YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
f(0) -> ttrue
f(1) -> ffalse
f(s(x:S)) -> f(x:S)
g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
if(ffalse,x:S,y:S) -> y:S
if(ttrue,x:S,y:S) -> x:S
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(s(x:S)) -> F(x:S)
 G(s(x:S),s(y:S)) -> F(x:S)
 G(s(x:S),s(y:S)) -> IF(f(x:S),s(x:S),s(y:S))
 G(x:S,c(y:S)) -> G(s(c(y:S)),y:S)
 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S))
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(x:S)) -> F(x:S)
 G(s(x:S),s(y:S)) -> F(x:S)
 G(s(x:S),s(y:S)) -> IF(f(x:S),s(x:S),s(y:S))
 G(x:S,c(y:S)) -> G(s(c(y:S)),y:S)
 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S))
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(x:S)) -> F(x:S)
->->-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
->->Cycle:
->->-> Pairs:
 G(x:S,c(y:S)) -> G(s(c(y:S)),y:S)
 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S))
->->-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(s(x:S)) -> F(x:S)
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
->Projection:
 pi(F) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 G(x:S,c(y:S)) -> G(s(c(y:S)),y:S)
 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S))
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
-> Usable rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 0
[g](X1,X2) = 2.X2 + 2
[if](X1,X2,X3) = X2 + 2.X3
[0] = 1
[1] = 0
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[false] = 0
[s](X) = 2
[true] = 0
[F](X) = 0
[G](X1,X2) = 2.X2
[IF](X1,X2,X3) = 0

Problem 1.2: 

SCC Processor:
-> Pairs:
 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S))
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S))
->->-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S))
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
-> Usable rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 2
[g](X1,X2) = X1 + 2.X2
[if](X1,X2,X3) = 2.X2 + X3
[0] = 2
[1] = 1
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[false] = 1
[s](X) = 1
[true] = 1
[F](X) = 0
[G](X1,X2) = 2.X2
[IF](X1,X2,X3) = 0

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S))
 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S))
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.