Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex1_GL02a_GM)

The rewrite relation of the following TRS is considered.

a__eq(0,0)	→	true	(1)
a__eq(s(X),s(Y))	→	a__eq(X,Y)	(2)
a__eq(X,Y)	→	false	(3)
a__inf(X)	→	cons(X,inf(s(X)))	(4)
a__take(0,X)	→	nil	(5)
a__take(s(X),cons(Y,L))	→	cons(Y,take(X,L))	(6)
a__length(nil)	→	0	(7)
a__length(cons(X,L))	→	s(length(L))	(8)
mark(eq(X1,X2))	→	a__eq(X1,X2)	(9)
mark(inf(X))	→	a__inf(mark(X))	(10)
mark(take(X1,X2))	→	a__take(mark(X1),mark(X2))	(11)
mark(length(X))	→	a__length(mark(X))	(12)
mark(0)	→	0	(13)
mark(true)	→	true	(14)
mark(s(X))	→	s(X)	(15)
mark(false)	→	false	(16)
mark(cons(X1,X2))	→	cons(X1,X2)	(17)
mark(nil)	→	nil	(18)
a__eq(X1,X2)	→	eq(X1,X2)	(19)
a__inf(X)	→	inf(X)	(20)
a__take(X1,X2)	→	take(X1,X2)	(21)
a__length(X)	→	length(X)	(22)

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^#(eq(X1,X2))	→	a__eq^#(X1,X2)	(23)
mark^#(length(X))	→	a__length^#(mark(X))	(24)
mark^#(inf(X))	→	mark^#(X)	(25)
mark^#(length(X))	→	mark^#(X)	(26)
mark^#(take(X1,X2))	→	mark^#(X2)	(27)
mark^#(take(X1,X2))	→	a__take^#(mark(X1),mark(X2))	(28)
a__eq^#(s(X),s(Y))	→	a__eq^#(X,Y)	(29)
mark^#(inf(X))	→	a__inf^#(mark(X))	(30)
mark^#(take(X1,X2))	→	mark^#(X1)	(31)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

mark^#(take(X1,X2))	→	mark^#(X1)	(31)
mark^#(inf(X))	→	mark^#(X)	(25)
mark^#(take(X1,X2))	→	mark^#(X2)	(27)
mark^#(length(X))	→	mark^#(X)	(26)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	0
[take(x₁, x₂)]	=	x₁ + x₂ + 1
[eq(x₁, x₂)]	=	0
[false]	=	0
[a__inf(x₁)]	=	0
[a__eq(x₁, x₂)]	=	0
[a__take^#(x₁, x₂)]	=	0
[true]	=	0
[mark^#(x₁)]	=	x₁ + 0
[0]	=	0
[nil]	=	0
[mark(x₁)]	=	0
[a__inf^#(x₁)]	=	0
[a__length^#(x₁)]	=	0
[cons(x₁, x₂)]	=	0
[a__take(x₁, x₂)]	=	0
[length(x₁)]	=	x₁ + 1
[inf(x₁)]	=	x₁ + 1
[a__length(x₁)]	=	0
[a__eq^#(x₁, x₂)]	=	0

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

mark^#(take(X1,X2))	→	mark^#(X1)	(31)
mark^#(inf(X))	→	mark^#(X)	(25)
mark^#(take(X1,X2))	→	mark^#(X2)	(27)
mark^#(length(X))	→	mark^#(X)	(26)

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

a__eq^#(s(X),s(Y))

→

a__eq^#(X,Y)

(29)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[take(x₁, x₂)]	=	1
[eq(x₁, x₂)]	=	0
[false]	=	0
[a__inf(x₁)]	=	0
[a__eq(x₁, x₂)]	=	0
[a__take^#(x₁, x₂)]	=	0
[true]	=	0
[mark^#(x₁)]	=	0
[0]	=	0
[nil]	=	0
[mark(x₁)]	=	0
[a__inf^#(x₁)]	=	0
[a__length^#(x₁)]	=	0
[cons(x₁, x₂)]	=	0
[a__take(x₁, x₂)]	=	0
[length(x₁)]	=	1
[inf(x₁)]	=	1
[a__length(x₁)]	=	0
[a__eq^#(x₁, x₂)]	=	x₁ + 0

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

a__eq^#(s(X),s(Y))

→

a__eq^#(X,Y)

(29)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.