPAL) = 1 Problem 1.1: SCC Processor: -> Pairs: A__U71(tt,P:S) -> A__ISPAL(P:S) A__ISPAL(V:S) -> A__ISNEPAL(V:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: A__U21(tt,V2:S) -> A__ISLIST(V2:S) A__U41(tt,V2:S) -> A__ISNELIST(V2:S) A__U51(tt,V2:S) -> A__ISLIST(V2:S) A__ISLIST(__(V1:S,V2:S)) -> A__U21(a__isList(V1:S),V2:S) A__ISLIST(__(V1:S,V2:S)) -> A__ISLIST(V1:S) A__ISLIST(V:S) -> A__ISNELIST(V:S) A__ISNELIST(__(V1:S,V2:S)) -> A__U41(a__isList(V1:S),V2:S) A__ISNELIST(__(V1:S,V2:S)) -> A__U51(a__isNeList(V1:S),V2:S) A__ISNELIST(__(V1:S,V2:S)) -> A__ISLIST(V1:S) A__ISNELIST(__(V1:S,V2:S)) -> A__ISNELIST(V1:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Projection: pi(A__U21) = 2 pi(A__U41) = 2 pi(A__U51) = 2 pi(A__ISLIST) = 1 pi(A__ISNELIST) = 1 Problem 1.2: SCC Processor: -> Pairs: A__U21(tt,V2:S) -> A__ISLIST(V2:S) A__U41(tt,V2:S) -> A__ISNELIST(V2:S) A__U51(tt,V2:S) -> A__ISLIST(V2:S) A__ISLIST(V:S) -> A__ISNELIST(V:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Reduction Pairs Processor: -> Pairs: A____(__(X:S,Y:S),Z:S) -> A____(mark(X:S),a____(mark(Y:S),mark(Z:S))) A____(__(X:S,Y:S),Z:S) -> A____(mark(Y:S),mark(Z:S)) A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = 2.X [a__U21](X1,X2) = 2.X1 [a__U22](X) = X [a__U31](X) = 2.X [a__U41](X1,X2) = 2.X1 [a__U42](X) = 2.X [a__U51](X1,X2) = 2.X1 [a__U52](X) = 2.X [a__U61](X) = 2.X + 1 [a__U71](X1,X2) = 2.X1 + 2.X2 + 2 [a__U72](X) = X [a__U81](X) = 2.X [a____](X1,X2) = 2.X1 + X2 + 2 [a__isList](X) = 0 [a__isNeList](X) = 0 [a__isNePal](X) = X + 1 [a__isPal](X) = 2.X + 2 [a__isQid](X) = 0 [mark](X) = X [U11](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X) = X [U31](X) = 2.X [U41](X1,X2) = 2.X1 [U42](X) = 2.X [U51](X1,X2) = 2.X1 [U52](X) = 2.X [U61](X) = 2.X + 1 [U71](X1,X2) = 2.X1 + 2.X2 + 2 [U72](X) = X [U81](X) = 2.X [__](X1,X2) = 2.X1 + X2 + 2 [a] = 1 [e] = 1 [fSNonEmpty] = 0 [i] = 1 [isList](X) = 0 [isNeList](X) = 0 [isNePal](X) = X + 1 [isPal](X) = 2.X + 2 [isQid](X) = 0 [nil] = 2 [o] = 1 [tt] = 0 [u] = 1 [A__U11](X) = 0 [A__U21](X1,X2) = 0 [A__U22](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X) = 0 [A__U71](X1,X2) = 0 [A__U72](X) = 0 [A__U81](X) = 0 [A____](X1,X2) = 2.X1 + X2 [A__ISLIST](X) = 0 [A__ISNELIST](X) = 0 [A__ISNEPAL](X) = 0 [A__ISPAL](X) = 0 [A__ISQID](X) = 0 [MARK](X) = X + 2 Problem 1.3: SCC Processor: -> Pairs: A____(__(X:S,Y:S),Z:S) -> A____(mark(Y:S),mark(Z:S)) A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A____(__(X:S,Y:S),Z:S) -> A____(mark(Y:S),mark(Z:S)) A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u Problem 1.3: Reduction Pairs Processor: -> Pairs: