Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PALINDROME_complete_FR)

The rewrite relation of the following TRS is considered.

__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(X,nil)	→	X	(2)
__(nil,X)	→	X	(3)
U11(tt,V)	→	U12(isNeList(activate(V)))	(4)
U12(tt)	→	tt	(5)
U21(tt,V1,V2)	→	U22(isList(activate(V1)),activate(V2))	(6)
U22(tt,V2)	→	U23(isList(activate(V2)))	(7)
U23(tt)	→	tt	(8)
U31(tt,V)	→	U32(isQid(activate(V)))	(9)
U32(tt)	→	tt	(10)
U41(tt,V1,V2)	→	U42(isList(activate(V1)),activate(V2))	(11)
U42(tt,V2)	→	U43(isNeList(activate(V2)))	(12)
U43(tt)	→	tt	(13)
U51(tt,V1,V2)	→	U52(isNeList(activate(V1)),activate(V2))	(14)
U52(tt,V2)	→	U53(isList(activate(V2)))	(15)
U53(tt)	→	tt	(16)
U61(tt,V)	→	U62(isQid(activate(V)))	(17)
U62(tt)	→	tt	(18)
U71(tt,V)	→	U72(isNePal(activate(V)))	(19)
U72(tt)	→	tt	(20)
and(tt,X)	→	activate(X)	(21)
isList(V)	→	U11(isPalListKind(activate(V)),activate(V))	(22)
isList(n__nil)	→	tt	(23)
isList(n____(V1,V2))	→	U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(24)
isNeList(V)	→	U31(isPalListKind(activate(V)),activate(V))	(25)
isNeList(n____(V1,V2))	→	U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(26)
isNeList(n____(V1,V2))	→	U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(27)
isNePal(V)	→	U61(isPalListKind(activate(V)),activate(V))	(28)
isNePal(n____(I,n____(P,I)))	→	and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))	(29)
isPal(V)	→	U71(isPalListKind(activate(V)),activate(V))	(30)
isPal(n__nil)	→	tt	(31)
isPalListKind(n__a)	→	tt	(32)
isPalListKind(n__e)	→	tt	(33)
isPalListKind(n__i)	→	tt	(34)
isPalListKind(n__nil)	→	tt	(35)
isPalListKind(n__o)	→	tt	(36)
isPalListKind(n__u)	→	tt	(37)
isPalListKind(n____(V1,V2))	→	and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(38)
isQid(n__a)	→	tt	(39)
isQid(n__e)	→	tt	(40)
isQid(n__i)	→	tt	(41)
isQid(n__o)	→	tt	(42)
isQid(n__u)	→	tt	(43)
nil	→	n__nil	(44)
__(X1,X2)	→	n____(X1,X2)	(45)
isPalListKind(X)	→	n__isPalListKind(X)	(46)
and(X1,X2)	→	n__and(X1,X2)	(47)
isPal(X)	→	n__isPal(X)	(48)
a	→	n__a	(49)
e	→	n__e	(50)
i	→	n__i	(51)
o	→	n__o	(52)
u	→	n__u	(53)
activate(n__nil)	→	nil	(54)
activate(n____(X1,X2))	→	__(activate(X1),activate(X2))	(55)
activate(n__isPalListKind(X))	→	isPalListKind(X)	(56)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(57)
activate(n__isPal(X))	→	isPal(X)	(58)
activate(n__a)	→	a	(59)
activate(n__e)	→	e	(60)
activate(n__i)	→	i	(61)
activate(n__o)	→	o	(62)
activate(n__u)	→	u	(63)
activate(X)	→	X	(64)

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.

