Certification Problem
Input (COPS 11)
We consider the TRS containing the following rules:
|
+(0,y) |
→ |
y |
(1) |
|
+(s(x),y) |
→ |
s(+(x,y)) |
(2) |
|
+(x,0) |
→ |
x |
(3) |
|
+(x,s(y)) |
→ |
s(+(x,y)) |
(4) |
The underlying signature is as follows:
{+/2, 0/0, s/1}Property / Task
Prove or disprove confluence.Answer / Result
Yes.Proof (by csi @ CoCo 2020)
1 Redundant Rules Transformation
To prove that the TRS is (non-)confluent, we show (non-)confluence of the following
modified system:
|
+(x,s(y)) |
→ |
s(+(x,y)) |
(4) |
|
+(x,0) |
→ |
x |
(3) |
|
+(s(x),y) |
→ |
s(+(x,y)) |
(2) |
|
+(0,y) |
→ |
y |
(1) |
All redundant rules that were added or removed can be
simulated in 2 steps
.
1.1 Critical Pair Closing System
Confluence is proven using the following terminating critical-pair-closing-system R:
|
+(s(x),y) |
→ |
s(+(x,y)) |
(2) |
|
+(x,s(y)) |
→ |
s(+(x,y)) |
(4) |
|
+(0,y) |
→ |
y |
(1) |
|
+(x,0) |
→ |
x |
(3) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1 + 4 |
| [+(x1, x2)] |
= |
1 · x1 + 1 · x2 + 2 |
| [0] |
= |
1 |
all of the following rules can be deleted.
|
+(0,y) |
→ |
y |
(1) |
|
+(x,0) |
→ |
x |
(3) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1 + 1 |
| [+(x1, x2)] |
= |
1 · x1 + 2 · x2 + 5 |
all of the following rules can be deleted.
|
+(x,s(y)) |
→ |
s(+(x,y)) |
(4) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1 + 1 |
| [+(x1, x2)] |
= |
2 · x1 + 4 · x2 + 5 |
all of the following rules can be deleted.
|
+(s(x),y) |
→ |
s(+(x,y)) |
(2) |
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.