YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
h(f(f(c)),b) |
→ |
f(h(h(h(c,h(f(h(c,f(b))),a)),b),c)) |
| c |
→ |
c |
|
f(f(h(h(f(a),a),c))) |
→ |
f(h(f(c),b)) |
|
h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) |
→ |
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 = h(f(f(c)),b) {1.1.1}←h(f(f(c)),b)→ε f(h(h(h(c,h(f(h(c,f(b))),a)),b),c)) = t can be joined as follows.
s
↔
t
-
The critical peak s = f(f(h(h(f(a),a),c))) {1.1.2}←f(f(h(h(f(a),a),c)))→ε f(h(f(c),b)) = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) {1.1.2.1.1.1.1, 1.1.2.1.1.1.2.1, 2.2}←h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c))→ε c = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) {1.1.2.1.1.1.1, 1.1.2.1.1.1.2.1}←h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c))→ε c = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) {1.1.2.1.1.1.1, 2.2}←h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c))→ε c = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) {1.1.2.1.1.1.1}←h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c))→ε c = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) {1.1.2.1.1.1.2.1, 2.2}←h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c))→ε c = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) {1.1.2.1.1.1.2.1}←h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c))→ε c = t can be joined as follows.
s
↔
t
-
The critical peak s = h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) {2.2}←h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c))→ε c = t can be joined as follows.
s
↔
t
The TRS C is chosen as:
|
h(f(f(c)),b) |
→ |
f(h(h(h(c,h(f(h(c,f(b))),a)),b),c)) |
|
f(f(h(h(f(a),a),c))) |
→ |
f(h(f(c),b)) |
|
h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) |
→ |
c |
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