YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S X3:S)
(RULES
active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
active(b) -> mark(a)
f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
mark(a) -> active(a)
mark(b) -> active(b)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(f(a,X:S,X:S)) -> F(X:S,b,b)
 ACTIVE(f(a,X:S,X:S)) -> MARK(f(X:S,b,b))
 ACTIVE(b) -> MARK(a)
 F(active(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S)
 F(mark(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,mark(X2:S),X3:S))
 MARK(f(X1:S,X2:S,X3:S)) -> F(X1:S,mark(X2:S),X3:S)
 MARK(f(X1:S,X2:S,X3:S)) -> MARK(X2:S)
 MARK(b) -> ACTIVE(b)
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(f(a,X:S,X:S)) -> F(X:S,b,b)
 ACTIVE(f(a,X:S,X:S)) -> MARK(f(X:S,b,b))
 ACTIVE(b) -> MARK(a)
 F(active(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S)
 F(mark(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,mark(X2:S),X3:S))
 MARK(f(X1:S,X2:S,X3:S)) -> F(X1:S,mark(X2:S),X3:S)
 MARK(f(X1:S,X2:S,X3:S)) -> MARK(X2:S)
 MARK(b) -> ACTIVE(b)
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(active(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S)
 F(mark(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
->->-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->->Cycle:
->->-> Pairs:
 ACTIVE(f(a,X:S,X:S)) -> MARK(f(X:S,b,b))
 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,mark(X2:S),X3:S))
 MARK(f(X1:S,X2:S,X3:S)) -> MARK(X2:S)
->->-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(active(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S)
 F(mark(X1:S),X2:S,X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Projection:
 pi(F) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
->->-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(X1:S,active(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,mark(X2:S),X3:S) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Projection:
 pi(F) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
->->-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(X1:S,X2:S,active(X3:S)) -> F(X1:S,X2:S,X3:S)
 F(X1:S,X2:S,mark(X3:S)) -> F(X1:S,X2:S,X3:S)
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Projection:
 pi(F) = 3

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(f(a,X:S,X:S)) -> MARK(f(X:S,b,b))
 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,mark(X2:S),X3:S))
 MARK(f(X1:S,X2:S,X3:S)) -> MARK(X2:S)
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
-> Usable rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[f](X1,X2,X3) = 2.X1 + X2 + X3 + 1
[mark](X) = X
[a] = 1
[b] = 1
[fSNonEmpty] = 0
[ACTIVE](X) = 2.X + 2
[F](X1,X2,X3) = 0
[MARK](X) = 2.X + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 ACTIVE(f(a,X:S,X:S)) -> MARK(f(X:S,b,b))
 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,mark(X2:S),X3:S))
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(f(a,X:S,X:S)) -> MARK(f(X:S,b,b))
 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,mark(X2:S),X3:S))
->->-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(f(a,X:S,X:S)) -> MARK(f(X:S,b,b))
 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,mark(X2:S),X3:S))
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
-> Usable rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[active](X) = [1 1;0 1].X + [1;0]
[f](X1,X2,X3) = [1 0;1 0].X1 + [1 0;1 0].X3 + [1;0]
[mark](X) = [1 1;0 1].X + [1;1]
[a] = [1;0]
[b] = [0;1]
[fSNonEmpty] = 0
[ACTIVE](X) = [1 0;1 1].X + [0;1]
[F](X1,X2,X3) = 0
[MARK](X) = [0 1;1 1].X + [1;1]

Problem 1.2: 

SCC Processor:
-> Pairs:
 MARK(f(X1:S,X2:S,X3:S)) -> ACTIVE(f(X1:S,mark(X2:S),X3:S))
-> Rules:
 active(f(a,X:S,X:S)) -> mark(f(X:S,b,b))
 active(b) -> mark(a)
 f(active(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(mark(X1:S),X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,active(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,mark(X2:S),X3:S) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,active(X3:S)) -> f(X1:S,X2:S,X3:S)
 f(X1:S,X2:S,mark(X3:S)) -> f(X1:S,X2:S,X3:S)
 mark(f(X1:S,X2:S,X3:S)) -> active(f(X1:S,mark(X2:S),X3:S))
 mark(a) -> active(a)
 mark(b) -> active(b)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.