isNePal^#(n____(I,n____(P,I)))	→	and^#(isQid(activate(I)),n__isPalListKind(activate(I)))	(65)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(66)
U51^#(tt,V1,V2)	→	isNeList^#(activate(V1))	(67)
activate^#(n__and(X1,X2))	→	and^#(activate(X1),X2)	(68)
isList^#(n____(V1,V2))	→	activate^#(V2)	(69)
isNeList^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(70)
isNeList^#(n____(V1,V2))	→	U51^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(71)
activate^#(n__o)	→	o^#	(72)
U51^#(tt,V1,V2)	→	U52^#(isNeList(activate(V1)),activate(V2))	(73)
U11^#(tt,V)	→	U12^#(isNeList(activate(V)))	(74)
activate^#(n__isPalListKind(X))	→	isPalListKind^#(X)	(75)
isList^#(n____(V1,V2))	→	activate^#(V2)	(69)
U11^#(tt,V)	→	activate^#(V)	(76)
activate^#(n__nil)	→	nil^#	(77)
activate^#(n__e)	→	e^#	(78)
isPal^#(V)	→	U71^#(isPalListKind(activate(V)),activate(V))	(79)
isNeList^#(V)	→	activate^#(V)	(80)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(66)
isPal^#(V)	→	isPalListKind^#(activate(V))	(81)
U61^#(tt,V)	→	isQid^#(activate(V))	(82)
U61^#(tt,V)	→	U62^#(isQid(activate(V)))	(83)
isNePal^#(V)	→	U61^#(isPalListKind(activate(V)),activate(V))	(84)
isNePal^#(V)	→	activate^#(V)	(85)
U42^#(tt,V2)	→	U43^#(isNeList(activate(V2)))	(86)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(P)	(87)
isNePal^#(V)	→	isPalListKind^#(activate(V))	(88)
U22^#(tt,V2)	→	U23^#(isList(activate(V2)))	(89)
isNeList^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(90)
U42^#(tt,V2)	→	activate^#(V2)	(91)
__^#(__(X,Y),Z)	→	__^#(Y,Z)	(92)
isNeList^#(V)	→	U31^#(isPalListKind(activate(V)),activate(V))	(93)
isPalListKind^#(n____(V1,V2))	→	activate^#(V2)	(94)
activate^#(n__i)	→	i^#	(95)
isPalListKind^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(96)
U22^#(tt,V2)	→	activate^#(V2)	(97)
U61^#(tt,V)	→	activate^#(V)	(98)
U52^#(tt,V2)	→	activate^#(V2)	(99)
U31^#(tt,V)	→	activate^#(V)	(100)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(101)
U31^#(tt,V)	→	isQid^#(activate(V))	(102)
isNeList^#(n____(V1,V2))	→	U41^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(103)
U71^#(tt,V)	→	isNePal^#(activate(V))	(104)
U51^#(tt,V1,V2)	→	activate^#(V2)	(105)
U11^#(tt,V)	→	isNeList^#(activate(V))	(106)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(66)
isPalListKind^#(n____(V1,V2))	→	activate^#(V1)	(107)
isList^#(n____(V1,V2))	→	U21^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(108)
U71^#(tt,V)	→	activate^#(V)	(109)
U22^#(tt,V2)	→	isList^#(activate(V2))	(110)
__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(111)
isNeList^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(90)
U21^#(tt,V1,V2)	→	U22^#(isList(activate(V1)),activate(V2))	(112)
isList^#(V)	→	U11^#(isPalListKind(activate(V)),activate(V))	(113)
activate^#(n__and(X1,X2))	→	activate^#(X1)	(114)
isNeList^#(V)	→	isPalListKind^#(activate(V))	(115)
isPal^#(V)	→	activate^#(V)	(116)
U41^#(tt,V1,V2)	→	U42^#(isList(activate(V1)),activate(V2))	(117)
activate^#(n____(X1,X2))	→	activate^#(X1)	(118)
and^#(tt,X)	→	activate^#(X)	(119)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(P)	(87)
U42^#(tt,V2)	→	isNeList^#(activate(V2))	(120)
isPalListKind^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(121)
U52^#(tt,V2)	→	U53^#(isList(activate(V2)))	(122)
U52^#(tt,V2)	→	isList^#(activate(V2))	(123)
isList^#(V)	→	activate^#(V)	(124)
isList^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(125)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(101)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(I)	(126)
U51^#(tt,V1,V2)	→	activate^#(V1)	(127)
isList^#(V)	→	activate^#(V)	(124)
U41^#(tt,V1,V2)	→	activate^#(V2)	(128)
U31^#(tt,V)	→	U32^#(isQid(activate(V)))	(129)
isList^#(n____(V1,V2))	→	activate^#(V1)	(130)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(66)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(101)
U41^#(tt,V1,V2)	→	activate^#(V1)	(131)
isNePal^#(V)	→	activate^#(V)	(85)
U21^#(tt,V1,V2)	→	activate^#(V2)	(132)
isList^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(133)
activate^#(n____(X1,X2))	→	__^#(activate(X1),activate(X2))	(134)
isList^#(V)	→	isPalListKind^#(activate(V))	(135)
U41^#(tt,V1,V2)	→	isList^#(activate(V1))	(136)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(101)
isNeList^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(70)
isList^#(n____(V1,V2))	→	activate^#(V1)	(130)
activate^#(n____(X1,X2))	→	activate^#(X2)	(137)
U21^#(tt,V1,V2)	→	activate^#(V1)	(138)
isNePal^#(n____(I,n____(P,I)))	→	isQid^#(activate(I))	(139)
U71^#(tt,V)	→	U72^#(isNePal(activate(V)))	(140)
isNePal^#(n____(I,n____(P,I)))	→	and^#(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))	(141)
isNeList^#(V)	→	activate^#(V)	(80)
activate^#(n__isPal(X))	→	isPal^#(X)	(142)
U21^#(tt,V1,V2)	→	isList^#(activate(V1))	(143)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(I)	(126)
activate^#(n__u)	→	u^#	(144)
isPal^#(V)	→	activate^#(V)	(116)
activate^#(n__a)	→	a^#	(145)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

