YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
H(I(x)) |
→ |
K(J(x)) |
|
J(x) |
→ |
K(J(x)) |
|
I(x) |
→ |
I(J(x)) |
|
J(x) |
→ |
J(K(J(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).
-
H(I(x))→K(J(x)) ↦ 0
-
J(x)→K(J(x)) ↦ 0
-
I(x)→I(J(x)) ↦ 1
-
J(x)→J(K(J(x))) ↦ 0
All critical pairs are joinable:
-
K(J(x))→K(J(K(J(x))))←J(K(J(x)))
-
H(I(J(x)))→K(J(J(x)))→K(J(K(J(x))))←K(J(x))
-
H(I(J(x)))→H(I(K(J(x))))→K(J(K(J(x))))←K(J(x))
-
J(K(J(x)))→K(J(K(J(x))))←K(J(x))
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)!