YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
f(a,a,b,b) |
→ |
f(c,c,c,c) |
| a |
→ |
b |
| a |
→ |
c |
| b |
→ |
a |
| b |
→ |
c |
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(b,b,a,a) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,a,a) ↔ f(c,c,a,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(b,b,a,c) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,a,c) ↔ f(c,c,a,c) ↔
t
-
The critical peak s = f(b,b,a,b) {1, 2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,b,a,b) ↔ f(a,a,a,b) ↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(b,b,c,a) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,c,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(b,b,c,c) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,c,c) ↔
t
-
The critical peak s = f(b,b,c,b) {1, 2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,c,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(b,b,b,a) {1, 2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,b,b,a) ↔ f(a,a,b,a) ↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(b,b,b,c) {1, 2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,b,c) ↔ f(c,c,b,c) ↔
t
-
The critical peak s = f(b,b,b,b) {1, 2}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,b,b,b) ↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(b,c,a,a) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(b,c,a,c) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,c) ↔
t
-
The critical peak s = f(b,c,a,b) {1, 2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(b,c,c,a) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(b,c,c,c) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(b,c,c,b) {1, 2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(b,c,b,a) {1, 2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(b,c,b,c) {1, 2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,c) ↔
t
-
The critical peak s = f(b,c,b,b) {1, 2}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(b,a,a,a) {1, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,a,a) ↔ f(a,a,b,a) ↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(b,a,a,c) {1, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,a,c) ↔ f(c,c,a,c) ↔
t
-
The critical peak s = f(b,a,a,b) {1, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,a,b) ↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(b,a,c,a) {1, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,c,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(b,a,c,c) {1, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,c,c) ↔
t
-
The critical peak s = f(b,a,c,b) {1, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,c,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(b,a,b,a) {1, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,b,a) ↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(b,a,b,c) {1, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,b,c) ↔ f(c,c,b,c) ↔
t
-
The critical peak s = f(b,a,b,b) {1}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(c,b,a,a) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(c,b,a,c) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,c) ↔
t
-
The critical peak s = f(c,b,a,b) {1, 2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(c,b,c,a) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(c,b,c,c) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(c,b,c,b) {1, 2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(c,b,b,a) {1, 2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(c,b,b,c) {1, 2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,c) ↔
t
-
The critical peak s = f(c,b,b,b) {1, 2}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(c,c,a,a) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(c,c,a,c) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(c,c,a,b) {1, 2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(c,c,c,a) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(c,c,c,c) {1, 2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(c,c,c,b) {1, 2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(c,c,b,a) {1, 2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(c,c,b,c) {1, 2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(c,c,b,b) {1, 2}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(c,a,a,a) {1, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(c,a,a,c) {1, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,c) ↔
t
-
The critical peak s = f(c,a,a,b) {1, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(c,a,c,a) {1, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(c,a,c,c) {1, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(c,a,c,b) {1, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(c,a,b,a) {1, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(c,a,b,c) {1, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,c) ↔
t
-
The critical peak s = f(c,a,b,b) {1}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(a,b,a,a) {2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,a,a) ↔ f(c,c,a,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(a,b,a,c) {2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,a,c) ↔ f(c,c,a,c) ↔
t
-
The critical peak s = f(a,b,a,b) {2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,a,b) ↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(a,b,c,a) {2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,c,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(a,b,c,c) {2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,c,c) ↔
t
-
The critical peak s = f(a,b,c,b) {2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,c,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(a,b,b,a) {2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,b,a) ↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(a,b,b,c) {2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,b,b,c) ↔ f(c,c,b,c) ↔
t
-
The critical peak s = f(a,b,b,b) {2}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(a,c,a,a) {2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(a,c,a,c) {2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,c) ↔
t
-
The critical peak s = f(a,c,a,b) {2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,a,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(a,c,c,a) {2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(a,c,c,c) {2, 3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(a,c,c,b) {2, 3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(a,c,b,a) {2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(a,c,b,c) {2, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,c) ↔
t
-
The critical peak s = f(a,c,b,b) {2}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,c,b,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(a,a,a,a) {3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,b,a) ↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(a,a,a,c) {3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,a,c) ↔ f(c,c,a,c) ↔
t
-
The critical peak s = f(a,a,a,b) {3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(a,a,c,a) {3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,c,a) ↔ f(c,c,c,a) ↔
t
-
The critical peak s = f(a,a,c,c) {3, 4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,c,c) ↔
t
-
The critical peak s = f(a,a,c,b) {3}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,c,b) ↔ f(c,c,c,b) ↔
t
-
The critical peak s = f(a,a,b,a) {4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(a,a,b,b) ↔
t
-
The critical peak s = f(a,a,b,c) {4}←f(a,a,b,b)→ε f(c,c,c,c) = t can be joined as follows.
s
↔ f(c,a,b,c) ↔ f(c,c,b,c) ↔
t
-
The critical peak s = c {ε}←a→ε b = t can be joined as follows.
s
↔
t
-
The critical peak s = b {ε}←a→ε c = t can be joined as follows.
s
↔
t
-
The critical peak s = c {ε}←b→ε a = t can be joined as follows.
s
↔
t
-
The critical peak s = a {ε}←b→ε c = t can be joined as follows.
s
↔
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