U41^#(tt,V1,V2)	→	U42^#(isList(activate(V1)),activate(V2))	(117)
isList^#(V)	→	U11^#(isPalListKind(activate(V)),activate(V))	(113)
U21^#(tt,V1,V2)	→	U22^#(isList(activate(V1)),activate(V2))	(112)
U21^#(tt,V1,V2)	→	isList^#(activate(V1))	(143)
U22^#(tt,V2)	→	isList^#(activate(V2))	(110)
isList^#(n____(V1,V2))	→	U21^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(108)
U11^#(tt,V)	→	isNeList^#(activate(V))	(106)
isNeList^#(n____(V1,V2))	→	U41^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(103)
U41^#(tt,V1,V2)	→	isList^#(activate(V1))	(136)
U52^#(tt,V2)	→	isList^#(activate(V2))	(123)
U51^#(tt,V1,V2)	→	U52^#(isNeList(activate(V1)),activate(V2))	(73)
isNeList^#(n____(V1,V2))	→	U51^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(71)
U42^#(tt,V2)	→	isNeList^#(activate(V2))	(120)
U51^#(tt,V1,V2)	→	isNeList^#(activate(V1))	(67)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[n__a]	=	1
[U72^#(x₁)]	=	0
[U32^#(x₁)]	=	0
[U21(x₁, x₂, x₃)]	=	12205
[isPalListKind^#(x₁)]	=	0
[U11(x₁, x₂)]	=	626
[n__o]	=	0
[isNeList(x₁)]	=	3
[isPal(x₁)]	=	x₁ + 12208
[U42(x₁, x₂)]	=	5
[e^#]	=	0
[u]	=	25984
[activate(x₁)]	=	x₁ + 0
[U71(x₁, x₂)]	=	12208
[and(x₁, x₂)]	=	x₂ + 0
[u^#]	=	0
[isNeList^#(x₁)]	=	x₁ + 12207
[U43(x₁)]	=	6
[activate^#(x₁)]	=	0
[n__i]	=	16021
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	0
[U43^#(x₁)]	=	0
[U23(x₁)]	=	12207
[isNePal(x₁)]	=	1
[U72(x₁)]	=	12208
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	0
[n__isPal(x₁)]	=	x₁ + 12208
[n__nil]	=	11582
[U52^#(x₁, x₂)]	=	x₂ + 23784
[U12(x₁)]	=	x₁ + 624
[isQid(x₁)]	=	12208
[n____(x₁, x₂)]	=	x₁ + x₂ + 11579
[n__e]	=	0
[o]	=	0
[n__isPalListKind(x₁)]	=	12208
[U42^#(x₁, x₂)]	=	x₂ + 12207
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 625
[isNePal^#(x₁)]	=	0
[nil]	=	11582
[n__u]	=	25984
[o^#]	=	0
[U62(x₁)]	=	x₁ + 1
[nil^#]	=	0
[isList^#(x₁)]	=	x₁ + 23783
[U32(x₁)]	=	5
[i]	=	16021
[U52(x₁, x₂)]	=	x₁ + 2
[U61(x₁, x₂)]	=	x₁ + 0
[U51^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 11577
[e]	=	0
[U11^#(x₁, x₂)]	=	x₁ + x₂ + 0
[U31(x₁, x₂)]	=	4
[U41^#(x₁, x₂, x₃)]	=	x₂ + x₃ + 23784
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 11576
[U22^#(x₁, x₂)]	=	x₂ + 23784
[tt]	=	12208
[n__and(x₁, x₂)]	=	x₂ + 0
[U71^#(x₁, x₂)]	=	0
[U22(x₁, x₂)]	=	12206
[U51(x₁, x₂, x₃)]	=	x₂ + x₃ + 4
[isPalListKind(x₁)]	=	12208
[U53(x₁)]	=	x₁ + 11586
[U41(x₁, x₂, x₃)]	=	x₃ + 4
[U31^#(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	0
[__(x₁, x₂)]	=	x₁ + x₂ + 11579
[U61^#(x₁, x₂)]	=	0
[i^#]	=	0

