YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S)
(RULES
active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
active(f(s(0))) -> mark(f(p(s(0))))
active(f(0)) -> mark(cons(0,f(s(0))))
active(f(X:S)) -> f(active(X:S))
active(p(s(X:S))) -> mark(X:S)
active(p(X:S)) -> p(active(X:S))
active(s(X:S)) -> s(active(X:S))
cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
f(mark(X:S)) -> mark(f(X:S))
f(ok(X:S)) -> ok(f(X:S))
p(mark(X:S)) -> mark(p(X:S))
p(ok(X:S)) -> ok(p(X:S))
proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
proper(f(X:S)) -> f(proper(X:S))
proper(p(X:S)) -> p(proper(X:S))
proper(s(X:S)) -> s(proper(X:S))
proper(0) -> ok(0)
s(mark(X:S)) -> mark(s(X:S))
s(ok(X:S)) -> ok(s(X:S))
top(mark(X:S)) -> top(proper(X:S))
top(ok(X:S)) -> top(active(X:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S)
 ACTIVE(f(X:S)) -> ACTIVE(X:S)
 ACTIVE(f(X:S)) -> F(active(X:S))
 ACTIVE(p(X:S)) -> ACTIVE(X:S)
 ACTIVE(p(X:S)) -> P(active(X:S))
 ACTIVE(s(X:S)) -> ACTIVE(X:S)
 ACTIVE(s(X:S)) -> S(active(X:S))
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
 F(mark(X:S)) -> F(X:S)
 F(ok(X:S)) -> F(X:S)
 P(mark(X:S)) -> P(X:S)
 P(ok(X:S)) -> P(X:S)
 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S))
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(f(X:S)) -> F(proper(X:S))
 PROPER(f(X:S)) -> PROPER(X:S)
 PROPER(p(X:S)) -> P(proper(X:S))
 PROPER(p(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> S(proper(X:S))
 S(mark(X:S)) -> S(X:S)
 S(ok(X:S)) -> S(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:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S)
 ACTIVE(f(X:S)) -> ACTIVE(X:S)
 ACTIVE(f(X:S)) -> F(active(X:S))
 ACTIVE(p(X:S)) -> ACTIVE(X:S)
 ACTIVE(p(X:S)) -> P(active(X:S))
 ACTIVE(s(X:S)) -> ACTIVE(X:S)
 ACTIVE(s(X:S)) -> S(active(X:S))
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
 F(mark(X:S)) -> F(X:S)
 F(ok(X:S)) -> F(X:S)
 P(mark(X:S)) -> P(X:S)
 P(ok(X:S)) -> P(X:S)
 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S))
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(f(X:S)) -> F(proper(X:S))
 PROPER(f(X:S)) -> PROPER(X:S)
 PROPER(p(X:S)) -> P(proper(X:S))
 PROPER(p(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> S(proper(X:S))
 S(mark(X:S)) -> S(X:S)
 S(ok(X:S)) -> S(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:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 S(mark(X:S)) -> S(X:S)
 S(ok(X:S)) -> S(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 P(mark(X:S)) -> P(X:S)
 P(ok(X:S)) -> P(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 F(mark(X:S)) -> F(X:S)
 F(ok(X:S)) -> F(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(f(X:S)) -> PROPER(X:S)
 PROPER(p(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> PROPER(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(f(X:S)) -> ACTIVE(X:S)
 ACTIVE(p(X:S)) -> ACTIVE(X:S)
 ACTIVE(s(X:S)) -> ACTIVE(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 S(mark(X:S)) -> S(X:S)
 S(ok(X:S)) -> S(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(S) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 P(mark(X:S)) -> P(X:S)
 P(ok(X:S)) -> P(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(P) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 F(mark(X:S)) -> F(X:S)
 F(ok(X:S)) -> F(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(F) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(CONS) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(f(X:S)) -> PROPER(X:S)
 PROPER(p(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> PROPER(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(PROPER) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(f(X:S)) -> ACTIVE(X:S)
 ACTIVE(p(X:S)) -> ACTIVE(X:S)
 ACTIVE(s(X:S)) -> ACTIVE(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(ACTIVE) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
-> Usable rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[active](X) = [1 0;0 1].X
[cons](X1,X2) = [1 0;1 0].X1
[f](X) = [1 0;1 0].X + [1;1]
[p](X) = [0 1;1 0].X + [1;0]
[proper](X) = [1 0;0 1].X
[s](X) = [0 1;1 0].X + [1;0]
[top](X) = 0
[0] = [0;1]
[fSNonEmpty] = 0
[mark](X) = [1 0;0 1].X + [1;1]
[ok](X) = [1 0;0 1].X
[ACTIVE](X) = 0
[CONS](X1,X2) = 0
[F](X) = 0
[P](X) = 0
[PROPER](X) = 0
[S](X) = 0
[TOP](X) = [0 1;1 1].X

Problem 1.7: 

SCC Processor:
-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
-> Usable rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = 2.X + 1
[cons](X1,X2) = 2.X1 + 1
[f](X) = 2.X + 1
[p](X) = X
[proper](X) = 0
[s](X) = 2.X + 1
[top](X) = 0
[0] = 1
[fSNonEmpty] = 0
[mark](X) = 2.X + 1
[ok](X) = 2.X + 2
[ACTIVE](X) = 0
[CONS](X1,X2) = 0
[F](X) = 0
[P](X) = 0
[PROPER](X) = 0
[S](X) = 0
[TOP](X) = 2.X

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(f(s(0))) -> mark(f(p(s(0))))
 active(f(0)) -> mark(cons(0,f(s(0))))
 active(f(X:S)) -> f(active(X:S))
 active(p(s(X:S))) -> mark(X:S)
 active(p(X:S)) -> p(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 f(mark(X:S)) -> mark(f(X:S))
 f(ok(X:S)) -> ok(f(X:S))
 p(mark(X:S)) -> mark(p(X:S))
 p(ok(X:S)) -> ok(p(X:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(p(X:S)) -> p(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(0) -> ok(0)
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 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.