Certification Problem

Input (COPS 37)

We consider the TRS containing the following rules:

f(a,x) → f(a,g(x)) (1)
a → b (2)
g(x) → x (3)

The underlying signature is as follows:

{f/2, a/0, g/1, b/0}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2021)

1 Critical Pair Closing System

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

g(x) → x (3)
a → b (2)

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[b] = 0
[a] = 1
[g(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
g(x) → x (3)
a → b (2)

1.1.1 R is empty

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