YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
| a1 |
→ |
b1 |
| a1 |
→ |
c1 |
| b1 |
→ |
b2 |
| c1 |
→ |
c2 |
| a2 |
→ |
b2 |
| a2 |
→ |
c2 |
| b2 |
→ |
b3 |
| c2 |
→ |
c3 |
| a3 |
→ |
b3 |
| a3 |
→ |
c3 |
| b3 |
→ |
b4 |
| c3 |
→ |
c4 |
| a4 |
→ |
b4 |
| a4 |
→ |
c4 |
| b4 |
→ |
b5 |
| c4 |
→ |
c5 |
| a5 |
→ |
b5 |
| a5 |
→ |
c5 |
| b5 |
→ |
b6 |
| c5 |
→ |
c6 |
| a6 |
→ |
b6 |
| a6 |
→ |
c6 |
| b6 |
→ |
b7 |
| c6 |
→ |
c7 |
| a7 |
→ |
b7 |
| a7 |
→ |
c7 |
| b7 |
→ |
b8 |
| c7 |
→ |
c8 |
| a8 |
→ |
b8 |
| a8 |
→ |
c8 |
| b8 |
→ |
b9 |
| c8 |
→ |
c9 |
| a9 |
→ |
b9 |
| a9 |
→ |
c9 |
| b9 |
→ |
b10 |
| c9 |
→ |
c10 |
| a10 |
→ |
b11 |
| b10 |
→ |
b11 |
| c10 |
→ |
b11 |
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).
-
a1→b1 ↦ 1
-
a1→c1 ↦ 1
-
b1→b2 ↦ 0
-
c1→c2 ↦ 0
-
a2→b2 ↦ 1
-
a2→c2 ↦ 1
-
b2→b3 ↦ 0
-
c2→c3 ↦ 0
-
a3→b3 ↦ 2
-
a3→c3 ↦ 1
-
b3→b4 ↦ 0
-
c3→c4 ↦ 0
-
a4→b4 ↦ 3
-
a4→c4 ↦ 1
-
b4→b5 ↦ 0
-
c4→c5 ↦ 0
-
a5→b5 ↦ 3
-
a5→c5 ↦ 1
-
b5→b6 ↦ 0
-
c5→c6 ↦ 0
-
a6→b6 ↦ 3
-
a6→c6 ↦ 1
-
b6→b7 ↦ 0
-
c6→c7 ↦ 0
-
a7→b7 ↦ 3
-
a7→c7 ↦ 1
-
b7→b8 ↦ 0
-
c7→c8 ↦ 0
-
a8→b8 ↦ 2
-
a8→c8 ↦ 7
-
b8→b9 ↦ 0
-
c8→c9 ↦ 0
-
a9→b9 ↦ 13
-
a9→c9 ↦ 1
-
b9→b10 ↦ 0
-
c9→c10 ↦ 0
-
a10→b11 ↦ 0
-
b10→b11 ↦ 1
-
c10→b11 ↦ 1
All critical pairs are joinable:
-
b1→b2→b3→b4→b5→b6→b7→b8→b9→b10→b11←c10←c9←c8←c7←c6←c5←c4←c3←c2←c1
-
c1→c2→c3→c4→c5→c6→c7→c8→c9→c10→b11←b10←b9←b8←b7←b6←b5←b4←b3←b2←b1
-
b2→b3→b4→b5→b6→b7→b8→b9→b10→b11←c10←c9←c8←c7←c6←c5←c4←c3←c2
-
c2→c3→c4→c5→c6→c7→c8→c9→c10→b11←b10←b9←b8←b7←b6←b5←b4←b3←b2
-
b3→b4→b5→b6→b7→b8→b9→b10→b11←c10←c9←c8←c7←c6←c5←c4←c3
-
c3→c4→c5→c6→c7→c8→c9→c10→b11←b10←b9←b8←b7←b6←b5←b4←b3
-
b4→b5→b6→b7→b8→b9→b10→b11←c10←c9←c8←c7←c6←c5←c4
-
c4→c5→c6→c7→c8→c9→c10→b11←b10←b9←b8←b7←b6←b5←b4
-
b5→b6→b7→b8→b9→b10→b11←c10←c9←c8←c7←c6←c5
-
c5→c6→c7→c8→c9→c10→b11←b10←b9←b8←b7←b6←b5
-
b6→b7→b8→b9→b10→b11←c10←c9←c8←c7←c6
-
c6→c7→c8→c9→c10→b11←b10←b9←b8←b7←b6
-
b7→b8→b9→b10→b11←c10←c9←c8←c7
-
c7→c8→c9→c10→b11←b10←b9←b8←b7
-
b8→b9→b10→b11←c10←c9←c8
-
c8→c9→c10→b11←b10←b9←b8
-
b9→b10→b11←c10←c9
-
c9→c10→b11←b10←b9
Tool configuration
csi
- version: csi 1.2.5 [hg: unknown]
- strategy:
(if left-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)!