Certification Problem

Input (COPS 690)

We consider the TRS containing the following rules:

h(f(f(c)),b) f(h(h(h(c,h(f(h(c,f(b))),a)),b),c)) (1)
c c (2)
f(f(h(h(f(a),a),c))) f(h(f(c),b)) (3)
h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) c (4)

The underlying signature is as follows:

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

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:

h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) c (4)
f(f(h(h(f(a),a),c))) f(h(f(c),b)) (3)
h(f(f(c)),b) f(h(h(h(c,h(f(h(c,f(b))),a)),b),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:

h(f(h(f(b),h(h(f(h(c,f(c))),b),a))),h(a,c)) c (4)
f(f(h(h(f(a),a),c))) f(h(f(c),b)) (3)
h(f(f(c)),b) f(h(h(h(c,h(f(h(c,f(b))),a)),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.