Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex5_7_Luc97_GM)

The rewrite relation of the following TRS is considered.

a__dbl(0)	→	0	(1)
a__dbl(s(X))	→	s(s(dbl(X)))	(2)
a__dbls(nil)	→	nil	(3)
a__dbls(cons(X,Y))	→	cons(dbl(X),dbls(Y))	(4)
a__sel(0,cons(X,Y))	→	mark(X)	(5)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(6)
a__indx(nil,X)	→	nil	(7)
a__indx(cons(X,Y),Z)	→	cons(sel(X,Z),indx(Y,Z))	(8)
a__from(X)	→	cons(X,from(s(X)))	(9)
a__dbl1(0)	→	01	(10)
a__dbl1(s(X))	→	s1(s1(a__dbl1(mark(X))))	(11)
a__sel1(0,cons(X,Y))	→	mark(X)	(12)
a__sel1(s(X),cons(Y,Z))	→	a__sel1(mark(X),mark(Z))	(13)
a__quote(0)	→	01	(14)
a__quote(s(X))	→	s1(a__quote(mark(X)))	(15)
a__quote(dbl(X))	→	a__dbl1(mark(X))	(16)
a__quote(sel(X,Y))	→	a__sel1(mark(X),mark(Y))	(17)
mark(dbl(X))	→	a__dbl(mark(X))	(18)
mark(dbls(X))	→	a__dbls(mark(X))	(19)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(20)
mark(indx(X1,X2))	→	a__indx(mark(X1),X2)	(21)
mark(from(X))	→	a__from(X)	(22)
mark(dbl1(X))	→	a__dbl1(mark(X))	(23)
mark(sel1(X1,X2))	→	a__sel1(mark(X1),mark(X2))	(24)
mark(quote(X))	→	a__quote(mark(X))	(25)
mark(0)	→	0	(26)
mark(s(X))	→	s(X)	(27)
mark(nil)	→	nil	(28)
mark(cons(X1,X2))	→	cons(X1,X2)	(29)
mark(01)	→	01	(30)
mark(s1(X))	→	s1(mark(X))	(31)
a__dbl(X)	→	dbl(X)	(32)
a__dbls(X)	→	dbls(X)	(33)
a__sel(X1,X2)	→	sel(X1,X2)	(34)
a__indx(X1,X2)	→	indx(X1,X2)	(35)
a__from(X)	→	from(X)	(36)
a__dbl1(X)	→	dbl1(X)	(37)
a__sel1(X1,X2)	→	sel1(X1,X2)	(38)
a__quote(X)	→	quote(X)	(39)

