Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PALINDROME_complete_Z)

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,__(P,I)))	→	and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(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)
a	→	n__a	(48)
e	→	n__e	(49)
i	→	n__i	(50)
o	→	n__o	(51)
u	→	n__u	(52)
activate(n__nil)	→	nil	(53)
activate(n____(X1,X2))	→	__(X1,X2)	(54)
activate(n__isPalListKind(X))	→	isPalListKind(X)	(55)
activate(n__and(X1,X2))	→	and(X1,X2)	(56)
activate(n__a)	→	a	(57)
activate(n__e)	→	e	(58)
activate(n__i)	→	i	(59)
activate(n__o)	→	o	(60)
activate(n__u)	→	u	(61)
activate(X)	→	X	(62)

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,__(P,I)))	→	and^#(isQid(activate(I)),n__isPalListKind(activate(I)))	(63)
__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(64)
U51^#(tt,V1,V2)	→	activate^#(V1)	(65)
activate^#(n__a)	→	a^#	(66)
isList^#(n____(V1,V2))	→	activate^#(V1)	(67)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(68)
isNeList^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(69)
isPal^#(V)	→	isPalListKind^#(activate(V))	(70)
U51^#(tt,V1,V2)	→	isNeList^#(activate(V1))	(71)
U51^#(tt,V1,V2)	→	U52^#(isNeList(activate(V1)),activate(V2))	(72)
U42^#(tt,V2)	→	U43^#(isNeList(activate(V2)))	(73)
U52^#(tt,V2)	→	activate^#(V2)	(74)
U61^#(tt,V)	→	isQid^#(activate(V))	(75)
isPal^#(V)	→	activate^#(V)	(76)
U22^#(tt,V2)	→	U23^#(isList(activate(V2)))	(77)
activate^#(n__o)	→	o^#	(78)
isPal^#(V)	→	activate^#(V)	(76)
U71^#(tt,V)	→	U72^#(isNePal(activate(V)))	(79)
U61^#(tt,V)	→	activate^#(V)	(80)
isNePal^#(V)	→	activate^#(V)	(81)
isList^#(V)	→	isPalListKind^#(activate(V))	(82)
U42^#(tt,V2)	→	isNeList^#(activate(V2))	(83)
isNePal^#(n____(I,__(P,I)))	→	isPal^#(activate(P))	(84)
isNePal^#(V)	→	activate^#(V)	(81)
U22^#(tt,V2)	→	activate^#(V2)	(85)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(68)
activate^#(n__and(X1,X2))	→	and^#(X1,X2)	(86)
U52^#(tt,V2)	→	isList^#(activate(V2))	(87)
isNeList^#(V)	→	activate^#(V)	(88)
isPalListKind^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(89)
activate^#(n__nil)	→	nil^#	(90)
activate^#(n__u)	→	u^#	(91)
isNePal^#(V)	→	isPalListKind^#(activate(V))	(92)
U71^#(tt,V)	→	isNePal^#(activate(V))	(93)
U11^#(tt,V)	→	activate^#(V)	(94)
U31^#(tt,V)	→	activate^#(V)	(95)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(96)
U31^#(tt,V)	→	isQid^#(activate(V))	(97)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(68)
isNeList^#(n____(V1,V2))	→	U41^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(98)
isPal^#(V)	→	U71^#(isPalListKind(activate(V)),activate(V))	(99)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(68)
