Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex1_Luc02b_GM)

The rewrite relation of the following TRS is considered.

a__from(X)	→	cons(mark(X),from(s(X)))	(1)
a__first(0,Z)	→	nil	(2)
a__first(s(X),cons(Y,Z))	→	cons(mark(Y),first(X,Z))	(3)
a__sel(0,cons(X,Z))	→	mark(X)	(4)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(5)
mark(from(X))	→	a__from(mark(X))	(6)
mark(first(X1,X2))	→	a__first(mark(X1),mark(X2))	(7)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(8)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(9)
mark(s(X))	→	s(mark(X))	(10)
mark(0)	→	0	(11)
mark(nil)	→	nil	(12)
a__from(X)	→	from(X)	(13)
a__first(X1,X2)	→	first(X1,X2)	(14)
a__sel(X1,X2)	→	sel(X1,X2)	(15)

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^#(from(X))	→	mark^#(X)	(16)
mark^#(first(X1,X2))	→	mark^#(X2)	(17)
mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(18)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(X)	(19)
a__first^#(s(X),cons(Y,Z))	→	mark^#(Y)	(20)
a__from^#(X)	→	mark^#(X)	(21)
mark^#(sel(X1,X2))	→	mark^#(X1)	(22)
mark^#(s(X))	→	mark^#(X)	(23)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(Z)	(24)
mark^#(first(X1,X2))	→	mark^#(X1)	(25)
a__sel^#(0,cons(X,Z))	→	mark^#(X)	(26)
mark^#(cons(X1,X2))	→	mark^#(X1)	(27)
mark^#(from(X))	→	a__from^#(mark(X))	(28)
a__sel^#(s(X),cons(Y,Z))	→	a__sel^#(mark(X),mark(Z))	(29)
mark^#(sel(X1,X2))	→	mark^#(X2)	(30)
mark^#(first(X1,X2))	→	a__first^#(mark(X1),mark(X2))	(31)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(first(X1,X2))	→	a__first^#(mark(X1),mark(X2))	(31)
mark^#(sel(X1,X2))	→	mark^#(X1)	(22)
mark^#(sel(X1,X2))	→	mark^#(X2)	(30)
a__from^#(X)	→	mark^#(X)	(21)
a__sel^#(s(X),cons(Y,Z))	→	a__sel^#(mark(X),mark(Z))	(29)
mark^#(from(X))	→	a__from^#(mark(X))	(28)
mark^#(cons(X1,X2))	→	mark^#(X1)	(27)
a__sel^#(0,cons(X,Z))	→	mark^#(X)	(26)
a__first^#(s(X),cons(Y,Z))	→	mark^#(Y)	(20)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(X)	(19)
mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(18)
mark^#(first(X1,X2))	→	mark^#(X2)	(17)
mark^#(first(X1,X2))	→	mark^#(X1)	(25)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(Z)	(24)
mark^#(s(X))	→	mark^#(X)	(23)
mark^#(from(X))	→	mark^#(X)	(16)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 0
[a__from^#(x₁)]	=	x₁ + 2441
[a__first^#(x₁, x₂)]	=	max(x₂ + 2440, 0)
[a__from(x₁)]	=	x₁ + 2441
[mark^#(x₁)]	=	x₁ + 1
[0]	=	1
[from(x₁)]	=	x₁ + 2441
[sel(x₁, x₂)]	=	max(x₁ + 23682, x₂ + 23683, 0)
[nil]	=	2440
[a__sel^#(x₁, x₂)]	=	max(x₁ + 23682, x₂ + 21240, 0)
[mark(x₁)]	=	x₁ + 0
[a__sel(x₁, x₂)]	=	max(x₁ + 23682, x₂ + 23683, 0)
[first(x₁, x₂)]	=	max(x₁ + 2438, x₂ + 2440, 0)
[a__first(x₁, x₂)]	=	max(x₁ + 2438, x₂ + 2440, 0)
[cons(x₁, x₂)]	=	max(x₁ + 2441, x₂ + 0, 0)

together with the usable rules

a__sel(0,cons(X,Z))	→	mark(X)	(4)
a__sel(X1,X2)	→	sel(X1,X2)	(15)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(8)
a__from(X)	→	cons(mark(X),from(s(X)))	(1)
a__first(s(X),cons(Y,Z))	→	cons(mark(Y),first(X,Z))	(3)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(5)
mark(s(X))	→	s(mark(X))	(10)
mark(first(X1,X2))	→	a__first(mark(X1),mark(X2))	(7)
a__first(X1,X2)	→	first(X1,X2)	(14)
mark(nil)	→	nil	(12)
mark(0)	→	0	(11)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(9)
a__from(X)	→	from(X)	(13)
mark(from(X))	→	a__from(mark(X))	(6)
a__first(0,Z)	→	nil	(2)

