YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
f(x1,g(x2)) |
→ |
f(x1,g(x1)) |
|
f(g(y1),y2) |
→ |
f(g(y1),g(y1)) |
|
g(a) |
→ |
g(b) |
| b |
→ |
a |
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(g(x0_1),g(x0_1)) {ε}←f(g(x0_1),g(y2))→ε f(g(x0_1),g(g(x0_1))) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(y1,g(b)) {2}←f(y1,g(a))→ε f(y1,g(y1)) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(g(y1),g(g(y1))) {ε}←f(g(y1),g(x0_2))→ε f(g(y1),g(y1)) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(g(b),y2) {1}←f(g(a),y2)→ε f(g(a),g(a)) = t can be joined as follows.
s
↔ f(g(b),g(b)) ↔ f(g(b),g(a)) ↔
t
The TRS C is chosen as:
|
f(x1,g(x2)) |
→ |
f(x1,g(x1)) |
|
f(g(y1),y2) |
→ |
f(g(y1),g(y1)) |
| b |
→ |
a |
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 = f(g(x0_1),g(x0_1)) {ε}←f(g(x0_1),g(y2))→ε f(g(x0_1),g(g(x0_1))) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(g(y1),g(g(y1))) {ε}←f(g(y1),g(x0_2))→ε f(g(y1),g(y1)) = t can be joined as follows.
s
↔
t
The TRS C is chosen as:
|
f(g(y1),y2) |
→ |
f(g(y1),g(y1)) |
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 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.
1.2.2.1 R is empty
There are no rules in the TRS R. Hence, R/S is relative terminating.
Confluence of C is proven as follows.
1.2.2.2 (Weakly) Orthogonal
Confluence is proven since the TRS is (weakly) orthogonal.
Tool configuration
Hakusan