isPalListKind^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(100)
isList^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(101)
isNeList^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(102)
isNePal^#(V)	→	U61^#(isPalListKind(activate(V)),activate(V))	(103)
U52^#(tt,V2)	→	U53^#(isList(activate(V2)))	(104)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(96)
isPalListKind^#(n____(V1,V2))	→	activate^#(V1)	(105)
isList^#(V)	→	activate^#(V)	(106)
isList^#(n____(V1,V2))	→	U21^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(107)
isNeList^#(V)	→	activate^#(V)	(88)
isNeList^#(V)	→	U31^#(isPalListKind(activate(V)),activate(V))	(108)
U41^#(tt,V1,V2)	→	U42^#(isList(activate(V1)),activate(V2))	(109)
activate^#(n__i)	→	i^#	(110)
activate^#(n____(X1,X2))	→	__^#(X1,X2)	(111)
isNePal^#(n____(I,__(P,I)))	→	activate^#(P)	(112)
U42^#(tt,V2)	→	activate^#(V2)	(113)
activate^#(n__e)	→	e^#	(114)
U11^#(tt,V)	→	U12^#(isNeList(activate(V)))	(115)
U11^#(tt,V)	→	isNeList^#(activate(V))	(116)
isNeList^#(n____(V1,V2))	→	U51^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(117)
isList^#(n____(V1,V2))	→	activate^#(V1)	(67)
U61^#(tt,V)	→	U62^#(isQid(activate(V)))	(118)
isNePal^#(n____(I,__(P,I)))	→	activate^#(I)	(119)
U51^#(tt,V1,V2)	→	activate^#(V2)	(120)
isNeList^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(102)
U41^#(tt,V1,V2)	→	activate^#(V2)	(121)
U31^#(tt,V)	→	U32^#(isQid(activate(V)))	(122)
isList^#(n____(V1,V2))	→	activate^#(V2)	(123)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(96)
__^#(__(X,Y),Z)	→	__^#(Y,Z)	(124)
U41^#(tt,V1,V2)	→	activate^#(V1)	(125)
isList^#(V)	→	U11^#(isPalListKind(activate(V)),activate(V))	(126)
U21^#(tt,V1,V2)	→	isList^#(activate(V1))	(127)
isList^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(128)
U21^#(tt,V1,V2)	→	activate^#(V2)	(129)
isList^#(V)	→	activate^#(V)	(106)
U41^#(tt,V1,V2)	→	isList^#(activate(V1))	(130)
and^#(tt,X)	→	activate^#(X)	(131)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(96)
isList^#(n____(V1,V2))	→	activate^#(V2)	(123)
activate^#(n__isPalListKind(X))	→	isPalListKind^#(X)	(132)
U21^#(tt,V1,V2)	→	U22^#(isList(activate(V1)),activate(V2))	(133)
isNePal^#(n____(I,__(P,I)))	→	isQid^#(activate(I))	(134)
U71^#(tt,V)	→	activate^#(V)	(135)
isNePal^#(n____(I,__(P,I)))	→	and^#(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P))))	(136)
U22^#(tt,V2)	→	isList^#(activate(V2))	(137)
isNePal^#(n____(I,__(P,I)))	→	activate^#(P)	(112)
isPalListKind^#(n____(V1,V2))	→	activate^#(V2)	(138)
isNePal^#(n____(I,__(P,I)))	→	activate^#(I)	(119)
isNeList^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(69)
isNeList^#(V)	→	isPalListKind^#(activate(V))	(139)
U21^#(tt,V1,V2)	→	activate^#(V1)	(140)

