Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
a__f(0) |
→ cons(0,f(s(0))) |
| 2: |
|
a__f(s(0)) |
→ a__f(a__p(s(0))) |
| 3: |
|
a__p(s(0)) |
→ 0 |
| 4: |
|
mark(f(X)) |
→ a__f(mark(X)) |
| 5: |
|
mark(p(X)) |
→ a__p(mark(X)) |
| 6: |
|
mark(0) |
→ 0 |
| 7: |
|
mark(cons(X1,X2)) |
→ cons(mark(X1),X2) |
| 8: |
|
mark(s(X)) |
→ s(mark(X)) |
| 9: |
|
a__f(X) |
→ f(X) |
| 10: |
|
a__p(X) |
→ p(X) |
|
There are 8 dependency pairs:
|
| 11: |
|
A__F(s(0)) |
→ A__F(a__p(s(0))) |
| 12: |
|
A__F(s(0)) |
→ A__P(s(0)) |
| 13: |
|
MARK(f(X)) |
→ A__F(mark(X)) |
| 14: |
|
MARK(f(X)) |
→ MARK(X) |
| 15: |
|
MARK(p(X)) |
→ A__P(mark(X)) |
| 16: |
|
MARK(p(X)) |
→ MARK(X) |
| 17: |
|
MARK(cons(X1,X2)) |
→ MARK(X1) |
| 18: |
|
MARK(s(X)) |
→ MARK(X) |
|
The approximated dependency graph contains 2 SCCs:
{11}
and {14,16-18}.
-
Consider the SCC {11}.
The usable rules are {3,10}.
By taking the AF π with
π(A__F) = 1
and π(a__p) = π(p) = π(s) = [ ] together with
the lexicographic path order with
precedence s ≻ a__p ≻ 0
and a__p ≻ p,
the rules in {3,10,11}
are strictly decreasing.
-
Consider the SCC {14,16-18}.
There are no usable rules.
By taking the AF π with
π(cons) = π(f) = π(MARK) = π(p) = 1 together with
the lexicographic path order with
empty precedence,
the rules in {14,16,17}
are weakly decreasing and
rule 18
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {14,16,17}.
By taking the AF π with
π(cons) = π(f) = π(MARK) = 1 together with
the lexicographic path order with
empty precedence,
the rules in {14,17}
are weakly decreasing and
rule 16
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {14,17}.
By taking the AF π with
π(cons) = π(MARK) = 1 together with
the lexicographic path order with
empty precedence,
rule 17
is weakly decreasing and
rule 14
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {17}.
By taking the AF π with
π(MARK) = 1
and π(cons) = [1] together with
the lexicographic path order with
empty precedence,
rule 17
is strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.02 seconds)
--- May 3, 2006