YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
b(w(x)) |
→ |
w(w(w(b(x)))) |
|
w(b(x)) |
→ |
b(x) |
|
b(b(x)) |
→ |
w(w(w(w(x)))) |
|
w(w(x)) |
→ |
w(x) |
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 = b(b(x1_1)) {1}←b(w(b(x1_1)))→ε w(w(w(b(b(x1_1))))) = t can be joined as follows.
s
↔ w(b(b(x1_1))) ↔ w(w(b(b(x1_1)))) ↔
t
-
The critical peak s = b(w(x1_1)) {1}←b(w(w(x1_1)))→ε w(w(w(b(w(x1_1))))) = t can be joined as follows.
s
↔ w(b(w(x1_1))) ↔ w(w(b(w(x1_1)))) ↔
t
-
The critical peak s = w(w(w(w(b(x1_1))))) {1}←w(b(w(x1_1)))→ε b(w(x1_1)) = t can be joined as follows.
s
↔ w(w(w(b(x1_1)))) ↔
t
-
The critical peak s = w(w(w(w(w(x1_1))))) {1}←w(b(b(x1_1)))→ε b(b(x1_1)) = t can be joined as follows.
s
↔ w(w(w(w(x1_1)))) ↔
t
-
The critical peak s = b(w(w(w(b(x1_1))))) {1}←b(b(w(x1_1)))→ε w(w(w(w(w(x1_1))))) = t can be joined as follows.
s
↔ b(w(w(b(x1_1)))) ↔ b(w(b(x1_1))) ↔ b(b(x1_1)) ↔ w(w(w(w(x1_1)))) ↔
t
-
The critical peak s = b(w(w(w(w(x1_1))))) {1}←b(b(b(x1_1)))→ε w(w(w(w(b(x1_1))))) = t can be joined as follows.
s
↔ b(w(w(w(x1_1)))) ↔ b(w(w(x1_1))) ↔ b(w(x1_1)) ↔ w(w(w(b(x1_1)))) ↔
t
-
The critical peak s = w(b(x1_1)) {1}←w(w(b(x1_1)))→ε w(b(x1_1)) = t can be joined as follows.
s
↔
t
-
The critical peak s = w(w(x1_1)) {1}←w(w(w(x1_1)))→ε w(w(x1_1)) = t can be joined as follows.
s
↔
t
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.
|
b(w(b(x1_1))) |
→ |
b(b(x1_1)) |
|
b(w(b(x1_1))) |
→ |
w(w(w(b(b(x1_1))))) |
|
b(w(w(x1_1))) |
→ |
b(w(x1_1)) |
|
b(w(w(x1_1))) |
→ |
w(w(w(b(w(x1_1))))) |
|
w(b(w(x1_1))) |
→ |
w(w(w(w(b(x1_1))))) |
|
w(b(w(x1_1))) |
→ |
b(w(x1_1)) |
|
w(b(b(x1_1))) |
→ |
w(w(w(w(w(x1_1))))) |
|
w(b(b(x1_1))) |
→ |
b(b(x1_1)) |
|
b(b(w(x1_1))) |
→ |
b(w(w(w(b(x1_1))))) |
|
b(b(w(x1_1))) |
→ |
w(w(w(w(w(x1_1))))) |
|
b(b(b(x1_1))) |
→ |
b(w(w(w(w(x1_1))))) |
|
b(b(b(x1_1))) |
→ |
w(w(w(w(b(x1_1))))) |
Relative termination of P / R is proven as follows.
1.1 Rule Removal
Using the
recursive path order with the following precedence and status
| prec(w) |
= |
0 |
|
stat(w) |
= |
lex
|
| prec(b) |
= |
1 |
|
stat(b) |
= |
lex
|
all rules of R could be removed.
Moreover,
all rules of S could be removed.
1.1.1 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 (Weakly) Orthogonal
Confluence is proven since the TRS is (weakly) orthogonal.
Tool configuration
Hakusan