1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

isPal^#(V)	→	U71^#(isPalListKind(activate(V)),activate(V))	(99)
U71^#(tt,V)	→	isNePal^#(activate(V))	(93)
isNePal^#(n____(I,__(P,I)))	→	isPal^#(activate(P))	(84)

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)]	=	0
[isPalListKind^#(x₁)]	=	0
[U11(x₁, x₂)]	=	0
[n__o]	=	46534
[isNeList(x₁)]	=	0
[isPal(x₁)]	=	0
[U42(x₁, x₂)]	=	0
[e^#]	=	0
[u]	=	282
[activate(x₁)]	=	x₁ + 1
[U71(x₁, x₂)]	=	0
[and(x₁, x₂)]	=	x₁ + x₂ + 0
[u^#]	=	0
[isNeList^#(x₁)]	=	0
[U43(x₁)]	=	0
[activate^#(x₁)]	=	0
[n__i]	=	14799
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	0
[U43^#(x₁)]	=	0
[U23(x₁)]	=	0
[isNePal(x₁)]	=	0
[U72(x₁)]	=	0
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	x₁ + 41078
[n__nil]	=	31098
[U52^#(x₁, x₂)]	=	0
[U12(x₁)]	=	0
[isQid(x₁)]	=	0
[n____(x₁, x₂)]	=	x₁ + x₂ + 20540
[n__e]	=	27428
[o]	=	46534
[n__isPalListKind(x₁)]	=	0
[U42^#(x₁, x₂)]	=	0
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	0
[isNePal^#(x₁)]	=	x₁ + 0
[nil]	=	31099
[n__u]	=	282
[o^#]	=	0
[U62(x₁)]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	0
[U32(x₁)]	=	0
[i]	=	14799
[U52(x₁, x₂)]	=	0
[U61(x₁, x₂)]	=	0
[U51^#(x₁, x₂, x₃)]	=	0
[e]	=	27428
[U11^#(x₁, x₂)]	=	0
[U31(x₁, x₂)]	=	0
[U41^#(x₁, x₂, x₃)]	=	0
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	0
[U22^#(x₁, x₂)]	=	0
[tt]	=	1
[n__and(x₁, x₂)]	=	x₁ + x₂ + 0
[U71^#(x₁, x₂)]	=	x₁ + x₂ + 41075
[U22(x₁, x₂)]	=	0
[U51(x₁, x₂, x₃)]	=	0
[isPalListKind(x₁)]	=	1
[U53(x₁)]	=	0
[U41(x₁, x₂, x₃)]	=	0
[U31^#(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	0
[__(x₁, x₂)]	=	x₁ + x₂ + 20540
[U61^#(x₁, x₂)]	=	0
[i^#]	=	0

together with the usable rules

i	→	n__i	(50)
activate(n____(X1,X2))	→	__(X1,X2)	(54)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
and(tt,X)	→	activate(X)	(21)
isPalListKind(n__o)	→	tt	(36)
isPalListKind(n__a)	→	tt	(32)
activate(n__o)	→	o	(60)
isPalListKind(n__i)	→	tt	(34)
nil	→	n__nil	(44)
isPalListKind(n__e)	→	tt	(33)
e	→	n__e	(49)
u	→	n__u	(52)
activate(X)	→	X	(62)
activate(n__and(X1,X2))	→	and(X1,X2)	(56)
__(X1,X2)	→	n____(X1,X2)	(45)
activate(n__a)	→	a	(57)
o	→	n__o	(51)
activate(n__isPalListKind(X))	→	isPalListKind(X)	(55)
activate(n__i)	→	i	(59)
isPalListKind(n____(V1,V2))	→	and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(38)
activate(n__u)	→	u	(61)
activate(n__e)	→	e	(58)
a	→	n__a	(48)
activate(n__nil)	→	nil	(53)
and(X1,X2)	→	n__and(X1,X2)	(47)
isPalListKind(n__u)	→	tt	(37)
isPalListKind(X)	→	n__isPalListKind(X)	(46)
isPalListKind(n__nil)	→	tt	(35)
__(X,nil)	→	X	(2)

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

isPal^#(V)	→	U71^#(isPalListKind(activate(V)),activate(V))	(99)
U71^#(tt,V)	→	isNePal^#(activate(V))	(93)
isNePal^#(n____(I,__(P,I)))	→	isPal^#(activate(P))	(84)

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

U41^#(tt,V1,V2)	→	U42^#(isList(activate(V1)),activate(V2))	(109)
isList^#(n____(V1,V2))	→	U21^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(107)
U22^#(tt,V2)	→	isList^#(activate(V2))	(137)
isNeList^#(n____(V1,V2))	→	U41^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(98)
U21^#(tt,V1,V2)	→	U22^#(isList(activate(V1)),activate(V2))	(133)
U41^#(tt,V1,V2)	→	isList^#(activate(V1))	(130)
U52^#(tt,V2)	→	isList^#(activate(V2))	(87)
U21^#(tt,V1,V2)	→	isList^#(activate(V1))	(127)
isList^#(V)	→	U11^#(isPalListKind(activate(V)),activate(V))	(126)
U42^#(tt,V2)	→	isNeList^#(activate(V2))	(83)
isNeList^#(n____(V1,V2))	→	U51^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(117)
U11^#(tt,V)	→	isNeList^#(activate(V))	(116)
U51^#(tt,V1,V2)	→	U52^#(isNeList(activate(V1)),activate(V2))	(72)
U51^#(tt,V1,V2)	→	isNeList^#(activate(V1))	(71)

