Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PALINDROME_nokinds-noand_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)	→	tt	(4)
U21(tt,V2)	→	U22(isList(activate(V2)))	(5)
U22(tt)	→	tt	(6)
U31(tt)	→	tt	(7)
U41(tt,V2)	→	U42(isNeList(activate(V2)))	(8)
U42(tt)	→	tt	(9)
U51(tt,V2)	→	U52(isList(activate(V2)))	(10)
U52(tt)	→	tt	(11)
U61(tt)	→	tt	(12)
U71(tt,P)	→	U72(isPal(activate(P)))	(13)
U72(tt)	→	tt	(14)
U81(tt)	→	tt	(15)
isList(V)	→	U11(isNeList(activate(V)))	(16)
isList(n__nil)	→	tt	(17)
isList(n____(V1,V2))	→	U21(isList(activate(V1)),activate(V2))	(18)
isNeList(V)	→	U31(isQid(activate(V)))	(19)
isNeList(n____(V1,V2))	→	U41(isList(activate(V1)),activate(V2))	(20)
isNeList(n____(V1,V2))	→	U51(isNeList(activate(V1)),activate(V2))	(21)
isNePal(V)	→	U61(isQid(activate(V)))	(22)
isNePal(n____(I,__(P,I)))	→	U71(isQid(activate(I)),activate(P))	(23)
isPal(V)	→	U81(isNePal(activate(V)))	(24)
isPal(n__nil)	→	tt	(25)
isQid(n__a)	→	tt	(26)
isQid(n__e)	→	tt	(27)
isQid(n__i)	→	tt	(28)
isQid(n__o)	→	tt	(29)
isQid(n__u)	→	tt	(30)
nil	→	n__nil	(31)
__(X1,X2)	→	n____(X1,X2)	(32)
a	→	n__a	(33)
e	→	n__e	(34)
i	→	n__i	(35)
o	→	n__o	(36)
u	→	n__u	(37)
activate(n__nil)	→	nil	(38)
activate(n____(X1,X2))	→	__(X1,X2)	(39)
activate(n__a)	→	a	(40)
activate(n__e)	→	e	(41)
activate(n__i)	→	i	(42)
activate(n__o)	→	o	(43)
activate(n__u)	→	u	(44)
activate(X)	→	X	(45)

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.

