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: quot(0,s(y)) -> 0
9: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
10: log(s(0)) -> 0
11: log(s(s(x))) -> s(log(s(quot(x,s(s(0))))))

There are 8 dependency pairs:

12: LE(s(x),s(y)) -> LE(x,y)
13: MINUS(s(x),y) -> IF_MINUS(le(s(x),y),s(x),y)
14: MINUS(s(x),y) -> LE(s(x),y)
15: IF_MINUS(false,s(x),y) -> MINUS(x,y)
16: QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))
17: QUOT(s(x),s(y)) -> MINUS(x,y)
18: LOG(s(s(x))) -> LOG(s(quot(x,s(s(0)))))
19: LOG(s(s(x))) -> QUOT(x,s(s(0)))

The approximated dependency graph contains 4 SCCs:
{12},
{13,15},
{16}
and {18}.

- 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 {13,15}.
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 13
is weakly decreasing and
the rules in {1-3,15}
are strictly decreasing.

- Consider the SCC {16}.
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) = [QUOT](x,y) = x + y + 1
and [if_minus](x,y,z) = y,
the rules in {4-7}
are weakly decreasing and
the rules in {1-3,16}
are strictly decreasing.

- Consider the SCC {18}.
The usable rules are {1-9}.
By taking the polynomial interpretation
[0] = [false] = [true] = 1,
[minus](x,y) = [quot](x,y) = x,
[LOG](x) = [s](x) = x + 1,
[le](x,y) = x + y + 1
and [if_minus](x,y,z) = y,
the rules in {4-9}
are weakly decreasing and
the rules in {1-3,18}
are strictly decreasing.

Hence the TRS is terminating.