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 2022)

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]	=	2
[g(x₁)]	=	1 · x₁ + 1
[h(x₁, x₂)]	=	1 · x₁ + 1 · x₂ + 1

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.