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

mark^#(first(X1,X2))	→	a__first^#(mark(X1),mark(X2))	(31)
mark^#(sel(X1,X2))	→	mark^#(X1)	(22)
mark^#(sel(X1,X2))	→	mark^#(X2)	(30)
a__from^#(X)	→	mark^#(X)	(21)
mark^#(from(X))	→	a__from^#(mark(X))	(28)
mark^#(cons(X1,X2))	→	mark^#(X1)	(27)
a__sel^#(0,cons(X,Z))	→	mark^#(X)	(26)
a__first^#(s(X),cons(Y,Z))	→	mark^#(Y)	(20)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(X)	(19)
mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(18)
mark^#(first(X1,X2))	→	mark^#(X2)	(17)
mark^#(first(X1,X2))	→	mark^#(X1)	(25)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(Z)	(24)
mark^#(from(X))	→	mark^#(X)	(16)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

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

→

a__sel^#(mark(X),mark(Z))

(29)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(mark)

in combination with the following Weighted Path Order with the following precedence and status

prec(s)	=	2	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__from^#)	=	0	status(a__from^#)	=	[]	list-extension(a__from^#)	=	Lex
prec(a__first^#)	=	0	status(a__first^#)	=	[]	list-extension(a__first^#)	=	Lex
prec(a__from)	=	1	status(a__from)	=	[]	list-extension(a__from)	=	Lex
prec(mark^#)	=	0	status(mark^#)	=	[]	list-extension(mark^#)	=	Lex
prec(0)	=	3	status(0)	=	[]	list-extension(0)	=	Lex
prec(from)	=	1	status(from)	=	[]	list-extension(from)	=	Lex
prec(sel)	=	1	status(sel)	=	[]	list-extension(sel)	=	Lex
prec(nil)	=	3	status(nil)	=	[]	list-extension(nil)	=	Lex
prec(a__sel^#)	=	3	status(a__sel^#)	=	[1, 2]	list-extension(a__sel^#)	=	Lex
prec(a__sel)	=	1	status(a__sel)	=	[]	list-extension(a__sel)	=	Lex
prec(first)	=	2	status(first)	=	[2]	list-extension(first)	=	Lex
prec(a__first)	=	2	status(a__first)	=	[2]	list-extension(a__first)	=	Lex
prec(cons)	=	0	status(cons)	=	[]	list-extension(cons)	=	Lex

and the following Max-polynomial interpretation

[s(x₁)]	=	x₁ + 0
[a__from^#(x₁)]	=	1
[a__first^#(x₁, x₂)]	=	x₁ + 1
[a__from(x₁)]	=	x₁ + 2244
[mark^#(x₁)]	=	1
[0]	=	2241
[from(x₁)]	=	x₁ + 2244
[sel(x₁, x₂)]	=	x₂ + 2
[nil]	=	2241
[a__sel^#(x₁, x₂)]	=	x₁ + x₂ + 2244
[a__sel(x₁, x₂)]	=	x₂ + 2
[first(x₁, x₂)]	=	x₂ + 2242
[a__first(x₁, x₂)]	=	x₂ + 2242
[cons(x₁, x₂)]	=	max(x₁ + 2243, x₂ + 0, 0)

together with the usable rules

a__sel(0,cons(X,Z))	→	mark(X)	(4)
a__sel(X1,X2)	→	sel(X1,X2)	(15)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(8)
a__from(X)	→	cons(mark(X),from(s(X)))	(1)
a__first(s(X),cons(Y,Z))	→	cons(mark(Y),first(X,Z))	(3)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(5)
mark(s(X))	→	s(mark(X))	(10)
mark(first(X1,X2))	→	a__first(mark(X1),mark(X2))	(7)
a__first(X1,X2)	→	first(X1,X2)	(14)
mark(nil)	→	nil	(12)
mark(0)	→	0	(11)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(9)
a__from(X)	→	from(X)	(13)
mark(from(X))	→	a__from(mark(X))	(6)
a__first(0,Z)	→	nil	(2)

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

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

→

a__sel^#(mark(X),mark(Z))

(29)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

mark^#(s(X))

→

mark^#(X)

(23)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[a__from^#(x₁)]	=	0
[a__first^#(x₁, x₂)]	=	1
[a__from(x₁)]	=	35496
[mark^#(x₁)]	=	x₁ + 0
[0]	=	35496
[from(x₁)]	=	x₁ + 35497
[sel(x₁, x₂)]	=	x₂ + 35497
[nil]	=	35497
[a__sel^#(x₁, x₂)]	=	0
[mark(x₁)]	=	35495
[a__sel(x₁, x₂)]	=	35496
[first(x₁, x₂)]	=	x₁ + x₂ + 35497
[a__first(x₁, x₂)]	=	35496
[cons(x₁, x₂)]	=	x₂ + 35496

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

mark^#(s(X))

→

mark^#(X)

(23)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.