A____(__(X:S,Y:S),Z:S) -> A____(mark(Y:S),mark(Z:S)) A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X1,X2) = 2.X1 + 2.X2 + 2 [a__U22](X) = X + 2 [a__U31](X) = 2.X [a__U41](X1,X2) = 2.X1 + 2.X2 + 2 [a__U42](X) = X [a__U51](X1,X2) = X1 + 2.X2 + 2 [a__U52](X) = X + 1 [a__U61](X) = X [a__U71](X1,X2) = 2.X1 + 2.X2 + 2 [a__U72](X) = X + 1 [a__U81](X) = 2.X [a____](X1,X2) = 2.X1 + X2 + 2 [a__isList](X) = 2.X + 2 [a__isNeList](X) = 2.X + 2 [a__isNePal](X) = X + 1 [a__isPal](X) = 2.X + 2 [a__isQid](X) = X + 1 [mark](X) = X [U11](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X) = X + 2 [U31](X) = 2.X [U41](X1,X2) = 2.X1 + 2.X2 + 2 [U42](X) = X [U51](X1,X2) = X1 + 2.X2 + 2 [U52](X) = X + 1 [U61](X) = X [U71](X1,X2) = 2.X1 + 2.X2 + 2 [U72](X) = X + 1 [U81](X) = 2.X [__](X1,X2) = 2.X1 + X2 + 2 [a] = 1 [e] = 1 [fSNonEmpty] = 0 [i] = 1 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isNePal](X) = X + 1 [isPal](X) = 2.X + 2 [isQid](X) = X + 1 [nil] = 2 [o] = 1 [tt] = 2 [u] = 1 [A__U11](X) = 0 [A__U21](X1,X2) = 0 [A__U22](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X) = 0 [A__U71](X1,X2) = 0 [A__U72](X) = 0 [A__U81](X) = 0 [A____](X1,X2) = 2.X1 + 2.X2 + 2 [A__ISLIST](X) = 0 [A__ISNELIST](X) = 0 [A__ISNEPAL](X) = 0 [A__ISPAL](X) = 0 [A__ISQID](X) = 0 [MARK](X) = 2.X Problem 1.3: SCC Processor: -> Pairs: A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u Problem 1.3: Reduction Pairs Processor: -> Pairs: A____(__(X:S,Y:S),Z:S) -> MARK(X:S) A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X1,X2) = 2.X1 + 2.X2 + 2 [a__U22](X) = X + 1 [a__U31](X) = X [a__U41](X1,X2) = 2.X1 + 2.X2 + 2 [a__U42](X) = X + 1 [a__U51](X1,X2) = X1 + 2.X2 + 2 [a__U52](X) = X + 2 [a__U61](X) = X [a__U71](X1,X2) = 2.X1 + 2.X2 + 2 [a__U72](X) = X + 2 [a__U81](X) = X [a____](X1,X2) = 2.X1 + X2 + 2 [a__isList](X) = 2.X + 2 [a__isNeList](X) = 2.X + 2 [a__isNePal](X) = 2.X + 2 [a__isPal](X) = 2.X + 2 [a__isQid](X) = 2.X + 2 [mark](X) = X [U11](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X) = X + 1 [U31](X) = X [U41](X1,X2) = 2.X1 + 2.X2 + 2 [U42](X) = X + 1 [U51](X1,X2) = X1 + 2.X2 + 2 [U52](X) = X + 2 [U61](X) = X [U71](X1,X2) = 2.X1 + 2.X2 + 2 [U72](X) = X + 2 [U81](X) = X [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [e] = 1 [fSNonEmpty] = 0 [i] = 2 [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 [nil] = 2 [o] = 1 [tt] = 2 [u] = 1 [A__U11](X) = 0 [A__U21](X1,X2) = 0 [A__U22](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X) = 0 [A__U71](X1,X2) = 0 [A__U72](X) = 0 [A__U81](X) = 0 [A____](X1,X2) = 2.X1 + 2.X2 + 2 [A__ISLIST](X) = 0 [A__ISNELIST](X) = 0 [A__ISNEPAL](X) = 0 [A__ISPAL](X) = 0 [A__ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.3: SCC