Certification Problem

Input (COPS 705)

We consider the TRS containing the following rules:

a	→	c	(1)
c	→	a	(2)
c	→	h(a,c)	(3)

The underlying signature is as follows:

{a/0, c/0, h/2}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2023)

1 Redundant Rules Transformation

To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:

c	→	h(a,c)	(3)
c	→	a	(2)
a	→	c	(1)
c	→	h(a,h(a,c))	(4)
c	→	h(a,a)	(5)
c	→	h(c,c)	(6)
c	→	c	(7)
a	→	h(a,c)	(8)
a	→	a	(9)

All redundant rules that were added or removed can be simulated in 2 steps .

1.1 Decreasing Diagrams

1.1.2 Rule Labeling

Confluence is proven, because all critical peaks can be joined decreasingly using the following rule labeling function (rules that are not shown have label 0).

c → h(a,c) (3)
↦ 1
c → a (2)
↦ 0
a → c (1)
↦ 2
c → h(a,h(a,c)) (4)
↦ 3
c → h(a,a) (5)
↦ 8
c → h(c,c) (6)
↦ 0
c → c (7)
↦ 0
a → h(a,c) (8)
↦ 0
a → a (9)
↦ 0

The critical pairs can be joined as follows. Here, ↔ is always chosen as an appropriate rewrite relation which is automatically inferred by the certifier.