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^#(s(X),cons(Y,Z))	→	mark^#(X)	(40)
mark^#(indx(X1,X2))	→	mark^#(X1)	(41)
mark^#(dbl(X))	→	mark^#(X)	(42)
a__sel1^#(s(X),cons(Y,Z))	→	mark^#(Z)	(43)
mark^#(dbl1(X))	→	mark^#(X)	(44)
mark^#(sel1(X1,X2))	→	a__sel1^#(mark(X1),mark(X2))	(45)
mark^#(sel1(X1,X2))	→	mark^#(X2)	(46)
a__quote^#(sel(X,Y))	→	mark^#(X)	(47)
a__quote^#(sel(X,Y))	→	a__sel1^#(mark(X),mark(Y))	(48)
mark^#(dbl1(X))	→	a__dbl1^#(mark(X))	(49)
a__quote^#(dbl(X))	→	mark^#(X)	(50)
a__sel1^#(0,cons(X,Y))	→	mark^#(X)	(51)
a__quote^#(dbl(X))	→	a__dbl1^#(mark(X))	(52)
a__quote^#(sel(X,Y))	→	mark^#(Y)	(53)
mark^#(sel(X1,X2))	→	mark^#(X2)	(54)
a__dbl1^#(s(X))	→	a__dbl1^#(mark(X))	(55)
mark^#(sel1(X1,X2))	→	mark^#(X1)	(56)
a__quote^#(s(X))	→	mark^#(X)	(57)
a__sel1^#(s(X),cons(Y,Z))	→	mark^#(X)	(58)
mark^#(indx(X1,X2))	→	a__indx^#(mark(X1),X2)	(59)
mark^#(from(X))	→	a__from^#(X)	(60)
mark^#(dbl(X))	→	a__dbl^#(mark(X))	(61)
mark^#(dbls(X))	→	mark^#(X)	(62)
mark^#(quote(X))	→	mark^#(X)	(63)
a__quote^#(s(X))	→	a__quote^#(mark(X))	(64)
mark^#(sel(X1,X2))	→	mark^#(X1)	(65)
mark^#(dbls(X))	→	a__dbls^#(mark(X))	(66)
a__sel^#(0,cons(X,Y))	→	mark^#(X)	(67)
mark^#(quote(X))	→	a__quote^#(mark(X))	(68)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(Z)	(69)
a__sel^#(s(X),cons(Y,Z))	→	a__sel^#(mark(X),mark(Z))	(70)
a__dbl1^#(s(X))	→	mark^#(X)	(71)
mark^#(s1(X))	→	mark^#(X)	(72)
a__sel1^#(s(X),cons(Y,Z))	→	a__sel1^#(mark(X),mark(Z))	(73)
mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(74)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(74)
a__quote^#(sel(X,Y))	→	mark^#(Y)	(53)
a__sel1^#(s(X),cons(Y,Z))	→	a__sel1^#(mark(X),mark(Z))	(73)
a__quote^#(dbl(X))	→	a__dbl1^#(mark(X))	(52)
a__sel1^#(0,cons(X,Y))	→	mark^#(X)	(51)
mark^#(s1(X))	→	mark^#(X)	(72)
a__quote^#(dbl(X))	→	mark^#(X)	(50)
mark^#(dbl1(X))	→	a__dbl1^#(mark(X))	(49)
a__dbl1^#(s(X))	→	mark^#(X)	(71)
a__sel^#(s(X),cons(Y,Z))	→	a__sel^#(mark(X),mark(Z))	(70)
a__quote^#(sel(X,Y))	→	a__sel1^#(mark(X),mark(Y))	(48)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(Z)	(69)
a__quote^#(sel(X,Y))	→	mark^#(X)	(47)
mark^#(quote(X))	→	a__quote^#(mark(X))	(68)
a__sel^#(0,cons(X,Y))	→	mark^#(X)	(67)
mark^#(sel1(X1,X2))	→	mark^#(X2)	(46)
mark^#(sel1(X1,X2))	→	a__sel1^#(mark(X1),mark(X2))	(45)
mark^#(dbl1(X))	→	mark^#(X)	(44)
a__quote^#(s(X))	→	a__quote^#(mark(X))	(64)
mark^#(sel(X1,X2))	→	mark^#(X1)	(65)
mark^#(quote(X))	→	mark^#(X)	(63)
a__sel1^#(s(X),cons(Y,Z))	→	mark^#(Z)	(43)
mark^#(dbls(X))	→	mark^#(X)	(62)
a__sel1^#(s(X),cons(Y,Z))	→	mark^#(X)	(58)
a__quote^#(s(X))	→	mark^#(X)	(57)
mark^#(sel1(X1,X2))	→	mark^#(X1)	(56)
a__dbl1^#(s(X))	→	a__dbl1^#(mark(X))	(55)
mark^#(dbl(X))	→	mark^#(X)	(42)
mark^#(indx(X1,X2))	→	mark^#(X1)	(41)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(X)	(40)
mark^#(sel(X1,X2))	→	mark^#(X2)	(54)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__quote^#(x₁)]	=	x₁ + 1
[a__dbl1(x₁)]	=	x₁ + 26435
[01]	=	1
[s(x₁)]	=	x₁ + 0
[a__from^#(x₁)]	=	0
[dbls(x₁)]	=	x₁ + 1
[a__indx(x₁, x₂)]	=	max(x₁ + 98343, x₂ + 98342, 0)
[a__from(x₁)]	=	x₁ + 39057
[a__indx^#(x₁, x₂)]	=	max(0)
[dbl(x₁)]	=	x₁ + 1
[indx(x₁, x₂)]	=	max(x₁ + 98343, x₂ + 98342, 0)
[a__dbl1^#(x₁)]	=	x₁ + 1
[a__dbl(x₁)]	=	x₁ + 1
[a__sel1(x₁, x₂)]	=	max(x₁ + 99938, x₂ + 66509, 0)
[a__dbls^#(x₁)]	=	0
[mark^#(x₁)]	=	x₁ + 0
[0]	=	42410
[a__quote(x₁)]	=	x₁ + 32287
[sel(x₁, x₂)]	=	max(x₁ + 98343, x₂ + 67650, 0)
[from(x₁)]	=	x₁ + 39057
[a__sel1^#(x₁, x₂)]	=	max(x₁ + 66508, x₂ + 35815, 0)
[nil]	=	98343
[dbl1(x₁)]	=	x₁ + 26435
[a__dbl^#(x₁)]	=	0
[a__sel^#(x₁, x₂)]	=	max(x₁ + 58794, x₂ + 28101, 0)
[mark(x₁)]	=	x₁ + 0
[a__sel(x₁, x₂)]	=	max(x₁ + 98343, x₂ + 67650, 0)
[a__dbls(x₁)]	=	x₁ + 1
[quote(x₁)]	=	x₁ + 32287
[cons(x₁, x₂)]	=	max(x₁ + 30692, x₂ + 0, 0)
[sel1(x₁, x₂)]	=	max(x₁ + 99938, x₂ + 66509, 0)
[s1(x₁)]	=	x₁ + 0

together with the usable rules

mark(dbl(X))	→	a__dbl(mark(X))	(18)
a__dbls(cons(X,Y))	→	cons(dbl(X),dbls(Y))	(4)
a__quote(s(X))	→	s1(a__quote(mark(X)))	(15)
a__indx(cons(X,Y),Z)	→	cons(sel(X,Z),indx(Y,Z))	(8)
a__dbl(0)	→	0	(1)
a__dbls(nil)	→	nil	(3)
a__quote(dbl(X))	→	a__dbl1(mark(X))	(16)
mark(indx(X1,X2))	→	a__indx(mark(X1),X2)	(21)
a__from(X)	→	from(X)	(36)
mark(0)	→	0	(26)
mark(dbls(X))	→	a__dbls(mark(X))	(19)
a__dbl(X)	→	dbl(X)	(32)
a__quote(sel(X,Y))	→	a__sel1(mark(X),mark(Y))	(17)
mark(s(X))	→	s(X)	(27)
a__sel(X1,X2)	→	sel(X1,X2)	(34)
mark(from(X))	→	a__from(X)	(22)
mark(nil)	→	nil	(28)
a__sel(0,cons(X,Y))	→	mark(X)	(5)
a__dbls(X)	→	dbls(X)	(33)
a__dbl1(0)	→	01	(10)
a__quote(X)	→	quote(X)	(39)
a__indx(nil,X)	→	nil	(7)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(20)
mark(quote(X))	→	a__quote(mark(X))	(25)
mark(01)	→	01	(30)
a__quote(0)	→	01	(14)
mark(s1(X))	→	s1(mark(X))	(31)
a__sel1(0,cons(X,Y))	→	mark(X)	(12)
mark(dbl1(X))	→	a__dbl1(mark(X))	(23)
mark(sel1(X1,X2))	→	a__sel1(mark(X1),mark(X2))	(24)
a__dbl1(s(X))	→	s1(s1(a__dbl1(mark(X))))	(11)
a__from(X)	→	cons(X,from(s(X)))	(9)
a__sel1(s(X),cons(Y,Z))	→	a__sel1(mark(X),mark(Z))	(13)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(6)
a__sel1(X1,X2)	→	sel1(X1,X2)	(38)
a__dbl1(X)	→	dbl1(X)	(37)
a__indx(X1,X2)	→	indx(X1,X2)	(35)
mark(cons(X1,X2))	→	cons(X1,X2)	(29)
a__dbl(s(X))	→	s(s(dbl(X)))	(2)

