Certification Problem

Input (COPS 212)

We consider the TRS containing the following rules:

The underlying signature is as follows:

{a/0, b/0, f/1}

Prove or disprove confluence.

No.

The system is not confluent due to the following forking derivations.

The two resulting terms cannot be joined for the following reason:

We take the usable rules of the first term (w.r.t. the TRS for the first term) and the usable rules of the second term (w.r.t. the TRS for the second term). Then the terms are not joinable w.r.t. the resulting TRSs.
The first mentioned term is different (not smaller than) the second one w.r.t. the following interpretation (or vice versa). Moreover, the (reversed) rules are (quasi)-models of the interpretation. The following interpretations form a model of the rules.
As carrier we take the set {0,1}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 2):

[b] = 1

[f(x₁)] = 1 + 1x₁

[b]	=	1
[f(x₁)]	=	1 + 1x₁