Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/MYNAT_nosorts_GM)

The rewrite relation of the following TRS is considered.

a__and(tt,X)	→	mark(X)	(1)
a__plus(N,0)	→	mark(N)	(2)
a__plus(N,s(M))	→	s(a__plus(mark(N),mark(M)))	(3)
a__x(N,0)	→	0	(4)
a__x(N,s(M))	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(5)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(6)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(7)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(8)
mark(tt)	→	tt	(9)
mark(0)	→	0	(10)
mark(s(X))	→	s(mark(X))	(11)
a__and(X1,X2)	→	and(X1,X2)	(12)
a__plus(X1,X2)	→	plus(X1,X2)	(13)
a__x(X1,X2)	→	x(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^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(15)
mark^#(plus(X1,X2))	→	mark^#(X1)	(16)
a__plus^#(N,s(M))	→	a__plus^#(mark(N),mark(M))	(17)
a__x^#(N,s(M))	→	a__x^#(mark(N),mark(M))	(18)
a__x^#(N,s(M))	→	mark^#(M)	(19)
a__x^#(N,s(M))	→	mark^#(N)	(20)
a__plus^#(N,s(M))	→	mark^#(N)	(21)
mark^#(plus(X1,X2))	→	mark^#(X2)	(22)
a__x^#(N,s(M))	→	mark^#(N)	(20)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(23)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(24)
mark^#(x(X1,X2))	→	mark^#(X2)	(25)
a__and^#(tt,X)	→	mark^#(X)	(26)
mark^#(and(X1,X2))	→	mark^#(X1)	(27)
a__plus^#(N,0)	→	mark^#(N)	(28)
a__x^#(N,s(M))	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(29)
a__plus^#(N,s(M))	→	mark^#(M)	(30)
mark^#(s(X))	→	mark^#(X)	(31)
mark^#(x(X1,X2))	→	mark^#(X1)	(32)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(x(X1,X2))	→	mark^#(X1)	(32)
mark^#(s(X))	→	mark^#(X)	(31)
a__plus^#(N,s(M))	→	mark^#(N)	(21)
a__plus^#(N,s(M))	→	mark^#(M)	(30)
a__x^#(N,s(M))	→	mark^#(N)	(20)
a__x^#(N,s(M))	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(29)
a__plus^#(N,0)	→	mark^#(N)	(28)
mark^#(and(X1,X2))	→	mark^#(X1)	(27)
a__and^#(tt,X)	→	mark^#(X)	(26)
a__x^#(N,s(M))	→	mark^#(M)	(19)
a__x^#(N,s(M))	→	a__x^#(mark(N),mark(M))	(18)
a__plus^#(N,s(M))	→	a__plus^#(mark(N),mark(M))	(17)
mark^#(x(X1,X2))	→	mark^#(X2)	(25)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(24)
mark^#(plus(X1,X2))	→	mark^#(X1)	(16)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(23)
a__x^#(N,s(M))	→	mark^#(N)	(20)
mark^#(plus(X1,X2))	→	mark^#(X2)	(22)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(15)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 591, 0)
[s(x₁)]	=	x₁ + 0
[and(x₁, x₂)]	=	max(x₁ + 16911, x₂ + 2, 0)
[a__x^#(x₁, x₂)]	=	max(x₁ + 15655, x₂ + 15064, 0)
[a__x(x₁, x₂)]	=	max(x₁ + 591, x₂ + 0, 0)
[x(x₁, x₂)]	=	max(x₁ + 591, x₂ + 0, 0)
[a__plus^#(x₁, x₂)]	=	max(x₁ + 15064, x₂ + 15655, 0)
[mark^#(x₁)]	=	x₁ + 15064
[0]	=	553
[a__and^#(x₁, x₂)]	=	max(x₁ + 15067, x₂ + 15065, 0)
[mark(x₁)]	=	x₁ + 0
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 591, 0)
[tt]	=	42199
[a__and(x₁, x₂)]	=	max(x₁ + 16911, x₂ + 2, 0)

together with the usable rules

a__x(N,0)	→	0	(4)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(8)
a__and(tt,X)	→	mark(X)	(1)
a__plus(N,s(M))	→	s(a__plus(mark(N),mark(M)))	(3)
a__x(N,s(M))	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(5)
mark(0)	→	0	(10)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(7)
a__x(X1,X2)	→	x(X1,X2)	(14)
a__and(X1,X2)	→	and(X1,X2)	(12)
mark(s(X))	→	s(mark(X))	(11)
mark(tt)	→	tt	(9)
a__plus(X1,X2)	→	plus(X1,X2)	(13)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(6)
a__plus(N,0)	→	mark(N)	(2)

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

mark^#(x(X1,X2))	→	mark^#(X1)	(32)
a__plus^#(N,s(M))	→	mark^#(M)	(30)
a__x^#(N,s(M))	→	mark^#(N)	(20)
mark^#(and(X1,X2))	→	mark^#(X1)	(27)
a__and^#(tt,X)	→	mark^#(X)	(26)
a__x^#(N,s(M))	→	mark^#(N)	(20)
mark^#(plus(X1,X2))	→	mark^#(X2)	(22)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(15)

