YES
(VAR y x)
(RULES 
le(0,y) -> true
le(s(x),0) -> false
le(s(x),s(y)) -> le(x,y)
minus(0,y) -> 0
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
if_minus(true,s(x),y) -> 0
if_minus(false,s(x),y) -> s(minus(x,y))
mod(0,y) -> 0
mod(s(x),0) -> 0
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
if_mod(true,s(x),s(y)) -> mod(minus(x,y),s(y))
if_mod(false,s(x),s(y)) -> s(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 _3_5b:

-> Dependency pairs:
nF_le(s(x),s(y)) -> nF_le(x,y)
nF_minus(s(x),y) -> nF_if_minus(le(s(x),y),s(x),y)
nF_minus(s(x),y) -> nF_le(s(x),y)
nF_if_minus(false,s(x),y) -> nF_minus(x,y)
nF_mod(s(x),s(y)) -> nF_if_mod(le(y,x),s(x),s(y))
nF_mod(s(x),s(y)) -> nF_le(y,x)
nF_if_mod(true,s(x),s(y)) -> nF_mod(minus(x,y),s(y))
nF_if_mod(true,s(x),s(y)) -> nF_minus(x,y)

-> Proof of termination for _3_5b_1_1:
-> -> Dependency pairs in cycle:
nF_mod(s(x),s(y)) -> nF_if_mod(le(y,x),s(x),s(y))
nF_if_mod(true,s(x),s(y)) -> nF_mod(minus(x,y),s(y))

UsableRules:
le(0,y) -> true
le(s(x),0) -> false
le(s(x),s(y)) -> le(x,y)
minus(0,y) -> 0
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
if_minus(true,s(x),y) -> 0
if_minus(false,s(x),y) -> s(minus(x,y))

Polynomial Interpretation:

[le](X1,X2) = X1 + X2
[0] = 1
[true] = 0
[s](X) = X + 1
[false] = 0
[minus](X1,X2) = X1
[if_minus](X1,X2,X3) = X2
[mod](X1,X2) = 0
[if_mod](X1,X2,X3) = 0
[nF_mod](X1,X2) = X1
[nF_if_mod](X1,X2,X3) = X2

TIME: 6.4284e-2


-> Proof of termination for _3_5b_1_2:
-> -> Dependency pairs in cycle:
nF_minus(s(x),y) -> nF_if_minus(le(s(x),y),s(x),y)
nF_if_minus(false,s(x),y) -> nF_minus(x,y)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for _3_5b_1_3:
-> -> Dependency pairs in cycle:
nF_le(s(x),s(y)) -> nF_le(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.