Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex7_BLR02_GM)

The rewrite relation of the following TRS is considered.

a__from(X)	→	cons(mark(X),from(s(X)))	(1)
a__head(cons(X,XS))	→	mark(X)	(2)
a__2nd(cons(X,XS))	→	a__head(mark(XS))	(3)
a__take(0,XS)	→	nil	(4)
a__take(s(N),cons(X,XS))	→	cons(mark(X),take(N,XS))	(5)
a__sel(0,cons(X,XS))	→	mark(X)	(6)
a__sel(s(N),cons(X,XS))	→	a__sel(mark(N),mark(XS))	(7)
mark(from(X))	→	a__from(mark(X))	(8)
mark(head(X))	→	a__head(mark(X))	(9)
mark(2nd(X))	→	a__2nd(mark(X))	(10)
mark(take(X1,X2))	→	a__take(mark(X1),mark(X2))	(11)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(12)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(13)
mark(s(X))	→	s(mark(X))	(14)
mark(0)	→	0	(15)
mark(nil)	→	nil	(16)
a__from(X)	→	from(X)	(17)
a__head(X)	→	head(X)	(18)
a__2nd(X)	→	2nd(X)	(19)
a__take(X1,X2)	→	take(X1,X2)	(20)
a__sel(X1,X2)	→	sel(X1,X2)	(21)

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.

a__sel^#(0,cons(X,XS))	→	mark^#(X)	(22)
mark^#(take(X1,X2))	→	a__take^#(mark(X1),mark(X2))	(23)
a__sel^#(s(N),cons(X,XS))	→	a__sel^#(mark(N),mark(XS))	(24)
mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(25)
mark^#(sel(X1,X2))	→	mark^#(X2)	(26)
mark^#(from(X))	→	mark^#(X)	(27)
mark^#(s(X))	→	mark^#(X)	(28)
a__sel^#(s(N),cons(X,XS))	→	mark^#(N)	(29)
a__2nd^#(cons(X,XS))	→	mark^#(XS)	(30)
mark^#(take(X1,X2))	→	mark^#(X1)	(31)
mark^#(sel(X1,X2))	→	mark^#(X1)	(32)
mark^#(head(X))	→	mark^#(X)	(33)
mark^#(from(X))	→	a__from^#(mark(X))	(34)
mark^#(2nd(X))	→	mark^#(X)	(35)
a__2nd^#(cons(X,XS))	→	a__head^#(mark(XS))	(36)
a__from^#(X)	→	mark^#(X)	(37)
mark^#(2nd(X))	→	a__2nd^#(mark(X))	(38)
mark^#(cons(X1,X2))	→	mark^#(X1)	(39)
a__head^#(cons(X,XS))	→	mark^#(X)	(40)
mark^#(take(X1,X2))	→	mark^#(X2)	(41)
a__sel^#(s(N),cons(X,XS))	→	mark^#(XS)	(42)
mark^#(head(X))	→	a__head^#(mark(X))	(43)
a__take^#(s(N),cons(X,XS))	→	mark^#(X)	(44)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a__take^#(s(N),cons(X,XS))	→	mark^#(X)	(44)
mark^#(head(X))	→	a__head^#(mark(X))	(43)
a__sel^#(s(N),cons(X,XS))	→	mark^#(N)	(29)
a__sel^#(s(N),cons(X,XS))	→	mark^#(XS)	(42)
mark^#(s(X))	→	mark^#(X)	(28)
mark^#(take(X1,X2))	→	mark^#(X2)	(41)
a__head^#(cons(X,XS))	→	mark^#(X)	(40)
mark^#(from(X))	→	mark^#(X)	(27)
mark^#(cons(X1,X2))	→	mark^#(X1)	(39)
mark^#(2nd(X))	→	a__2nd^#(mark(X))	(38)
a__from^#(X)	→	mark^#(X)	(37)
mark^#(sel(X1,X2))	→	mark^#(X2)	(26)
mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(25)
a__sel^#(s(N),cons(X,XS))	→	a__sel^#(mark(N),mark(XS))	(24)
a__2nd^#(cons(X,XS))	→	a__head^#(mark(XS))	(36)
mark^#(2nd(X))	→	mark^#(X)	(35)
mark^#(take(X1,X2))	→	a__take^#(mark(X1),mark(X2))	(23)
mark^#(from(X))	→	a__from^#(mark(X))	(34)
mark^#(head(X))	→	mark^#(X)	(33)
mark^#(sel(X1,X2))	→	mark^#(X1)	(32)
mark^#(take(X1,X2))	→	mark^#(X1)	(31)
a__sel^#(0,cons(X,XS))	→	mark^#(X)	(22)
a__2nd^#(cons(X,XS))	→	mark^#(XS)	(30)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 0
[a__head^#(x₁)]	=	x₁ + 47562
[a__from^#(x₁)]	=	x₁ + 47563
[a__from(x₁)]	=	x₁ + 35658
[take(x₁, x₂)]	=	max(x₁ + 26287, x₂ + 26287, 0)
[2nd(x₁)]	=	x₁ + 2999
[a__2nd(x₁)]	=	x₁ + 2999
[a__take^#(x₁, x₂)]	=	max(x₂ + 47563, 0)
[a__2nd^#(x₁)]	=	x₁ + 47563
[mark^#(x₁)]	=	x₁ + 47562
[0]	=	1
[from(x₁)]	=	x₁ + 35658
[sel(x₁, x₂)]	=	max(x₁ + 20977, x₂ + 0, 0)
[nil]	=	26288
[a__sel^#(x₁, x₂)]	=	max(x₁ + 68539, x₂ + 47562, 0)
[mark(x₁)]	=	x₁ + 0
[a__sel(x₁, x₂)]	=	max(x₁ + 20977, x₂ + 0, 0)
[head(x₁)]	=	x₁ + 1098
[cons(x₁, x₂)]	=	max(x₁ + 20978, x₂ + 0, 0)
[a__take(x₁, x₂)]	=	max(x₁ + 26287, x₂ + 26287, 0)
[a__head(x₁)]	=	x₁ + 1098

