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) |
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
All critical pairs are joinable:
-
+(+(x27,+(x28,y)),z)→+(+(+(x27,x28),y),z)←+(+(x27,x28),+(y,z))
-
+(x30,+(x31,+(y,z)))→+(+(x30,x31),+(y,z))←+(+(+(x30,x31),y),z)
-
+(x,+(x33,+(x34,z)))→+(x,+(+(x33,x34),z))←+(+(x,+(x33,x34)),z)
-
+(+(+(x,y),x37),x38)→+(+(x,y),+(x37,x38))←+(x,+(y,+(x37,x38)))
-
+(+(+(x,x40),x41),z)→+(+(x,+(x40,x41)),z)←+(x,+(+(x40,x41),z))
-
+(x,+(+(y,x43),x44))→+(x,+(y,+(x43,x44)))←+(+(x,y),+(x43,x44))
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)!