YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
a(a(x)) |
→ |
a(b(a(x))) |
|
b(a(b(x))) |
→ |
a(c(a(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 = a(a(b(a(x1_1)))) {1}←a(a(a(x1_1)))→ε a(b(a(a(x1_1)))) = t can be joined as follows.
s
↔ a(b(a(b(a(x1_1))))) ↔
t
-
The critical peak s = b(a(a(c(a(x2_1))))) {1.1}←b(a(b(a(b(x2_1)))))→ε a(c(a(a(b(x2_1))))) = t can be joined as follows.
s
↔ b(a(b(a(c(a(x2_1)))))) ↔ a(c(a(a(c(a(x2_1)))))) ↔ a(c(a(b(a(b(x2_1)))))) ↔
t
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.
|
a(a(a(x1_1))) |
→ |
a(a(b(a(x1_1)))) |
|
a(a(a(x1_1))) |
→ |
a(b(a(a(x1_1)))) |
|
b(a(b(a(b(x2_1))))) |
→ |
b(a(a(c(a(x2_1))))) |
|
b(a(b(a(b(x2_1))))) |
→ |
a(c(a(a(b(x2_1))))) |
Relative termination of P / R is proven as follows.
1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [b(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
the
rules
|
a(a(a(x1_1))) |
→ |
a(a(b(a(x1_1)))) |
|
a(a(a(x1_1))) |
→ |
a(b(a(a(x1_1)))) |
|
b(a(b(a(b(x2_1))))) |
→ |
a(c(a(a(b(x2_1))))) |
remain in R.
Moreover,
the
rules
|
a(a(x)) |
→ |
a(b(a(x))) |
|
b(a(b(x))) |
→ |
a(c(a(x))) |
remain in S.
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [b(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
the
rules
|
a(a(a(x1_1))) |
→ |
a(a(b(a(x1_1)))) |
|
a(a(a(x1_1))) |
→ |
a(b(a(a(x1_1)))) |
remain in R.
Moreover,
the
rules
|
a(a(x)) |
→ |
a(b(a(x))) |
|
b(a(b(x))) |
→ |
a(c(a(x))) |
remain in S.
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [b(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
the
rules
|
a(a(a(x1_1))) |
→ |
a(a(b(a(x1_1)))) |
|
a(a(a(x1_1))) |
→ |
a(b(a(a(x1_1)))) |
remain in R.
Moreover,
the
rule
remains in S.
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the naturals
| [b(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
all rules of R could be removed.
Moreover,
all rules of S could be removed.
1.1.1.1.1.1 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 (Weakly) Orthogonal
Confluence is proven since the TRS is (weakly) orthogonal.
Tool configuration
Hakusan