Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex15_Luc98_GM)

The rewrite relation of the following TRS is considered.

a__and(true,X)	→	mark(X)	(1)
a__and(false,Y)	→	false	(2)
a__if(true,X,Y)	→	mark(X)	(3)
a__if(false,X,Y)	→	mark(Y)	(4)
a__add(0,X)	→	mark(X)	(5)
a__add(s(X),Y)	→	s(add(X,Y))	(6)
a__first(0,X)	→	nil	(7)
a__first(s(X),cons(Y,Z))	→	cons(Y,first(X,Z))	(8)
a__from(X)	→	cons(X,from(s(X)))	(9)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(10)
mark(if(X1,X2,X3))	→	a__if(mark(X1),X2,X3)	(11)
mark(add(X1,X2))	→	a__add(mark(X1),X2)	(12)
mark(first(X1,X2))	→	a__first(mark(X1),mark(X2))	(13)
mark(from(X))	→	a__from(X)	(14)
mark(true)	→	true	(15)
mark(false)	→	false	(16)
mark(0)	→	0	(17)
mark(s(X))	→	s(X)	(18)
mark(nil)	→	nil	(19)
mark(cons(X1,X2))	→	cons(X1,X2)	(20)
a__and(X1,X2)	→	and(X1,X2)	(21)
a__if(X1,X2,X3)	→	if(X1,X2,X3)	(22)
a__add(X1,X2)	→	add(X1,X2)	(23)
a__first(X1,X2)	→	first(X1,X2)	(24)
a__from(X)	→	from(X)	(25)

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^#(first(X1,X2))	→	mark^#(X1)	(26)
mark^#(add(X1,X2))	→	a__add^#(mark(X1),X2)	(27)
a__and^#(true,X)	→	mark^#(X)	(28)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(29)
a__add^#(0,X)	→	mark^#(X)	(30)
a__if^#(true,X,Y)	→	mark^#(X)	(31)
a__if^#(false,X,Y)	→	mark^#(Y)	(32)
mark^#(add(X1,X2))	→	mark^#(X1)	(33)
mark^#(and(X1,X2))	→	mark^#(X1)	(34)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(35)
mark^#(first(X1,X2))	→	mark^#(X2)	(36)
mark^#(first(X1,X2))	→	a__first^#(mark(X1),mark(X2))	(37)
mark^#(from(X))	→	a__from^#(X)	(38)
mark^#(if(X1,X2,X3))	→	a__if^#(mark(X1),X2,X3)	(39)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(if(X1,X2,X3))	→	a__if^#(mark(X1),X2,X3)	(39)
a__if^#(false,X,Y)	→	mark^#(Y)	(32)
a__if^#(true,X,Y)	→	mark^#(X)	(31)
mark^#(first(X1,X2))	→	mark^#(X2)	(36)
a__add^#(0,X)	→	mark^#(X)	(30)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(29)
a__and^#(true,X)	→	mark^#(X)	(28)
mark^#(add(X1,X2))	→	a__add^#(mark(X1),X2)	(27)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(35)
mark^#(and(X1,X2))	→	mark^#(X1)	(34)
mark^#(add(X1,X2))	→	mark^#(X1)	(33)
mark^#(first(X1,X2))	→	mark^#(X1)	(26)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 28107
[a__from^#(x₁)]	=	0
[a__first^#(x₁, x₂)]	=	0
[a__from(x₁)]	=	x₁ + 51641
[and(x₁, x₂)]	=	x₁ + x₂ + 2
[a__add^#(x₁, x₂)]	=	x₂ + 1
[a__add(x₁, x₂)]	=	54393
[false]	=	1
[true]	=	28101
[mark^#(x₁)]	=	x₁ + 0
[0]	=	1143
[if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 283
[a__and^#(x₁, x₂)]	=	x₂ + 1
[from(x₁)]	=	x₁ + 23535
[nil]	=	24867
[mark(x₁)]	=	x₁ + 28105
[first(x₁, x₂)]	=	x₁ + x₂ + 8946
[a__first(x₁, x₂)]	=	8945
[a__if^#(x₁, x₂, x₃)]	=	x₂ + x₃ + 282
[cons(x₁, x₂)]	=	x₁ + 51642
[add(x₁, x₂)]	=	x₁ + x₂ + 26287
[a__and(x₁, x₂)]	=	x₁ + 3
[a__if(x₁, x₂, x₃)]	=	28389

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

mark^#(if(X1,X2,X3))	→	a__if^#(mark(X1),X2,X3)	(39)
a__if^#(false,X,Y)	→	mark^#(Y)	(32)
a__if^#(true,X,Y)	→	mark^#(X)	(31)
mark^#(first(X1,X2))	→	mark^#(X2)	(36)
a__add^#(0,X)	→	mark^#(X)	(30)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(29)
a__and^#(true,X)	→	mark^#(X)	(28)
mark^#(add(X1,X2))	→	a__add^#(mark(X1),X2)	(27)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(35)
mark^#(and(X1,X2))	→	mark^#(X1)	(34)
mark^#(add(X1,X2))	→	mark^#(X1)	(33)
mark^#(first(X1,X2))	→	mark^#(X1)	(26)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.