Certification Problem

Input (COPS 982)

We consider the TRS containing the following rules:

a(x) → x (1)
a(a(x)) → b(c(x)) (2)
b(x) → x (3)
c(x) → x (4)
c(b(x)) → b(a(c(x))) (5)

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 Critical Pair Closing System

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

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

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[c(x1)] = 1 · x1 + 1
[a(x1)] = 1 · x1 + 1
[b(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
a(x) → x (1)
c(x) → x (4)
b(x) → x (3)

1.1.1 R is empty

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