together with the usable rules

e	→	n__e	(50)
activate(n__nil)	→	nil	(54)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
and(tt,X)	→	activate(X)	(21)
isPalListKind(n__o)	→	tt	(36)
activate(n__u)	→	u	(63)
U71(tt,V)	→	U72(isNePal(activate(V)))	(19)
isPalListKind(n__a)	→	tt	(32)
activate(n__e)	→	e	(60)
isPalListKind(n__i)	→	tt	(34)
nil	→	n__nil	(44)
isPalListKind(n__e)	→	tt	(33)
activate(X)	→	X	(64)
isQid(n__a)	→	tt	(39)
U72(tt)	→	tt	(20)
a	→	n__a	(49)
o	→	n__o	(52)
isPal(V)	→	U71(isPalListKind(activate(V)),activate(V))	(30)
activate(n__o)	→	o	(62)
activate(n__isPalListKind(X))	→	isPalListKind(X)	(56)
isPal(n__nil)	→	tt	(31)
__(X1,X2)	→	n____(X1,X2)	(45)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(57)
i	→	n__i	(51)
isQid(n__e)	→	tt	(40)
activate(n____(X1,X2))	→	__(activate(X1),activate(X2))	(55)
activate(n__a)	→	a	(59)
isPalListKind(n____(V1,V2))	→	and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(38)
activate(n__i)	→	i	(61)
activate(n__isPal(X))	→	isPal(X)	(58)
isPal(X)	→	n__isPal(X)	(48)
u	→	n__u	(53)
and(X1,X2)	→	n__and(X1,X2)	(47)
isPalListKind(n__u)	→	tt	(37)
isQid(n__i)	→	tt	(41)
isQid(n__o)	→	tt	(42)
isPalListKind(X)	→	n__isPalListKind(X)	(46)
isPalListKind(n__nil)	→	tt	(35)
isQid(n__u)	→	tt	(43)
__(X,nil)	→	X	(2)

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

U41^#(tt,V1,V2)	→	U42^#(isList(activate(V1)),activate(V2))	(117)
isList^#(V)	→	U11^#(isPalListKind(activate(V)),activate(V))	(113)
U21^#(tt,V1,V2)	→	isList^#(activate(V1))	(143)
U22^#(tt,V2)	→	isList^#(activate(V2))	(110)
isList^#(n____(V1,V2))	→	U21^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(108)
U11^#(tt,V)	→	isNeList^#(activate(V))	(106)
isNeList^#(n____(V1,V2))	→	U41^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(103)
U41^#(tt,V1,V2)	→	isList^#(activate(V1))	(136)
U52^#(tt,V2)	→	isList^#(activate(V2))	(123)
U51^#(tt,V1,V2)	→	U52^#(isNeList(activate(V1)),activate(V2))	(73)
isNeList^#(n____(V1,V2))	→	U51^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(71)
U51^#(tt,V1,V2)	→	isNeList^#(activate(V1))	(67)

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

isPal^#(V)	→	activate^#(V)	(116)
isPal^#(V)	→	activate^#(V)	(116)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(I)	(126)
activate^#(n__and(X1,X2))	→	activate^#(X1)	(114)
U71^#(tt,V)	→	activate^#(V)	(109)
isPalListKind^#(n____(V1,V2))	→	activate^#(V1)	(107)
activate^#(n__isPal(X))	→	isPal^#(X)	(142)
U71^#(tt,V)	→	isNePal^#(activate(V))	(104)
isNePal^#(n____(I,n____(P,I)))	→	and^#(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))	(141)
activate^#(n____(X1,X2))	→	activate^#(X2)	(137)
U61^#(tt,V)	→	activate^#(V)	(98)
isPalListKind^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(96)
isPalListKind^#(n____(V1,V2))	→	activate^#(V2)	(94)
isNePal^#(V)	→	activate^#(V)	(85)
isNePal^#(V)	→	isPalListKind^#(activate(V))	(88)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(P)	(87)
isNePal^#(V)	→	activate^#(V)	(85)
isNePal^#(V)	→	U61^#(isPalListKind(activate(V)),activate(V))	(84)
isPal^#(V)	→	isPalListKind^#(activate(V))	(81)
isPal^#(V)	→	U71^#(isPalListKind(activate(V)),activate(V))	(79)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(I)	(126)
activate^#(n__isPalListKind(X))	→	isPalListKind^#(X)	(75)
isPalListKind^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(121)
and^#(tt,X)	→	activate^#(X)	(119)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(P)	(87)
activate^#(n__and(X1,X2))	→	and^#(activate(X1),X2)	(68)
isNePal^#(n____(I,n____(P,I)))	→	and^#(isQid(activate(I)),n__isPalListKind(activate(I)))	(65)
activate^#(n____(X1,X2))	→	activate^#(X1)	(118)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[n__a]	=	1
[U72^#(x₁)]	=	0
[U32^#(x₁)]	=	0
[U21(x₁, x₂, x₃)]	=	max(0)
[isPalListKind^#(x₁)]	=	x₁ + 6
[U11(x₁, x₂)]	=	max(0)
[n__o]	=	1143
[isNeList(x₁)]	=	0
[isPal(x₁)]	=	x₁ + 6
[U42(x₁, x₂)]	=	max(0)
[e^#]	=	0
[u]	=	23626
[activate(x₁)]	=	x₁ + 0
[U71(x₁, x₂)]	=	max(x₁ + 0, x₂ + 3, 0)
[and(x₁, x₂)]	=	max(x₁ + 1, x₂ + 0, 0)
[u^#]	=	0
[isNeList^#(x₁)]	=	0
[U43(x₁)]	=	0
[activate^#(x₁)]	=	x₁ + 4
[n__i]	=	21258
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	max(0)
[U43^#(x₁)]	=	0
[U23(x₁)]	=	0
[isNePal(x₁)]	=	1
[U72(x₁)]	=	3
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	x₁ + 9
[n__isPal(x₁)]	=	x₁ + 6
[n__nil]	=	52429
[U52^#(x₁, x₂)]	=	max(0)
[U12(x₁)]	=	0
[isQid(x₁)]	=	2
[n____(x₁, x₂)]	=	max(x₁ + 26290, x₂ + 0, 0)
[n__e]	=	15922
[o]	=	1143
[n__isPalListKind(x₁)]	=	x₁ + 2
[U42^#(x₁, x₂)]	=	max(0)
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	0
[isNePal^#(x₁)]	=	x₁ + 7
[nil]	=	52429
[n__u]	=	23626
[o^#]	=	0
[U62(x₁)]	=	1
[nil^#]	=	0
[isList^#(x₁)]	=	0
[U32(x₁)]	=	0
[i]	=	21258
[U52(x₁, x₂)]	=	max(0)
[U61(x₁, x₂)]	=	max(0)
[U51^#(x₁, x₂, x₃)]	=	max(0)
[e]	=	15922
[U11^#(x₁, x₂)]	=	max(0)
[U31(x₁, x₂)]	=	max(0)
[U41^#(x₁, x₂, x₃)]	=	max(0)
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	max(0)
[U22^#(x₁, x₂)]	=	max(0)
[tt]	=	2
[n__and(x₁, x₂)]	=	max(x₁ + 1, x₂ + 0, 0)
[U71^#(x₁, x₂)]	=	max(x₁ + 6, x₂ + 8, 0)
[U22(x₁, x₂)]	=	max(0)
[U51(x₁, x₂, x₃)]	=	max(0)
[isPalListKind(x₁)]	=	x₁ + 2
[U53(x₁)]	=	0
[U41(x₁, x₂, x₃)]	=	max(0)
[U31^#(x₁, x₂)]	=	max(0)
[and^#(x₁, x₂)]	=	max(x₁ + 1, x₂ + 4, 0)
[__(x₁, x₂)]	=	max(x₁ + 26290, x₂ + 0, 0)
[U61^#(x₁, x₂)]	=	max(x₁ + 3, x₂ + 5, 0)
[i^#]	=	0

