YES

Problem 1: 

(VAR v_NonEmpty:S I:S P:S X:S X1:S X2:S Y:S Z:S)
(RULES
U11(mark(X:S)) -> mark(U11(X:S))
U11(ok(X:S)) -> ok(U11(X:S))
U12(mark(X:S)) -> mark(U12(X:S))
U12(ok(X:S)) -> ok(U12(X:S))
__(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(U11(tt)) -> mark(U12(tt))
active(U11(X:S)) -> U11(active(X:S))
active(U12(tt)) -> mark(tt)
active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
active(isNePal(X:S)) -> isNePal(active(X:S))
isNePal(mark(X:S)) -> mark(isNePal(X:S))
isNePal(ok(X:S)) -> ok(isNePal(X:S))
proper(U11(X:S)) -> U11(proper(X:S))
proper(U12(X:S)) -> U12(proper(X:S))
proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
proper(isNePal(X:S)) -> isNePal(proper(X:S))
proper(nil) -> ok(nil)
proper(tt) -> ok(tt)
top(mark(X:S)) -> top(proper(X:S))
top(ok(X:S)) -> top(active(X:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U11#(mark(X:S)) -> U11#(X:S)
 U11#(ok(X:S)) -> U11#(X:S)
 U12#(mark(X:S)) -> U12#(X:S)
 U12#(ok(X:S)) -> U12#(X:S)
 __#(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(U11(X:S)) -> U11#(active(X:S))
 ACTIVE(U11(X:S)) -> ACTIVE(X:S)
 ACTIVE(U12(X:S)) -> U12#(active(X:S))
 ACTIVE(U12(X:S)) -> ACTIVE(X: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(isNePal(X:S)) -> ACTIVE(X:S)
 ACTIVE(isNePal(X:S)) -> ISNEPAL(active(X:S))
 ISNEPAL(mark(X:S)) -> ISNEPAL(X:S)
 ISNEPAL(ok(X:S)) -> ISNEPAL(X:S)
 PROPER(U11(X:S)) -> U11#(proper(X:S))
 PROPER(U11(X:S)) -> PROPER(X:S)
 PROPER(U12(X:S)) -> U12#(proper(X:S))
 PROPER(U12(X:S)) -> PROPER(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(isNePal(X:S)) -> ISNEPAL(proper(X:S))
 PROPER(isNePal(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:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(mark(X:S)) -> U11#(X:S)
 U11#(ok(X:S)) -> U11#(X:S)
 U12#(mark(X:S)) -> U12#(X:S)
 U12#(ok(X:S)) -> U12#(X:S)
 __#(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(U11(X:S)) -> U11#(active(X:S))
 ACTIVE(U11(X:S)) -> ACTIVE(X:S)
 ACTIVE(U12(X:S)) -> U12#(active(X:S))
 ACTIVE(U12(X:S)) -> ACTIVE(X: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(isNePal(X:S)) -> ACTIVE(X:S)
 ACTIVE(isNePal(X:S)) -> ISNEPAL(active(X:S))
 ISNEPAL(mark(X:S)) -> ISNEPAL(X:S)
 ISNEPAL(ok(X:S)) -> ISNEPAL(X:S)
 PROPER(U11(X:S)) -> U11#(proper(X:S))
 PROPER(U11(X:S)) -> PROPER(X:S)
 PROPER(U12(X:S)) -> U12#(proper(X:S))
 PROPER(U12(X:S)) -> PROPER(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(isNePal(X:S)) -> ISNEPAL(proper(X:S))
 PROPER(isNePal(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:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ISNEPAL(mark(X:S)) -> ISNEPAL(X:S)
 ISNEPAL(ok(X:S)) -> ISNEPAL(X:S)
->->-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 U12#(mark(X:S)) -> U12#(X:S)
 U12#(ok(X:S)) -> U12#(X:S)
->->-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 U11#(mark(X:S)) -> U11#(X:S)
 U11#(ok(X:S)) -> U11#(X:S)
->->-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(X:S)) -> ACTIVE(X:S)
 ACTIVE(U12(X:S)) -> ACTIVE(X:S)
 ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(isNePal(X:S)) -> ACTIVE(X:S)
->->-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 PROPER(U11(X:S)) -> PROPER(X:S)
 PROPER(U12(X:S)) -> PROPER(X:S)
 PROPER(__(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(__(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(isNePal(X:S)) -> PROPER(X:S)
->->-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))