together with the usable rules

a__head(X)	→	head(X)	(18)
a__take(0,XS)	→	nil	(4)
mark(0)	→	0	(15)
mark(from(X))	→	a__from(mark(X))	(8)
a__from(X)	→	cons(mark(X),from(s(X)))	(1)
a__2nd(cons(X,XS))	→	a__head(mark(XS))	(3)
mark(nil)	→	nil	(16)
a__sel(X1,X2)	→	sel(X1,X2)	(21)
a__2nd(X)	→	2nd(X)	(19)
a__from(X)	→	from(X)	(17)
a__take(s(N),cons(X,XS))	→	cons(mark(X),take(N,XS))	(5)
mark(2nd(X))	→	a__2nd(mark(X))	(10)
a__sel(s(N),cons(X,XS))	→	a__sel(mark(N),mark(XS))	(7)
a__take(X1,X2)	→	take(X1,X2)	(20)
mark(s(X))	→	s(mark(X))	(14)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(12)
mark(take(X1,X2))	→	a__take(mark(X1),mark(X2))	(11)
mark(head(X))	→	a__head(mark(X))	(9)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(13)
a__sel(0,cons(X,XS))	→	mark(X)	(6)
a__head(cons(X,XS))	→	mark(X)	(2)

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

a__take^#(s(N),cons(X,XS))	→	mark^#(X)	(44)
mark^#(head(X))	→	a__head^#(mark(X))	(43)
a__sel^#(s(N),cons(X,XS))	→	mark^#(N)	(29)
mark^#(take(X1,X2))	→	mark^#(X2)	(41)
a__head^#(cons(X,XS))	→	mark^#(X)	(40)
mark^#(from(X))	→	mark^#(X)	(27)
mark^#(cons(X1,X2))	→	mark^#(X1)	(39)
mark^#(2nd(X))	→	a__2nd^#(mark(X))	(38)
a__from^#(X)	→	mark^#(X)	(37)
a__2nd^#(cons(X,XS))	→	a__head^#(mark(XS))	(36)
mark^#(2nd(X))	→	mark^#(X)	(35)
mark^#(take(X1,X2))	→	a__take^#(mark(X1),mark(X2))	(23)
mark^#(from(X))	→	a__from^#(mark(X))	(34)
mark^#(head(X))	→	mark^#(X)	(33)
mark^#(sel(X1,X2))	→	mark^#(X1)	(32)
mark^#(take(X1,X2))	→	mark^#(X1)	(31)
a__sel^#(0,cons(X,XS))	→	mark^#(X)	(22)
a__2nd^#(cons(X,XS))	→	mark^#(XS)	(30)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a__sel^#(s(N),cons(X,XS))	→	mark^#(XS)	(42)
a__sel^#(s(N),cons(X,XS))	→	a__sel^#(mark(N),mark(XS))	(24)
mark^#(s(X))	→	mark^#(X)	(28)
mark^#(sel(X1,X2))	→	mark^#(X2)	(26)
mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(25)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 0
[a__head^#(x₁)]	=	x₁ + 47562
[a__from^#(x₁)]	=	x₁ + 47563
[a__from(x₁)]	=	x₁ + 38104
[take(x₁, x₂)]	=	max(x₁ + 1, x₂ + 1, 0)
[2nd(x₁)]	=	x₁ + 42594
[a__2nd(x₁)]	=	x₁ + 42594