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

mark^#(x(X1,X2))	→	mark^#(X2)	(25)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(24)
a__plus^#(N,s(M))	→	mark^#(N)	(21)
a__plus^#(N,s(M))	→	a__plus^#(mark(N),mark(M))	(17)
a__x^#(N,s(M))	→	mark^#(M)	(19)
a__x^#(N,s(M))	→	a__x^#(mark(N),mark(M))	(18)
a__x^#(N,s(M))	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(29)
mark^#(plus(X1,X2))	→	mark^#(X1)	(16)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(23)
mark^#(s(X))	→	mark^#(X)	(31)
a__plus^#(N,0)	→	mark^#(N)	(28)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	x₁ + 0
[s(x₁)]	=	x₁ + 0
[and(x₁, x₂)]	=	x₂ + 1
[a__x^#(x₁, x₂)]	=	x₂ + 3
[a__x(x₁, x₂)]	=	x₂ + 1
[x(x₁, x₂)]	=	x₂ + 1
[a__plus^#(x₁, x₂)]	=	x₁ + 2
[mark^#(x₁)]	=	x₁ + 2
[0]	=	1
[a__and^#(x₁, x₂)]	=	2
[mark(x₁)]	=	x₁ + 0
[plus(x₁, x₂)]	=	x₁ + 0
[tt]	=	25074
[a__and(x₁, x₂)]	=	x₂ + 1

together with the usable rules

a__x(N,0)	→	0	(4)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(8)
a__and(tt,X)	→	mark(X)	(1)
a__plus(N,s(M))	→	s(a__plus(mark(N),mark(M)))	(3)
a__x(N,s(M))	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(5)
mark(0)	→	0	(10)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(7)
a__x(X1,X2)	→	x(X1,X2)	(14)
a__and(X1,X2)	→	and(X1,X2)	(12)
mark(s(X))	→	s(mark(X))	(11)
mark(tt)	→	tt	(9)
a__plus(X1,X2)	→	plus(X1,X2)	(13)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(6)
a__plus(N,0)	→	mark(N)	(2)

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

mark^#(x(X1,X2))	→	mark^#(X2)	(25)
a__x^#(N,s(M))	→	mark^#(M)	(19)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(24)
a__plus^#(N,s(M))	→	mark^#(N)	(21)
a__plus^#(N,s(M))	→	a__plus^#(mark(N),mark(M))	(17)
a__x^#(N,s(M))	→	a__x^#(mark(N),mark(M))	(18)
a__x^#(N,s(M))	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(29)
mark^#(plus(X1,X2))	→	mark^#(X1)	(16)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(23)
mark^#(s(X))	→	mark^#(X)	(31)
a__plus^#(N,0)	→	mark^#(N)	(28)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(and)	=	2
π(mark)	=	1
π(a__and)	=	2

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

prec(a__plus)	=	1	status(a__plus)	=	[1, 2]	list-extension(a__plus)	=	Lex
prec(s)	=	0	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__x^#)	=	2	status(a__x^#)	=	[1, 2]	list-extension(a__x^#)	=	Lex
prec(a__x)	=	2	status(a__x)	=	[1, 2]	list-extension(a__x)	=	Lex
prec(x)	=	2	status(x)	=	[1, 2]	list-extension(x)	=	Lex
prec(a__plus^#)	=	1	status(a__plus^#)	=	[1, 2]	list-extension(a__plus^#)	=	Lex
prec(mark^#)	=	1	status(mark^#)	=	[1]	list-extension(mark^#)	=	Lex
prec(0)	=	3	status(0)	=	[]	list-extension(0)	=	Lex
prec(a__and^#)	=	0	status(a__and^#)	=	[1, 2]	list-extension(a__and^#)	=	Lex
prec(plus)	=	1	status(plus)	=	[1, 2]	list-extension(plus)	=	Lex
prec(tt)	=	4	status(tt)	=	[]	list-extension(tt)	=	Lex

and the following Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[s(x₁)]	=	x₁ + 0
[a__x^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__plus^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[mark^#(x₁)]	=	x₁ + 0
[0]	=	0
[a__and^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[tt]	=	0

together with the usable rules

a__x(N,0)	→	0	(4)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(8)
a__and(tt,X)	→	mark(X)	(1)
a__plus(N,s(M))	→	s(a__plus(mark(N),mark(M)))	(3)
a__x(N,s(M))	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(5)
mark(0)	→	0	(10)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(7)
a__x(X1,X2)	→	x(X1,X2)	(14)
a__and(X1,X2)	→	and(X1,X2)	(12)
mark(s(X))	→	s(mark(X))	(11)
mark(tt)	→	tt	(9)
a__plus(X1,X2)	→	plus(X1,X2)	(13)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(6)
a__plus(N,0)	→	mark(N)	(2)

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

mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(24)
a__plus^#(N,s(M))	→	mark^#(N)	(21)
a__plus^#(N,s(M))	→	a__plus^#(mark(N),mark(M))	(17)
a__x^#(N,s(M))	→	a__x^#(mark(N),mark(M))	(18)
a__x^#(N,s(M))	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(29)
mark^#(plus(X1,X2))	→	mark^#(X1)	(16)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(23)
mark^#(s(X))	→	mark^#(X)	(31)
a__plus^#(N,0)	→	mark^#(N)	(28)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.