YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
F(H(x),y) |
→ |
F(H(x),I(I(y))) |
|
F(x,G(y)) |
→ |
F(I(x),G(y)) |
|
I(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(I(H(y1)),G(x0_2)) {ε}←F(H(y1),G(x0_2))→ε F(H(y1),I(I(G(x0_2)))) = t can be joined as follows.
s
↔ F(H(y1),G(x0_2)) ↔ F(H(y1),I(G(x0_2))) ↔
t
-
The critical peak s = F(H(x0_1),I(I(G(y2)))) {ε}←F(H(x0_1),G(y2))→ε F(I(H(x0_1)),G(y2)) = t can be joined as follows.
s
↔ F(H(x0_1),G(y2)) ↔
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