activate^#(n__i)	→	i^#	(46)
U41^#(tt,V2)	→	U42^#(isNeList(activate(V2)))	(47)
isNePal^#(V)	→	isQid^#(activate(V))	(48)
isNeList^#(n____(V1,V2))	→	isNeList^#(activate(V1))	(49)
isList^#(n____(V1,V2))	→	U21^#(isList(activate(V1)),activate(V2))	(50)
__^#(__(X,Y),Z)	→	__^#(Y,Z)	(51)
__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(52)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(53)
isList^#(n____(V1,V2))	→	isList^#(activate(V1))	(54)
isList^#(n____(V1,V2))	→	activate^#(V1)	(55)
isList^#(V)	→	U11^#(isNeList(activate(V)))	(56)
U71^#(tt,P)	→	U72^#(isPal(activate(P)))	(57)
isList^#(V)	→	activate^#(V)	(58)
isNePal^#(n____(I,__(P,I)))	→	isQid^#(activate(I))	(59)
isPal^#(V)	→	U81^#(isNePal(activate(V)))	(60)
isPal^#(V)	→	activate^#(V)	(61)
U21^#(tt,V2)	→	U22^#(isList(activate(V2)))	(62)
U51^#(tt,V2)	→	activate^#(V2)	(63)
U41^#(tt,V2)	→	activate^#(V2)	(64)
isNePal^#(n____(I,__(P,I)))	→	U71^#(isQid(activate(I)),activate(P))	(65)
isNeList^#(V)	→	U31^#(isQid(activate(V)))	(66)
isNePal^#(n____(I,__(P,I)))	→	activate^#(I)	(67)
isNePal^#(V)	→	activate^#(V)	(68)
isList^#(n____(V1,V2))	→	activate^#(V2)	(69)
U21^#(tt,V2)	→	isList^#(activate(V2))	(70)
activate^#(n____(X1,X2))	→	__^#(X1,X2)	(71)
U71^#(tt,P)	→	isPal^#(activate(P))	(72)
isList^#(V)	→	isNeList^#(activate(V))	(73)
isPal^#(V)	→	isNePal^#(activate(V))	(74)
isNeList^#(V)	→	activate^#(V)	(75)
activate^#(n__a)	→	a^#	(76)
U41^#(tt,V2)	→	isNeList^#(activate(V2))	(77)
isNePal^#(V)	→	U61^#(isQid(activate(V)))	(78)
U51^#(tt,V2)	→	isList^#(activate(V2))	(79)
isNeList^#(n____(V1,V2))	→	U51^#(isNeList(activate(V1)),activate(V2))	(80)
activate^#(n__u)	→	u^#	(81)
isNeList^#(n____(V1,V2))	→	isList^#(activate(V1))	(82)
isNeList^#(V)	→	isQid^#(activate(V))	(83)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(53)
U21^#(tt,V2)	→	activate^#(V2)	(84)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(85)
U51^#(tt,V2)	→	U52^#(isList(activate(V2)))	(86)
isNeList^#(n____(V1,V2))	→	U41^#(isList(activate(V1)),activate(V2))	(87)
activate^#(n__e)	→	e^#	(88)
activate^#(n__o)	→	o^#	(89)
U71^#(tt,P)	→	activate^#(P)	(90)
isNePal^#(n____(I,__(P,I)))	→	activate^#(P)	(91)
activate^#(n__nil)	→	nil^#	(92)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(85)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

isNePal^#(n____(I,__(P,I)))	→	U71^#(isQid(activate(I)),activate(P))	(65)
isPal^#(V)	→	isNePal^#(activate(V))	(74)
U71^#(tt,P)	→	isPal^#(activate(P))	(72)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	27543
[n__a]	=	27543
[U72^#(x₁)]	=	0
[U21(x₁, x₂)]	=	0
[U11(x₁)]	=	0
[n__o]	=	58155
[isNeList(x₁)]	=	0
[isPal(x₁)]	=	0
[U42(x₁)]	=	0
[e^#]	=	0
[u]	=	27543
[activate(x₁)]	=	x₁ + 1
[U71(x₁, x₂)]	=	0
[U81^#(x₁)]	=	0
[u^#]	=	0
[isNeList^#(x₁)]	=	0
[activate^#(x₁)]	=	0
[n__i]	=	33396
[__^#(x₁, x₂)]	=	0
[isNePal(x₁)]	=	0
[U72(x₁)]	=	0
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	x₁ + 48078
[n__nil]	=	1
[U52^#(x₁)]	=	0
[isQid(x₁)]	=	x₁ + 20537
[n____(x₁, x₂)]	=	x₁ + x₂ + 282
[n__e]	=	27543
[o]	=	58155
[U42^#(x₁)]	=	0
[isList(x₁)]	=	0
[isNePal^#(x₁)]	=	x₁ + 48076
[nil]	=	1
[n__u]	=	27543
[o^#]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	0
[i]	=	33396
[U52(x₁)]	=	0
[U61(x₁)]	=	0
[U51^#(x₁, x₂)]	=	0
[e]	=	27543
[U11^#(x₁)]	=	0
[U31(x₁)]	=	0
[U41^#(x₁, x₂)]	=	0
[a^#]	=	0
[U21^#(x₁, x₂)]	=	0
[U81(x₁)]	=	0
[U22^#(x₁)]	=	0
[tt]	=	48080
[U71^#(x₁, x₂)]	=	x₁ + x₂ + 0
[U22(x₁)]	=	0
[U51(x₁, x₂)]	=	0
[U41(x₁, x₂)]	=	0
[U31^#(x₁)]	=	0
[__(x₁, x₂)]	=	x₁ + x₂ + 282
[U61^#(x₁)]	=	0
[i^#]	=	0

