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 __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: __#(mark(X1:S),X2:S) -> __#(X1:S,X2:S) __#(ok(X1:S),ok(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) ACTIVE(__(__(X:S,Y:S),Z:S)) -> __#(X:S,__(Y:S,Z:S)) ACTIVE(__(__(X:S,Y:S),Z:S)) -> __#(Y:S,Z:S) ACTIVE(__(X1:S,X2:S)) -> __#(active(X1:S),X2:S) ACTIVE(__(X1:S,X2:S)) -> __#(X1:S,active(X2:S)) ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(and(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(and(X1:S,X2:S)) -> AND(active(X1:S),X2:S) ACTIVE(isList(__(V1:S,V2:S))) -> AND(isList(V1:S),isList(V2:S)) ACTIVE(isList(__(V1:S,V2:S))) -> ISLIST(V1:S) ACTIVE(isList(__(V1:S,V2:S))) -> ISLIST(V2:S) ACTIVE(isList(V:S)) -> ISNELIST(V:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> AND(isList(V1:S),isNeList(V2:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> AND(isNeList(V1:S),isList(V2:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISLIST(V1:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISLIST(V2:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISNELIST(V1:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISNELIST(V2:S) ACTIVE(isNeList(V:S)) -> ISQID(V:S) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> AND(isQid(I:S),isPal(P:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> ISPAL(P:S) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> ISQID(I:S) ACTIVE(isNePal(V:S)) -> ISQID(V:S) ACTIVE(isPal(V:S)) -> ISNEPAL(V:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(ok(X1:S),ok(X2:S)) -> AND(X1:S,X2:S) ISLIST(ok(X:S)) -> ISLIST(X:S) ISNELIST(ok(X:S)) -> ISNELIST(X:S) ISNEPAL(ok(X:S)) -> ISNEPAL(X:S) ISPAL(ok(X:S)) -> ISPAL(X:S) ISQID(ok(X:S)) -> ISQID(X:S) PROPER(__(X1:S,X2:S)) -> __#(proper(X1:S),proper(X2:S)) PROPER(__(X1:S,X2:S)) -> PROPER(X1:S) PROPER(__(X1:S,X2:S)) -> PROPER(X2:S) PROPER(and(X1:S,X2:S)) -> AND(proper(X1:S),proper(X2:S)) PROPER(and(X1:S,X2:S)) -> PROPER(X1:S) PROPER(and(X1:S,X2:S)) -> PROPER(X2:S) PROPER(isList(X:S)) -> ISLIST(proper(X:S)) PROPER(isList(X:S)) -> PROPER(X:S) PROPER(isNeList(X:S)) -> ISNELIST(proper(X:S)) PROPER(isNeList(X:S)) -> PROPER(X:S) PROPER(isNePal(X:S)) -> ISNEPAL(proper(X:S)) PROPER(isNePal(X:S)) -> PROPER(X:S) PROPER(isPal(X:S)) -> ISPAL(proper(X:S)) PROPER(isPal(X:S)) -> PROPER(X:S) PROPER(isQid(X:S)) -> ISQID(proper(X:S)) PROPER(isQid(X:S)) -> PROPER(X:S) TOP(mark(X:S)) -> PROPER(X:S) TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> ACTIVE(X:S) TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1: SCC Processor: -> Pairs: __#(mark(X1:S),X2:S) -> __#(X1:S,X2:S) __#(ok(X1:S),ok(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) ACTIVE(__(__(X:S,Y:S),Z:S)) -> __#(X:S,__(Y:S,Z:S)) ACTIVE(__(__(X:S,Y:S),Z:S)) -> __#(Y:S,Z:S) ACTIVE(__(X1:S,X2:S)) -> __#(active(X1:S),X2:S) ACTIVE(__(X1:S,X2:S)) -> __#(X1:S,active(X2:S)) ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(and(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(and(X1:S,X2:S)) -> AND(active(X1:S),X2:S) ACTIVE(isList(__(V1:S,V2:S))) -> AND(isList(V1:S),isList(V2:S)) ACTIVE(isList(__(V1:S,V2:S))) -> ISLIST(V1:S) ACTIVE(isList(__(V1:S,V2:S))) -> ISLIST(V2:S) ACTIVE(isList(V:S)) -> ISNELIST(V:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> AND(isList(V1:S),isNeList(V2:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> AND(isNeList(V1:S),isList(V2:S)) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISLIST(V1:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISLIST(V2:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISNELIST(V1:S) ACTIVE(isNeList(__(V1:S,V2:S))) -> ISNELIST(V2:S) ACTIVE(isNeList(V:S)) -> ISQID(V:S) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> AND(isQid(I:S),isPal(P:S)) