YES

Problem 1: 

(VAR v_NonEmpty:S f:S x:S xs:S y:S)
(RULES
app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
app(app(map,f:S),nil) -> nil
app(add,0) -> id
app(id,x:S) -> x:S
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(app(add,app(s,x:S)),y:S) -> APP(app(add,x:S),y:S)
 APP(app(add,app(s,x:S)),y:S) -> APP(add,x:S)
 APP(app(add,app(s,x:S)),y:S) -> APP(s,app(app(add,x:S),y:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(f:S,x:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
-> Rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(add,app(s,x:S)),y:S) -> APP(app(add,x:S),y:S)
 APP(app(add,app(s,x:S)),y:S) -> APP(add,x:S)
 APP(app(add,app(s,x:S)),y:S) -> APP(s,app(app(add,x:S),y:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(f:S,x:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
-> Rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(add,app(s,x:S)),y:S) -> APP(app(add,x:S),y:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
->->-> Rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S

Problem 1: 

Subterm Processor:
-> Pairs:
 APP(app(add,app(s,x:S)),y:S) -> APP(app(add,x:S),y:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
-> Rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S
->Projection:
 pi(APP) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(add,app(s,x:S)),y:S) -> APP(app(add,x:S),y:S)
-> Rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(add,app(s,x:S)),y:S) -> APP(app(add,x:S),y:S)
->->-> Rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S

Problem 1: 

Reduction Pairs Processor:
-> Pairs:
 APP(app(add,app(s,x:S)),y:S) -> APP(app(add,x:S),y:S)
-> Rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S
-> Usable rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = X1.X2 + 2.X1 + 2
[0] = 0
[add] = 1
[cons] = 0
[fSNonEmpty] = 0
[id] = 2
[map] = 2
[nil] = 2
[s] = 1
[APP](X1,X2) = X1.X2 + 2.X1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(add,app(s,x:S)),y:S) -> app(s,app(app(add,x:S),y:S))
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(add,0) -> id
 app(id,x:S) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.