Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PALINDROME_nokinds_noand_GM)

The rewrite relation of the following TRS is considered.

a____(__(X,Y),Z)	→	a____(mark(X),a____(mark(Y),mark(Z)))	(1)
a____(X,nil)	→	mark(X)	(2)
a____(nil,X)	→	mark(X)	(3)
a__U11(tt)	→	tt	(4)
a__U21(tt,V2)	→	a__U22(a__isList(V2))	(5)
a__U22(tt)	→	tt	(6)
a__U31(tt)	→	tt	(7)
a__U41(tt,V2)	→	a__U42(a__isNeList(V2))	(8)
a__U42(tt)	→	tt	(9)
a__U51(tt,V2)	→	a__U52(a__isList(V2))	(10)
a__U52(tt)	→	tt	(11)
a__U61(tt)	→	tt	(12)
a__U71(tt,P)	→	a__U72(a__isPal(P))	(13)
a__U72(tt)	→	tt	(14)
a__U81(tt)	→	tt	(15)
a__isList(V)	→	a__U11(a__isNeList(V))	(16)
a__isList(nil)	→	tt	(17)
a__isList(__(V1,V2))	→	a__U21(a__isList(V1),V2)	(18)
a__isNeList(V)	→	a__U31(a__isQid(V))	(19)
a__isNeList(__(V1,V2))	→	a__U41(a__isList(V1),V2)	(20)
a__isNeList(__(V1,V2))	→	a__U51(a__isNeList(V1),V2)	(21)
a__isNePal(V)	→	a__U61(a__isQid(V))	(22)
a__isNePal(__(I,__(P,I)))	→	a__U71(a__isQid(I),P)	(23)
a__isPal(V)	→	a__U81(a__isNePal(V))	(24)
a__isPal(nil)	→	tt	(25)
a__isQid(a)	→	tt	(26)
a__isQid(e)	→	tt	(27)
a__isQid(i)	→	tt	(28)
a__isQid(o)	→	tt	(29)
a__isQid(u)	→	tt	(30)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(31)
mark(U11(X))	→	a__U11(mark(X))	(32)
mark(U21(X1,X2))	→	a__U21(mark(X1),X2)	(33)
mark(U22(X))	→	a__U22(mark(X))	(34)
mark(isList(X))	→	a__isList(X)	(35)
mark(U31(X))	→	a__U31(mark(X))	(36)
mark(U41(X1,X2))	→	a__U41(mark(X1),X2)	(37)
mark(U42(X))	→	a__U42(mark(X))	(38)
mark(isNeList(X))	→	a__isNeList(X)	(39)
mark(U51(X1,X2))	→	a__U51(mark(X1),X2)	(40)
mark(U52(X))	→	a__U52(mark(X))	(41)
mark(U61(X))	→	a__U61(mark(X))	(42)
mark(U71(X1,X2))	→	a__U71(mark(X1),X2)	(43)
mark(U72(X))	→	a__U72(mark(X))	(44)
mark(isPal(X))	→	a__isPal(X)	(45)
mark(U81(X))	→	a__U81(mark(X))	(46)
mark(isQid(X))	→	a__isQid(X)	(47)
mark(isNePal(X))	→	a__isNePal(X)	(48)
mark(nil)	→	nil	(49)
mark(tt)	→	tt	(50)
mark(a)	→	a	(51)
mark(e)	→	e	(52)
mark(i)	→	i	(53)
mark(o)	→	o	(54)
mark(u)	→	u	(55)
a____(X1,X2)	→	__(X1,X2)	(56)
a__U11(X)	→	U11(X)	(57)
a__U21(X1,X2)	→	U21(X1,X2)	(58)
a__U22(X)	→	U22(X)	(59)
a__isList(X)	→	isList(X)	(60)
a__U31(X)	→	U31(X)	(61)
a__U41(X1,X2)	→	U41(X1,X2)	(62)
a__U42(X)	→	U42(X)	(63)
a__isNeList(X)	→	isNeList(X)	(64)
a__U51(X1,X2)	→	U51(X1,X2)	(65)
a__U52(X)	→	U52(X)	(66)
a__U61(X)	→	U61(X)	(67)
a__U71(X1,X2)	→	U71(X1,X2)	(68)
a__U72(X)	→	U72(X)	(69)
a__isPal(X)	→	isPal(X)	(70)
a__U81(X)	→	U81(X)	(71)
a__isQid(X)	→	isQid(X)	(72)
a__isNePal(X)	→	isNePal(X)	(73)

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.

