YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
H(F(x,y)) |
→ |
F(H(R(x)),y) |
|
F(x,K(y,z)) |
→ |
G(P(y),Q(z,x)) |
|
H(Q(x,y)) |
→ |
Q(x,H(R(y))) |
|
Q(x,H(R(y))) |
→ |
H(Q(x,y)) |
|
H(G(x,y)) |
→ |
G(x,H(y)) |
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 = H(G(P(x1_2),Q(x1_3,y1))) {1}←H(F(y1,K(x1_2,x1_3)))→ε F(H(R(y1)),K(x1_2,x1_3)) = t can be joined as follows.
s
↔ G(P(x1_2),H(Q(x1_3,y1))) ↔ G(P(x1_2),Q(x1_3,H(R(y1)))) ↔
t
-
The critical peak s = H(H(Q(y1,x1_2))) {1}←H(Q(y1,H(R(x1_2))))→ε Q(y1,H(R(H(R(x1_2))))) = t can be joined as follows.
s
↔ H(Q(y1,H(R(x1_2)))) ↔
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.
|
H(F(y1,K(x1_2,x1_3))) |
→ |
H(G(P(x1_2),Q(x1_3,y1))) |
|
H(F(y1,K(x1_2,x1_3))) |
→ |
F(H(R(y1)),K(x1_2,x1_3)) |
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
| [Q(x1, x2)] |
= |
· x1 + · x2 +
|
| [F(x1, x2)] |
= |
· x1 + · x2 +
|
| [P(x1)] |
= |
· x1 +
|
| [G(x1, x2)] |
= |
· x1 + · x2 +
|
| [K(x1, x2)] |
= |
· x1 + · x2 +
|
| [R(x1)] |
= |
· x1 +
|
| [H(x1)] |
= |
· x1 +
|
all rules of R could be removed.
Moreover,
the
rules
|
H(F(x,y)) |
→ |
F(H(R(x)),y) |
|
H(Q(x,y)) |
→ |
Q(x,H(R(y))) |
|
Q(x,H(R(y))) |
→ |
H(Q(x,y)) |
remain in S.
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