Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PALINDROME_nokinds_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)
and(tt,X)	→	activate(X)	(4)
isList(V)	→	isNeList(activate(V))	(5)
isList(n__nil)	→	tt	(6)
isList(n____(V1,V2))	→	and(isList(activate(V1)),n__isList(activate(V2)))	(7)
isNeList(V)	→	isQid(activate(V))	(8)
isNeList(n____(V1,V2))	→	and(isList(activate(V1)),n__isNeList(activate(V2)))	(9)
isNeList(n____(V1,V2))	→	and(isNeList(activate(V1)),n__isList(activate(V2)))	(10)
isNePal(V)	→	isQid(activate(V))	(11)
isNePal(n____(I,__(P,I)))	→	and(isQid(activate(I)),n__isPal(activate(P)))	(12)
isPal(V)	→	isNePal(activate(V))	(13)
isPal(n__nil)	→	tt	(14)
isQid(n__a)	→	tt	(15)
isQid(n__e)	→	tt	(16)
isQid(n__i)	→	tt	(17)
isQid(n__o)	→	tt	(18)
isQid(n__u)	→	tt	(19)
nil	→	n__nil	(20)
__(X1,X2)	→	n____(X1,X2)	(21)
isList(X)	→	n__isList(X)	(22)
isNeList(X)	→	n__isNeList(X)	(23)
isPal(X)	→	n__isPal(X)	(24)
a	→	n__a	(25)
e	→	n__e	(26)
i	→	n__i	(27)
o	→	n__o	(28)
u	→	n__u	(29)
activate(n__nil)	→	nil	(30)
activate(n____(X1,X2))	→	__(X1,X2)	(31)
activate(n__isList(X))	→	isList(X)	(32)
activate(n__isNeList(X))	→	isNeList(X)	(33)
activate(n__isPal(X))	→	isPal(X)	(34)
activate(n__a)	→	a	(35)
activate(n__e)	→	e	(36)
activate(n__i)	→	i	(37)
activate(n__o)	→	o	(38)
activate(n__u)	→	u	(39)
activate(X)	→	X	(40)

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^#	(41)
isList^#(V)	→	activate^#(V)	(42)
isNeList^#(V)	→	isQid^#(activate(V))	(43)
and^#(tt,X)	→	activate^#(X)	(44)
isNeList^#(n____(V1,V2))	→	and^#(isList(activate(V1)),n__isNeList(activate(V2)))	(45)
isNePal^#(n____(I,__(P,I)))	→	isQid^#(activate(I))	(46)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(47)
isNePal^#(V)	→	activate^#(V)	(48)
isNeList^#(n____(V1,V2))	→	isNeList^#(activate(V1))	(49)
isNeList^#(n____(V1,V2))	→	and^#(isNeList(activate(V1)),n__isList(activate(V2)))	(50)
isNePal^#(n____(I,__(P,I)))	→	and^#(isQid(activate(I)),n__isPal(activate(P)))	(51)
activate^#(n__isList(X))	→	isList^#(X)	(52)
isNePal^#(n____(I,__(P,I)))	→	activate^#(I)	(53)
activate^#(n__isPal(X))	→	isPal^#(X)	(54)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(55)
isList^#(n____(V1,V2))	→	activate^#(V1)	(56)
isNeList^#(n____(V1,V2))	→	isList^#(activate(V1))	(57)
isNePal^#(V)	→	isQid^#(activate(V))	(58)
__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(59)
isPal^#(V)	→	activate^#(V)	(60)
isList^#(V)	→	isNeList^#(activate(V))	(61)
activate^#(n__u)	→	u^#	(62)
__^#(__(X,Y),Z)	→	__^#(Y,Z)	(63)
activate^#(n__isNeList(X))	→	isNeList^#(X)	(64)
activate^#(n__nil)	→	nil^#	(65)
activate^#(n__e)	→	e^#	(66)
isList^#(n____(V1,V2))	→	isList^#(activate(V1))	(67)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(47)
isNeList^#(V)	→	activate^#(V)	(68)
isList^#(n____(V1,V2))	→	activate^#(V2)	(69)
activate^#(n____(X1,X2))	→	__^#(X1,X2)	(70)
activate^#(n__o)	→	o^#	(71)
activate^#(n__a)	→	a^#	(72)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(55)
isNePal^#(n____(I,__(P,I)))	→	activate^#(P)	(73)
isPal^#(V)	→	isNePal^#(activate(V))	(74)
isList^#(n____(V1,V2))	→	and^#(isList(activate(V1)),n__isList(activate(V2)))	(75)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

