YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
-(0,0) |
→ |
0 |
|
-(s(x),0) |
→ |
s(x) |
|
-(x,s(y)) |
→ |
-(d(x),y) |
|
d(s(x)) |
→ |
x |
|
-(s(x),s(y)) |
→ |
-(x,y) |
|
-(d(x),y) |
→ |
-(x,s(y)) |
Proof
1 Compositional Parallel Critical Pair Systems
All parallel critical pairs of the TRS R are joinable by R.
This can be seen as follows:
The parallel 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 = -(x0_1,y2) {ε}←-(s(x0_1),s(y2))→ε -(d(s(x0_1)),y2) = t can be joined as follows.
s
↔
t
-
The critical peak s = -(x0_1,s(s(y2))) {ε}←-(d(x0_1),s(y2))→ε -(d(d(x0_1)),y2) = t can be joined as follows.
s
↔ -(d(x0_1),s(y2)) ↔
t
-
The critical peak s = -(d(s(y1)),y2) {ε}←-(s(y1),s(y2))→ε -(y1,y2) = t can be joined as follows.
s
↔
t
-
The critical peak s = -(d(d(y1)),x0_2) {ε}←-(d(y1),s(x0_2))→ε -(y1,s(s(x0_2))) = t can be joined as follows.
s
↔ -(d(y1),s(x0_2)) ↔
t
-
The critical peak s = -(x1_1,y2) {1}←-(d(s(x1_1)),y2)→ε -(s(x1_1),s(y2)) = t can be joined as follows.
s
↔
t
The TRS C is chosen as:
|
-(s(x),s(y)) |
→ |
-(x,y) |
|
-(d(x),y) |
→ |
-(x,s(y)) |
Consequently, PCPS(R,C) is included in the following TRS P where
steps are used to show that certain pairs are C-convertible.
There are no rules.
Relative termination of P / R is proven as follows.
1.1 R is empty
There are no rules in the TRS R. Hence, R/S is relative terminating.
Confluence of C is proven as follows.
1.2 Compositional Parallel Critical Pair Systems
All parallel critical pairs of the TRS R are joinable by R.
This can be seen as follows:
The parallel critical pairs can be joined as follows. Here,
↔ is always chosen as an appropriate rewrite relation which
is automatically inferred by the certifier.
The TRS C is chosen as:
There are no rules.
Consequently, PCPS(R,C) is included in the following TRS P where
steps are used to show that certain pairs are C-convertible.
There are no rules.
Relative termination of P / R is proven as follows.
1.2.1 R is empty
There are no rules in the TRS R. Hence, R/S is relative terminating.
Confluence of C is proven as follows.
1.2.2 (Weakly) Orthogonal
Confluence is proven since the TRS is (weakly) orthogonal.
Tool configuration
Hakusan