Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
div(0,y) |
→ 0 |
| 2: |
|
div(x,y) |
→ quot(x,y,y) |
| 3: |
|
quot(0,s(y),z) |
→ 0 |
| 4: |
|
quot(s(x),s(y),z) |
→ quot(x,y,z) |
| 5: |
|
quot(x,0,s(z)) |
→ s(div(x,s(z))) |
|
There are 3 dependency pairs:
|
| 6: |
|
DIV(x,y) |
→ QUOT(x,y,y) |
| 7: |
|
QUOT(s(x),s(y),z) |
→ QUOT(x,y,z) |
| 8: |
|
QUOT(x,0,s(z)) |
→ DIV(x,s(z)) |
|
The approximated dependency graph contains one SCC:
{6-8}.
-
Consider the SCC {6-8}.
There are no usable rules.
By taking the AF π with
π(DIV) = π(QUOT) = 1 together with
the lexicographic path order with
empty precedence,
the rules in {6,8}
are weakly decreasing and
rule 7
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {6,8}.
By taking the AF π with
π(DIV) = π(QUOT) = 2
and π(s) = [ ] together with
the lexicographic path order with
precedence 0 ≻ s,
rule 6
is weakly decreasing and
rule 8
is strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.01 seconds)
--- May 4, 2006