YES

Problem 1: 

(VAR v_NonEmpty:S X:S)
(RULES
active(f(g(X:S))) -> mark(g(X:S))
active(c) -> mark(f(g(c)))
f(ok(X:S)) -> ok(f(X:S))
g(ok(X:S)) -> ok(g(X:S))
proper(f(X:S)) -> f(proper(X:S))
proper(g(X:S)) -> g(proper(X:S))
proper(c) -> ok(c)
top(mark(X:S)) -> top(proper(X:S))
top(ok(X:S)) -> top(active(X:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(ok(X:S)) -> F(X:S)
 G(ok(X:S)) -> G(X:S)
 PROPER(f(X:S)) -> F(proper(X:S))
 PROPER(f(X:S)) -> PROPER(X:S)
 PROPER(g(X:S)) -> G(proper(X:S))
 PROPER(g(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:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))

Problem 1: 

SCC Processor:
-> Pairs:
 F(ok(X:S)) -> F(X:S)
 G(ok(X:S)) -> G(X:S)
 PROPER(f(X:S)) -> F(proper(X:S))
 PROPER(f(X:S)) -> PROPER(X:S)
 PROPER(g(X:S)) -> G(proper(X:S))
 PROPER(g(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:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(ok(X:S)) -> G(X:S)
->->-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 F(ok(X:S)) -> F(X:S)
->->-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 PROPER(f(X:S)) -> PROPER(X:S)
 PROPER(g(X:S)) -> PROPER(X:S)
->->-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 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(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 G(ok(X:S)) -> G(X:S)
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(G) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 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:
 F(ok(X:S)) -> F(X:S)
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(F) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 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:
 PROPER(f(X:S)) -> PROPER(X:S)
 PROPER(g(X:S)) -> PROPER(X:S)
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(PROPER) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 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: 

Reduction Pairs Processor:
-> Pairs:
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
-> Usable rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[f](X) = 1
[g](X) = 0
[proper](X) = X
[top](X) = 0
[c] = 2
[fSNonEmpty] = 0
[mark](X) = X + 1
[ok](X) = X
[ACTIVE](X) = 0
[F](X) = 0
[G](X) = 0
[PROPER](X) = 0
[TOP](X) = 2.X

Problem 1.4: 

SCC Processor:
-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 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(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
-> Usable rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X + 1
[f](X) = X + 2
[g](X) = X
[proper](X) = 0
[top](X) = 0
[c] = 1
[fSNonEmpty] = 0
[mark](X) = 2
[ok](X) = X + 2
[ACTIVE](X) = 0
[F](X) = 0
[G](X) = 0
[PROPER](X) = 0
[TOP](X) = X

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(ok(X:S)) -> ok(f(X:S))
 g(ok(X:S)) -> ok(g(X:S))
 proper(f(X:S)) -> f(proper(X:S))
 proper(g(X:S)) -> g(proper(X:S))
 proper(c) -> ok(c)
 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.