mark^#(U71(X1,X2))	→	a__U71^#(mark(X1),X2)	(74)
mark^#(U22(X))	→	mark^#(X)	(75)
a__isNeList^#(__(V1,V2))	→	a__U41^#(a__isList(V1),V2)	(76)
mark^#(U21(X1,X2))	→	mark^#(X1)	(77)
a__isNeList^#(V)	→	a__U31^#(a__isQid(V))	(78)
mark^#(U22(X))	→	a__U22^#(mark(X))	(79)
a__isNePal^#(V)	→	a__isQid^#(V)	(80)
a__U21^#(tt,V2)	→	a__isList^#(V2)	(81)
a__isNeList^#(V)	→	a__isQid^#(V)	(82)
a__isList^#(V)	→	a__isNeList^#(V)	(83)
mark^#(U31(X))	→	a__U31^#(mark(X))	(84)
a____^#(__(X,Y),Z)	→	mark^#(Y)	(85)
a__U41^#(tt,V2)	→	a__isNeList^#(V2)	(86)
mark^#(U72(X))	→	a__U72^#(mark(X))	(87)
mark^#(U81(X))	→	mark^#(X)	(88)
a____^#(__(X,Y),Z)	→	mark^#(Z)	(89)
a____^#(__(X,Y),Z)	→	a____^#(mark(X),a____(mark(Y),mark(Z)))	(90)
a__isNePal^#(V)	→	a__U61^#(a__isQid(V))	(91)
a__isNeList^#(__(V1,V2))	→	a__isNeList^#(V1)	(92)
mark^#(isQid(X))	→	a__isQid^#(X)	(93)
mark^#(U52(X))	→	a__U52^#(mark(X))	(94)
mark^#(U41(X1,X2))	→	a__U41^#(mark(X1),X2)	(95)
a____^#(__(X,Y),Z)	→	a____^#(mark(Y),mark(Z))	(96)
a__isNePal^#(__(I,__(P,I)))	→	a__isQid^#(I)	(97)
a__isNePal^#(__(I,__(P,I)))	→	a__U71^#(a__isQid(I),P)	(98)
mark^#(U11(X))	→	mark^#(X)	(99)
mark^#(U41(X1,X2))	→	mark^#(X1)	(100)
mark^#(isNeList(X))	→	a__isNeList^#(X)	(101)
a____^#(__(X,Y),Z)	→	mark^#(X)	(102)
mark^#(isNePal(X))	→	a__isNePal^#(X)	(103)
a__isList^#(__(V1,V2))	→	a__U21^#(a__isList(V1),V2)	(104)
a__isNeList^#(__(V1,V2))	→	a__isList^#(V1)	(105)
a__isList^#(V)	→	a__U11^#(a__isNeList(V))	(106)
mark^#(U31(X))	→	mark^#(X)	(107)
mark^#(isPal(X))	→	a__isPal^#(X)	(108)
a__U71^#(tt,P)	→	a__isPal^#(P)	(109)
mark^#(U61(X))	→	a__U61^#(mark(X))	(110)
mark^#(U72(X))	→	mark^#(X)	(111)
mark^#(U52(X))	→	mark^#(X)	(112)
a__U51^#(tt,V2)	→	a__isList^#(V2)	(113)
mark^#(U81(X))	→	a__U81^#(mark(X))	(114)
mark^#(U11(X))	→	a__U11^#(mark(X))	(115)
mark^#(__(X1,X2))	→	mark^#(X2)	(116)
a__isPal^#(V)	→	a__isNePal^#(V)	(117)
mark^#(U21(X1,X2))	→	a__U21^#(mark(X1),X2)	(118)
mark^#(__(X1,X2))	→	mark^#(X1)	(119)
a__U41^#(tt,V2)	→	a__U42^#(a__isNeList(V2))	(120)
mark^#(U51(X1,X2))	→	a__U51^#(mark(X1),X2)	(121)
a__U51^#(tt,V2)	→	a__U52^#(a__isList(V2))	(122)
a__U71^#(tt,P)	→	a__U72^#(a__isPal(P))	(123)
a__isList^#(__(V1,V2))	→	a__isList^#(V1)	(124)
mark^#(__(X1,X2))	→	a____^#(mark(X1),mark(X2))	(125)
a__U21^#(tt,V2)	→	a__U22^#(a__isList(V2))	(126)
a__isPal^#(V)	→	a__U81^#(a__isNePal(V))	(127)
mark^#(U42(X))	→	mark^#(X)	(128)
mark^#(U71(X1,X2))	→	mark^#(X1)	(129)
a____^#(X,nil)	→	mark^#(X)	(130)
a____^#(nil,X)	→	mark^#(X)	(131)
mark^#(U61(X))	→	mark^#(X)	(132)
mark^#(U42(X))	→	a__U42^#(mark(X))	(133)
mark^#(isList(X))	→	a__isList^#(X)	(134)
a__isNeList^#(__(V1,V2))	→	a__U51^#(a__isNeList(V1),V2)	(135)
mark^#(U51(X1,X2))	→	mark^#(X1)	(136)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

