YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
W(W(x)) |
→ |
W(x) |
|
B(I(x)) |
→ |
W(x) |
|
W(B(x)) |
→ |
B(x) |
|
F(H(x),y) |
→ |
F(H(x),G(y)) |
|
F(x,I(y)) |
→ |
F(G(x),I(y)) |
|
G(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 = W(W(x1_1)) {1}←W(W(W(x1_1)))→ε W(W(x1_1)) = t can be joined as follows.
s
↔
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(B(I(x1_1)))→ε B(I(x1_1)) = t can be joined as follows.
s
↔ W(x1_1) ↔
t
-
The critical peak s = F(G(H(y1)),I(x0_2)) {ε}←F(H(y1),I(x0_2))→ε F(H(y1),G(I(x0_2))) = t can be joined as follows.
s
↔ F(H(y1),I(x0_2)) ↔
t
-
The critical peak s = F(H(x0_1),G(I(y2))) {ε}←F(H(x0_1),I(y2))→ε F(G(H(x0_1)),I(y2)) = t can be joined as follows.
s
↔ F(H(x0_1),I(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.
|
W(B(I(x1_1))) |
→ |
W(W(x1_1)) |
|
W(B(I(x1_1))) |
→ |
B(I(x1_1)) |
Relative termination of P / R is proven as follows.
1.1 Rule Removal
Using the
linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the naturals
| [F(x1, x2)] |
= |
· x1 + · x2 +
|
| [B(x1)] |
= |
· x1 +
|
| [H(x1)] |
= |
· x1 +
|
| [G(x1)] |
= |
· x1 +
|
| [I(x1)] |
= |
· x1 +
|
| [W(x1)] |
= |
· x1 +
|
the
rule
|
W(B(I(x1_1))) |
→ |
B(I(x1_1)) |
remains in R.
Moreover,
the
rules
|
W(W(x)) |
→ |
W(x) |
|
W(B(x)) |
→ |
B(x) |
|
F(H(x),y) |
→ |
F(H(x),G(y)) |
|
F(x,I(y)) |
→ |
F(G(x),I(y)) |
|
G(x) |
→ |
x |
remain in S.
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the naturals
| [F(x1, x2)] |
= |
· x1 + · x2 +
|
| [B(x1)] |
= |
· x1 +
|
| [H(x1)] |
= |
· x1 +
|
| [G(x1)] |
= |
· x1 +
|
| [I(x1)] |
= |
· x1 +
|
| [W(x1)] |
= |
· x1 +
|
all rules of R could be removed.
Moreover,
the
rules
|
F(H(x),y) |
→ |
F(H(x),G(y)) |
|
F(x,I(y)) |
→ |
F(G(x),I(y)) |
|
G(x) |
→ |
x |
remain in S.
1.1.1.1 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 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