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)]	=	x₁ + x₂ + x₃ + 31902
[isPalListKind^#(x₁)]	=	0
[U11(x₁, x₂)]	=	x₁ + x₂ + 15946
[n__o]	=	1
[isNeList(x₁)]	=	x₁ + 15946
[isPal(x₁)]	=	0
[U42(x₁, x₂)]	=	x₁ + x₂ + 15946
[e^#]	=	0
[u]	=	1
[activate(x₁)]	=	x₁ + 2
[U71(x₁, x₂)]	=	0
[and(x₁, x₂)]	=	x₁ + x₂ + 0
[u^#]	=	0
[isNeList^#(x₁)]	=	x₁ + 0
[U43(x₁)]	=	x₁ + 0
[activate^#(x₁)]	=	0
[n__i]	=	1
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	0
[U43^#(x₁)]	=	0
[U23(x₁)]	=	x₁ + 0
[isNePal(x₁)]	=	0
[U72(x₁)]	=	0
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	x₁ + 41078
[n__nil]	=	1
[U52^#(x₁, x₂)]	=	x₁ + x₂ + 7
[U12(x₁)]	=	x₁ + 0
[isQid(x₁)]	=	x₁ + 0
[n____(x₁, x₂)]	=	x₁ + x₂ + 15963
[n__e]	=	1
[o]	=	1
[n__isPalListKind(x₁)]	=	0
[U42^#(x₁, x₂)]	=	x₁ + x₂ + 1
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 15950
[isNePal^#(x₁)]	=	x₁ + 0
[nil]	=	3
[n__u]	=	1
[o^#]	=	0
[U62(x₁)]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	x₁ + 6
[U32(x₁)]	=	2
[i]	=	1
[U52(x₁, x₂)]	=	2
[U61(x₁, x₂)]	=	0
[U51^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 15956
[e]	=	1
[U11^#(x₁, x₂)]	=	x₁ + x₂ + 1
[U31(x₁, x₂)]	=	x₁ + x₂ + 0
[U41^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 15954
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 15960
[U22^#(x₁, x₂)]	=	x₁ + x₂ + 7
[tt]	=	2
[n__and(x₁, x₂)]	=	x₁ + x₂ + 0
[U71^#(x₁, x₂)]	=	x₁ + x₂ + 41075
[U22(x₁, x₂)]	=	x₁ + x₂ + 15950
[U51(x₁, x₂, x₃)]	=	x₁ + x₂ + 10
[isPalListKind(x₁)]	=	2
[U53(x₁)]	=	2
[U41(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 31898
[U31^#(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	0
[__(x₁, x₂)]	=	x₁ + x₂ + 15963
[U61^#(x₁, x₂)]	=	0
[i^#]	=	0