mark^#(U51(X1,X2))	→	mark^#(X1)	(136)
mark^#(U31(X))	→	mark^#(X)	(107)
a____^#(__(X,Y),Z)	→	mark^#(X)	(102)
mark^#(U41(X1,X2))	→	mark^#(X1)	(100)
mark^#(U61(X))	→	mark^#(X)	(132)
mark^#(U11(X))	→	mark^#(X)	(99)
a____^#(nil,X)	→	mark^#(X)	(131)
a____^#(X,nil)	→	mark^#(X)	(130)
mark^#(U71(X1,X2))	→	mark^#(X1)	(129)
mark^#(U42(X))	→	mark^#(X)	(128)
a____^#(__(X,Y),Z)	→	a____^#(mark(Y),mark(Z))	(96)
mark^#(__(X1,X2))	→	a____^#(mark(X1),mark(X2))	(125)
a____^#(__(X,Y),Z)	→	a____^#(mark(X),a____(mark(Y),mark(Z)))	(90)
a____^#(__(X,Y),Z)	→	mark^#(Z)	(89)
mark^#(U81(X))	→	mark^#(X)	(88)
mark^#(__(X1,X2))	→	mark^#(X1)	(119)
a____^#(__(X,Y),Z)	→	mark^#(Y)	(85)
mark^#(__(X1,X2))	→	mark^#(X2)	(116)
mark^#(U52(X))	→	mark^#(X)	(112)
mark^#(U72(X))	→	mark^#(X)	(111)
mark^#(U21(X1,X2))	→	mark^#(X1)	(77)
mark^#(U22(X))	→	mark^#(X)	(75)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[a__isNePal^#(x₁)]	=	0
[U21(x₁, x₂)]	=	x₁ + x₂ + 62952
[a__U72^#(x₁)]	=	0
[a__U71^#(x₁, x₂)]	=	0
[U11(x₁)]	=	x₁ + 25907
[a__U31^#(x₁)]	=	0
[a__isNeList^#(x₁)]	=	0
[isNeList(x₁)]	=	x₁ + 90561
[isPal(x₁)]	=	x₁ + 133103
[U42(x₁)]	=	x₁ + 1
[u]	=	13941
[U71(x₁, x₂)]	=	x₁ + x₂ + 89312
[a__isPal^#(x₁)]	=	0
[a__U51^#(x₁, x₂)]	=	0
[a__U22(x₁)]	=	x₁ + 1
[isNePal(x₁)]	=	x₁ + 56347
[U72(x₁)]	=	x₁ + 9726
[a__U11^#(x₁)]	=	0
[a__U31(x₁)]	=	x₁ + 37045
[a__U51(x₁, x₂)]	=	x₁ + x₂ + 62952
[a____^#(x₁, x₂)]	=	x₁ + x₂ + 62951
[a__U81(x₁)]	=	x₁ + 16908
[isQid(x₁)]	=	x₁ + 53516
[a____(x₁, x₂)]	=	x₁ + x₂ + 62952
[a__U42^#(x₁)]	=	0
[a__U41(x₁, x₂)]	=	x₁ + x₂ + 37045
[a__isList(x₁)]	=	x₁ + 116468
[a__isNeList(x₁)]	=	x₁ + 90561
[o]	=	1
[a__U21^#(x₁, x₂)]	=	0
[a__U81^#(x₁)]	=	0
[a__U61^#(x₁)]	=	0
[mark^#(x₁)]	=	x₁ + 0
[isList(x₁)]	=	x₁ + 116468
[a__U21(x₁, x₂)]	=	x₁ + x₂ + 62952
[nil]	=	1
[a__isQid^#(x₁)]	=	0
[a__U52^#(x₁)]	=	0
[mark(x₁)]	=	x₁ + 0
[a__U72(x₁)]	=	x₁ + 9726
[a__isNePal(x₁)]	=	x₁ + 56347
[a__U11(x₁)]	=	x₁ + 25907
[a__U42(x₁)]	=	x₁ + 1
[a__U52(x₁)]	=	x₁ + 1
[i]	=	26058
[U52(x₁)]	=	x₁ + 1
[U61(x₁)]	=	x₁ + 2831
[e]	=	843
[a__U22^#(x₁)]	=	0
[U31(x₁)]	=	x₁ + 37045
[a__U71(x₁, x₂)]	=	x₁ + x₂ + 89312
[a__isPal(x₁)]	=	x₁ + 133103
[a__U61(x₁)]	=	x₁ + 2831
[U81(x₁)]	=	x₁ + 16908
[a__U41^#(x₁, x₂)]	=	0
[tt]	=	53517
[a__isQid(x₁)]	=	x₁ + 53516
[a__isList^#(x₁)]	=	0
[U22(x₁)]	=	x₁ + 1
[U51(x₁, x₂)]	=	x₁ + x₂ + 62952
[U41(x₁, x₂)]	=	x₁ + x₂ + 37045
[__(x₁, x₂)]	=	x₁ + x₂ + 62952

together with the usable rules

a__isList(__(V1,V2))	→	a__U21(a__isList(V1),V2)	(18)
mark(tt)	→	tt	(50)
a__U11(tt)	→	tt	(4)
a__U81(tt)	→	tt	(15)
a__U41(tt,V2)	→	a__U42(a__isNeList(V2))	(8)
mark(o)	→	o	(54)
a____(__(X,Y),Z)	→	a____(mark(X),a____(mark(Y),mark(Z)))	(1)
a____(nil,X)	→	mark(X)	(3)
a__isList(V)	→	a__U11(a__isNeList(V))	(16)
a__isNeList(__(V1,V2))	→	a__U51(a__isNeList(V1),V2)	(21)
mark(U31(X))	→	a__U31(mark(X))	(36)
a__U71(X1,X2)	→	U71(X1,X2)	(68)
a__isQid(a)	→	tt	(26)
a__U42(X)	→	U42(X)	(63)
a__isNeList(V)	→	a__U31(a__isQid(V))	(19)
mark(U11(X))	→	a__U11(mark(X))	(32)
a__isList(nil)	→	tt	(17)
a__isList(X)	→	isList(X)	(60)
a__isQid(e)	→	tt	(27)
mark(U22(X))	→	a__U22(mark(X))	(34)
a__isNePal(V)	→	a__U61(a__isQid(V))	(22)
a__isQid(i)	→	tt	(28)
a__U51(X1,X2)	→	U51(X1,X2)	(65)
mark(U72(X))	→	a__U72(mark(X))	(44)
a__U21(tt,V2)	→	a__U22(a__isList(V2))	(5)
a__isQid(X)	→	isQid(X)	(72)
mark(U21(X1,X2))	→	a__U21(mark(X1),X2)	(33)
a__isNeList(X)	→	isNeList(X)	(64)
a__U51(tt,V2)	→	a__U52(a__isList(V2))	(10)
mark(isNeList(X))	→	a__isNeList(X)	(39)
a__U31(tt)	→	tt	(7)
a__isNeList(__(V1,V2))	→	a__U41(a__isList(V1),V2)	(20)
a__isPal(nil)	→	tt	(25)
mark(nil)	→	nil	(49)
mark(e)	→	e	(52)
a__isQid(u)	→	tt	(30)
a__U41(X1,X2)	→	U41(X1,X2)	(62)
a__U72(tt)	→	tt	(14)
a____(X1,X2)	→	__(X1,X2)	(56)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(31)
a__U61(tt)	→	tt	(12)
a__U72(X)	→	U72(X)	(69)
mark(isPal(X))	→	a__isPal(X)	(45)
a__isNePal(__(I,__(P,I)))	→	a__U71(a__isQid(I),P)	(23)
a__isPal(X)	→	isPal(X)	(70)
a__isPal(V)	→	a__U81(a__isNePal(V))	(24)
a__U11(X)	→	U11(X)	(57)
a__U52(tt)	→	tt	(11)
a__U42(tt)	→	tt	(9)
a__U71(tt,P)	→	a__U72(a__isPal(P))	(13)
mark(a)	→	a	(51)
mark(U51(X1,X2))	→	a__U51(mark(X1),X2)	(40)
a__U61(X)	→	U61(X)	(67)
mark(u)	→	u	(55)
a__U22(X)	→	U22(X)	(59)
a__U22(tt)	→	tt	(6)
mark(U42(X))	→	a__U42(mark(X))	(38)
a__U31(X)	→	U31(X)	(61)
a__U21(X1,X2)	→	U21(X1,X2)	(58)
mark(isNePal(X))	→	a__isNePal(X)	(48)
a__U81(X)	→	U81(X)	(71)
mark(i)	→	i	(53)
mark(isQid(X))	→	a__isQid(X)	(47)
a__isNePal(X)	→	isNePal(X)	(73)
mark(U41(X1,X2))	→	a__U41(mark(X1),X2)	(37)
mark(U52(X))	→	a__U52(mark(X))	(41)
mark(U61(X))	→	a__U61(mark(X))	(42)
mark(U81(X))	→	a__U81(mark(X))	(46)
a__U52(X)	→	U52(X)	(66)
mark(isList(X))	→	a__isList(X)	(35)
a__isQid(o)	→	tt	(29)
mark(U71(X1,X2))	→	a__U71(mark(X1),X2)	(43)
a____(X,nil)	→	mark(X)	(2)

