YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
|
+(x,+(y,z)) |
→ |
+(+(x,y),z) |
|
+(x,y) |
→ |
+(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).
-
+(+(x,y),z)→+(x,+(y,z)) ↦ 0
-
+(x,+(y,z))→+(+(x,y),z) ↦ 0
-
+(x,y)→+(y,x) ↦ 0
All critical pairs are joinable:
-
+(+(x43,+(x44,y)),z)→+(+(+(x43,x44),y),z)←+(+(x43,x44),+(y,z))
-
+(x46,+(x47,+(y,z)))→+(+(x46,x47),+(y,z))←+(+(+(x46,x47),y),z)
-
+(x,+(x49,+(x50,z)))→+(x,+(+(x49,x50),z))←+(+(x,+(x49,x50)),z)
-
+(x52,+(x53,y))→+(+(x52,x53),y)←+(y,+(x52,x53))
-
+(+(+(x,y),x56),x57)→+(+(x,y),+(x56,x57))←+(x,+(y,+(x56,x57)))
-
+(+(+(x,x59),x60),z)→+(+(x,+(x59,x60)),z)←+(x,+(+(x59,x60),z))
-
+(x,+(+(y,x62),x63))→+(x,+(y,+(x62,x63)))←+(+(x,y),+(x62,x63))
-
+(+(x,x65),x66)→+(x,+(x65,x66))←+(+(x65,x66),x)
-
+(z,+(x,y))→+(+(x,y),z)←+(x,+(y,z))
-
+(+(y,x),z)→+(+(x,y),z)←+(x,+(y,z))
-
+(+(y,z),x)→+(x,+(y,z))←+(+(x,y),z)
-
+(x,+(z,y))→+(x,+(y,z))←+(+(x,y),z)
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)!