YES

Problem 1: 

(VAR v_NonEmpty:S f:S g:S x:S)
(RULES
app(app(app(comp,f:S),g:S),x:S) -> app(f:S,app(g:S,x:S))
app(twice,f:S) -> app(app(comp,f:S),f:S)
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(app(app(comp,f:S),g:S),x:S) -> app(f:S,app(g:S,x:S))
 app(twice,f:S) -> app(app(comp,f:S),f: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(app(comp,f:S),g:S),x:S) -> APP(f:S,app(g:S,x:S))
 APP(app(app(comp,f:S),g:S),x:S) -> APP(g:S,x:S)
-> Rules:
 app(app(app(comp,f:S),g:S),x:S) -> app(f:S,app(g:S,x:S))
 app(twice,f:S) -> app(app(comp,f:S),f:S)

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(app(comp,f:S),g:S),x:S) -> APP(f:S,app(g:S,x:S))
 APP(app(app(comp,f:S),g:S),x:S) -> APP(g:S,x:S)
-> Rules:
 app(app(app(comp,f:S),g:S),x:S) -> app(f:S,app(g:S,x:S))
 app(twice,f:S) -> app(app(comp,f:S),f:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(app(comp,f:S),g:S),x:S) -> APP(f:S,app(g:S,x:S))
 APP(app(app(comp,f:S),g:S),x:S) -> APP(g:S,x:S)
->->-> Rules:
 app(app(app(comp,f:S),g:S),x:S) -> app(f:S,app(g:S,x:S))
 app(twice,f:S) -> app(app(comp,f:S),f:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 APP(app(app(comp,f:S),g:S),x:S) -> APP(f:S,app(g:S,x:S))
 APP(app(app(comp,f:S),g:S),x:S) -> APP(g:S,x:S)
-> Rules:
 app(app(app(comp,f:S),g:S),x:S) -> app(f:S,app(g:S,x:S))
 app(twice,f:S) -> app(app(comp,f:S),f:S)
->Projection:
 pi(APP) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(app(comp,f:S),g:S),x:S) -> app(f:S,app(g:S,x:S))
 app(twice,f:S) -> app(app(comp,f:S),f:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.