[a__take^#(x₁, x₂)]	=	max(x₂ + 47563, 0)
[a__2nd^#(x₁)]	=	x₁ + 47563
[mark^#(x₁)]	=	x₁ + 841
[0]	=	1
[from(x₁)]	=	x₁ + 38104
[sel(x₁, x₂)]	=	max(x₁ + 2, x₂ + 1, 0)
[nil]	=	2
[a__sel^#(x₁, x₂)]	=	max(x₂ + 842, 0)
[mark(x₁)]	=	x₁ + 0
[a__sel(x₁, x₂)]	=	max(x₁ + 2, x₂ + 1, 0)
[head(x₁)]	=	x₁ + 32869
[cons(x₁, x₂)]	=	max(x₁ + 1, x₂ + 0, 0)
[a__take(x₁, x₂)]	=	max(x₁ + 1, x₂ + 1, 0)
[a__head(x₁)]	=	x₁ + 32869

together with the usable rules

a__head(X)	→	head(X)	(18)
a__take(0,XS)	→	nil	(4)
mark(0)	→	0	(15)
mark(from(X))	→	a__from(mark(X))	(8)
a__from(X)	→	cons(mark(X),from(s(X)))	(1)
a__2nd(cons(X,XS))	→	a__head(mark(XS))	(3)
mark(nil)	→	nil	(16)
a__sel(X1,X2)	→	sel(X1,X2)	(21)
a__2nd(X)	→	2nd(X)	(19)
a__from(X)	→	from(X)	(17)
a__take(s(N),cons(X,XS))	→	cons(mark(X),take(N,XS))	(5)
mark(2nd(X))	→	a__2nd(mark(X))	(10)
a__sel(s(N),cons(X,XS))	→	a__sel(mark(N),mark(XS))	(7)
a__take(X1,X2)	→	take(X1,X2)	(20)
mark(s(X))	→	s(mark(X))	(14)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(12)
mark(take(X1,X2))	→	a__take(mark(X1),mark(X2))	(11)
mark(head(X))	→	a__head(mark(X))	(9)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(13)
a__sel(0,cons(X,XS))	→	mark(X)	(6)
a__head(cons(X,XS))	→	mark(X)	(2)

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

a__sel^#(s(N),cons(X,XS))	→	mark^#(XS)	(42)
mark^#(sel(X1,X2))	→	mark^#(X2)	(26)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

mark^#(s(X))

→

mark^#(X)

(28)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[a__head^#(x₁)]	=	0
[a__from^#(x₁)]	=	0
[a__from(x₁)]	=	x₁ + 1
[take(x₁, x₂)]	=	2
[2nd(x₁)]	=	x₁ + 31603
[a__2nd(x₁)]	=	31602
[a__take^#(x₁, x₂)]	=	0
[a__2nd^#(x₁)]	=	0
[mark^#(x₁)]	=	x₁ + 0
[0]	=	31602
[from(x₁)]	=	2
[sel(x₁, x₂)]	=	2
[nil]	=	31604
[a__sel^#(x₁, x₂)]	=	0
[mark(x₁)]	=	31601
[a__sel(x₁, x₂)]	=	x₂ + 1
[head(x₁)]	=	3
[cons(x₁, x₂)]	=	x₁ + 31598
[a__take(x₁, x₂)]	=	x₁ + 1
[a__head(x₁)]	=	x₁ + 2

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

mark^#(s(X))

→

mark^#(X)

(28)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

a__sel^#(s(N),cons(X,XS))

→

a__sel^#(mark(N),mark(XS))

(24)

