YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__F(a,b,X:S) -> A__F(X:S,X:S,mark(X:S))
 A__F(a,b,X:S) -> MARK(X:S)
 MARK(c) -> A__C
 MARK(f(X1:S,X2:S,X3:S)) -> A__F(X1:S,X2:S,mark(X3:S))
 MARK(f(X1:S,X2:S,X3:S)) -> MARK(X3:S)
-> Rules:
 a__c -> a
 a__c -> b
 a__c -> c
 a__f(a,b,X:S) -> a__f(X:S,X:S,mark(X:S))
 a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 mark(a) -> a
 mark(b) -> b
 mark(c) -> a__c
 mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,mark(X3:S))

Problem 1: 

SCC Processor:
-> Pairs:
 A__F(a,b,X:S) -> A__F(X:S,X:S,mark(X:S))
 A__F(a,b,X:S) -> MARK(X:S)
 MARK(c) -> A__C
 MARK(f(X1:S,X2:S,X3:S)) -> A__F(X1:S,X2:S,mark(X3:S))
 MARK(f(X1:S,X2:S,X3:S)) -> MARK(X3:S)
-> Rules:
 a__c -> a
 a__c -> b
 a__c -> c
 a__f(a,b,X:S) -> a__f(X:S,X:S,mark(X:S))
 a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 mark(a) -> a
 mark(b) -> b
 mark(c) -> a__c
 mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,mark(X3:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__F(a,b,X:S) -> MARK(X:S)
 MARK(f(X1:S,X2:S,X3:S)) -> A__F(X1:S,X2:S,mark(X3:S))
 MARK(f(X1:S,X2:S,X3:S)) -> MARK(X3:S)
->->-> Rules:
 a__c -> a
 a__c -> b
 a__c -> c
 a__f(a,b,X:S) -> a__f(X:S,X:S,mark(X:S))
 a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 mark(a) -> a
 mark(b) -> b
 mark(c) -> a__c
 mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,mark(X3:S))

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Subterm Processor:
-> Pairs:
 MARK(f(X1:S,X2:S,X3:S)) -> MARK(X3:S)
-> Rules:
 a__c -> a
 a__c -> b
 a__c -> c
 a__f(a,b,X:S) -> a__f(X:S,X:S,mark(X:S))
 a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 mark(a) -> a
 mark(b) -> b
 mark(c) -> a__c
 mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,mark(X3:S))
->Projection:
 pi(MARK) = 1

Problem 1: 

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

The problem is finite.