Processor: -> Pairs: A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u Problem 1.3: Reduction Pairs Processor: -> Pairs: A____(__(X:S,Y:S),Z:S) -> MARK(Y:S) A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = 2.X [a__U21](X1,X2) = 2.X1 [a__U22](X) = 2.X [a__U31](X) = 2.X [a__U41](X1,X2) = 2.X1 [a__U42](X) = 2.X [a__U51](X1,X2) = 2.X1 [a__U52](X) = 2.X [a__U61](X) = 2.X [a__U71](X1,X2) = 2.X1 + 2.X2 + 2 [a__U72](X) = X [a__U81](X) = 2.X + 2 [a____](X1,X2) = 2.X1 + X2 + 2 [a__isList](X) = 0 [a__isNeList](X) = 0 [a__isNePal](X) = X [a__isPal](X) = 2.X + 2 [a__isQid](X) = 0 [mark](X) = X [U11](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X) = 2.X [U31](X) = 2.X [U41](X1,X2) = 2.X1 [U42](X) = 2.X [U51](X1,X2) = 2.X1 [U52](X) = 2.X [U61](X) = 2.X [U71](X1,X2) = 2.X1 + 2.X2 + 2 [U72](X) = X [U81](X) = 2.X + 2 [__](X1,X2) = 2.X1 + X2 + 2 [a] = 1 [e] = 1 [fSNonEmpty] = 0 [i] = 0 [isList](X) = 0 [isNeList](X) = 0 [isNePal](X) = X [isPal](X) = 2.X + 2 [isQid](X) = 0 [nil] = 0 [o] = 0 [tt] = 0 [u] = 2 [A__U11](X) = 0 [A__U21](X1,X2) = 0 [A__U22](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X) = 0 [A__U71](X1,X2) = 0 [A__U72](X) = 0 [A__U81](X) = 0 [A____](X1,X2) = 2.X1 + 2.X2 + 1 [A__ISLIST](X) = 0 [A__ISNELIST](X) = 0 [A__ISNEPAL](X) = 0 [A__ISPAL](X) = 0 [A__ISQID](X) = 0 [MARK](X) = 2.X + 1 Problem 1.3: SCC Processor: -> Pairs: A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u Problem 1.3: Reduction Pairs Processor: -> Pairs: A____(__(X:S,Y:S),Z:S) -> MARK(Z:S) A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X1,X2) = 2.X1 + 2.X2 + 2 [a__U22](X) = X + 2 [a__U31](X) = X [a__U41](X1,X2) = 2.X1 + 2.X2 + 1 [a__U42](X) = X + 2 [a__U51](X1,X2) = X1 + 2.X2 + 2 [a__U52](X) = X + 2 [a__U61](X) = 2.X + 1 [a__U71](X1,X2) = 2.X1 + 2.X2 + 2 [a__U72](X) = X + 2 [a__U81](X) = X [a____](X1,X2) = 2.X1 + X2 + 2 [a__isList](X) = 2.X [a__isNeList](X) = 2.X [a__isNePal](X) = 2.X + 2 [a__isPal](X) = 2.X + 2 [a__isQid](X) = X [mark](X) = X [U11](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X) = X + 2 [U31](X) = X [U41](X1,X2) = 2.X1 + 2.X2 + 1 [U42](X) = X + 2 [U51](X1,X2) = X1 + 2.X2 + 2 [U52](X) = X + 2 [U61](X) = 2.X + 1 [U71](X1,X2) = 2.X1 + 2.X2 + 2 [U72](X) = X + 2 [U81](X) = X [__](X1,X2) = 2.X1 + X2 + 2 [a] = 2 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [isList](X) = 2.X [isNeList](X) = 2.X [isNePal](X) = 2.X + 2 [isPal](X) = 2.X + 2 [isQid](X) = X [nil] = 2 [o] = 2 [tt] = 2 [u] = 2 [A__U11](X) = 0 [A__U21](X1,X2) = 0 [A__U22](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X) = 0 [A__U71](X1,X2) = 0 [A__U72](X) = 0 [A__U81](X) = 0 [A____](X1,X2) = 2.X1 + 2.X2 [A__ISLIST](X) = 0 [A__ISNELIST](X) = 0 [A__ISNEPAL](X) = 0 [A__ISPAL](X) = 0 [A__ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.3: SCC Processor: -> Pairs: A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u Problem 1.3: Reduction Pairs Processor: -> Pairs: A____(nil,X:S) -> MARK(X:S) A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X1,X2) = 2.X1 + 2.X2 + 1 [a__U22](X) = X + 2 [a__U31](X) = X [a__U41](X1,X2) = 2.X1 + 2.X2 + 2 [a__U42](X) = X + 2 [a__U51](X1,X2) = 2.X1 + 2.X2 + 2 [a__U52](X) = X + 1 [a__U61](X) = X [a__U71](X1,X2) = 2.X1 + 2.X2 + 2 [a__U72](X) = X + 2 [a__U81](X) = X [a____](X1,X2) = 2.X1 + X2 + 2 [a__isList](X) = 2.X + 2 [a__isNeList](X) = 2.X + 2 [a__isNePal](X) = 2.X + 2 [a__isPal](X) = 2.X + 2 [a__isQid](X) = 2.X + 2 [mark](X) = X [U11](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 1 [U22](X) = X + 2 [U31](X) = X [U41](X1,X2) = 2.X1 + 2.X2 + 2 [U42](X) = X + 2 [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X) = X + 1 [U61](X) = X [U71](X1,X2) = 2.X1 + 2.X2 + 2 [U72](X) = X + 2 [U81](X) = X [__](X1,X2) = 2.X1 + X2 + 2 [a] = 1 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [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 [nil] = 2 [o] = 1 [tt] = 2 [u] = 1 [A__U11](X) = 0 [A__U21](X1,X2) = 0 [A__U22](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X) = 0 [A__U71](X1,X2) = 0 [A__U72](X) = 0 [A__U81](X) = 0 [A____](X1,X2) = 2.X1 + 2.X2 [A__ISLIST](X) = 0 [A__ISNELIST](X) = 0 [A__ISNEPAL](X) = 0 [A__ISPAL](X) = 0 [A__ISQID](X) = 0 [MARK](X) = 2.X + 2 Problem 1.3: SCC Processor: -> Pairs: A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u Problem 1.3: Reduction Pairs Processor: -> Pairs: A____(X:S,nil) -> MARK(X:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X1,X2) = X1 + 2.X2 + 1 [a__U22](X) = X [a__U31](X) = X [a__U41](X1,X2) = 2.X1 + 2.X2 + 2 [a__U42](X) = X + 2 [a__U51](X1,X2) = 2.X1 + 2.X2 + 1 [a__U52](X) = X + 2 [a__U61](X) = X [a__U71](X1,X2) = 2.X1 + 2.X2 + 2 [a__U72](X) = X + 2 [a__U81](X) = X [a____](X1,X2) = 2.X1 + X2 + 2 [a__isList](X) = 2.X + 2 [a__isNeList](X) = 2.X + 2 [a__isNePal](X) = 2.X + 2 [a__isPal](X) = 2.X + 2 [a__isQid](X) = 2 [mark](X) = X [U11](X) = X [U21](X1,X2) = X1 + 2.X2 + 1 [U22](X) = X [U31](X) = X [U41](X1,X2) = 2.X1 + 2.X2 + 2 [U42](X) = X + 2 [U51](X1,X2) = 2.X1 + 2.X2 + 1 [U52](X) = X + 2 [U61](X) = X [U71](X1,X2) = 2.X1 + 2.X2 + 2 [U72](X) = X + 2 [U81](X) = X [__](X1,X2) = 2.X1 + X2 + 2 [a] = 0 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isNePal](X) = 2.X + 2 [isPal](X) = 2.X + 2 [isQid](X) = 2 [nil] = 2 [o] = 1 [tt] = 2 [u] = 1 [A__U11](X) = 0 [A__U21](X1,X2) = 0 [A__U22](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X) = 