Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex15_Luc98_iGM)

The rewrite relation of the following TRS is considered.

active(and(true,X))	→	mark(X)	(1)
active(and(false,Y))	→	mark(false)	(2)
active(if(true,X,Y))	→	mark(X)	(3)
active(if(false,X,Y))	→	mark(Y)	(4)
active(add(0,X))	→	mark(X)	(5)
active(add(s(X),Y))	→	mark(s(add(X,Y)))	(6)
active(first(0,X))	→	mark(nil)	(7)
active(first(s(X),cons(Y,Z)))	→	mark(cons(Y,first(X,Z)))	(8)
active(from(X))	→	mark(cons(X,from(s(X))))	(9)
mark(and(X1,X2))	→	active(and(mark(X1),X2))	(10)
mark(true)	→	active(true)	(11)
mark(false)	→	active(false)	(12)
mark(if(X1,X2,X3))	→	active(if(mark(X1),X2,X3))	(13)
mark(add(X1,X2))	→	active(add(mark(X1),X2))	(14)
mark(0)	→	active(0)	(15)
mark(s(X))	→	active(s(X))	(16)
mark(first(X1,X2))	→	active(first(mark(X1),mark(X2)))	(17)
mark(nil)	→	active(nil)	(18)
mark(cons(X1,X2))	→	active(cons(X1,X2))	(19)
mark(from(X))	→	active(from(X))	(20)
and(mark(X1),X2)	→	and(X1,X2)	(21)
and(X1,mark(X2))	→	and(X1,X2)	(22)
and(active(X1),X2)	→	and(X1,X2)	(23)
and(X1,active(X2))	→	and(X1,X2)	(24)
if(mark(X1),X2,X3)	→	if(X1,X2,X3)	(25)
if(X1,mark(X2),X3)	→	if(X1,X2,X3)	(26)
if(X1,X2,mark(X3))	→	if(X1,X2,X3)	(27)
if(active(X1),X2,X3)	→	if(X1,X2,X3)	(28)
if(X1,active(X2),X3)	→	if(X1,X2,X3)	(29)
if(X1,X2,active(X3))	→	if(X1,X2,X3)	(30)
add(mark(X1),X2)	→	add(X1,X2)	(31)
add(X1,mark(X2))	→	add(X1,X2)	(32)
add(active(X1),X2)	→	add(X1,X2)	(33)
add(X1,active(X2))	→	add(X1,X2)	(34)
s(mark(X))	→	s(X)	(35)
s(active(X))	→	s(X)	(36)
first(mark(X1),X2)	→	first(X1,X2)	(37)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)
cons(X1,active(X2))	→	cons(X1,X2)	(44)
from(mark(X))	→	from(X)	(45)
from(active(X))	→	from(X)	(46)

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.

