Certification Problem

Input (COPS 691)

We consider the TRS containing the following rules:

h(a,a) → f(c) (1)
a → a (2)
b → h(c,a) (3)

The underlying signature is as follows:

{h/2, a/0, f/1, c/0, b/0}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2022)

1 Redundant Rules Transformation

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

b → h(c,a) (3)
h(a,a) → f(c) (1)

All redundant rules that were added or removed can be simulated in 1 steps .

1.1 Redundant Rules Transformation

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

b → h(c,a) (3)
h(a,a) → f(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.