Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PALINDROME_nosorts_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__and(tt,X)	→	mark(X)	(4)
a__isNePal(__(I,__(P,I)))	→	tt	(5)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(6)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(7)
mark(isNePal(X))	→	a__isNePal(mark(X))	(8)
mark(nil)	→	nil	(9)
mark(tt)	→	tt	(10)
a____(X1,X2)	→	__(X1,X2)	(11)
a__and(X1,X2)	→	and(X1,X2)	(12)
a__isNePal(X)	→	isNePal(X)	(13)

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^#(__(X1,X2))	→	a____^#(mark(X1),mark(X2))	(14)
mark^#(and(X1,X2))	→	mark^#(X1)	(15)
mark^#(isNePal(X))	→	a__isNePal^#(mark(X))	(16)
a____^#(__(X,Y),Z)	→	mark^#(X)	(17)
a____^#(__(X,Y),Z)	→	mark^#(Y)	(18)
a____^#(__(X,Y),Z)	→	mark^#(Z)	(19)
mark^#(isNePal(X))	→	mark^#(X)	(20)
a____^#(nil,X)	→	mark^#(X)	(21)
a____^#(__(X,Y),Z)	→	a____^#(mark(Y),mark(Z))	(22)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(23)
mark^#(__(X1,X2))	→	mark^#(X1)	(24)
a____^#(X,nil)	→	mark^#(X)	(25)
a____^#(__(X,Y),Z)	→	a____^#(mark(X),a____(mark(Y),mark(Z)))	(26)
a__and^#(tt,X)	→	mark^#(X)	(27)
mark^#(__(X1,X2))	→	mark^#(X2)	(28)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(__(X1,X2))	→	mark^#(X2)	(28)
mark^#(isNePal(X))	→	mark^#(X)	(20)
a__and^#(tt,X)	→	mark^#(X)	(27)
a____^#(__(X,Y),Z)	→	mark^#(Z)	(19)
a____^#(__(X,Y),Z)	→	a____^#(mark(X),a____(mark(Y),mark(Z)))	(26)
a____^#(X,nil)	→	mark^#(X)	(25)
mark^#(__(X1,X2))	→	mark^#(X1)	(24)
a____^#(__(X,Y),Z)	→	mark^#(Y)	(18)
a____^#(__(X,Y),Z)	→	mark^#(X)	(17)
mark^#(and(X1,X2))	→	mark^#(X1)	(15)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(23)
a____^#(__(X,Y),Z)	→	a____^#(mark(Y),mark(Z))	(22)
a____^#(nil,X)	→	mark^#(X)	(21)
mark^#(__(X1,X2))	→	a____^#(mark(X1),mark(X2))	(14)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__isNePal^#(x₁)]	=	0
[and(x₁, x₂)]	=	x₁ + x₂ + 1
[isNePal(x₁)]	=	x₁ + 2438
[a____^#(x₁, x₂)]	=	x₁ + x₂ + 0
[a____(x₁, x₂)]	=	x₁ + x₂ + 1
[mark^#(x₁)]	=	x₁ + 0
[a__and^#(x₁, x₂)]	=	x₁ + x₂ + 0
[nil]	=	8856
[mark(x₁)]	=	x₁ + 0
[a__isNePal(x₁)]	=	x₁ + 2438
[tt]	=	1
[a__and(x₁, x₂)]	=	x₁ + x₂ + 1
[__(x₁, x₂)]	=	x₁ + x₂ + 1

together with the usable rules

a__and(tt,X)	→	mark(X)	(4)
mark(isNePal(X))	→	a__isNePal(mark(X))	(8)
a____(__(X,Y),Z)	→	a____(mark(X),a____(mark(Y),mark(Z)))	(1)
a____(nil,X)	→	mark(X)	(3)
a__isNePal(__(I,__(P,I)))	→	tt	(5)
mark(tt)	→	tt	(10)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(7)
a__and(X1,X2)	→	and(X1,X2)	(12)
a____(X1,X2)	→	__(X1,X2)	(11)
mark(nil)	→	nil	(9)
a__isNePal(X)	→	isNePal(X)	(13)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(6)
a____(X,nil)	→	mark(X)	(2)

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

mark^#(__(X1,X2))	→	mark^#(X2)	(28)
mark^#(isNePal(X))	→	mark^#(X)	(20)
a__and^#(tt,X)	→	mark^#(X)	(27)
a____^#(__(X,Y),Z)	→	mark^#(Z)	(19)
a____^#(X,nil)	→	mark^#(X)	(25)
mark^#(__(X1,X2))	→	mark^#(X1)	(24)
a____^#(__(X,Y),Z)	→	mark^#(Y)	(18)
a____^#(__(X,Y),Z)	→	mark^#(X)	(17)
mark^#(and(X1,X2))	→	mark^#(X1)	(15)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(23)
a____^#(__(X,Y),Z)	→	a____^#(mark(Y),mark(Z))	(22)
a____^#(nil,X)	→	mark^#(X)	(21)
mark^#(__(X1,X2))	→	a____^#(mark(X1),mark(X2))	(14)

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)))

(26)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__isNePal^#(x₁)]	=	0
[and(x₁, x₂)]	=	x₁ + x₂ + 1
[isNePal(x₁)]	=	x₁ + 2438
[a____^#(x₁, x₂)]	=	x₁ + 0
[a____(x₁, x₂)]	=	x₁ + x₂ + 1
[mark^#(x₁)]	=	0
[a__and^#(x₁, x₂)]	=	0
[nil]	=	50349
[mark(x₁)]	=	x₁ + 0
[a__isNePal(x₁)]	=	x₁ + 2438
[tt]	=	1507
[a__and(x₁, x₂)]	=	x₁ + x₂ + 1
[__(x₁, x₂)]	=	x₁ + x₂ + 1

together with the usable rules

a__and(tt,X)	→	mark(X)	(4)
mark(isNePal(X))	→	a__isNePal(mark(X))	(8)
a____(__(X,Y),Z)	→	a____(mark(X),a____(mark(Y),mark(Z)))	(1)
a____(nil,X)	→	mark(X)	(3)
a__isNePal(__(I,__(P,I)))	→	tt	(5)
mark(tt)	→	tt	(10)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(7)
a__and(X1,X2)	→	and(X1,X2)	(12)
a____(X1,X2)	→	__(X1,X2)	(11)
mark(nil)	→	nil	(9)
a__isNePal(X)	→	isNePal(X)	(13)
mark(__(X1,X2))	→	a____(mark(X1),mark(X2))	(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)))

(26)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.