YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
+(0,y) |
→ |
y |
|
+(s(x),y) |
→ |
s(+(x,y)) |
|
+(x,0) |
→ |
x |
|
+(x,s(y)) |
→ |
s(+(x,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,y)→y ↦ 0
-
+(s(x),y)→s(+(x,y)) ↦ 0
-
+(x,0)→x ↦ 0
-
+(x,s(y))→s(+(x,y)) ↦ 0
All critical pairs are joinable:
- 0
-
s(y)←s(+(0,y))
-
s(+(x52,0))→s(x52)
-
s(+(x54,s(y)))→s(s(+(x54,y)))←s(+(s(x54),y))
- 0
-
s(x)←s(+(x,0))
-
s(+(0,x59))→s(x59)
-
s(+(s(x),x61))→s(s(+(x,x61)))←s(+(x,s(x61)))
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)!