together with the usable rules

e	→	n__e	(50)
activate(n__nil)	→	nil	(54)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
and(tt,X)	→	activate(X)	(21)
isPalListKind(n__o)	→	tt	(36)
activate(n__u)	→	u	(63)
U71(tt,V)	→	U72(isNePal(activate(V)))	(19)
isPalListKind(n__a)	→	tt	(32)
activate(n__e)	→	e	(60)
isPalListKind(n__i)	→	tt	(34)
nil	→	n__nil	(44)
isPalListKind(n__e)	→	tt	(33)
activate(X)	→	X	(64)
isQid(n__a)	→	tt	(39)
U72(tt)	→	tt	(20)
a	→	n__a	(49)
o	→	n__o	(52)
isPal(V)	→	U71(isPalListKind(activate(V)),activate(V))	(30)
activate(n__o)	→	o	(62)
activate(n__isPalListKind(X))	→	isPalListKind(X)	(56)
isPal(n__nil)	→	tt	(31)
__(X1,X2)	→	n____(X1,X2)	(45)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(57)
i	→	n__i	(51)
isQid(n__e)	→	tt	(40)
activate(n____(X1,X2))	→	__(activate(X1),activate(X2))	(55)
activate(n__a)	→	a	(59)
isPalListKind(n____(V1,V2))	→	and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(38)
activate(n__i)	→	i	(61)
activate(n__isPal(X))	→	isPal(X)	(58)
isPal(X)	→	n__isPal(X)	(48)
u	→	n__u	(53)
and(X1,X2)	→	n__and(X1,X2)	(47)
isPalListKind(n__u)	→	tt	(37)
isQid(n__i)	→	tt	(41)
isQid(n__o)	→	tt	(42)
isPalListKind(X)	→	n__isPalListKind(X)	(46)
isPalListKind(n__nil)	→	tt	(35)
isQid(n__u)	→	tt	(43)
__(X,nil)	→	X	(2)

