YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
f(x) |
→ |
g(x) |
|
f(x) |
→ |
h(f(x)) |
|
h(f(x)) |
→ |
h(g(x)) |
|
g(x) |
→ |
h(g(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 = h(f(y1)) {ε}←f(y1)→ε g(y1) = t can be joined as follows.
s
↔ h(g(y1)) ↔
t
-
The critical peak s = g(y1) {ε}←f(y1)→ε h(f(y1)) = t can be joined as follows.
s
↔ h(g(y1)) ↔
t
-
The critical peak s = h(g(y1)) {1}←h(f(y1))→ε h(g(y1)) = t can be joined as follows.
s
↔
t
-
The critical peak s = h(h(f(y1))) {1}←h(f(y1))→ε h(g(y1)) = t can be joined as follows.
s
↔ h(h(g(y1))) ↔
t
The TRS C is chosen as:
|
h(f(x)) |
→ |
h(g(x)) |
|
g(x) |
→ |
h(g(x)) |
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