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

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 Ex1SimplyTyped:

-> Dependency pairs:
nF_app(app(add,app(s,x)),y) -> nF_app(s,app(app(add,x),y))
nF_app(app(add,app(s,x)),y) -> nF_app(app(add,x),y)
nF_app(app(add,app(s,x)),y) -> nF_app(add,x)
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(app(cons,app(f,x)),app(app(map,f),xs))
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(cons,app(f,x))
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(f,x)
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(app(map,f),xs)
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(map,f)

-> Proof of termination for Ex1SimplyTyped_1:
-> -> Dependency pairs in cycle:
nF_app(app(add,app(s,x)),y) -> nF_app(app(add,x),y)
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(f,x)
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(app(map,f),xs)
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(app(cons,app(f,x)),app(app(map,f),xs))

UsableRules:
app(id,x) -> x
app(add,0) -> id
app(app(add,app(s,x)),y) -> app(s,app(app(add,x),y))
app(app(map,f),nil) -> nil
app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))

Polynomial Interpretation:

[app](X1,X2) = X1.X2 + X1
[id] = 1
[add] = 1
[0] = 1
[s] = 1
[map] = 1
[nil] = 1
[cons] = 0
[nF_app](X1,X2) = X1

TIME: 0.304075

-> -> Dependency pairs in cycle:
nF_app(app(add,app(s,x)),y) -> nF_app(app(add,x),y)
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(app(map,f),xs)
nF_app(app(map,f),app(app(cons,x),xs)) -> nF_app(f,x)

Dependency pairs oriented using subterm criterion.

-> -> Dependency pairs in cycle:
nF_app(app(add,app(s,x)),y) -> nF_app(app(add,x),y)

UsableRules:
app(id,x) -> x
app(add,0) -> id
app(app(add,app(s,x)),y) -> app(s,app(app(add,x),y))
app(app(map,f),nil) -> nil
app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))

Polynomial Interpretation:

[app](X1,X2) = X1.X2 + X1
[id] = 1
[add] = 1
[0] = 1
[s] = 1
[map] = 0
[nil] = 0
[cons] = 0
[nF_app](X1,X2) = X1

TIME: 0.101063


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: Simple mixed
Coeffs in polynomials: No rationals
Delta: automatic



Termination was proved succesfully.