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

mark^#(U51(X1,X2))	→	mark^#(X1)	(136)
mark^#(U31(X))	→	mark^#(X)	(107)
a____^#(__(X,Y),Z)	→	mark^#(X)	(102)
mark^#(U41(X1,X2))	→	mark^#(X1)	(100)
mark^#(U61(X))	→	mark^#(X)	(132)
mark^#(U11(X))	→	mark^#(X)	(99)
a____^#(nil,X)	→	mark^#(X)	(131)
a____^#(X,nil)	→	mark^#(X)	(130)
mark^#(U71(X1,X2))	→	mark^#(X1)	(129)
mark^#(U42(X))	→	mark^#(X)	(128)
a____^#(__(X,Y),Z)	→	a____^#(mark(Y),mark(Z))	(96)
mark^#(__(X1,X2))	→	a____^#(mark(X1),mark(X2))	(125)
a____^#(__(X,Y),Z)	→	mark^#(Z)	(89)
mark^#(U81(X))	→	mark^#(X)	(88)
mark^#(__(X1,X2))	→	mark^#(X1)	(119)
a____^#(__(X,Y),Z)	→	mark^#(Y)	(85)
mark^#(__(X1,X2))	→	mark^#(X2)	(116)
mark^#(U52(X))	→	mark^#(X)	(112)
mark^#(U72(X))	→	mark^#(X)	(111)
mark^#(U21(X1,X2))	→	mark^#(X1)	(77)
mark^#(U22(X))	→	mark^#(X)	(75)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a____^#(__(X,Y),Z)

→

a____^#(mark(X),a____(mark(Y),mark(Z)))

(90)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[a__isNePal^#(x₁)]	=	0
[U21(x₁, x₂)]	=	14098
[a__U72^#(x₁)]	=	0
[a__U71^#(x₁, x₂)]	=	0
[U11(x₁)]	=	x₁ + 32527
[a__U31^#(x₁)]	=	0
[a__isNeList^#(x₁)]	=	0
[isNeList(x₁)]	=	x₁ + 28959
[isPal(x₁)]	=	x₁ + 2
[U42(x₁)]	=	13214
[u]	=	1
[U71(x₁, x₂)]	=	x₁ + 1
[a__isPal^#(x₁)]	=	0
[a__U51^#(x₁, x₂)]	=	0
[a__U22(x₁)]	=	1
[isNePal(x₁)]	=	x₁ + 1
[U72(x₁)]	=	1
[a__U11^#(x₁)]	=	0
[a__U31(x₁)]	=	x₁ + 28959
[a__U51(x₁, x₂)]	=	16704
[a____^#(x₁, x₂)]	=	x₁ + 62951
[a__U81(x₁)]	=	x₁ + 1
[isQid(x₁)]	=	x₁ + 0
[a____(x₁, x₂)]	=	x₁ + x₂ + 61548
[a__U42^#(x₁)]	=	0
[a__U41(x₁, x₂)]	=	x₁ + x₂ + 29021
[a__isList(x₁)]	=	x₁ + 61486
[a__isNeList(x₁)]	=	x₁ + 28959
[o]	=	1
[a__U21^#(x₁, x₂)]	=	0
[a__U81^#(x₁)]	=	0
[a__U61^#(x₁)]	=	0
[mark^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 61486
[a__U21(x₁, x₂)]	=	14098
[nil]	=	1
[a__isQid^#(x₁)]	=	0
[a__U52^#(x₁)]	=	0
[mark(x₁)]	=	x₁ + 0
[a__U72(x₁)]	=	1
[a__isNePal(x₁)]	=	x₁ + 1
[a__U11(x₁)]	=	x₁ + 32527
[a__U42(x₁)]	=	13214
[a__U52(x₁)]	=	16704
[i]	=	1
[U52(x₁)]	=	16704
[U61(x₁)]	=	1
[e]	=	33177
[a__U22^#(x₁)]	=	0
[U31(x₁)]	=	x₁ + 28959
[a__U71(x₁, x₂)]	=	x₁ + 1
[a__isPal(x₁)]	=	x₁ + 2
[a__U61(x₁)]	=	1
[U81(x₁)]	=	x₁ + 1
[a__U41^#(x₁, x₂)]	=	0
[tt]	=	0
[a__isQid(x₁)]	=	x₁ + 0
[a__isList^#(x₁)]	=	0
[U22(x₁)]	=	1
[U51(x₁, x₂)]	=	16704
[U41(x₁, x₂)]	=	x₁ + x₂ + 29021
[__(x₁, x₂)]	=	x₁ + x₂ + 61548