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

isPal^#(V)	→	activate^#(V)	(116)
isPal^#(V)	→	activate^#(V)	(116)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(I)	(126)
activate^#(n__and(X1,X2))	→	activate^#(X1)	(114)
U71^#(tt,V)	→	activate^#(V)	(109)
isPalListKind^#(n____(V1,V2))	→	activate^#(V1)	(107)
activate^#(n__isPal(X))	→	isPal^#(X)	(142)
U71^#(tt,V)	→	isNePal^#(activate(V))	(104)
isNePal^#(n____(I,n____(P,I)))	→	and^#(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))	(141)
U61^#(tt,V)	→	activate^#(V)	(98)
isPalListKind^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(96)
isPalListKind^#(n____(V1,V2))	→	activate^#(V2)	(94)
isNePal^#(V)	→	activate^#(V)	(85)
isNePal^#(V)	→	isPalListKind^#(activate(V))	(88)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(P)	(87)
isNePal^#(V)	→	activate^#(V)	(85)
isNePal^#(V)	→	U61^#(isPalListKind(activate(V)),activate(V))	(84)
isPal^#(V)	→	isPalListKind^#(activate(V))	(81)
isPal^#(V)	→	U71^#(isPalListKind(activate(V)),activate(V))	(79)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(I)	(126)
isNePal^#(n____(I,n____(P,I)))	→	activate^#(P)	(87)
isNePal^#(n____(I,n____(P,I)))	→	and^#(isQid(activate(I)),n__isPalListKind(activate(I)))	(65)
activate^#(n____(X1,X2))	→	activate^#(X1)	(118)

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

and^#(tt,X)	→	activate^#(X)	(119)
activate^#(n__isPalListKind(X))	→	isPalListKind^#(X)	(75)
activate^#(n__and(X1,X2))	→	and^#(activate(X1),X2)	(68)
activate^#(n____(X1,X2))	→	activate^#(X2)	(137)
isPalListKind^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(121)

1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[n__a]	=	2
[U72^#(x₁)]	=	0
[U32^#(x₁)]	=	0
[U21(x₁, x₂, x₃)]	=	7968
[isPalListKind^#(x₁)]	=	26034
[U11(x₁, x₂)]	=	x₁ + 4386
[n__o]	=	2
[isNeList(x₁)]	=	633
[isPal(x₁)]	=	x₁ + 1
[U42(x₁, x₂)]	=	7972
[e^#]	=	0
[u]	=	1
[activate(x₁)]	=	0
[U71(x₁, x₂)]	=	x₂ + 2
[and(x₁, x₂)]	=	x₂ + 1
[u^#]	=	0
[isNeList^#(x₁)]	=	12207
[U43(x₁)]	=	7970
[activate^#(x₁)]	=	x₁ + 12208
[n__i]	=	16022
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	0
[U43^#(x₁)]	=	0
[U23(x₁)]	=	7970
[isNePal(x₁)]	=	1
[U72(x₁)]	=	3
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	12208
[n__isPal(x₁)]	=	2
[n__nil]	=	2
[U52^#(x₁, x₂)]	=	23784
[U12(x₁)]	=	x₁ + 12831
[isQid(x₁)]	=	7971
[n____(x₁, x₂)]	=	x₂ + 3581
[n__e]	=	2
[o]	=	1
[n__isPalListKind(x₁)]	=	13826
[U42^#(x₁, x₂)]	=	12207
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 4386
[isNePal^#(x₁)]	=	12208
[nil]	=	1
[n__u]	=	2
[o^#]	=	0
[U62(x₁)]	=	x₁ + 2
[nil^#]	=	0
[isList^#(x₁)]	=	23783
[U32(x₁)]	=	635
[i]	=	16021
[U52(x₁, x₂)]	=	x₁ + 2
[U61(x₁, x₂)]	=	x₁ + 1
[U51^#(x₁, x₂, x₃)]	=	11577
[e]	=	1
[U11^#(x₁, x₂)]	=	0
[U31(x₁, x₂)]	=	634
[U41^#(x₁, x₂, x₃)]	=	23784
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	11576
[U22^#(x₁, x₂)]	=	23784
[tt]	=	7971
[n__and(x₁, x₂)]	=	x₂ + 14023
[U71^#(x₁, x₂)]	=	12208
[U22(x₁, x₂)]	=	7969
[U51(x₁, x₂, x₃)]	=	x₃ + 634
[isPalListKind(x₁)]	=	1
[U53(x₁)]	=	x₁ + 3588
[U41(x₁, x₂, x₃)]	=	x₁ + x₃ + 0
[U31^#(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	x₂ + 12208
[__(x₁, x₂)]	=	x₁ + 3580
[U61^#(x₁, x₂)]	=	1
[i^#]	=	0

together with the usable rules

isQid(n__a)	→	tt	(39)
isQid(n__e)	→	tt	(40)
isQid(n__i)	→	tt	(41)
isQid(n__o)	→	tt	(42)
isQid(n__u)	→	tt	(43)

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

activate^#(n__and(X1,X2))	→	and^#(activate(X1),X2)	(68)
activate^#(n____(X1,X2))	→	activate^#(X2)	(137)

could be deleted.

