Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex6_Luc98_GM)

The rewrite relation of the following TRS is considered.

a__first(0,X)	→	nil	(1)
a__first(s(X),cons(Y,Z))	→	cons(mark(Y),first(X,Z))	(2)
a__from(X)	→	cons(mark(X),from(s(X)))	(3)
mark(first(X1,X2))	→	a__first(mark(X1),mark(X2))	(4)
mark(from(X))	→	a__from(mark(X))	(5)
mark(0)	→	0	(6)
mark(nil)	→	nil	(7)
mark(s(X))	→	s(mark(X))	(8)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(9)
a__first(X1,X2)	→	first(X1,X2)	(10)
a__from(X)	→	from(X)	(11)

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)	(12)
mark^#(s(X))	→	mark^#(X)	(13)
mark^#(first(X1,X2))	→	mark^#(X2)	(14)
mark^#(first(X1,X2))	→	a__first^#(mark(X1),mark(X2))	(15)
a__from^#(X)	→	mark^#(X)	(16)
mark^#(from(X))	→	a__from^#(mark(X))	(17)
a__first^#(s(X),cons(Y,Z))	→	mark^#(Y)	(18)
mark^#(first(X1,X2))	→	mark^#(X1)	(19)
mark^#(from(X))	→	mark^#(X)	(20)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(from(X))	→	mark^#(X)	(20)
mark^#(first(X1,X2))	→	mark^#(X1)	(19)
a__first^#(s(X),cons(Y,Z))	→	mark^#(Y)	(18)
mark^#(from(X))	→	a__from^#(mark(X))	(17)
mark^#(first(X1,X2))	→	mark^#(X2)	(14)
mark^#(s(X))	→	mark^#(X)	(13)
a__from^#(X)	→	mark^#(X)	(16)
mark^#(first(X1,X2))	→	a__first^#(mark(X1),mark(X2))	(15)
mark^#(cons(X1,X2))	→	mark^#(X1)	(12)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[a__from^#(x₁)]	=	x₁ + 1
[a__first^#(x₁, x₂)]	=	x₁ + x₂ + 0
[a__from(x₁)]	=	x₁ + 2
[mark^#(x₁)]	=	x₁ + 0
[0]	=	29604
[from(x₁)]	=	x₁ + 2
[nil]	=	11798
[mark(x₁)]	=	x₁ + 0
[first(x₁, x₂)]	=	x₁ + x₂ + 1
[a__first(x₁, x₂)]	=	x₁ + x₂ + 1
[cons(x₁, x₂)]	=	x₁ + 2

together with the usable rules

mark(first(X1,X2))	→	a__first(mark(X1),mark(X2))	(4)
mark(s(X))	→	s(mark(X))	(8)
a__first(0,X)	→	nil	(1)
a__from(X)	→	cons(mark(X),from(s(X)))	(3)
mark(from(X))	→	a__from(mark(X))	(5)
a__first(X1,X2)	→	first(X1,X2)	(10)
mark(nil)	→	nil	(7)
a__from(X)	→	from(X)	(11)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(9)
mark(0)	→	0	(6)
a__first(s(X),cons(Y,Z))	→	cons(mark(Y),first(X,Z))	(2)

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

mark^#(from(X))	→	mark^#(X)	(20)
mark^#(first(X1,X2))	→	mark^#(X1)	(19)
a__first^#(s(X),cons(Y,Z))	→	mark^#(Y)	(18)
mark^#(from(X))	→	a__from^#(mark(X))	(17)
mark^#(first(X1,X2))	→	mark^#(X2)	(14)
mark^#(s(X))	→	mark^#(X)	(13)
a__from^#(X)	→	mark^#(X)	(16)
mark^#(first(X1,X2))	→	a__first^#(mark(X1),mark(X2))	(15)
mark^#(cons(X1,X2))	→	mark^#(X1)	(12)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.