1.1.1.1.1.1.2 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)	=	1	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__head^#)	=	0	status(a__head^#)	=	[]	list-extension(a__head^#)	=	Lex
prec(a__from^#)	=	0	status(a__from^#)	=	[]	list-extension(a__from^#)	=	Lex
prec(a__from)	=	2	status(a__from)	=	[]	list-extension(a__from)	=	Lex
prec(take)	=	4	status(take)	=	[]	list-extension(take)	=	Lex
prec(2nd)	=	5	status(2nd)	=	[]	list-extension(2nd)	=	Lex
prec(a__2nd)	=	5	status(a__2nd)	=	[]	list-extension(a__2nd)	=	Lex
prec(a__take^#)	=	0	status(a__take^#)	=	[]	list-extension(a__take^#)	=	Lex
prec(a__2nd^#)	=	0	status(a__2nd^#)	=	[]	list-extension(a__2nd^#)	=	Lex
prec(mark^#)	=	0	status(mark^#)	=	[]	list-extension(mark^#)	=	Lex
prec(0)	=	3	status(0)	=	[]	list-extension(0)	=	Lex
prec(from)	=	2	status(from)	=	[]	list-extension(from)	=	Lex
prec(sel)	=	4	status(sel)	=	[]	list-extension(sel)	=	Lex
prec(nil)	=	0	status(nil)	=	[]	list-extension(nil)	=	Lex
prec(a__sel^#)	=	2	status(a__sel^#)	=	[1, 2]	list-extension(a__sel^#)	=	Lex
prec(a__sel)	=	4	status(a__sel)	=	[]	list-extension(a__sel)	=	Lex
prec(head)	=	4	status(head)	=	[]	list-extension(head)	=	Lex
prec(cons)	=	0	status(cons)	=	[]	list-extension(cons)	=	Lex
prec(a__take)	=	4	status(a__take)	=	[]	list-extension(a__take)	=	Lex
prec(a__head)	=	4	status(a__head)	=	[]	list-extension(a__head)	=	Lex

and the following Max-polynomial interpretation

[s(x₁)]	=	x₁ + 0
[a__head^#(x₁)]	=	1
[a__from^#(x₁)]	=	1
[a__from(x₁)]	=	x₁ + 27232
[take(x₁, x₂)]	=	x₂ + 1326
[2nd(x₁)]	=	x₁ + 0
[a__2nd(x₁)]	=	x₁ + 0
[a__take^#(x₁, x₂)]	=	1
[a__2nd^#(x₁)]	=	1
[mark^#(x₁)]	=	1
[0]	=	1058
[from(x₁)]	=	x₁ + 27232
[sel(x₁, x₂)]	=	x₂ + 1
[nil]	=	1057
[a__sel^#(x₁, x₂)]	=	x₁ + x₂ + 1
[a__sel(x₁, x₂)]	=	x₂ + 1
[head(x₁)]	=	x₁ + 0
[cons(x₁, x₂)]	=	max(x₁ + 1325, x₂ + 0, 0)
[a__take(x₁, x₂)]	=	x₂ + 1326
[a__head(x₁)]	=	x₁ + 0

together with the usable rules

a__head(X)	→	head(X)	(18)
a__take(0,XS)	→	nil	(4)
mark(0)	→	0	(15)
mark(from(X))	→	a__from(mark(X))	(8)
a__from(X)	→	cons(mark(X),from(s(X)))	(1)
a__2nd(cons(X,XS))	→	a__head(mark(XS))	(3)
mark(nil)	→	nil	(16)
a__sel(X1,X2)	→	sel(X1,X2)	(21)
a__2nd(X)	→	2nd(X)	(19)
a__from(X)	→	from(X)	(17)
a__take(s(N),cons(X,XS))	→	cons(mark(X),take(N,XS))	(5)
mark(2nd(X))	→	a__2nd(mark(X))	(10)
a__sel(s(N),cons(X,XS))	→	a__sel(mark(N),mark(XS))	(7)
a__take(X1,X2)	→	take(X1,X2)	(20)
mark(s(X))	→	s(mark(X))	(14)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(12)
mark(take(X1,X2))	→	a__take(mark(X1),mark(X2))	(11)
mark(head(X))	→	a__head(mark(X))	(9)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(13)
a__sel(0,cons(X,XS))	→	mark(X)	(6)
a__head(cons(X,XS))	→	mark(X)	(2)

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

a__sel^#(s(N),cons(X,XS))

→

a__sel^#(mark(N),mark(XS))

(24)

could be deleted.

1.1.1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.