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: pred(s(x)) -> x
5: minus(x,0) -> x
6: minus(x,s(y)) -> pred(minus(x,y))
7: mod(0,y) -> 0
8: mod(s(x),0) -> 0
9: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
10: if_mod(true,s(x),s(y)) -> mod(minus(x,y),s(y))
11: if_mod(false,s(x),s(y)) -> s(x)

There are 7 dependency pairs:

12: LE(s(x),s(y)) -> LE(x,y)
13: MINUS(x,s(y)) -> PRED(minus(x,y))
14: MINUS(x,s(y)) -> MINUS(x,y)
15: MOD(s(x),s(y)) -> IF_MOD(le(y,x),s(x),s(y))
16: MOD(s(x),s(y)) -> LE(y,x)
17: IF_MOD(true,s(x),s(y)) -> MOD(minus(x,y),s(y))
18: IF_MOD(true,s(x),s(y)) -> MINUS(x,y)

The approximated dependency graph contains 3 SCCs:
{12},
{14}
and {15,17}.

- Consider the SCC {12}.
There are no usable rules.
By taking the polynomial interpretation
[s](x) = x + 1
and [LE](x,y) = x + y + 1,
rule 12
is strictly decreasing.

- Consider the SCC {14}.
There are no usable rules.
By taking the polynomial interpretation
[s](x) = x + 1
and [MINUS](x,y) = x + y + 1,
rule 14
is strictly decreasing.

- Consider the SCC {15,17}.
The usable rules are {1-6}.
By taking the polynomial interpretation
[0] = [false] = [true] = 1,
[minus](x,y) = [pred](x) = x,
[s](x) = x + 1,
[le](x,y) = [MOD](x,y) = x + y + 1
and [IF_MOD](x,y,z) = y + z + 1,
the rules in {5,6,15}
are weakly decreasing and
the rules in {1-4,17}
are strictly decreasing.

Hence the TRS is terminating.