together with the usable rules

__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
o	→	n__o	(36)
isQid(n__a)	→	tt	(26)
__(X1,X2)	→	n____(X1,X2)	(32)
isQid(n__e)	→	tt	(27)
e	→	n__e	(34)
isQid(n__i)	→	tt	(28)
activate(n__u)	→	u	(44)
a	→	n__a	(33)
activate(n____(X1,X2))	→	__(X1,X2)	(39)
isQid(n__u)	→	tt	(30)
nil	→	n__nil	(31)
activate(X)	→	X	(45)
activate(n__a)	→	a	(40)
activate(n__nil)	→	nil	(38)
u	→	n__u	(37)
activate(n__e)	→	e	(41)
activate(n__i)	→	i	(42)
i	→	n__i	(35)
isQid(n__o)	→	tt	(29)
activate(n__o)	→	o	(43)
__(X,nil)	→	X	(2)

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

isNePal^#(n____(I,__(P,I)))	→	U71^#(isQid(activate(I)),activate(P))	(65)
isPal^#(V)	→	isNePal^#(activate(V))	(74)
U71^#(tt,P)	→	isPal^#(activate(P))	(72)

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

U21^#(tt,V2)	→	isList^#(activate(V2))	(70)
isNeList^#(n____(V1,V2))	→	U41^#(isList(activate(V1)),activate(V2))	(87)
isNeList^#(n____(V1,V2))	→	isList^#(activate(V1))	(82)
isNeList^#(n____(V1,V2))	→	U51^#(isNeList(activate(V1)),activate(V2))	(80)
U51^#(tt,V2)	→	isList^#(activate(V2))	(79)
isList^#(n____(V1,V2))	→	isList^#(activate(V1))	(54)
U41^#(tt,V2)	→	isNeList^#(activate(V2))	(77)
isList^#(n____(V1,V2))	→	U21^#(isList(activate(V1)),activate(V2))	(50)
isList^#(V)	→	isNeList^#(activate(V))	(73)
isNeList^#(n____(V1,V2))	→	isNeList^#(activate(V1))	(49)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[n__a]	=	1
[U72^#(x₁)]	=	0
[U21(x₁, x₂)]	=	x₁ + x₂ + 6705
[U11(x₁)]	=	x₁ + 0
[n__o]	=	22078
[isNeList(x₁)]	=	x₁ + 6704
[isPal(x₁)]	=	0
[U42(x₁)]	=	x₁ + 0
[e^#]	=	0
[u]	=	10206
[activate(x₁)]	=	x₁ + 1
[U71(x₁, x₂)]	=	0
[U81^#(x₁)]	=	0
[u^#]	=	0
[isNeList^#(x₁)]	=	x₁ + 0
[activate^#(x₁)]	=	0
[n__i]	=	11885
[__^#(x₁, x₂)]	=	0
[isNePal(x₁)]	=	0
[U72(x₁)]	=	0
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	x₁ + 48078
[n__nil]	=	1
[U52^#(x₁)]	=	0
[isQid(x₁)]	=	x₁ + 6703
[n____(x₁, x₂)]	=	x₁ + x₂ + 6710
[n__e]	=	27543
[o]	=	22079
[U42^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 6705
[isNePal^#(x₁)]	=	x₁ + 48076
[nil]	=	1
[n__u]	=	10205
[o^#]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	x₁ + 2
[i]	=	11886
[U52(x₁)]	=	x₁ + 3
[U61(x₁)]	=	0
[U51^#(x₁, x₂)]	=	x₁ + x₂ + 3
[e]	=	27543
[U11^#(x₁)]	=	0
[U31(x₁)]	=	x₁ + 0
[U41^#(x₁, x₂)]	=	x₁ + x₂ + 2
[a^#]	=	0
[U21^#(x₁, x₂)]	=	x₁ + x₂ + 3
[U81(x₁)]	=	0
[U22^#(x₁)]	=	0
[tt]	=	1
[U71^#(x₁, x₂)]	=	x₁ + x₂ + 0
[U22(x₁)]	=	x₁ + 0
[U51(x₁, x₂)]	=	x₁ + x₂ + 6708
[U41(x₁, x₂)]	=	x₁ + x₂ + 6707
[U31^#(x₁)]	=	0
[__(x₁, x₂)]	=	x₁ + x₂ + 6711
[U61^#(x₁)]	=	0
[i^#]	=	0

