YES
(VAR X Y)
(RULES 
minus(X,0) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
div(0,s(Y)) -> 0
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
)

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 gm:

-> Dependency pairs:
nF_minus(s(X),s(Y)) -> nF_p(minus(X,Y))
nF_minus(s(X),s(Y)) -> nF_minus(X,Y)
nF_div(s(X),s(Y)) -> nF_div(minus(X,Y),s(Y))
nF_div(s(X),s(Y)) -> nF_minus(X,Y)

-> Proof of termination for gm_1_1:
-> -> Dependency pairs in cycle:
nF_div(s(X),s(Y)) -> nF_div(minus(X,Y),s(Y))

UsableRules:
minus(X,0) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X

Polynomial Interpretation:

[minus](X1,X2) = X1
[0] = 0
[s](X) = X + 1
[p](X) = X
[div](X1,X2) = 0
[nF_div](X1,X2) = X1

TIME: 7.7429e-2


-> Proof of termination for gm_1_2:
-> -> 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.