Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_Zan97_GM)

The rewrite relation of the following TRS is considered.

a__f(X)	→	cons(mark(X),f(g(X)))	(1)
a__g(0)	→	s(0)	(2)
a__g(s(X))	→	s(s(a__g(mark(X))))	(3)
a__sel(0,cons(X,Y))	→	mark(X)	(4)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(5)
mark(f(X))	→	a__f(mark(X))	(6)
mark(g(X))	→	a__g(mark(X))	(7)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(8)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(9)
mark(0)	→	0	(10)
mark(s(X))	→	s(mark(X))	(11)
a__f(X)	→	f(X)	(12)
a__g(X)	→	g(X)	(13)
a__sel(X1,X2)	→	sel(X1,X2)	(14)

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^#(f(X))	→	mark^#(X)	(15)
mark^#(g(X))	→	mark^#(X)	(16)
mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(17)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(Z)	(18)
a__g^#(s(X))	→	mark^#(X)	(19)
a__f^#(X)	→	mark^#(X)	(20)
mark^#(sel(X1,X2))	→	mark^#(X1)	(21)
a__sel^#(s(X),cons(Y,Z))	→	a__sel^#(mark(X),mark(Z))	(22)
a__g^#(s(X))	→	a__g^#(mark(X))	(23)
mark^#(g(X))	→	a__g^#(mark(X))	(24)
a__sel^#(0,cons(X,Y))	→	mark^#(X)	(25)
mark^#(cons(X1,X2))	→	mark^#(X1)	(26)
mark^#(f(X))	→	a__f^#(mark(X))	(27)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(X)	(28)
mark^#(sel(X1,X2))	→	mark^#(X2)	(29)
mark^#(s(X))	→	mark^#(X)	(30)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(s(X))	→	mark^#(X)	(30)
mark^#(sel(X1,X2))	→	mark^#(X1)	(21)
mark^#(sel(X1,X2))	→	mark^#(X2)	(29)
a__f^#(X)	→	mark^#(X)	(20)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(X)	(28)
mark^#(f(X))	→	a__f^#(mark(X))	(27)
mark^#(cons(X1,X2))	→	mark^#(X1)	(26)
a__sel^#(0,cons(X,Y))	→	mark^#(X)	(25)
a__g^#(s(X))	→	mark^#(X)	(19)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(Z)	(18)
mark^#(sel(X1,X2))	→	a__sel^#(mark(X1),mark(X2))	(17)
mark^#(g(X))	→	mark^#(X)	(16)
mark^#(g(X))	→	a__g^#(mark(X))	(24)
a__g^#(s(X))	→	a__g^#(mark(X))	(23)
a__sel^#(s(X),cons(Y,Z))	→	a__sel^#(mark(X),mark(Z))	(22)
mark^#(f(X))	→	mark^#(X)	(15)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__g(x₁)]	=	x₁ + 0
[s(x₁)]	=	x₁ + 0
[a__g^#(x₁)]	=	x₁ + 0
[a__f(x₁)]	=	x₁ + 2242
[f(x₁)]	=	x₁ + 2242
[mark^#(x₁)]	=	x₁ + 0
[0]	=	1
[sel(x₁, x₂)]	=	max(x₁ + 28884, x₂ + 1, 0)
[a__sel^#(x₁, x₂)]	=	max(x₁ + 28884, x₂ + 1, 0)
[mark(x₁)]	=	x₁ + 0
[a__sel(x₁, x₂)]	=	max(x₁ + 28884, x₂ + 1, 0)
[cons(x₁, x₂)]	=	max(x₁ + 1, x₂ + 0, 0)
[a__f^#(x₁)]	=	x₁ + 2241
[g(x₁)]	=	x₁ + 0

together with the usable rules

a__sel(0,cons(X,Y))	→	mark(X)	(4)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(8)
a__f(X)	→	cons(mark(X),f(g(X)))	(1)
a__g(s(X))	→	s(s(a__g(mark(X))))	(3)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(5)
mark(0)	→	0	(10)
mark(g(X))	→	a__g(mark(X))	(7)
a__sel(X1,X2)	→	sel(X1,X2)	(14)
a__f(X)	→	f(X)	(12)
mark(s(X))	→	s(mark(X))	(11)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(9)
a__g(X)	→	g(X)	(13)
mark(f(X))	→	a__f(mark(X))	(6)
a__g(0)	→	s(0)	(2)

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

mark^#(sel(X1,X2))	→	mark^#(X1)	(21)
mark^#(sel(X1,X2))	→	mark^#(X2)	(29)
a__f^#(X)	→	mark^#(X)	(20)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(X)	(28)
mark^#(f(X))	→	a__f^#(mark(X))	(27)
mark^#(cons(X1,X2))	→	mark^#(X1)	(26)
a__sel^#(0,cons(X,Y))	→	mark^#(X)	(25)
a__sel^#(s(X),cons(Y,Z))	→	mark^#(Z)	(18)
mark^#(f(X))	→	mark^#(X)	(15)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

a__g^#(s(X))	→	mark^#(X)	(19)
a__g^#(s(X))	→	a__g^#(mark(X))	(23)
mark^#(g(X))	→	mark^#(X)	(16)
mark^#(g(X))	→	a__g^#(mark(X))	(24)
mark^#(s(X))	→	mark^#(X)	(30)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__g^#)	=	1
π(mark^#)	=	1
π(mark)	=	1

