Certification Problem
Input (COPS 128)
We consider the TRS containing the following rules:
|
+(0,y) |
→ |
y |
(1) |
|
+(x,s(y)) |
→ |
s(+(x,y)) |
(2) |
|
+(x,y) |
→ |
+(y,x) |
(3) |
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,y) |
→ |
+(y,x) |
(3) |
|
+(x,s(y)) |
→ |
s(+(x,y)) |
(2) |
|
+(0,y) |
→ |
y |
(1) |
|
+(y,0) |
→ |
y |
(4) |
|
+(s(y),x) |
→ |
s(+(x,y)) |
(5) |
|
+(s(x32),x) |
→ |
s(+(x,x32)) |
(6) |
All redundant rules that were added or removed can be
simulated in 2 steps
.
1.1 Decreasing Diagrams
1.1.2 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).
-
↦ 0
-
|
+(x,s(y)) |
→ |
s(+(x,y)) |
(2) |
↦ 1
-
↦ 0
-
↦ 0
-
|
+(s(y),x) |
→ |
s(+(x,y)) |
(5) |
↦ 1
-
|
+(s(x32),x) |
→ |
s(+(x,x32)) |
(6) |
↦ 3
The critical pairs can be joined as follows. Here,
↔ is always chosen as an appropriate rewrite relation which
is automatically inferred by the certifier.
-
The critical peak s = +(s(y),x)←→ε s(+(x,y)) = t can be joined as follows.
s
↔
t
-
The critical peak s = +(y,0)←→ε y = t can be joined as follows.
s
↔
t
-
The critical peak s = +(0,y)←→ε y = t can be joined as follows.
s
↔
t
-
The critical peak s = +(x,s(y))←→ε s(+(x,y)) = t can be joined as follows.
s
↔
t
-
The critical peak s = +(x,s(x32))←→ε s(+(x,x32)) = t can be joined as follows.
s
↔
t
-
The critical peak s = s(+(x,x158))←→ε +(s(x158),x) = t can be joined as follows.
s
↔ s(+(x,x158)) ↔
t
-
The critical peak s = s(+(0,x160))←→ε s(x160) = t can be joined as follows.
s
↔
t
-
The critical peak s = s(+(s(y),x162))←→ε s(+(s(x162),y)) = t can be joined as follows.
s
↔ s(s(+(x162,y))) ↔ s(s(+(y,x162))) ↔
t
-
The critical peak s = s(+(s(y),x162))←→ε s(+(s(x162),y)) = t can be joined as follows.
s
↔ s(+(x162,s(y))) ↔ s(s(+(x162,y))) ↔ s(s(+(y,x162))) ↔
t
-
The critical peak s = s(+(s(y),x162))←→ε s(+(s(x162),y)) = t can be joined as follows.
s
↔ s(s(+(x162,y))) ↔ s(s(+(y,x162))) ↔
t
-
The critical peak s = s(+(s(y),x162))←→ε s(+(s(x162),y)) = t can be joined as follows.
s
↔ s(s(+(x162,y))) ↔ s(s(+(y,x162))) ↔ s(+(y,s(x162))) ↔
t
-
The critical peak s = s(+(s(x32),x164))←→ε s(+(s(x164),x32)) = t can be joined as follows.
s
↔ s(s(+(x164,x32))) ↔ s(s(+(x32,x164))) ↔
t
-
The critical peak s = s(+(s(x32),x164))←→ε s(+(s(x164),x32)) = t can be joined as follows.
s
↔ s(+(x164,s(x32))) ↔ s(s(+(x164,x32))) ↔ s(s(+(x32,x164))) ↔
t
-
The critical peak s = s(+(s(x32),x164))←→ε s(+(s(x164),x32)) = t can be joined as follows.
s
↔ s(s(+(x164,x32))) ↔ s(s(+(x32,x164))) ↔
t
-
The critical peak s = s(+(s(x32),x164))←→ε s(+(s(x164),x32)) = t can be joined as follows.
s
↔ s(s(+(x164,x32))) ↔ s(s(+(x32,x164))) ↔ s(+(x32,s(x164))) ↔
t
-
The critical peak s = y←→ε +(y,0) = t can be joined as follows.
s
↔ y ↔
t
-
The critical peak s = s(y)←→ε s(+(0,y)) = t can be joined as follows.
s
↔ s(y) ↔
t
-
The critical peak s = 0←→ε 0 = t can be joined as follows.
s
↔
t
-
The critical peak s = x←→ε +(0,x) = t can be joined as follows.
s
↔ x ↔
t
-
The critical peak s = 0←→ε 0 = t can be joined as follows.
s
↔
t
-
The critical peak s = s(y)←→ε s(+(0,y)) = t can be joined as follows.
s
↔ s(y) ↔
t
-
The critical peak s = s(x32)←→ε s(+(0,x32)) = t can be joined as follows.
s
↔ s(x32) ↔
t
-
The critical peak s = s(+(y,x172))←→ε +(y,s(x172)) = t can be joined as follows.
s
↔ s(+(y,x172)) ↔
t
-
The critical peak s = s(+(s(y),x174))←→ε s(+(s(x174),y)) = t can be joined as follows.
s
↔ s(s(+(x174,y))) ↔ s(s(+(y,x174))) ↔
t
-
The critical peak s = s(+(s(y),x174))←→ε s(+(s(x174),y)) = t can be joined as follows.
s
↔ s(+(x174,s(y))) ↔ s(s(+(x174,y))) ↔ s(s(+(y,x174))) ↔
t
-
The critical peak s = s(+(s(y),x174))←→ε s(+(s(x174),y)) = t can be joined as follows.
s
↔ s(s(+(x174,y))) ↔ s(s(+(y,x174))) ↔
t
-
The critical peak s = s(+(s(y),x174))←→ε s(+(s(x174),y)) = t can be joined as follows.
s
↔ s(s(+(x174,y))) ↔ s(s(+(y,x174))) ↔ s(+(y,s(x174))) ↔
t
-
The critical peak s = s(+(0,x176))←→ε s(x176) = t can be joined as follows.
s
↔
t
-
The critical peak s = s(+(y,x178))←→ε +(y,s(x178)) = t can be joined as follows.
s
↔ s(+(y,x178)) ↔
t
-
The critical peak s = s(+(s(y),x180))←→ε s(+(s(x180),y)) = t can be joined as follows.
s
↔ s(s(+(x180,y))) ↔ s(s(+(y,x180))) ↔
t
-
The critical peak s = s(+(s(y),x180))←→ε s(+(s(x180),y)) = t can be joined as follows.
s
↔ s(+(x180,s(y))) ↔ s(s(+(x180,y))) ↔ s(s(+(y,x180))) ↔
t
-
The critical peak s = s(+(s(y),x180))←→ε s(+(s(x180),y)) = t can be joined as follows.
s
↔ s(s(+(x180,y))) ↔ s(s(+(y,x180))) ↔
t
-
The critical peak s = s(+(s(y),x180))←→ε s(+(s(x180),y)) = t can be joined as follows.
s
↔ s(s(+(x180,y))) ↔ s(s(+(y,x180))) ↔ s(+(y,s(x180))) ↔
t
-
The critical peak s = s(+(0,x182))←→ε s(x182) = t can be joined as follows.
s
↔
t
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