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

mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(74)
a__quote^#(sel(X,Y))	→	mark^#(Y)	(53)
a__quote^#(dbl(X))	→	a__dbl1^#(mark(X))	(52)
a__sel1^#(0,cons(X,Y))	→	mark^#(X)	(51)
a__quote^#(dbl(X))	→	mark^#(X)	(50)
mark^#(dbl1(X))	→	a__dbl1^#(mark(X))	(49)
a__dbl1^#(s(X))	→	mark^#(X)	(71)
a__quote^#(sel(X,Y))	→	a__sel1^#(mark(X),mark(Y))	(48)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(Z)	(69)
a__quote^#(sel(X,Y))	→	mark^#(X)	(47)
mark^#(quote(X))	→	a__quote^#(mark(X))	(68)
a__sel^#(0,cons(X,Y))	→	mark^#(X)	(67)
mark^#(sel1(X1,X2))	→	mark^#(X2)	(46)
mark^#(sel1(X1,X2))	→	a__sel1^#(mark(X1),mark(X2))	(45)
mark^#(dbl1(X))	→	mark^#(X)	(44)
mark^#(sel(X1,X2))	→	mark^#(X1)	(65)
mark^#(quote(X))	→	mark^#(X)	(63)
a__sel1^#(s(X),cons(Y,Z))	→	mark^#(Z)	(43)
mark^#(dbls(X))	→	mark^#(X)	(62)
a__sel1^#(s(X),cons(Y,Z))	→	mark^#(X)	(58)
a__quote^#(s(X))	→	mark^#(X)	(57)
mark^#(sel1(X1,X2))	→	mark^#(X1)	(56)
mark^#(dbl(X))	→	mark^#(X)	(42)
mark^#(indx(X1,X2))	→	mark^#(X1)	(41)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(X)	(40)
mark^#(sel(X1,X2))	→	mark^#(X2)	(54)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 5 components.

The 1^st component contains the pair

a__quote^#(s(X))

→

a__quote^#(mark(X))

(64)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__quote^#)	=	1
π(a__from^#)	=	1
π(dbls)	=	1
π(mark)	=	1
π(a__dbls)	=	1
π(s1)	=	1

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

prec(a__dbl1)	=	6	status(a__dbl1)	=	[1]	list-extension(a__dbl1)	=	Lex
prec(01)	=	5	status(01)	=	[]	list-extension(01)	=	Lex
prec(s)	=	4	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__indx)	=	2	status(a__indx)	=	[]	list-extension(a__indx)	=	Lex
prec(a__from)	=	1	status(a__from)	=	[1]	list-extension(a__from)	=	Lex
prec(a__indx^#)	=	0	status(a__indx^#)	=	[2, 1]	list-extension(a__indx^#)	=	Lex
prec(dbl)	=	6	status(dbl)	=	[1]	list-extension(dbl)	=	Lex
prec(indx)	=	2	status(indx)	=	[]	list-extension(indx)	=	Lex
prec(a__dbl1^#)	=	0	status(a__dbl1^#)	=	[]	list-extension(a__dbl1^#)	=	Lex
prec(a__dbl)	=	6	status(a__dbl)	=	[1]	list-extension(a__dbl)	=	Lex
prec(a__sel1)	=	2	status(a__sel1)	=	[]	list-extension(a__sel1)	=	Lex
prec(a__dbls^#)	=	0	status(a__dbls^#)	=	[]	list-extension(a__dbls^#)	=	Lex
prec(mark^#)	=	0	status(mark^#)	=	[]	list-extension(mark^#)	=	Lex
prec(0)	=	7	status(0)	=	[]	list-extension(0)	=	Lex
prec(a__quote)	=	3	status(a__quote)	=	[1]	list-extension(a__quote)	=	Lex
prec(sel)	=	0	status(sel)	=	[]	list-extension(sel)	=	Lex
prec(from)	=	1	status(from)	=	[1]	list-extension(from)	=	Lex
prec(a__sel1^#)	=	0	status(a__sel1^#)	=	[1]	list-extension(a__sel1^#)	=	Lex
prec(nil)	=	7	status(nil)	=	[]	list-extension(nil)	=	Lex
prec(dbl1)	=	6	status(dbl1)	=	[1]	list-extension(dbl1)	=	Lex
prec(a__dbl^#)	=	0	status(a__dbl^#)	=	[]	list-extension(a__dbl^#)	=	Lex
prec(a__sel^#)	=	0	status(a__sel^#)	=	[1]	list-extension(a__sel^#)	=	Lex
prec(a__sel)	=	0	status(a__sel)	=	[]	list-extension(a__sel)	=	Lex
prec(quote)	=	3	status(quote)	=	[1]	list-extension(quote)	=	Lex
prec(cons)	=	1	status(cons)	=	[]	list-extension(cons)	=	Lex
prec(sel1)	=	2	status(sel1)	=	[]	list-extension(sel1)	=	Lex