together with the usable rules

isList(n____(V1,V2))	→	U21(isList(activate(V1)),activate(V2))	(18)
U11(tt)	→	tt	(4)
U41(tt,V2)	→	U42(isNeList(activate(V2)))	(8)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
isList(V)	→	U11(isNeList(activate(V)))	(16)
isNeList(n____(V1,V2))	→	U51(isNeList(activate(V1)),activate(V2))	(21)
o	→	n__o	(36)
isQid(n__a)	→	tt	(26)
isNeList(V)	→	U31(isQid(activate(V)))	(19)
__(X1,X2)	→	n____(X1,X2)	(32)
isList(n__nil)	→	tt	(17)
isQid(n__e)	→	tt	(27)
e	→	n__e	(34)
isQid(n__i)	→	tt	(28)
activate(n__u)	→	u	(44)
U21(tt,V2)	→	U22(isList(activate(V2)))	(5)
a	→	n__a	(33)
U51(tt,V2)	→	U52(isList(activate(V2)))	(10)
activate(n____(X1,X2))	→	__(X1,X2)	(39)
U31(tt)	→	tt	(7)
isNeList(n____(V1,V2))	→	U41(isList(activate(V1)),activate(V2))	(20)
isQid(n__u)	→	tt	(30)
nil	→	n__nil	(31)
activate(X)	→	X	(45)
U52(tt)	→	tt	(11)
U42(tt)	→	tt	(9)
activate(n__a)	→	a	(40)
U22(tt)	→	tt	(6)
activate(n__nil)	→	nil	(38)
u	→	n__u	(37)
activate(n__e)	→	e	(41)
activate(n__i)	→	i	(42)
i	→	n__i	(35)
isQid(n__o)	→	tt	(29)
activate(n__o)	→	o	(43)
__(X,nil)	→	X	(2)

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

U21^#(tt,V2)	→	isList^#(activate(V2))	(70)
isNeList^#(n____(V1,V2))	→	U41^#(isList(activate(V1)),activate(V2))	(87)
isNeList^#(n____(V1,V2))	→	isList^#(activate(V1))	(82)
isNeList^#(n____(V1,V2))	→	U51^#(isNeList(activate(V1)),activate(V2))	(80)
U51^#(tt,V2)	→	isList^#(activate(V2))	(79)
isList^#(n____(V1,V2))	→	isList^#(activate(V1))	(54)
U41^#(tt,V2)	→	isNeList^#(activate(V2))	(77)
isList^#(n____(V1,V2))	→	U21^#(isList(activate(V1)),activate(V2))	(50)
isList^#(V)	→	isNeList^#(activate(V))	(73)
isNeList^#(n____(V1,V2))	→	isNeList^#(activate(V1))	(49)

