YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
F(H(x),y) |
→ |
F(H(x),I(I(y))) |
|
F(x,G(y)) |
→ |
F(I(x),G(y)) |
|
I(x) |
→ |
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).
-
F(H(x),y)→F(H(x),I(I(y))) ↦ 0
-
F(x,G(y))→F(I(x),G(y)) ↦ 1
-
I(x)→x ↦ 0
All critical pairs are joinable:
-
F(H(x27),I(I(G(y))))←F(H(x27),G(y))←F(I(H(x27)),G(y))
-
F(H(x27),I(I(G(y))))→F(H(x27),I(G(y)))→F(H(x27),G(y))←F(I(H(x27)),G(y))
-
F(I(H(x)),G(x30))→F(H(x),G(x30))←F(H(x),I(G(x30)))←F(H(x),I(I(G(x30))))
-
F(I(H(x)),G(x30))→F(H(x),G(x30))→F(H(x),I(I(G(x30))))
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)!