and the following Max-polynomial interpretation

[a__dbl1(x₁)]	=	x₁ + 15923
[01]	=	2
[s(x₁)]	=	x₁ + 0
[a__indx(x₁, x₂)]	=	x₂ + 4
[a__from(x₁)]	=	x₁ + 4
[a__indx^#(x₁, x₂)]	=	x₁ + x₂ + 1
[dbl(x₁)]	=	x₁ + 0
[indx(x₁, x₂)]	=	x₂ + 4
[a__dbl1^#(x₁)]	=	1
[a__dbl(x₁)]	=	x₁ + 0
[a__sel1(x₁, x₂)]	=	x₂ + 2
[a__dbls^#(x₁)]	=	1
[mark^#(x₁)]	=	1
[0]	=	1
[a__quote(x₁)]	=	x₁ + 15924
[sel(x₁, x₂)]	=	x₂ + 0
[from(x₁)]	=	x₁ + 4
[a__sel1^#(x₁, x₂)]	=	x₁ + 1
[nil]	=	3
[dbl1(x₁)]	=	x₁ + 15923
[a__dbl^#(x₁)]	=	1
[a__sel^#(x₁, x₂)]	=	x₁ + 1
[a__sel(x₁, x₂)]	=	x₂ + 0
[quote(x₁)]	=	x₁ + 15924
[cons(x₁, x₂)]	=	max(x₁ + 3, x₂ + 0, 0)
[sel1(x₁, x₂)]	=	x₂ + 2

together with the usable rules

mark(dbl(X))	→	a__dbl(mark(X))	(18)
a__dbls(cons(X,Y))	→	cons(dbl(X),dbls(Y))	(4)
a__quote(s(X))	→	s1(a__quote(mark(X)))	(15)
a__indx(cons(X,Y),Z)	→	cons(sel(X,Z),indx(Y,Z))	(8)
a__dbl(0)	→	0	(1)
a__dbls(nil)	→	nil	(3)
a__quote(dbl(X))	→	a__dbl1(mark(X))	(16)
mark(indx(X1,X2))	→	a__indx(mark(X1),X2)	(21)
a__from(X)	→	from(X)	(36)
mark(0)	→	0	(26)
mark(dbls(X))	→	a__dbls(mark(X))	(19)
a__dbl(X)	→	dbl(X)	(32)
a__quote(sel(X,Y))	→	a__sel1(mark(X),mark(Y))	(17)
mark(s(X))	→	s(X)	(27)
a__sel(X1,X2)	→	sel(X1,X2)	(34)
mark(from(X))	→	a__from(X)	(22)
mark(nil)	→	nil	(28)
a__sel(0,cons(X,Y))	→	mark(X)	(5)
a__dbls(X)	→	dbls(X)	(33)
a__dbl1(0)	→	01	(10)
a__quote(X)	→	quote(X)	(39)
a__indx(nil,X)	→	nil	(7)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(20)
mark(quote(X))	→	a__quote(mark(X))	(25)
mark(01)	→	01	(30)
a__quote(0)	→	01	(14)
mark(s1(X))	→	s1(mark(X))	(31)
a__sel1(0,cons(X,Y))	→	mark(X)	(12)
mark(dbl1(X))	→	a__dbl1(mark(X))	(23)
mark(sel1(X1,X2))	→	a__sel1(mark(X1),mark(X2))	(24)
a__dbl1(s(X))	→	s1(s1(a__dbl1(mark(X))))	(11)
a__from(X)	→	cons(X,from(s(X)))	(9)
a__sel1(s(X),cons(Y,Z))	→	a__sel1(mark(X),mark(Z))	(13)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(6)
a__sel1(X1,X2)	→	sel1(X1,X2)	(38)
a__dbl1(X)	→	dbl1(X)	(37)
a__indx(X1,X2)	→	indx(X1,X2)	(35)
mark(cons(X1,X2))	→	cons(X1,X2)	(29)
a__dbl(s(X))	→	s(s(dbl(X)))	(2)

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

a__quote^#(s(X))

→

a__quote^#(mark(X))

(64)

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^#(s1(X))

→

mark^#(X)

(72)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__quote^#(x₁)]	=	2
[a__dbl1(x₁)]	=	x₁ + 1
[01]	=	4
[s(x₁)]	=	1
[a__from^#(x₁)]	=	0
[dbls(x₁)]	=	21653
[a__indx(x₁, x₂)]	=	x₁ + x₂ + 0
[a__from(x₁)]	=	14008
[a__indx^#(x₁, x₂)]	=	0
[dbl(x₁)]	=	1
[indx(x₁, x₂)]	=	1
[a__dbl1^#(x₁)]	=	2
[a__dbl(x₁)]	=	x₁ + 2
[a__sel1(x₁, x₂)]	=	x₁ + x₂ + 0
[a__dbls^#(x₁)]	=	0
[mark^#(x₁)]	=	x₁ + 2
[0]	=	1
[a__quote(x₁)]	=	x₁ + 2
[sel(x₁, x₂)]	=	x₁ + x₂ + 1
[from(x₁)]	=	14003
[a__sel1^#(x₁, x₂)]	=	2
[nil]	=	1
[dbl1(x₁)]	=	24339
[a__dbl^#(x₁)]	=	0
[a__sel^#(x₁, x₂)]	=	0
[mark(x₁)]	=	x₁ + 4
[a__sel(x₁, x₂)]	=	x₁ + 0
[a__dbls(x₁)]	=	x₁ + 21652
[quote(x₁)]	=	1
[cons(x₁, x₂)]	=	x₁ + x₂ + 6
[sel1(x₁, x₂)]	=	1
[s1(x₁)]	=	x₁ + 27857

