Certification Problem

Input (COPS 223)

We consider the TRS containing the following rules:

f(f(a)) → f(f(f(a))) (1)
f(f(a)) → f(a) (2)

The underlying signature is as follows:

{f/1, a/0}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2020)

1 Critical Pair Closing System

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

f(f(a)) → f(a) (2)

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[f(x1)] = 1 · x1 + 5
[a] = 3
all of the following rules can be deleted.
f(f(a)) → f(a) (2)

1.1.1 R is empty

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