together with the usable rules

a__isList(__(V1,V2))	→	a__U21(a__isList(V1),V2)	(18)
mark(tt)	→	tt	(50)
a__U11(tt)	→	tt	(4)
a__U81(tt)	→	tt	(15)
a__U41(tt,V2)	→	a__U42(a__isNeList(V2))	(8)
mark(o)	→	o	(54)
a____(__(X,Y),Z)	→	a____(mark(X),a____(mark(Y),mark(Z)))	(1)
a____(nil,X)	→	mark(X)	(3)
a__isList(V)	→	a__U11(a__isNeList(V))	(16)
a__isNeList(__(V1,V2))	→	a__U51(a__isNeList(V1),V2)	(21)
mark(U31(X))	→	a__U31(mark(X))	(36)
a__U71(X1,X2)	→	U71(X1,X2)	(68)
a__isQid(a)	→	tt	(26)
a__U42(X)	→	U42(X)	(63)
a__isNeList(V)	→	a__U31(a__isQid(V))	(19)
mark(U11(X))	→	a__U11(mark(X))	(32)
a__isList(nil)	→	tt	(17)
a__isList(X)	→	isList(X)	(60)
a__isQid(e)	→	tt	(27)
mark(U22(X))	→	a__U22(mark(X))	(34)
a__isNePal(V)	→	a__U61(a__isQid(V))	(22)
a__isQid(i)	→	tt	(28)
a__U51(X1,X2)	→	U51(X1,X2)	(65)
mark(U72(X))	→	a__U72(mark(X))	(44)
a__U21(tt,V2)	→	a__U22(a__isList(V2))	(5)
a__isQid(X)	→	isQid(X)	(72)
mark(U21(X1,X2))	→	a__U21(mark(X1),X2)	(33)
a__isNeList(X)	→	isNeList(X)	(64)
a__U51(tt,V2)	→	a__U52(a__isList(V2))	(10)
mark(isNeList(X))	→	a__isNeList(X)	(39)
a__U31(tt)	→	tt	(7)
a__isNeList(__(V1,V2))	→	a__U41(a__isList(V1),V2)	(20)
a__isPal(nil)	→	tt	(25)
mark(nil)	→	nil	(49)
mark(e)	→	e	(52)
a__isQid(u)	→	tt	(30)
a__U41(X1,X2)	→	U41(X1,X2)	(62)
a__U72(tt)	→	tt	(14)
a____(X1,X2)	→	__(X1,X2)	(56)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(31)
a__U61(tt)	→	tt	(12)
a__U72(X)	→	U72(X)	(69)
mark(isPal(X))	→	a__isPal(X)	(45)
a__isNePal(__(I,__(P,I)))	→	a__U71(a__isQid(I),P)	(23)
a__isPal(X)	→	isPal(X)	(70)
a__isPal(V)	→	a__U81(a__isNePal(V))	(24)
a__U11(X)	→	U11(X)	(57)
a__U52(tt)	→	tt	(11)
a__U42(tt)	→	tt	(9)
a__U71(tt,P)	→	a__U72(a__isPal(P))	(13)
mark(a)	→	a	(51)
mark(U51(X1,X2))	→	a__U51(mark(X1),X2)	(40)
a__U61(X)	→	U61(X)	(67)
mark(u)	→	u	(55)
a__U22(X)	→	U22(X)	(59)
a__U22(tt)	→	tt	(6)
mark(U42(X))	→	a__U42(mark(X))	(38)
a__U31(X)	→	U31(X)	(61)
a__U21(X1,X2)	→	U21(X1,X2)	(58)
mark(isNePal(X))	→	a__isNePal(X)	(48)
a__U81(X)	→	U81(X)	(71)
mark(i)	→	i	(53)
mark(isQid(X))	→	a__isQid(X)	(47)
a__isNePal(X)	→	isNePal(X)	(73)
mark(U41(X1,X2))	→	a__U41(mark(X1),X2)	(37)
mark(U52(X))	→	a__U52(mark(X))	(41)
mark(U61(X))	→	a__U61(mark(X))	(42)
mark(U81(X))	→	a__U81(mark(X))	(46)
a__U52(X)	→	U52(X)	(66)
mark(isList(X))	→	a__isList(X)	(35)
a__isQid(o)	→	tt	(29)
mark(U71(X1,X2))	→	a__U71(mark(X1),X2)	(43)
a____(X,nil)	→	mark(X)	(2)

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

a____^#(__(X,Y),Z)

→

a____^#(mark(X),a____(mark(Y),mark(Z)))

(90)