together with the usable rules

i	→	n__i	(50)
U11(tt,V)	→	U12(isNeList(activate(V)))	(4)
U52(tt,V2)	→	U53(isList(activate(V2)))	(15)
U23(tt)	→	tt	(8)
activate(n____(X1,X2))	→	__(X1,X2)	(54)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
U53(tt)	→	tt	(16)
and(tt,X)	→	activate(X)	(21)
isPalListKind(n__o)	→	tt	(36)
isNeList(n____(V1,V2))	→	U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(26)
isPalListKind(n__a)	→	tt	(32)
activate(n__o)	→	o	(60)
isNeList(n____(V1,V2))	→	U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(27)
isPalListKind(n__i)	→	tt	(34)
isList(V)	→	U11(isPalListKind(activate(V)),activate(V))	(22)
nil	→	n__nil	(44)
U12(tt)	→	tt	(5)
isPalListKind(n__e)	→	tt	(33)
U32(tt)	→	tt	(10)
U22(tt,V2)	→	U23(isList(activate(V2)))	(7)
isNeList(V)	→	U31(isPalListKind(activate(V)),activate(V))	(25)
e	→	n__e	(49)
u	→	n__u	(52)
activate(X)	→	X	(62)
U51(tt,V1,V2)	→	U52(isNeList(activate(V1)),activate(V2))	(14)
activate(n__and(X1,X2))	→	and(X1,X2)	(56)
U42(tt,V2)	→	U43(isNeList(activate(V2)))	(12)
__(X1,X2)	→	n____(X1,X2)	(45)
isList(n__nil)	→	tt	(23)
isList(n____(V1,V2))	→	U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(24)
activate(n__a)	→	a	(57)
U41(tt,V1,V2)	→	U42(isList(activate(V1)),activate(V2))	(11)
U31(tt,V)	→	U32(isQid(activate(V)))	(9)
U43(tt)	→	tt	(13)
o	→	n__o	(51)
activate(n__isPalListKind(X))	→	isPalListKind(X)	(55)
activate(n__i)	→	i	(59)
U21(tt,V1,V2)	→	U22(isList(activate(V1)),activate(V2))	(6)
isPalListKind(n____(V1,V2))	→	and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(38)
activate(n__u)	→	u	(61)
activate(n__e)	→	e	(58)
a	→	n__a	(48)
activate(n__nil)	→	nil	(53)
and(X1,X2)	→	n__and(X1,X2)	(47)
isPalListKind(n__u)	→	tt	(37)
isPalListKind(X)	→	n__isPalListKind(X)	(46)
isPalListKind(n__nil)	→	tt	(35)
__(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))	(109)
isList^#(n____(V1,V2))	→	U21^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(107)
U22^#(tt,V2)	→	isList^#(activate(V2))	(137)
isNeList^#(n____(V1,V2))	→	U41^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(98)
U21^#(tt,V1,V2)	→	U22^#(isList(activate(V1)),activate(V2))	(133)
U41^#(tt,V1,V2)	→	isList^#(activate(V1))	(130)
U52^#(tt,V2)	→	isList^#(activate(V2))	(87)
U21^#(tt,V1,V2)	→	isList^#(activate(V1))	(127)
isList^#(V)	→	U11^#(isPalListKind(activate(V)),activate(V))	(126)
U42^#(tt,V2)	→	isNeList^#(activate(V2))	(83)
isNeList^#(n____(V1,V2))	→	U51^#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(117)
U11^#(tt,V)	→	isNeList^#(activate(V))	(116)
U51^#(tt,V1,V2)	→	U52^#(isNeList(activate(V1)),activate(V2))	(72)
U51^#(tt,V1,V2)	→	isNeList^#(activate(V1))	(71)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

