YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
f(a,a) |
→ |
g(f(a,a)) |
| a |
→ |
b |
|
f(b,x) |
→ |
g(f(x,x)) |
|
f(x,b) |
→ |
g(f(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 = f(b,b) {1, 2}←f(a,a)→ε g(f(a,a)) = t can be joined as follows.
s
↔ g(f(b,b)) ↔ g(f(b,a)) ↔
t
-
The critical peak s = f(b,a) {1}←f(a,a)→ε g(f(a,a)) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(a,b) {2}←f(a,a)→ε g(f(a,a)) = t can be joined as follows.
s
↔
t
-
The critical peak s = g(f(b,b)) {ε}←f(b,b)→ε g(f(b,b)) = t can be joined as follows.
s
↔
t
-
The critical peak s = g(f(b,b)) {ε}←f(b,b)→ε g(f(b,b)) = t can be joined as follows.
s
↔
t
The TRS C is chosen as:
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