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> ISPAL(P:S) ACTIVE(isNePal(__(I:S,__(P:S,I:S)))) -> ISQID(I:S) ACTIVE(isNePal(V:S)) -> ISQID(V:S) ACTIVE(isPal(V:S)) -> ISNEPAL(V:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(ok(X1:S),ok(X2:S)) -> AND(X1:S,X2:S) ISLIST(ok(X:S)) -> ISLIST(X:S) ISNELIST(ok(X:S)) -> ISNELIST(X:S) ISNEPAL(ok(X:S)) -> ISNEPAL(X:S) ISPAL(ok(X:S)) -> ISPAL(X:S) ISQID(ok(X:S)) -> ISQID(X:S) PROPER(__(X1:S,X2:S)) -> __#(proper(X1:S),proper(X2:S)) PROPER(__(X1:S,X2:S)) -> PROPER(X1:S) PROPER(__(X1:S,X2:S)) -> PROPER(X2:S) PROPER(and(X1:S,X2:S)) -> AND(proper(X1:S),proper(X2:S)) PROPER(and(X1:S,X2:S)) -> PROPER(X1:S) PROPER(and(X1:S,X2:S)) -> PROPER(X2:S) PROPER(isList(X:S)) -> ISLIST(proper(X:S)) PROPER(isList(X:S)) -> PROPER(X:S) PROPER(isNeList(X:S)) -> ISNELIST(proper(X:S)) PROPER(isNeList(X:S)) -> PROPER(X:S) PROPER(isNePal(X:S)) -> ISNEPAL(proper(X:S)) PROPER(isNePal(X:S)) -> PROPER(X:S) PROPER(isPal(X:S)) -> ISPAL(proper(X:S)) PROPER(isPal(X:S)) -> PROPER(X:S) PROPER(isQid(X:S)) -> ISQID(proper(X:S)) PROPER(isQid(X:S)) -> PROPER(X:S) TOP(mark(X:S)) -> PROPER(X:S) TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> ACTIVE(X:S) TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISQID(ok(X:S)) -> ISQID(X:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ISPAL(ok(X:S)) -> ISPAL(X:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ISNEPAL(ok(X:S)) -> ISNEPAL(X:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ISNELIST(ok(X:S)) -> ISNELIST(X:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ISLIST(ok(X:S)) -> ISLIST(X:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(ok(X1:S),ok(X2:S)) -> AND(X1:S,X2:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: __#(mark(X1:S),X2:S) -> __#(X1:S,X2:S) __#(ok(X1:S),ok(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(and(X1:S,X2:S)) -> ACTIVE(X1:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: PROPER(__(X1:S,X2:S)) -> PROPER(X1:S) PROPER(__(X1:S,X2:S)) -> PROPER(X2:S) PROPER(and(X1:S,X2:S)) -> PROPER(X1:S) PROPER(and(X1:S,X2:S)) -> PROPER(X2:S) PROPER(isList(X:S)) -> PROPER(X:S) PROPER(isNeList(X:S)) -> PROPER(X:S) PROPER(isNePal(X:S)) -> PROPER(X:S) PROPER(isPal(X:S)) -> PROPER(X:S) PROPER(isQid(X:S)) -> PROPER(X:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> TOP(active(X:S)) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) The problem is decomposed in 10 subproblems. Problem 1.1: Subterm Processor: -> Pairs: ISQID(ok(X:S)) -> ISQID(X:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ISQID) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: ISPAL(ok(X:S)) -> ISPAL(X:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ISPAL) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: ISNEPAL(ok(X:S)) -> ISNEPAL(X:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ISNEPAL) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: ISNELIST(ok(X:S)) -> ISNELIST(X:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ISNELIST) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: ISLIST(ok(X:S)) -> ISLIST(X:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ISLIST) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(ok(X1:S),ok(X2:S)) -> AND(X1:S,X2:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(AND) = 1 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: __#(mark(X1:S),X2:S) -> __#(X1:S,X2:S) __#(ok(X1:S),ok(X2:S)) -> __#(X1:S,X2:S) __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(__#) = 1 Problem 1.7: SCC Processor: -> Pairs: __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1.7: Subterm Processor: -> Pairs: __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(__#) = 2 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Subterm Processor: -> Pairs: ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(and(X1:S,X2:S)) -> ACTIVE(X1:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ACTIVE) = 1 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.9: Subterm Processor: -> Pairs: PROPER(__(X1:S,X2:S)) -> PROPER(X1:S) PROPER(__(X1:S,X2:S)) -> PROPER(X2:S) PROPER(and(X1:S,X2:S)) -> PROPER(X1:S) PROPER(and(X1:S,X2:S)) -> PROPER(X2:S) PROPER(isList(X:S)) -> PROPER(X:S) PROPER(isNeList(X:S)) -> PROPER(X:S) PROPER(isNePal(X:S)) -> PROPER(X:S) PROPER(isPal(X:S)) -> PROPER(X:S) PROPER(isQid(X:S)) -> PROPER(X:S) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(PROPER) = 1 Problem 1.9: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.10: Reduction Pairs Processor: -> Pairs: TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) -> Usable rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = X1 + X2 + 1 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 1 [isNePal](X) = 2.X + 1 [isPal](X) = 2.X + 2 [isQid](X) = 2.X [proper](X) = X [top](X) = 0 [a] = 2 [e] = 2 [fSNonEmpty] = 0 [i] = 2 [mark](X) = X + 1 [nil] = 2 [o] = 2 [ok](X) = X [tt] = 2 [u] = 2 [__#](X1,X2) = 0 [ACTIVE](X) = 0 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [PROPER](X) = 0 [TOP](X) = 2.X Problem 1.10: SCC Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TOP(ok(X:S)) -> TOP(active(X:S)) ->->-> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1.10: Reduction Pairs Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) -> Usable rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + 2.X2 + 2 [active](X) = 2.X [and](X1,X2) = X1 + 2.X2 [isList](X) = 2.X + 2 [isNeList](X) = 2.X + 2 [isNePal](X) = 2.X + 2 [isPal](X) = 2.X + 2 [isQid](X) = 2.X + 1 [proper](X) = 0 [top](X) = 0 [a] = 2 [e] = 2 [fSNonEmpty] = 0 [i] = 1 [mark](X) = 0 [nil] = 2 [o] = 2 [ok](X) = 2.X + 2 [tt] = 0 [u] = 2 [__#](X1,X2) = 0 [ACTIVE](X) = 0 [AND](X1,X2) = 0 [ISLIST](X) = 0 [ISNELIST](X) = 0 [ISNEPAL](X) = 0 [ISPAL](X) = 0 [ISQID](X) = 0 [PROPER](X) = 0 [TOP](X) = 2.X Problem 1.10: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1:S),X2:S) -> mark(__(X1:S,X2:S)) __(ok(X1:S),ok(X2:S)) -> ok(__(X1:S,X2:S)) __(X1:S,mark(X2:S)) -> mark(__(X1:S,X2:S)) active(__(__(X:S,Y:S),Z:S)) -> mark(__(X:S,__(Y:S,Z:S))) active(__(nil,X:S)) -> mark(X:S) active(__(X:S,nil)) -> mark(X:S) active(__(X1:S,X2:S)) -> __(active(X1:S),X2:S) active(__(X1:S,X2:S)) -> __(X1:S,active(X2:S)) active(and(tt,X:S)) -> mark(X:S) active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) active(isList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isList(V2:S))) active(isList(nil)) -> mark(tt) active(isList(V:S)) -> mark(isNeList(V:S)) active(isNeList(__(V1:S,V2:S))) -> mark(and(isList(V1:S),isNeList(V2:S))) active(isNeList(__(V1:S,V2:S))) -> mark(and(isNeList(V1:S),isList(V2:S))) active(isNeList(V:S)) -> mark(isQid(V:S)) active(isNePal(__(I:S,__(P:S,I:S)))) -> mark(and(isQid(I:S),isPal(P:S))) active(isNePal(V:S)) -> mark(isQid(V:S)) active(isPal(nil)) -> mark(tt) active(isPal(V:S)) -> mark(isNePal(V:S)) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) isList(ok(X:S)) -> ok(isList(X:S)) isNeList(ok(X:S)) -> ok(isNeList(X:S)) isNePal(ok(X:S)) -> ok(isNePal(X:S)) isPal(ok(X:S)) -> ok(isPal(X:S)) isQid(ok(X:S)) -> ok(isQid(X:S)) proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S)) proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) proper(isList(X:S)) -> isList(proper(X:S)) proper(isNeList(X:S)) -> isNeList(proper(X:S)) proper(isNePal(X:S)) -> isNePal(proper(X:S)) proper(isPal(X:S)) -> isPal(proper(X:S)) proper(isQid(X:S)) -> isQid(proper(X:S)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(nil) -> ok(nil) proper(o) -> ok(o) proper(tt) -> ok(tt) proper(u) -> ok(u) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.