isPalListKind^#(n____(V1,V2))	→	activate^#(V1)	(105)
isPalListKind^#(n____(V1,V2))	→	activate^#(V2)	(138)
isPalListKind^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(100)
activate^#(n__isPalListKind(X))	→	isPalListKind^#(X)	(132)
and^#(tt,X)	→	activate^#(X)	(131)
isPalListKind^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(89)
activate^#(n__and(X1,X2))	→	and^#(X1,X2)	(86)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	2
[n__a]	=	2
[U72^#(x₁)]	=	0
[U32^#(x₁)]	=	0
[U21(x₁, x₂, x₃)]	=	15951
[isPalListKind^#(x₁)]	=	x₁ + 0
[U11(x₁, x₂)]	=	x₁ + x₂ + 0
[n__o]	=	26816
[isNeList(x₁)]	=	x₁ + 1
[isPal(x₁)]	=	0
[U42(x₁, x₂)]	=	x₁ + x₂ + 4
[e^#]	=	0
[u]	=	3825
[activate(x₁)]	=	x₁ + 0
[U71(x₁, x₂)]	=	0
[and(x₁, x₂)]	=	x₂ + 0
[u^#]	=	0
[isNeList^#(x₁)]	=	x₁ + 0
[U43(x₁)]	=	x₁ + 0
[activate^#(x₁)]	=	x₁ + 1
[n__i]	=	30738
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	0
[U43^#(x₁)]	=	0
[U23(x₁)]	=	2
[isNePal(x₁)]	=	0
[U72(x₁)]	=	0
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	x₁ + 41078
[n__nil]	=	978
[U52^#(x₁, x₂)]	=	x₂ + 7
[U12(x₁)]	=	x₁ + 0
[isQid(x₁)]	=	1
[n____(x₁, x₂)]	=	x₁ + x₂ + 2
[n__e]	=	32529
[o]	=	26816
[n__isPalListKind(x₁)]	=	x₁ + 0
[U42^#(x₁, x₂)]	=	x₂ + 1
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 1
[isNePal^#(x₁)]	=	x₁ + 0
[nil]	=	978
[n__u]	=	3825
[o^#]	=	0
[U62(x₁)]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	x₁ + 6
[U32(x₁)]	=	2
[i]	=	30738
[U52(x₁, x₂)]	=	2
[U61(x₁, x₂)]	=	0
[U51^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 15956
[e]	=	32529
[U11^#(x₁, x₂)]	=	x₁ + x₂ + 1
[U31(x₁, x₂)]	=	x₁ + x₂ + 0
[U41^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 15954
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 15960
[U22^#(x₁, x₂)]	=	x₂ + 7
[tt]	=	2
[n__and(x₁, x₂)]	=	x₂ + 0
[U71^#(x₁, x₂)]	=	x₁ + x₂ + 41075
[U22(x₁, x₂)]	=	x₁ + 15950
[U51(x₁, x₂, x₃)]	=	x₂ + 3
[isPalListKind(x₁)]	=	x₁ + 0
[U53(x₁)]	=	2
[U41(x₁, x₂, x₃)]	=	x₁ + 3
[U31^#(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	x₂ + 1
[__(x₁, x₂)]	=	x₁ + x₂ + 2
[U61^#(x₁, x₂)]	=	0
[i^#]	=	0

together with the usable rules

i	→	n__i	(50)
U52(tt,V2)	→	U53(isList(activate(V2)))	(15)
U23(tt)	→	tt	(8)
activate(n____(X1,X2))	→	__(X1,X2)	(54)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
U53(tt)	→	tt	(16)
and(tt,X)	→	activate(X)	(21)
isPalListKind(n__o)	→	tt	(36)
isPalListKind(n__a)	→	tt	(32)
activate(n__o)	→	o	(60)
isNeList(n____(V1,V2))	→	U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))	(27)
isPalListKind(n__i)	→	tt	(34)
nil	→	n__nil	(44)
U12(tt)	→	tt	(5)
isPalListKind(n__e)	→	tt	(33)
U32(tt)	→	tt	(10)
U22(tt,V2)	→	U23(isList(activate(V2)))	(7)
e	→	n__e	(49)
u	→	n__u	(52)
activate(X)	→	X	(62)
U51(tt,V1,V2)	→	U52(isNeList(activate(V1)),activate(V2))	(14)
activate(n__and(X1,X2))	→	and(X1,X2)	(56)
__(X1,X2)	→	n____(X1,X2)	(45)
activate(n__a)	→	a	(57)
U31(tt,V)	→	U32(isQid(activate(V)))	(9)
U43(tt)	→	tt	(13)
o	→	n__o	(51)
activate(n__isPalListKind(X))	→	isPalListKind(X)	(55)
activate(n__i)	→	i	(59)
isPalListKind(n____(V1,V2))	→	and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(38)
activate(n__u)	→	u	(61)
activate(n__e)	→	e	(58)
a	→	n__a	(48)
activate(n__nil)	→	nil	(53)
and(X1,X2)	→	n__and(X1,X2)	(47)
isPalListKind(n__u)	→	tt	(37)
isPalListKind(X)	→	n__isPalListKind(X)	(46)
isPalListKind(n__nil)	→	tt	(35)
__(X,nil)	→	X	(2)

