Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
0(#) |
→ # |
| 2: |
|
x + # |
→ x |
| 3: |
|
# + x |
→ x |
| 4: |
|
0(x) + 0(y) |
→ 0(x + y) |
| 5: |
|
0(x) + 1(y) |
→ 1(x + y) |
| 6: |
|
1(x) + 0(y) |
→ 1(x + y) |
| 7: |
|
1(x) + 1(y) |
→ 0((x + y) + 1(#)) |
| 8: |
|
# * x |
→ # |
| 9: |
|
0(x) * y |
→ 0(x * y) |
| 10: |
|
1(x) * y |
→ 0(x * y) + y |
| 11: |
|
sum(nil) |
→ 0(#) |
| 12: |
|
sum(cons(x,l)) |
→ x + sum(l) |
| 13: |
|
prod(nil) |
→ 1(#) |
| 14: |
|
prod(cons(x,l)) |
→ x * prod(l) |
|
There are 17 dependency pairs:
|
| 15: |
|
0(x) +# 0(y) |
→ 0#(x + y) |
| 16: |
|
0(x) +# 0(y) |
→ x +# y |
| 17: |
|
0(x) +# 1(y) |
→ x +# y |
| 18: |
|
1(x) +# 0(y) |
→ x +# y |
| 19: |
|
1(x) +# 1(y) |
→ 0#((x + y) + 1(#)) |
| 20: |
|
1(x) +# 1(y) |
→ (x + y) +# 1(#) |
| 21: |
|
1(x) +# 1(y) |
→ x +# y |
| 22: |
|
0(x) *# y |
→ 0#(x * y) |
| 23: |
|
0(x) *# y |
→ x *# y |
| 24: |
|
1(x) *# y |
→ 0(x * y) +# y |
| 25: |
|
1(x) *# y |
→ 0#(x * y) |
| 26: |
|
1(x) *# y |
→ x *# y |
| 27: |
|
SUM(nil) |
→ 0#(#) |
| 28: |
|
SUM(cons(x,l)) |
→ x +# sum(l) |
| 29: |
|
SUM(cons(x,l)) |
→ SUM(l) |
| 30: |
|
PROD(cons(x,l)) |
→ x *# prod(l) |
| 31: |
|
PROD(cons(x,l)) |
→ PROD(l) |
|
The approximated dependency graph contains 4 SCCs:
{16-18,20,21},
{23,26},
{31}
and {29}.
-
Consider the SCC {16-18,20,21}.
The usable rules are {1-7}.
The constraints could not be solved.
-
Consider the SCC {23,26}.
There are no usable rules.
By taking the AF π with
π(*#) = π(0) = 1 together with
the lexicographic path order with
empty precedence,
rule 23
is weakly decreasing and
rule 26
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {23}.
By taking the AF π with
π(*#) = 1 together with
the lexicographic path order with
empty precedence,
rule 23
is strictly decreasing.
-
Consider the SCC {31}.
There are no usable rules.
By taking the AF π with
π(PROD) = 1
and π(cons) = [2] together with
the lexicographic path order with
empty precedence,
rule 31
is strictly decreasing.
-
Consider the SCC {29}.
There are no usable rules.
By taking the AF π with
π(SUM) = 1
and π(cons) = [2] together with
the lexicographic path order with
empty precedence,
rule 29
is strictly decreasing.
Tyrolean Termination Tool (0.05 seconds)
--- May 3, 2006