YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
max(x,0) |
→ |
x |
|
max(0,y) |
→ |
y |
|
max(s(x),s(y)) |
→ |
s(max(x,y)) |
|
max(x,y) |
→ |
max(y,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).
-
max(x,0)→x ↦ 1
-
max(0,y)→y ↦ 0
-
max(s(x),s(y))→s(max(x,y)) ↦ 0
-
max(x,y)→max(y,x) ↦ 2
All critical pairs are joinable:
- 0
-
x←max(0,x)
- 0
-
y←max(y,0)
-
s(max(x54,x55))→s(max(x55,x54))←max(s(x55),s(x54))
-
max(0,x)→x
-
max(y,0)→y
-
max(s(y),s(x))→s(max(y,x))←s(max(x,y))
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)!