YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
a(x) |
→ |
x |
|
a(b(x)) |
→ |
c(b(b(a(a(x))))) |
|
b(x) |
→ |
c(x) |
|
c(c(x)) |
→ |
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 = c(b(b(a(a(x0_1))))) {ε}←a(b(x0_1))→ε b(x0_1) = t can be joined as follows.
s
↔ c(c(b(a(a(x0_1))))) ↔ b(a(a(x0_1))) ↔ b(a(x0_1)) ↔
t
-
The critical peak s = b(y1) {ε}←a(b(y1))→ε c(b(b(a(a(y1))))) = t can be joined as follows.
s
↔ b(a(y1)) ↔ b(a(a(y1))) ↔ c(c(b(a(a(y1))))) ↔
t
-
The critical peak s = a(c(y1)) {1}←a(b(y1))→ε c(b(b(a(a(y1))))) = t can be joined as follows.
s
↔ c(y1) ↔ c(a(y1)) ↔ c(a(a(y1))) ↔ b(a(a(y1))) ↔ c(c(b(a(a(y1))))) ↔
t
-
The critical peak s = c(x1_1) {1}←c(c(c(x1_1)))→ε c(x1_1) = t can be joined as follows.
s
↔
t
The TRS C is chosen as:
|
a(x) |
→ |
x |
|
b(x) |
→ |
c(x) |
|
c(c(x)) |
→ |
x |
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 critical peak s = c(x1_1) {1}←c(c(c(x1_1)))→ε c(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.
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