YES
(VAR x y u z)
(RULES 
perfectp(0) -> false
perfectp(s(x)) -> f(x,s(0),s(x),s(x))
f(0,y,0,u) -> true
f(0,y,s(z),u) -> false
f(s(x),0,z,u) -> f(x,u,minus(z,s(x)),u)
f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
)

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 perfect:

-> Dependency pairs:
nF_perfectp(s(x)) -> nF_f(x,s(0),s(x),s(x))
nF_f(s(x),0,z,u) -> nF_f(x,u,minus(z,s(x)),u)
nF_f(s(x),s(y),z,u) -> nF_f(s(x),minus(y,x),z,u)
nF_f(s(x),s(y),z,u) -> nF_f(x,u,z,u)

-> Proof of termination for perfect_1:
-> -> Dependency pairs in cycle:
nF_f(s(x),s(y),z,u) -> nF_f(x,u,z,u)
nF_f(s(x),0,z,u) -> nF_f(x,u,minus(z,s(x)),u)

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.