YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
H(F(x,y)) |
→ |
F(H(R(x)),y) |
|
F(x,K(y,z)) |
→ |
G(P(y),Q(z,x)) |
|
H(Q(x,y)) |
→ |
Q(x,H(R(y))) |
|
Q(x,H(R(y))) |
→ |
H(Q(x,y)) |
|
H(G(x,y)) |
→ |
G(x,H(y)) |
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(F(x,y))→F(H(R(x)),y) ↦ 1
-
F(x,K(y,z))→G(P(y),Q(z,x)) ↦ 1
-
H(Q(x,y))→Q(x,H(R(y))) ↦ 0
-
Q(x,H(R(y)))→H(Q(x,y)) ↦ 0
-
H(G(x,y))→G(x,H(y)) ↦ 0
All critical pairs are joinable:
-
H(G(P(x125),Q(x126,x)))→G(P(x125),H(Q(x126,x)))←G(P(x125),Q(x126,H(R(x))))←F(H(R(x)),K(x125,x126))
-
H(G(P(x125),Q(x126,x)))→G(P(x125),H(Q(x126,x)))→G(P(x125),Q(x126,H(R(x))))←F(H(R(x)),K(x125,x126))
-
H(H(Q(x,x128)))→H(Q(x,H(R(x128))))←Q(x,H(R(H(R(x128)))))
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)!