in combination with the following Weighted Path Order with the following precedence and status

prec(a__g)	=	4	status(a__g)	=	[1]	list-extension(a__g)	=	Lex
prec(s)	=	3	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__f)	=	0	status(a__f)	=	[]	list-extension(a__f)	=	Lex
prec(f)	=	0	status(f)	=	[]	list-extension(f)	=	Lex
prec(0)	=	2	status(0)	=	[]	list-extension(0)	=	Lex
prec(sel)	=	0	status(sel)	=	[]	list-extension(sel)	=	Lex
prec(a__sel^#)	=	0	status(a__sel^#)	=	[1]	list-extension(a__sel^#)	=	Lex
prec(a__sel)	=	0	status(a__sel)	=	[]	list-extension(a__sel)	=	Lex
prec(cons)	=	0	status(cons)	=	[]	list-extension(cons)	=	Lex
prec(a__f^#)	=	0	status(a__f^#)	=	[]	list-extension(a__f^#)	=	Lex
prec(g)	=	4	status(g)	=	[1]	list-extension(g)	=	Lex

and the following Max-polynomial interpretation

[a__g(x₁)]	=	x₁ + 0
[s(x₁)]	=	x₁ + 0
[a__f(x₁)]	=	x₁ + 1
[f(x₁)]	=	x₁ + 1
[0]	=	0
[sel(x₁, x₂)]	=	x₁ + x₂ + 18588
[a__sel^#(x₁, x₂)]	=	x₁ + 1
[a__sel(x₁, x₂)]	=	x₁ + x₂ + 18588
[cons(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__f^#(x₁)]	=	1
[g(x₁)]	=	x₁ + 0

together with the usable rules

a__sel(0,cons(X,Y))	→	mark(X)	(4)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(8)
a__f(X)	→	cons(mark(X),f(g(X)))	(1)
a__g(s(X))	→	s(s(a__g(mark(X))))	(3)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(5)
mark(0)	→	0	(10)
mark(g(X))	→	a__g(mark(X))	(7)
a__sel(X1,X2)	→	sel(X1,X2)	(14)
a__f(X)	→	f(X)	(12)
mark(s(X))	→	s(mark(X))	(11)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(9)
a__g(X)	→	g(X)	(13)
mark(f(X))	→	a__f(mark(X))	(6)
a__g(0)	→	s(0)	(2)

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

a__g^#(s(X))	→	mark^#(X)	(19)
a__g^#(s(X))	→	a__g^#(mark(X))	(23)
mark^#(g(X))	→	mark^#(X)	(16)
mark^#(g(X))	→	a__g^#(mark(X))	(24)
mark^#(s(X))	→	mark^#(X)	(30)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

a__sel^#(s(X),cons(Y,Z))

→

a__sel^#(mark(X),mark(Z))

(22)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__g^#)	=	1
π(mark^#)	=	1
π(a__sel^#)	=	1
π(mark)	=	1

in combination with the following Weighted Path Order with the following precedence and status

prec(a__g)	=	4	status(a__g)	=	[1]	list-extension(a__g)	=	Lex
prec(s)	=	3	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__f)	=	2	status(a__f)	=	[]	list-extension(a__f)	=	Lex
prec(f)	=	2	status(f)	=	[]	list-extension(f)	=	Lex
prec(0)	=	2	status(0)	=	[]	list-extension(0)	=	Lex
prec(sel)	=	2	status(sel)	=	[]	list-extension(sel)	=	Lex
prec(a__sel)	=	2	status(a__sel)	=	[]	list-extension(a__sel)	=	Lex
prec(cons)	=	2	status(cons)	=	[]	list-extension(cons)	=	Lex
prec(a__f^#)	=	0	status(a__f^#)	=	[]	list-extension(a__f^#)	=	Lex
prec(g)	=	4	status(g)	=	[1]	list-extension(g)	=	Lex

and the following Max-polynomial interpretation

[a__g(x₁)]	=	x₁ + 0
[s(x₁)]	=	x₁ + 0
[a__f(x₁)]	=	x₁ + 25075
[f(x₁)]	=	x₁ + 25075
[0]	=	0
[sel(x₁, x₂)]	=	x₂ + 36069
[a__sel(x₁, x₂)]	=	x₂ + 36069
[cons(x₁, x₂)]	=	max(x₁ + 12457, x₂ + 0, 0)
[a__f^#(x₁)]	=	1
[g(x₁)]	=	x₁ + 0

together with the usable rules

a__sel(0,cons(X,Y))	→	mark(X)	(4)
mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(8)
a__f(X)	→	cons(mark(X),f(g(X)))	(1)
a__g(s(X))	→	s(s(a__g(mark(X))))	(3)
a__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(5)
mark(0)	→	0	(10)
mark(g(X))	→	a__g(mark(X))	(7)
a__sel(X1,X2)	→	sel(X1,X2)	(14)
a__f(X)	→	f(X)	(12)
mark(s(X))	→	s(mark(X))	(11)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(9)
a__g(X)	→	g(X)	(13)
mark(f(X))	→	a__f(mark(X))	(6)
a__g(0)	→	s(0)	(2)

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

a__sel^#(s(X),cons(Y,Z))

→

a__sel^#(mark(X),mark(Z))

(22)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.