Certification Problem

Input (COPS 709)

We consider the TRS containing the following rules:

b → c (1)
a → a (2)

The underlying signature is as follows:

{b/0, c/0, a/0}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2021)

1 Persistent Decomposition (Many-Sorted)

Confluence is proven, because the maximal systems induced by the sorts in the following many-sorted sort attachment are confluent.
b : 0
c : 0
a : 1
The subsystems are

(1.1)

b → c (1)

(1.2)

a → a (2)

1.1 Redundant Rules Transformation

To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:

b → c (1)

All redundant rules that were added or removed can be simulated in 2 steps .

1.1.1 Critical Pair Closing System

Confluence is proven using the following terminating critical-pair-closing-system R:

There are no rules.

1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.

1.2 Redundant Rules Transformation

To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:

There are no rules.

All redundant rules that were added or removed can be simulated in 4 steps .

1.2.1 Parallel Closed

Confluence is proven since the TRS is (almost) parallel closed. The joins can be performed using 1 parallel step(s).