YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
a(a(x)) |
→ |
a(b(a(x))) |
|
b(a(b(x))) |
→ |
a(c(a(x))) |
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(a(x))→a(b(a(x))) ↦ 0
-
b(a(b(x)))→a(c(a(x))) ↦ 2
All critical pairs are joinable:
-
a(a(b(a(x11))))→a(b(a(b(a(x11)))))←a(b(a(a(x11))))
-
b(a(a(c(a(x12)))))→b(a(b(a(c(a(x12))))))→a(c(a(a(c(a(x12))))))←a(c(a(b(a(b(x12))))))←a(c(a(a(b(x12)))))
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)!