YES
(VAR f g x)
(RULES 
app(app(app(compose,f),g),x) -> app(f,app(g,x))
)

The TRS is an overlay system and all critical pairs are trivial, thus termination of innermost rewriting is equivalent to termination of rewriting.

Proving termination of innermost rewriting for Ex2_6_1Composition:

-> Dependency pairs:
nF_app(app(app(compose,f),g),x) -> nF_app(f,app(g,x))
nF_app(app(app(compose,f),g),x) -> nF_app(g,x)

-> Proof of termination for Ex2_6_1Composition_1:
-> -> Dependency pairs in cycle:
nF_app(app(app(compose,f),g),x) -> nF_app(f,app(g,x))
nF_app(app(app(compose,f),g),x) -> nF_app(g,x)

Termination proved: Cycles verify subterm criterion.

SETTINGS:
Base ordering: Polynomial ordering
Proof mode: SCCs in DG + base ordering
Upper bound for coeffs: 1
Rationals below 1 for all non-replacing args: No
Polynomial interpretation: Linear
Coeffs in polynomials: No rationals
Delta: automatic



Termination was proved succesfully.