YES

Problem 1: 

(VAR v_NonEmpty:S a:S b:S c:S n:S t:S x:S)
(RULES
f(t:S,x:S) -> f'(t:S,g(x:S))
f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
foldB(t:S,0) -> t:S
foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
foldC(t:S,0) -> t:S
foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
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''(foldB(triple(s(a:S),0,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDB(triple(s(a:S),0,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)) -> FOLDC(triple(a:S,b:S,0),c:S)
 FOLDB(t:S,s(n:S)) -> F(foldB(t:S,n:S),B)
 FOLDB(t:S,s(n:S)) -> FOLDB(t:S,n:S)
 FOLDC(t:S,s(n:S)) -> F(foldC(t:S,n:S),C)
 FOLDC(t:S,s(n:S)) -> FOLDC(t:S,n:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 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''(foldB(triple(s(a:S),0,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDB(triple(s(a:S),0,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)) -> FOLDC(triple(a:S,b:S,0),c:S)
 FOLDB(t:S,s(n:S)) -> F(foldB(t:S,n:S),B)
 FOLDB(t:S,s(n:S)) -> FOLDB(t:S,n:S)
 FOLDC(t:S,s(n:S)) -> F(foldC(t:S,n:S),C)
 FOLDC(t:S,s(n:S)) -> FOLDC(t:S,n:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 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''(foldB(triple(s(a:S),0,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDB(triple(s(a:S),0,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)) -> FOLDC(triple(a:S,b:S,0),c:S)
 FOLDB(t:S,s(n:S)) -> F(foldB(t:S,n:S),B)
 FOLDB(t:S,s(n:S)) -> FOLDB(t:S,n:S)
 FOLDC(t:S,s(n:S)) -> F(foldC(t:S,n:S),C)
 FOLDC(t:S,s(n:S)) -> FOLDC(t:S,n:S)
->->-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 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''(foldB(triple(s(a:S),0,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDB(triple(s(a:S),0,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)) -> FOLDC(triple(a:S,b:S,0),c:S)
 FOLDB(t:S,s(n:S)) -> F(foldB(t:S,n:S),B)
 FOLDB(t:S,s(n:S)) -> FOLDB(t:S,n:S)
 FOLDC(t:S,s(n:S)) -> F(foldC(t:S,n:S),C)
 FOLDC(t:S,s(n:S)) -> FOLDC(t:S,n:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 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''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 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 + X2 + 2
[f'](X1,X2) = X1 + X2 + 2
[f''](X) = X + 2
[foldB](X1,X2) = X1 + 2.X2
[foldC](X1,X2) = X1 + 2.X2
[g](X) = X
[0] = 0
[A] = 1
[B] = 2
[C] = 2
[s](X) = X + 2
[triple](X1,X2,X3) = 2.X2 + 2.X3 + 2
[F](X1,X2) = X1 + 2.X2 + 2
[F'](X1,X2) = X1 + 2.X2 + 1
[F''](X) = X + 2
[FOLDB](X1,X2) = X1 + 2.X2 + 2
[FOLDC](X1,X2) = X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 F'(triple(a:S,b:S,c:S),A) -> F''(foldB(triple(s(a:S),0,c:S),b:S))
 F'(triple(a:S,b:S,c:S),A) -> FOLDB(triple(s(a:S),0,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)) -> FOLDC(triple(a:S,b:S,0),c:S)
 FOLDB(t:S,s(n:S)) -> F(foldB(t:S,n:S),B)
 FOLDB(t:S,s(n:S)) -> FOLDB(t:S,n:S)
 FOLDC(t:S,s(n:S)) -> F(foldC(t:S,n:S),C)
 FOLDC(t:S,s(n:S)) -> FOLDC(t:S,n:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 FOLDC(t:S,s(n:S)) -> FOLDC(t:S,n:S)
->->-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
->->Cycle:
->->-> Pairs:
 FOLDB(t:S,s(n:S)) -> FOLDB(t:S,n:S)
->->-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 FOLDC(t:S,s(n:S)) -> FOLDC(t:S,n:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
->Projection:
 pi(FOLDC) = 2

Problem 1.1: 

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''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 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.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 FOLDB(t:S,s(n:S)) -> FOLDB(t:S,n:S)
-> Rules:
 f(t:S,x:S) -> f'(t:S,g(x:S))
 f'(triple(a:S,b:S,c:S),A) -> f''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 g(A) -> A
 g(B) -> A
 g(B) -> B
 g(C) -> A
 g(C) -> B
 g(C) -> C
->Projection:
 pi(FOLDB) = 2

Problem 1.2: 

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''(foldB(triple(s(a:S),0,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,s(c:S))
 f''(triple(a:S,b:S,c:S)) -> foldC(triple(a:S,b:S,0),c:S)
 foldB(t:S,0) -> t:S
 foldB(t:S,s(n:S)) -> f(foldB(t:S,n:S),B)
 foldC(t:S,0) -> t:S
 foldC(t:S,s(n:S)) -> f(foldC(t:S,n:S),C)
 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.