1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

and^#(tt,X)	→	activate^#(X)	(119)
activate^#(n__isPalListKind(X))	→	isPalListKind^#(X)	(75)
isPalListKind^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(121)

1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[n__a]	=	1
[U72^#(x₁)]	=	0
[U32^#(x₁)]	=	0
[U21(x₁, x₂, x₃)]	=	x₁ + x₃ + 0
[isPalListKind^#(x₁)]	=	x₁ + 3
[U11(x₁, x₂)]	=	4386
[n__o]	=	1
[isNeList(x₁)]	=	1
[isPal(x₁)]	=	19459
[U42(x₁, x₂)]	=	x₁ + 2
[e^#]	=	0
[u]	=	1
[activate(x₁)]	=	x₁ + 0
[U71(x₁, x₂)]	=	19459
[and(x₁, x₂)]	=	x₁ + x₂ + 1
[u^#]	=	0
[isNeList^#(x₁)]	=	12207
[U43(x₁)]	=	0
[activate^#(x₁)]	=	x₁ + 0
[n__i]	=	0
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	0
[U43^#(x₁)]	=	0
[U23(x₁)]	=	3
[isNePal(x₁)]	=	1
[U72(x₁)]	=	19459
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	12208
[n__isPal(x₁)]	=	19459
[n__nil]	=	1
[U52^#(x₁, x₂)]	=	23784
[U12(x₁)]	=	x₁ + 5492
[isQid(x₁)]	=	x₁ + 4
[n____(x₁, x₂)]	=	x₁ + x₂ + 5
[n__e]	=	0
[o]	=	1
[n__isPalListKind(x₁)]	=	x₁ + 4
[U42^#(x₁, x₂)]	=	12207
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 1
[isNePal^#(x₁)]	=	12208
[nil]	=	1
[n__u]	=	1
[o^#]	=	0
[U62(x₁)]	=	x₁ + 0
[nil^#]	=	0
[isList^#(x₁)]	=	23783
[U32(x₁)]	=	3
[i]	=	0
[U52(x₁, x₂)]	=	x₁ + 4
[U61(x₁, x₂)]	=	x₂ + 2
[U51^#(x₁, x₂, x₃)]	=	11577
[e]	=	0
[U11^#(x₁, x₂)]	=	0
[U31(x₁, x₂)]	=	2
[U41^#(x₁, x₂, x₃)]	=	23784
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	11576
[U22^#(x₁, x₂)]	=	23784
[tt]	=	4
[n__and(x₁, x₂)]	=	x₁ + x₂ + 1
[U71^#(x₁, x₂)]	=	12208
[U22(x₁, x₂)]	=	2
[U51(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0
[isPalListKind(x₁)]	=	x₁ + 4
[U53(x₁)]	=	x₁ + 3588
[U41(x₁, x₂, x₃)]	=	x₃ + 2
[U31^#(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	x₂ + 1
[__(x₁, x₂)]	=	x₁ + x₂ + 5
[U61^#(x₁, x₂)]	=	1
[i^#]	=	0

together with the usable rules

e	→	n__e	(50)
activate(n__nil)	→	nil	(54)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
and(tt,X)	→	activate(X)	(21)
isPalListKind(n__o)	→	tt	(36)
activate(n__u)	→	u	(63)
U71(tt,V)	→	U72(isNePal(activate(V)))	(19)
isPalListKind(n__a)	→	tt	(32)
activate(n__e)	→	e	(60)
isPalListKind(n__i)	→	tt	(34)
nil	→	n__nil	(44)
isPalListKind(n__e)	→	tt	(33)
activate(X)	→	X	(64)
isQid(n__a)	→	tt	(39)
U72(tt)	→	tt	(20)
a	→	n__a	(49)
o	→	n__o	(52)
isPal(V)	→	U71(isPalListKind(activate(V)),activate(V))	(30)
activate(n__o)	→	o	(62)
activate(n__isPalListKind(X))	→	isPalListKind(X)	(56)
isPal(n__nil)	→	tt	(31)
__(X1,X2)	→	n____(X1,X2)	(45)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(57)
i	→	n__i	(51)
isQid(n__e)	→	tt	(40)
activate(n____(X1,X2))	→	__(activate(X1),activate(X2))	(55)
activate(n__a)	→	a	(59)
isPalListKind(n____(V1,V2))	→	and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(38)
activate(n__i)	→	i	(61)
activate(n__isPal(X))	→	isPal(X)	(58)
isPal(X)	→	n__isPal(X)	(48)
u	→	n__u	(53)
and(X1,X2)	→	n__and(X1,X2)	(47)
isPalListKind(n__u)	→	tt	(37)
isQid(n__i)	→	tt	(41)
isQid(n__o)	→	tt	(42)
isPalListKind(X)	→	n__isPalListKind(X)	(46)
isPalListKind(n__nil)	→	tt	(35)
isQid(n__u)	→	tt	(43)
__(X,nil)	→	X	(2)

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

and^#(tt,X)	→	activate^#(X)	(119)
activate^#(n__isPalListKind(X))	→	isPalListKind^#(X)	(75)
isPalListKind^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(121)

could be deleted.