cons^#(active(X1),X2)	→	cons^#(X1,X2)	(47)
add^#(X1,active(X2))	→	add^#(X1,X2)	(48)
mark^#(add(X1,X2))	→	mark^#(X1)	(49)
mark^#(and(X1,X2))	→	active^#(and(mark(X1),X2))	(50)
mark^#(first(X1,X2))	→	first^#(mark(X1),mark(X2))	(51)
and^#(X1,mark(X2))	→	and^#(X1,X2)	(52)
if^#(active(X1),X2,X3)	→	if^#(X1,X2,X3)	(53)
mark^#(and(X1,X2))	→	mark^#(X1)	(54)
mark^#(first(X1,X2))	→	mark^#(X1)	(55)
and^#(mark(X1),X2)	→	and^#(X1,X2)	(56)
mark^#(first(X1,X2))	→	mark^#(X2)	(57)
active^#(first(s(X),cons(Y,Z)))	→	cons^#(Y,first(X,Z))	(58)
active^#(if(false,X,Y))	→	mark^#(Y)	(59)
add^#(active(X1),X2)	→	add^#(X1,X2)	(60)
cons^#(X1,mark(X2))	→	cons^#(X1,X2)	(61)
active^#(first(s(X),cons(Y,Z)))	→	first^#(X,Z)	(62)
active^#(if(true,X,Y))	→	mark^#(X)	(63)
cons^#(X1,active(X2))	→	cons^#(X1,X2)	(64)
if^#(X1,mark(X2),X3)	→	if^#(X1,X2,X3)	(65)
first^#(X1,active(X2))	→	first^#(X1,X2)	(66)
first^#(X1,mark(X2))	→	first^#(X1,X2)	(67)
active^#(add(s(X),Y))	→	s^#(add(X,Y))	(68)
active^#(first(s(X),cons(Y,Z)))	→	mark^#(cons(Y,first(X,Z)))	(69)
from^#(mark(X))	→	from^#(X)	(70)
and^#(active(X1),X2)	→	and^#(X1,X2)	(71)
mark^#(first(X1,X2))	→	active^#(first(mark(X1),mark(X2)))	(72)
active^#(add(s(X),Y))	→	add^#(X,Y)	(73)
if^#(mark(X1),X2,X3)	→	if^#(X1,X2,X3)	(74)
active^#(and(true,X))	→	mark^#(X)	(75)
mark^#(if(X1,X2,X3))	→	active^#(if(mark(X1),X2,X3))	(76)
mark^#(nil)	→	active^#(nil)	(77)
if^#(X1,X2,active(X3))	→	if^#(X1,X2,X3)	(78)
s^#(active(X))	→	s^#(X)	(79)
add^#(X1,mark(X2))	→	add^#(X1,X2)	(80)
mark^#(false)	→	active^#(false)	(81)
active^#(from(X))	→	s^#(X)	(82)
cons^#(mark(X1),X2)	→	cons^#(X1,X2)	(83)
active^#(add(0,X))	→	mark^#(X)	(84)
active^#(add(s(X),Y))	→	mark^#(s(add(X,Y)))	(85)
active^#(first(0,X))	→	mark^#(nil)	(86)
from^#(active(X))	→	from^#(X)	(87)
if^#(X1,X2,mark(X3))	→	if^#(X1,X2,X3)	(88)
mark^#(add(X1,X2))	→	add^#(mark(X1),X2)	(89)
and^#(X1,active(X2))	→	and^#(X1,X2)	(90)
first^#(active(X1),X2)	→	first^#(X1,X2)	(91)
mark^#(add(X1,X2))	→	active^#(add(mark(X1),X2))	(92)
mark^#(0)	→	active^#(0)	(93)
active^#(from(X))	→	mark^#(cons(X,from(s(X))))	(94)
mark^#(from(X))	→	active^#(from(X))	(95)
active^#(from(X))	→	from^#(s(X))	(96)
add^#(mark(X1),X2)	→	add^#(X1,X2)	(97)
mark^#(and(X1,X2))	→	and^#(mark(X1),X2)	(98)
mark^#(true)	→	active^#(true)	(99)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(100)
if^#(X1,active(X2),X3)	→	if^#(X1,X2,X3)	(101)
active^#(and(false,Y))	→	mark^#(false)	(102)
mark^#(s(X))	→	active^#(s(X))	(103)
first^#(mark(X1),X2)	→	first^#(X1,X2)	(104)
mark^#(if(X1,X2,X3))	→	if^#(mark(X1),X2,X3)	(105)
s^#(mark(X))	→	s^#(X)	(106)
mark^#(cons(X1,X2))	→	active^#(cons(X1,X2))	(107)
active^#(from(X))	→	cons^#(X,from(s(X)))	(108)

1.1 Dependency Graph Processor

The dependency pairs are split into 8 components.

