YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
h(f,a,a) |
→ |
h(g,a,a) |
|
h(g,a,a) |
→ |
h(f,a,a) |
| a |
→ |
a' |
|
h(x,a',y) |
→ |
h(x,y,y) |
| g |
→ |
f |
| f |
→ |
g |
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,a',a') {1, 2, 3}←h(f,a,a)→ε h(g,a,a) = t can be joined as follows.
s
↔ h(g,a',a) ↔
t
-
The critical peak s = h(g,a',a) {1, 2}←h(f,a,a)→ε h(g,a,a) = t can be joined as follows.
s
↔
t
-
The critical peak s = h(g,a,a') {1, 3}←h(f,a,a)→ε h(g,a,a) = t can be joined as follows.
s
↔
t
-
The critical peak s = h(g,a,a) {1}←h(f,a,a)→ε h(g,a,a) = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f,a',a') {2, 3}←h(f,a,a)→ε h(g,a,a) = t can be joined as follows.
s
↔ h(f,a',a) ↔ h(f,a,a) ↔
t
-
The critical peak s = h(f,a',a) {2}←h(f,a,a)→ε h(g,a,a) = t can be joined as follows.
s
↔ h(f,a,a) ↔
t
-
The critical peak s = h(f,a,a') {3}←h(f,a,a)→ε h(g,a,a) = t can be joined as follows.
s
↔ h(f,a,a) ↔
t
-
The critical peak s = h(f,a',a') {1, 2, 3}←h(g,a,a)→ε h(f,a,a) = t can be joined as follows.
s
↔ h(f,a',a) ↔
t
-
The critical peak s = h(f,a',a) {1, 2}←h(g,a,a)→ε h(f,a,a) = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f,a,a') {1, 3}←h(g,a,a)→ε h(f,a,a) = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f,a,a) {1}←h(g,a,a)→ε h(f,a,a) = t can be joined as follows.
s
↔
t
-
The critical peak s = h(g,a',a') {2, 3}←h(g,a,a)→ε h(f,a,a) = t can be joined as follows.
s
↔ h(g,a',a) ↔ h(g,a,a) ↔
t
-
The critical peak s = h(g,a',a) {2}←h(g,a,a)→ε h(f,a,a) = t can be joined as follows.
s
↔ h(g,a,a) ↔
t
-
The critical peak s = h(g,a,a') {3}←h(g,a,a)→ε h(f,a,a) = t can be joined as follows.
s
↔ h(g,a,a) ↔
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.
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