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

a__isNeList^#(__(V1,V2))	→	a__U51^#(a__isNeList(V1),V2)	(135)
a__isList^#(__(V1,V2))	→	a__U21^#(a__isList(V1),V2)	(104)
a__isNeList^#(__(V1,V2))	→	a__isList^#(V1)	(105)
a__isList^#(__(V1,V2))	→	a__isList^#(V1)	(124)
a__isNeList^#(__(V1,V2))	→	a__isNeList^#(V1)	(92)
a__U41^#(tt,V2)	→	a__isNeList^#(V2)	(86)
a__isList^#(V)	→	a__isNeList^#(V)	(83)
a__U21^#(tt,V2)	→	a__isList^#(V2)	(81)
a__U51^#(tt,V2)	→	a__isList^#(V2)	(113)
a__isNeList^#(__(V1,V2))	→	a__U41^#(a__isList(V1),V2)	(76)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[a__isNePal^#(x₁)]	=	0
[U21(x₁, x₂)]	=	1
[a__U72^#(x₁)]	=	0
[a__U71^#(x₁, x₂)]	=	0
[U11(x₁)]	=	1
[a__U31^#(x₁)]	=	0
[a__isNeList^#(x₁)]	=	x₁ + 0
[isNeList(x₁)]	=	x₁ + 1
[isPal(x₁)]	=	x₁ + 40513
[U42(x₁)]	=	1
[u]	=	21633
[U71(x₁, x₂)]	=	x₁ + 9
[a__isPal^#(x₁)]	=	0
[a__U51^#(x₁, x₂)]	=	x₁ + x₂ + 2
[a__U22(x₁)]	=	1
[isNePal(x₁)]	=	x₁ + 1
[U72(x₁)]	=	5
[a__U11^#(x₁)]	=	0
[a__U31(x₁)]	=	x₁ + 1
[a__U51(x₁, x₂)]	=	4
[a____^#(x₁, x₂)]	=	x₁ + 62951
[a__U81(x₁)]	=	x₁ + 40512
[isQid(x₁)]	=	x₁ + 0
[a____(x₁, x₂)]	=	x₁ + x₂ + 4
[a__U42^#(x₁)]	=	0
[a__U41(x₁, x₂)]	=	x₁ + x₂ + 4
[a__isList(x₁)]	=	1
[a__isNeList(x₁)]	=	x₁ + 1
[o]	=	1
[a__U21^#(x₁, x₂)]	=	x₂ + 2
[a__U81^#(x₁)]	=	0
[a__U61^#(x₁)]	=	0
[mark^#(x₁)]	=	0
[isList(x₁)]	=	1
[a__U21(x₁, x₂)]	=	1
[nil]	=	1
[a__isQid^#(x₁)]	=	0
[a__U52^#(x₁)]	=	0
[mark(x₁)]	=	x₁ + 0
[a__U72(x₁)]	=	5
[a__isNePal(x₁)]	=	x₁ + 1
[a__U11(x₁)]	=	1
[a__U42(x₁)]	=	1
[a__U52(x₁)]	=	4
[i]	=	13140
[U52(x₁)]	=	4
[U61(x₁)]	=	1
[e]	=	1
[a__U22^#(x₁)]	=	0
[U31(x₁)]	=	x₁ + 1
[a__U71(x₁, x₂)]	=	x₁ + 9
[a__isPal(x₁)]	=	x₁ + 40513
[a__U61(x₁)]	=	1
[U81(x₁)]	=	x₁ + 40512
[a__U41^#(x₁, x₂)]	=	x₂ + 3
[tt]	=	0
[a__isQid(x₁)]	=	x₁ + 0
[a__isList^#(x₁)]	=	x₁ + 1
[U22(x₁)]	=	1
[U51(x₁, x₂)]	=	4
[U41(x₁, x₂)]	=	x₁ + x₂ + 4
[__(x₁, x₂)]	=	x₁ + x₂ + 4

