YES

Problem 1: 

(VAR v_NonEmpty:S k:S l:S x:S y:S z:S)
(RULES
a(h,h,x:S) -> s(x:S)
a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
app(nil,k:S) -> k:S
app(l:S,nil) -> l:S
sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
sum(cons(x:S,nil)) -> cons(x:S,nil)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(s(x:S),h,z:S) -> A(x:S,z:S,z:S)
 A(x:S,s(y:S),h) -> A(x:S,y:S,s(h))
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
 A(x:S,s(y:S),s(z:S)) -> A(x:S,y:S,a(x:S,s(y:S),z:S))
 APP(cons(x:S,l:S),k:S) -> APP(l:S,k:S)
 SUM(cons(x:S,cons(y:S,l:S))) -> A(x:S,y:S,h)
 SUM(cons(x:S,cons(y:S,l:S))) -> SUM(cons(a(x:S,y:S,h),l:S))
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)

Problem 1: 

SCC Processor:
-> Pairs:
 A(s(x:S),h,z:S) -> A(x:S,z:S,z:S)
 A(x:S,s(y:S),h) -> A(x:S,y:S,s(h))
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
 A(x:S,s(y:S),s(z:S)) -> A(x:S,y:S,a(x:S,s(y:S),z:S))
 APP(cons(x:S,l:S),k:S) -> APP(l:S,k:S)
 SUM(cons(x:S,cons(y:S,l:S))) -> A(x:S,y:S,h)
 SUM(cons(x:S,cons(y:S,l:S))) -> SUM(cons(a(x:S,y:S,h),l:S))
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(cons(x:S,l:S),k:S) -> APP(l:S,k:S)
->->-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->->Cycle:
->->-> Pairs:
 A(s(x:S),h,z:S) -> A(x:S,z:S,z:S)
 A(x:S,s(y:S),h) -> A(x:S,y:S,s(h))
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
 A(x:S,s(y:S),s(z:S)) -> A(x:S,y:S,a(x:S,s(y:S),z:S))
->->-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->->Cycle:
->->-> Pairs:
 SUM(cons(x:S,cons(y:S,l:S))) -> SUM(cons(a(x:S,y:S,h),l:S))
->->-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 APP(cons(x:S,l:S),k:S) -> APP(l:S,k:S)
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Projection:
 pi(APP) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 A(s(x:S),h,z:S) -> A(x:S,z:S,z:S)
 A(x:S,s(y:S),h) -> A(x:S,y:S,s(h))
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
 A(x:S,s(y:S),s(z:S)) -> A(x:S,y:S,a(x:S,s(y:S),z:S))
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Projection:
 pi(A) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 A(x:S,s(y:S),h) -> A(x:S,y:S,s(h))
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
 A(x:S,s(y:S),s(z:S)) -> A(x:S,y:S,a(x:S,s(y:S),z:S))
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(x:S,s(y:S),h) -> A(x:S,y:S,s(h))
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
 A(x:S,s(y:S),s(z:S)) -> A(x:S,y:S,a(x:S,s(y:S),z:S))
->->-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)

Problem 1.2: 

Subterm Processor:
-> Pairs:
 A(x:S,s(y:S),h) -> A(x:S,y:S,s(h))
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
 A(x:S,s(y:S),s(z:S)) -> A(x:S,y:S,a(x:S,s(y:S),z:S))
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Projection:
 pi(A) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
->->-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)

Problem 1.2: 

Subterm Processor:
-> Pairs:
 A(x:S,s(y:S),s(z:S)) -> A(x:S,s(y:S),z:S)
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Projection:
 pi(A) = 3

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 SUM(cons(x:S,cons(y:S,l:S))) -> SUM(cons(a(x:S,y:S,h),l:S))
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
-> Usable rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X1,X2,X3) = X3
[cons](X1,X2) = 2.X2 + 2
[h] = 1
[s](X) = X
[SUM](X) = 2.X

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(h,h,x:S) -> s(x:S)
 a(s(x:S),h,z:S) -> a(x:S,z:S,z:S)
 a(x:S,s(y:S),h) -> a(x:S,y:S,s(h))
 a(x:S,s(y:S),s(z:S)) -> a(x:S,y:S,a(x:S,s(y:S),z:S))
 app(cons(x:S,l:S),k:S) -> cons(x:S,app(l:S,k:S))
 app(nil,k:S) -> k:S
 app(l:S,nil) -> l:S
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(a(x:S,y:S,h),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.