0 [A__U71](X1,X2) = 0 [A__U72](X) = 0 [A__U81](X) = 0 [A____](X1,X2) = 2.X1 + X2 [A__ISLIST](X) = 0 [A__ISNELIST](X) = 0 [A__ISNEPAL](X) = 0 [A__ISPAL](X) = 0 [A__ISQID](X) = 0 [MARK](X) = 2.X Problem 1.3: SCC Processor: -> Pairs: MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> A____(mark(X1:S),mark(X2:S)) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u Problem 1.3: Subterm Processor: -> Pairs: MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S)) -> MARK(X1:S) MARK(U72(X:S)) -> MARK(X:S) MARK(U81(X:S)) -> MARK(X:S) MARK(__(X1:S,X2:S)) -> MARK(X1:S) MARK(__(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Projection: pi(MARK) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt,V2:S) -> a__U22(a__isList(V2:S)) a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) a__U22(tt) -> tt a__U22(X:S) -> U22(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNeList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt) -> tt a__U61(X:S) -> U61(X:S) a__U71(tt,P:S) -> a__U72(a__isPal(P:S)) a__U71(X1:S,X2:S) -> U71(X1:S,X2:S) a__U72(tt) -> tt a__U72(X:S) -> U72(X:S) a__U81(tt) -> tt a__U81(X:S) -> U81(X:S) a____(__(X:S,Y:S),Z:S) -> a____(mark(X:S),a____(mark(Y:S),mark(Z:S))) a____(nil,X:S) -> mark(X:S) a____(X:S,nil) -> mark(X:S) a____(X1:S,X2:S) -> __(X1:S,X2:S) a__isList(__(V1:S,V2:S)) -> a__U21(a__isList(V1:S),V2:S) a__isList(nil) -> tt a__isList(V:S) -> a__U11(a__isNeList(V:S)) a__isList(X:S) -> isList(X:S) a__isNeList(__(V1:S,V2:S)) -> a__U41(a__isList(V1:S),V2:S) a__isNeList(__(V1:S,V2:S)) -> a__U51(a__isNeList(V1:S),V2:S) a__isNeList(V:S) -> a__U31(a__isQid(V:S)) a__isNeList(X:S) -> isNeList(X:S) a__isNePal(__(I:S,__(P:S,I:S))) -> a__U71(a__isQid(I:S),P:S) a__isNePal(V:S) -> a__U61(a__isQid(V:S)) a__isNePal(X:S) -> isNePal(X:S) a__isPal(nil) -> tt a__isPal(V:S) -> a__U81(a__isNePal(V:S)) a__isPal(X:S) -> isPal(X:S) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt a__isQid(X:S) -> isQid(X:S) mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) mark(U22(X:S)) -> a__U22(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X:S)) -> a__U61(mark(X:S)) mark(U71(X1:S,X2:S)) -> a__U71(mark(X1:S),X2:S) mark(U72(X:S)) -> a__U72(mark(X:S)) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(__(X1:S,X2:S)) -> a____(mark(X1:S),mark(X2:S)) mark(a) -> a mark(e) -> e mark(i) -> i mark(isList(X:S)) -> a__isList(X:S) mark(isNeList(X:S)) -> a__isNeList(X:S) mark(isNePal(X:S)) -> a__isNePal(X:S) mark(isPal(X:S)) -> a__isPal(X:S) mark(isQid(X:S)) -> a__isQid(X:S) mark(nil) -> nil mark(o) -> o mark(tt) -> tt mark(u) -> u ->Strongly Connected Components: There is no strongly connected component The problem is finite.