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

mark^#(s1(X))

→

mark^#(X)

(72)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

a__dbl1^#(s(X))

→

a__dbl1^#(mark(X))

(55)

1.1.1.1.3 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__quote^#)	=	1
π(a__from^#)	=	1
π(a__dbl1^#)	=	1
π(mark)	=	1
π(s1)	=	1

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

prec(a__dbl1)	=	5	status(a__dbl1)	=	[]	list-extension(a__dbl1)	=	Lex
prec(01)	=	4	status(01)	=	[]	list-extension(01)	=	Lex
prec(s)	=	5	status(s)	=	[1]	list-extension(s)	=	Lex
prec(dbls)	=	5	status(dbls)	=	[]	list-extension(dbls)	=	Lex
prec(a__indx)	=	2	status(a__indx)	=	[1]	list-extension(a__indx)	=	Lex
prec(a__from)	=	2	status(a__from)	=	[]	list-extension(a__from)	=	Lex
prec(a__indx^#)	=	0	status(a__indx^#)	=	[2, 1]	list-extension(a__indx^#)	=	Lex
prec(dbl)	=	6	status(dbl)	=	[1]	list-extension(dbl)	=	Lex
prec(indx)	=	2	status(indx)	=	[1]	list-extension(indx)	=	Lex
prec(a__dbl)	=	6	status(a__dbl)	=	[1]	list-extension(a__dbl)	=	Lex
prec(a__sel1)	=	3	status(a__sel1)	=	[]	list-extension(a__sel1)	=	Lex
prec(a__dbls^#)	=	0	status(a__dbls^#)	=	[]	list-extension(a__dbls^#)	=	Lex
prec(mark^#)	=	0	status(mark^#)	=	[]	list-extension(mark^#)	=	Lex
prec(0)	=	7	status(0)	=	[]	list-extension(0)	=	Lex
prec(a__quote)	=	3	status(a__quote)	=	[]	list-extension(a__quote)	=	Lex
prec(sel)	=	0	status(sel)	=	[]	list-extension(sel)	=	Lex
prec(from)	=	2	status(from)	=	[]	list-extension(from)	=	Lex
prec(a__sel1^#)	=	0	status(a__sel1^#)	=	[1]	list-extension(a__sel1^#)	=	Lex
prec(nil)	=	4	status(nil)	=	[]	list-extension(nil)	=	Lex
prec(dbl1)	=	5	status(dbl1)	=	[]	list-extension(dbl1)	=	Lex
prec(a__dbl^#)	=	0	status(a__dbl^#)	=	[]	list-extension(a__dbl^#)	=	Lex
prec(a__sel^#)	=	0	status(a__sel^#)	=	[1]	list-extension(a__sel^#)	=	Lex
prec(a__sel)	=	0	status(a__sel)	=	[]	list-extension(a__sel)	=	Lex
prec(a__dbls)	=	5	status(a__dbls)	=	[]	list-extension(a__dbls)	=	Lex
prec(quote)	=	3	status(quote)	=	[]	list-extension(quote)	=	Lex
prec(cons)	=	1	status(cons)	=	[1]	list-extension(cons)	=	Lex
prec(sel1)	=	3	status(sel1)	=	[]	list-extension(sel1)	=	Lex

and the following Max-polynomial interpretation

[a__dbl1(x₁)]	=	15924
[01]	=	2
[s(x₁)]	=	x₁ + 0
[dbls(x₁)]	=	x₁ + 0
[a__indx(x₁, x₂)]	=	x₁ + x₂ + 0
[a__from(x₁)]	=	x₁ + 36231
[a__indx^#(x₁, x₂)]	=	x₁ + x₂ + 1
[dbl(x₁)]	=	x₁ + 0
[indx(x₁, x₂)]	=	x₁ + x₂ + 0
[a__dbl(x₁)]	=	x₁ + 0
[a__sel1(x₁, x₂)]	=	x₂ + 12618
[a__dbls^#(x₁)]	=	1
[mark^#(x₁)]	=	1
[0]	=	1
[a__quote(x₁)]	=	x₁ + 15925
[sel(x₁, x₂)]	=	x₂ + 0
[from(x₁)]	=	x₁ + 36231
[a__sel1^#(x₁, x₂)]	=	x₁ + 1
[nil]	=	33694
[dbl1(x₁)]	=	15924
[a__dbl^#(x₁)]	=	1
[a__sel^#(x₁, x₂)]	=	x₁ + 1
[a__sel(x₁, x₂)]	=	x₂ + 0
[a__dbls(x₁)]	=	x₁ + 0
[quote(x₁)]	=	x₁ + 15925
[cons(x₁, x₂)]	=	max(x₁ + 36230, x₂ + 0, 0)
[sel1(x₁, x₂)]	=	x₂ + 12618

