Certification Problem

Input (COPS 578)

We consider the TRS containing the following rules:

s(p(x)) → x (1)
p(s(x)) → x (2)
+(x,0) → x (3)
+(x,s(y)) → s(+(x,y)) (4)
+(x,p(y)) → p(+(x,y)) (5)
-(0,0) → 0 (6)
-(x,s(y)) → p(-(x,y)) (7)
-(x,p(y)) → s(-(x,y)) (8)
-(p(x),y) → p(-(x,y)) (9)
-(s(x),y) → s(-(x,y)) (10)

The underlying signature is as follows:

{s/1, p/1, +/2, 0/0, -/2}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2022)

1 Locally confluent and terminating

Confluence is proven by showing local confluence and termination.

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[-(x1, x2)] = 2 · x1 + 2 · x2 + 6
[s(x1)] = 1 · x1 + 3
[+(x1, x2)] = 4 · x1 + 4 · x2 + 3
[p(x1)] = 1 · x1 + 4
[0] = 1
all of the following rules can be deleted.
s(p(x)) → x (1)
p(s(x)) → x (2)
+(x,0) → x (3)
+(x,s(y)) → s(+(x,y)) (4)
+(x,p(y)) → p(+(x,y)) (5)
-(0,0) → 0 (6)
-(x,s(y)) → p(-(x,y)) (7)
-(x,p(y)) → s(-(x,y)) (8)
-(p(x),y) → p(-(x,y)) (9)
-(s(x),y) → s(-(x,y)) (10)

1.1.1 R is empty

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

1.2 Local Confluence Proof

All critical pairs are joinable which can be seen by computing normal forms of all critical pairs.