YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
f(g(x)) |
→ |
f(h(x,x)) |
|
g(a) |
→ |
g(g(a)) |
|
h(a,a) |
→ |
g(g(a)) |
Proof
1 Decreasing Diagrams
1.1 Relative Termination Proof
The duplicating rules (R) terminate relative to the other rules (S).
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [f(x1)] |
= |
· x1 +
|
| [a] |
= |
|
| [g(x1)] |
= |
· x1 +
|
| [h(x1, x2)] |
= |
· x1 + · x2 +
|
the
rule
remains in R.
Moreover,
the
rule
remains in S.
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the naturals
| [f(x1)] |
= |
· x1 +
|
| [a] |
= |
|
| [g(x1)] |
= |
· x1 +
|
| [h(x1, x2)] |
= |
· x1 + · x2 +
|
all rules of R could be removed.
Moreover,
the
rule
remains in S.
1.1.1.1.1 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).
-
f(g(x))→f(h(x,x)) ↦ 0
-
g(a)→g(g(a)) ↦ 2
-
h(a,a)→g(g(a)) ↦ 0
All critical pairs are joinable:
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)!