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

isPalListKind^#(n____(V1,V2))	→	activate^#(V1)	(105)
isPalListKind^#(n____(V1,V2))	→	activate^#(V2)	(138)
isPalListKind^#(n____(V1,V2))	→	and^#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))	(100)
activate^#(n__isPalListKind(X))	→	isPalListKind^#(X)	(132)
isPalListKind^#(n____(V1,V2))	→	isPalListKind^#(activate(V1))	(89)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

and^#(tt,X)	→	activate^#(X)	(131)
activate^#(n__and(X1,X2))	→	and^#(X1,X2)	(86)

1.1.3.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	0
[n__a]	=	1
[U72^#(x₁)]	=	0
[U32^#(x₁)]	=	0
[U21(x₁, x₂, x₃)]	=	x₁ + 15949
[isPalListKind^#(x₁)]	=	0
[U11(x₁, x₂)]	=	x₁ + x₂ + 0
[n__o]	=	1
[isNeList(x₁)]	=	x₁ + 1
[isPal(x₁)]	=	0
[U42(x₁, x₂)]	=	x₁ + x₂ + 1
[e^#]	=	0
[u]	=	0
[activate(x₁)]	=	x₁ + 0
[U71(x₁, x₂)]	=	0
[and(x₁, x₂)]	=	x₁ + 2
[u^#]	=	0
[isNeList^#(x₁)]	=	0
[U43(x₁)]	=	x₁ + 0
[activate^#(x₁)]	=	x₁ + 0
[n__i]	=	1
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	0
[U43^#(x₁)]	=	0
[U23(x₁)]	=	2
[isNePal(x₁)]	=	0
[U72(x₁)]	=	0
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	41078
[n__nil]	=	979
[U52^#(x₁, x₂)]	=	7
[U12(x₁)]	=	x₁ + 0
[isQid(x₁)]	=	1
[n____(x₁, x₂)]	=	x₁ + x₂ + 1
[n__e]	=	1
[o]	=	0
[n__isPalListKind(x₁)]	=	11453
[U42^#(x₁, x₂)]	=	1
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 1
[isNePal^#(x₁)]	=	0
[nil]	=	978
[n__u]	=	1
[o^#]	=	0
[U62(x₁)]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	6
[U32(x₁)]	=	2
[i]	=	2
[U52(x₁, x₂)]	=	2
[U61(x₁, x₂)]	=	0
[U51^#(x₁, x₂, x₃)]	=	15956
[e]	=	0
[U11^#(x₁, x₂)]	=	1
[U31(x₁, x₂)]	=	x₁ + 0
[U41^#(x₁, x₂, x₃)]	=	15954
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	15960
[U22^#(x₁, x₂)]	=	7
[tt]	=	2
[n__and(x₁, x₂)]	=	x₂ + 2
[U71^#(x₁, x₂)]	=	41075
[U22(x₁, x₂)]	=	x₁ + 15950
[U51(x₁, x₂, x₃)]	=	x₂ + 2
[isPalListKind(x₁)]	=	x₁ + 0
[U53(x₁)]	=	2
[U41(x₁, x₂, x₃)]	=	x₁ + 0
[U31^#(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	x₂ + 1
[__(x₁, x₂)]	=	2
[U61^#(x₁, x₂)]	=	0
[i^#]	=	0

together with the usable rules

U52(tt,V2)	→	U53(isList(activate(V2)))	(15)
U23(tt)	→	tt	(8)
U53(tt)	→	tt	(16)
U12(tt)	→	tt	(5)
U32(tt)	→	tt	(10)
U22(tt,V2)	→	U23(isList(activate(V2)))	(7)
U51(tt,V1,V2)	→	U52(isNeList(activate(V1)),activate(V2))	(14)
U31(tt,V)	→	U32(isQid(activate(V)))	(9)
U43(tt)	→	tt	(13)

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

and^#(tt,X)	→	activate^#(X)	(131)
activate^#(n__and(X1,X2))	→	and^#(X1,X2)	(86)

could be deleted.