isList^#(n____(V1,V2))	→	and^#(isList(activate(V1)),n__isList(activate(V2)))	(75)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(55)
isPal^#(V)	→	isNePal^#(activate(V))	(74)
activate^#(n__isPal(X))	→	isPal^#(X)	(54)
isNePal^#(n____(I,__(P,I)))	→	activate^#(I)	(53)
isNePal^#(n____(I,__(P,I)))	→	activate^#(P)	(73)
activate^#(n__isList(X))	→	isList^#(X)	(52)
isNePal^#(n____(I,__(P,I)))	→	and^#(isQid(activate(I)),n__isPal(activate(P)))	(51)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(55)
isNeList^#(n____(V1,V2))	→	and^#(isNeList(activate(V1)),n__isList(activate(V2)))	(50)
isNeList^#(n____(V1,V2))	→	isNeList^#(activate(V1))	(49)
isList^#(n____(V1,V2))	→	activate^#(V2)	(69)
isNeList^#(V)	→	activate^#(V)	(68)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(47)
isNePal^#(V)	→	activate^#(V)	(48)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(47)
isList^#(n____(V1,V2))	→	isList^#(activate(V1))	(67)
isNeList^#(n____(V1,V2))	→	and^#(isList(activate(V1)),n__isNeList(activate(V2)))	(45)
activate^#(n__isNeList(X))	→	isNeList^#(X)	(64)
isList^#(V)	→	isNeList^#(activate(V))	(61)
isPal^#(V)	→	activate^#(V)	(60)
and^#(tt,X)	→	activate^#(X)	(44)
isNeList^#(n____(V1,V2))	→	isList^#(activate(V1))	(57)
isList^#(V)	→	activate^#(V)	(42)
isList^#(n____(V1,V2))	→	activate^#(V1)	(56)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	1
[n__a]	=	1
[n__o]	=	2438
[isNeList(x₁)]	=	x₁ + 2
[isPal(x₁)]	=	x₁ + 39471
[e^#]	=	0
[u]	=	10451
[activate(x₁)]	=	x₁ + 39468
[and(x₁, x₂)]	=	x₁ + x₂ + 39467
[u^#]	=	0
[n__isNeList(x₁)]	=	x₁ + 2
[isNeList^#(x₁)]	=	x₁ + 1
[activate^#(x₁)]	=	x₁ + 0
[n__i]	=	11798
[__^#(x₁, x₂)]	=	0
[n__isList(x₁)]	=	x₁ + 39471
[isNePal(x₁)]	=	x₁ + 1
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	x₁ + 39470
[n__isPal(x₁)]	=	x₁ + 39471
[n__nil]	=	14219
[isQid(x₁)]	=	1
[n____(x₁, x₂)]	=	x₁ + x₂ + 157874
[n__e]	=	32286
[o]	=	2438
[isList(x₁)]	=	x₁ + 39471
[isNePal^#(x₁)]	=	x₁ + 1
[nil]	=	14219
[n__u]	=	10451
[o^#]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	x₁ + 39470
[i]	=	11798
[e]	=	32286
[a^#]	=	0
[tt]	=	1
[and^#(x₁, x₂)]	=	x₂ + 78935
[__(x₁, x₂)]	=	x₁ + x₂ + 157874
[i^#]	=	0

