YES

Problem 1: 

(VAR v_NonEmpty:S k:S l:S x:S y:S)
(RULES
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
plus(0,y:S) -> y:S
plus(s(x:S),y:S) -> s(plus(x:S,y:S))
pred(cons(s(x:S),nil)) -> cons(x:S,nil)
sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
sum(cons(x:S,nil)) -> cons(x:S,nil)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(cons(x:S,l:S),k:S) -> APP(l:S,k:S)
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 SUM(app(l:S,cons(x:S,cons(y:S,k:S)))) -> APP(l:S,sum(cons(x:S,cons(y:S,k:S))))
 SUM(app(l:S,cons(x:S,cons(y:S,k:S)))) -> SUM(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 SUM(app(l:S,cons(x:S,cons(y:S,k:S)))) -> SUM(cons(x:S,cons(y:S,k:S)))
 SUM(plus(cons(0,x:S),cons(y:S,l:S))) -> PRED(sum(cons(s(x:S),cons(y:S,l:S))))
 SUM(plus(cons(0,x:S),cons(y:S,l:S))) -> SUM(cons(s(x:S),cons(y:S,l:S)))
 SUM(cons(x:S,cons(y:S,l:S))) -> PLUS(x:S,y:S)
 SUM(cons(x:S,cons(y:S,l:S))) -> SUM(cons(plus(x:S,y:S),l:S))
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)

Problem 1: 

SCC Processor:
-> Pairs:
 APP(cons(x:S,l:S),k:S) -> APP(l:S,k:S)
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 SUM(app(l:S,cons(x:S,cons(y:S,k:S)))) -> APP(l:S,sum(cons(x:S,cons(y:S,k:S))))
 SUM(app(l:S,cons(x:S,cons(y:S,k:S)))) -> SUM(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 SUM(app(l:S,cons(x:S,cons(y:S,k:S)))) -> SUM(cons(x:S,cons(y:S,k:S)))
 SUM(plus(cons(0,x:S),cons(y:S,l:S))) -> PRED(sum(cons(s(x:S),cons(y:S,l:S))))
 SUM(plus(cons(0,x:S),cons(y:S,l:S))) -> SUM(cons(s(x:S),cons(y:S,l:S)))
 SUM(cons(x:S,cons(y:S,l:S))) -> PLUS(x:S,y:S)
 SUM(cons(x:S,cons(y:S,l:S))) -> SUM(cons(plus(x:S,y:S),l:S))
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
->->-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->->Cycle:
->->-> Pairs:
 SUM(cons(x:S,cons(y:S,l:S))) -> SUM(cons(plus(x:S,y:S),l:S))
->->-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->->Cycle:
->->-> Pairs:
 APP(cons(x:S,l:S),k:S) -> APP(l:S,k:S)
->->-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->->Cycle:
->->-> Pairs:
 SUM(app(l:S,cons(x:S,cons(y:S,k:S)))) -> SUM(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
->->-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),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: 

Reduction Pair Processor:
-> Pairs:
 SUM(cons(x:S,cons(y:S,l:S))) -> SUM(cons(plus(x:S,y:S),l:S))
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
-> Usable rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X2
[0] = 0
[cons](X1,X2) = 2.X1 + X2 + 2
[s](X) = X
[SUM](X) = 2.X

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),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: 

Subterm Processor:
-> Pairs:
 APP(cons(x:S,l:S),k:S) -> APP(l:S,k:S)
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Projection:
 pi(APP) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),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 1.4: 

Reduction Pair Processor:
-> Pairs:
 SUM(app(l:S,cons(x:S,cons(y:S,k:S)))) -> SUM(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
-> Usable rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 2.X1 + 2.X2 + 2
[plus](X1,X2) = X2 + 2
[pred](X) = 2
[sum](X) = 2
[0] = 2
[cons](X1,X2) = X2 + 2
[nil] = 0
[s](X) = 2
[SUM](X) = 2.X

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 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
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 pred(cons(s(x:S),nil)) -> cons(x:S,nil)
 sum(app(l:S,cons(x:S,cons(y:S,k:S)))) -> sum(app(l:S,sum(cons(x:S,cons(y:S,k:S)))))
 sum(plus(cons(0,x:S),cons(y:S,l:S))) -> pred(sum(cons(s(x:S),cons(y:S,l:S))))
 sum(cons(x:S,cons(y:S,l:S))) -> sum(cons(plus(x:S,y:S),l:S))
 sum(cons(x:S,nil)) -> cons(x:S,nil)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.