YES
Confluence Proof
Confluence Proof
by Hakusan
Input
The rewrite relation of the following TRS is considered.
|
-(+(x,y)) |
→ |
+(-(x),-(y)) |
|
+(-(x),-(y)) |
→ |
-(+(x,y)) |
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 = -(-(+(x1_1,x1_2))) {1}←-(+(-(x1_1),-(x1_2)))→ε +(-(-(x1_1)),-(-(x1_2))) = t can be joined as follows.
s
↔ -(+(-(x1_1),-(x1_2))) ↔
t
-
The critical peak s = +(+(-(x1_1),-(x1_2)),+(-(x2_1),-(x2_2))) {1, 2}←+(-(+(x1_1,x1_2)),-(+(x2_1,x2_2)))→ε -(+(+(x1_1,x1_2),+(x2_1,x2_2))) = t can be joined as follows.
s
↔ +(+(-(x1_1),-(x1_2)),-(+(x2_1,x2_2))) ↔ +(-(+(x1_1,x1_2)),-(+(x2_1,x2_2))) ↔
t
-
The critical peak s = +(+(-(x1_1),-(x1_2)),-(y2)) {1}←+(-(+(x1_1,x1_2)),-(y2))→ε -(+(+(x1_1,x1_2),y2)) = t can be joined as follows.
s
↔ +(-(+(x1_1,x1_2)),-(y2)) ↔
t
-
The critical peak s = +(-(y1),+(-(x2_1),-(x2_2))) {2}←+(-(y1),-(+(x2_1,x2_2)))→ε -(+(y1,+(x2_1,x2_2))) = t can be joined as follows.
s
↔ +(-(y1),-(+(x2_1,x2_2))) ↔
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