The critical peak s = h(a,c)←→_ε a = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(a,c)←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ t
The critical peak s = h(a,c)←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,h(a,h(a,c))) ↔ t
The critical peak s = h(a,c)←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,h(a,a)) ↔ t
The critical peak s = h(a,c)←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,h(c,c)) ↔ t
The critical peak s = h(a,c)←→_ε h(a,a) = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(a,c)←→_ε h(a,a) = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = h(a,c)←→_ε h(a,a) = t can be joined as follows.
s ↔ t
The critical peak s = h(a,c)←→_ε h(c,c) = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(a,c)←→_ε h(c,c) = t can be joined as follows.
s ↔ t
The critical peak s = h(a,c)←→_ε h(c,c) = t can be joined as follows.
s ↔ h(h(a,c),c) ↔ t
The critical peak s = h(a,c)←→_ε c = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(a,c)←→_ε c = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = h(a,c)←→_ε c = t can be joined as follows.
s ↔ h(a,a) ↔ t
The critical peak s = h(a,c)←→_ε c = t can be joined as follows.
s ↔ h(c,c) ↔ t
The critical peak s = a←→_ε h(a,c) = t can be joined as follows.
s ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ c ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ h(a,h(a,h(a,c))) ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ h(a,h(a,h(a,c))) ↔ h(a,h(a,h(a,c))) ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ h(a,h(a,h(a,c))) ↔ h(a,h(a,a)) ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ h(a,h(a,h(a,c))) ↔ h(a,h(a,h(a,a))) ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ h(a,h(a,h(a,c))) ↔ h(a,h(a,h(c,c))) ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ h(a,h(a,a)) ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ h(a,h(a,a)) ↔ h(a,h(a,a)) ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ h(a,h(c,c)) ↔ t
The critical peak s = a←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(a,c) ↔ h(a,h(c,c)) ↔ h(a,h(c,c)) ↔ t
The critical peak s = a←→_ε h(a,a) = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = a←→_ε h(c,c) = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = a←→_ε c = t can be joined as follows.
s ↔ a ↔ t
The critical peak s = a←→_ε c = t can be joined as follows.
s ↔ t
The critical peak s = a←→_ε c = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = c←→_ε h(a,c) = t can be joined as follows.
s ↔ t
The critical peak s = c←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = c←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,a) ↔ t
The critical peak s = c←→_ε h(a,c) = t can be joined as follows.
s ↔ h(c,c) ↔ t
The critical peak s = c←→_ε a = t can be joined as follows.
s ↔ c ↔ t
The critical peak s = c←→_ε a = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = c←→_ε a = t can be joined as follows.
s ↔ t
The critical peak s = h(a,h(a,c))←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = h(a,h(a,c))←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,h(a,h(a,c))) ↔ t
The critical peak s = h(a,h(a,c))←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,h(a,a)) ↔ t
The critical peak s = h(a,h(a,c))←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,h(c,c)) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ c ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(a,h(a,c))) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(a,h(a,c))) ↔ h(a,h(a,h(a,c))) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(a,a)) ↔ h(a,h(a,h(a,c))) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(a,h(a,a))) ↔ h(a,h(a,h(a,c))) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(a,h(c,c))) ↔ h(a,h(a,h(a,c))) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(a,a)) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(a,a)) ↔ h(a,h(a,a)) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(c,c)) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε a = t can be joined as follows.
s ↔ h(a,h(c,c)) ↔ h(a,h(c,c)) ↔ h(a,c) ↔ t
The critical peak s = h(a,h(a,c))←→_ε h(a,a) = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = h(a,h(a,c))←→_ε h(c,c) = t can be joined as follows.
s ↔ h(c,h(a,c)) ↔ t
The critical peak s = h(a,h(a,c))←→_ε c = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = h(a,a)←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,a) ↔ t
The critical peak s = h(a,a)←→_ε h(a,c) = t can be joined as follows.
s ↔ t
The critical peak s = h(a,a)←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = h(a,a)←→_ε a = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(a,a)←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ t
The critical peak s = h(a,a)←→_ε h(c,c) = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(a,a)←→_ε h(c,c) = t can be joined as follows.
s ↔ h(c,a) ↔ t
The critical peak s = h(a,a)←→_ε c = t can be joined as follows.
s ↔ h(a,a) ↔ t
The critical peak s = h(a,a)←→_ε c = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(a,a)←→_ε c = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = h(c,c)←→_ε h(a,c) = t can be joined as follows.
s ↔ h(c,c) ↔ t
The critical peak s = h(c,c)←→_ε h(a,c) = t can be joined as follows.
s ↔ h(h(a,c),c) ↔ t
The critical peak s = h(c,c)←→_ε h(a,c) = t can be joined as follows.
s ↔ t
The critical peak s = h(c,c)←→_ε a = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(c,c)←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ h(c,h(a,c)) ↔ t
The critical peak s = h(c,c)←→_ε h(a,a) = t can be joined as follows.
s ↔ h(c,a) ↔ t
The critical peak s = h(c,c)←→_ε h(a,a) = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(c,c)←→_ε c = t can be joined as follows.
s ↔ h(c,c) ↔ t
The critical peak s = h(c,c)←→_ε c = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = c←→_ε h(a,c) = t can be joined as follows.
s ↔ t
The critical peak s = c←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = c←→_ε h(a,c) = t can be joined as follows.
s ↔ h(a,a) ↔ t
The critical peak s = c←→_ε h(a,c) = t can be joined as follows.
s ↔ h(c,c) ↔ t
The critical peak s = c←→_ε a = t can be joined as follows.
s ↔ c ↔ t
The critical peak s = c←→_ε a = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = c←→_ε a = t can be joined as follows.
s ↔ t
The critical peak s = c←→_ε h(a,h(a,c)) = t can be joined as follows.
s ↔ t
The critical peak s = c←→_ε h(a,a) = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = c←→_ε h(a,a) = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = c←→_ε h(a,a) = t can be joined as follows.
s ↔ t
The critical peak s = c←→_ε h(c,c) = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = c←→_ε h(c,c) = t can be joined as follows.
s ↔ t
The critical peak s = h(a,c)←→_ε c = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = h(a,c)←→_ε c = t can be joined as follows.
s ↔ h(a,h(a,c)) ↔ t
The critical peak s = h(a,c)←→_ε c = t can be joined as follows.
s ↔ h(a,a) ↔ t
The critical peak s = h(a,c)←→_ε c = t can be joined as follows.
s ↔ h(c,c) ↔ t
The critical peak s = h(a,c)←→_ε a = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = a←→_ε c = t can be joined as follows.
s ↔ a ↔ t
The critical peak s = a←→_ε c = t can be joined as follows.
s ↔ t
The critical peak s = a←→_ε c = t can be joined as follows.
s ↔ h(a,c) ↔ t
The critical peak s = a←→_ε h(a,c) = t can be joined as follows.
s ↔ t