Certification Problem

Input (COPS 614)

We consider the TRS containing the following rules:

+(0,y)	→	y	(1)
+(s(x),y)	→	s(+(x,y))	(2)
+(p(x),y)	→	+(x,p(y))	(3)
p(s(x))	→	s(p(x))	(4)
s(p(x))	→	p(s(x))	(5)
p(s(0))	→	0	(6)

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2023)

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t₀	=	+(p(s(0)),f5)
	→	+(0,f5)
	=	t₁

t₀	=	+(p(s(0)),f5)
	→	+(s(0),p(f5))
	=	t₁

The two resulting terms cannot be joined for the following reason:

The reachable terms of these two terms are approximated via the following two tree automata, and the tree automata have an empty intersection.
- Automaton 1
  - final states:
    
    {11}
  - transitions:
    
    12 → 11
    
    f5 → 12
    
    +(13,12) → 11
    
    0 → 13
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {14}
  - transitions:
    
    16 → 107
    
    f5 → 15
    
    s(17) → 18
    
    s(107) → 14
    
    s(15) → 277
    
    +(18,16) → 14
    
    +(17,16) → 107
    
    0 → 17
    
    p(15) → 16
    
    p(277) → 14
  The automaton is closed under rewriting as it is compatible.