YES

Problem 1: 

(VAR v_NonEmpty:S a:S b:S c:S t:S x:S y:S z:S)
(RULES
f(t:S,x:S) -> f'(t:S,g(x:S))
f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
foldf(x:S,nil) -> x:S
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(t:S,x:S) -> F'(t:S,g(x:S))
 F(t:S,x:S) -> G(x:S)
 F'(triple(a:S,b:S,c:S),A) -> F''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDF(triple(cons(A,a:S),nil,c:S),b:S)
 F'(triple(a:S,b:S,c:S),B) -> F(triple(a:S,b:S,c:S),A)
 F''(triple(a:S,b:S,c:S)) -> FOLDF(triple(a:S,b:S,nil),c:S)
 FOLDF(x:S,cons(y:S,z:S)) -> F(foldf(x:S,z:S),y:S)
 FOLDF(x:S,cons(y:S,z:S)) -> FOLDF(x:S,z:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
 f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
 f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
 foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
 foldf(x:S,nil) -> x:S
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C

Problem 1: 

SCC Processor:
-> Pairs:
 F(t:S,x:S) -> F'(t:S,g(x:S))
 F(t:S,x:S) -> G(x:S)
 F'(triple(a:S,b:S,c:S),A) -> F''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDF(triple(cons(A,a:S),nil,c:S),b:S)
 F'(triple(a:S,b:S,c:S),B) -> F(triple(a:S,b:S,c:S),A)
 F''(triple(a:S,b:S,c:S)) -> FOLDF(triple(a:S,b:S,nil),c:S)
 FOLDF(x:S,cons(y:S,z:S)) -> F(foldf(x:S,z:S),y:S)
 FOLDF(x:S,cons(y:S,z:S)) -> FOLDF(x:S,z:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
 f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
 f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
 foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
 foldf(x:S,nil) -> x:S
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(t:S,x:S) -> F'(t:S,g(x:S))
 F'(triple(a:S,b:S,c:S),A) -> F''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDF(triple(cons(A,a:S),nil,c:S),b:S)
 F'(triple(a:S,b:S,c:S),B) -> F(triple(a:S,b:S,c:S),A)
 F''(triple(a:S,b:S,c:S)) -> FOLDF(triple(a:S,b:S,nil),c:S)
 FOLDF(x:S,cons(y:S,z:S)) -> F(foldf(x:S,z:S),y:S)
 FOLDF(x:S,cons(y:S,z:S)) -> FOLDF(x:S,z:S)
->->-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
 f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
 f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
 foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
 foldf(x:S,nil) -> x:S
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 F(t:S,x:S) -> F'(t:S,g(x:S))
 F'(triple(a:S,b:S,c:S),A) -> F''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDF(triple(cons(A,a:S),nil,c:S),b:S)
 F'(triple(a:S,b:S,c:S),B) -> F(triple(a:S,b:S,c:S),A)
 F''(triple(a:S,b:S,c:S)) -> FOLDF(triple(a:S,b:S,nil),c:S)
 FOLDF(x:S,cons(y:S,z:S)) -> F(foldf(x:S,z:S),y:S)
 FOLDF(x:S,cons(y:S,z:S)) -> FOLDF(x:S,z:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
 f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
 f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
 foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
 foldf(x:S,nil) -> x:S
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
-> Usable rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
 f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
 f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
 foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
 foldf(x:S,nil) -> x:S
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X1,X2) = X1 + 2.X2 + 2
[f'](X1,X2) = X1 + 2.X2 + 2
[f''](X) = X + 2
[foldf](X1,X2) = X1 + X2
[g](X) = X
[A] = 0
[B] = 2
[C] = 2
[cons](X1,X2) = 2.X1 + X2 + 2
[nil] = 0
[triple](X1,X2,X3) = 2.X2 + X3 + 2
[F](X1,X2) = X1 + 2.X2 + 2
[F'](X1,X2) = X1 + 2.X2 + 1
[F''](X) = X + 1
[FOLDF](X1,X2) = X1 + X2 + 1

Problem 1: 

SCC Processor:
-> Pairs:
 F'(triple(a:S,b:S,c:S),A) -> F''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDF(triple(cons(A,a:S),nil,c:S),b:S)
 F'(triple(a:S,b:S,c:S),B) -> F(triple(a:S,b:S,c:S),A)
 F''(triple(a:S,b:S,c:S)) -> FOLDF(triple(a:S,b:S,nil),c:S)
 FOLDF(x:S,cons(y:S,z:S)) -> F(foldf(x:S,z:S),y:S)
 FOLDF(x:S,cons(y:S,z:S)) -> FOLDF(x:S,z:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
 f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
 f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
 foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
 foldf(x:S,nil) -> x:S
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 FOLDF(x:S,cons(y:S,z:S)) -> FOLDF(x:S,z:S)
->->-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
 f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
 f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
 foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
 foldf(x:S,nil) -> x:S
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C

Problem 1: 

Subterm Processor:
-> Pairs:
 FOLDF(x:S,cons(y:S,z:S)) -> FOLDF(x:S,z:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
 f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
 f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
 foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
 foldf(x:S,nil) -> x:S
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
->Projection:
 pi(FOLDF) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldf(triple(cons(A,a:S),nil,c:S),b:S))
 f'(triple(a:S,b:S,c:S),B) -> f(triple(a:S,b:S,c:S),A)
 f'(triple(a:S,b:S,c:S),C) -> triple(a:S,b:S,cons(C,c:S))
 f''(triple(a:S,b:S,c:S)) -> foldf(triple(a:S,b:S,nil),c:S)
 foldf(x:S,cons(y:S,z:S)) -> f(foldf(x:S,z:S),y:S)
 foldf(x:S,nil) -> x:S
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.