1.1.2.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(111)
__^#(__(X,Y),Z)	→	__^#(Y,Z)	(92)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[n__a]	=	1
[U72^#(x₁)]	=	0
[U32^#(x₁)]	=	0
[U21(x₁, x₂, x₃)]	=	x₁ + x₃ + 0
[isPalListKind^#(x₁)]	=	x₁ + 3
[U11(x₁, x₂)]	=	4386
[n__o]	=	1
[isNeList(x₁)]	=	1
[isPal(x₁)]	=	4
[U42(x₁, x₂)]	=	x₁ + 2
[e^#]	=	0
[u]	=	1
[activate(x₁)]	=	x₁ + 0
[U71(x₁, x₂)]	=	4
[and(x₁, x₂)]	=	x₁ + x₂ + 22227
[u^#]	=	0
[isNeList^#(x₁)]	=	12207
[U43(x₁)]	=	0
[activate^#(x₁)]	=	0
[n__i]	=	0
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	x₁ + 0
[U43^#(x₁)]	=	0
[U23(x₁)]	=	3
[isNePal(x₁)]	=	1
[U72(x₁)]	=	4
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	12208
[n__isPal(x₁)]	=	4
[n__nil]	=	1
[U52^#(x₁, x₂)]	=	23784
[U12(x₁)]	=	x₁ + 5492
[isQid(x₁)]	=	x₁ + 4
[n____(x₁, x₂)]	=	x₁ + x₂ + 22231
[n__e]	=	0
[o]	=	1
[n__isPalListKind(x₁)]	=	x₁ + 4
[U42^#(x₁, x₂)]	=	12207
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 2
[isNePal^#(x₁)]	=	12208
[nil]	=	1
[n__u]	=	1
[o^#]	=	0
[U62(x₁)]	=	x₁ + 0
[nil^#]	=	0
[isList^#(x₁)]	=	23783
[U32(x₁)]	=	3
[i]	=	0
[U52(x₁, x₂)]	=	x₁ + 4
[U61(x₁, x₂)]	=	x₂ + 2
[U51^#(x₁, x₂, x₃)]	=	11577
[e]	=	0
[U11^#(x₁, x₂)]	=	0
[U31(x₁, x₂)]	=	2
[U41^#(x₁, x₂, x₃)]	=	23784
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	11576
[U22^#(x₁, x₂)]	=	23784
[tt]	=	4
[n__and(x₁, x₂)]	=	x₁ + x₂ + 22227
[U71^#(x₁, x₂)]	=	12208
[U22(x₁, x₂)]	=	2
[U51(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0
[isPalListKind(x₁)]	=	x₁ + 4
[U53(x₁)]	=	x₁ + 3588
[U41(x₁, x₂, x₃)]	=	x₃ + 3
[U31^#(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	x₂ + 1
[__(x₁, x₂)]	=	x₁ + x₂ + 22231
[U61^#(x₁, x₂)]	=	1
[i^#]	=	0

together with the usable rules

e	→	n__e	(50)
activate(n__nil)	→	nil	(54)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
and(tt,X)	→	activate(X)	(21)
isPalListKind(n__o)	→	tt	(36)
activate(n__u)	→	u	(63)
U71(tt,V)	→	U72(isNePal(activate(V)))	(19)
isPalListKind(n__a)	→	tt	(32)
activate(n__e)	→	e	(60)
isPalListKind(n__i)	→	tt	(34)
nil	→	n__nil	(44)
isPalListKind(n__e)	→	tt	(33)
activate(X)	→	X	(64)
isQid(n__a)	→	tt	(39)
U72(tt)	→	tt	(20)
a	→	n__a	(49)
o	→	n__o	(52)
isPal(V)	→	U71(isPalListKind(activate(V)),activate(V))	(30)
activate(n__o)	→	o	(62)
activate(n__isPalListKind(X))	→	isPalListKind(X)	(56)
isPal(n__nil)	→	tt	(31)
__(X1,X2)	→	n____(X1,X2)	(45)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(57)
i	→	n__i	(51)
isQid(n__e)	→	tt	(40)
activate(n____(X1,X2))	→	__(activate(X1),activate(X2))	(55)
activate(n__a)	→	a	(59)
isPalListKind(n____(V1,V2))	→	and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(38)
activate(n__i)	→	i	(61)
activate(n__isPal(X))	→	isPal(X)	(58)
isPal(X)	→	n__isPal(X)	(48)
u	→	n__u	(53)
and(X1,X2)	→	n__and(X1,X2)	(47)
isPalListKind(n__u)	→	tt	(37)
isQid(n__i)	→	tt	(41)
isQid(n__o)	→	tt	(42)
isPalListKind(X)	→	n__isPalListKind(X)	(46)
isPalListKind(n__nil)	→	tt	(35)
isQid(n__u)	→	tt	(43)
__(X,nil)	→	X	(2)

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

__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(111)
__^#(__(X,Y),Z)	→	__^#(Y,Z)	(92)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.