YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
| a |
→ |
b |
| a |
→ |
c |
| a |
→ |
e |
| b |
→ |
d |
| c |
→ |
a |
| d |
→ |
a |
| d |
→ |
e |
|
g(x) |
→ |
h(a) |
|
h(x) |
→ |
e |
Proof
1 Decreasing Diagrams
1.1 Relative Termination Proof
The duplicating rules (R) terminate relative to the other rules (S).
1.1.1 R is empty
There are no rules in the TRS R. Hence, R/S is relative terminating.
1.2 Rule Labeling
Confluence is proven, because all critical peaks can be joined decreasingly
using the following rule labeling function (rules that are not shown have label 0).
-
a→b ↦ 1
-
a→c ↦ 1
-
a→e ↦ 1
-
b→d ↦ 0
-
c→a ↦ 0
-
d→a ↦ 2
-
d→e ↦ 1
-
g(x)→h(a) ↦ 0
-
h(x)→e ↦ 0
All critical pairs are joinable:
-
b←a←c
-
b→d→a←c
-
b→d→e←a←c
-
b→d→e
-
c→a←d←b
-
c→a→b
-
c→a→e←d←b
-
c→a→e
-
e←d←b
-
e←a←c
-
a→e
-
e←a
Tool configuration
csi
- version: csi 1.2.5 [hg: unknown]
- strategy:
(if linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)!