together with the usable rules

isQid(n__o)	→	tt	(18)
and(tt,X)	→	activate(X)	(4)
isQid(n__a)	→	tt	(15)
isNeList(V)	→	isQid(activate(V))	(8)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
isQid(n__e)	→	tt	(16)
__(X1,X2)	→	n____(X1,X2)	(21)
activate(n__e)	→	e	(36)
e	→	n__e	(26)
isQid(n__u)	→	tt	(19)
activate(n__isList(X))	→	isList(X)	(32)
isQid(n__i)	→	tt	(17)
i	→	n__i	(27)
activate(n__isPal(X))	→	isPal(X)	(34)
isList(X)	→	n__isList(X)	(22)
o	→	n__o	(28)
isList(V)	→	isNeList(activate(V))	(5)
activate(n__isNeList(X))	→	isNeList(X)	(33)
isNeList(n____(V1,V2))	→	and(isNeList(activate(V1)),n__isList(activate(V2)))	(10)
activate(n__u)	→	u	(39)
isList(n____(V1,V2))	→	and(isList(activate(V1)),n__isList(activate(V2)))	(7)
nil	→	n__nil	(20)
a	→	n__a	(25)
activate(n__nil)	→	nil	(30)
isPal(n__nil)	→	tt	(14)
activate(n____(X1,X2))	→	__(X1,X2)	(31)
isNePal(n____(I,__(P,I)))	→	and(isQid(activate(I)),n__isPal(activate(P)))	(12)
isNeList(X)	→	n__isNeList(X)	(23)
isPal(X)	→	n__isPal(X)	(24)
isNePal(V)	→	isQid(activate(V))	(11)
isNeList(n____(V1,V2))	→	and(isList(activate(V1)),n__isNeList(activate(V2)))	(9)
isPal(V)	→	isNePal(activate(V))	(13)
activate(X)	→	X	(40)
isList(n__nil)	→	tt	(6)
activate(n__o)	→	o	(38)
activate(n__i)	→	i	(37)
activate(n__a)	→	a	(35)
u	→	n__u	(29)
__(X,nil)	→	X	(2)

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

isList^#(n____(V1,V2))	→	and^#(isList(activate(V1)),n__isList(activate(V2)))	(75)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(55)
isPal^#(V)	→	isNePal^#(activate(V))	(74)
activate^#(n__isPal(X))	→	isPal^#(X)	(54)
isNePal^#(n____(I,__(P,I)))	→	activate^#(I)	(53)
isNePal^#(n____(I,__(P,I)))	→	activate^#(P)	(73)
activate^#(n__isList(X))	→	isList^#(X)	(52)
isNePal^#(n____(I,__(P,I)))	→	and^#(isQid(activate(I)),n__isPal(activate(P)))	(51)
isNeList^#(n____(V1,V2))	→	activate^#(V1)	(55)
isNeList^#(n____(V1,V2))	→	and^#(isNeList(activate(V1)),n__isList(activate(V2)))	(50)
isNeList^#(n____(V1,V2))	→	isNeList^#(activate(V1))	(49)
isList^#(n____(V1,V2))	→	activate^#(V2)	(69)
isNeList^#(V)	→	activate^#(V)	(68)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(47)
isNePal^#(V)	→	activate^#(V)	(48)
isNeList^#(n____(V1,V2))	→	activate^#(V2)	(47)
isList^#(n____(V1,V2))	→	isList^#(activate(V1))	(67)
isNeList^#(n____(V1,V2))	→	and^#(isList(activate(V1)),n__isNeList(activate(V2)))	(45)
activate^#(n__isNeList(X))	→	isNeList^#(X)	(64)
isList^#(V)	→	isNeList^#(activate(V))	(61)
isPal^#(V)	→	activate^#(V)	(60)
and^#(tt,X)	→	activate^#(X)	(44)
isNeList^#(n____(V1,V2))	→	isList^#(activate(V1))	(57)
isList^#(V)	→	activate^#(V)	(42)
isList^#(n____(V1,V2))	→	activate^#(V1)	(56)

