Certification Problem
Input (COPS 492)
The rewrite relation of the following conditional TRS is considered.
|
f(x) |
→ |
a |
| a ≈ x
|
|
f(x) |
→ |
b |
| b ≈ x
|
Property / Task
Prove or disprove confluence.Answer / Result
Yes.Proof (by ConCon @ CoCo 2020)
1 Quasi-reductive SDTRS where all CCPs are joinable
The given strongly deterministic oriented 3-CTRS is quasi-reductive and all CCPs are joinable.
1.1 Quasi-Reductive CTRS
The given CTRS is quasi-reductive
1.1.1 Unraveling
To prove that the CTRS is quasi-reductive, we show termination of the following
unraveled system.
| For |
f(x)aax we get |
| For |
f(x)bbx we get |
1.1.1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(U2) |
= |
6 |
|
weight(U2) |
= |
1 |
|
|
|
| prec(b) |
= |
0 |
|
weight(b) |
= |
3 |
|
|
|
| prec(U1) |
= |
4 |
|
weight(U1) |
= |
1 |
|
|
|
| prec(a) |
= |
1 |
|
weight(a) |
= |
3 |
|
|
|
| prec(f) |
= |
7 |
|
weight(f) |
= |
4 |
|
|
|
all of the following rules can be deleted.
|
f(x) |
→ |
U1(a,x) |
(1) |
|
U1(x,x) |
→ |
a |
(2) |
|
f(x) |
→ |
U2(b,x) |
(3) |
|
U2(x,x) |
→ |
b |
(4) |
1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.
1.2 All CCPs are joinable
A CCP is joinable if it is context-joinable, infeasible, or unfeasible.
-
The
1st
CCP
b
=
a | bzaz
is context-joinable.
-
The
2nd
CCP
a
=
b | azbz
is context-joinable.