Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex3_12_Luc96a_FR)

The rewrite relation of the following TRS is considered.

from(X)	→	cons(X,n__from(n__s(X)))	(1)
sel(0,cons(X,Y))	→	X	(2)
sel(s(X),cons(Y,Z))	→	sel(X,activate(Z))	(3)
from(X)	→	n__from(X)	(4)
s(X)	→	n__s(X)	(5)
activate(n__from(X))	→	from(activate(X))	(6)
activate(n__s(X))	→	s(activate(X))	(7)
activate(X)	→	X	(8)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

activate^#(n__from(X))	→	activate^#(X)	(9)
sel^#(s(X),cons(Y,Z))	→	activate^#(Z)	(10)
sel^#(s(X),cons(Y,Z))	→	sel^#(X,activate(Z))	(11)
activate^#(n__s(X))	→	s^#(activate(X))	(12)
activate^#(n__from(X))	→	from^#(activate(X))	(13)
activate^#(n__s(X))	→	activate^#(X)	(14)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

sel^#(s(X),cons(Y,Z))

→

sel^#(X,activate(Z))

(11)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[activate(x₁)]	=	8856
[n__from(x₁)]	=	8858
[activate^#(x₁)]	=	0
[n__s(x₁)]	=	2
[0]	=	0
[sel^#(x₁, x₂)]	=	x₁ + 0
[from(x₁)]	=	8857
[sel(x₁, x₂)]	=	0
[s^#(x₁)]	=	0
[from^#(x₁)]	=	0
[cons(x₁, x₂)]	=	x₁ + 8858

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

sel^#(s(X),cons(Y,Z))

→

sel^#(X,activate(Z))

(11)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

activate^#(n__s(X))	→	activate^#(X)	(14)
activate^#(n__from(X))	→	activate^#(X)	(9)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[activate(x₁)]	=	8856
[n__from(x₁)]	=	x₁ + 8858
[activate^#(x₁)]	=	x₁ + 0
[n__s(x₁)]	=	x₁ + 0
[0]	=	0
[sel^#(x₁, x₂)]	=	0
[from(x₁)]	=	8857
[sel(x₁, x₂)]	=	0
[s^#(x₁)]	=	0
[from^#(x₁)]	=	0
[cons(x₁, x₂)]	=	x₁ + 8858

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

activate^#(n__from(X))

→

activate^#(X)

(9)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

activate^#(n__s(X))

→

activate^#(X)

(14)

1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[activate(x₁)]	=	8856
[n__from(x₁)]	=	x₁ + 8858
[activate^#(x₁)]	=	x₁ + 0
[n__s(x₁)]	=	x₁ + 1
[0]	=	0
[sel^#(x₁, x₂)]	=	0
[from(x₁)]	=	8857
[sel(x₁, x₂)]	=	0
[s^#(x₁)]	=	0
[from^#(x₁)]	=	0
[cons(x₁, x₂)]	=	x₁ + 8858

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

activate^#(n__s(X))

→

activate^#(X)

(14)

could be deleted.

1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.