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

__^#(__(X,Y),Z)	→	__^#(Y,Z)	(63)
__^#(__(X,Y),Z)	→	__^#(X,__(Y,Z))	(59)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	2
[n__a]	=	1
[n__o]	=	42193
[isNeList(x₁)]	=	x₁ + 1
[isPal(x₁)]	=	x₁ + 39471
[e^#]	=	0
[u]	=	46572
[activate(x₁)]	=	x₁ + 1
[and(x₁, x₂)]	=	x₁ + x₂ + 0
[u^#]	=	0
[n__isNeList(x₁)]	=	x₁ + 1
[isNeList^#(x₁)]	=	x₁ + 1
[activate^#(x₁)]	=	0
[n__i]	=	15922
[__^#(x₁, x₂)]	=	x₁ + x₂ + 0
[n__isList(x₁)]	=	x₁ + 2
[isNePal(x₁)]	=	x₁ + 39467
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	39470
[n__isPal(x₁)]	=	x₁ + 39471
[n__nil]	=	1
[isQid(x₁)]	=	1
[n____(x₁, x₂)]	=	x₁ + x₂ + 4
[n__e]	=	47262
[o]	=	42194
[isList(x₁)]	=	x₁ + 2
[isNePal^#(x₁)]	=	x₁ + 1
[nil]	=	1
[n__u]	=	46572
[o^#]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	x₁ + 39470
[i]	=	15923
[e]	=	47263
[a^#]	=	0
[tt]	=	1
[and^#(x₁, x₂)]	=	x₂ + 78935
[__(x₁, x₂)]	=	x₁ + x₂ + 4
[i^#]	=	0

together with the usable rules

isQid(n__o)	→	tt	(18)
and(tt,X)	→	activate(X)	(4)
isQid(n__a)	→	tt	(15)
isNeList(V)	→	isQid(activate(V))	(8)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
isQid(n__e)	→	tt	(16)
__(X1,X2)	→	n____(X1,X2)	(21)
activate(n__e)	→	e	(36)
e	→	n__e	(26)
isQid(n__u)	→	tt	(19)
activate(n__isList(X))	→	isList(X)	(32)
isQid(n__i)	→	tt	(17)
i	→	n__i	(27)
activate(n__isPal(X))	→	isPal(X)	(34)
isList(X)	→	n__isList(X)	(22)
o	→	n__o	(28)
isList(V)	→	isNeList(activate(V))	(5)
activate(n__isNeList(X))	→	isNeList(X)	(33)
isNeList(n____(V1,V2))	→	and(isNeList(activate(V1)),n__isList(activate(V2)))	(10)
activate(n__u)	→	u	(39)
isList(n____(V1,V2))	→	and(isList(activate(V1)),n__isList(activate(V2)))	(7)
nil	→	n__nil	(20)
a	→	n__a	(25)
activate(n__nil)	→	nil	(30)
isPal(n__nil)	→	tt	(14)
activate(n____(X1,X2))	→	__(X1,X2)	(31)
isNePal(n____(I,__(P,I)))	→	and(isQid(activate(I)),n__isPal(activate(P)))	(12)
isNeList(X)	→	n__isNeList(X)	(23)
isPal(X)	→	n__isPal(X)	(24)
isNePal(V)	→	isQid(activate(V))	(11)
isNeList(n____(V1,V2))	→	and(isList(activate(V1)),n__isNeList(activate(V2)))	(9)
isPal(V)	→	isNePal(activate(V))	(13)
activate(X)	→	X	(40)
isList(n__nil)	→	tt	(6)
activate(n__o)	→	o	(38)
activate(n__i)	→	i	(37)
activate(n__a)	→	a	(35)
u	→	n__u	(29)
__(X,nil)	→	X	(2)

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

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

→

__^#(Y,Z)

(63)

