Consider the TRS R consisting of the rewrite rules

1: le(0,y) -> true
2: le(s(x),0) -> false
3: le(s(x),s(y)) -> le(x,y)
4: minus(0,y) -> 0
5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
6: if_minus(true,s(x),y) -> 0
7: if_minus(false,s(x),y) -> s(minus(x,y))
8: mod(0,y) -> 0
9: mod(s(x),0) -> 0
10: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
11: if_mod(true,s(x),s(y)) -> mod(minus(x,y),s(y))
12: if_mod(false,s(x),s(y)) -> s(x)

There are 8 dependency pairs:

13: LE(s(x),s(y)) -> LE(x,y)
14: MINUS(s(x),y) -> IF_MINUS(le(s(x),y),s(x),y)
15: MINUS(s(x),y) -> LE(s(x),y)
16: IF_MINUS(false,s(x),y) -> MINUS(x,y)
17: MOD(s(x),s(y)) -> IF_MOD(le(y,x),s(x),s(y))
18: MOD(s(x),s(y)) -> LE(y,x)
19: IF_MOD(true,s(x),s(y)) -> MOD(minus(x,y),s(y))
20: IF_MOD(true,s(x),s(y)) -> MINUS(x,y)

The approximated dependency graph contains 3 SCCs:
{13},
{14,16}
and {17,19}.

- Consider the SCC {13}.
There are no usable rules.
By taking the polynomial interpretation
[s](x) = x + 1
and [LE](x,y) = x + y + 1,
rule 13
is strictly decreasing.

- Consider the SCC {14,16}.
The usable rules are {1-3}.
By taking the polynomial interpretation
[0] = [false] = [true] = 1,
[s](x) = x + 1,
[le](x,y) = [MINUS](x,y) = x + y + 1
and [IF_MINUS](x,y,z) = y + z + 1,
rule 14
is weakly decreasing and
the rules in {1-3,16}
are strictly decreasing.

- Consider the SCC {17,19}.
The usable rules are {1-7}.
By taking the polynomial interpretation
[0] = [false] = [true] = 1,
[minus](x,y) = x,
[s](x) = x + 1,
[le](x,y) = [MOD](x,y) = x + y + 1,
[if_minus](x,y,z) = y
and [IF_MOD](x,y,z) = y + z + 1,
the rules in {4-7,17}
are weakly decreasing and
the rules in {1-3,19}
are strictly decreasing.

Hence the TRS is terminating.