Certification Problem

Input (COPS 979)

We consider the TRS containing the following rules:

a(x) b(x) (1)
a(b(x)) b(a(c(a(x)))) (2)
b(x) c(x) (3)
c(c(x)) x (4)

The underlying signature is as follows:

{a/1, b/1, c/1}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2023)

1 Redundant Rules Transformation

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

a(x) b(x) (1)
b(x) c(x) (3)
c(c(x)) x (4)

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

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 R is empty

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