together with the usable rules

mark(dbl(X))	→	a__dbl(mark(X))	(18)
a__dbls(cons(X,Y))	→	cons(dbl(X),dbls(Y))	(4)
a__quote(s(X))	→	s1(a__quote(mark(X)))	(15)
a__indx(cons(X,Y),Z)	→	cons(sel(X,Z),indx(Y,Z))	(8)
a__dbl(0)	→	0	(1)
a__dbls(nil)	→	nil	(3)
a__quote(dbl(X))	→	a__dbl1(mark(X))	(16)
mark(indx(X1,X2))	→	a__indx(mark(X1),X2)	(21)
a__from(X)	→	from(X)	(36)
mark(0)	→	0	(26)
mark(dbls(X))	→	a__dbls(mark(X))	(19)
a__dbl(X)	→	dbl(X)	(32)
a__quote(sel(X,Y))	→	a__sel1(mark(X),mark(Y))	(17)
mark(s(X))	→	s(X)	(27)
a__sel(X1,X2)	→	sel(X1,X2)	(34)
mark(from(X))	→	a__from(X)	(22)
mark(nil)	→	nil	(28)
a__sel(0,cons(X,Y))	→	mark(X)	(5)
a__dbls(X)	→	dbls(X)	(33)
a__dbl1(0)	→	01	(10)
a__quote(X)	→	quote(X)	(39)
a__indx(nil,X)	→	nil	(7)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(20)
mark(quote(X))	→	a__quote(mark(X))	(25)
mark(01)	→	01	(30)
a__quote(0)	→	01	(14)
mark(s1(X))	→	s1(mark(X))	(31)
a__sel1(0,cons(X,Y))	→	mark(X)	(12)
mark(dbl1(X))	→	a__dbl1(mark(X))	(23)
mark(sel1(X1,X2))	→	a__sel1(mark(X1),mark(X2))	(24)
a__dbl1(s(X))	→	s1(s1(a__dbl1(mark(X))))	(11)
a__from(X)	→	cons(X,from(s(X)))	(9)
a__sel1(s(X),cons(Y,Z))	→	a__sel1(mark(X),mark(Z))	(13)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(6)
a__sel1(X1,X2)	→	sel1(X1,X2)	(38)
a__dbl1(X)	→	dbl1(X)	(37)
a__indx(X1,X2)	→	indx(X1,X2)	(35)
mark(cons(X1,X2))	→	cons(X1,X2)	(29)
a__dbl(s(X))	→	s(s(dbl(X)))	(2)

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

a__dbl1^#(s(X))

→

a__dbl1^#(mark(X))

(55)

could be deleted.

1.1.1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

a__sel1^#(s(X),cons(Y,Z))

→

a__sel1^#(mark(X),mark(Z))

(73)

1.1.1.1.4 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__quote^#)	=	1
π(a__from^#)	=	1
π(a__dbl1^#)	=	1
π(a__sel1^#)	=	1
π(mark)	=	1
π(s1)	=	1

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

prec(a__dbl1)	=	5	status(a__dbl1)	=	[]	list-extension(a__dbl1)	=	Lex
prec(01)	=	4	status(01)	=	[]	list-extension(01)	=	Lex
prec(s)	=	5	status(s)	=	[1]	list-extension(s)	=	Lex
prec(dbls)	=	5	status(dbls)	=	[]	list-extension(dbls)	=	Lex
prec(a__indx)	=	2	status(a__indx)	=	[1]	list-extension(a__indx)	=	Lex
prec(a__from)	=	2	status(a__from)	=	[]	list-extension(a__from)	=	Lex
prec(a__indx^#)	=	0	status(a__indx^#)	=	[2, 1]	list-extension(a__indx^#)	=	Lex
prec(dbl)	=	6	status(dbl)	=	[1]	list-extension(dbl)	=	Lex
prec(indx)	=	2	status(indx)	=	[1]	list-extension(indx)	=	Lex
prec(a__dbl)	=	6	status(a__dbl)	=	[1]	list-extension(a__dbl)	=	Lex
prec(a__sel1)	=	3	status(a__sel1)	=	[]	list-extension(a__sel1)	=	Lex
prec(a__dbls^#)	=	0	status(a__dbls^#)	=	[]	list-extension(a__dbls^#)	=	Lex
prec(mark^#)	=	0	status(mark^#)	=	[]	list-extension(mark^#)	=	Lex
prec(0)	=	7	status(0)	=	[]	list-extension(0)	=	Lex
prec(a__quote)	=	3	status(a__quote)	=	[]	list-extension(a__quote)	=	Lex
prec(sel)	=	0	status(sel)	=	[]	list-extension(sel)	=	Lex
prec(from)	=	2	status(from)	=	[]	list-extension(from)	=	Lex
prec(nil)	=	4	status(nil)	=	[]	list-extension(nil)	=	Lex
prec(dbl1)	=	5	status(dbl1)	=	[]	list-extension(dbl1)	=	Lex
prec(a__dbl^#)	=	0	status(a__dbl^#)	=	[]	list-extension(a__dbl^#)	=	Lex
prec(a__sel^#)	=	0	status(a__sel^#)	=	[1]	list-extension(a__sel^#)	=	Lex
prec(a__sel)	=	0	status(a__sel)	=	[]	list-extension(a__sel)	=	Lex
prec(a__dbls)	=	5	status(a__dbls)	=	[]	list-extension(a__dbls)	=	Lex
prec(quote)	=	3	status(quote)	=	[]	list-extension(quote)	=	Lex
prec(cons)	=	1	status(cons)	=	[1]	list-extension(cons)	=	Lex
prec(sel1)	=	3	status(sel1)	=	[]	list-extension(sel1)	=	Lex