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

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

→

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

(59)

1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	3
[n__a]	=	1
[n__o]	=	46441
[isNeList(x₁)]	=	x₁ + 1
[isPal(x₁)]	=	x₁ + 39471
[e^#]	=	0
[u]	=	46572
[activate(x₁)]	=	x₁ + 2
[and(x₁, x₂)]	=	x₁ + x₂ + 1
[u^#]	=	0
[n__isNeList(x₁)]	=	x₁ + 1
[isNeList^#(x₁)]	=	x₁ + 1
[activate^#(x₁)]	=	0
[n__i]	=	23612
[__^#(x₁, x₂)]	=	x₁ + 0
[n__isList(x₁)]	=	x₁ + 1
[isNePal(x₁)]	=	x₁ + 39466
[isQid^#(x₁)]	=	0
[isPal^#(x₁)]	=	39470
[n__isPal(x₁)]	=	x₁ + 39471
[n__nil]	=	1
[isQid(x₁)]	=	1
[n____(x₁, x₂)]	=	x₁ + x₂ + 8
[n__e]	=	47262
[o]	=	46443
[isList(x₁)]	=	x₁ + 3
[isNePal^#(x₁)]	=	x₁ + 1
[nil]	=	1
[n__u]	=	46572
[o^#]	=	0
[nil^#]	=	0
[isList^#(x₁)]	=	x₁ + 39470
[i]	=	23614
[e]	=	47264
[a^#]	=	0
[tt]	=	1
[and^#(x₁, x₂)]	=	x₂ + 78935
[__(x₁, x₂)]	=	x₁ + x₂ + 8
[i^#]	=	0

together with the usable rules

isQid(n__o)	→	tt	(18)
and(tt,X)	→	activate(X)	(4)
isQid(n__a)	→	tt	(15)
isNeList(V)	→	isQid(activate(V))	(8)
__(__(X,Y),Z)	→	__(X,__(Y,Z))	(1)
__(nil,X)	→	X	(3)
isQid(n__e)	→	tt	(16)
__(X1,X2)	→	n____(X1,X2)	(21)
activate(n__e)	→	e	(36)
e	→	n__e	(26)
isQid(n__u)	→	tt	(19)
activate(n__isList(X))	→	isList(X)	(32)
isQid(n__i)	→	tt	(17)
i	→	n__i	(27)
activate(n__isPal(X))	→	isPal(X)	(34)
isList(X)	→	n__isList(X)	(22)
o	→	n__o	(28)
isList(V)	→	isNeList(activate(V))	(5)
activate(n__isNeList(X))	→	isNeList(X)	(33)
isNeList(n____(V1,V2))	→	and(isNeList(activate(V1)),n__isList(activate(V2)))	(10)
activate(n__u)	→	u	(39)
isList(n____(V1,V2))	→	and(isList(activate(V1)),n__isList(activate(V2)))	(7)
nil	→	n__nil	(20)
a	→	n__a	(25)
activate(n__nil)	→	nil	(30)
isPal(n__nil)	→	tt	(14)
activate(n____(X1,X2))	→	__(X1,X2)	(31)
isNePal(n____(I,__(P,I)))	→	and(isQid(activate(I)),n__isPal(activate(P)))	(12)
isNeList(X)	→	n__isNeList(X)	(23)
isPal(X)	→	n__isPal(X)	(24)
isNePal(V)	→	isQid(activate(V))	(11)
isNeList(n____(V1,V2))	→	and(isList(activate(V1)),n__isNeList(activate(V2)))	(9)
isPal(V)	→	isNePal(activate(V))	(13)
activate(X)	→	X	(40)
isList(n__nil)	→	tt	(6)
activate(n__o)	→	o	(38)
activate(n__i)	→	i	(37)
activate(n__a)	→	a	(35)
u	→	n__u	(29)
__(X,nil)	→	X	(2)

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

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

→

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

(59)

could be deleted.

1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.