The 1^st component contains the pair

mark^#(cons(X1,X2))	→	active^#(cons(X1,X2))	(107)
mark^#(if(X1,X2,X3))	→	active^#(if(mark(X1),X2,X3))	(76)
active^#(and(true,X))	→	mark^#(X)	(75)
mark^#(first(X1,X2))	→	active^#(first(mark(X1),mark(X2)))	(72)
mark^#(s(X))	→	active^#(s(X))	(103)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(100)
active^#(first(s(X),cons(Y,Z)))	→	mark^#(cons(Y,first(X,Z)))	(69)
mark^#(from(X))	→	active^#(from(X))	(95)
active^#(from(X))	→	mark^#(cons(X,from(s(X))))	(94)
active^#(if(true,X,Y))	→	mark^#(X)	(63)
mark^#(add(X1,X2))	→	active^#(add(mark(X1),X2))	(92)
active^#(if(false,X,Y))	→	mark^#(Y)	(59)
mark^#(first(X1,X2))	→	mark^#(X2)	(57)
mark^#(first(X1,X2))	→	mark^#(X1)	(55)
mark^#(and(X1,X2))	→	mark^#(X1)	(54)
active^#(add(s(X),Y))	→	mark^#(s(add(X,Y)))	(85)
active^#(add(0,X))	→	mark^#(X)	(84)
mark^#(and(X1,X2))	→	active^#(and(mark(X1),X2))	(50)
mark^#(add(X1,X2))	→	mark^#(X1)	(49)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + x₂ + 1
[false]	=	1
[true]	=	1
[mark^#(x₁)]	=	x₁ + 1
[if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 1
[0]	=	29922
[from(x₁)]	=	x₁ + 3
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	0
[nil]	=	2737
[mark(x₁)]	=	x₁ + 0
[first(x₁, x₂)]	=	x₁ + x₂ + 2998
[from^#(x₁)]	=	0
[active(x₁)]	=	x₁ + 0
[cons(x₁, x₂)]	=	1
[if^#(x₁, x₂, x₃)]	=	0
[active^#(x₁)]	=	x₁ + 0
[add^#(x₁, x₂)]	=	0
[add(x₁, x₂)]	=	x₁ + x₂ + 31893
[and^#(x₁, x₂)]	=	0

together with the usable rules

mark(nil)	→	active(nil)	(18)
active(if(false,X,Y))	→	mark(Y)	(4)
mark(0)	→	active(0)	(15)
active(first(s(X),cons(Y,Z)))	→	mark(cons(Y,first(X,Z)))	(8)
active(and(true,X))	→	mark(X)	(1)
active(if(true,X,Y))	→	mark(X)	(3)
mark(s(X))	→	active(s(X))	(16)
and(mark(X1),X2)	→	and(X1,X2)	(21)
s(active(X))	→	s(X)	(36)
if(X1,mark(X2),X3)	→	if(X1,X2,X3)	(26)
mark(cons(X1,X2))	→	active(cons(X1,X2))	(19)
add(X1,mark(X2))	→	add(X1,X2)	(32)
mark(first(X1,X2))	→	active(first(mark(X1),mark(X2)))	(17)
if(X1,X2,mark(X3))	→	if(X1,X2,X3)	(27)
add(X1,active(X2))	→	add(X1,X2)	(34)
and(X1,mark(X2))	→	and(X1,X2)	(22)
if(active(X1),X2,X3)	→	if(X1,X2,X3)	(28)
cons(X1,active(X2))	→	cons(X1,X2)	(44)
active(add(0,X))	→	mark(X)	(5)
add(active(X1),X2)	→	add(X1,X2)	(33)
mark(and(X1,X2))	→	active(and(mark(X1),X2))	(10)
first(active(X1),X2)	→	first(X1,X2)	(39)
active(first(0,X))	→	mark(nil)	(7)
mark(from(X))	→	active(from(X))	(20)
if(mark(X1),X2,X3)	→	if(X1,X2,X3)	(25)
if(X1,X2,active(X3))	→	if(X1,X2,X3)	(30)
mark(add(X1,X2))	→	active(add(mark(X1),X2))	(14)
add(mark(X1),X2)	→	add(X1,X2)	(31)
mark(false)	→	active(false)	(12)
from(mark(X))	→	from(X)	(45)
and(active(X1),X2)	→	and(X1,X2)	(23)
and(X1,active(X2))	→	and(X1,X2)	(24)
mark(true)	→	active(true)	(11)
active(from(X))	→	mark(cons(X,from(s(X))))	(9)
mark(if(X1,X2,X3))	→	active(if(mark(X1),X2,X3))	(13)
first(X1,active(X2))	→	first(X1,X2)	(40)
active(add(s(X),Y))	→	mark(s(add(X,Y)))	(6)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
from(active(X))	→	from(X)	(46)
s(mark(X))	→	s(X)	(35)
if(X1,active(X2),X3)	→	if(X1,X2,X3)	(29)
cons(active(X1),X2)	→	cons(X1,X2)	(43)
active(and(false,Y))	→	mark(false)	(2)