and the following Max-polynomial interpretation

[a__dbl1(x₁)]	=	28226
[01]	=	2
[s(x₁)]	=	x₁ + 0
[dbls(x₁)]	=	x₁ + 0
[a__indx(x₁, x₂)]	=	x₁ + x₂ + 0
[a__from(x₁)]	=	x₁ + 17890
[a__indx^#(x₁, x₂)]	=	x₁ + x₂ + 1
[dbl(x₁)]	=	x₁ + 0
[indx(x₁, x₂)]	=	x₁ + x₂ + 0
[a__dbl(x₁)]	=	x₁ + 0
[a__sel1(x₁, x₂)]	=	x₂ + 17888
[a__dbls^#(x₁)]	=	1
[mark^#(x₁)]	=	1
[0]	=	1
[a__quote(x₁)]	=	x₁ + 28227
[sel(x₁, x₂)]	=	x₂ + 0
[from(x₁)]	=	x₁ + 17890
[nil]	=	38476
[dbl1(x₁)]	=	28226
[a__dbl^#(x₁)]	=	1
[a__sel^#(x₁, x₂)]	=	x₁ + 1
[a__sel(x₁, x₂)]	=	x₂ + 0
[a__dbls(x₁)]	=	x₁ + 0
[quote(x₁)]	=	x₁ + 28227
[cons(x₁, x₂)]	=	max(x₁ + 17889, x₂ + 0, 0)
[sel1(x₁, x₂)]	=	x₂ + 17888

together with the usable rules

mark(dbl(X))	→	a__dbl(mark(X))	(18)
a__dbls(cons(X,Y))	→	cons(dbl(X),dbls(Y))	(4)
a__quote(s(X))	→	s1(a__quote(mark(X)))	(15)
a__indx(cons(X,Y),Z)	→	cons(sel(X,Z),indx(Y,Z))	(8)
a__dbl(0)	→	0	(1)
a__dbls(nil)	→	nil	(3)
a__quote(dbl(X))	→	a__dbl1(mark(X))	(16)
mark(indx(X1,X2))	→	a__indx(mark(X1),X2)	(21)
a__from(X)	→	from(X)	(36)
mark(0)	→	0	(26)
mark(dbls(X))	→	a__dbls(mark(X))	(19)
a__dbl(X)	→	dbl(X)	(32)
a__quote(sel(X,Y))	→	a__sel1(mark(X),mark(Y))	(17)
mark(s(X))	→	s(X)	(27)
a__sel(X1,X2)	→	sel(X1,X2)	(34)
mark(from(X))	→	a__from(X)	(22)
mark(nil)	→	nil	(28)
a__sel(0,cons(X,Y))	→	mark(X)	(5)
a__dbls(X)	→	dbls(X)	(33)
a__dbl1(0)	→	01	(10)
a__quote(X)	→	quote(X)	(39)
a__indx(nil,X)	→	nil	(7)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(20)
mark(quote(X))	→	a__quote(mark(X))	(25)
mark(01)	→	01	(30)
a__quote(0)	→	01	(14)
mark(s1(X))	→	s1(mark(X))	(31)
a__sel1(0,cons(X,Y))	→	mark(X)	(12)
mark(dbl1(X))	→	a__dbl1(mark(X))	(23)
mark(sel1(X1,X2))	→	a__sel1(mark(X1),mark(X2))	(24)
a__dbl1(s(X))	→	s1(s1(a__dbl1(mark(X))))	(11)
a__from(X)	→	cons(X,from(s(X)))	(9)
a__sel1(s(X),cons(Y,Z))	→	a__sel1(mark(X),mark(Z))	(13)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(6)
a__sel1(X1,X2)	→	sel1(X1,X2)	(38)
a__dbl1(X)	→	dbl1(X)	(37)
a__indx(X1,X2)	→	indx(X1,X2)	(35)
mark(cons(X1,X2))	→	cons(X1,X2)	(29)
a__dbl(s(X))	→	s(s(dbl(X)))	(2)

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

a__sel1^#(s(X),cons(Y,Z))

→

a__sel1^#(mark(X),mark(Z))

(73)

could be deleted.

1.1.1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 5^th component contains the pair

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

→

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

(70)

1.1.1.1.5 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__quote^#)	=	1
π(a__from^#)	=	1
π(a__dbl1^#)	=	1
π(a__sel1^#)	=	1
π(mark)	=	1
π(s1)	=	1