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

__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(52)
__^#(__(X,Y),Z)	→	__^#(Y,Z)	(51)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[n__a]	=	1
[U72^#(x₁)]	=	0
[U21(x₁, x₂)]	=	x₁ + x₂ + 4
[U11(x₁)]	=	x₁ + 0
[n__o]	=	1
[isNeList(x₁)]	=	x₁ + 3
[isPal(x₁)]	=	0
[U42(x₁)]	=	x₁ + 0
[e^#]	=	0
[u]	=	14099
[activate(x₁)]	=	x₁ + 1
[U71(x₁, x₂)]	=	0
[U81^#(x₁)]	=	0
[u^#]	=	0
[isNeList^#(x₁)]	=	x₁ + 0
[activate^#(x₁)]	=	0
[n__i]	=	44560
[__^#(x₁, x₂)]	=	x₁ + 0
[isNePal(x₁)]	=	0
[U72(x₁)]	=	0
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	x₁ + 48078
[n__nil]	=	1
[U52^#(x₁)]	=	0
[isQid(x₁)]	=	x₁ + 0
[n____(x₁, x₂)]	=	x₁ + x₂ + 6
[n__e]	=	10282
[o]	=	2
[U42^#(x₁)]	=	0
[isList(x₁)]	=	x₁ + 4
[isNePal^#(x₁)]	=	x₁ + 48076
[nil]	=	1
[n__u]	=	14098
[o^#]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	x₁ + 2
[i]	=	44561
[U52(x₁)]	=	x₁ + 0
[U61(x₁)]	=	0
[U51^#(x₁, x₂)]	=	x₁ + x₂ + 3
[e]	=	10282
[U11^#(x₁)]	=	0
[U31(x₁)]	=	x₁ + 0
[U41^#(x₁, x₂)]	=	x₁ + x₂ + 2
[a^#]	=	0
[U21^#(x₁, x₂)]	=	x₁ + x₂ + 3
[U81(x₁)]	=	0
[U22^#(x₁)]	=	0
[tt]	=	1
[U71^#(x₁, x₂)]	=	x₁ + x₂ + 0
[U22(x₁)]	=	x₁ + 0
[U51(x₁, x₂)]	=	x₁ + x₂ + 4
[U41(x₁, x₂)]	=	x₁ + x₂ + 3
[U31^#(x₁)]	=	0
[__(x₁, x₂)]	=	x₁ + x₂ + 6
[U61^#(x₁)]	=	0
[i^#]	=	0

together with the usable rules

isList(n____(V1,V2))	→	U21(isList(activate(V1)),activate(V2))	(18)
U11(tt)	→	tt	(4)
U41(tt,V2)	→	U42(isNeList(activate(V2)))	(8)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
isList(V)	→	U11(isNeList(activate(V)))	(16)
isNeList(n____(V1,V2))	→	U51(isNeList(activate(V1)),activate(V2))	(21)
o	→	n__o	(36)
isQid(n__a)	→	tt	(26)
isNeList(V)	→	U31(isQid(activate(V)))	(19)
__(X1,X2)	→	n____(X1,X2)	(32)
isList(n__nil)	→	tt	(17)
isQid(n__e)	→	tt	(27)
e	→	n__e	(34)
isQid(n__i)	→	tt	(28)
activate(n__u)	→	u	(44)
U21(tt,V2)	→	U22(isList(activate(V2)))	(5)
a	→	n__a	(33)
U51(tt,V2)	→	U52(isList(activate(V2)))	(10)
activate(n____(X1,X2))	→	__(X1,X2)	(39)
U31(tt)	→	tt	(7)
isNeList(n____(V1,V2))	→	U41(isList(activate(V1)),activate(V2))	(20)
isQid(n__u)	→	tt	(30)
nil	→	n__nil	(31)
activate(X)	→	X	(45)
U52(tt)	→	tt	(11)
U42(tt)	→	tt	(9)
activate(n__a)	→	a	(40)
U22(tt)	→	tt	(6)
activate(n__nil)	→	nil	(38)
u	→	n__u	(37)
activate(n__e)	→	e	(41)
activate(n__i)	→	i	(42)
i	→	n__i	(35)
isQid(n__o)	→	tt	(29)
activate(n__o)	→	o	(43)
__(X,nil)	→	X	(2)

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

__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(52)
__^#(__(X,Y),Z)	→	__^#(Y,Z)	(51)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.