YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
a(x) |
→ |
x |
|
a(a(x)) |
→ |
b(c(x)) |
|
b(x) |
→ |
x |
|
c(x) |
→ |
x |
|
c(b(x)) |
→ |
b(a(c(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(x)→x ↦ 2
-
a(a(x))→b(c(x)) ↦ 3
-
b(x)→x ↦ 0
-
c(x)→x ↦ 2
-
c(b(x))→b(a(c(x))) ↦ 3
All critical pairs are joinable:
-
a(x)→x←c(x)←b(c(x))
-
a(x)→x←b(x)←b(c(x))
-
a(x)→x←c(x)←b(c(x))
-
a(x)→x←b(x)←b(c(x))
-
b(c(x38))→c(x38)→x38←a(x38)
-
b(c(x38))→b(x38)→x38←a(x38)
-
a(b(c(x39)))→b(c(x39))←b(c(a(x39)))
-
c(x)←b(c(x))←b(a(c(x)))
-
c(x)←a(c(x))←b(a(c(x)))
-
b(x)←b(c(x))←b(a(c(x)))
-
b(x)←b(a(x))←b(a(c(x)))
-
b(a(c(x42)))→b(c(x42))→b(x42)
-
b(a(c(x42)))→b(a(x42))→b(x42)
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)!