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

prec(a__dbl1)	=	6	status(a__dbl1)	=	[]	list-extension(a__dbl1)	=	Lex
prec(01)	=	5	status(01)	=	[]	list-extension(01)	=	Lex
prec(s)	=	6	status(s)	=	[1]	list-extension(s)	=	Lex
prec(dbls)	=	6	status(dbls)	=	[]	list-extension(dbls)	=	Lex
prec(a__indx)	=	3	status(a__indx)	=	[1]	list-extension(a__indx)	=	Lex
prec(a__from)	=	3	status(a__from)	=	[]	list-extension(a__from)	=	Lex
prec(a__indx^#)	=	0	status(a__indx^#)	=	[2, 1]	list-extension(a__indx^#)	=	Lex
prec(dbl)	=	7	status(dbl)	=	[1]	list-extension(dbl)	=	Lex
prec(indx)	=	3	status(indx)	=	[1]	list-extension(indx)	=	Lex
prec(a__dbl)	=	7	status(a__dbl)	=	[1]	list-extension(a__dbl)	=	Lex
prec(a__sel1)	=	4	status(a__sel1)	=	[]	list-extension(a__sel1)	=	Lex
prec(a__dbls^#)	=	0	status(a__dbls^#)	=	[]	list-extension(a__dbls^#)	=	Lex
prec(mark^#)	=	0	status(mark^#)	=	[]	list-extension(mark^#)	=	Lex
prec(0)	=	8	status(0)	=	[]	list-extension(0)	=	Lex
prec(a__quote)	=	4	status(a__quote)	=	[]	list-extension(a__quote)	=	Lex
prec(sel)	=	0	status(sel)	=	[]	list-extension(sel)	=	Lex
prec(from)	=	3	status(from)	=	[]	list-extension(from)	=	Lex
prec(nil)	=	5	status(nil)	=	[]	list-extension(nil)	=	Lex
prec(dbl1)	=	6	status(dbl1)	=	[]	list-extension(dbl1)	=	Lex
prec(a__dbl^#)	=	0	status(a__dbl^#)	=	[]	list-extension(a__dbl^#)	=	Lex
prec(a__sel^#)	=	2	status(a__sel^#)	=	[1, 2]	list-extension(a__sel^#)	=	Lex
prec(a__sel)	=	0	status(a__sel)	=	[]	list-extension(a__sel)	=	Lex
prec(a__dbls)	=	6	status(a__dbls)	=	[]	list-extension(a__dbls)	=	Lex
prec(quote)	=	4	status(quote)	=	[]	list-extension(quote)	=	Lex
prec(cons)	=	1	status(cons)	=	[1]	list-extension(cons)	=	Lex
prec(sel1)	=	4	status(sel1)	=	[]	list-extension(sel1)	=	Lex

and the following Max-polynomial interpretation

[a__dbl1(x₁)]	=	3
[01]	=	2
[s(x₁)]	=	x₁ + 0
[dbls(x₁)]	=	x₁ + 0
[a__indx(x₁, x₂)]	=	x₁ + x₂ + 0
[a__from(x₁)]	=	x₁ + 7581
[a__indx^#(x₁, x₂)]	=	x₁ + x₂ + 1
[dbl(x₁)]	=	x₁ + 0
[indx(x₁, x₂)]	=	x₁ + x₂ + 0
[a__dbl(x₁)]	=	x₁ + 0
[a__sel1(x₁, x₂)]	=	x₂ + 1
[a__dbls^#(x₁)]	=	1
[mark^#(x₁)]	=	1
[0]	=	1
[a__quote(x₁)]	=	x₁ + 28227
[sel(x₁, x₂)]	=	x₂ + 0
[from(x₁)]	=	x₁ + 7581
[nil]	=	43324
[dbl1(x₁)]	=	3
[a__dbl^#(x₁)]	=	1
[a__sel^#(x₁, x₂)]	=	x₁ + x₂ + 1
[a__sel(x₁, x₂)]	=	x₂ + 0
[a__dbls(x₁)]	=	x₁ + 0
[quote(x₁)]	=	x₁ + 28227
[cons(x₁, x₂)]	=	max(x₁ + 2, x₂ + 0, 0)
[sel1(x₁, x₂)]	=	x₂ + 1

together with the usable rules

mark(dbl(X))	→	a__dbl(mark(X))	(18)
a__dbls(cons(X,Y))	→	cons(dbl(X),dbls(Y))	(4)
a__quote(s(X))	→	s1(a__quote(mark(X)))	(15)
a__indx(cons(X,Y),Z)	→	cons(sel(X,Z),indx(Y,Z))	(8)
a__dbl(0)	→	0	(1)
a__dbls(nil)	→	nil	(3)
a__quote(dbl(X))	→	a__dbl1(mark(X))	(16)
mark(indx(X1,X2))	→	a__indx(mark(X1),X2)	(21)
a__from(X)	→	from(X)	(36)
mark(0)	→	0	(26)
mark(dbls(X))	→	a__dbls(mark(X))	(19)
a__dbl(X)	→	dbl(X)	(32)
a__quote(sel(X,Y))	→	a__sel1(mark(X),mark(Y))	(17)
mark(s(X))	→	s(X)	(27)
a__sel(X1,X2)	→	sel(X1,X2)	(34)
mark(from(X))	→	a__from(X)	(22)
mark(nil)	→	nil	(28)
a__sel(0,cons(X,Y))	→	mark(X)	(5)
a__dbls(X)	→	dbls(X)	(33)
a__dbl1(0)	→	01	(10)
a__quote(X)	→	quote(X)	(39)
a__indx(nil,X)	→	nil	(7)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(20)
mark(quote(X))	→	a__quote(mark(X))	(25)
mark(01)	→	01	(30)
a__quote(0)	→	01	(14)
mark(s1(X))	→	s1(mark(X))	(31)
a__sel1(0,cons(X,Y))	→	mark(X)	(12)
mark(dbl1(X))	→	a__dbl1(mark(X))	(23)
mark(sel1(X1,X2))	→	a__sel1(mark(X1),mark(X2))	(24)
a__dbl1(s(X))	→	s1(s1(a__dbl1(mark(X))))	(11)
a__from(X)	→	cons(X,from(s(X)))	(9)
a__sel1(s(X),cons(Y,Z))	→	a__sel1(mark(X),mark(Z))	(13)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(6)
a__sel1(X1,X2)	→	sel1(X1,X2)	(38)
a__dbl1(X)	→	dbl1(X)	(37)
a__indx(X1,X2)	→	indx(X1,X2)	(35)
mark(cons(X1,X2))	→	cons(X1,X2)	(29)
a__dbl(s(X))	→	s(s(dbl(X)))	(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))

(70)

could be deleted.

1.1.1.1.5.1 Dependency Graph Processor

The dependency pairs are split into 0 components.