together with the usable rules

a__isList(__(V1,V2))	→	a__U21(a__isList(V1),V2)	(18)
mark(tt)	→	tt	(50)
a__U11(tt)	→	tt	(4)
a__U81(tt)	→	tt	(15)
a__U41(tt,V2)	→	a__U42(a__isNeList(V2))	(8)
mark(o)	→	o	(54)
a____(__(X,Y),Z)	→	a____(mark(X),a____(mark(Y),mark(Z)))	(1)
a____(nil,X)	→	mark(X)	(3)
a__isList(V)	→	a__U11(a__isNeList(V))	(16)
a__isNeList(__(V1,V2))	→	a__U51(a__isNeList(V1),V2)	(21)
mark(U31(X))	→	a__U31(mark(X))	(36)
a__U71(X1,X2)	→	U71(X1,X2)	(68)
a__isQid(a)	→	tt	(26)
a__U42(X)	→	U42(X)	(63)
a__isNeList(V)	→	a__U31(a__isQid(V))	(19)
mark(U11(X))	→	a__U11(mark(X))	(32)
a__isList(nil)	→	tt	(17)
a__isList(X)	→	isList(X)	(60)
a__isQid(e)	→	tt	(27)
mark(U22(X))	→	a__U22(mark(X))	(34)
a__isNePal(V)	→	a__U61(a__isQid(V))	(22)
a__isQid(i)	→	tt	(28)
a__U51(X1,X2)	→	U51(X1,X2)	(65)
mark(U72(X))	→	a__U72(mark(X))	(44)
a__U21(tt,V2)	→	a__U22(a__isList(V2))	(5)
a__isQid(X)	→	isQid(X)	(72)
mark(U21(X1,X2))	→	a__U21(mark(X1),X2)	(33)
a__isNeList(X)	→	isNeList(X)	(64)
a__U51(tt,V2)	→	a__U52(a__isList(V2))	(10)
mark(isNeList(X))	→	a__isNeList(X)	(39)
a__U31(tt)	→	tt	(7)
a__isNeList(__(V1,V2))	→	a__U41(a__isList(V1),V2)	(20)
a__isPal(nil)	→	tt	(25)
mark(nil)	→	nil	(49)
mark(e)	→	e	(52)
a__isQid(u)	→	tt	(30)
a__U41(X1,X2)	→	U41(X1,X2)	(62)
a__U72(tt)	→	tt	(14)
a____(X1,X2)	→	__(X1,X2)	(56)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(31)
a__U61(tt)	→	tt	(12)
a__U72(X)	→	U72(X)	(69)
mark(isPal(X))	→	a__isPal(X)	(45)
a__isNePal(__(I,__(P,I)))	→	a__U71(a__isQid(I),P)	(23)
a__isPal(X)	→	isPal(X)	(70)
a__isPal(V)	→	a__U81(a__isNePal(V))	(24)
a__U11(X)	→	U11(X)	(57)
a__U52(tt)	→	tt	(11)
a__U42(tt)	→	tt	(9)
a__U71(tt,P)	→	a__U72(a__isPal(P))	(13)
mark(a)	→	a	(51)
mark(U51(X1,X2))	→	a__U51(mark(X1),X2)	(40)
a__U61(X)	→	U61(X)	(67)
mark(u)	→	u	(55)
a__U22(X)	→	U22(X)	(59)
a__U22(tt)	→	tt	(6)
mark(U42(X))	→	a__U42(mark(X))	(38)
a__U31(X)	→	U31(X)	(61)
a__U21(X1,X2)	→	U21(X1,X2)	(58)
mark(isNePal(X))	→	a__isNePal(X)	(48)
a__U81(X)	→	U81(X)	(71)
mark(i)	→	i	(53)
mark(isQid(X))	→	a__isQid(X)	(47)
a__isNePal(X)	→	isNePal(X)	(73)
mark(U41(X1,X2))	→	a__U41(mark(X1),X2)	(37)
mark(U52(X))	→	a__U52(mark(X))	(41)
mark(U61(X))	→	a__U61(mark(X))	(42)
mark(U81(X))	→	a__U81(mark(X))	(46)
a__U52(X)	→	U52(X)	(66)
mark(isList(X))	→	a__isList(X)	(35)
a__isQid(o)	→	tt	(29)
mark(U71(X1,X2))	→	a__U71(mark(X1),X2)	(43)
a____(X,nil)	→	mark(X)	(2)

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

a__isNeList^#(__(V1,V2))	→	a__U51^#(a__isNeList(V1),V2)	(135)
a__isList^#(__(V1,V2))	→	a__U21^#(a__isList(V1),V2)	(104)
a__isNeList^#(__(V1,V2))	→	a__isList^#(V1)	(105)
a__isList^#(__(V1,V2))	→	a__isList^#(V1)	(124)
a__isNeList^#(__(V1,V2))	→	a__isNeList^#(V1)	(92)
a__U41^#(tt,V2)	→	a__isNeList^#(V2)	(86)
a__isList^#(V)	→	a__isNeList^#(V)	(83)
a__U21^#(tt,V2)	→	a__isList^#(V2)	(81)
a__U51^#(tt,V2)	→	a__isList^#(V2)	(113)
a__isNeList^#(__(V1,V2))	→	a__U41^#(a__isList(V1),V2)	(76)

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

a__isNePal^#(__(I,__(P,I)))	→	a__U71^#(a__isQid(I),P)	(98)
a__isPal^#(V)	→	a__isNePal^#(V)	(117)
a__U71^#(tt,P)	→	a__isPal^#(P)	(109)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[a__isNePal^#(x₁)]	=	x₁ + 0
[U21(x₁, x₂)]	=	1
[a__U72^#(x₁)]	=	0
[a__U71^#(x₁, x₂)]	=	x₂ + 2
[U11(x₁)]	=	1
[a__U31^#(x₁)]	=	0
[a__isNeList^#(x₁)]	=	0
[isNeList(x₁)]	=	x₁ + 1
[isPal(x₁)]	=	x₁ + 43940
[U42(x₁)]	=	3685
[u]	=	1
[U71(x₁, x₂)]	=	x₁ + 7373
[a__isPal^#(x₁)]	=	x₁ + 1
[a__U51^#(x₁, x₂)]	=	x₁ + 0
[a__U22(x₁)]	=	1
[isNePal(x₁)]	=	x₁ + 1
[U72(x₁)]	=	7373
[a__U11^#(x₁)]	=	0
[a__U31(x₁)]	=	x₁ + 1
[a__U51(x₁, x₂)]	=	1
[a____^#(x₁, x₂)]	=	x₁ + 0
[a__U81(x₁)]	=	x₁ + 43939
[isQid(x₁)]	=	x₁ + 0
[a____(x₁, x₂)]	=	x₁ + x₂ + 3686
[a__U42^#(x₁)]	=	0
[a__U41(x₁, x₂)]	=	x₁ + x₂ + 3685
[a__isList(x₁)]	=	1
[a__isNeList(x₁)]	=	x₁ + 1
[o]	=	1
[a__U21^#(x₁, x₂)]	=	0
[a__U81^#(x₁)]	=	0
[a__U61^#(x₁)]	=	0
[mark^#(x₁)]	=	0
[isList(x₁)]	=	1
[a__U21(x₁, x₂)]	=	1
[nil]	=	1
[a__isQid^#(x₁)]	=	0
[a__U52^#(x₁)]	=	0
[mark(x₁)]	=	x₁ + 0
[a__U72(x₁)]	=	7373
[a__isNePal(x₁)]	=	x₁ + 1
[a__U11(x₁)]	=	1
[a__U42(x₁)]	=	3685
[a__U52(x₁)]	=	1
[i]	=	1
[U52(x₁)]	=	1
[U61(x₁)]	=	1
[e]	=	1
[a__U22^#(x₁)]	=	0
[U31(x₁)]	=	x₁ + 1
[a__U71(x₁, x₂)]	=	x₁ + 7373
[a__isPal(x₁)]	=	x₁ + 43940
[a__U61(x₁)]	=	1
[U81(x₁)]	=	x₁ + 43939
[a__U41^#(x₁, x₂)]	=	0
[tt]	=	0
[a__isQid(x₁)]	=	x₁ + 0
[a__isList^#(x₁)]	=	0
[U22(x₁)]	=	1
[U51(x₁, x₂)]	=	1
[U41(x₁, x₂)]	=	x₁ + x₂ + 3685
[__(x₁, x₂)]	=	x₁ + x₂ + 3686

