YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
-(0,0) |
→ |
0 |
|
-(s(x),0) |
→ |
s(x) |
|
-(x,s(y)) |
→ |
-(d(x),y) |
|
d(s(x)) |
→ |
x |
|
-(s(x),s(y)) |
→ |
-(x,y) |
|
-(d(x),y) |
→ |
-(x,s(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).
-
-(0,0)→0 ↦ 0
-
-(s(x),0)→s(x) ↦ 0
-
-(x,s(y))→-(d(x),y) ↦ 0
-
d(s(x))→x ↦ 0
-
-(s(x),s(y))→-(x,y) ↦ 1
-
-(d(x),y)→-(x,s(y)) ↦ 2
All critical pairs are joinable:
-
-(d(s(x)),y)→-(x,y)
-
-(d(d(x)),x125)→-(d(x),s(x125))←-(x,s(s(x125)))
-
-(x126,y)←-(s(x126),s(y))
-
-(x127,y)←-(d(s(x127)),y)
-
-(x129,s(s(y)))→-(d(x129),s(y))←-(d(d(x129)),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)!