1.1.3.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

__^#(__(X,Y),Z)	→	__^#(Y,Z)	(124)
__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(64)

1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	0
[n__a]	=	1
[U72^#(x₁)]	=	0
[U32^#(x₁)]	=	0
[U21(x₁, x₂, x₃)]	=	x₁ + 15949
[isPalListKind^#(x₁)]	=	0
[U11(x₁, x₂)]	=	x₁ + x₂ + 0
[n__o]	=	1
[isNeList(x₁)]	=	x₁ + 2
[isPal(x₁)]	=	0
[U42(x₁, x₂)]	=	x₁ + x₂ + 2
[e^#]	=	0
[u]	=	0
[activate(x₁)]	=	x₁ + 0
[U71(x₁, x₂)]	=	0
[and(x₁, x₂)]	=	x₁ + 3
[u^#]	=	0
[isNeList^#(x₁)]	=	0
[U43(x₁)]	=	x₁ + 2
[activate^#(x₁)]	=	0
[n__i]	=	1
[U23^#(x₁)]	=	0
[U53^#(x₁)]	=	0
[__^#(x₁, x₂)]	=	x₁ + 0
[U43^#(x₁)]	=	0
[U23(x₁)]	=	2
[isNePal(x₁)]	=	0
[U72(x₁)]	=	0
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	41078
[n__nil]	=	2
[U52^#(x₁, x₂)]	=	7
[U12(x₁)]	=	x₁ + 0
[isQid(x₁)]	=	1
[n____(x₁, x₂)]	=	x₁ + x₂ + 2
[n__e]	=	1
[o]	=	0
[n__isPalListKind(x₁)]	=	11453
[U42^#(x₁, x₂)]	=	1
[U12^#(x₁)]	=	0
[U62^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 1
[isNePal^#(x₁)]	=	0
[nil]	=	0
[n__u]	=	1
[o^#]	=	0
[U62(x₁)]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	6
[U32(x₁)]	=	2
[i]	=	2
[U52(x₁, x₂)]	=	2
[U61(x₁, x₂)]	=	0
[U51^#(x₁, x₂, x₃)]	=	15956
[e]	=	0
[U11^#(x₁, x₂)]	=	1
[U31(x₁, x₂)]	=	x₁ + 0
[U41^#(x₁, x₂, x₃)]	=	15954
[a^#]	=	0
[U21^#(x₁, x₂, x₃)]	=	15960
[U22^#(x₁, x₂)]	=	7
[tt]	=	2
[n__and(x₁, x₂)]	=	x₂ + 3
[U71^#(x₁, x₂)]	=	41075
[U22(x₁, x₂)]	=	x₁ + 15950
[U51(x₁, x₂, x₃)]	=	x₂ + 4
[isPalListKind(x₁)]	=	x₁ + 0
[U53(x₁)]	=	2
[U41(x₁, x₂, x₃)]	=	x₁ + 1
[U31^#(x₁, x₂)]	=	0
[and^#(x₁, x₂)]	=	x₂ + 1
[__(x₁, x₂)]	=	x₁ + x₂ + 1
[U61^#(x₁, x₂)]	=	0
[i^#]	=	0

together with the usable rules

U52(tt,V2)	→	U53(isList(activate(V2)))	(15)
U23(tt)	→	tt	(8)
U53(tt)	→	tt	(16)
U12(tt)	→	tt	(5)
U32(tt)	→	tt	(10)
U22(tt,V2)	→	U23(isList(activate(V2)))	(7)
U51(tt,V1,V2)	→	U52(isNeList(activate(V1)),activate(V2))	(14)
U31(tt,V)	→	U32(isQid(activate(V)))	(9)
U43(tt)	→	tt	(13)

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

__^#(__(X,Y),Z)	→	__^#(Y,Z)	(124)
__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(64)

could be deleted.

1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 0 components.