YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
f(f(f(x))) |
→ |
a |
|
f(f(a)) |
→ |
a |
|
f(a) |
→ |
a |
|
f(f(g(g(x)))) |
→ |
f(a) |
|
g(f(a)) |
→ |
a |
|
g(a) |
→ |
a |
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 = f(a) {1}←f(f(f(f(x1_1))))→ε a = t can be joined as follows.
s
↔
t
-
The critical peak s = f(a) {1}←f(f(f(a)))→ε a = t can be joined as follows.
s
↔
t
-
The critical peak s = f(f(a)) {1}←f(f(f(g(g(x1_1)))))→ε a = t can be joined as follows.
s
↔
t
-
The critical peak s = f(f(a)) {1.1}←f(f(f(f(f(x2_1)))))→ε a = t can be joined as follows.
s
↔
t
-
The critical peak s = f(f(a)) {1.1}←f(f(f(f(a))))→ε a = t can be joined as follows.
s
↔
t
-
The critical peak s = f(f(a)) {1.1}←f(f(f(a)))→ε a = t can be joined as follows.
s
↔
t
-
The critical peak s = f(f(f(a))) {1.1}←f(f(f(f(g(g(x2_1))))))→ε a = t can be joined as follows.
s
↔
t
-
The critical peak s = f(a) {1}←f(f(a))→ε a = t can be joined as follows.
s
↔
t
-
The critical peak s = f(f(g(a))) {1.1.1}←f(f(g(g(f(a)))))→ε f(a) = t can be joined as follows.
s
↔ f(f(a)) ↔ a ↔
t
-
The critical peak s = f(f(g(a))) {1.1.1}←f(f(g(g(a))))→ε f(a) = t can be joined as follows.
s
↔ f(f(a)) ↔ a ↔
t
-
The critical peak s = g(a) {1}←g(f(a))→ε a = 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.
|
f(f(f(f(x1_1)))) |
→ |
f(a) |
|
f(f(f(f(x1_1)))) |
→ |
a |
|
f(f(f(a))) |
→ |
f(a) |
|
f(f(f(a))) |
→ |
a |
|
f(f(f(g(g(x1_1))))) |
→ |
f(f(a)) |
|
f(f(f(g(g(x1_1))))) |
→ |
a |
|
f(f(f(f(f(x2_1))))) |
→ |
f(f(a)) |
|
f(f(f(f(f(x2_1))))) |
→ |
a |
|
f(f(f(f(a)))) |
→ |
f(f(a)) |
|
f(f(f(f(a)))) |
→ |
a |
|
f(f(f(a))) |
→ |
f(f(a)) |
|
f(f(f(f(g(g(x2_1)))))) |
→ |
f(f(f(a))) |
|
f(f(f(f(g(g(x2_1)))))) |
→ |
a |
|
f(f(a)) |
→ |
f(a) |
|
f(f(a)) |
→ |
a |
|
f(f(g(g(f(a))))) |
→ |
f(f(g(a))) |
|
f(f(g(g(f(a))))) |
→ |
f(a) |
|
f(f(g(g(a)))) |
→ |
f(f(g(a))) |
|
f(f(g(g(a)))) |
→ |
f(a) |
|
g(f(a)) |
→ |
g(a) |
|
g(f(a)) |
→ |
a |
Relative termination of P / R is proven as follows.
1.1 Rule Removal
Using the
recursive path order with the following precedence and status
| prec(g) |
= |
2 |
|
stat(g) |
= |
lex
|
| prec(a) |
= |
0 |
|
stat(a) |
= |
lex
|
| prec(f) |
= |
0 |
|
stat(f) |
= |
lex
|
all rules of R could be removed.
Moreover,
all rules of S could be removed.
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