YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
foo(0(x)) |
→ |
0(s(p(p(p(s(s(s(p(s(x)))))))))) |
|
foo(s(x)) |
→ |
p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x)))))))))))))))))))))))))) |
|
bar(0(x)) |
→ |
0(p(s(s(s(x))))) |
|
bar(s(x)) |
→ |
p(s(p(p(s(s(foo(s(p(p(s(s(x)))))))))))) |
|
p(p(s(x))) |
→ |
p(x) |
|
p(s(x)) |
→ |
x |
|
p(0(x)) |
→ |
0(s(s(s(s(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 = p(y1) {1}←p(p(s(y1)))→ε p(y1) = t can be joined as follows.
s
↔
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.
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 (Weakly) Orthogonal
Confluence is proven since the TRS is (weakly) orthogonal.
Tool configuration
Hakusan