Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PALINDROME_nosorts_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)	→	a__U12(tt)	(4)
a__U12(tt)	→	tt	(5)
a__isNePal(__(I,__(P,I)))	→	a__U11(tt)	(6)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(7)
mark(U11(X))	→	a__U11(mark(X))	(8)
mark(U12(X))	→	a__U12(mark(X))	(9)
mark(isNePal(X))	→	a__isNePal(mark(X))	(10)
mark(nil)	→	nil	(11)
mark(tt)	→	tt	(12)
a____(X1,X2)	→	__(X1,X2)	(13)
a__U11(X)	→	U11(X)	(14)
a__U12(X)	→	U12(X)	(15)
a__isNePal(X)	→	isNePal(X)	(16)

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.

a__isNePal^#(__(I,__(P,I)))	→	a__U11^#(tt)	(17)
mark^#(__(X1,X2))	→	mark^#(X1)	(18)
a____^#(__(X,Y),Z)	→	a____^#(mark(Y),mark(Z))	(19)
mark^#(isNePal(X))	→	mark^#(X)	(20)
a____^#(__(X,Y),Z)	→	a____^#(mark(X),a____(mark(Y),mark(Z)))	(21)
a____^#(__(X,Y),Z)	→	mark^#(X)	(22)
a____^#(__(X,Y),Z)	→	mark^#(Y)	(23)
mark^#(__(X1,X2))	→	mark^#(X2)	(24)
a____^#(nil,X)	→	mark^#(X)	(25)
mark^#(__(X1,X2))	→	a____^#(mark(X1),mark(X2))	(26)
mark^#(U11(X))	→	mark^#(X)	(27)
mark^#(U11(X))	→	a__U11^#(mark(X))	(28)
mark^#(U12(X))	→	a__U12^#(mark(X))	(29)
a____^#(X,nil)	→	mark^#(X)	(30)
mark^#(isNePal(X))	→	a__isNePal^#(mark(X))	(31)
a____^#(__(X,Y),Z)	→	mark^#(Z)	(32)
mark^#(U12(X))	→	mark^#(X)	(33)
a__U11^#(tt)	→	a__U12^#(tt)	(34)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(U12(X))	→	mark^#(X)	(33)
a____^#(__(X,Y),Z)	→	mark^#(Y)	(23)
a____^#(__(X,Y),Z)	→	mark^#(Z)	(32)
a____^#(__(X,Y),Z)	→	mark^#(X)	(22)
a____^#(X,nil)	→	mark^#(X)	(30)
a____^#(__(X,Y),Z)	→	a____^#(mark(X),a____(mark(Y),mark(Z)))	(21)
mark^#(isNePal(X))	→	mark^#(X)	(20)
a____^#(__(X,Y),Z)	→	a____^#(mark(Y),mark(Z))	(19)
mark^#(U11(X))	→	mark^#(X)	(27)
mark^#(__(X1,X2))	→	mark^#(X1)	(18)
mark^#(__(X1,X2))	→	a____^#(mark(X1),mark(X2))	(26)
a____^#(nil,X)	→	mark^#(X)	(25)
mark^#(__(X1,X2))	→	mark^#(X2)	(24)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__isNePal^#(x₁)]	=	0
[U11(x₁)]	=	x₁ + 9709
[isNePal(x₁)]	=	x₁ + 10451
[a__U11^#(x₁)]	=	0
[a____^#(x₁, x₂)]	=	x₁ + x₂ + 0
[U12(x₁)]	=	x₁ + 9709
[a____(x₁, x₂)]	=	x₁ + x₂ + 1
[a__U12^#(x₁)]	=	0
[mark^#(x₁)]	=	x₁ + 0
[nil]	=	5854
[mark(x₁)]	=	x₁ + 0
[a__isNePal(x₁)]	=	x₁ + 10451
[a__U11(x₁)]	=	x₁ + 9709
[a__U12(x₁)]	=	x₁ + 9709
[tt]	=	399
[__(x₁, x₂)]	=	x₁ + x₂ + 1

together with the usable rules

a__U11(tt)	→	a__U12(tt)	(4)
a__U12(X)	→	U12(X)	(15)
mark(U11(X))	→	a__U11(mark(X))	(8)
a____(__(X,Y),Z)	→	a____(mark(X),a____(mark(Y),mark(Z)))	(1)
a____(nil,X)	→	mark(X)	(3)
a__isNePal(X)	→	isNePal(X)	(16)
a__U12(tt)	→	tt	(5)
mark(isNePal(X))	→	a__isNePal(mark(X))	(10)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(7)
a__U11(X)	→	U11(X)	(14)
mark(tt)	→	tt	(12)
mark(nil)	→	nil	(11)
mark(U12(X))	→	a__U12(mark(X))	(9)
a____(X1,X2)	→	__(X1,X2)	(13)
a__isNePal(__(I,__(P,I)))	→	a__U11(tt)	(6)
a____(X,nil)	→	mark(X)	(2)

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

mark^#(U12(X))	→	mark^#(X)	(33)
a____^#(__(X,Y),Z)	→	mark^#(Y)	(23)
a____^#(__(X,Y),Z)	→	mark^#(Z)	(32)
a____^#(__(X,Y),Z)	→	mark^#(X)	(22)
a____^#(X,nil)	→	mark^#(X)	(30)
mark^#(isNePal(X))	→	mark^#(X)	(20)
a____^#(__(X,Y),Z)	→	a____^#(mark(Y),mark(Z))	(19)
mark^#(U11(X))	→	mark^#(X)	(27)
mark^#(__(X1,X2))	→	mark^#(X1)	(18)
mark^#(__(X1,X2))	→	a____^#(mark(X1),mark(X2))	(26)
a____^#(nil,X)	→	mark^#(X)	(25)
mark^#(__(X1,X2))	→	mark^#(X2)	(24)

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)))

(21)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__isNePal^#(x₁)]	=	0
[U11(x₁)]	=	x₁ + 9709
[isNePal(x₁)]	=	x₁ + 29372
[a__U11^#(x₁)]	=	0
[a____^#(x₁, x₂)]	=	x₁ + 0
[U12(x₁)]	=	x₁ + 9709
[a____(x₁, x₂)]	=	x₁ + x₂ + 10225
[a__U12^#(x₁)]	=	0
[mark^#(x₁)]	=	0
[nil]	=	1
[mark(x₁)]	=	x₁ + 0
[a__isNePal(x₁)]	=	x₁ + 29372
[a__U11(x₁)]	=	x₁ + 9709
[a__U12(x₁)]	=	x₁ + 9709
[tt]	=	40113
[__(x₁, x₂)]	=	x₁ + x₂ + 10225

together with the usable rules

a__U11(tt)	→	a__U12(tt)	(4)
a__U12(X)	→	U12(X)	(15)
mark(U11(X))	→	a__U11(mark(X))	(8)
a____(__(X,Y),Z)	→	a____(mark(X),a____(mark(Y),mark(Z)))	(1)
a____(nil,X)	→	mark(X)	(3)
a__isNePal(X)	→	isNePal(X)	(16)
a__U12(tt)	→	tt	(5)
mark(isNePal(X))	→	a__isNePal(mark(X))	(10)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(7)
a__U11(X)	→	U11(X)	(14)
mark(tt)	→	tt	(12)
mark(nil)	→	nil	(11)
mark(U12(X))	→	a__U12(mark(X))	(9)
a____(X1,X2)	→	__(X1,X2)	(13)
a__isNePal(__(I,__(P,I)))	→	a__U11(tt)	(6)
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)))

(21)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.