together with the usable rules

a__isList(__(V1,V2))	→	a__U21(a__isList(V1),V2)	(18)
mark(tt)	→	tt	(50)
a__U11(tt)	→	tt	(4)
a__U81(tt)	→	tt	(15)
a__U41(tt,V2)	→	a__U42(a__isNeList(V2))	(8)
mark(o)	→	o	(54)
a____(__(X,Y),Z)	→	a____(mark(X),a____(mark(Y),mark(Z)))	(1)
a____(nil,X)	→	mark(X)	(3)
a__isList(V)	→	a__U11(a__isNeList(V))	(16)
a__isNeList(__(V1,V2))	→	a__U51(a__isNeList(V1),V2)	(21)
mark(U31(X))	→	a__U31(mark(X))	(36)
a__U71(X1,X2)	→	U71(X1,X2)	(68)
a__isQid(a)	→	tt	(26)
a__U42(X)	→	U42(X)	(63)
a__isNeList(V)	→	a__U31(a__isQid(V))	(19)
mark(U11(X))	→	a__U11(mark(X))	(32)
a__isList(nil)	→	tt	(17)
a__isList(X)	→	isList(X)	(60)
a__isQid(e)	→	tt	(27)
mark(U22(X))	→	a__U22(mark(X))	(34)
a__isNePal(V)	→	a__U61(a__isQid(V))	(22)
a__isQid(i)	→	tt	(28)
a__U51(X1,X2)	→	U51(X1,X2)	(65)
mark(U72(X))	→	a__U72(mark(X))	(44)
a__U21(tt,V2)	→	a__U22(a__isList(V2))	(5)
a__isQid(X)	→	isQid(X)	(72)
mark(U21(X1,X2))	→	a__U21(mark(X1),X2)	(33)
a__isNeList(X)	→	isNeList(X)	(64)
a__U51(tt,V2)	→	a__U52(a__isList(V2))	(10)
mark(isNeList(X))	→	a__isNeList(X)	(39)
a__U31(tt)	→	tt	(7)
a__isNeList(__(V1,V2))	→	a__U41(a__isList(V1),V2)	(20)
a__isPal(nil)	→	tt	(25)
mark(nil)	→	nil	(49)
mark(e)	→	e	(52)
a__isQid(u)	→	tt	(30)
a__U41(X1,X2)	→	U41(X1,X2)	(62)
a__U72(tt)	→	tt	(14)
a____(X1,X2)	→	__(X1,X2)	(56)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(31)
a__U61(tt)	→	tt	(12)
a__U72(X)	→	U72(X)	(69)
mark(isPal(X))	→	a__isPal(X)	(45)
a__isNePal(__(I,__(P,I)))	→	a__U71(a__isQid(I),P)	(23)
a__isPal(X)	→	isPal(X)	(70)
a__isPal(V)	→	a__U81(a__isNePal(V))	(24)
a__U11(X)	→	U11(X)	(57)
a__U52(tt)	→	tt	(11)
a__U42(tt)	→	tt	(9)
a__U71(tt,P)	→	a__U72(a__isPal(P))	(13)
mark(a)	→	a	(51)
mark(U51(X1,X2))	→	a__U51(mark(X1),X2)	(40)
a__U61(X)	→	U61(X)	(67)
mark(u)	→	u	(55)
a__U22(X)	→	U22(X)	(59)
a__U22(tt)	→	tt	(6)
mark(U42(X))	→	a__U42(mark(X))	(38)
a__U31(X)	→	U31(X)	(61)
a__U21(X1,X2)	→	U21(X1,X2)	(58)
mark(isNePal(X))	→	a__isNePal(X)	(48)
a__U81(X)	→	U81(X)	(71)
mark(i)	→	i	(53)
mark(isQid(X))	→	a__isQid(X)	(47)
a__isNePal(X)	→	isNePal(X)	(73)
mark(U41(X1,X2))	→	a__U41(mark(X1),X2)	(37)
mark(U52(X))	→	a__U52(mark(X))	(41)
mark(U61(X))	→	a__U61(mark(X))	(42)
mark(U81(X))	→	a__U81(mark(X))	(46)
a__U52(X)	→	U52(X)	(66)
mark(isList(X))	→	a__isList(X)	(35)
a__isQid(o)	→	tt	(29)
mark(U71(X1,X2))	→	a__U71(mark(X1),X2)	(43)
a____(X,nil)	→	mark(X)	(2)

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

a__isNePal^#(__(I,__(P,I)))	→	a__U71^#(a__isQid(I),P)	(98)
a__isPal^#(V)	→	a__isNePal^#(V)	(117)
a__U71^#(tt,P)	→	a__isPal^#(P)	(109)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.