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

mark^#(cons(X1,X2))	→	active^#(cons(X1,X2))	(107)
mark^#(if(X1,X2,X3))	→	active^#(if(mark(X1),X2,X3))	(76)
active^#(and(true,X))	→	mark^#(X)	(75)
mark^#(first(X1,X2))	→	active^#(first(mark(X1),mark(X2)))	(72)
mark^#(s(X))	→	active^#(s(X))	(103)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(100)
active^#(first(s(X),cons(Y,Z)))	→	mark^#(cons(Y,first(X,Z)))	(69)
mark^#(from(X))	→	active^#(from(X))	(95)
active^#(from(X))	→	mark^#(cons(X,from(s(X))))	(94)
active^#(if(true,X,Y))	→	mark^#(X)	(63)
mark^#(add(X1,X2))	→	active^#(add(mark(X1),X2))	(92)
active^#(if(false,X,Y))	→	mark^#(Y)	(59)
mark^#(first(X1,X2))	→	mark^#(X2)	(57)
mark^#(first(X1,X2))	→	mark^#(X1)	(55)
mark^#(and(X1,X2))	→	mark^#(X1)	(54)
active^#(add(s(X),Y))	→	mark^#(s(add(X,Y)))	(85)
active^#(add(0,X))	→	mark^#(X)	(84)
mark^#(and(X1,X2))	→	active^#(and(mark(X1),X2))	(50)
mark^#(add(X1,X2))	→	mark^#(X1)	(49)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

if^#(X1,X2,active(X3))	→	if^#(X1,X2,X3)	(78)
if^#(mark(X1),X2,X3)	→	if^#(X1,X2,X3)	(74)
if^#(X1,active(X2),X3)	→	if^#(X1,X2,X3)	(101)
if^#(X1,mark(X2),X3)	→	if^#(X1,X2,X3)	(65)
if^#(X1,X2,mark(X3))	→	if^#(X1,X2,X3)	(88)
if^#(active(X1),X2,X3)	→	if^#(X1,X2,X3)	(53)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + 30344
[false]	=	1
[true]	=	1
[mark^#(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₃ + 20585
[0]	=	0
[from(x₁)]	=	x₁ + 61131
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	0
[nil]	=	2737
[mark(x₁)]	=	x₁ + 30347
[first(x₁, x₂)]	=	1
[from^#(x₁)]	=	0
[active(x₁)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₂ + 6879
[if^#(x₁, x₂, x₃)]	=	x₁ + 0
[active^#(x₁)]	=	0
[add^#(x₁, x₂)]	=	0
[add(x₁, x₂)]	=	x₁ + 3308
[and^#(x₁, x₂)]	=	0

together with the usable rules

cons(X1,active(X2))	→	cons(X1,X2)	(44)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)

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

if^#(mark(X1),X2,X3)	→	if^#(X1,X2,X3)	(74)
if^#(active(X1),X2,X3)	→	if^#(X1,X2,X3)	(53)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

if^#(X1,mark(X2),X3)	→	if^#(X1,X2,X3)	(65)
if^#(X1,X2,mark(X3))	→	if^#(X1,X2,X3)	(88)
if^#(X1,X2,active(X3))	→	if^#(X1,X2,X3)	(78)
if^#(X1,active(X2),X3)	→	if^#(X1,X2,X3)	(101)

1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + 19265
[false]	=	1
[true]	=	1
[mark^#(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₃ + 28019
[0]	=	0
[from(x₁)]	=	x₁ + 1
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	0
[nil]	=	18826
[mark(x₁)]	=	x₁ + 30347
[first(x₁, x₂)]	=	1
[from^#(x₁)]	=	0
[active(x₁)]	=	x₁ + 2326
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	x₂ + x₃ + 0
[active^#(x₁)]	=	0
[add^#(x₁, x₂)]	=	0
[add(x₁, x₂)]	=	x₁ + 22115
[and^#(x₁, x₂)]	=	0

together with the usable rules

cons(X1,active(X2))	→	cons(X1,X2)	(44)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)

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

if^#(X1,mark(X2),X3)	→	if^#(X1,X2,X3)	(65)
if^#(X1,X2,mark(X3))	→	if^#(X1,X2,X3)	(88)
if^#(X1,X2,active(X3))	→	if^#(X1,X2,X3)	(78)
if^#(X1,active(X2),X3)	→	if^#(X1,X2,X3)	(101)

could be deleted.

