YES
(VAR y x u z)
(RULES 
minus(0,y) -> 0
minus(s(x),0) -> s(x)
minus(s(x),s(y)) -> minus(x,y)
le(0,y) -> true
le(s(x),0) -> false
le(s(x),s(y)) -> le(x,y)
if(true,x,y) -> x
if(false,x,y) -> y
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 perfect2:

-> Dependency pairs:
nF_minus(s(x),s(y)) -> nF_minus(x,y)
nF_le(s(x),s(y)) -> nF_le(x,y)
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),0,z,u) -> nF_minus(z,s(x))
nF_f(s(x),s(y),z,u) -> nF_if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
nF_f(s(x),s(y),z,u) -> nF_le(x,y)
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_minus(y,x)
nF_f(s(x),s(y),z,u) -> nF_f(x,u,z,u)

-> Proof of termination for perfect2_1_1:
-> -> Dependency pairs in cycle:
nF_f(s(x),s(y),z,u) -> nF_f(x,u,z,u)
nF_f(s(x),s(y),z,u) -> nF_f(s(x),minus(y,x),z,u)
nF_f(s(x),0,z,u) -> nF_f(x,u,minus(z,s(x)),u)

Dependency pairs oriented using subterm criterion.

-> -> Dependency pairs in cycle:
nF_f(s(x),s(y),z,u) -> nF_f(s(x),minus(y,x),z,u)

UsableRules:
minus(0,y) -> 0
minus(s(x),0) -> s(x)
minus(s(x),s(y)) -> minus(x,y)

Polynomial Interpretation:

[minus](X1,X2) = X1
[0] = 1
[s](X) = X + 1
[le](X1,X2) = 0
[true] = 0
[false] = 0
[if](X1,X2,X3) = 0
[perfectp](X) = 0
[f](X1,X2,X3,X4) = 0
[nF_f](X1,X2,X3,X4) = X2

TIME: 5.3332e-2


-> Proof of termination for perfect2_1_2:
-> -> Dependency pairs in cycle:
nF_le(s(x),s(y)) -> nF_le(x,y)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for perfect2_1_3:
-> -> Dependency pairs in cycle:
nF_minus(s(x),s(y)) -> nF_minus(x,y)

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.