YES

Problem 1: 

(VAR v_NonEmpty:S M:S N:S X:S X1:S X2:S)
(RULES
active(and(tt,X:S)) -> mark(X:S)
active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
active(plus(N:S,0)) -> mark(N:S)
and(active(X1:S),X2:S) -> and(X1:S,X2:S)
and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
mark(s(X:S)) -> active(s(mark(X:S)))
mark(0) -> active(0)
mark(tt) -> active(tt)
plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
s(active(X:S)) -> s(X:S)
s(mark(X:S)) -> s(X:S)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(plus(N:S,s(M:S))) -> PLUS(N:S,M:S)
 ACTIVE(plus(N:S,s(M:S))) -> S(plus(N:S,M:S))
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S)
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S))
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> S(mark(X:S))
 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(plus(N:S,s(M:S))) -> PLUS(N:S,M:S)
 ACTIVE(plus(N:S,s(M:S))) -> S(plus(N:S,M:S))
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S)
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S))
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> S(mark(X:S))
 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
->->-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
->->-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
->->-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(S) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(PLUS) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
->->-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.2: 

Subterm Processor:
-> Pairs:
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(PLUS) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(AND) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
->->-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.3: 

Subterm Processor:
-> Pairs:
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(AND) = 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[and](X1,X2) = 2.X1 + X2 + 2
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X + 2
[0] = 2
[fSNonEmpty] = 0
[tt] = 2
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.4: 

SCC Processor:
-> Pairs:
 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[and](X1,X2) = X1 + 2.X2 + 2
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2
[s](X) = X + 1
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.4: 

SCC Processor:
-> Pairs:
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(plus(N:S,0)) -> MARK(N:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2 + 2
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[ACTIVE](X) = 2.X + 1
[AND](X1,X2) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.4: 

SCC Processor:
-> Pairs:
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.4: 

Subterm Processor:
-> Pairs:
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(MARK) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(and(tt,X:S)) -> mark(X:S)
 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
 active(plus(N:S,0)) -> mark(N:S)
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.