Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_7_77_Bor03_GM)

The rewrite relation of the following TRS is considered.

a__zeros	→	cons(0,zeros)	(1)
a__tail(cons(X,XS))	→	mark(XS)	(2)
mark(zeros)	→	a__zeros	(3)
mark(tail(X))	→	a__tail(mark(X))	(4)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(5)
mark(0)	→	0	(6)
a__zeros	→	zeros	(7)
a__tail(X)	→	tail(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.

mark^#(cons(X1,X2))	→	mark^#(X1)	(9)
mark^#(tail(X))	→	mark^#(X)	(10)
mark^#(zeros)	→	a__zeros^#	(11)
a__tail^#(cons(X,XS))	→	mark^#(XS)	(12)
mark^#(tail(X))	→	a__tail^#(mark(X))	(13)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(tail(X))	→	a__tail^#(mark(X))	(13)
a__tail^#(cons(X,XS))	→	mark^#(XS)	(12)
mark^#(tail(X))	→	mark^#(X)	(10)
mark^#(cons(X1,X2))	→	mark^#(X1)	(9)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zeros]	=	1
[tail(x₁)]	=	x₁ + 17223
[mark^#(x₁)]	=	x₁ + 11797
[0]	=	17221
[a__zeros^#]	=	0
[mark(x₁)]	=	x₁ + 29019
[cons(x₁, x₂)]	=	x₁ + x₂ + 11798
[a__tail(x₁)]	=	x₁ + 17223
[a__tail^#(x₁)]	=	x₁ + 0
[a__zeros]	=	29020

together with the usable rules

mark(tail(X))	→	a__tail(mark(X))	(4)
a__tail(X)	→	tail(X)	(8)
a__zeros	→	cons(0,zeros)	(1)
mark(zeros)	→	a__zeros	(3)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(5)
a__zeros	→	zeros	(7)
mark(0)	→	0	(6)
a__tail(cons(X,XS))	→	mark(XS)	(2)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

mark^#(tail(X))	→	a__tail^#(mark(X))	(13)
a__tail^#(cons(X,XS))	→	mark^#(XS)	(12)
mark^#(tail(X))	→	mark^#(X)	(10)
mark^#(cons(X1,X2))	→	mark^#(X1)	(9)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.