YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
f(g(x,a,b)) |
→ |
x |
|
g(f(h(c,d)),x,y) |
→ |
h(k1(x),k2(y)) |
|
k1(a) |
→ |
c |
|
k2(b) |
→ |
d |
|
f(h(k1(a),k2(b))) |
→ |
f(h(c,d)) |
|
f(h(c,k2(b))) |
→ |
f(h(c,d)) |
|
f(h(k1(a),d)) |
→ |
f(h(c,d)) |
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(g(x,a,b))→x ↦ 1
-
g(f(h(c,d)),x,y)→h(k1(x),k2(y)) ↦ 0
-
k1(a)→c ↦ 1
-
k2(b)→d ↦ 1
-
f(h(k1(a),k2(b)))→f(h(c,d)) ↦ 0
-
f(h(c,k2(b)))→f(h(c,d)) ↦ 0
-
f(h(k1(a),d))→f(h(c,d)) ↦ 0
All critical pairs are joinable:
-
f(h(k1(a),k2(b)))→f(h(c,d))
-
f(h(c,k2(b)))→f(h(c,d))
-
f(h(c,d))
-
f(h(k1(a),d))→f(h(c,d))
-
f(h(c,d))
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)!