1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

from^#(mark(X))	→	from^#(X)	(70)
from^#(active(X))	→	from^#(X)	(87)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + 2748
[false]	=	4744
[true]	=	1
[mark^#(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₃ + 5596
[0]	=	0
[from(x₁)]	=	x₁ + 1
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	0
[nil]	=	31203
[mark(x₁)]	=	x₁ + 30347
[first(x₁, x₂)]	=	1
[from^#(x₁)]	=	x₁ + 0
[active(x₁)]	=	x₁ + 2326
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[active^#(x₁)]	=	0
[add^#(x₁, x₂)]	=	0
[add(x₁, x₂)]	=	x₁ + 1
[and^#(x₁, x₂)]	=	0

together with the usable rules

cons(X1,active(X2))	→	cons(X1,X2)	(44)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)

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

from^#(mark(X))	→	from^#(X)	(70)
from^#(active(X))	→	from^#(X)	(87)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

and^#(active(X1),X2)	→	and^#(X1,X2)	(71)
and^#(X1,active(X2))	→	and^#(X1,X2)	(90)
and^#(mark(X1),X2)	→	and^#(X1,X2)	(56)
and^#(X1,mark(X2))	→	and^#(X1,X2)	(52)

1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + 1
[false]	=	1
[true]	=	1
[mark^#(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₃ + 36319
[0]	=	0
[from(x₁)]	=	x₁ + 1
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	0
[nil]	=	36554
[mark(x₁)]	=	x₁ + 36322
[first(x₁, x₂)]	=	1
[from^#(x₁)]	=	0
[active(x₁)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₂ + 32968
[if^#(x₁, x₂, x₃)]	=	0
[active^#(x₁)]	=	0
[add^#(x₁, x₂)]	=	0
[add(x₁, x₂)]	=	x₁ + 1
[and^#(x₁, x₂)]	=	x₁ + 0

together with the usable rules

cons(X1,active(X2))	→	cons(X1,X2)	(44)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)

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

and^#(active(X1),X2)	→	and^#(X1,X2)	(71)
and^#(mark(X1),X2)	→	and^#(X1,X2)	(56)

could be deleted.

1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

and^#(X1,mark(X2))	→	and^#(X1,X2)	(52)
and^#(X1,active(X2))	→	and^#(X1,X2)	(90)

1.1.4.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + 1
[false]	=	1
[true]	=	1
[mark^#(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₃ + 1
[0]	=	0
[from(x₁)]	=	x₁ + 1
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	0
[nil]	=	47804
[mark(x₁)]	=	x₁ + 4
[first(x₁, x₂)]	=	15620
[from^#(x₁)]	=	0
[active(x₁)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[active^#(x₁)]	=	0
[add^#(x₁, x₂)]	=	0
[add(x₁, x₂)]	=	x₁ + 2
[and^#(x₁, x₂)]	=	x₂ + 0

together with the usable rules

cons(X1,active(X2))	→	cons(X1,X2)	(44)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)

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

and^#(X1,mark(X2))	→	and^#(X1,X2)	(52)
and^#(X1,active(X2))	→	and^#(X1,X2)	(90)

could be deleted.

1.1.4.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 5^th component contains the pair

first^#(mark(X1),X2)	→	first^#(X1,X2)	(104)
first^#(X1,mark(X2))	→	first^#(X1,X2)	(67)
first^#(X1,active(X2))	→	first^#(X1,X2)	(66)
first^#(active(X1),X2)	→	first^#(X1,X2)	(91)

1.1.5 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + 1
[false]	=	1
[true]	=	1
[mark^#(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₃ + 1
[0]	=	0
[from(x₁)]	=	x₁ + 1
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	x₁ + x₂ + 0
[nil]	=	26170
[mark(x₁)]	=	x₁ + 4
[first(x₁, x₂)]	=	1
[from^#(x₁)]	=	0
[active(x₁)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[active^#(x₁)]	=	0
[add^#(x₁, x₂)]	=	0
[add(x₁, x₂)]	=	x₁ + 2
[and^#(x₁, x₂)]	=	0

together with the usable rules

cons(X1,active(X2))	→	cons(X1,X2)	(44)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)

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

first^#(mark(X1),X2)	→	first^#(X1,X2)	(104)
first^#(X1,mark(X2))	→	first^#(X1,X2)	(67)
first^#(X1,active(X2))	→	first^#(X1,X2)	(66)
first^#(active(X1),X2)	→	first^#(X1,X2)	(91)

could be deleted.

