Certification Problem

Input (COPS 42)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{f/1, g/1, h/2, 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:

h(a,a) → g(g(a)) (3)

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[a] = 6
[h(x1, x2)] = 1 · x1 + 4 · x2 + 1
[g(x1)] = 1 · x1 + 0
all of the following rules can be deleted.
h(a,a) → g(g(a)) (3)

1.1.1 R is empty

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