YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
f(a,b) |
→ |
c |
| a |
→ |
a' |
| b |
→ |
b' |
| c |
→ |
f(a',b) |
| c |
→ |
f(a,b') |
| c |
→ |
f(a,b) |
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).
-
f(a,b)→c ↦ 4
-
a→a' ↦ 1
-
b→b' ↦ 0
-
c→f(a',b) ↦ 0
-
c→f(a,b') ↦ 3
-
c→f(a,b) ↦ 2
All critical pairs are joinable:
-
f(a',b)←c
-
f(a,b')←c
-
f(a',b)→f(a',b')←f(a,b')
-
f(a',b)←f(a,b)
-
f(a,b')→f(a',b')←f(a',b)
-
f(a,b')←f(a,b)
-
f(a,b)→f(a',b)
-
f(a,b)→f(a,b')
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)!