Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex15_Luc98_Z)

The rewrite relation of the following TRS is considered.

and(true,X)	→	activate(X)	(1)
and(false,Y)	→	false	(2)
if(true,X,Y)	→	activate(X)	(3)
if(false,X,Y)	→	activate(Y)	(4)
add(0,X)	→	activate(X)	(5)
add(s(X),Y)	→	s(n__add(activate(X),activate(Y)))	(6)
first(0,X)	→	nil	(7)
first(s(X),cons(Y,Z))	→	cons(activate(Y),n__first(activate(X),activate(Z)))	(8)
from(X)	→	cons(activate(X),n__from(n__s(activate(X))))	(9)
add(X1,X2)	→	n__add(X1,X2)	(10)
first(X1,X2)	→	n__first(X1,X2)	(11)
from(X)	→	n__from(X)	(12)
s(X)	→	n__s(X)	(13)
activate(n__add(X1,X2))	→	add(X1,X2)	(14)
activate(n__first(X1,X2))	→	first(X1,X2)	(15)
activate(n__from(X))	→	from(X)	(16)
activate(n__s(X))	→	s(X)	(17)
activate(X)	→	X	(18)

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.

add^#(s(X),Y)	→	activate^#(X)	(19)
activate^#(n__add(X1,X2))	→	add^#(X1,X2)	(20)
first^#(s(X),cons(Y,Z))	→	activate^#(X)	(21)
activate^#(n__from(X))	→	from^#(X)	(22)
and^#(true,X)	→	activate^#(X)	(23)
first^#(s(X),cons(Y,Z))	→	activate^#(Y)	(24)
first^#(s(X),cons(Y,Z))	→	activate^#(Z)	(25)
add^#(0,X)	→	activate^#(X)	(26)
if^#(true,X,Y)	→	activate^#(X)	(27)
from^#(X)	→	activate^#(X)	(28)
if^#(false,X,Y)	→	activate^#(Y)	(29)
add^#(s(X),Y)	→	activate^#(Y)	(30)
add^#(s(X),Y)	→	s^#(n__add(activate(X),activate(Y)))	(31)
activate^#(n__s(X))	→	s^#(X)	(32)
activate^#(n__first(X1,X2))	→	first^#(X1,X2)	(33)
from^#(X)	→	activate^#(X)	(28)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

from^#(X)	→	activate^#(X)	(28)
first^#(s(X),cons(Y,Z))	→	activate^#(Z)	(25)
activate^#(n__first(X1,X2))	→	first^#(X1,X2)	(33)
first^#(s(X),cons(Y,Z))	→	activate^#(Y)	(24)
add^#(s(X),Y)	→	activate^#(Y)	(30)
activate^#(n__from(X))	→	from^#(X)	(22)
first^#(s(X),cons(Y,Z))	→	activate^#(X)	(21)
activate^#(n__add(X1,X2))	→	add^#(X1,X2)	(20)
from^#(X)	→	activate^#(X)	(28)
add^#(0,X)	→	activate^#(X)	(26)
add^#(s(X),Y)	→	activate^#(X)	(19)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[n__first(x₁, x₂)]	=	x₁ + x₂ + 21239
[activate(x₁)]	=	0
[and(x₁, x₂)]	=	0
[n__from(x₁)]	=	x₁ + 2
[activate^#(x₁)]	=	x₁ + 0
[false]	=	0
[n__add(x₁, x₂)]	=	x₁ + x₂ + 2438
[true]	=	0
[n__s(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	0
[0]	=	17221
[from(x₁)]	=	0
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	x₁ + x₂ + 0
[nil]	=	0
[first(x₁, x₂)]	=	0
[from^#(x₁)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₁ + x₂ + 44083
[if^#(x₁, x₂, x₃)]	=	0
[add^#(x₁, x₂)]	=	x₁ + x₂ + 2437
[add(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	0

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

from^#(X)	→	activate^#(X)	(28)
first^#(s(X),cons(Y,Z))	→	activate^#(Z)	(25)
activate^#(n__first(X1,X2))	→	first^#(X1,X2)	(33)
first^#(s(X),cons(Y,Z))	→	activate^#(Y)	(24)
add^#(s(X),Y)	→	activate^#(Y)	(30)
activate^#(n__from(X))	→	from^#(X)	(22)
first^#(s(X),cons(Y,Z))	→	activate^#(X)	(21)
activate^#(n__add(X1,X2))	→	add^#(X1,X2)	(20)
from^#(X)	→	activate^#(X)	(28)
add^#(0,X)	→	activate^#(X)	(26)
add^#(s(X),Y)	→	activate^#(X)	(19)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.