YES

Problem 1: 

(VAR v_NonEmpty:S m:S n:S x:S y:S)
(RULES
sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
sum(nil,y:S) -> y:S
weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
weight(cons(n:S,nil)) -> n:S
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S)
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
 WEIGHT(cons(n:S,cons(m:S,x:S))) -> SUM(cons(n:S,cons(m:S,x:S)),cons(0,x:S))
 WEIGHT(cons(n:S,cons(m:S,x:S))) -> WEIGHT(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S

Problem 1: 

SCC Processor:
-> Pairs:
 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S)
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
 WEIGHT(cons(n:S,cons(m:S,x:S))) -> SUM(cons(n:S,cons(m:S,x:S)),cons(0,x:S))
 WEIGHT(cons(n:S,cons(m:S,x:S))) -> WEIGHT(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S)
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
->->-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S
->->Cycle:
->->-> Pairs:
 WEIGHT(cons(n:S,cons(m:S,x:S))) -> WEIGHT(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
->->-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S)
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[sum](X1,X2) = 0
[weight](X) = 0
[0] = 2
[cons](X1,X2) = 2.X2 + 2
[fSNonEmpty] = 0
[nil] = 0
[s](X) = 2.X + 2
[SUM](X1,X2) = 2.X1
[WEIGHT](X) = 0

Problem 1.1: 

SCC Processor:
-> Pairs:
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
->->-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[sum](X1,X2) = 0
[weight](X) = 0
[0] = 0
[cons](X1,X2) = 2.X1
[fSNonEmpty] = 0
[nil] = 0
[s](X) = 2.X + 2
[SUM](X1,X2) = 2.X1
[WEIGHT](X) = 0

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 WEIGHT(cons(n:S,cons(m:S,x:S))) -> WEIGHT(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S
-> Usable rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[sum](X1,X2) = X2 + 1
[weight](X) = 0
[0] = 1
[cons](X1,X2) = 2.X1 + 2.X2 + 2
[fSNonEmpty] = 0
[nil] = 0
[s](X) = 0
[SUM](X1,X2) = 0
[WEIGHT](X) = 2.X

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S)))
 weight(cons(n:S,nil)) -> n:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.