1.1.5.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 6^th component contains the pair

add^#(X1,mark(X2))	→	add^#(X1,X2)	(80)
add^#(mark(X1),X2)	→	add^#(X1,X2)	(97)
add^#(active(X1),X2)	→	add^#(X1,X2)	(60)
add^#(X1,active(X2))	→	add^#(X1,X2)	(48)

1.1.6 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + 1
[false]	=	1
[true]	=	1
[mark^#(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₃ + 1
[0]	=	0
[from(x₁)]	=	x₁ + 55691
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	0
[nil]	=	32570
[mark(x₁)]	=	x₁ + 4
[first(x₁, x₂)]	=	11668
[from^#(x₁)]	=	0
[active(x₁)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[active^#(x₁)]	=	0
[add^#(x₁, x₂)]	=	x₁ + x₂ + 0
[add(x₁, x₂)]	=	x₁ + 1
[and^#(x₁, x₂)]	=	0

together with the usable rules

cons(X1,active(X2))	→	cons(X1,X2)	(44)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)

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

add^#(X1,mark(X2))	→	add^#(X1,X2)	(80)
add^#(mark(X1),X2)	→	add^#(X1,X2)	(97)
add^#(active(X1),X2)	→	add^#(X1,X2)	(60)
add^#(X1,active(X2))	→	add^#(X1,X2)	(48)

could be deleted.

1.1.6.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 7^th component contains the pair

cons^#(X1,active(X2))	→	cons^#(X1,X2)	(64)
cons^#(X1,mark(X2))	→	cons^#(X1,X2)	(61)
cons^#(mark(X1),X2)	→	cons^#(X1,X2)	(83)
cons^#(active(X1),X2)	→	cons^#(X1,X2)	(47)

1.1.7 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	x₁ + x₂ + 0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + 1
[false]	=	1
[true]	=	1
[mark^#(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₃ + 1
[0]	=	0
[from(x₁)]	=	x₁ + 1
[s^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	0
[nil]	=	60104
[mark(x₁)]	=	x₁ + 4
[first(x₁, x₂)]	=	1
[from^#(x₁)]	=	0
[active(x₁)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[active^#(x₁)]	=	0
[add^#(x₁, x₂)]	=	0
[add(x₁, x₂)]	=	x₁ + 1
[and^#(x₁, x₂)]	=	0

together with the usable rules

cons(X1,active(X2))	→	cons(X1,X2)	(44)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)

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

cons^#(X1,active(X2))	→	cons^#(X1,X2)	(64)
cons^#(X1,mark(X2))	→	cons^#(X1,X2)	(61)
cons^#(mark(X1),X2)	→	cons^#(X1,X2)	(83)
cons^#(active(X1),X2)	→	cons^#(X1,X2)	(47)

could be deleted.

1.1.7.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 8^th component contains the pair

s^#(active(X))	→	s^#(X)	(79)
s^#(mark(X))	→	s^#(X)	(106)

1.1.8 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[cons^#(x₁, x₂)]	=	0
[s(x₁)]	=	1
[and(x₁, x₂)]	=	x₁ + 1
[false]	=	1
[true]	=	1
[mark^#(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₃ + 1
[0]	=	0
[from(x₁)]	=	x₁ + 1
[s^#(x₁)]	=	x₁ + 0
[first^#(x₁, x₂)]	=	0
[nil]	=	33651
[mark(x₁)]	=	x₁ + 4
[first(x₁, x₂)]	=	11495
[from^#(x₁)]	=	0
[active(x₁)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[active^#(x₁)]	=	0
[add^#(x₁, x₂)]	=	0
[add(x₁, x₂)]	=	x₁ + 1
[and^#(x₁, x₂)]	=	0

together with the usable rules

cons(X1,active(X2))	→	cons(X1,X2)	(44)
first(active(X1),X2)	→	first(X1,X2)	(39)
first(X1,active(X2))	→	first(X1,X2)	(40)
first(X1,mark(X2))	→	first(X1,X2)	(38)
first(mark(X1),X2)	→	first(X1,X2)	(37)
cons(mark(X1),X2)	→	cons(X1,X2)	(41)
cons(X1,mark(X2))	→	cons(X1,X2)	(42)
cons(active(X1),X2)	→	cons(X1,X2)	(43)

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

s^#(active(X))	→	s^#(X)	(79)
s^#(mark(X))	→	s^#(X)	(106)

could be deleted.

1.1.8.1 Dependency Graph Processor

The dependency pairs are split into 0 components.