The problem is decomposed in 7 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ISNEPAL(mark(X:S)) -> ISNEPAL(X:S)
 ISNEPAL(ok(X:S)) -> ISNEPAL(X:S)
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(ISNEPAL) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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:
 __#(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:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(__#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S)
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))

Problem 1.2: 

Subterm Processor:
-> Pairs:
 __#(X1:S,mark(X2:S)) -> __#(X1:S,X2:S)
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(__#) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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:
 U12#(mark(X:S)) -> U12#(X:S)
 U12#(ok(X:S)) -> U12#(X:S)
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(U12#) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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:
 U11#(mark(X:S)) -> U11#(X:S)
 U11#(ok(X:S)) -> U11#(X:S)
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(U11#) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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:
 ACTIVE(U11(X:S)) -> ACTIVE(X:S)
 ACTIVE(U12(X:S)) -> ACTIVE(X:S)
 ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(__(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(isNePal(X:S)) -> ACTIVE(X:S)
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(ACTIVE) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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:
 PROPER(U11(X:S)) -> PROPER(X:S)
 PROPER(U12(X:S)) -> PROPER(X:S)
 PROPER(__(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(__(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(isNePal(X:S)) -> PROPER(X:S)
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(PROPER) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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: 

Reduction Pairs Processor:
-> Pairs:
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
-> Usable rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X) = 2.X + 2
[U12](X) = X + 2
[__](X1,X2) = 2.X1 + X2 + 2
[active](X) = X
[isNePal](X) = 2.X + 2
[proper](X) = X
[top](X) = 0
[fSNonEmpty] = 0
[mark](X) = X + 2
[nil] = 1
[ok](X) = X
[tt] = 2
[U11#](X) = 0
[U12#](X) = 0
[__#](X1,X2) = 0
[ACTIVE](X) = 0
[ISNEPAL](X) = 0
[PROPER](X) = 0
[TOP](X) = 2.X

Problem 1.7: 

SCC Processor:
-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
-> Usable rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X) = 2.X + 2
[U12](X) = 2.X + 2
[__](X1,X2) = X1 + X2
[active](X) = 2.X + 1
[isNePal](X) = 2.X + 2
[proper](X) = 0
[top](X) = 0
[fSNonEmpty] = 0
[mark](X) = X + 1
[nil] = 2
[ok](X) = 2.X + 2
[tt] = 1
[U11#](X) = 0
[U12#](X) = 0
[__#](X1,X2) = 0
[ACTIVE](X) = 0
[ISNEPAL](X) = 0
[PROPER](X) = 0
[TOP](X) = 2.X

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X:S)) -> mark(U11(X:S))
 U11(ok(X:S)) -> ok(U11(X:S))
 U12(mark(X:S)) -> mark(U12(X:S))
 U12(ok(X:S)) -> ok(U12(X:S))
 __(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(U11(tt)) -> mark(U12(tt))
 active(U11(X:S)) -> U11(active(X:S))
 active(U12(tt)) -> mark(tt)
 active(U12(X:S)) -> U12(active(X: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(isNePal(__(I:S,__(P:S,I:S)))) -> mark(U11(tt))
 active(isNePal(X:S)) -> isNePal(active(X:S))
 isNePal(mark(X:S)) -> mark(isNePal(X:S))
 isNePal(ok(X:S)) -> ok(isNePal(X:S))
 proper(U11(X:S)) -> U11(proper(X:S))
 proper(U12(X:S)) -> U12(proper(X:S))
 proper(__(X1:S,X2:S)) -> __(proper(X1:S),proper(X2:S))
 proper(isNePal(X:S)) -> isNePal(proper(X:S))
 proper(nil) -> ok(nil)
 proper(tt) -> ok(tt)
 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.